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this note is on inverse spectral theory for the schrdinger operator on a flat two - dimensional torus with electric and magnetic potentials . this problem can be remarkably rigid . for generic flat tori , if the variation of the magnetic field is strictly less than its mean , and the total magnetic flux on the torus is @xmath0 , then the spectrum of the schrdinger operator determines both the electric and magnetic fields . this is in marked contrast to both the schrdinger operator without a magnetic field ( see [ 3 ] ) and the case of a magnetic field of mean zero ( see [ 1 ] ) . in both those problems there are large families of isospectral fields , and rigidity results are much more difficult to obtain ( see also [ 2 ] ) . the observation that there can be spectral rigidity when the total flux is @xmath0 is due to guillemin ( [ 5 ] ) . here we give a short proof of the slightly stronger result stated above . instead of thinking of the hamiltonian as acting on functions with values in a line bundle over the torus @xmath1 , we think of the hamiltonian as acting on functions on @xmath2 which are invariant with respect to the magnetic translations " associated to @xmath3 . however , these two settings are completely equivalent . our assumption that the variation of the magnetic field @xmath4 is strictly less than its mean @xmath5 takes the simple form @xmath6 for all @xmath7 . the spectrum of the laplacian plus lower order perturbations on flat tori has the feature that there are large families of spectral invariants corresponding to sets of geodesics with a fixed length . in analogy with results on @xmath8 guillemin proposed the name band invariants " for these families . the nice feature of the problem discussed here is that only the simplest of the band invariants are needed to prove rigidity . the first complete solution of an inverse spectral problem was mark krein s definitive analysis of the weighted string " , [ 9 ] , [ 10 ] . since that time many other inverse spectral problems in one space dimension have been solved ( see [ 11 ] ) . in higher dimensions it is widely believed that , modulo natural symmetries and deformations like gauge transformation , most problems will be spectrally rigid . however , so far there have been relatively few settings where this has been proven ( for instance those in [ 6 ] and [ 14 ] ) and many interesting examples where it fails ( see [ 12 ] and [ 4 ] ) . this should remain an active field of research for many years to come , and one can reasonably say that it began with the work of mark grigorevich krein . .2 in we begin with the smooth magnetic field @xmath9 , periodic with respect to the lattice @xmath3 in two dimensions , expanded in a fourier series in terms of the dual lattice @xmath10 @xmath11 for this magnetic field we introduce the magnetic potential @xmath12 with @xmath13 , chosen to be as periodic as possible , i.e. @xmath14 we also have a mean zero periodic electric field which is the gradient of the mean zero periodic potential @xmath15 the quantum hamiltonian for an electron in these fields ( with all physical constants set to 1 ) is @xmath16 .1 in let @xmath17 be a fundamental domain for @xmath3 . to define the domain of @xmath18 as an operator in @xmath19 we will use magnetic translation operators " ( see [ 13 ] ) . letting @xmath20 and @xmath21 be a basis for @xmath3 and the corresponding dual basis for @xmath10 , define for linearly independent vectors @xmath22 and @xmath23 @xmath24 then the commutator @xmath25 $ ] is given by @xmath26u(x)=(e^{iv_2\cdot e_1}-e^{iv_1\cdot e_2})e^{i(v_1+v_2)\cdot x}u(x+e_1+e_2),\ ] ] and the periodicity of @xmath27 and @xmath28 implies that the commutator @xmath29 $ ] is given by @xmath30u(x)=e^{iv_j\cdot x}((i\partial_x+a(x)+a^0(e_j))^2-(i\partial_x+a(x)-v_j)^2)u(x+e_j).\ ] ] thus , in order for the @xmath31 s to commute with @xmath18 we require @xmath32 , and in order for the @xmath31 s to commute with each other we require @xmath33 for some integer @xmath34 . note that this implies @xmath35 and @xmath36 and @xmath37 , and @xmath38 is the total magnetic flux . hence the assumption @xmath39 is equivalent to nonzero flux , and it implies @xmath40 . defining the domain of @xmath18 to be the subspace of @xmath41 such that @xmath42 , we make @xmath18 a self - adjoint operator in @xmath19 . .1 in as in many previous works we will look for spectral invariants for @xmath18 by studying the wave trace . letting @xmath43 be the distribution kernel for the fundamental solution for the initial value problem @xmath44 the distribution kernel for the corresponding initial value problem in @xmath45 is @xmath46 where the operators @xmath31 act on the @xmath7 variable . note that , since the principal part of @xmath47 is @xmath48 , @xmath49 when @xmath50 and the sum in is has only a finite number of nonzero terms for @xmath51 in a bounded interval . thus @xmath25=0 $ ] implies @xmath52 . the fundamental spectral invariant for this problem is the distribution trace of the operator @xmath53 corresponding to the kernel @xmath54 . conventionally ( with all terms to be interpreted in distribution sense ) this is written @xmath55 .1 in to avoid degeneracies in the contributions to @xmath56 from the terms in ( 1 ) , we assume that vectors in @xmath3 have distinct lengths , i.e. @xmath57 since @xmath43 is singular as a distribution in @xmath58 only when @xmath59 , it now follows that the singularity of @xmath56 at @xmath60 comes from just two terms @xmath61dx . % \eqno{(2)}\ ] ] to determine the spectral invariants coming from the leading terms in the expansion of this singularity it is convenient to use the hadamard - hrmander expansion [ 7 ] , [ 8 ] for @xmath43 . beginning with the forward fundamental solution , @xmath62 , defined by @xmath63 and @xmath64 for @xmath65 one has @xmath66 where @xmath67 is chosen so that @xmath68 for @xmath69 and @xmath70 is the forward fundamental solution for @xmath48 . in two space dimensions this means @xmath71 for @xmath72 , @xmath73 for @xmath65 . for @xmath74 the distribution @xmath75 is defined by @xmath76 for @xmath77 and @xmath78 for @xmath79 . hence the coefficients @xmath80 are determined by the recursion @xmath81 where @xmath18 acts in the variable @xmath7 . solving this with the requirement that @xmath82 , we have where @xmath85 is determined by @xmath86 . the fundamental solution @xmath43 is given by @xmath87 .1 in we define @xmath88 and @xmath89e^{i(2a^0(x)\cdot d-\int_0 ^ 1d\cdot a^1(x+sd)ds)}dx.\ ] ] from - one sees that @xmath90 and @xmath91 are spectral invariants for @xmath18 . however , the periodicity implies that @xmath92 and @xmath93 . .1 in the rest of this article is devoted to studying @xmath94 and @xmath95 . we have @xmath96 , @xmath97 and gcd(@xmath98)=1 . let @xmath99 . then we have @xmath100 since @xmath101 when @xmath102 , the terms in the fourier series for @xmath27 which contribute to @xmath94 have @xmath103 , and this implies @xmath104 hence , @xmath105 , and @xmath94 reduces to @xmath106 defining @xmath107 ( note that @xmath108 ) , we have @xmath109 extending @xmath110 to a basis for @xmath111 , and letting @xmath112 be the dual basis for @xmath3 , we make the change of variables @xmath113 , and choose @xmath114 then we have @xmath115 where @xmath116 is the jacobian factor , and only depends on @xmath117 . since we have this spectral invariant for all @xmath118 , it follows that @xmath119 is a spectral invariant for any function @xmath120 which can be expanded in terms of @xmath121 , i.e. for any @xmath122 which has period @xmath123 . since @xmath128 , the hypotheses imply that the derivative of @xmath129 is strictly positive and the inverse function @xmath130 to @xmath131 is defined on the range of @xmath132 for @xmath133 $ ] . since @xmath134 has period 1 , the range is @xmath135 $ ] . letting @xmath120 in tend to the @xmath110-function at @xmath136 , the limit of is @xmath137 if @xmath138 for @xmath133 $ ] . if @xmath139 for @xmath133 $ ] then the limit of is @xmath140 , where @xmath141 , and @xmath142 . in other words taking these limits we recover a function of period 1 in @xmath136 which agrees with @xmath137 on @xmath143 . thus we recover @xmath144 modulo an additive constant , and we obtain @xmath145 by taking the derivative . since we can carry out this argument for all prime elements @xmath146 , we recover the full fourier expansion of @xmath9 . we now turn to the recovery of @xmath28 . the preceding analysis shows that , keeping the same @xmath147 as above , the spectral invariant @xmath95 , modulo terms determined by @xmath86 , reduces to @xmath148 where @xmath149 this immediately gives the following : since we are assuming the hypotheses of theorem [ theo:2.1 ] , we have the function @xmath130 and can make the substitution @xmath150 in . that gives @xmath151 but , since @xmath152 , we can extend @xmath130 smoothly to the whole line by defining @xmath153 . thus , since @xmath154 has period 1 in @xmath155 , we have @xmath156 since we have this spectral invariant for @xmath157 , we recover the fourier series of @xmath158 , and , hence , since @xmath130 is determined by @xmath159 , we have @xmath154 . as before , since we can carry out this argument for all prime elements @xmath146 , we recover the full fourier expansion of @xmath28 . if @xmath160 , for the lattice @xmath3 , then @xmath124 for the lattice @xmath161 generated by @xmath162 and @xmath163 when @xmath164 . so if @xmath4 and @xmath165 are periodic with respect to @xmath161 , theorems 1 and 2 apply in the sense that the spectrum of @xmath18 on the torus @xmath166 determines @xmath4 and @xmath165 . note that @xmath4 and @xmath165 will automatically be periodic with respect to @xmath161 when @xmath167 . equation with periodic vector potential , comm . . phys . * 125 * ( 1989 ) , 263 - 300 g.eskin , j.ralston , inverse spectral problems in rectangular domains , commun . 32*(2007 ) , 971 - 1000 . g.eskin , j.ralston , e.trubowitz , on isospectral periodic potentials in @xmath168 , i and ii , commun . pure and appl . 37*(1984 ) , 647 - 676,715 - 753 . c.gordon , survey of isospectral manifolds , _ handbook of differential geometry_,volume 1 ( 2000 ) , north holland , 747 - 778 . v.guillemin , inverse spectral results on two - dimensional tori , journal of the ams , * 3*(1990),375 - 387 . v.guillemin , d.kazdhan , some inverse spectral results for negatively curved 2-manifolds , topology * 19*(1980 ) , 301 - 312 . j.hadamard , _ le problme de cauchy et les equations aux drives partielles linaires hyperboliques , _ hermann , paris ( 1932 ) l.hrmander , _ the analysis of linear partial differential operators , iii , _ springer - verlag , vienna ( 1985 ) m.krein , solution of the inverse sturm - liouville problem ( russian ) , doklady akad . nauk sssr ( n.s.)*76*(1951 ) , 21 - 24 . m.krein , determination of the density of a nonhomogeneous cord by its frequency spectrum.doklady akad . nauk sssr ( n.s.)*76*(1951 ) , 345 - 348 . marchenko , _ operatory shturma liuvilli i ikh prilozheni _ , naukova dumka , kiev ( 1977 ) t.sunada , riemannian coverings and isospectral manifolds , ann . of math . * 121*(1985 ) , 169 - 186 . j.zak , dynamics of electrons in solids in external fields , phys . rev . * 168*(1968 ) , 686 - 695 s.zelditch , spectral determination of analytic , bi - axisymmetric plane domains , geometric funct . 10*(2000 ) , 628 - 677 .

we give a simple proof of guillemin s theorem on the determination of the magnetic field on the torus by the spectrum of the corresponding schrdinger operator .

w75 n is a massive star forming region with an integrated iras luminosity of @xmath10 ( moore , mountain , & yamashita 1991 ; moore et al . 1988,1991 ) . the w75 n cloud is located at a distance of 2 kpc ( dickel , wendker , & bieritz 1969 ) , just @xmath11 north of the massive outflow system dr 21 powered by a cluster of ob stars ( e.g. garden et al . 1991 and references therein ) . both dr 21 and w75 n are part of the cygnus - x complex of dense molecular clouds . haschick et al . ( 1981 ) identified three regions of ionized gas in w75 n at a resolution of @xmath12 : w75 n ( a ) , w75 n ( b ) , and w75 n ( c ) . hunter et al . ( 1994 ) later resolved w75 n ( b ) with @xmath13 resolution into three regions : ba , bb , and bc . torelles et al . ( 1997 ) then imaged w75 n ( b ) at @xmath14 resolution , and detected ba and bb ( which they called vla 1 & vla 3 ) , along with another weaker , and more compact hii region , vla 2 . a parsec - scale molecular outflow originates near the cluster of ultracompact hii ( uc hii ) regions in w75 n ( b ) . the mass of the co outflow has been estimated to be 50 m to 500 m based on single - dish , co observations ( e.g fischer et al . 1985 ; hunter et al . 1994 ; davis et al . 1998a , b ; ridge & moore 2001 ) . the uc hii regions have a combined @xmath15 of @xmath16 l and most are in a protostellar phase based on the presence of oh , , & methanol masers , and compact millimeter continuum emission ( baart et al . 1986 ; hunter et al . 1994 ; torrelles et al . 1997 ; minier , conway , & booth 2000 , 2001 ; shepherd 2001 ; hutawarakorn , cohen , & brebner 2002 ; slysh et al . 2002 ; watson et al . 2002 ) . several studies have assumed the flow is dominated by a single massive star : the central source in the uc hii region vla 1 ( ba ) because the position angles of the ionized gas and the co emission are similar ( hunter et al . 1994 ; torrelles et al . 1997 ; davis et al . 1998a , b ) . shepherd ( 2001 ) suggested vla 3 ( bb ) and , perhaps , vla 2 may be the primary powering sources based on the presence of compact millimeter continuum emission . more recently , hutawarakorn et al . ( 2002 ) suggested vla 2 is the dominant source powering the outflow based on oh maser emission . given the sheer number of interpretations , it is clear that w75 n is a confused region . assuming a primary driving source for the co outflow , davis et al . ( 1998b ) suggested that the co red - shifted lobe and morphology supported a jet - driven , bow - shock entrainment scenario in which a steady , over - dense molecular jet , developed to explain highly - collimated outflows from low - mass protostars , was applied to w75 n ( lada & fich 1996 ; smith et al . 1997 ; & suttner et al . the proposed model implied a jet radius of 0.03 pc at 1.3 pc from the star with a jet opening angle of about 2.6 ( richer et al . if a powerful , well - collimated jet was being driven by an ob protostar in w75 n , it would provide strong constraints on outflow / accretion theories for luminous protostars ( see , e.g. , shang et al . 2002 ; cabrit , ferreira , & raga 1999 ; knigl 1999 ; shu et al . 2000 ; knigl & pudritz 2000 ) . to obtain a better understanding of the number of sources driving outflows and the energetics of the flow(s ) , we have made interferometric mosaics of the w75 n region in co(j=10 ) and millimeter continuum using the owens valley radio observatory and obtained images at near - infrared wavelengths using the telescopio nazionale galileo to compare the morphology of the shocked gas & infrared nebulosity with the co emission . observations in 2.7 mm continuum and line were made with the owens valley radio observatory ( ovro ) array of six 10.4 m telescopes between 1999 march 15 and 1999 december 4 . projected baselines ranging from 15 to 115 meters provided sensitivity to structures up to about @xmath17 . the final @xmath18 mosaic images of both line and continuum emission are made up of 17 fields with primary beam @xmath19 ( fwhm ) spaced @xmath20 apart . the total integration time on source was approximately 3.25 hours / pointing center . cryogenically cooled sis receivers operating at 4 k produced typical single sideband system temperatures of 200 to 600 k. the gain calibrator was the quasar bl lac and the bandpass calibrators were 3c 454.3 and 3c 345 . observations of uranus , neptune , or 3c 273 provided the flux density calibration scale with an estimated uncertainty of @xmath21% . calibration was carried out using the caltech mma data reduction package ( scoville et al . images were produced using the miriad software package ( sault et al . 1995 ) and deconvolved with a maximum - entropy - based algorithm designed for mosaic images ( cornwell & braun 1988 ) . the co @xmath22 data at 115.27 ghz were convolved with a @xmath23 taper resulting in a synthesized beam of @xmath24 ( fwhm ) at p.a . the spectral resolution was 2.6 and the rms noise was 0.13 . the spectral band pass was centered on the local standard of rest velocity ( @xmath5 ) of @xmath26 ( the assumed systemic velocity of the w75 n cloud ) , taken from the cs(j=76 ) emission peak ( hunter et al . 1994 ) . simultaneous 2.7 mm continuum observations were made in a 1 ghz bandwidth channel with central frequency 112.77 ghz . the @xmath22 data were convolved with a @xmath27 taper resulting in a synthesized beam @xmath28 ( fwhm ) at p.a . the rms noise was 3.6 mjy beam@xmath30 . an additional on - source integration time of 4.6 hours was obtained with ovro centered on the position of w75 n : mm 1 ( @xmath31 @xmath32 ) . observations were made on 1999 march 29 and 2001 march 18 . baselines between 35 and 240 meters provided sensitivity to structures up to @xmath27 . the co @xmath22 data ( spectral resolution 2.6 ) were convolved with a @xmath33 taper resulting in a synthesized beam of @xmath34 ( fwhm ) at p.a . the final rms noise was 50 mjy beam@xmath30 in each channel . observations were made in and with the kitt peak 12 m telescope on 2000 may 9 using the sis 3 mm receiver with 1 mhz filter banks centered on @xmath36 to give a velocity resolution of 2.6 and a total bandwidth of 650 . system temperatures ranged from 230 k for to 360 k for . the half - power beam width ( hpbw ) at 115 ghz is about @xmath37 . single dish spectra were obtained at three positions ( j2000 coordinates ) : centered on w75 n : mm 1 ( @xmath38 @xmath39 ) ; and in the south - east and north - west outflow lobes ( @xmath38 @xmath40 and @xmath38 @xmath40 , respectively ) . the data were reduced with the nrao unipops software package . the resulting spectra were used to estimate the optical depth in as a function of velocity and position in the w75 n region . near - infrared observations of the w75 n region were made on 2000 june 16 , at the 3.5 m telescopio nazionale galileo ( tng ) at the roque de los muchachos observatory on the spanish island of la palma . the arnica nir imager ( lisi et al . @xcite ; hunt et al . @xcite ) was used to obtain images in the h@xmath41 narrow band filter and in the k@xmath2 broad band filter . arnica is equipped with a hgcdte @xmath42 nicmos3 infrared array . the pixel scale , when coupled with the tng , is 0.355 arcsec / pixel , and the corresponding field of view is @xmath43 arcmin@xmath44 per frame . the seeing at the time of the observations was about 09 . to search for emission beyond the co emission , a 14 pointing mosaic pattern was employed covering an area of approximately 9@xmath452 arcmin@xmath44 that was roughly aligned with the co flow . the mosaic was repeated several times , dithering the telescope by a few pixels each time , until the desired integration time was achieved . the final integration times per sky position was 8 minutes in k@xmath2-band and 30 minutes in the h@xmath41 filter . data reduction and analysis were performed using the iraf software package . following standard flat - fielding and sky subtraction , the individual images were registered and the final mosaic was produced . the k@xmath2-band observations were calibrated using standard stars from the arnica list ( hunt et al . the h@xmath41 mosaic was calibrated assuming that a set of stars have the same flux density in the narrow- and broad - band filters . the broad - band mosaic was then used to subtract the continuum emission from the h@xmath41 mosaic . integrated line fluxes are then estimated assuming the width of the h@xmath41 filter as measured by vanzi et al . ( @xcite ) . the calibration accuracy is expected to be within 20% . accurate ( @xmath46 ) astrometry was derived for both mosaics using stellar positions from the 2mass second incremental data release . additional observations of two @xmath474@xmath48 fields centered north - east and south - west of w75 n were obtained in 2002 august 21 using the tng near infrared camera spectrograph ( nics , baffa et al . each of the two fields was observed through the h@xmath41 ( @xmath49=2.12 @xmath50 m ) and [ feii ] ( @xmath49=1.64 @xmath50 m ) narrow - band filters and in two narrow - band continuum filters , k@xmath51 and h@xmath51 ; a detailed characterization of all these filters can be found in ghinassi et al . ( @xcite ) . the observations were reduced and astrometrically calibrated following the procedure outlined above . the weather conditions were not photometric during the observations so the data could not be flux calibrated . line - only images were obtained by subtracting the narrow - band continuum images from the line@xmath52continuum images . the subtraction was not perfect on strong stellar sources or on stars with very red or very blue spectra . infrared reflection nebulosity is associated with two distinct regions of ionized gas ( fig . 1 ) : w75 n ( a ) at position @xmath31 @xmath53 and w75 n ( b ) at position @xmath31 @xmath53 ( e.g. haschick et al . 1981 , moore et al . 1988 ) . shock - excited emission ( figs . 2 & 3 ) is present in the north - east ( near @xmath31 @xmath53 ) and along the co emission boundaries ( see also davis et al . 1998a , b ) . our images also show that faint , patchy emission extends nearly an arcminute ( 0.6 pc at a distance of 2 kpc ) beyond the south - west co flow ( @xmath31 @xmath53 ) . the continuum subtracted h@xmath41 mosaic is shown in fig . 2 . following the nomenclature used by davis et al . ( 1998a , b ) for the south - west portion of the h@xmath41 flow , all previously known h@xmath41 knots and filaments are marked with solid lines . these are labeled sw a to sw h in the south - west flow , c a and c b in the central region , and ne a to ne f in the north - east flow . additionally , figs . 2 & 3 reveal faint diffuse h@xmath41 emission beyond the tip of the south - west flow and within the north - east flow . these new features are marked with dashed lines and labeled sw i to sw k and ne g to ne faint , diffuse h@xmath41 features as well as the filamentary structure of the emission in knots ne d , ne e , & ne f are confirmed by observations obtained in 2002 august . figure 3 presents an overlay of the continuum subtracted [ feii ] emission ( contours ) on the h@xmath41 ( greyscale ) images from the 2002 august observations . [ feii ] line emission is only detected close to the uc hii regions in w75 n b and near the exciting star of w75 n a. no [ feii ] emission is detected in the outer flow regions . the non - detection of [ feii ] far from the protostars does not appear to be due to higher extinction since we clearly detect 2.12@xmath50 m emission beyond the co outflow boundaries and we detect [ feii ] emission near the cloud core where the column density is higher . a chain of h@xmath41 knots , apparently unrelated to the main flow , are detected near @xmath31 @xmath54 ( fig . this jet is outside of the co and infrared mosaic fields . the knots may be associated with a jet from a young star north of the main complex discussed in this paper . a high - velocity co outflow , centered near w75 n ( b ) , measures 3 pc from end - to - end ( projected length ) and extends well beyond the infrared reflection nebula ( figs . 1 & 4 ) . red and blue - shifted co emission exists both in the north - east and south - west . the co mosaic did not include areas to the north - west and south - east so it is unclear if high - velocity co exists in these regions . the boundaries and flux density of the co outflow are well determined on the red - shifted side of the line , however , at velocities between 0 and 5.6 the dr 21 cloud ( @xmath55 ) confuses the identification of the outflow structure . nine millimeter continuum peaks showing the locations of warm dust emission are identified in fig . 5 & table 1 ( w75 n : mm 1 through w75 n : mm 9 ) . mm 1 mm 4 are near the origin of the outflow activity and lie @xmath56 from the w75 n ( b ) reflection nebulosity to the north and west . mm 5 is associated with the more extended hii region , w75 n ( a ) , while mm 6 mm 9 are not associated with any previously known sources . infrared counterparts do not exist for the millimeter sources ( except mm 5 ) suggesting that these sources are too deeply embedded to be detected at 2@xmath50 m . figure 5 also shows the @xmath0 resolution image from shepherd ( 2001 ) for comparison . the @xmath0 resolution resolved the individual millimeter cores mm 14 but resolved out the more extended emission associated with mm 5 and mm 6 . millimeter cores mm 79 were outside of the primary beam of the shepherd ( 2001 ) observations and thus , were not detected . figure 5 also compares the mm 5 millimeter source with narrow - band @xmath52continuum emission and k@xmath2 broad - band emission in w75 n ( a ) . the central star of w75 n ( a ) ( spectral type b0.5 ; haschick et al . 1981 ) is clearly visible in the infrared and is surrounded by a @xmath57 shell of thermal dust emission . diffuse reflection nebulosity is centered on the star and a wisp of emission is visible just east of the star ( @xmath31 @xmath53 ) . compact , high - velocity co emission appears to originate from the uc hii regions vla 1 ( ba ) & vla 3 ( bb ) and from mm 2 ( figs . 6 , 7 , & 8) . the @xmath58 resolution is not sufficient to determine if vla 2 , located only @xmath59 north of vla 3 ( bb ) , is also associated with high - velocity co gas . a detailed discussion of each of the proposed outflows is given below . * the outflow from vla 1 ( ba ) : * vla 1 ( ba ) is a thermal jet source associated with and oh masers ( baart et al . 1986 , torrelles et al . it is embedded in the mm 1 core detected in 1 & 3 mm continuum emission ( shepherd 2001 ) . the spectral type of the powering source is unknown since the observed centimeter continuum emission is likely due to the ionized jet rather than emission from an ionization - bounded uc hii region . red - shifted emission to the north - east of mm 1 can be traced to the jet - like ionized flow from vla 1 ( ba ) . figure 7 presents a @xmath27 image of the integrated co emission from the flow as well as a position - velocity diagram from a slice along the proposed flow axis ( p.a . a ridge of co emission extends to the north - east with projected velocity greater than 25 ( relative to @xmath5 ) almost an arcminute from the uc hii region . high - velocity co emission is also centered on the uc hii regions ( position offset @xmath60 in the pv diagram of fig . 7 ) . a @xmath0 resolution image ( fig . 6 , top left ) shows that this red - shifted emission near the base of the outflow appears to be produced by vla 1 ( ba ) as well as vla 3 ( bb ) and possibly vla 2 . the @xmath0 resolution of fig . 6 resolves out much of the extended emission in the flow leaving only compact clumps visible along the flow axis . the p.a . is similar to the elongation of the ionized emission in vla 1 ( ba ) ( p.a . @xmath61 ) and a line of masers detected along the jet ( torrelles et al . the molecular flow is also seen in fig . 1 as a well - collimated , red - shifted lobe extending to the north - east from the mm 1 core . the outflow is shown as one - sided in figs . 6 & 7 because only one side of the flow is detected . the counterflow may exist however it may be too extended to image at this high - resolution or it may be expanding into a less dense medium that would not create appreciable co emission . * the outflow from vla 3 ( bb ) : * vla 3 ( bb ) is a compact uc hii region in mm 1 with a central star of spectral type b0.5 to b0 . it is associated with a single maser and compact 1 & 3 mm continuum emission . a lower limit on the mass of warm gas and dust within 2000 au of the protostar is 5 m ( shepherd 2001 ) . figure 6 illustrates that compact red - shifted emission to the east and blue - shifted emission to the west of mm 1 can be traced to vla 3 ( bb ) with p.a . @xmath62 . there are no obvious features in the extended emission that correspond to an outflow with this orientation , however , the co mosaic did not extend to the north - west or south - east so a large - scale outflow could have been missed ( fig . 1 ) . the molecular gas morphology does not seem to be correlated with that of the ionized gas or masers near the source : the ionized gas is slightly elongated along p.a . 149and the maser is located near the southern boundary of the ionized gas . * the outflow from mm 2 : * mm 2 is a molecular core identified by compact , warm dust emission at 1 & 3 mm and maser emission ( torrelles et al 1997 , shepherd 2001 ) . the mass of the core is @xmath63 m. no ionized gas has been detected indicating that the spectral type is less than a b2 star or high accretion is preventing the formation of a uc hii region . high - velocity , blue - shifted emission ( 13 to 36 projected velocity relative to @xmath5 ) can be traced from the mm 2 core to the south - east ( p.a . figure 8 presents a @xmath27 image of the integrated emission from the flow as well as a position - velocity diagram from a slice along the proposed flow axis . the velocity of the jet relative to @xmath64 increases away from the position of mm 2 and remains well collimated . diffuse emission at velocities greater than @xmath65 is due to the w75 n & dr 21 clouds . figure 6 shows the more compact emission in the flow . three clumps of high - velocity gas are detected extending away from mm 2 along with faint emission at the location of the core . the compact clumps appear to trace a shell of dense gas surrounding the outflow axis . the molecular outflow is identified as one - sided because the red - shifted counterflow was not detected . the diffuse millimeter core mm 4 is located along the axis of the proposed mm 2 outflow . although no or oh maser emission has been detected toward mm 4 , nor has centimeter or 1 mm continuum emission been detected , there is diffuse warm dust emission traced by 3 mm continuum . thus , mm 4 may harbor an embedded protostar . assuming the mm 4 core is heated internally , the lack of maser activity , compact warm dust emission at 1 mm , and the absence of ionized gas emission at centimeter wavelengths suggest that mm 4 is a low - mass protostar ( shepherd 2001 ) . with the current resolution and sensitivity , we can not determine if the mm 4 protostar is contributing to the observed flow dynamics . the proposed position angles of the outflowing gas ( illustrated by arrows in figs . 6 , 7 , & 8) are 51 for vla 1 ( ba ) , 101 for vla 3 ( bb ) , and 124 for mm 2 . the position angle of the parsec - scale outflow detected in and co ( fig . 1 ) is 62.5 . although the orientation of the vla 1 ( ba ) outflow is similar to that of the parsec - scale flow , it does not appear likely that vla 1 ( ba ) is the powering source . assuming the vla 1 ( ba ) flow is symmetric , a blue - shifted counterflow is expected in the south - west , not a red - shifted flow . thus , our observations do not not identify the source responsible for the 3 pc outflow which dominates the large - scale morphology and kinematics of the region . the mass associated with co line emission is calculated following the method proposed by scoville et al . the co excitation temperature near the millimeter continuum emission varies from about 35 to 75 k with @xmath66 k being the median value near the mm 1 peak ( davis et al . rotational temperatures derived from ch@xmath67cn in the cloud core vary from 47 to 78 k , consistent with the davis et al . estimates ( kalenski et al . 2000 ) . we assume the gas is in lte , at a temperature of 50 k , with [ co]/ [ ] = @xmath68 , and [ co]/ [ ] = 71 at the galacto - centric distance of 8.5 kpc ( wilson & rood 1994 ) . the co optical depth as a function of velocity and position is calculated using single dish and spectra taken at three positions within the w75 n region ( fig . we assume is optically thin at all velocities which is probably valid in the line wings ; however , is likely to be optically thick near the line core . in channels where no emission is detected , we assume the co is optically thin . the co channel images ( fig . 4 ) show that the emission near @xmath5 is almost entirely resolved out by the interferometer . if high - velocity ( @xmath4 ) structures exist that are larger than the largest angular scale that can be imaged ( @xmath69 ) , then our mass estimate represents a lower limit . because multiple , overlapping flows are present , it is not possible to obtain a reasonable estimate of the inclination of each flow . thus , we assume an inclination angle of 45 which minimizes errors introduced by inclination effects . table 2 summarizes the physical properties of the molecular gas in the combined outflows originating within mm 1 and from the blue - shifted mm 2 outflow lobe . the total flow mass @xmath70 is given by @xmath71 where @xmath72 is the flow mass in velocity channel @xmath73 corrected for optical depth . the momentum @xmath74 is given by @xmath75 and the kinetic energy e by @xmath76 where @xmath77 is the central velocity of the channel relative to @xmath5 . the characteristic flow timescale @xmath78 is @xmath79 , where the intensity - weighted velocity @xmath80 is given by @xmath81 ( cabrit & bertout 1990 ) and @xmath82 is the flow radius . the mass outflow rate @xmath83 is @xmath84 and the force f is @xmath85 . assuming d=2 kpc , the total molecular mass in outflowing gas ( @xmath86 relative to @xmath5 ) is @xmath87 m. the values presented in table 2 are derived assuming all flows have the same systemic velocity . this is a reasonable assumption for the driving sources of the combined mm 1 outflows since the observed uc hii regions are embedded within the same molecular clump and have a projected separation of only 0.5 to @xmath88 ( 1000 - 2000 au at a distance of 2 kpc ) . the source driving the outflow from the mm 2 molecular core has a projected separation of about @xmath23 ( 10,000 au or 0.05 pc ) from mm 1 . it is possible that mm 2 could have a slightly different systemic velocity from mm 1 that we can not detect in our images or single dish spectra . if the systemic velocity of the mm 2 core is different by a factor of @xmath89 from the assumed velocity of 10 , then the error in the momentum estimates will be proportional to @xmath90 and mechanical energy to @xmath91 . the combined mm 1 outflows have a total mass of at least 165 m and energy , e @xmath92 ergs . the mm 2 outflow has a total mass @xmath93 m , e @xmath94 ergs . the co mosaic did not extend in the south - east direction of the mm 2 flow , thus , the full outflow was not imaged and age and @xmath95 should be considered a lower limit . despite the uncertainties , the flow masses and energies are consistent with those for outflows driven by young , early b stars . this is in agreement with the estimated spectral types of the stars powering the uc hii regions in mm 1 ( b2 to o9 ; hunter et al . 1994 ; torrelles et al . 1997 ; shepherd 2001 ; slysh et al . 2002 ) . the mass of gas and dust associated with warm dust being heated by the central protostars is estimated from the millimeter continuum emission using @xmath96 where d is the distance to the source , @xmath97 is the continuum flux density due to thermal dust emission at frequency @xmath98 , and @xmath99 is the planck function at temperature t@xmath100 ( hildebrand 1983 ) . assuming a gas - to - dust ratio of 100 , the dust opacity per gram of gas is taken to be @xmath101 cm@xmath44 g@xmath30 where @xmath103 is the opacity index ( see kramer et al . 1998 ; and the discussion in shepherd & watson 2002 ) . this value of @xmath104 agrees with those derived by hildebrand ( 1983 ) and kramer et al . ( 1998 ) to within a factor of 2 . the opacity index @xmath105 appears to be appropriate between wavelengths of 650 microns and 2.7 mm for sub - micron to millimeter - sized grains expected in warm molecular clouds and young disks ( pollack et al . we assume the emission is optically thin and the temperature of the dust can be characterized by a single value . using values of @xmath106 k and @xmath105 , we find the total mass of gas and dust associated with the 2.7 mm continuum emission is approximately 475 m ( table 1 ) . our results are consistent with those of shepherd ( 2001 ) and watson et al . ( 2002 ) to within the errors . the total molecular mass of outflowing gas from the mm 1 and mm 2 combined flows ( @xmath86 relative to @xmath5 ) is @xmath3 m. hunter et al . ( 1994 ) found @xmath107 with a rough scaling performed to take into account an optical depth correction . however , their image covered only the inner region of the flow so their estimate should be considered a lower limit . based on single dish observations of co(j=32 ) , davis et al ( 1998a ) estimated a total flow mass of @xmath108 , uncorrected for optical depth effects . this mass estimate is extremely high for an optically thin approximation . examination of their fig . 11 , t@xmath109 vs. ( @xmath110 ) , shows that the blue - shifted lobe has an order of magnitude increase in the integrated flux at the velocity of dr 21 ( 2.5 ) . it appears that their single - dish map may have been significantly contaminated by emission from the dr 21 cloud , which introduced uncertainties in the mass and kinematics estimates . ridge & moore ( 2001 ) estimated the outflow mass of the red - shifted lobe only to be 273 based on a co(j=21 ) single dish image corrected for optical depth . the mass of blue - shifted gas was not estimated by ridge & moore due to the contamination by the dr 21 cloud . this value is significantly higher than our estimate and may be due to missing extended emission in the interferometer image , especially at low velocities . despite this problem , interferometric imaging also provided benefits : it was easier to distinguish between outflow gas and the dr 21 cloud and to identify flows from multiple sources in the cluster . the total cloud core mass of w75 n has been estimated to be 18002500 based on observations at submillimeter wavelengths ( moore , mountain , & yamashita , 1991 ) and 1200 based on cs(j=76 ) emission ( hunter et al . 1994 ) . the gravitational binding energy of the cloud , @xmath111 , is 12@xmath112 ergs , where we take the radius @xmath113 pc ( moore et al . 1991 ) and @xmath114 is a constant which depends on the mass distribution ( @xmath115 for @xmath116 ) . more than 10% of the molecular cloud is participating in the outflow and the combined outflow energy is roughly half the gravitation binding energy of the cloud . the observed w75 n outflows are injecting a significant amount of mechanical energy into the cloud core and may help prevent further collapse of the cloud . our co(j=10 ) images suggest that high - velocity gas is associated with at least two uc hii regions : vla 1 ( ba ) and vla 3 ( bb ) and an embedded source in the millimeter core mm 2 . the position angles of the individuals outflows are not aligned , ranging from 51 to 124 . the morphology is diffuse and patchy both in the north - east and south - west . the irregular morphology of the infrared reflection nebula with fingers of nebulosity radiating out from the mm 1/mm 2 millimeter cores supports the conclusion that multiple energetic outflows are carving large cavities in the molecular cloud . low surface brightness emission extends well beyond the co outflow while [ feii ] emission is only detected close to the protostellar cluster . it is generally believed that [ feii ] line emission associated with low - mass outflows requires the presence of fast , dissociative shocks that disrupt dust grains and release heavy elements just behind a jump - shock ( j - shock ) boundary . emission , on the other hand , appears to be produced in slow , non - dissociative j - type shocks ( e.g. hollenbach & mckee 1989 ; smith 1994 ; gredel 1994 ; beck - winchatz et al . continuous - shocks ( c - shocks ) can not easily produce emission from ionized species such as [ feii ] nor can they produce the observed column densities typically seen in toward herbig haro objects from low - mass protostars ( gredel 1994 ) . in fact , nisini et al . ( 2002 ) find that there appears to be no correlation between and [ feii ] emission in outflows from low - mass ysos which supports the interpretation that physically different mechanisms are responsible for producing and [ feii ] emission . in a sample of herbig haro objects produced by jets from low - mass protostars , both and [ feii ] is found toward all sources and the morphology of and [ feii ] emission is similar on large scales although it differs in the detail ( gredel 1994 ; reipurth et al . these observations indicate that jets from low - mass protostars produce both fast , dissociative regions where [ feii ] emissions arises and slower , non - dissociative regions where emission arises . in contrast , [ feii ] emission toward w75 n is only detected close to the central sources and does not show a jet - like morphology as in outflows from low- and intermediate - mass young stellar objects ( e.g. lorenzetti et al . 2002 ; nisini et al . 2002 ; reipurth et al . the outflows from w75 n appear to exhibit only slow , non - dissociative j - type shocks which produce copious emission throughout the outflow region but the fast , dissociative shocks responsible for [ feii ] emission are absent in the outer regions of the flow . instead , the diffuse [ feii ] line emission in w75 n is coincident with the brightest k@xmath2-band reflection nebulosity . one possibility may be that the [ feii ] emission traces photo - dissociation regions ( pdrs ) along cloud surfaces illuminated by the massive protostars in the mm 1 core . this situation is also observed in the orion bar pdr ( e.g. walmsley et al . 2000 ) suggesting that the w75 n nebula may exhibit similar excitation conditions to those in orion . the and [ feii ] line emission in w75 n does not conform to what is expected for shock - excited emission resulting from the interaction between a well - collimated jet and diffuse molecular gas . in this respect , the physical characteristics of the w75 n flows differ from their low - mass counterparts which produce collimated jets observed in both and [ feii ] emission . many previous authors have assumed that only vla 1 ( ba ) was in an outflow stage based on the elongated morphology of the ionized gas , the presence of maser emission along the uc hii region axis , and because the position angles of the ionized gas and the co emission were similar . maser emission is also associated with vla 2 and vla 3 ( bb ) as well as mm 3 and mm 2 , however , outflowing material could not be traced to specific sources . assuming a single primary driving source for the co gas , davis et al . ( 1998b ) suggested that the south - west co red - shifted lobe and morphology supports a bow - shock entrainment scenario for a molecular outflow driven by a jet from a single massive star . our co and millimeter continuum observations do not support this theory that a single source drives the high - velocity co gas . further , our infrared observations suggest that the w75 n outflows are not likely to be scaled - up versions of jet - driven outflows from low - mass protostars . a question remains unanswered by this work : what source powers the 3 pc flow at p.a . the flow mass is @xmath117 m , the dynamical age is roughly @xmath118 years , and the mass loss rate @xmath119 to @xmath120 m@xmath121 yr@xmath30 . the flow parameters are consistent with those produced by an early b protostar . hutawarakorn et al . ( 2002 ) have modeled the oh maser position - velocity data and find evidence for a massive disk centered on vla 2 ( m@xmath122 m with p.a . = 155 , roughly perpendicular to the outflow axis ) . a high - velocity , time - variable oh maser cluster is coincident with vla 2 suggesting an outflow origin . further , recent observations with the very long baseline array ( vlba ) show that a clump of strong maser emission with high velocity dispersion is centered on vla 2 ( torrelles et al . 2003 ) . thus , the oh and maser activity suggests vla 2 is producing a powerful outflow . although our observations did not have adequate resolution to isolate high - velocity gas toward vla 2 , we have determined that vla 1 ( ba ) , vla 3 ( bb ) , and mm 2 are not likely to drive the 3 pc flow that dominates the region dynamics . thus , it is possible that vla 2 may drive the large - scale flow . follow - up observations at a resolution less than @xmath88 will be required to determine if , in fact , vla 2 drives the 3 pc flow . based on the size and velocity of the co outflows from the w75 n ( b ) uc hii regions , the region is @xmath123 years old . w75 n ( a ) is more evolved than the sources in mm 1 and the exciting star of w75 n ( a ) has no detectable high - velocity gas associated with it . the star , detected in the infrared , is centered within a shell of warm dust emission and an extended hii region ( hashick et al . 1981 ) . figure 10 shows a color - color diagram using data from the 2mass point source catalog for stars within @xmath124 of mm 1 that were detected at all three bands . the locus of main - sequence stars is represented by the thick , curved line ( bessell & brett 1988 , koornneef 1983 ) while the two diagonal lines show reddening vectors up to @xmath125 of dust ( adopting the @xmath126 extinction law from cardelli , clayton , & mathis 1989 ) . sources within the reddening vectors have colors consistent with main - sequence stars reddened by foreground dust . those to the right of the reddening vectors demonstrate excess emission at @xmath127 m , consistent with the presence of circumstellar material . the infrared colors of the w75 n ( a ) exciting star are consistent with those of a main - sequence star reddened by foreground dust . in comparison , the two bright stars to the south - east and south - west of mm 1 ( irs 2 and irs 3 ) have excess emission at @xmath127 m consistent with the presence of circumstellar material . the protostars within mm 1 and mm 2 are not detectable at infrared wavelengths . n represents a region of clustered star formation which appears to be forming mid to early - b stars which exist at a range of developmental stages . vla 1 ( ba ) appears to have a well - collimated , outflow based on the ionized gas morphology imaged by torrelles et al . ( 1997 ) and the presence of a relatively well - collimated , red - shifted co lobe which extends about 0.5 pc north - east of the source . however , the spectral type of the protostar is unknown since the ionized gas appears to be due to thermal jet emission . the well - collimated outflow which appears to be produced by the embedded source in mm 2 is not detected at centimeter wavelengths . either the powering source is not an early - b star ( e.g. it does not have sufficient ionizing radiation to produce a detectable uc hii region ) or accretion onto the protostar is sufficiently high that it prevents the formation of a uc hii region ( see e.g. churchwell 1999 and references therein ) . there is no evidence for well - collimated flows ( collimation ratios , length / width , @xmath128 ) from the remaining embedded sources ( early - b protostars ) or in the large - scale morphology of the co , , & [ feii ] emission . the lack of highly - collimated flows from the known , early - b protostars in w75 n suggests that it may be difficult for massive stars to collimate outflowing material . although a few mid to early - b protostars appear to be powering ionized jets , their molecular outflows tend to be complex and poorly collimated ( see , e.g. , the review by shepherd 2002 and references therein ) . to our knowledge , there is no well - collimated _ molecular _ outflow powered by a massive protostar ( spectral type early - b to o ) and most do not appear to have ionized outflow components that are well - collimated ( e.g. ridge & moore 2001 , shepherd , claussen , & kurtz 2001 ; churchwell 1999 ) . poorly collimated flows could be due to several factors : * confusion from multiple outflow sources in a cluster ( e.g. w75 n : this work ; or dr 21 : garden et al . 1991 ) ; * large flow precession angles ( e.g. pv ceph : reipurth , bally , & devine 1997 , gomez , kenyon , & whitney 1997 ; or iras 20126@xmath524104 : shepherd et al . 2000 ) ; * the presence of a strong wide - angle wind ( e.g. orion i : greenhill et al . 1998 ; or g192.163.82 : shepherd , claussen & kurtz 2001 ) ; and/or * the molecular flow represents only the truncated base of a much larger flow ( e.g. hh 8081 : yamashita et al . 1989 , rodrguez et al . 1994 ; or g192.163.82 : devine et al . 1999 ) . our observations of w75 n supports the interpretation that massive protostars may not be able to produce well - collimated molecular outflows . this conclusion does not rule out the possibility that an underlying , neutral jet may still exist as part of the outflow from the massive protostars in w75 n. high - resolution observations in shock tracers such as sio(j=10,v=0 ) or sio(j=21,v=0 ) may be able to determine whether collimated neutral jets are present in w75 n. w75 n represents an example of clustered , massive star formation . the cluster covers a wide range of evolutionary stages ; from stars with no apparent circumstellar material to deeply embedded protostars actively powering massive outflows . the co outflow measures more than 3 pc from end - to - end and is produced by at least four individual sources . emission extends well - beyond the co boundaries while [ feii ] emission is only located close to the protostellar cluster . the co , & [ feii ] morphology does not conform to what is expected for shock - excited emission resulting from the interaction between a well - collimated jet and diffuse molecular gas . the irregular morphology of the infrared reflection nebula with fingers of nebulosity radiating out from the millimeter cores supports the conclusion that multiple energetic outflows are carving large cavities in the molecular cloud . more than 10% of the molecular cloud is outflowing material and the combined outflow energy is roughly half the gravitational binding energy of the cloud . thus , the observed w75 n outflows are injecting a significant amount of mechanical energy into the cloud core and may help prevent further collapse of the cloud . research at the owens valley radio observatory is supported by the national science foundation through nsf grant number ast 99 - 81546 . star formation research at owens valley is also supported by nasa s origins of solar systems program , grant nagw-4030 , and by the norris planetary origins project . this paper is partly based on observations made with the italian telescopio nazionale galileo ( tng ) operated on the island of la palma by the centro galileo galilei of the inaf ( istituto nazionale di astrofisica ) at the spanish observatorio del roque de los muchachos of the instituto de astrofisica de canarias . the arnica and nics observations were performed in service mode by the tng staff , we especially acknowledge the help of francesca ghinassi , juan carlos guerra , and antonio magazz . 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 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36.0 . + mm 2 outflow emission measured between 26.4 and 2.2 . * figure 1 . * integrated co red - shifted ( red lines ) and blue - shifted ( blue lines ) emission contours from 36.0 to 17.8 and 0.4 to 26.4 , respectively . the images have an rms of 23.5 with a peak of 56.0 in the red - shifted emission image and 100.6 in the blue - shifted emission image . contours begin at 10% of the peak emission and continue at increments of 20% . the synthesized beam is @xmath24 at p.a . @xmath130 . line @xmath52 continuum emission is shown as grey scale displayed as the square root of the intensity with a peak of @xmath131 erg @xmath132s@xmath30arcsec@xmath133 . w75 n ( a ) is located at position @xmath31 @xmath53 and w75 n ( b ) at position @xmath31 @xmath53 . uc hii regions embedded in the core of mm 1 are shown as filled triangles while the millimeter cores mm 29 are shown as open circles . the large open circle ( mm 5 ) is coincident with the infrared emission associated with w75 n a. the solid black line delineates the boundaries of the co mosaic . * figure 2 . * continuum - subtracted line mosaic of w75 n. the image is displayed on a linear scale from 0.6 to a peak intensity of @xmath134 erg @xmath132 s@xmath30 . individual knots of emission are labeled ne a through i , c a & b , and sw a through k. features observed by davis et al . ( 1998a , b ) are identified by solid lines , previously undetected features which are more diffuse are shown as dashed lines . large dashed boxes outline the fields shown in fig . figure 3 . * continuum - subtracted emission shown in grey scale with [ feii ] emission shown as contours . the [ feii ] contours begin at @xmath135 and continue with a spacing of @xmath136 . the left panel shows the central and north - east outflow regions . the strong , diffuse [ feii ] emission is coincident with the w75 n ( a ) and ( b ) reflection nebulae . the right panel shows the south - west outflow . [ feii ] contours coincident with point sources are due to imperfect continuum subtraction . * figures 4a & 4b . * co channel images at 2.6 spectral resolution between 32.1 and @xmath137 . the central velocity is indicated in the upper right of each panel . the lsr velocity is 10 . the rms is 0.13 and the peak emission is 16.9 . in the first 12 and last 12 panels , contours are plotted from @xmath138 , 8 , 12 , 16 , 20 @xmath139 and continue with a spacing of @xmath140 . in the central 6 panels ( 8.7 to 4.3 ) , contours begin at @xmath141 and continue with a spacing of @xmath142 . panels at velocities 32.1 & 27.7 show the synthesized beam in the bottom right corner ( @xmath24 at p.a . @xmath130 ) and a scale size of 0.9 pc . the plus symbol in each panel represents the location of the peak emission in mm 1 . no other emission was detected outside of this velocity range . * figure 5 . * the bottom left image shows continuum emission at 2.7 mm . no other continuum sources were detected within the mosaic field . the image has an rms of 3.6 . contours begin at @xmath143 and continue with a spacing of @xmath144 . the greyscale is plotted on a linear scale from 7.2 to 265 . the synthesized beam in the lower right corner is @xmath28 at p.a . uc hii regions in the center of mm 1 are identified by filled triangles , mm 24 are shown as open circles . a scale size of 0.15 pc is represented by a bar in the lower left corner . the bottom right inset shows 3.3 mm continuum emission obtained with @xmath145 resolution ( figure 1 from shepherd 2001 ) . upper panels show the mm 5 millimeter source compared with narrow - band @xmath52continuum emission and wide - band 2.12@xmath50 m emission in w75 n ( a ) ( from fig . * figure 6 . * the relationship between compact , high velocity co emission , infrared emission , millimeter continuum peaks , and uc hii regions in w75 n ( b ) . red- and blue - shifted co emission ( upper panels ) is plotted from 20.4 to 36.0 and 5.6 to 23.8 , respectively . the rms in both images is 0.5 ; contours are plotted from 3 , 2 , 3 , 4 @xmath139 and continue at spacings of 1 @xmath139 . millimeter core positions for mm 24 are shown as filled circles , uc hii regions embedded in mm 1 are represented as filled triangles . proposed outflows are identified by arrows from uc hii regions vla 1 ( ba ) , vla 3 ( bb ) , and mm 2 . the synthesized beam ( @xmath146 at p.a . @xmath147 ) is shown in the top right image and a scale size of 0.1 pc is represented by a bar in the top left image . the bottom two images show the k@xmath2 and emission relative to the proposed outflows . both images are displayed as the square root of the intensity . * figure 7 . * * top : * integrated emission ( zeroth moment ) from 17.8 to 36.0 . the rms in the image is 1.2 , contours are plotted from 4 to @xmath142 with increments of @xmath148 and then from 20 to @xmath149 with increments of @xmath135 . greyscale is plotted from @xmath150 to a peak of 56.0 . millimeter cores are identified as filled circles , uc hii regions by filled triangles . the synthesized beam in the bottom left corner is @xmath151 at p.a . @xmath130 . * bottom * position - velocity plot along the length of the high - velocity , red - shifted outflow . contours are plotted at 2 , 5 , 10 , 15 , 20 , 30 , 40 , 50 , 70 , & 90% of the peak . * figure 8 . * * top : * integrated emission ( zeroth moment ) from 5.6 to 26.4 . the rms in the image is 0.96 , contours are plotted from 4 to @xmath142 with increments of @xmath148 , greyscale is plotted from @xmath150 to a peak of 20.0 . millimeter cores are identified as filled circles , uc hii regions by filled triangles . the synthesized beam in the bottom right corner is @xmath151 at p.a . @xmath130 . * bottom * position - velocity plot along the length of the high - velocity , blue - shifted outflow . contours are plotted at 3 , 5 , 7 , 9 , 20 , 40 , 60 , 80 , & 100% of the peak . * * co(j=10 ) optical depth as a function of velocity based on single - dish observations with the kitt peak 12 m telescope . optical depth is derived for three positions within the outflow : in the north - east lobe ( top ) ; centered on the mm 1 core ( center ) ; and in the south - west lobe ( bottom ) . co emission was measured in each channel image of the interferometer mosaic and an optical depth correction was made to the mass estimate based on the location and velocity of the emission . * figure 10 . * * left : * three color image using data from the two micron all sky survey ( 2mass ) . the image is @xmath152 on a side . the j - band image at 1.25@xmath50 m is shown as blue , h - band at 1.65@xmath50 m is green , & k@xmath2-band at 2.17@xmath50 m is red . the @xmath52 symbol represents the location of the mm 1 peak . * right : * a color - color diagram using data from the 2mass point source catalog for stars within @xmath124 of mm 1 that were detected at all three bands . the locus of main - sequence stars is represented by the thick , curved line while the two diagonal lines show reddening vectors up to @xmath125 of dust .

we present @xmath0 to @xmath1 resolution 3 mm continuum and line emission and near infrared k@xmath2 , , and [ feii ] images toward the massive star forming region w75 n. the co emission uncovers a complex morphology of multiple , overlapping outflows . a total flow mass of @xmath3 m extends 3 pc from end - to - end and is being driven by at least four late to early - b protostars . more than 10% of the molecular cloud has been accelerated to high velocities by the molecular flows ( @xmath4 relative to @xmath5 ) and the mechanical energy in the outflowing gas is roughly half the gravitational binding energy of the cloud . the w75 n cluster members represent a range of evolutionary stages , from stars with no apparent circumstellar material to deeply embedded protostars that are actively powering massive outflows . nine cores of millimeter - wavelength emission highlight the locations of embedded protostars in w75 n. the total mass of gas & dust associated with the millimeter cores ranges from 340 m to 11 m. the infrared reflection nebula and shocked emission have multiple peaks and extensions which , again , suggests the presence of several outflows . diffuse emission extends about 0.6 parsecs beyond the outer boundaries of the co emission while the [ feii ] emission is only detected close to the protostars . the infrared line emission morphology suggests that only slow , non - dissociative j - type shocks exist throughout the pc - scale outflows . fast , dissociative shocks , common in jet - driven low - mass outflows , are absent in w75 n. thus , the energetics of the outflows from the late to early b protostars in w75 n differ from their low - mass counterparts they do not appear to be simply scaled - up versions of low - mass outflows . # 1#2#3#1@xmath6#2@xmath7#3@xmath8 # 1#2#3@xmath9 = 500 = 500 =

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the primordial universe can be used as a laboratory to set constraints on quantum gravity . in the framework of loop quantum cosmology , we show that such a proposal for quantum gravity not only solves for the big bang singularity issu but also naturally generates inflation . thanks to a quantitative computation of the amount of gravity waves produced in the loopy early universe , we show that future cosmological datas on the polarized anisotropies of the cosmic microwave background can be used to probe lqc model of the universe . [ [ methodological - introduction ] ] methodological introduction + + + + + + + + + + + + + + + + + + + + + + + + + + + + building a quantum theory of gravity is probably the most outstanding problem of modern physics , but remains also one of the most difficult task . facing such a great challenge and retrospectively , looking at the many attempts developped since the last decades one could be tempted to consider that searching for quantum gravity is a never ending quest and that the wise attitude would be to withdraw such a challenge . however , quantum gravity is not _ optional _ ! several heuristic arguments pin down the necessity of a successful building of quantum gravity @xcite . more importantly , the singularity theorems derived by penrose and hawking @xcite shows that general relativity ontologically fails : a way out reconciling quantum mechanics and general relativity is mandatory ! a new physical paradigm is needed and , to be successful , such a potentially new paradigm requires a well - established theoretical framework . on the one hand , this assumes a well - posed theoretical problem namely how to quantize general relativity or , alternatively , how to gravitize quantum field theory ? many tentative theories have been proposed such as string theory @xcite and loop quantum gravity ( lqg ) @xcite to mention the most popular ones , and though some of them have made great progresses , none of them is fully setted up . on the other hand , this also implies a new kind of paradigmatic experiments or observations . this is a mandatory step to discriminate theoretical proposals which are deeply on the wrong track from those which are not , as well as to guide theoretical developments . this well - posed phenomenological and experimental problem is the second main difficulty of quantum gravity as quantum gravitational effects should pop up at an energy scale of @xmath0 gev , far beyond any high energy physics experiments . fortunately , the failure of general relativity precisely points towards those phenomena potentially probing quantum gravity : space - time singularities provide the territory for quantum gravity and one therefore has to search for those phenomena built up of singular space - times . this qualifies the primordial universe , being the neighborhood of the big bang singularity , as an ideal territory for probing quantum gravity ! observing quantum gravity is therefore splitted into two questions . how a given proposal for quantum gravity affects the physics of the primordial universe _ and _ how can we probe the physics of the primordial universe ? apparently , the second question points towards observation . but , as a theoretical question , this should not be overlooked since it strongly determined the precise implementation of the first question the main requirement for a quantum gravity proposal is to solve for the pathological big bang singularities and this can be checked by computing the global evolution of the universe in a quantum cosmological setting . however , it should be stressed out that such a global evolution can not be probed directly as one can not extract himself from the universe to see how it evolves in its primary ages . we are stuck inside the universe and , from the inside , we are able to probe the physics of the early universe via the cosmic microwave background ( cmb ) anisotropies . the origin of such anisotropies are _ cosmological perturbations _ produced during a phase of accelerated expansion dubbed _ cosmic inflation_. inflation is a key ingredient of the standard cosmological paradigm as it solves for the horizon and flatness problems and provides a mechanism for generating perturbations , the primordial seeds for galaxies and large scale structures formation . moreover , inflation could be a high energetic phenomenon ( its energy scale could be as high as @xmath1 gev ) and is therefore potentially affected by quantum gravity . nevertheless , the inflationnary scenario is not free of any problems as it is very difficult to generate such a phase without invoking speculative physics or fine tuning . bridging a link between loop quantum cosmology ( lqc ) the cosmological implementation of lqg @xcite and inflation in the early universe allows us to solve for the big bang singularity _ and _ to naturally trig a phase of inflation . in addition , a quantitative computation of cosmological perturbations produced in such a _ loopy _ universe makes the early universe a potential laboratory for testing quantum gravity . our methodology is the following . the evolution of the universe is derived in the framework of _ effective _ lqc which includes first order quantum corrections coming from holonomies . on top of that modified cosmological background , we will quantitatively compute the amount of cosmological perturbations of tensor type ( _ i.e. _ primordial gravity waves ) and investigate how it impacts on the @xmath2-mode angular power spectrum of cmb anisotropies . such an angular power spectrum is the main observable and can finally be used to forecast how future cmb observations could be used to probe lqc . we consider the early universe to be filled with a massive scalar field @xmath3 as matter sources and our time variable is cosmic time @xmath4 . [ [ background - evolution ] ] background evolution + + + + + + + + + + + + + + + + + + + + + with holonomy corrections at first order , the modifed friedman and klein - gordon equations are ( denoting @xmath5 the hubble parameter and @xmath6 the mass of the scalar field ) : @xmath7 where @xmath8 is a critical energy density which can not be overcomed and which encodes lqc corrections . because of @xmath9 , the universe is not singular as the big bang is now replaced by a big bounce . the basical history of the universe is thus a contracting phase , followed by an expanding phase . the _ regular _ transition from the contracting phase to the expanding one is ensured by the quantum corrections and the bounce occurs at @xmath10 . more interestingly , this peculiar evolution of the universe naturally leads to inflation ! indeed , a phase of accelerated expansion can start right after the bounce if the scalar field is in the appropriate energy state for the slow - roll conditions to be fulfilled . for a massive field , this is translated into @xmath11 with @xmath12 denoting some time _ after _ the bounce . this condition is not easily met in the standard cosmological scenario . but in the lqc bouncing universe , the hubble parameter @xmath13 is negative valued during contraction . it therefore acts as an _ anti - friction _ term making @xmath3 to climb up its potential . right after the bounce , it appears that such a field is precisely in the appropriate energy setting for a sufficiently long phase of inflation to start @xcite . the key point about this scenario has been raised by ashtekar and sloan who showed that generating a phase of inflation in the lqc universe with the mandatory amount of at least 60 e - folds is very close to unity , making inflation rather natural in this framework @xcite . [ [ cmb - power - spectra ] ] cmb power spectra + + + + + + + + + + + + + + + + + + to derive the amount of gravity waves produced at the end of inflation , one needs to solve the following equation describing tensor perturbations evolving on top of the modified flrw background : @xmath14 where the last term , acting as a time - dependant mass term , encodes the quantum deformation of the background . this equation of motion @xcite has been derived from an algebra which is anomaly - free at all orders and can be safely used throughout the entire history of the bouncing universe . the main characteristics of a bouncy power spectrum for tensor modes are the following . the ir ( large scales ) part is @xmath15 suppressed . this is due to the freezing of very large - scale modes in the minkowski vacuum . those modes indeed exit the horizon long _ before _ the bounce and naturally exhibit a quadratic spectrum . the uv part is identical to the standard prediction . small scales indeed experience a history basically similar to that of the big bang scenario . they exit the horizon during inflation and reenter later , leading to the standard nearly scale - invariant spectrum . intermediate scales , around @xmath16 , exhibit both a bump of amplitude @xmath17 and damped oscillations shape to its nearly scale - invariant shape ) is a phenomenological parameter related to @xmath6 and to the fraction of potential energy at the time of the bounce . similarly , @xmath17 is a phenomenological parameter related to @xmath6 ( see @xcite for further details ) . ] . this is mostly due to the fact that all modes are inevitably in causal contact at the bounce ( the hubble parameter vanishes , therefore leading to an infinite hubble radius ) . those characteristics have been fully determined by numerically solving the equations of motion of tensor perturbations propagating in the lqc - corrected , \{bouncing+inflationary } universe @xcite . @xmath2-mode angular power spectrum ( left panel ) and detectable region of the parameters @xmath18 describing a loopy universe ( right panel ) . figure adapted from @xcite . ] using a boltzmann code , such a primordial tensor power spectrum serves as an input for computing the footprints of primordial gravity waves on the cmb @xmath2-mode angular power spectrum ( see left panel of fig . [ fig ] ) . as compared to the standard prediction , the distortion of lqc @xmath2-mode power spectrum depends on the value of @xmath19 @xcite . ( roughly speaking , the suppression of power at large scales can be seen in the cmb only for values of @xmath19 corresponding to scales _ smaller _ than the hubble horizon today . ) if @xmath20 corresponding to a length scale greater than the hubble scale today , the tail of the bump for @xmath21 translates into a slight bump at large angular scales ( _ i.e. _ small values of the multipole @xmath22 ) . however , for @xmath23 , the @xmath2-mode angular spectrum shows , as compared to the standard prediction , a suppression at large angular scales due to the suppression at large length scales in the primordial spectrum and a bump at intermediate angular scales @xmath24 . finally , if @xmath19 is greater than @xmath25 mpc@xmath26 , the suppression is effective up to @xmath22-values of a few hundreds and the primordial part of the @xmath2-mode is therefore systematically pushed below the lensing - induced @xmath2-mode . using those angular power spectra as an observational probe and considering both the instrumental noise as expected for a future dedicated @xmath2-mode experimentk - arcmin . , the beam width to 8 arcmin . and the sky fraction to 70% . ] and the lensing - induced @xmath2-mode as a foreground masking the detection of primordial @xmath2-mode , one can forecast some constraints to be set on the fundamental parameters describing the lqc model of the universe @xcite . the detectable values at 1-@xmath27 of the mass of the scalar field @xmath6 and its fraction of potential energy at the bounce , denoted @xmath28 , are depicted by the blue band in the right panel of fig . [ fig ] . this roughly corresponds to a detectable range of @xmath19 from @xmath29 mpc@xmath26 to @xmath30 mpc@xmath26 considering _ degeneracies _ with other cosmological parameters . the upper part is not detectable as it corresponds to @xmath31 making the @xmath2-mode power spectrum _ undistorted _ as compared to the standard general relativistic prediction . the lower part is not detectable as it corresponds to @xmath32 making the primordial @xmath2-mode systematically smaller than the lensing - induced part . though measurements of the lqc parameters is not possible in this second case , a discrimination with pure general relativity is still possible as the suppression induced by the bounce is seen via the masking of the primordial @xmath2-mode . however in the precise case of lqc , the three parameters @xmath33 , @xmath19 and @xmath17 are fully determined by the _ two _ parameters @xmath6 and @xmath28 , thus breaking the degeneracy between @xmath19 and @xmath33 ( see @xcite for details ) . ] ; that is via its _ non - detection_. [ [ conclusion ] ] conclusion + + + + + + + + + + + lqg offers an appealing approach to build a quantum theory of gravity . its implementation to the symmetry reduced case of flrw metric describing our whole universe shows that the big bang singularity is cured and that inflation is _ naturally _ triggered . moreover , thanks to a quantitative computation of gravity waves produced in the early universe , subsequently impacting on the cmb polarized anisotropies of @xmath2-type , a possible window to constraint those loopy models of the universe using cosmological datas is now opened . as stressed out in @xcite , the case of lqc illustrates a possible future way to test for quantum gravity using cosmological / astronomical observations . the author warmly thanks his collaborators aurlien barrau , thomas cailleteau and jakub mielczarek .

there is a growing consensus among physicists that the classical notion of spacetime has to be drastically revised in order to find a consistent formulation combining quantum mechanics and gravity @xcite . one such nontrivial attempt comprises of replacing functions of continuous spacetime coordinates with functions over noncommutative algebra @xcite . dynamics on such noncommutative spacetime ( noncommutative theories ) are of interest for a variety of reasons among the physicists for a long time . let us start with introducing such noncommutative theories . the idea of extension of noncommutativity to the coordinates was first suggested by heisenberg as a possible solution for removing the infinite quantities of field theories before the renormalization procedure was developed and had gained acceptance . they seem to have first appeared in a letter from heisenberg to peierls in 1930 @xcite . the first paper on the subject was published in 1947 by hartland snyder who sought to use it to regularize quantum field theories @xcite . the success of renormalization theory however drained interest from the subject for some time . + the renewed interest by particle physics community started to grow after the paper by seiberg and witten @xcite . arguments combining quantum uncertainties with classical gravity also provide an alternative motivation for their study @xcite , and these theories can provide a self - consistent deformation of ordinary quantum field theories at small distances , yielding non - locality @xcite , or create a framework for finite truncation of quantum field theories while preserving symmetries @xcite . also , the emergence of noncommutative field theories in string theory @xcite has provided a considerable impetus to their investigation . the noncommutativity of the coordinates can be described by the following commutation relation @xmath6=i\theta_{\mu\nu}\ ] ] where @xmath7 is an anti - symmetric tensor . the simplest case corresponds to @xmath7 being constant . the field theories on such spacetimes are described by elements of unital algebra ( associative , but not commutative ) generated by @xmath8 s . there is a sense in which the algebra over the noncommutative space @xmath9 and the algebra over commutative space @xmath10 ( @xmath11 being the dimension of the space ) have the same topology . this algebra can be considered as a deformation of the algebra over @xmath10 such that the elements of the algebra have the same addition law , but a different ( noncommutative ) multiplication law which reduces to the commutative point - wise multiplication in the limit @xmath12 . this noncommutative multiplication law is often denoted by @xmath13 or _ the star product _ to distinguish it from the ordinary pointwise multiplication of functions . let us give an example of such star product . if we denote by @xmath14 the pointwise multiplication map @xmath15 then the star product is generally given by @xmath16 for instance , @xmath17 corresponding to the moyal product is given by @xmath18 one can easily check that the following is satisfied : @xmath19_\star = x_\mu\star x_\nu - x_\nu\star x_\mu = i\theta_{\mu\nu}.\ ] ] noncommutativity of the spacetime coordinates apparently conflicts with poincar invariance , as one can see in ( [ basic_comm ] ) . we circumvent this problem by twisting " the poincar symmetry @xcite . it goes as follows . first of all one needs to define the action of a poincar group element on a tensor product space , which is generally given by the _ coproduct_. the usual coproduct used is as follows : @xmath20 @xmath21 being an element of the poincar group . we twist the coproduct to @xmath22 action of a poincar element on star product of two functions is given by @xmath23 putting @xmath24 one can check that the commutation relation ( [ star_comm ] ) remains invariant under the action of poincar group . this modification in coproduct changes the standard hopf algebra structure associated with the poincar group ( the poincar - hopf algebra ) to the twisted poincar - hopf algebra . this twist in turn give rise to twisted statistics leading to variety of interesting consequences @xcite . we will not be discussing them here to avoid any deviation from the desired topic . in this thesis we study the gauge theories on noncommutative moyal space @xmath25 . we start by studying the implications of the construction of so - called generalized bose operators ( gbo ) @xcite in noncommutative gauge theories . gbos correspond to the reducible representation of the oscillator algebra @xmath26 = 1.\ ] ] we discuss them in detail in chapter [ instanton ] . the noncommutative theory discussed here contains commutative time . having studied gauge theories with space - space noncommutativity through the construction of gbos we proceed to understand the implications of noncommutative time coordinate . there have been even claims that quantum field theories based on noncommutative time coordinate are nonunitary @xcite . in contrast , in a series of fundamental papers , doplicher et al . @xcite have studied ( [ basic_comm ] ) in complete generality , without assuming that @xmath27 and developed unitary quantum field theories which are ultraviolet finite to all orders . in this thesis we study a simple quantum mechanical model of forced harmonic oscillator in @xmath28 dimensional noncommutative spacetime to get the essence of the effect of noncommutative time coordinate . for this we follow the unitary formulation of quantum mechanics with time - space noncommutativity discussed in @xcite . there are kinds of noncommutative spaces other than the moyal space . in particular , we study the thermodynamics in the so - called doubly / deformed special relativistic ( dsr ) theories which are related to a noncommutative space known as @xmath1-minkowski space @xcite . dsr ( doubly / deformed special relativity ) aims to search for an alternate relativistic theory which keeps a length / energy scale ( the planck scale ) and a velocity scale ( the speed of light scale ) invariant . attempts to combine gravity with quantum mechanics in search of the theory for quantum gravity always seem to give rise to the planck length @xmath29 that provides the scale at which the quantum effects of gravity will show up @xcite . the existence of such a length scale is in conflict with the equivalence principle because observers in different inertial frames will not agree on @xmath30 due to the lorentz - fitzgerald contraction . to understand in a better way consider the following : if the planck scale is to play the role of the threshold for the discrete spacetime , it may come in the quantities involving the metric ( for example the action ) . for equivalence principle to hold , i.e. , for physics to be independent of the inertial frames the action and hence the planck scale has to remain invariant under the relativistic transformation . it has been shown that it is possible to still have equivalence principle by deforming special relativity ( sr ) . these classes of theories fall under the name doubly / deformed special relativity ( dsr ) @xcite . in dsr , apart from the constancy of speed - of - light scale , the planck length @xmath30 or equivalently planck energy @xmath1 is also constant under coordinate transformation from one inertial frame to another . it was first proposed by g. amelino - camelia @xcite . to proceed one can take lessons from the transition from galilean to lorentz transformations . the galilean transformation is linear in velocity and hence it does not keep any velocity scale invariant . to have an invariant velocity scale we needed to accept the lorentz transformation which is nonlinear in velocity . thus if one wants to have planck length / energy scale invariant the relativistic transformation has to be nonlinear in position / momentum space . one also keep in mind the possibility of the symmetry group being different from the usual lorentz group . this symmetry group must act nonlinearly in the position / momentum space representation to keep planck length / energy invariant . in the special relativistic limit , planck length / energy @xmath31 the symmetry group has to go to the lorentz group . j. magueijo and l. smolin have argued that the symmetry group for the dsr transformations remains to be the lorentz group @xcite . but in dsr prescription they act nonlinearly on the position / momentum space . they have also suggested the following nonlinear representation of the lorentz transformation on momentum space @xmath32 with @xmath33 . the above transformation corresponds to the boost along @xmath34-axis . note that they also satisfy the group multiplication law of the usual lorentz group , i.e. , @xmath35 the transformation keeps the following mass invariant @xmath36 we call it the _ invariant mass_. note that unlike the special relativistic ( sr ) case , this is not same as the rest mass energy . there have been attempts to study the representation on the position space too @xcite . but there is no proper understanding of its relation to the momentum space representation . in this thesis we have used the momentum space representation given by ( [ p0_trans ] ) ( [ p2_trans ] ) which in turn implies the modification in the dispersion relation @xmath37 consequences of the modified dispersion relations on the thermodynamics are being studied extensively to infer the effect of planck scale physics @xcite . the effect of modified dispersion relations in loop - quantum - gravity on black hole thermodynamics was studied in @xcite . the same as a lorentz violating phenomena on the thermodynamics of macroscopic systems ( like white dwarfs ) @xcite and as a noncommutative phenomena on cosmology and astrophysical systems @xcite have also been studied . moreover , photon gas thermodynamics in the context of modified dispersion relations @xcite and dsr @xcite are being investigated . in @xcite the effect comes solely because of the presence of a maximum energy scale as the photon dispersion relation remains unmodified . in this thesis we study the thermodynamics of an ideal gas in the above described dsr scenario . the organization of the thesis is as follows : chapter [ instanton ] starts with describing the gbos . they correspond to reducible representations of the harmonic oscillator algebra . we demonstrate their relevance in the construction of topologically non - trivial solutions in noncommutative gauge theories , focusing our attention to flux tubes , vortices , and instantons . our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the gbo . when used in conjunction with the noncommutative adhm construction , we find that these new instantons are in general not unitarily equivalent to the ones currently known in literature . chapter [ qo ] studies the time dependent transitions of quantum forced harmonic oscillator ( qfho ) in noncommutative @xmath2 perturbatively to linear order in the noncommutativity @xmath3 . we show that the poisson distribution gets modified , and that the vacuum state evolves into a `` squeezed '' state rather than a coherent state . the time evolutions of uncertainties in position and momentum in vacuum are also studied and imply interesting consequences for modelling nonlinear phenomena in quantum optics . in chapter [ dsr ] we study thermodynamics of an ideal gas in doubly special relativity . new type of special functions ( which we call incomplete modified bessel functions ) emerge . we obtain a series solution for the partition function and derive thermodynamic quantities . we observe that dsr thermodynamics is non - perturbative in the sr and massless limits . the equation of state found makes the _ pressure vs energy density _ graph stiffer . we conclude our results in the last chapter . we take this opportunity to mention that we have corrected some typos in the notations and some results in the published version of our papers @xcite and @xcite . while the other corrections are trivial ( or they come in the middle of calculations ) we must mention the corrected results in ( [ uncertainties_x1])-([uncertainties_x1x2 ] ) . detailed investigations of noncommutative gauge theories have led to the discovery of localized static classical solutions in noncommutative spaces @xcite . among the models of gauge theories in noncommutative spaces , one of the simplest is the abelian - higgs model possessing vortex like solutions @xcite . another interesting class of solutions in noncommutative euclidean space are instantons in @xmath38 yang - mills theories . nekrasov and schwarz developed a generalization of adhm construction as given in @xcite to find these noncommutative instantons @xcite . the @xmath38 yang - mills instantons in @xmath39 and @xmath40 were further studied in @xcite . pedagogical reviews can be found in @xcite . apart from these , there are multitudes of other solutions in noncommutative gauge fields like merons @xcite , flux tubes @xcite , monopoles @xcite , dyons @xcite , skyrmions @xcite , false vacuum bubbles @xcite , to name just a few . in this chapter we present a new construction of such topological objects , based on an analysis of the _ reducible _ representations of the standard harmonic oscillator algebra . our method gives rise to new instanton solutions ( i.e. not gauge equivalent to the known adhm instantons ) on a noncommutative space , and in the process provides a simple interpretation for the instanton number : it simply counts " the number of copies of the basic irreducible representation . our construction relies on operators called generalized bose operators ( gbo ) @xcite which provide an explicit realization of the reducible representations of the oscillator algebra , and are well - known in the quantum optics literature ( see for examples @xcite ) . as a warm - up , we first study the significance of gbos in constructing fluxes and vortices with higher winding numbers , and then discuss instanton solutions in noncommutative yang mills theories . the chapter is organized as follows . we start with a brief review of gbos and their representations in section [ bg_review ] . in section [ static_soln ] , we discuss the flux tube solutions of @xcite in the language of gbos and then we go on to show the relevance of these operators in noncommutative nielsen - olesen vortices . section [ instantons ] discusses the noncommutative instantons . using the gbo in conjunction with the adhm construction , we construct a class of new instantons and compute their topological charges . notations and formalism of the theories discussed in sections [ sec : flux_sec ] , [ nov ] and [ instantons ] are all independent to each other and they should not be mixed . brandt and greenberg @xcite give a construction of generalized bose operators that change the number of quanta of the standard bose operator @xmath41 by 2 ( or more generally by a positive integer @xmath0 ) . we briefly recall their construction in this section . consider the infinite - dimensional hilbert space @xmath42 spanned by a complete orthonormal basis @xmath43 labeled by a non - negative integer @xmath44 . vectors in @xmath42 are of the form @xmath45 such that @xmath46 . we formally write @xmath47 the standard bosonic annihilation operator @xmath41 acts on this basis as @xmath48 the annihilation operator is unbounded , and hence comes with a domain of definition : @xmath49 its adjoint @xmath50 satisfies @xmath51 and has the same domain @xmath52 . the number operator @xmath53 is defined with a domain @xmath54 the basis vectors @xmath55 are eigenstates of @xmath56 : @xmath57 on @xmath58 , the operators @xmath41 and @xmath50 satisfy @xmath59=1 . \label{basiccomm}\ ] ] while @xmath56 counts the number of quanta in a state , the @xmath41 and @xmath50 destroy and create respectively a single quantum . they satisfy the following commutation relations @xmath60 = -a , \quad \left[n , a^\dagger\right ] = a^\dagger\ ] ] we denote this irreducible representation of the oscillator algebra by @xmath61 . this , up to a unitary equivalence , is the unique irreducible representation @xcite . the hilbert space @xmath42 can be split into two disjoint subspaces @xmath62 and @xmath63 given by @xmath64 @xmath65 such that @xmath66 the projection operators @xmath67 project onto the subspaces @xmath62 and @xmath63 respectively . one can define the operators @xmath68 and its adjoint @xmath69 on the subspaces @xmath70 by@xmath71 @xmath72 with domain of closure @xmath73 . on the domain @xmath74 we have @xmath75=1.\ ] ] thus @xmath76 , @xmath77 and @xmath61 are isomorphic to each other . in other words , there exist unitary operators @xmath78 such that @xmath79 using the projection operators @xmath80 , one can define an operator @xmath81 as @xmath82 on @xmath42 whose action on the basis vectors @xmath83 is @xmath84 notice that both @xmath85 and @xmath86 are annihilated by @xmath81 . the operator @xmath81 satisfies the commutation relation @xmath87 = -2b.\ ] ] the adjoint of @xmath81 satisfies @xmath88 and @xmath89 = 2b^\dagger.\ ] ] a new number operator can be defined as @xmath90 which has the same eigenstates @xmath83 , but each eigenvalue is two - fold degenerate ! we denote these eigenvalues by @xmath91 where we define @xmath92 as @xmath93 @xmath94 we rewrite the action of @xmath81 and @xmath95 in a compact form @xmath96 the operators @xmath81 and @xmath95 have @xmath52 as their domain of closure and they satisfy @xmath97 = 1\ ] ] in the domain @xmath58 . hence ( @xmath98 ) forms a reducible representation of the oscillator algebra characterized by ( [ basiccomm ] ) having ( [ irreducible_decomp ] ) as its irreducible decomposition . one can generalize the above construction and formulate an operator @xmath99 which lowers a state @xmath83 by @xmath100steps . first of all we note that @xmath42 can be split as @xmath101 for an integer @xmath0 where @xmath102 s are defined as @xmath103 we define projection operators @xmath104 as @xmath105 which project onto the subspace @xmath102 . in each subspace @xmath102 , one can define operators @xmath106 and their adjoints @xmath107 that satisfy @xmath108=1\ ] ] and hence correspond to the uir of the oscillator algebra . they act on the states as @xmath109 we can easily check that @xmath110= \left[b_i^{(k)\dagger},\lambda_i^{(k)}\right]=0\ ] ] a reducible representation is given by @xmath111 they satisfy the oscillator algebra commutation relation @xmath112=1\ ] ] on the domain @xmath58 . thus @xmath113 for @xmath114 are isomorphic to @xmath61 and hence @xmath115 forms a reducible representation of the oscillator algebra . discussions from ( [ h+ ] ) to ( [ basic_comm_k=2 ] ) represent the case @xmath116 , the simplest non - trivial example of this construction . henceforth we will use @xmath81 for @xmath117 . an explicit expression for @xmath81 is @xcite @xmath118 before we end this section , let us point out a minor generalization of the brandt - greenberg construction . under any unitary transformation @xmath78 defined on @xmath119 that transforms @xmath68 as @xmath120 , the fundamental commutation relation ( [ comm_b+- ] ) remains unchanged . in particular , if we choose @xmath121 then we find that @xmath122 i.e. , we get the translated " annihilation operator . one can construct a reducible representation using @xmath123 and @xmath124 as @xmath125 here @xmath81 gets translated by different amounts in different subspaces @xmath70 and the `` translated '' operator @xmath126 is unitarily related to @xmath81 as @xmath127 more generally , using ( [ b_red ] ) we can write @xmath128 and the unitary operator is @xmath129 though minor , this generalization will play a role in the construction of noncommutative multi - instantons . there exist other possibilities as well . for example , choosing @xmath130 gives @xmath131 the above is the well known squeezed annihilation operator . a reducible representation may be constructed : @xmath132 and more generally , @xmath133 we discussed this squeezing " operators just to give an example and will not study it in detail as they are not directly used further . fields in noncommutative moyal space are generally interpreted in terms of the simple harmonic oscillator algebra elements . having found the reducible representation of this algebra in terms of the so - called gbos , we try to seek nontrivial solutions of field theories on noncommutative spaces in terms of these operators . as we are interested in exploring the relevance of the gbos @xmath99 in noncommutative field theories , we start with the following two simple situations where this relevance is most visible : * the flux tube solution in @xmath134dimensional pure gauge theory @xcite * the vortex solution in @xmath135dimensional abelian higgs model @xcite . we find a new interpretation for the already known solutions of the above theories in terms of the gbos . [ sec : flux_sec ] consider pure @xmath136 gauge theory in @xmath137-dimensional spacetime with only spatial noncommutativity . this theory incorporates magnetic flux tube solutions @xcite which are important in the context of monopoles and strings discussed in @xcite . we will search for non - trivial solutions of the static equation of motion . these solutions do not possess a smooth @xmath138 limit @xcite , implying that they have no commutative counterpart , i.e , the origin of this effect is entirely due the noncommutativity of the underlying space . the noncommutativity is only among the space coordiantes ( time is commutative ) @xmath139 = i\theta^{ij } ; \quad \quad i , j=1,2,3.\ ] ] the anti - symmetry of the real @xmath3-matrix , i.e. , @xmath140 guarantees a choice of axes in which the noncommutativity becomes @xmath141=i\theta , \quad [ \hat{x}^1,\hat{x}^3]=0 , \quad [ \hat{x}^2,\hat{x}^3]=0\ ] ] so that only the @xmath142 plane is noncommutative . on a noncommutative space , functions " are elements of the noncommutative algebra generated by the operators @xmath143 . derivatives in the @xmath144 and @xmath145 directions are defined via the adjoint action @xmath146 , \quad \partial_{x^2}f=-\frac{i}{\theta}\left[\hat{x}^1,f\right ] \label{derivative}\ ] ] while the derivatives in the @xmath147 and @xmath148 directions are the same as in the commutative case . we can define a set of complex ( noncommuting ) variables @xmath34 and @xmath149 and a set of creation - annihilation operators as @xmath150 here @xmath41 and @xmath50 satisfy ( [ basiccomm ] ) . with this convention , the derivatives with respect to the complex coordinates are given as @xmath151 , \quad \partial_{\bar{z}}f=\frac{1}{\sqrt{\theta } } \left[a , f\right].\ ] ] integration on @xmath152 plane is replaced by trace over fock space @xmath42 : @xmath153 where @xmath83 s are the number eigenstates and the factor @xmath154 ensures the proper commutative limit ( @xmath155 ) . we will follow the construction of gauge theories on a noncommutative space as given in @xcite . in accordance with the commutative theory the field strength is defined as @xmath156\ ] ] where the noncommutative covariant derivative @xmath157 is defined as @xmath158 here @xmath159 is an anti - hermitian operator field @xmath160 the gauge transformation is such that it transforms @xmath157 covariantly @xmath161 noncommutative gauge theory has an alternative ( and possibly more natural ) formulation in terms of @xmath162 rather than @xmath163 . in terms of @xmath162 , for a pure gauge field , one works with the action @xmath164 this formulation of noncommutative gauge theory is classically equivalent to the standard formulation where action is given by @xmath165 with standard field strength given by @xmath166.\ ] ] the derivatives for @xmath167 are defined in ( [ derivative ] ) . one can verify @xmath168 the anti - symmetric levi - civita tensor is defined as @xmath169 this also implies @xmath170 now it is just a matter of simple algebra to see that @xmath171 while each term of @xmath172 is a commutator or a total derivative and hence will contribute only to boundary terms , the term with @xmath173 is only an infinite irrelevant term . as the variations in @xmath157 and @xmath159 are same , both @xmath174 and @xmath175 give the same equations of motion and hence are classically equivalent ! for static , magnetic configurations @xmath176 and with the choice @xmath177 , the equations of motion in terms of @xmath178 s reduce to @xmath179=0 , \quad \quad \mathrm{with } \quad \mathcal{d}=\frac{1}{\sqrt{2 } } ( \mathcal{d}_{1}+i\mathcal{d}_{2 } ) , \quad \mathcal{\bar{d}}=\frac{1}{\sqrt{2 } } ( \mathcal{d}_{1}-i\mathcal{d}_{2 } ) = -\mathcal{d}^\dagger . \label{eom_flux}\ ] ] it is easy to check that the standard vacuum " configuration corresponding to @xmath180 ( and hence @xmath181 ) will satisfy the equation of motion ( [ eom_flux ] ) and corresponds to @xmath182 we can construct solutions about this vacuum by taking a rotationally invariant ansatz @xmath183 and it can be shown that there exists a solution of the form ( see appendix [ flux_tube_calculation ] for a detailed calculation ) @xmath184 with @xmath56 being the number operator as given in ( [ number_operator ] ) . this solution corresponds to @xmath185 the standard field strength becomes @xmath186 thus it represents a classical localized static circular magnetic flux tube in @xmath187 direction centred about origin of the @xmath188 plane with @xmath189 related to its radius . the total magnetic flux @xmath190 gets quantized @xmath191 the choice @xmath192 corresponds to the vacuum configuration and has zero magnetic flux . by virtue of the relation ( [ basic_comm_k=2 ] ) , @xmath193 is also a vacuum solution ( though reducible ) of ( [ eom_flux ] ) . again , we can start with the ansatz @xmath194 to construct the following solution ( see appendix [ flux_tube_calculation ] for a detailed calculation ) @xmath195 here @xmath196 and @xmath197 . this solution can be re - written in the form @xmath198 with @xmath199 @xmath200 this solution is same as the higher moment solution obtained in @xcite starting with the ansatz @xmath201 . again , for the choice @xmath202 and @xmath203 , this solution reduces to ( [ d_b ] ) . furthermore , the rotationally invariant ansatz of @xmath204 about the reducible vacuum " @xmath205 for @xmath206 gives the following solution of the equation of motion @xmath207 with @xmath208 . these solutions are also same as the higher moment solutions obtained from the ansatz @xmath209 in @xcite . further computation of the magnetic field and other things for the above solution can be found in @xcite . these solutions with the gbo represent static magnetic flux tubes with localized flux , with size of the tube in the @xmath210-th irreducible part of the fock space related to @xmath211 . thus the radial profile of the magnetic field is determined by the set @xmath212 . let us try to understand the above construction in the following manner : the flux tube solution using irreducible representation @xmath61 can be described by a single integral index @xmath189 which is related to the extension of the magnetic field in the fock space ( see ( [ f_mu_nu ] ) ) . the use of reducible representation @xmath213 separates the fock space in @xmath0-parts , resulting in @xmath0 numbers of indices @xmath214 each of which is related to the extension of magnetic field in the corresponding subspace of the fock space . increase in the number of indices ( charges ) clearly expand the moduli space of the static magnetic flux tube solutions ! these solutions are nonperturbative in @xmath3 as the reducibility of the fock space and hence the notion of gbos are typically attributed to noncommutative spaces ! in this section we saw that there already exist certain solutions of noncommutative gauge theory which can be re - written in terms of the gbos . this fact shows the importance of gbos and motivates us to seek such solutions in other noncommutative gauge theories . the abelian higgs model in noncommutative spaces is of some interest because of its simplicity as a noncommutative gauge theory and the existence of vortex solutions . various topologically non - trivial vortex solutions in this context have been studied in detail . an interesting class of vortex solutions in this theory is studied in @xcite , which are analogous to the nielsen - olesen vortices in the commutative space @xcite . the model is in @xmath135dimensions , and consists of a complex higgs field @xmath190 which is a left gauge module ( the gauge fields multiply the complex higgs field @xmath190 from left and @xmath215 from right ) . minimizing the static noncommutative energy functional , bogomolnyi equations are generalized to the noncommutative space and @xmath216-expansion is done in large @xmath3 limit . the equations are then solved order by order and the corrections to the leading order equation converge rapidly . in the large distance limit ( which is the commutative limit in this case ) , the solution reduces to the nielsen - olesen vortex solution in ordinary ( commutative ) abelian higgs model . here we will give a brief formalism of the standard noncommutative theory . for extensive calculation one can refer to @xcite . the noncommutativity is same as that in section ( [ sec : flux_sec ] ) , with the only difference that now the space is @xmath217dimensional , so the direction @xmath187 is absent : @xmath141=i\theta.\ ] ] the notations defined in ( [ derivative ] ) , ( [ z_zbar ] ) , ( [ derivative_z ] ) and ( [ integration_trace ] ) remain intact . the energy functional in the static configuration is given by @xcite @xmath218 \label{energy_func}\ ] ] where @xmath178 is a covariant derivative with the gauge field @xmath219 and is given by @xmath220 note the left and right actions of the gauge fields on @xmath190 and @xmath215 respectively . the magnetic field @xmath221 is defined as @xmath222\ ] ] with the abuse of notation a dimensionful quantity @xmath223 ( @xmath224 and @xmath225 are the coefficients of gauge coupling and the self - coupling of the higgs fields respectively ) has been set to @xmath226 . for more details see @xcite . @xmath227 is the topological term defined as @xmath228 + b.\ ] ] where @xmath229 takes the values @xmath34 and @xmath149 . here @xmath230 with the convention @xmath231 . it can be shown that @xmath232 corresponds to the topological charge . our prime interest is to study the bogomolnyi equations . minimizing ( [ energy_func ] ) we get the following operator equations : @xmath233 which are the noncommutative bogomolnyi equations . now one can do a @xmath216 expansion of the higgs and the gauge fields , in the large @xmath3 limit @xmath234 the factor of @xmath235 is used for scaling the variables @xmath219 as it is a @xmath226-form . this makes sure that the fields @xmath236 and @xmath237 are dimensionless . from the definition of the magnetic field we see @xmath238 with @xmath239+\left[a^\dagger,\bar{a}_\infty\right]\right ) - \left[a_\infty , \bar{a}_\infty\right ] \label{b_infinity } \\ b_{-1 } & = & i\left(\left[a , a_{-1}\right]+\left[a^\dagger,\bar{a}_{-1}\right]\right ) - \left(\left[a_\infty,\bar{a}_{-1}\right]+\left[a_{-1},\bar{a}_\infty\right]\right).\end{aligned}\ ] ] with this expansion , we can get the leading order @xmath240 bogomolnyi equation as @xmath241 this equation admits a solution @xcite @xmath242 which represents an @xmath44-vortex at origin . a more general solution is discussed in @xcite which represents @xmath44 single vortices at @xmath44 different points in the noncommutative plane and ( [ witten_vortex ] ) is a special case of that general solution . but for our discussion , ( [ witten_vortex ] ) is sufficient and due to its simple form , computation and understanding becomes easier . for the solution ( [ witten_vortex ] ) we can check @xmath243 the next order bogomolnyi equations become @xmath244=i\bar{a}_\infty \phi_\infty , \quad \left[a^\dagger,\bar{\phi}_\infty\right ] = i\bar{\phi}_\infty a_\infty \label{eom_sublead}\ ] ] which can be solved to get ( for details see appendix [ det_gauge_field ] ) @xmath245 @xmath56 being the number operator ( [ number_operator ] ) . in the coherent state @xmath246 ( @xmath247 ) , the expectation of the field @xmath236 is @xmath248 the phase dependence is @xmath249 , which comes solely from @xmath250 as the other factor @xmath251 @xmath252 is purely real , signifying a vortex in the noncommutative plane . the large distance behavior is given by the large @xmath253 limit or equivalently large @xmath254 limit @xcite . the coherent state expectations in this limit becomes ( for details see appendix [ l_d_b ] ) @xmath255 which is exactly like the commutative @xmath44-nielsen - olesen vortex . it is interesting to note that the leading order magnetic field is ( for details see [ appendix_b ] ) @xmath256 which means that the magnetic field of the solution is localized and the magnetic fluxes are confined . the flux ( trace of the field ) is quantized and is characterized by the integer @xmath44 . we try to seek the vortex solutions in terms of the gbos . one can easily check that @xmath257 satisfies ( [ eom_lead ] ) . this gives ( details in appendix [ det_gauge_field ] ) @xmath258 the new solutions satisfy @xmath259 the expectation value in the coherent state @xmath246 ( eigenstate of @xmath41 ) gives a phase dependence of @xmath260 ( as @xmath261 can always be reduced to the form @xmath262 ) , a characteristic feature of @xmath263 vortex in noncommutative plane . in the large @xmath253 limit it gives the large distance behavior : @xmath264 which is exactly the commutative @xmath263 nielsen - olesen vortex . one can also calculate the magnetic field for the new solution to be ( see appendix [ appendix_b ] ) . @xmath265 till now all the calculated properties of the new solutions matched exactly with those of the witten s solution ( [ witten_vortex ] ) of vortex number @xmath263 . this stimulates us to compare the expression of the new solution ( [ new_vortex ] ) with that of the witten s solution . for the simplicity of expression and better understanding of the underlying algebra , we take @xmath266 in ( [ new_vortex ] ) . using the explicit expression of the gbo @xmath81 , the new vortex solution can be written as @xmath267 @xmath268 being the reducible number operator ( [ new_number_operator ] ) . further simplification can be done and the expression ( [ reduction1 ] ) for the new vortex reduces to @xmath269 the eigenvalues of the projection operators @xmath270 are @xmath271 and @xmath226 and they never contribute simultaneously . owing to this fact ( also keep in mind that @xmath80 commute with @xmath56 ) the expression ( [ reduction2 ] ) simplifies to @xmath272 which is same as the @xmath273 witten s vortex . this calculation can be generalized for any @xmath44 and it can be always shown that @xmath44-new vortex solution is same as the @xmath263-witten vortex for all @xmath44 . it is also easy to show that @xmath274 is also a solution of ( [ eom_lead ] ) and this solution is same as the @xmath275-witten vortex . note that the leading order gauge field and hence the magnetic field are determined uniquely by ( [ expression_gauge_field ] ) and ( [ b_infinity ] ) for a given higgs field . hence equality of @xmath276 to @xmath277 with vortex number @xmath275 ensures that the new gauge fields and the magnetic field are also equal to the old ones . instantons are localized finite action solutions of the classical euclidean field equations of a theory ( for a review see @xcite ) . the finite action condition is satisfied only if the lagrangian density of the theory vanishes at boundary . this in turn can lead to different topological configurations of the field characterized by its `` topological charge '' . for yang - mills theories , the instantons are further classified as self - dual ( sd ) or anti - self - dual ( asd ) with their topological charges having opposite signs . a simple prescription to construct ( anti- ) self - dual instantons in the yang - mills theory is given in @xcite . let us first review this construction . we will not distinguish between lower and upper indices in this section as the space is euclidean . in order to describe charge @xmath0 instantons with gauge group @xmath38 on @xmath278 one starts with the following data : 1 . a pair of complex hermitian vector spaces @xmath279 and @xmath280 . 2 . the operators @xmath281 , @xmath282 , @xmath283 , which must obey the equations @xmath284+[b_2,b_2^\dagger]+ii^\dagger - j^\dagger j = 0 , \quad [ b_1,b_2]+ij = 0.\ ] ] for @xmath285 , define an operator @xmath286 as @xmath287 for anti - self - dual instantons and by @xmath288 for self - dual instantons . given the matrices @xmath289 and @xmath290 obeying all the conditions above , the actual instanton solution , @xmath291 is determined by the following rather explicit formulae : @xmath292 ( @xmath293 ) where @xmath294 is the normalized mode of the operator @xmath295 @xmath296 here @xmath297 is derivative with respect to the spacetime coordinates @xmath298 which are related to the @xmath34-coordinates as @xmath299 for given adhm data and the zero mode condition ( [ zero_mode ] ) , the following completeness relation has to be satisfied @xmath300 it has been shown in @xcite that ( [ completeness ] ) can be satisfied even for noncommutative spaces . note that the fields @xmath301 are anti - hermitian , consistent with @xmath302 . the field strength @xmath303 and its dual @xmath304 are given as @xmath305 , \quad \tilde{f}_{\alpha\beta } = \frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}f_{\gamma\delta}.\ ] ] the instantons found by the adhm construction satisfy both the yang - mills equation of motion and the ( anti- ) self - duality condition @xmath306 and has topological charge given by @xmath307 the trace is over the space @xmath280 . to study instantons on a noncommutative @xmath278 , we will use the notation outlined below . the 4-dimensional noncommutative euclidean space is defined by the following noncommutative coordinates : @xmath308 = i\theta^{\alpha\beta } , \quad \alpha,\beta=1,2,3,4,\ ] ] and @xmath309 is a constant anti - symmetric @xmath310 matrix . we denote the algebra generated by these @xmath311 s by @xmath312 . there are three distinct cases one may consider : 1 . @xmath3 has rank 0 ( @xmath313 @xmath314 @xmath315 ) . in this case @xmath312 is isomorphic to the algebra of functions on the ordinary @xmath278 . this space may be denoted by @xmath316 . @xmath3 has rank 2 . in this case @xmath312 is the algebra of functions on the ordinary @xmath317 times the noncommutative @xmath317 , which may be denoted by @xmath318 . without loss of generality , we can choose @xmath319.\ ] ] let us define a system of complex coordinates and a set of operators as @xmath320 which reduces the algebra ( [ commutation_x ] ) to @xmath321=\theta , \quad [ \bar{z}_{2},z_{2}]=0 , \quad [ a_1,a_1^{\dagger}]=1 , \quad [ a_2,a_2^{\dagger}]=0 . \label{commutation_z1}\ ] ] here @xmath322 and @xmath323 are like annihilation and creation operators respectively while @xmath324 are ordinary complex numbers . we can define a number operator by @xmath325 . the fock space on which the elements of @xmath312 act , consists of states denoted by @xmath326 . here @xmath327 denotes the eigenvalues of the number operator @xmath328 and can take only non - negative integral values , while @xmath329 can be any complex number and denotes the eigenvalues of @xmath329 . 3 . @xmath3 has rank 4 . in this case @xmath312 is the noncommutative @xmath278 . we choose @xmath3 to be of the form given by @xmath330= \left [ \begin{array}{cccc } 0 & -\theta & 0 & 0 \\ \theta & 0 & 0 & 0 \\ 0 & 0 & 0 & -\theta\\ 0&0 & \theta & 0 \end{array } \right],\ ] ] where we have assumed @xmath331 . again , we can define a system of complex coordinates and a set of operators as in ( [ complex_coordinates ] ) but now the algebra ( [ commutation_x ] ) becomes @xmath321=[\bar{z}_{2},z_{2}]=\theta , \quad [ a_1,a_1^{\dagger}]=[a_2,a_2^{\dagger}]=1 . \label{commutation_z2}\ ] ] the fock space on which the elements of @xmath312 act consists of states denoted by @xmath332 . here @xmath327 and @xmath333 denote the eigenvalues of the number operators @xmath325 and @xmath334 respectively which can take only non - negative integral values . as already mentioned in section [ sec : flux_sec ] , differentiation in the noncommutative space is implimented as an adjoint as in ( [ derivative]),i.e . , @xmath335\ ] ] where @xmath336 is the inverse of the matrix @xmath337 , i.e. , @xmath338 and is given by @xmath339 differentiation with respect to the complex coordinates are @xmath340 , \quad \partial_{\bar{z}_a } f = -\frac{1}{\sqrt{\theta } } \left[a_a^\dagger , f\right ] ; \quad \quad a=1,2.\ ] ] again , the integration is implemented by a suitable trace . adhm construction for instantons has been generalized to a noncommutative space in @xcite . the construction effectively remains same as in the commutative case , only change being the replacement of 0 in the right hand side of the first equation of ( [ adhm_condition ] ) by the noncommutative parameter @xmath3 for the case of @xmath341 and by @xmath342 for the case of @xmath343 respectively : @xmath344+[b_2,b_2^\dagger]+ii^\dagger - j^\dagger j = \theta , \quad [ b_1,b_2]+ij = 0\ ] ] for @xmath341 as in @xcite and @xmath345+[b_2,b_2^\dagger]+ii^\dagger - j^\dagger j = 2\theta , \quad [ b_1,b_2]+ij = 0\ ] ] for @xmath343 . the yang - mills gauge connection @xmath346 is given by @xmath347\ ] ] the gauge field , as in section [ sec : flux_sec ] , is related to the gauge connection by @xmath348 and is given by @xmath349 both @xmath346 and @xmath350 are again anti - hermitian . in @xmath341 , the components of the gauge field along the commutative directions will be given by ( [ gauge_field ] ) , while those along the noncommutative axes by @xmath350 . let us first discuss the usual single anti - self - dual @xmath136 instanton solutions ( @xmath351 ) in @xmath352 @xcite and in @xmath343 @xcite . for @xmath353 , @xmath354 and @xmath290 are all complex numbers . as the noncommutative space ( described by the coordinates @xmath34 ) has translational invariance , we can always choose the origin in such a way that @xmath355 and @xmath356 in @xmath357 can be taken to be zero . thus ( [ adhm_condition_r2 ] ) or ( [ adhm_condition_r4 ] ) ensures that either @xmath358 or @xmath290 is zero . @xmath359 gives @xmath360 while @xmath361 gives @xmath362 for @xmath363 the two choices are related by a rotation in the plane of complex coordinates @xmath364 , namely @xmath365 . we choose @xmath361 without the loss of any generality . now let us discuss the two cases seperately . [ [ section ] ] here we get @xmath366 from the equation ( [ adhm_condition_r2 ] ) . the phase in @xmath358 does not effect the solution for the gauge field and hence has been taken to be zero . the operator ( [ d_asd ] ) for anti - self - dual instantons becomes @xmath367 and its normalized zero mode solution is given by @xmath368 with @xmath369 and @xmath370 note that the inverse of the operator @xmath371 is well defined , but that of @xmath372 is not since @xmath373 , @xmath374 is a zero mode of @xmath372 . [ [ section-1 ] ] in this case , ( [ adhm_condition_r4 ] ) gives @xmath375 and the operator in ( [ d_asd ] ) becomes @xmath376 the zero mode solution is again a 3-element column matrix @xmath377 where we write @xmath378 and @xmath379 . then ( [ zero_mode ] ) becomes @xmath380 where @xmath381 . but this operator does not have an inverse since it has a zero mode @xmath382 and hence finding @xmath383 and @xmath384 is a bit tricky . we define a shift operator @xmath175 such that @xmath385 note that although the inverse of @xmath386 is not defined otherwise , it is well defined when sandwiched between @xmath175 and @xmath387 . now we can solve for @xmath384 and @xmath383 : @xmath388 which satisfy ( [ eqn_v_xi_1 ] ) . thus we get @xmath389 we can define the components of the gauge field in terms of the complex coordinates as @xmath390 then ( [ instanton_field_connection ] ) translates to @xmath391 with @xmath392 also ( [ instanton_field ] ) translates into @xmath393 the solution for @xmath394 @xmath136 asd instanton becomes @xmath395 where the shift operator @xmath175 , written explicitly , is @xmath396 the field strengths are given by @xmath397 + i\tilde{\theta}_{\beta\gamma}\left[x^\gamma , a_{x^\alpha}\right ] + \left[a_{x^\alpha } , a_{x^\beta}\right ] = -i\tilde{\theta}_{\alpha\beta } + \left[\hat{d}_{x^\alpha } , \hat{d}_{x^\beta}\right]\ ] ] their dual are define as @xmath398 the asd condition @xmath399 translates to @xmath400 where the field strengths , in our notation of complex coordinates , are given by @xmath401 + \frac{1}{\sqrt{\theta}}\left[a_b^\dagger , a_{a}\right ] + \left[a_a , a_{\bar{b}}\right ] & = \frac{1}{\theta}\delta_{ab } + \left [ \hat{d}_a , \hat{d}_{\bar{b } } \right ] , \\ f_{ab } = & \frac{1}{\sqrt{\theta}}\left[a_a , a_{b}\right ] - \frac{1}{\sqrt{\theta}}\left[a_b , a_{a}\right ] + \left[a_a , a_{b}\right ] & = \left [ \hat{d}_a , \hat{d}_{b } \right ] , \\ f_{\bar{a}\bar{b } } = & - \frac{1}{\sqrt{\theta}}\left[a_a^\dagger , a_{\bar{b}}\right ] + \frac{1}{\sqrt{\theta}}\left[a_b^\dagger , a_{\bar{a}}\right ] + \left[a_{\bar{a}},a_{\bar{b}}\right ] & = \left [ \hat{d}_{\bar{a } } , \hat{d}_{\bar{b } } \right ] \\ & f_{\bar{b}a } = -f_{a\bar{b } } & \end{aligned}\ ] ] the equations of motion in the @xmath402 in terms of the fields @xmath350 are @xmath403=0.\ ] ] the solution ( [ usual_instanton ] ) satisfies both the asd condition and the equations of motion . we can try to get new solutions for noncommutative instantons by using the gbos . [ [ section-2 ] ] let us first define two operators @xmath81 and @xmath404 @xmath405 where @xmath328 is the number operator and @xmath406 and @xmath407 are the projection operators corresponding to @xmath322 . here @xmath81 is the generalized operator defined in ( [ expression_b ] ) . we can easily show that @xmath408 the zero - mode solution @xmath409 is given by @xmath410 but this solution is not normalized i.e. @xmath411 . usually the single instanton solution in @xmath318 with @xmath136 gauge group is normalized as @xcite @xmath412 but in our solution this technique can not be used because in this case as @xmath413 vanishes when it operates on the state @xmath85 and the inverse of @xmath414 does not exist . we fix this problem by defining @xmath415 where @xmath416 is a projection operator , and @xmath417 is a shift operator projecting out the vacuum . the operator @xmath418 can be written as @xmath419 it should be noted that the new solution in @xmath318 is completely non - singular . [ [ section-3 ] ] we claim the new solution to be @xmath420 where @xmath421 and @xmath422 here again the operator @xmath423 is not well - defined otherwise ( as @xmath424 ) , but is well - defined when sandwiched between @xmath425 and @xmath426 ( defined below ) which projects out the states @xmath427 , @xmath428 , @xmath429 and @xmath430 : @xmath431 the explicit form for the new shift operator is as follows @xmath432 the operator @xmath433 ( corresponding to @xmath434 ) is defined as in ( [ expression_b ] ) . we can check that this new solution satisfies the asd condition ( [ asd_r4 ] ) and the equations of motion ( [ eom_r4 ] ) ( for details see appendices [ new_soln_eom ] and [ new_soln_asd ] ) . the topological charge of this new solution can be shown to be 4 times the charge of the usual single asd instanton : @xmath435 ( see appendix [ topo_charge ] ) . this solution is different from the usual @xmath436 instanton for @xmath136 gauge group despite the topological charge being the same . we can understand this by observing that the difference @xmath437 between the numbers of @xmath41 s and @xmath50 s in the two solutions is not same : @xmath438 for the usual adhm solutions irrespective of its charge and the gauge group , whereas @xmath439 for the new solution ( coming solely because of the operator @xmath81 in the expression of @xmath440 given by ( [ gauge_field_r4_new ] ) ) . ( the number of @xmath41 s in @xmath377 is equal to that of @xmath50 s in @xmath441 and vice - versa . ) the new solution can not be reduced to the usual instanton by a unitary transformation and hence represents gauge inequivalent configuration . if we use the operator @xmath442 defined in ( [ translation_operator ] ) , we can construct an instanton solution by repeating the steps we have outlined above . this instanton also has charge @xmath443 as the trace of an operator is invariant under unitary transformation . then the four complex parameters @xmath444 can be thought as characterizing the `` locations '' of four instantons with charge @xmath445 . it is easy to see that in the coincident limit @xmath446 , we recover ( [ gauge_field_r4_new ] ) . we can use the above technique to find a new solution in terms of the gbo @xmath447 : @xmath448 with @xmath425 satisfying @xmath449 this solution represents an asd instanton with charge @xmath450 . again , it is gauge inequivalent to the @xmath451 instanton known in the literature . we could as well have used a different shift operator given by @xmath452 their actions are given by @xmath453 @xmath454 other multi instanton solutions can be constructed using the reducible representations involving squeezed operators ( [ reducible_squeeze ] ) . our construction of multi instantonic solutions using reducible representations of the standard harmonic oscillator algebra may also be generalized for 4k - dimensional instantons as discussed in @xcite . this exercise will be left as a future work . in this section we found multi instantons with charge @xmath455 ( @xmath456 non - negative integers ) which are not gauge equivalent to known solutions . the charge of the newly found multi instantons has an explicit relation with the representation theory labels @xmath457 and @xmath458 . thus the instanton number does have the information about the reducibility of the space along with the topological nature of the solutions . using the `` translated '' @xmath81 operators ( [ translation_operator ] ) we could construct multi instantons that depend explicitly on @xmath459 number of complex parameters . while the full moduli space of noncommutative multi instantons is still not well understood , we hope that this identification contributes partially to this question . in the previous chapter we discussed the theories in noncommutative moyal spacetime where time coordinate remained to be commutative . in such situations where only the spatial coordinates do not commute with each other , the quantum theory is conceptually straightforward ( but nonetheless may display novel phenomena ) @xcite . in this chapter we will concentrate on understanding some implications of quantum mechanics with time - space noncommutativity , specifically we will work with the moyal plane @xmath460 . we will use the formalism of unitary quantum mechanics on this space as developed by balachandran et . @xcite ( see also @xcite ) . when time and space do not commute with each other it is not unreasonable to expect that the dynamics of the time dependent processes get altered . we will verify this explicitly in the context of a simple model of the forced harmonic oscillator ( fho ) with the forcing term switched on only for a finite duration of time . in the commutative case this is a much studied model . we will compute deviations from the commutative case to leading order in @xmath3 . these deviations suggest that time - space noncommutativity can capture certain nonlinear effects seen in quantum optics . this chapter is organized as follows : in section [ bala_paper ] we will briefly review the formulation of unitary quantum mechanics on @xmath460 @xcite . in section [ problem ] we will solve the problem of the fho perturbatively in @xmath3 and compute corrections to the transition probabilities between simple harmonic oscillator ( sho ) states . these corrections suggest the noncoherent nature of the time - evolved vacuum state and are the reminiscent of those seen in nonlinear quantum optics @xcite . to flesh out this analogy better we study the time - evolution of uncertainties in position and momentum in section [ uncertainties ] . encouraged by these results we , in section [ quantum_optics ] , suggest a correspondence between the nonlinearity in quantum optics and the quantum mechanics on @xmath460 . the noncommutative space @xmath460 is described by the coordinates @xmath461 s satisfying @xmath462=i\theta\varepsilon_{\mu\nu}\mbox { with } \varepsilon_{\mu\nu}=-\varepsilon_{\nu\mu}\mbox { and } \varepsilon_{01}=1,\ ] ] where @xmath463 and @xmath464 can take values 0,1 . without loss of generality we can take @xmath465 , as its sign can always be flipped by changing @xmath466 to @xmath467 . let @xmath468 be the unital algebra generated by @xmath469 and @xmath466 . we associate to each @xmath470 , its left and right representations @xmath471 and @xmath472 : @xmath473 unless stated , we work with the left representation . + for a quantum theory , what we need are : ( 1 ) a suitable inner product on @xmath474 ; ( 2 ) a schrdinger constraint on @xmath474 ; and ( 3 ) a self - adjoint ( with respect to the inner product defined ) hamiltonian @xmath475 and observables which act on the constrained subspace of @xmath474 . _ 1 . the inner product : _ + there are several suitable inner products and they are all equivalent to each other as discussed in @xcite . here we discuss only one such example . using the commutator ( [ commutator ] ) any @xmath476 can always be written as @xmath477 . we associate a symbol @xmath478 corresponding to each such @xmath479 given by @xmath480 note that @xmath481 and @xmath482 and hence @xmath478 are purely commutative . the inner product is defined as @xmath483 _ 2 . the schrdinger constraint and time evolution : _ + the operators @xmath484 and @xmath485 , given by @xmath486 generate time and space translations respectively . the adjoint action is defined as @xmath487.\ ] ] it can be easily checked that the canonical commutation relations are satisfied : @xmath488= -i\eta_{\mu\nu } \mbox { with } \eta_{\mu\nu}=\eta_{\nu\mu}\mbox { and } \eta_{01}=0 , \eta_{00}=1,\eta_{11}=-1\ ] ] the hamiltonian @xmath475 , in general , may depend on @xmath489 and @xmath485 . the possible dependence of @xmath490 and @xmath491 can be bypassed by @xmath492 also , there is no dependence on @xmath484 assumed in the line of the commutative case where there is never such dependence of h on @xmath493 . now note that the inner product ( [ s - inner - product ] ) has an explicit dependence on the parameter @xmath148 and hence there exist more than one null vectors with respect to this inner product ( actually any vector which vanishes at @xmath494 is a null vector ) . but this fact need not bother us as we are only interested in those states that satisfy the schrdinger constraint @xmath495 the hamiltonian @xmath475 depends on @xmath489 and @xmath485 . since @xmath496 commutes with @xmath497 and @xmath485 we will choose @xmath496 as `` time '' . it is easy to write down the formal solution of the schrdinger constraint and find the time evolution . the time evolution is given by @xmath498 ( or equivalently by @xmath499 ) . thus the amount of time - translation is always commutative , though the time - operator itself is noncommutative . the time evolved wave functions satisfying the schrdinger constraint are of the form @xmath500 , where @xmath501\right ) \right|_{x_{0 } = \hat{x}_{0}^{r } } \,\ , . \label{formula_psi_2}\ ] ] as @xmath496 commutes with @xmath497 and @xmath485 , the time dependence of the time evolved wave function given above will mimic the same for the commutative case @xmath502 . we know in the commutative case a wave function which vanishes at some finite time @xmath148 , will vanish for all times . hence the only null vector satisfying the schrdinger constraint for the case of nonzero @xmath3 , is @xmath503 and there are no other non - trivial null vectors ! _ 3 . the spectral map : _ + consider a time - independent hamiltonian @xmath504 . the corresponding commutative hamiltonian is @xmath505 , with eigenfunctions @xmath506 and eigenvalues e. the spectrum of the corresponding noncommutative @xmath475 will be given by @xmath507 with the same eigenvalues @xmath508 as @xmath509 . here @xmath510 has been obtained by replacing @xmath482 with @xmath466 in @xmath511 . let us recall the dynamics of a qfho in ordinary spacetime . for a detailed discussion one can look into the section 14.6 of @xcite . the hamiltonian of this system is given by @xmath513 where @xmath14 is the mass of the particle and @xmath514 is the angular frequency of the oscillator . we are interested in real functions obeying @xmath515 at @xmath516 the hamiltonian is simple harmonic and we assume the system to be in one of the eigenstates of this sho hamiltonian . at @xmath517 the hamiltonian again becomes simple harmonic and we try to find the probability ( the transition probability ) for the system to be in any arbitrary eigenstate of the sho hamiltonian subjected to the fact that the system was in some already given eigenstate at @xmath516 . for this what we do is the following : * first we assume our system to be in an eigenstate @xmath518 at @xmath519 . * the state @xmath518 evolves under the sho hamiltonian from @xmath519 to @xmath520 . * at @xmath520 the interaction gets switched on . * the system then evolves under the full hamiltonian ( [ cfho ] ) from @xmath520 to @xmath521 . * at @xmath521 the interaction gets switched off . * the system again evolves under the sho hamiltonian from @xmath521 to @xmath522 . * we find the inner product of the final state we get at @xmath522 with the eigenstate @xmath523 . this gives the transition amplitude @xmath524 while its absolute square gives the transition probability @xmath525 . the generalization of the above hamiltonian in @xmath512 is @xmath526+g(\hat{x}_0)\hat{p}_1 = \hat{h}_0+\hat{h}_i , \label{fho}\ ] ] with @xmath527+g(\hat{x}_0)\hat{p}_1 . \label{h_0_i}\ ] ] as @xmath469 and @xmath485 commute with each other , the ordering does not matter in the last term . + to define the transitions for the above hamiltonian consider the time evolution by an amount @xmath357 . the functions @xmath528 and @xmath529 have the properties of vanishing in the far past and the far future , i.e. , @xmath530 we shall find the transition probabilities ( @xmath531 ) for an sho state `` @xmath44 '' at initial time ( @xmath532 ) to go to some other sho state `` @xmath229 '' at final time ( @xmath533 ) after evolving under the hamiltonian ( [ fho ] ) . the spectral map tells us that the energy spectrum of the sho hamiltonian in @xmath512 is same as that of the commutative one , i.e. , @xmath534 where @xmath535 is the eigenfunctions of the commutative sho hamiltonian . the orthonarmality of the eigenfunctions @xmath536 with respect to the inner product defined in section [ bala_paper ] can easily be checked and is shown explicitly in appendix [ orthonormality ] . the transition probabilities for our problem can be found by computing the same for the commutative hamiltonian obtained after replacing @xmath537 in the hamiltonian ( [ fho ] ) . here @xmath148 has come in place of the `` time '' @xmath496 which commutes with @xmath466 and @xmath485 . also , we have retained the fundamental constant @xmath538 while it was taken to be @xmath226 in section [ bala_paper ] . to linear order in @xmath3 we obtain the following commutative hamiltonian @xmath539 with @xmath540 @xmath541 and @xmath542 the function @xmath543 is related to @xmath544 and @xmath545 as @xmath546 also , @xmath41 and @xmath50 are the annihilation and creation operators respectively defined as @xmath547 the nonlinearity in the hamiltonian ( [ h ] ) is purely due to the noncommutativity . this provokes us to model certain types of nonlinear phenomena in quantum optics by the noncommutativity between time and space coordinates . this analogy will be further studied in section [ quantum_optics ] . let us now continue with calculating the transition amplitude which is given by @xmath548 where @xmath549 and @xmath550 are the time evolution operators from time @xmath148 to time @xmath551 for the hamiltonians @xmath552 and @xmath553 respectively , i.e. , @xmath554,\ ] ] the latter one being the time - ordered exponential . this gives @xmath555}\langle\phi_m|u(t_2,t_1)|\phi_n\rangle . \label{a_}\ ] ] the state @xmath556 evolves according to the schrdinger equation for the hamiltonian ( [ h ] ) @xmath557 with the initial condition @xmath558 . if we define the green s operator function @xmath559 as @xmath560 then solution of the schrdinger equation ( [ schro ] ) will be @xmath561 which in turn gives the born series @xmath562 here @xmath563 is the solution of the homogeneous equation @xmath564 which is nothing but the schrdinger equation for sho . @xmath565 has been found in the appendix [ greens ] ( see ( [ g_sol ] ) ) . note that the @xmath566-function in the expression of the @xmath565 restricts the integration over @xmath567 in ( [ born ] ) within the limit of @xmath568 to @xmath569 ( @xmath570 ) . thus , at @xmath520 the integrations are only in the intervals when the interaction was switched off , i.e. , @xmath571 . hence , we get @xmath572 ) with the initial condition ( [ phi_t1 ] ) is given by @xmath573 ) we get @xmath574 for @xmath521 we get @xmath575 we put this back in ( [ a _ ] ) to get @xmath576 with @xmath577 and @xmath578 for @xmath579 . here @xmath580 } \langle \phi_m | g(t_2,t_0 ) h_i ( t_0 ) \nonumber \\ & & g(t_0,t_1 ) h_i(t_1 ) ... g(t_{j-2},t_{j-1 } ) h_i(t_{j-1 } ) | \phi(t_{j-1})\rangle \nonumber \\\end{aligned}\ ] ] we put the expressions for all @xmath581 s as found in ( [ g_sol ] ) to get @xmath582 } \langle \phi_m|h_i^{int}(t_0)h_i^{int}(t_1 ) ... \nonumber \\ & & h_i^{int}(t_{j-1 } ) e^{\frac{i}{\hbar}h_0 t_{j-1}}|\phi(t_{j-1})\rangle.\end{aligned}\ ] ] the @xmath583 s are defined as @xmath584 we put @xmath585 ( see ( [ hom_sol ] ) ) to get @xmath586 writing @xmath587 ( see ( [ h ] ) ) and hence @xmath588 to separate out the @xmath3-dependent and independent parts up to linear order in @xmath3 , gives @xmath589|\phi_n\rangle , \label{amn}\ ] ] with @xmath590 \label{a0},\end{aligned}\ ] ] @xmath591&= & -\frac{i}{\hbar}\int_{-\infty}^{+\infty}dt_0\theta ( t_2-t_0)h_{i1}^{int}(t_0 ) \nonumber \\ & & + \left(-\frac{i}{\hbar}\right)^2\int_{-\infty}^{+\infty}dt_0\int_{-\infty}^{+\infty}dt_1\theta ( t_2-t_0)\theta ( t_0-t_1 ) \nonumber \\ & & \hspace{2 cm}.[h_{i1}^{int}(t_0)h_{i0}^{int}(t_1 ) + h_{i0}^{int}(t_0 ) h_{i1}^{int}(t_1 ) ] \nonumber \\ & & + \left(-\frac{i}{\hbar}\right)^3\int_{-\infty}^{+\infty}dt_0\int_{-\infty}^{+\infty}dt_1\int_{-\infty}^{+\infty } dt_2\theta ( t_2-t_0)\theta ( t_0-t_1)\theta ( t_1-t_2 ) \nonumber \\ & & \,\,\,\,\ , \quad \quad \quad .\left[h_{i1}^{int}(t_0)h_{i0}^{int}(t_1)h_{i0}^{int}(t_2 ) + h_{i0}^{int}(t_0)h_{i1}^{int}(t_1)h_{i0}^{int}(t_2 ) \right . \nonumber \\ & & \left . \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + h_{i0}^{int}(t_0)h_{i0}^{int}(t_1)h_{i1}^{int}(t_2)\right ] \nonumber \\ & & + .... \label{a1_series}\end{aligned}\ ] ] @xmath592 and @xmath593 are again defined in accordance to ( [ int ] ) . the arguments of @xmath594 do not come in the formal expression as the integrands of all the integrals vanish for @xmath595 . the expression for @xmath596 can be simplified to ( see appendix [ simplify_a1 ] ) @xmath597^{-1}h_{i1}^{int}(t_0)a^{(0)}(t_0,t_1).\ ] ] where @xmath598 with arbitrary arguments is defined in ( [ gint_sol ] ) . putting this in equation ( [ amn ] ) we get @xmath599^{-1}h_{i1}^{int}(t_0)a^{(0)}(t_0,t_1)\right]|\phi_n\rangle . \nonumber \\ \label{amn_a0}\end{aligned}\ ] ] here we have removed the arguments of @xmath594 as they do nt come in it s formal expression . for the next few steps we are going to use the following identity extensively : @xmath600+\frac{\lambda^2}{2!}\left[a,[a , b]\right]+\frac{\lambda^3}{3!}\left[a,\left[a,[a , b]\right]\right]+ ... \label{identity}\ ] ] first of all to find @xmath601 and @xmath602 we calculate the following commutators : @xmath603 & = & \hbar\omega \left(-z^*(t)a+z(t)a^{\dagger}\right ) \\ \left[h_0,[h_0,h_{i0}(t)]\right ] & = & \hbar^2\omega^2 h_{i0}(t ) \\ & & ... \rm{and\,\,\,so\,\,\,on}. \nonumber \\ \left[h_0,h_{i1}(t)\right ] & = & -2i\omega\sqrt{\frac{m_0\hbar\omega}{2}}\left(z^{*\prime}(t)a^2+z^{\prime}(t)a^{\dagger 2}\right ) \\ \left[h_0,[h_0,h_{i1}(t)]\right ] & = & 4\hbar\omega^2 \left(h_{i1}(t)+m_0g^{\prime}(t)h_0\right ) \\ \left[h_0,\left[h_0,[h_0,h_{i1}(t)]\right]\right ] & = & 4\hbar\omega^2[h_0,h_{i1}(t ) ] \\ & & ... \rm{and\,\,\,so\,\,\,on}. \nonumber\end{aligned}\ ] ] now , using above commutators in the identity ( [ identity ] ) gives @xmath604 and @xmath605 we put the expression of @xmath606 given above to find @xmath607 ( see ( [ gint_sol ] ) ) and follow the discussions given in pages 338 - 340 of @xcite to eliminate the time ordering . this gives @xmath608 with @xmath609\ ] ] and @xmath610 the limit @xmath611 gives the expression for @xmath612 . now to simplify the integrand in ( [ amn_a0 ] ) we find the following commutators @xmath613 & \\ & = & \\ & \frac{2}{\hbar}\sqrt{\frac{m_0\hbar\omega}{2}}\left [ \left ( -\sqrt{\frac{m_0\hbar\omega}{2}}g^{\prime}(t_0)\xi^*(t_0,t_1)+iz^{*\prime}(t_0)\xi(t_0,t_1)e^{-2i\omega t_0}\right)a\right.&\\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ , \left.+\left ( -\sqrt{\frac{m_0\hbar\omega}{2}}g^{\prime}(t_0)\xi(t_0,t_1)-iz^{\prime}(t_0)\xi^*(t_0,t_1)e^{2i\omega t_0}\right)a^{\dagger}\right ] & \\\end{aligned}\ ] ] and @xmath614 \right ] & \\ & = & \\ & -2m_0\omega g^{\prime}(t_0)|\xi(t_0,t_1)|^2+\frac{i}{\hbar}\sqrt{2m_0\hbar\omega}\left\ { z^{*\prime}(t_0)\xi^2(t_0,t_1)e^{-2i\omega t_0}-z^{\prime}(t_0)\xi^{*2}(t_0,t_1)e^{2i\omega t_0}\right\ } & \\\end{aligned}\ ] ] all higher order commutators vanish ! thus we get @xmath615^{-1}h_{i1}^{int}(t_0)a^{(0)}(t_0,t_1 ) & = & \alpha_1(t_0,t_1)\mathbf{i } \nonumber \\ & & + \alpha_2(t_0,t_1)a+\alpha_2^*(t_0,t_1)a^{\dagger } \nonumber \\ & & + \alpha_3(t_0,t_1)a^2+\alpha_3^*(t_0,t_1)a^{\dagger 2 } \nonumber \\ & & + \alpha_4(t_0,t_1)a^{\dagger}a \label{a0_h_a0}\end{aligned}\ ] ] with @xmath616 \label{alpha1 } \\ \alpha_2(t_0,t_1 ) & = & -m_0\omega g^{\prime}(t_0)\xi^*(t_0,t_1 ) \nonumber \\ & & + \frac{2i}{\hbar}\sqrt{\frac{m_0\hbar\omega}{2}}z^{*\prime}(t_0)\xi(t_0,t_1)e^{-2i\omega t_0 } \label{alpha2 } \\ \alpha_3(t_0,t_1 ) & = & \frac{i}{\hbar}\sqrt{\frac{m_0\hbar\omega}{2}}z^{*\prime}(t_0)e^{-2i\omega t_0 } \label{alpha3 } \\ \alpha_4(t_0,t_1 ) & = & -m_0\omega g^{\prime}(t_0 ) \label{alpha4}\end{aligned}\ ] ] finally we get the expression of the transition amplitude as @xmath617 , \label{amn_final}\end{aligned}\ ] ] with @xmath618 \nonumber \\ & = & { \rm real } \\ \beta_1 & = & -m_0\omega \int_{-\infty}^{+\infty}d\tau g^{\prime}(\tau)|\xi(\tau)|^2 + \frac{i}{\hbar}\sqrt{\frac{m_0\hbar\omega}{2 } } \nonumber \\ & & \quad \quad \quad \quad \quad \quad \int_{-\infty}^{+\infty}d\tau \left [ z^{*\prime}(\tau)\xi^2(\tau)e^{-2i\omega \tau}-z^{\prime}(\tau)\xi^{*2}(\tau)e^{2i\omega \tau}\right ] \label{beta1}\\ \beta_2 & = & -m_0\omega \int_{-\infty}^{+\infty}d\tau g^{\prime}(\tau)\xi^*(\tau ) + \frac{2i}{\hbar}\sqrt{\frac{m_0\hbar\omega}{2}}\int_{-\infty}^{+\infty}d\tau z^{*\prime}(\tau)\xi(\tau)e^{-2i\omega \tau } \\ \beta_3 & = & \frac{i}{\hbar}\sqrt{\frac{m_0\hbar\omega}{2}}\int_{-\infty}^{+\infty}d\tau z^{*\prime}(\tau)e^{-2i\omega \tau}\end{aligned}\ ] ] here the function @xmath619 is given as @xmath620 @xmath621 s are the matrix elements of the displacement operator @xmath622 given by @xcite @xmath623 @xmath624 are the associated laguerre polynomials . also , the limits of the integrations have been extended to @xmath568 and @xmath625 as the integrands are zero in the extended region . note that the contribution from @xmath626 given in ( [ alpha4 ] ) to the transition amplitude vanishes by virtue of ( [ bc ] ) . the transition probability is given by @xmath627 as usual . the arguments have been omitted as the formal expression of transition probability does not contain the times @xmath628,@xmath629;@xmath630,@xmath631 . for vacuum @xmath633 as the initial state , the transition amplitude is @xmath634 . \label{am0}\end{aligned}\ ] ] we have used the explicit forms of the following associated laguerre polynomials : @xmath635\end{aligned}\ ] ] the transition probability becomes ( upto linear order in @xmath3 ) @xmath636 , \label{pm0}\ ] ] with @xmath637 note that as @xmath638 , the @xmath3-correction starts dominating and in this case the expansion upto linear order in @xmath3 is no more meaningful ! hence , the above result is valid only for those @xmath229-values which are far smaller than @xmath639 ( in the unit @xmath640 ) . for @xmath138 , the transition probability becomes the well known poisson distribution as expected . + as a specific example let us work with the functions @xmath544 and @xmath545 of the form ( see figure [ fig_fg ] ) @xmath641 \\ g(t ) & = & g_0\left[\theta ( t+t)-\theta ( t - t)\right ] \end{array } \right\rbrace \quad\quad ; t>0.\ ] ] this gives @xmath642 , \quad z_0 = \sqrt{\frac{\hbar}{2m_0\omega}}\left[f_0+im_0\omega g_0\right]\ ] ] their derivatives are @xmath643 , \quad \mbox{{\rm for } } x = f , g , z.\ ] ] for these functions we calculate @xmath619 to be @xmath644 we again calculate the following quantities : @xmath645\ ] ] @xmath646 @xmath647 which further give @xmath648\ ] ] and @xmath649 thus we get @xmath650 the choices @xmath651 , @xmath652 , @xmath653 , @xmath654 and @xmath655 ( in natural units , i.e. , @xmath656 ) in commutative case ( @xmath502 ) give the following poisson distribution : @xmath657 , while for nonzero @xmath3 the probability distribution modifies to @xmath658 $ ] . the @xmath3-correction becomes of the order of the @xmath3-independent part when @xmath229 approaches the value @xmath659 . hence , our result is valid only in the region where @xmath660 . note that @xmath661 rather than @xmath235 because the @xmath662 is identically zero for the choices taken . we choose @xmath663 ( @xmath664 ) and get @xmath665.\ ] ] this deformed distribution along with the poisson distribution is shown in figure [ fig_distribution ] . such deformation of the poisson distribution suggests that the vacuum does not evolve to be a coherent state anymore . to explore this further let us look at the time - evolution of position and momentum uncertainties . the expectation value of any operator @xmath668 in a state @xmath669 at any time @xmath148 is defined to be @xmath670 also , @xmath671 hence the time evolution of the expectation value of an operator is given by that of the state in which it is being calculated . for the qfho in @xmath512 the time evolution of any operator @xmath668 will be given by @xmath672\right\rangle,\ ] ] where @xmath553 is the hamiltonian ( [ h ] ) . the uncertainty in any observable @xmath668 is given by @xmath673 for an observable not having an explicit dependence on time we will have @xmath674\rangle - 2\langle\hat{\mathcal{o}}\rangle\langle[h(t),\hat{\mathcal{o}}]\rangle\right)\ ] ] to find the evolution of @xmath675 and @xmath676 we calculate the following commutators @xmath677 = i\left(2\theta g'(t)-\frac{\hbar}{m_0}\right)p + i\theta f'(t)x - i\hbar g(t)\ ] ] @xmath678 = i\left(2\theta g'(t)-\frac{\hbar}{m_0}\right)(xp+px ) + 2i\theta f'(t ) x^2 - 2i\hbar g(t)x\ ] ] @xmath679 = i\hbar m_0\omega^2 x - i\theta f'(t ) p + i\hbar f(t)\ ] ] @xmath680 = i\hbar m_0\omega^2 ( xp+px ) - 2i\theta f'(t ) p^2 + 2i\hbar f(t ) p\ ] ] thus the evolution of @xmath675 and @xmath676 is @xmath681 let us define @xmath682 using @xmath683 = 2i\left(2\theta g'(t)-\frac{\hbar}{m_0}\right)p^2 + 2i\hbar m_0 \omega^2 x^2 + 2i\hbar f(t ) x - 2i\hbar g(t ) p\ ] ] we find the following first order coupled equations @xmath684 for vacuum as the initial state , the initial conditions for the above are @xmath685 our strategy for solving these equations is simple . we do so perturbatively in @xmath3 . we write the perturbative expansion of the solution up to linear order in @xmath3 @xmath686 splitting the equations in the part independent of @xmath3 and the part which is linear in @xmath3 we get @xmath687 with initial conditions @xmath688 and @xmath689 with initial conditions @xmath690 the solutions of the @xmath3 independent part with the given initial conditions can be easily found to be @xmath691 for all time @xmath148 . the results are also obvious from the fact that for the commutative case ( @xmath502 ) the time - evolved vacuum state is nothing but a coherent state in which the uncertainties are constants and are given by the above values . putting above in the equations for the @xmath3-part , we get @xmath692 the first two equations give @xmath693=0\ ] ] whose solution satisfying the initial conditions is given by @xmath694 we put this in ( [ d_xp ] ) to get @xmath695 differentiating ( [ d_x ] ) once and using the above equation we get the decoupled equation in @xmath696 as @xmath697 with the initial conditions @xmath698 here , the second condition comes from the equation ( [ d_x ] ) and the initial conditions of @xmath699 and @xmath700 . the solution is @xmath701 using above in ( [ d_1 ] ) and ( [ d_x ] ) respectively we get @xmath702g(\tau ) \\ \delta_{xp}^{(1)}(t ) & = & m_0\omega \left[\frac{2}{m_0 } \displaystyle{\int_{-\infty}^t}d\tau \cos\{2\omega(\tau - t)\}f(\tau)\right . \nonumber \\ & & \quad \quad \quad \left . - 2\omega\displaystyle{\int_{-\infty}^t}d\tau \sin\{2\omega(\tau - t)\}g(\tau ) - g(t)\right]\end{aligned}\ ] ] thus , the uncertainties upto linear order in @xmath3 are @xmath703 \label{uncertainties_x_final2}\end{aligned}\ ] ] @xmath704 \label{uncertainties_p_final2}\end{aligned}\ ] ] @xmath705\ ] ] the fundamental uncertainty product ( to linear order in @xmath3 ) is @xmath706 thus the vacuum state evolves to a `` squeezed state '' rather than a coherent state as in the commutative case @xcite . the uncertainties in the commutative case depend only on the product @xmath707 . but , their @xmath3-corrections change with @xmath514 even if @xmath707 is kept constant . also , the squeezing effect is oscillatory in time as is obvious from the @xmath3-dependent terms in ( [ uncertainties_x_final2 ] ) , ( [ uncertainties_p_final2 ] ) and ( [ uncertainties_xp_final2 ] ) . for the specific forms of @xmath544 and @xmath545 of ( [ fg_expression ] ) we get @xmath708 & ; \,\,-t < t < t \\ % \begin{array}{l } \sqrt{\frac{\hbar}{2m_0\omega}}-\frac{\theta}{2}\sqrt{\frac{1}{2\hbar m_0\omega } } \left[2f_0\left(\cos\{2\omega(t+t)\}-\cos\{2\omega(t - t)\}\right)\right . \\ % \hspace*{3 cm } \quad \quad \quad + \left . 2m_0g_0\omega \left(\sin\{2\omega(t+t)\}-\sin\{2\omega(t - t)\}\right)\right ] % \quad \quad % \end{array } & ; \,\,t > t \end{array}\right.\ ] ] @xmath709 \quad & ; \,\ , -t < % \begin{array}{l } \sqrt{\frac{m_0\hbar\omega}{2}}+\frac{\theta}{2}\sqrt{\frac{m_0\omega}{2\hbar } } \left[2f_0\left(\cos\{2\omega(t+t)\}-\cos\{2\omega(t - t)\}\right)\right . \\ \quad \quad \quad + \left . 2m_0g_0\omega \left(\sin\{2\omega(t+t)\}-\sin\{2\omega(t - t)\}\right)\right ] \quad \quad % \end{array } & ; \,\ , t > t \end{array}\right.\ ] ] and @xmath710 & ; \,\ , -t < t < t \\ % \begin{array}{l } \frac{\theta}{2 } \left[2f_0\left(\sin\{2\omega(t+t)\}-\sin\{2\omega(t - t)\}\right)\right . \\ \quad \quad \quad \left . -2m_0\omega g_0\left(\cos\{2\omega(t+t)\}-\cos\{2\omega(t - t)\}\right)\right ] & ; \,\ , t > t. \end{array } % \end{array } \right.\ ] ] for the choice of parameters @xmath651 , @xmath652 , @xmath653 , @xmath711 , @xmath655 and @xmath663 in natural units ( @xmath712 ) we get @xmath713 @xmath714 @xmath715 figures [ fig_deltaxdeltap ] and [ fig_deltaxp ] show the time - dependence of the different uncertainties . the discontinuities at @xmath716 is simply the manifestation of the fact that the functions @xmath544 and @xmath545 themselves are discontinuous at these times . before the interaction was switched on , the uncertainties were having values equal to those for the vacuum state . during the time of nonvanishing interaction ( and even after the interaction gets switched off ! ) , they oscillate with frequency equal to twice that of the oscillator . in quantum optics a monochromatic ( single - mode ) coherent light field is usually described by the harmonic oscillator coherent states @xcite . it has also been shown that a coherent state ( in particular the vacuum state ) remains to be coherent under the fho hamiltonian @xcite . the annihilation and creation operators for photons are related to the field quadratures @xmath717 and @xmath718 by @xmath719 @xmath717 and @xmath718 being hermitian . the commutation @xmath720=1 $ ] translates to @xmath721=\frac{i}{2}$ ] . the coherent state has different uncertainties as @xmath722 , @xmath723 and @xmath724 @xmath725 @xmath726 which is the minimum . also , the photon count ( probability for having a certain number of photons ) in the coherent state is given by the transition probabilities of the corresponding number eigenstate and the profile is poissonian . + the fho hamiltonian @xmath727 ( @xmath543 is related with @xmath544 and @xmath545 by ( [ z ] ) ) with the effective noncommutativity between time and the field quadrature @xmath717 of the form @xmath728=i\sqrt{\frac{m_0\omega}{2\hbar}}\theta \label{qo_com}\ ] ] will allow us to use the calculation of the previous sections . the photon count will be given by ( [ pm0 ] ) , while the uncertainties in the field quadratures @xmath729 , @xmath730 and @xmath731 will get modified as @xmath732 \label{uncertainties_x1}\end{aligned}\ ] ] @xmath733 \label{uncertainties_x2}\end{aligned}\ ] ] @xmath734 \label{uncertainties_x1x2}\end{aligned}\ ] ] we further study the correlation among the photons . the time - evolved vacuum state @xmath735 will give @xmath736 @xmath737 being the average number of photons in state @xmath738 . also @xmath739 this , to linear order in @xmath3 , gives the 2nd order correlation among photons with zero time delay to be equal to ( see appendix [ correlation ] ) @xmath740 for the case @xmath741 , the photons try to bunch together while for @xmath742 , they anti - bunch @xcite . for the functions ( [ fg_expression ] ) , we get @xmath743 which implies that the bunching or anti - bunching will depend only on the sign of the factor @xmath744 . for the choices taken in figures [ fig_distribution ] , [ fig_deltaxdeltap ] @xmath745 [ fig_deltaxp ] , @xmath746 and hence no bunching or anti - bunching occurs . as a future work one can try to formulate the scattering process in higher dimensions and study its implications in quantum optics . the correspondence found between noncommutativity and quantum optics also encourages one to study such possibilities in other forms of time - sapce noncommutativity . as an example one can start with assuming the spacetime dependent noncommutative parameter @xmath3 @xcite in dsr , apart from the constancy of speed - of - light scale , the planck length @xmath30 or equivalently planck energy @xmath1 is also constant under coordinate transformation from one inertial frame to another . this leads to modification in the dispersion relation . consequences of the modified dispersion relations on the thermodynamics are being studied extensively to infer the effect of planck scale physics @xcite . the present chapter aims to study the thermodynamics of an ideal gas consisting of massive particles in dsr scenario . both the modification in the dispersion relation of the constituent particles and the presence of a maximum energy scale are expected to contribute to new effects . dsr transformations can be of several type . we follow the formulation of @xcite where the modified dispersion relation becomes @xmath747 as @xmath748 , the energy of a particle with a given momentum decreases in dsr with respect to that in sr . this has consequence on the thermodynamics as we will see in @xmath749 [ thermodynamics ] . the parameter @xmath229 can be called `` invariant mass '' as it remains invariant under a dsr transformation . note that in contrary to the sr case , @xmath229 is no more the rest mass energy of the particle . to get the rest mass energy @xmath14 , we put @xmath750 in ( [ ms ] ) . we get two expressions for @xmath14 , namely @xmath751 the two solutions are connected by the redefinition of the parameter @xmath752 . henceforth , without any loss of generality we use @xmath753 the physical world is characterized by @xmath754 @xcite . in this sub - planck regime ( @xmath755 ) , the positivity of rest mass ( @xmath756 ) restricts the range of the invariant mass to @xmath757 . thus , in ( [ ms ] ) , we have @xmath758 and @xmath757 . we study the thermodynamics of an ideal gas in dsr setup . we obtain a series solution for the partition function and compute the various thermodynamic quantities . we show that our results go to the standard results in the sr limit ( @xmath759 ) @xcite as well as in the massless dsr limit @xcite . we consider a gaseous system of non - interacting particles obeying maxwell - boltzmann statistics whose macrostate is denoted by @xmath760 where @xmath56 is the number of particles in the system confined in volume @xmath761 at a temperature @xmath762 . in the canonical ensemble the thermodynamics of this system is derived from its partition function @xcite @xmath763,\ ] ] where @xmath764 and @xmath765 denotes sum over all the energy eigenvalues of the system . the total energy @xmath508 of the system can be written in terms of single particle energy @xmath766 e=_n _ , where @xmath767 is the number of particles in the single - particle energy state @xmath766 and satisfy the following condition _ n_&=&n . [ cond ] we can rewrite @xmath768 as @xmath769,\ ] ] where @xmath770 is the statistical weight factor appropriate to the distribution set @xmath771 . the summation @xmath772 goes over all distribution sets that conform to the above restrictive condition ( [ cond ] ) . for maxwell - boltzmann statistics , it can be shown @xcite z_nv , t=1n![z_1v , t]^n , where @xmath773 is the single particle partition function given by z_1v , t=_. [ zsingle ] while for ordinary spacetime , it is easy to show that in the large volume limit one can replace the sum by an integral @xcite _ d^3p , [ replace ] for more exotic spacetimes the measure of integration is expected to get modified d^3p f ( ) d^3p . [ modmeas ] hence putting together ( [ zsingle ] ) , ( [ replace ] ) , ( [ modmeas ] ) and taking @xmath774 we get z_1v , t&= & _ p=0^d^3p f ( ) . [ z1_modmeas ] note that in accordance with standard practice , we have subtracted the rest mass @xmath14 from the relativistic energy @xmath766 of the particle . although there have been few attempts@xcite , the form of @xmath775 is far from settled . assuming isotropy of spacetime we may take @xmath776 . for a possible deformation of the integration measure , @xmath777 should be expandable in taylor series in @xmath778 @xmath779 with @xmath780 since in the limit @xmath781 , @xmath782 . hence @xmath783 becomes z_1v , t&= & _ p=0^d^3p _ n=0^ ( ) ^n [ z1_final1 ] + & = & m_0-^n z_1 ^ 0v , t , [ z1_final2 ] where @xmath784 is the single particle partition function with unmodified measure z_1 ^ 0v , t=_p=0^d^3p . [ z10 ] the derivation of ( [ z1_final2 ] ) from ( [ z1_final1 ] ) involves two steps . firstly , the interchange of the summation and integration which is allowed if ( see theorem 1.38 of @xcite ) @xmath785 = \displaystyle{\sum_{n=0}^{\infty } } \frac{|a_{n}|}{n ! \kappa^{n } } \l m_{0}-\frac{\partial}{\partial \beta}\r^{n } z_1 ^ 0\l v , t\r < \infty . \label{condition_modmeas_expansion}\ ] ] secondly , writing @xmath786 & = & 4\pi \int_{0}^{\kappa}dp\,\ , p^{2 } \e^{n } \exp[-\beta \l\e - m_0\r ] \nonumber \\ & = & 4\pi \l m_{0}-\frac{\partial}{\partial \beta}\r^{n } \int_{0}^{\kappa}dp\,\ , p^{2 } \exp[-\beta \l\e - m_0\r ] \nonumber \\ & = & \l m_{0}-\frac{\partial}{\partial \beta}\r^{n } \int_{p=0}^{\kappa}d^3p\,\ , \exp[-\beta \l\e - m_0\r]\end{aligned}\ ] ] since the integrand @xmath787 $ ] remains to be continuous and bounded for @xmath788 , \beta\in [ 0,\infty]$ ] ( see @xmath789 5.12 of @xcite ) . hence our problem has boiled down to solving the integral in ( [ z10 ] ) where @xmath766 and @xmath790 are related by the modified dsr dispersion relation given in ( [ ms ] ) . the solution of @xmath791 in the massless case obtained in @xcite is z_1ml^0=(2-e^-(^2 ^ 2 + 2 + 2 ) ) . [ dsrphoton ] the term with @xmath792 makes @xmath793 non - analytic at @xmath794 . we anticipate that even when @xmath795 , @xmath791 continues to be non - analytic at @xmath794 and hence does not admit a straightforward taylor series expansion in @xmath796 . thus in order to find the leading order deviation of dsr thermodynamics from the sr case , one would require a non - trivial series expansion . the dispersion relation ( [ ms ] ) gives @xmath797^{1/2}\ ] ] @xmath798d\e\ ] ] hence changing the variable from @xmath790 to @xmath766 in ( [ z10 ] ) we get @xmath799\int_{m_0}^{\kappa}\left[\e+ \frac{m^2}{\kappa}\l1-\frac{\e}{\kappa}\r\right ] \left[\e^2-m^2\l1-\frac{\e}{\kappa}\r^2\right]^{1/2}\exp[-\beta \e]d\e . \label{z_e}\ ] ] we now consider three different regions of values of @xmath229 : [ [ case - i - mkappa ] ] case i : @xmath800 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + this case is equivalent to the case when the rest mass energy @xmath801 in this case the dispersion relation ( [ ms ] ) become @xmath802 also the partition function ( [ z_e ] ) reduces to @xmath803\int_{\kappa/2}^{\kappa}d\e(2\e-\kappa)^{1/2}\exp\left[-\beta \e\right]\nonumber\\ & = & \frac{2\sqrt 2v}{\l2\pi\r^2}\l\frac{\kappa}{\beta}\r^{3/2}\gamma\l\frac{3}{2},\frac{\beta\kappa}{2}\r , \label{zcase2}\end{aligned}\ ] ] where @xmath804 is the incomplete gamma function ( see ( 6.5.2 ) of @xcite ) . we get a very simple analytical form for @xmath805 . [ [ case - ii - kappaminfty ] ] case ii : @xmath806 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + this case is equivalent to the case when @xmath807 we change the variable in ( [ z_e ] ) from @xmath766 to @xmath808-\frac{m}{\kappa}$ ] which gives @xmath809^{3/2}}\exp\left[\beta m_0-\frac{\beta^{\prime}m^2}{\kappa}\right ] \int_{-1}^{-\kappa /m}dt \,\,\,\,\,\,t ( 1-t^2)^{1/2}\exp\left[-\beta^{\prime}mt\right]\nonumber\\ & = & -\frac{2vm^3}{\l2\pi\r^2\left[\l\frac{m}{\kappa}\r^2 - 1\right]^{3/2}}\exp\left[\beta m_0-\frac{\beta^{\prime}m^2}{\kappa}\right ] \left[i^*\left(\beta^{\prime}m,1\right)-i^*\left(\beta^{\prime}m,\frac{\kappa}{m}\right)\right ] , \label{istar}\end{aligned}\ ] ] where @xmath810 and @xmath811 . \label{istar}\ ] ] we define incomplete modified bessel function @xmath812 of order @xmath464 i_(z , y)=^ _ -y^1(1-t^2)^-dt , such that @xmath813.\ ] ] in particular for @xmath814 , using ( 3.387 ( 1 ) ) of @xcite and ( 9.6.26 ) of @xcite we get @xmath815 where @xmath816 is the 2nd order modified bessel function . let us consider * a very interesting case * of @xmath817 @xmath818 . in this limit one gets from ( [ ms ] ) = 1-^2p.thus the total energy @xmath508 of the system becomes @xmath819 and the thermodynamics simplifies . entropy can be computed by counting the total number of microstates @xmath820 available to the system _ n==v_p=0^^n = ^n , where @xmath821 is the total number of microstates available for a single particle . thus the entropy @xmath175 of the system is s=. [ s ] the first law of thermodynamics in this case becomes de =- pdv+dn . note that the usual term @xmath822 has been dropped as from ( [ s ] ) it is evident that @xmath175 is a function of @xmath56 and @xmath761 alone . the pressure of the system is zero as @xmath823 while the chemical potential is @xmath824 equation ( [ z10 ] ) can now be easily integrated to give z_1 ^ 0= , [ istarminf ] which gives the limiting behaviour of @xmath825 using ( [ istar ] ) and ( 9.6.7 ) of @xcite i^* , . [ [ case - iii-0mkappa ] ] case iii : @xmath826 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + this case is equivalent to the case when @xmath827 we change the variable in ( [ z_e ] ) from @xmath766 to @xmath828+\frac{m}{\kappa}$ ] to get @xmath829^{3/2}}\exp\left[\beta m_0+\frac{\beta^{\prime \prime}m^2}{\kappa}\right ] \int_1^{\kappa /m}dt \,\,\,\,\,\,t ( t^2 - 1)^{1/2}\exp\left[-\beta^{\prime \prime}mt\right]\nonumber\\ & = & \frac{2vm^3}{\l2\pi\r^2\left[1-\l\frac{m}{\kappa}\r^2\right]^{3/2}}\exp\left[\beta m_0+\frac{\beta^{\prime \prime}m^2}{\kappa}\right ] \left[k^*\left(\beta^{\prime \prime}m,1\right)-k^*\left(\beta^{\prime \prime}m,\frac{\kappa}{m}\right)\right ] , \nonumber \\ \label{z}\end{aligned}\ ] ] where @xmath830 and @xmath831 . \label{kstar}\ ] ] as in case ii , we define incomplete modified bessel function @xmath832 of order @xmath464 k_(z , y)=^_y^(t^2 - 1)^-dt , such that @xmath833.\ ] ] in particular for @xmath814 , using ( 9.6.23 ) and ( 9.6.26 ) of @xcite we get @xmath834 where @xmath835 is the 2nd order modified bessel function . we shall now obtain the series solution of @xmath836 . we rewrite ( [ kstar ] ) as @xmath837 inside the integral @xmath838 and for @xmath839 ( which is a valid assumption for the case of our interest ) the factor @xmath840 can be expanded in series of @xmath841 to get @xmath842 \label{kstar_series}\ ] ] with @xmath843 and @xmath844 where @xmath845 now the integral and the summation in ( [ kstar_series ] ) can be interchanged if ( see theorem 1.38 of @xcite ) @xmath846 now as @xmath847 is @xmath848 for all @xmath849 we have @xmath850 this allows us to interchange the summation and the integral if the final series is converging . thus we get @xmath851 if the above is a converging series ( see appendix [ kstar_convergence ] for convergence of @xmath852 ) . here @xmath853 for @xmath854 . + @xmath855 and @xmath856 can be easily calculated to be @xmath857 @xmath858 now , changing the variable to @xmath859 in @xmath860 for @xmath861 we get @xmath862 taking @xmath863 as first function , if we do the integration by parts again and again we finally get @xmath864 here @xmath865 is the exponential integral ( see ( 5.1.1 ) of @xcite ) . a similar attempt to obtain the series solution of @xmath866 fails ! although we obtain the solutions of @xmath791 in three different regions of values of @xmath229 , @xmath791 can be shown to be smooth in @xmath229 ( see appendix [ app1 ] ) and hence we do not expect any phase transition like thermodynamic discontinuity as we vary @xmath229 . we use the continuity of @xmath791 to obtain the limiting behaviour of @xmath867 and @xmath868 as @xmath869 . from ( [ zcase2 ] ) , ( [ istar ] ) , ( [ istar_i2 ] ) and ( 9.7.1 ) of @xcite we obtain the leading order behaviour of @xmath825 as @xmath870 to be i^*^m,^3/2e^2 , with @xmath871 . for @xmath872 , using ( [ zcase2 ] ) , ( [ z ] ) , ( [ kstar_k2 ] ) and ( 9.7.2 ) of @xcite the leading order behaviour of @xmath873 turns out to be @xmath874,\ ] ] where @xmath875 . having obtained the series solution of @xmath791 in case iii , we shall now obtain the leading order corrections from the massless and the sr cases . [ [ leading - order - deviation - from - the - massless - case ] ] leading order deviation from the massless case + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + thermodynamics of a photon gas in dsr with dispersion relation ( [ ms ] ) and unmodified measure has been worked out in @xcite . here we calculate the deviation of single particle partition function from that of a photon gas . on expanding @xmath791 in @xmath876 with @xmath877 ( assuming @xmath1 to be finite ) and using ( 9.6.10 ) and ( 9.6.11 ) of @xcite , we get z_1 ^ 0=z_1 ml^0+z_1 ml corr^0 , where @xmath878 is the single particle partition function of photon gas in dsr scenario with unmodified measure@xcite and @xmath879 is @xmath880 : z_1ml^0&=&(2-e^-(^2 ^ 2 + 2 + 2 ) ) , + z_1mlcorr^0&= & - ( ) ^4+(^5)+z_1ml^0 + ( ^2 ) . [ zm0 ] note that the correction due to mass of the constituent particle is non - perturbative in nature as the first term in @xmath881 which contains @xmath882 is the non - analytic piece and does not allow a trivial taylor series expansion at @xmath883 . we can rewrite ( [ z1_final2 ] ) as z_1=z_1 ml+z_1 ml corr , where @xmath884 @xmath885 the above leading order behaviours have been plotted in fig [ fg.z ] . for our choice of parameters they match with the numerical plots up to @xmath886 . [ [ leading - order - deviation - from - the - sr - case ] ] leading order deviation from the sr case + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + on expanding @xmath791 in @xmath876 with @xmath781 ( assuming @xmath887 to be finite ) , we get z_1 ^ 0=z_1 sr^0+z_1 sr corr^0 , where @xmath888 is the single particle partition function in sr and @xmath889 is @xmath880 . @xmath890 @xmath891 note that the dsr correction is non - perturbative in nature as the first term in @xmath889 which contains @xmath892 is the non - analytic piece and does not allow a taylor series expansion at @xmath883 . this is a novel feature in dsr as we know that sr thermodynamics is perturbative in the non - relativistic limit : z_1sr^0&=&u^1/2e^uk_2(u ) + & & v^3/21++^2 + & & = z_1nr1++^2 , where @xmath893 and @xmath894 is the single particle partition function in the non - relativistic case . we can rewrite ( [ z1_final2 ] ) as z_1=z_1 sr+z_1 sr corr , where z_1 sr&=&z_1 sr^0 , + z_1 sr corr&=&z^0_1 sr corr + z_1 sr^0- . the above leading order behaviours have been plotted in fig [ fg.z ] . for our choice of parameters they match with the numerical plots up to @xmath895 . having obtained the leading order correction to @xmath773 due to dsr , we shall now compute its effect on the various thermodynamic quantities . the free energy @xmath896 , pressure @xmath897 , entropy @xmath175 , internal energy @xmath898 , internal energy density @xmath899 and heat capacity @xmath900 are defined as f & = & -1(z_n(v,,m_0 ) ) = - n\{(z_1n)+1 } + p & = & -(fv)_n , t + s & = & -(ft)_v , n + u & = & f+ts + & = & + c_v & = & ( ut)_n , v + + + + the above quantities for case i can be found by using ( [ zcase2 ] ) and they have been plotted in fig [ fg.tq ] . now we shall obtain the leading order thermodynamics for case iii . if we denote the free energy , pressure , entropy , internal energy , internal energy density and heat capacity obtained in the sr or massless cases by @xmath901 , @xmath902 and @xmath903 respectively , and write @xmath904 , where @xmath905 or @xmath906 , we have @xmath907-{\beta n\over z_1}{\partial z_1\over \partial \beta}\nonumber\\ & = & s_{0}+{nz_{1corr}\over z_{10}}-{\beta n\over z_{10 } } \left({\partial z_{1corr}\over \partial \beta}-{z_{1corr}\over z_{10}}{\partial z_{10}\over \partial \beta}\right)+ \mathcal{o}\l\left({z_{1corr}\over z_{10}}\right)^2\r \\ u&=&u_{0}\l1-{z_{1corr}\over z_{10}}\r-{n\over z_{10}}{\partial z_{1corr}\over \partial \beta}+ \mathcal{o}\l\left({z_{1corr}\over z_{10}}\right)^2\r \\ c_v&=&c_{v0}\l1-\frac{z_{1corr}}{z_{10}}\r \nonumber \\ & & -\beta^2\left[- { u_{0}\over z_{10}}{\partial z_{1corr}\over \partial \beta}+{z_{1corr}\over z_{10}^2}u_{0}{\partial z_{10}\over \partial \beta}+ { n\over z_{10}^2}{\partial z_{10}\over \partial \beta}{\partial z_{1corr}\over \partial \beta}-{n\over z_{10}}{\partial^2 z_{1corr}\over \partial^2 \beta}\right]\nonumber\\ & & + \mathcal{o}\l\left({z_{1corr}\over z_{10}}\right)^2\r \\ \rho&=&\rho_{0}\left(1-{z_{1corr}\over z_{10}}\right)-{n\over z_{10}}{\partial z_{1corr}\over \partial \beta}+ \mathcal{o}\l\left({z_{1corr}\over z_{10}}\right)^2\r\end{aligned}\ ] ] where @xmath908 is the number density . the correction to @xmath896 depends on the ratio of @xmath909 and @xmath910 and is independent of the volume @xmath761 of the system . note that this is true to all orders . hence , the pressure @xmath897 of the system which is defined as @xmath911 gets no correction : @xmath912 the equation of state in sr is ( see @xmath913 and @xmath914 of @xcite ) p_sr=_sr3-m+m , which gives the following dsr equation of state @xmath915 in fig [ fg.tq ] , we have plotted the various thermodynamic quantities for cases i and iii as a function of @xmath762 and compared them with the sr case . the @xmath897 vs @xmath899 plots have been obtained by varying @xmath762 keeping all other parameters fixed . the qualitative natures of the plots for different cases are same . in case of @xmath175 there are two competing effects : while the cutoff tries to reduce @xmath175 by limiting the number of accessible states , the modified dispersion tries to increase @xmath175 by enhancing the boltzmann weight @xmath916 ( note that @xmath917 for a given momentum state @xmath790 , the change being more for greater value of the parameter @xmath229 ) . at low temperatures , the latter is dominant and @xmath918 . for our choice of parameters this is clearly visible for the plot of @xmath175 in case i. in the high @xmath762 regime , the cutoff effect comes into play and @xmath919 . the cutoff also saturates @xmath898 as @xmath762 increases , and @xmath920 , resulting in a steeper equation of state . here we make an interesting observation . there have been attempts to define velocity in dsr @xcite . if we adopt the usual definition for the speed of sound @xmath921 , then we observe that @xmath922 grows without any bound . possibility of such scenarios has been discussed in @xcite . for given choice of parameters in case of quantities like @xmath175 and @xmath898 , the leading order behaviours for case iii match with the numerical plots up to @xmath923 , while in case of @xmath924 which contains second order derivatives of the partition function with respect to @xmath762 , the leading order behaviours match with the numerical plots up to @xmath925 . note that the leading order behaviours have been obtained assuming @xmath926 to be finite and @xmath927 which in turn implies @xmath928 . hence as @xmath762 increases , the leading order plots depart from their numerical counterparts . in chapter [ instanton ] we described static classical solutions of noncommutative gauge theories in various spacetime dimensions and showed that the gbos are significant in constructing solutions with higher topological numbers . we started with describing the static magnetic flux tube solutions of higher moment found by polychronakos in terms of gbos . in doing so we get a better understanding of the winding numbers @xmath211 s . they correspond to the radii of the flux tube in different irreducible parts of the full fock space . the increase in the degrees of freedom ( higher number of required @xmath211 s ) due to use of reducible representation of the oscillator algebra cause to expand the solution space of the flux tube solitons . on the other hand , the vortices with higher winding numbers correspond to known solutions . the case of multi instantons is different . the multi instantons with charge @xmath455 ( @xmath456 non - negative integers ) are not gauge equivalent to known solutions . another significant result of this study is an explicit relation between the instanton number and the representation theory labels @xmath457 and @xmath458 . the charge of a multi instanton does have the information about the reducibility of the space along with the topological nature of the solutions . using the `` translated '' @xmath81 operators ( [ translation_operator ] ) we have constructed multi instantons that depend explicitly on @xmath459 complex parameters . while the full moduli space of noncommutative multi instantons is still not well understood , we hope that this identification contributes partially to this question . though we have only considered a few cases , there is actually a large variety of situations in noncommutative gauge theories where gbos may be used . in particular we expect that this procedure may shade new light on merons , monopoles , dyons , skyrmions etc . we plan to revisit some of these questions in future . in chapter [ qo ] we studied the effect of noncommutative time coordinate . to do so we took a simple quantum mechanical system with time - space noncommutativity in 1 + 1 dimension . we developed a formalism to compute the transitions between asymptotic states of the quantum mechanical system with noncommutative time . we found that for a free hamiltonian in @xmath512 which is independent of time , the transitions are equal to the same for a different hamiltonian in @xmath2 found after the replacements ( [ replace ] ) . the time evolution of an operator and its expectation value ( and hence also its uncertainty ) can also be found in a similar manner . specifically , for fho the transition probabilities get modified and is given by ( [ amn_final ] ) and ( [ pmn ] ) . the poissonian distribution for the `` vacuum to any state transition '' also gets modified and is given by ( [ pm0 ] ) . the study of uncertainties in position and momentum says that the time - evolved state is no more coherent . it gets some squeezing effect due to the noncommutativity , keeping the product of the uncertainties minimum . these uncertainties are explicitly found and is given in ( [ uncertainties_x_final2 ] ) , ( [ uncertainties_p_final2 ] ) and ( [ uncertainties_xp_final2 ] ) . the leading order corrections in these uncertainties are oscillatory in time and they depend independently on the mass of the particle @xmath14 and the frequency of the oscillator @xmath514 ( note that the commutative uncertainties depend only on the product @xmath707 ) . these results suggest a possible modelling of the noncommutativity for the nonlinear phenomena in quantum optics . the noncommutativity results in the following nonlinear effects : 1 . the photon - count gets modified from the usual poisson distribution . the uncertainties in the field quadratures change keeping the product minimum ( the squeezing effect ) . 3 . the second order correlation function @xmath929 gets modified producing new effects like bunching or anti - bunching of photons depending on the value of @xmath930 . all these observations suggest that the noncommutativity produces incoherency in the otherwise coherent field . in chapter [ dsr ] we used the dispersion relation ( [ ms ] ) and have considered the modified phase space measure ( the modification being isotropic and expandable in taylor series ) . we considered three cases separately ( @xmath931 ) . the single particle partition function has been shown to be smooth in @xmath932 ( see appendix [ app1 ] ) . for the case @xmath800 , a simple analytical form for the partition function has been obtained ( see ( [ zcase2 ] ) ) while a series solution for the partition function has been obtained for @xmath933 ( see ( [ z ] ) and ( [ kstar_mr ] ) ) . in doing so , new type of special functions ( incomplete modified bessel functions ) emerged . we observed that dsr thermodynamics is non - perturbative in the sr and massless limits . using the leading order solutions , we derived thermodynamic quantities like the free energy , pressure , entropy , internal energy and heat capacity ( see fig.[fg.tq ] ) . a stiffer equation of state has been found . we start with the ansatz @xmath934 the equation of motion ( [ eom_flux ] ) reduces to @xmath935 we write @xmath936 in the basis of number states @xmath937 to get @xmath938 we try to find a solution for which @xmath939 satisfies the first of the above equations , i.e. , @xmath940 for @xmath941 and the second of the above equations for @xmath942 . this choice leads to the localized flux tube solutions as discussed in the section [ sec : flux_sec ] . the most general solution of this type is given by @xmath943 for the choice @xmath944 we get @xmath945 which corresponds to the solution ( [ polychronakos_sol ] ) . let us now start with the ansatz @xmath946 the equation of motion reduces to @xmath947 we expand @xmath948 in the number state basis , i.e. , @xmath949 to get @xmath950 for @xmath951 and @xmath952 for @xmath953 . there exists a solution of the form @xmath954 for @xmath951 and @xmath955 for @xmath953 . here @xmath956 and @xmath957 are even and odd integers respectively . for the choice @xmath958 we get @xmath959 which corresponds to the solution ( [ flux_tube_sol_b ] ) . [ [ determination - of - a_infty - and - bara_infty - det_gauge_field ] ] determination of @xmath237 and @xmath960 [ det_gauge_field ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ we multiply @xmath961 to the first equation of ( [ eom_sublead ] ) from right and @xmath236 to the second equation of ( [ eom_sublead ] ) from left and use ( [ eom_lead ] ) to get @xmath962 for the solutions ( [ witten_vortex ] ) we get @xmath963 here we have used the identity @xmath964 in a similar fashion we get the expression for @xmath960 . the same technique can be used to compute @xmath965 and @xmath966 as in ( [ new_gauge_field ] ) . let @xmath246 be the coherent states of the operator @xmath41 , where @xmath514 is a complex number labeling the states . in the large @xmath253 limit , the expectation value @xmath967 and @xmath968 gives the large distance behavior of the solution : @xmath969 for the gauge field , @xmath970 similar thing can done for @xmath971 and @xmath966 to get ( [ large_distance2 ] ) . using ( [ expression_gauge_field ] ) in the expression of @xmath972 ( [ b_infinity ] ) we get @xmath973 \\ & = & \phi_\infty \left(a\bar{\phi}_\infty\phi_\infty a^\dagger - a^\dagger \bar{\phi}_\infty\phi_\infty a\right ) \bar{\phi}_\infty -1\end{aligned}\ ] ] using ( [ phibar_phi ] ) one can calculate @xmath974 finally for the solution ( [ witten_vortex ] ) we get the magnetic field as given in ( [ b_witten ] ) . similar technique can be used to calculate @xmath975 . the equations of motion are @xmath976=0 , \quad \alpha,\beta=1,2,3,4.\ ] ] there are four equations for different values of @xmath977 . we will show only for @xmath978 and other follow similarly . for @xmath978 the equation becomes @xmath979+[\hat{d}_{x^3 } , [ \hat{d}_{x^3},\hat{d}_{x^1}]]+[\hat{d}_{x^4 } , [ \hat{d}_{x^4},\hat{d}_{x^1}]]=0.\ ] ] in terms of the complex coordinates , @xmath980+[\hat{d}_2,[\hat{d}_{\bar{2}},\hat{d}_1]]\right)+\left([\hat{d}_{\bar{1}},[\hat{d}_{\bar{1}},\hat{d}_1]]- [ \hat{d}_{\bar{2}},[\hat{d}_{2},\hat{d}_{\bar{1}}]]\right)=0.\ ] ] now using the explicit expression for the solutions ( [ gauge_field_r4_new ] ) we get @xmath981&= & s_{new } b_1^\dagger \sqrt{\frac{m+3}{m+1 } } \sqrt{\frac{m}{m+2 } } \left ( \frac{m+2}{m } \frac{m-1}{m+1 } m_1 - \frac{m}{m+2 } \frac{m+3}{m+1 } ( m_1 + 1)\right.\nonumber \\ & & \left . - \frac{m}{m+2 } \frac{m+3}{m+1 } ( m_1 + 1)+ \frac{m+1}{m+3 } \frac{m+4}{m+2 } ( m_1 + 2)\right ) s_{new}^\dagger,\end{aligned}\ ] ] @xmath982&= & -s_{new } b_1^\dagger \sqrt{\frac{m+3}{m+1 } } \sqrt{\frac{m}{m+2 } } \left ( \frac{m+2}{m } \frac{m-1}{m+1 } m_2 - \frac{m}{m+2 } \frac{m+3}{m+1}m_2\right . \nonumber \\ & & \left . - \frac{m}{m+2 } \frac{m+3}{m+1 } ( m_2 + 1)+ \frac{m+1}{m+3 } \frac{m+4}{m+2 } ( m_2 + 2)\right ) s_{new}^\dagger.\end{aligned}\ ] ] adding above two we get @xmath983+[\hat{d}_{\bar{2 } } , [ \hat{d}_{2},\hat{d}_{\bar{1}}]]=0.\ ] ] similarly we can show @xmath984-[\hat{d}_{\bar{2}},[\hat{d}_{2},\hat{d}_{\bar{1}}]]=0 $ ] and hence the equation of motion is satisfied . the commutator between @xmath440 and @xmath985 is @xmath986 = -\frac{1}{\theta}s_{new}\left(\frac{m}{m+2}\frac{m+3}{m+1 } ( m_{a}+1)-\frac{m+2}{m}\frac{m-1}{m+1}m_{a}\right)s_{new}^\dagger.\ ] ] summing over @xmath987 we get @xmath988+[\hat{d}_2 , \hat{d}_{\bar{2}}]= -\frac{1}{\theta}s_{new}\left ( m\frac{m+3}{m+1}-(m+2)\frac{m-1}{m+1}\right)s_{new}^\dagger = -\frac{2}{\theta}.\ ] ] therefore , @xmath989+[\hat{d}_2 , \hat{d}_{\bar{2}}]=0 $ ] . similarly one can show @xmath990 . the solutions ( [ usual_instanton ] ) can be simplified to @xmath993 an arbitrary commutator becomes @xmath994 = -\frac{4}{\theta}s a_b^\dagger a_a\frac{1}{n(n+1)(n+2 ) } s^\dagger -\frac{\delta_{ab}}{\theta}s \frac{n(n+3)}{(n+1)(n+2 ) } s^\dagger\ ] ] and the components of the field strength ( and their products ) can be calculated to be @xmath995 again using ( [ d_comm_alpha_beta ] ) we get @xmath996 multiplying the above two and simplifying gives @xmath997 thus we get @xmath998 again the solutions ( [ gauge_field_r4_new ] ) can be written as @xmath999 then the commutators evaluate to @xmath1000 = -\frac{4}{\theta}s_{new } b_b^\dagger b_a\frac{1}{m(m+1)(m+2 ) } s_{new}^\dagger -\frac{\delta_{ab}}{\theta}s_{new } \frac{m(m+3)}{(m+1)(m+2 ) } s_{new}^\dagger.\ ] ] the product of the field strength becomes @xmath1001 hence the charge is @xmath1002 with @xmath1003 now for all even @xmath44 s we have @xmath1004 . hence any absolutely convergent series over @xmath327 ( or @xmath333 ) whose terms depend only on @xmath1005 and @xmath1006 can be broken into equal sums to give @xmath1007 hence @xmath1008 again @xmath1009 for all even @xmath44 s . thus the charge becomes @xmath1010.\end{aligned}\ ] ] redefining @xmath1011 and @xmath1012 , we get @xmath1013 the eigenfunctions of the noncommutative sho hamiltonian @xmath1014 given by ( [ h_0_i ] ) are given in ( [ en_psin ] ) . if we define @xmath1015 such that @xmath1016 we can write @xmath1017 e^{ik_1\hat{x}_1}e^{ik_0\hat{x}_0}\ ] ] hence @xmath1018 e^{ik_1x_1}e^{ik_0x_0 } \nonumber \\ & = & \phi_n ( x_1 ) e^{-i\omega\left(n+\frac{1}{2}\right)x_0},\end{aligned}\ ] ] @xmath535 being the orthonormal eigenfuntions in the commutative case @xmath502 . the inner product of two eigenfunctions become @xmath1019 expanding @xmath559 as a fourier integral we get @xmath1020 with @xmath1021 thus @xmath1022 here , @xmath1023 has been introduced to avoid the pole on the contour ( real axis ) . after finding the integral inside the summation using the complex analysis we get the green s function to be @xmath1024 where @xmath1025 is the heaviside step function . to simplify ( [ a1_series ] ) what we do is to find the differential equation for @xmath1026 with independent variable @xmath148 and solve it with proper initial conditions . the differential equation has been found to be @xmath1027 which is a first order equation and hence @xmath1026 is unique if an initial condition is given . the initial condition comes from the fact that @xmath1028 and @xmath1026 must become identity and zero respectively for @xmath520 , i.e. , no interaction . thus , @xmath1029 . to solve the equation we define the green s operator function @xmath1030 as @xmath1031 now generalizing solution ( [ g_sol ] ) for the time - dependent case we get @xmath1032 = -\frac{i}{\hbar}\theta ( t - t_0)a^{(0)}(t , t_0 ) . \label{gint_sol}\ ] ] it can be easily checked that the above expression for the @xmath1030 satisfies the corresponding differential equation . the solution for @xmath1026 is then given by @xmath1033 where @xmath1034 is the solution of the homogeneous equation @xmath1035 the @xmath566-function in the expression of @xmath1030 and the fact that interaction was off before @xmath630 , with the initial condition for @xmath1029 , gives the initial condition for @xmath1034 , i.e. , @xmath1036 . the only solution of the homogeneous equation with this initial condition is @xmath1037 . thus we get @xmath1038 now , putting the expression of @xmath1030 above and introducing @xmath1039^{-1}$ ] before @xmath1040 , we get ( [ a1_a0 ] ) . here we have also used the following property of @xmath1028 : @xmath1041 an operator corresponding to the detection of a photon by a detector should be proportional to the annihilation operator @xmath41 ( say @xmath1042 ) @xcite . hence if @xmath1043 is the initial state of the radiation field , the state after the detection of one photon is @xmath1044 . the amplitude for going to the final state @xmath1045 is given by @xmath1046 . the corresponding probability is @xmath1047 . thus the probability of detection of one photon in the state @xmath1043 @xmath1048 similarly , probability of detection of two photons with a time delay of @xmath357 is @xmath1049 the 2nd order correlation function with a time delay @xmath357 is defined as @xmath1050 for @xmath1051 @xmath1052 for a coherent state it can be calculated to be equal to 1 . @xmath852 given by ( [ kstar_mr],[m0],[m1],[mr_series ] ) is convergent if the following two series are convergent : @xmath1053 and @xmath1054 @xmath175 can be easily proved to be absolutely convergent using ratio test . for @xmath1055 first consider the following double series : @xmath1056 let us first test the convergence of @xmath1057 ( see theorem ( 2.7 ) of @xcite ) . the row series @xmath1058 ( for a fixed @xmath1059 ) and the column series @xmath1060 ( for a fixed @xmath0 ) are defined as @xmath1061 @xmath1062 the ratio tests for @xmath1058 and @xmath1060 show that they are absolutely convergent ( for @xmath839 ) . also @xmath1063 . hence , @xmath1057 is absolutely convergent . now @xmath1064 as @xmath1065 is convergent ( or in other words @xmath1066 is absolutely convergent ) we must have @xmath1067 to be convergent ( or in other words @xmath1055 to be absolutely convergent ) . thus the series expansion of @xmath852 is absolutely convergent . we shall show that @xmath791 is continuous in @xmath14 for @xmath1068 $ ] . after integrating over the angular coordinates ( [ z10 ] ) gives @xmath1069 } = \frac{2v}{\l2\pi\r^2}\int_{0}^{\kappa}dp \,\,\ , f(p , m_0 ) . \label{z10_m0}\ ] ] the integrand @xmath1070}$ ] is a continuous bounded function of @xmath790 and @xmath14 in the range @xmath1068 , p\in [ 0,\kappa]$ ] . thus @xmath1071 is a continuous function of @xmath14 as the function @xmath1072 , where @xmath1073 is the upper bound of @xmath1074 in the range @xmath1068 , p\in [ 0,\kappa]$ ] , satisfies @xmath1075 for all @xmath1068 , p\in [ 0,\kappa]$ ] and is integrable as @xmath1076 ( see lemma 1 in @xmath789 5.12 of @xcite ) . + the derivative of the integrand with respect to @xmath14 is given by @xmath1077 it has 2 poles ( and also branch points ) in the complex @xmath790-plane at @xmath1078 . we note that the poles and the branch points remain to be at the same positions for all higher order derivatives of @xmath1079 with respect to @xmath14 . for @xmath1080 both the poles are at @xmath750 and as @xmath14 increases the poles separate towards the imaginary axis . they keep on moving on the imaginary axis till they reach @xmath1081 at @xmath1082 . after that they start to come closer to each other on the real line and finally at @xmath1083 they stop at @xmath1084 . note that for all @xmath1085 the poles are never on the contour of integration ( the real line from @xmath750 to @xmath1086 ) and the functions @xmath1087 remain to be bounded . this ( by the same argument as given in the case of the continuity of @xmath1071 ) ensures the infinite - order differentiability of @xmath1071 in @xmath1085 and the derivatives can be found by using the leibniz rule ( see lemma 2 in @xmath789 5.12 of @xcite ) . note that the fact that we are not being able to say about the differentiability of @xmath791 at @xmath1080 could be a relic of the non - analytic part in ( [ zm0 ] ) . d. bahns , s. doplicher , k. fredenhagen and g. piacitelli , commun . phys . * 237 * , 221 ( 2003 ) [ arxiv : hep - th/0301100 ] . d. bahns , s. doplicher , k. fredenhagen and g. piacitelli , phys . d * 71 * , 025022 ( 2005 ) [ arxiv : hep - th/0408204 ] . h. grosse , c. klimcik and p. presnajder , int . j. theor . * 35 * , 231 ( 1996 ) [ arxiv : hep - th/9505175 ] . h. grosse , c. klimcik and p. presnajder , commun . phys . * 185 * , 155 ( 1997 ) [ arxiv : hep - th/9507074 ] . h. grosse , c. klimcik and p. presnajder , commun . phys . * 178 * , 507 ( 1996 ) [ arxiv : hep - th/9510083 ] . h. grosse , c. klimcik and p. presnajder , commun . phys . * 180 * , 429 ( 1996 ) [ arxiv : hep - th/9602115 ] . s. baez , a. p. balachandran , b. ydri and s. vaidya , commun . * 208 * , 787 ( 2000 ) [ arxiv : hep - th/9811169 ] . a. p. balachandran and s. vaidya , int . j. mod . phys . a * 16 * , 17 ( 2001 ) [ arxiv : hep - th/9910129 ] . d. bak , k. m. lee and j. h. park , phys . d * 63 * , 125010 ( 2001 ) [ arxiv : hep - th/0011099 ] . g. s. lozano , e. f. moreno and f. a. schaposnik , phys . b * 504 * , 117 ( 2001 ) [ arxiv : hep - th/0011205 ] . m. f. atiyah , n. j. hitchin , v. g. drinfeld and yu . i. manin , phys . lett . * 65a * , 185 ( 1978 ) .

_ there is a growing consensus among physicists that the classical notion of spacetime has to be drastically revised in order to find a consistent formulation of quantum mechanics and gravity . one such nontrivial attempt comprises of replacing functions of continuous spacetime coordinates with functions over noncommutative algebra . dynamics on such noncommutative spacetimes ( noncommutative theories ) are of great interest for a variety of reasons among the physicists . additionally arguments combining quantum uncertainties with classical gravity provide an alternative motivation for their study , and it is hoped that these theories can provide a self - consistent deformation of ordinary quantum field theories at small distances , yielding non - locality , or create a framework for finite truncation of quantum field theories while preserving symmetries . in this thesis we study the gauge theories on noncommutative moyal space . we find new static solitons and instantons in terms of the so - called generalized bose operators ( gbo ) . gbos are constructed to describe reducible representation of the oscillator algebra . they create / annihilate @xmath0-quanta , @xmath0 being a positive integer . we start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of gbos . the nielsen - olesen vortex solutions found in terms of these operators also reduce to the ones known in the literature . on the other hand , we find a class of new instanton solutions which are unitarily inequivalent to the ones found from adhm construction on noncommutative space . the charge of the instanton has a description in terms of the index representing the reducibility of the fock space representation , i.e. , @xmath0 . after studying the static soliton solutions in noncommutative minkowski space and the instanton solutions in noncommutative euclidean space we go on to study the implications of the time - space noncommutativity in minkowski space . to understand it properly we study the time - dependent transitions of a forced harmonic oscillator in noncommutative 1 + 1 dimensional spacetime . we also provide an interpretation of our results in the context of non - linear quantum optics . we then shift to the so - called dsr theories which are related to a different kind of noncommutative ( @xmath1-minkowski ) space . dsr ( doubly / deformed special relativity ) aims to search for an alternate relativistic theory which keeps a length / energy scale ( the planck scale ) and a velocity scale ( the speed of light scale ) invariant . we study thermodynamics of an ideal gas in such a scenario . _ in first chapter we introduce the subjects of the noncommutative quantum theories and the dsr . chapter [ instanton ] starts with describing the gbos . they correspond to reducible representations of the harmonic oscillator algebra . we demonstrate their relevance in the construction of topologically non - trivial solutions in noncommutative gauge theories , focusing our attention to flux tubes , vortices , and instantons . our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the gbo . when used in conjunction with the noncommutative adhm construction , we find that these new instantons are in general not unitarily equivalent to the ones currently known in literature . chapter [ qo ] studies the time dependent transitions of quantum forced harmonic oscillator ( qfho ) in noncommutative @xmath2 perturbatively to linear order in the noncommutativity @xmath3 . we show that the poisson distribution gets modified , and that the vacuum state evolves into a `` squeezed '' state rather than a coherent state . the time evolutions of uncertainties in position and momentum in vacuum are also studied and imply interesting consequences for modelling nonlinear phenomena in quantum optics . in chapter [ dsr ] we study thermodynamics of an ideal gas in doubly special relativity . we obtain a series solution for the partition function and derive thermodynamic quantities . we observe that dsr thermodynamics is non - perturbative in the sr and massless limits . a stiffer equation of state is found . we conclude our results in the last chapter . to + _ bhaiji ( shri dhiraj kumar mishra ) + and + _ pappaji ( shri karunakar dutt ) _ _ i hereby declare that the work presented in this thesis entitled `` topics in noncommutative gauge theories and deformed relativistic theories '' is the result of the investigations carried out by me under the supervision of prof . sachindeo vaidya at the centre for high energy physics , indian institute of science , bangalore , india , and that it has not been submitted elsewhere for the conferment of any degree or diploma of any institute or university . keeping with the general practice , due acknowledgements have been made wherever the work described is based on other investigations . + dated : nitin chandra + sachindeo vaidya + associate professor + centre for high energy physics , + indian institute of science , + bangalore - 560012 , + karnataka , india + first and foremost i thank my phd adviser sachindeo vaidya , whose guidance made my ph.d . possible . despite of my slow progress in research , he has been very patient , cooperative and helpful . timely completion of this thesis is a result of his support and encouragement . i also acknowledge my sincere thanks to manu paranjape of universit de montreal . i simply ca nt explain in words how wonderful person he is . his friendly behaviour is the prime reason for my stay in montreal being a wonderful and memorable one . i thank iisc and udem for giving me an opportunity to pursue my research and learn a lot in the process . i also thank canadian commonwealth scholarship program and iisc for providing the financial assistance . i thank all chep members , especially the student friends for all the fun including baba and chicken parties . thanks to all my friends who were also my collaborators . i learnt a lot about independent research while working with sandeep on the problem of dsr . thanks to all the friends from iisc including those in physics , bihari samiti , my rkm friends , integrated phd batch - mates , b - mess friends for making this 7 year - long stay enjoyable . i also want to thank all my friends outside iisc in bangalore for the memorable moments i spent with them . presence of my friend sp makes me forget all my worries . i am thankful to all my childhood friends who kept contact with me throughout these years . i thank pappu bhaiya and his family . they supported me as local guardians throughout my stay in bangalore . i dedicate my thesis to bhaiji and pappaji . bhaiji is the person with whom i discussed all my curiosities in science when i was a child . pappaji taught me english when i needed it most . finally and most importantly i am grateful to my parents and family . i am forever indebted to their love and support which gives me greatest satisfaction and happiness . 1 . _ time dependent transitions with time - space noncommutativity and its implications in quantum optics _ + n. chandra + j. phys . a a * 45 * , 015307 ( 2012 ) [ arxiv:1104.1059 [ hep - th ] ] . 2 . _ thermodynamics of ideal gas in doubly special relativity _ + n. chandra and s. chatterjee + phys . rev . d * 85 * , 045012 ( 2012 ) [ arxiv:1108.0896 [ gr - qc ] ] . 3 . _ noncommutative vortices and instantons from generalized bose operators _ + n. acharyya , n. chandra and s. vaidya + jhep * 1112 * , 110 ( 2011 ) [ arxiv:1109.3703 [ hep - th ] ] . the metric for @xmath4 minkowski space has been taken to be @xmath5 where the first element comes for time coordinate . the abbreviations used in the thesis are * gbo : _ generalized bose operator _ * uir : _ unitary irreducible representation _ * adhm : _ atiyah , drinfeld , hitchin and manin _ * sd : _ self dual _ * asd : _ anti self dual _ * sho : _ simple harmonic oscillator _ * fho : _ forced harmonic oscillator _ * qfho : _ quantum forced harmonic oscillator _ * sr : _ special relativity _ * dsr : _ doubly / deformed special relativity _

let @xmath0 be a right - angled bounded convex polytope in hyperbolic space @xmath6 , and let @xmath2 be the group generated by reflections across codimension - one faces . for any torsion - free subgroup @xmath7 of finite index , the quotient @xmath4 is a closed hyperbolic manifold which is an orbifold cover of @xmath0 . we call this manifold a _ small cover of @xmath0 _ ( as in @xcite ) if the index of @xmath5 is minimal . examples of small covers include the first closed hyperbolic @xmath8-manifold to appear in the literature @xcite as well as its generalizations @xcite . the point of this note is to show that if @xmath0 is regular then any small cover of @xmath0 is uniquely determined up to isometry ( and up to homeomorphism when @xmath9 ) by @xmath5 ( modulo symmetries of @xmath0 ) . in fact , there are only two right - angled regular hyperbolic polytopes with dimension @xmath10 , the dodecahedron and the @xmath1-cell . we conclude the paper by showing that up to homeomorphism there are exactly @xmath11 small covers of the dodecahedron , and that there is a unique small cover of the @xmath1-cell with minimal complexity ( in the sense of section [ s : examples ] , below ) . there are other hyperbolic manifolds based on the dodecahedron @xcite and the @xmath1-cell @xcite appearing in the literature , but these are obtained by identifying faces of a single copy of the polytope and require that it have dihedrals angles of @xmath12 . analogous constructions fail for right - angled realizations of these polytopes , and small covers provide a natural alternative . let @xmath0 be an @xmath13-dimensional right - angled convex polytope in @xmath6 , and let @xmath14 denote the set of facets ( i.e. , codimension - one faces ) of @xmath0 . for each @xmath15 , we let @xmath16 denote the reflection across @xmath17 , and we let @xmath2 be the group generated by @xmath18 . defining relations for @xmath2 are @xmath19 for all @xmath17 and @xmath20 whenever @xmath21 . following @xcite , we call an epimorphism @xmath22 a _ characteristic function _ if whenever @xmath23 are facets that all meet at a vertex , the images @xmath24 form a @xmath25-basis . if @xmath5 is a torsion free subgroup of @xmath2 , then its index is @xmath26 . if the index is equal to @xmath27 , then @xmath5 is normal and is the kernel of a characteristic function @xmath22 . we consider the reflection tiling of @xmath6 corresponding to @xmath0 and @xmath2 . if @xmath28 is a codimension-@xmath29 cell in this tiling , then there are exactly @xmath30 maximal cells containing @xmath28 , and the stabilizer @xmath31 is isomorphic to @xmath32 . if @xmath5 is torsion - free , then the natural map @xmath33 must be injective ; hence , the index of @xmath5 is at least @xmath27 . if the index is equal to @xmath27 , then for any @xmath34-cell @xmath35 , the map @xmath36 is a bijection . we let @xmath37 be the composition @xmath38 it is not _ a priori _ a homomorphism , but since the defining relations for @xmath2 also hold for the images @xmath39 , there is an induced epimorphism @xmath40 defined by @xmath41 , for all @xmath42 . it is clear that if @xmath37 and @xmath43 agree on all but one element in the stabilizer of a cell , then they must also agree on the entire stabilizer . using this fact and the fact that @xmath37 and @xmath43 agree on the generators of @xmath2 , it follows that @xmath44 . in particular , @xmath37 is an epimorphism and @xmath5 is its kernel . that @xmath37 is a characteristic function follows from the fact that @xmath45 must be a bijection _ for every @xmath35_. let @xmath22 be a characteristic function , and let @xmath46 . the _ small cover of @xmath0 associated to @xmath37 _ is the closed hyperbolic manifold @xmath47 . small covers are functorial in the following sense . [ prop : subface ] let @xmath48 be a small cover of @xmath0 . for any face @xmath17 of @xmath0 , let @xmath49 be the hyperbolic subspace spanned by @xmath17 and let @xmath50 be the image of @xmath51 in @xmath48 ( that is , @xmath52 ) . then @xmath50 is a totally geodesic submanifold which is itself a small cover of the face @xmath17 . the stabilizer @xmath53 fixes @xmath51 pointwise and is mapped isomorphically by @xmath37 onto a codimension-@xmath29 subspace @xmath54 ( where @xmath55 ) . each facet of @xmath17 can be expressed uniquely as the intersection @xmath56 where @xmath57 is a facet of @xmath0 that is orthogonal to @xmath17 , and we let @xmath58 denote the subgroup of @xmath2 generated by the corresponding reflections @xmath59 . the characteristic function @xmath37 induces a characteristic function @xmath60 for the polytope @xmath17 with kernel @xmath61 . it follows that @xmath62 . let @xmath63 ( @xmath64 ) denote the symmetry group of @xmath0 , and let @xmath65 . we say that two small covers @xmath48 and @xmath66 are _ equivalent _ if @xmath67 for some @xmath68 and @xmath69 . any small cover @xmath48 has a natural ( totally geodesic ) cell decomposition induced by the tiling of @xmath6 . [ prop : equivalence ] two small covers @xmath48 and @xmath66 are equivalent if and only if they are isomorphic as cell complexes . it is clear that equivalent small covers are isomorphic as cell complexes . conversely , suppose @xmath70 is an isomorphism of cell complexes . then the lift @xmath71 is an automorphism of the tiling of @xmath6 such that @xmath72 . since the automorphism group of the tiling is a semi - direct product of @xmath2 and @xmath63 , there exists an @xmath68 such that @xmath73 for some element @xmath74 . it follows that @xmath75 ( they are both normal ) , hence @xmath67 for some @xmath69 . let @xmath0 be a right - angled hyperbolic polytope as above . @xmath0 is _ regular _ if its symmetry group acts transitively on the simplices in its barycentric subdivision . any such simplex must be a hyperbolic coxeter simplex , and by examining the standard list of these simplices ( bourbaki @xcite , p. 133 ) one can show that @xmath0 must be either a polygon with @xmath76 sides , the dodecahedron , or the ( @xmath77-dimensional ) @xmath1-cell . an important property of these regular polytopes is the following : [ lem : connectingpath ] let @xmath0 be a regular right - angled hyperbolic polytope , and let @xmath78 be the length of an edge in @xmath0 . then any path in @xmath0 that connects non - adjacent facets has length @xmath79 with equality holding only if the connecting path is an edge of @xmath0 . let @xmath80 be a connecting path with endpoints lying on the ( disjoint ) facets @xmath17 and @xmath57 . because @xmath0 is regular , it has a `` cubical '' decomposition obtained by cutting each edge with the orthogonal hyperplane through its midpoint ( see figure [ fig : pminpath.fig ] ) . we define two open neighborhoods @xmath81 and @xmath82 of the facet @xmath17 as follows . @xmath81 is the union of the interiors of all cubes that intersect @xmath17 , and @xmath82 is the set of all points of @xmath0 whose distance to @xmath17 is less than @xmath83 . we claim that @xmath84 . to see this , it suffices to note that if @xmath85 is an edge meeting @xmath17 othogonally at a vertex and @xmath86 is the hyperplane bisecting @xmath85 , then @xmath82 lies entirely on one side of @xmath86 ( since the edge @xmath85 , being a common perpendicular to @xmath86 and @xmath17 , realizes the shortest distance between the corresponding hyperplanes ) . similarly , if @xmath87 and @xmath88 are the corresponding neighborhoods of @xmath57 , we have @xmath89 . since @xmath17 and @xmath57 are not adjacent , @xmath90 and , hence , @xmath91 . this means @xmath92 and since the edge @xmath85 is the _ unique _ path of minimal length joining @xmath17 to @xmath86 , the equality @xmath93 is only possible if @xmath80 is an edge of @xmath0 . to see that the lemma can fail without the regularity hypothesis , let @xmath0 be any right - angled @xmath13-dimensional polytope and let @xmath94 be a minimal length edge . by gluing @xmath95 copies of @xmath0 together around the edge @xmath85 , one obtains a right - angled polytope @xmath96 with @xmath85 now being an interior connecting path that is as short as any edge of @xmath96 . we are now in a position to prove the main theorem . the argument is based on the fact that an isometry must preserve the set of minimal length closed geodesics . to simplify the exposition , we call a closed geodesic in a small cover an _ edge loop _ if it is of the form @xmath97 ( proposition [ prop : subface ] ) for some edge @xmath94 . [ thm : main ] let @xmath0 be an @xmath13-dimensional right - angled regular hyperbolic polytope . then two small covers of @xmath0 are isometric if and only if they are equivalent . first we show that if @xmath98 is any small cover of @xmath0 , then any closed geodesic of minimal length must be an edge loop . let @xmath80 be a closed geodesic in @xmath98 , and let @xmath17 be a codimension - one cell in @xmath98 that intersects @xmath80 transversely . lifting @xmath80 to the universal cover @xmath6 , we obtain a geodesic segment @xmath99 that connects two lifts @xmath100 and @xmath101 of the cell @xmath17 ( see figure [ fig : ucover.fig ] ) . let @xmath102 and @xmath103 be the hyperplanes spanned by @xmath100 and @xmath101 , respectively . since @xmath102 and @xmath103 are both mapped to the face @xmath17 under the projection @xmath104 , they are hyperparallel , thus have a ( unique ) common perpendicular @xmath105 . moreover , @xmath105 must pass through at least two copies of the tile @xmath0 . it follows from lemma [ lem : connectingpath ] that the length of @xmath105 is @xmath106 with equality holding only if @xmath105 is the lift of an edge . since the geodesic @xmath99 is at least as long as @xmath105 with equality holding only if @xmath107 , the closed geodesic @xmath80 will have minimal length only if it is an edge loop . now suppose @xmath70 is an isometry . since @xmath108 takes minimal length closed geodesics to minimal length closed goedesics , it must take edge loops to edge loops . the @xmath34-cells in a small cover can be characterized as the points where edge loops intersect , thus @xmath108 must take @xmath34-cells to @xmath34-cells and , therefore , @xmath109-cells to @xmath109-cells . since any cell of dimension @xmath110 in a small cover can be characterized as the convex hull of its bounding @xmath109-cells , the isometry @xmath108 must take cells to cells . thus , by proposition [ prop : equivalence ] , @xmath48 and @xmath66 are equivalent . mostow rigidity gives the following : if @xmath0 is the dodecahedron or the @xmath1-cell , then two small covers of @xmath0 are homeomorphic if and only if they are equivalent . in this section , we describe an algorithm for enumerating equivalence classes of small covers for general @xmath0 , and apply it to the dodecahedron and @xmath1-cell . the complexity of the algorithm becomes unfeasible for the @xmath1-cell , so instead we apply it to a restricted class of characteristic functions . * the algorithm . * let @xmath0 be @xmath13-dimensional , and let @xmath111 be any ordering of the facets such that the first @xmath13 facets @xmath23 all meet at a vertex . given any characteristic function @xmath22 , we let @xmath112 be the image @xmath113 of the fundamental reflection across the @xmath114th facet . by definition a @xmath115-tuple @xmath116 determines a characteristic function if and only if the @xmath112 s corresponding to facets meeting at a vertex form a basis for @xmath117 . we call such a @xmath118 a _ labeling of @xmath0 _ and each @xmath112 a _ label_. we say a labeling is _ normalized _ if the first @xmath13 labels form the standard basis for @xmath117 , and we let @xmath119 denote the set of normalized labelings of @xmath0 . the following algorithm determines @xmath119 . 1 . assume by induction that @xmath123 have been chosen , and let @xmath124 be the list @xmath121 with all elements of the following form removed : @xmath125 where that @xmath126 for @xmath127 and @xmath128 . 2 . if @xmath129 we know that there are no ( unrecorded ) labelings that begin @xmath130 . in this case , we back up until we find a nonempty @xmath131 with @xmath132 ( and @xmath133 as large as possible ) . we let @xmath134 be the first element in @xmath135 and go back to step 1 , using @xmath133 instead of @xmath114 . if @xmath136 , we let @xmath137 be the first element in the list @xmath124 and remove it from @xmath124 . if @xmath138 , we go back to step 1 , using @xmath139 instead of @xmath114 . if @xmath140 , we add the full labeling @xmath118 to @xmath119 and repeat step 2 . the group of symmetries @xmath141 acts on @xmath119 , and the equivalence classes of small covers are in bijection with @xmath141-orbits . to determine these orbits , we choose a set of generators for @xmath141 and form the graph whose vertices are elements of @xmath119 and whose edges join any pair of vertices that differ by a generator of @xmath141 one then applies any standard graph algorithm to determine the connected components of this graph . using the algorithm , we obtain @xmath145 normalized labelings of @xmath142 . forming the graph on this set @xmath146 with respect to the standard generating set @xmath147 for @xmath143 , we obtain @xmath11 connected components . representative labelings are given in table [ tab : dodec ] . to keep the data concise , we have used decimal equivalents for @xmath148 ( for example , @xmath149 and @xmath150 ) . the ordering we use for the facets is shown in figure [ fig : dodec.fig ] . xxx n. bourbaki . _ groupes et algbres de lie , _ chapters 4 - 6 . masson , paris , 1981 . . a hyperbolic @xmath77-manifold . _ 93 ( 1985 ) , no . 2 , 325328 . m. davis and t. januszkiewicz . convex polytopes , coxeter orbifolds and torus actions . _ duke math . j. _ 62 ( 1991 ) , no . 2 , 417451 . f. lbell . beispiele geschlossener dreidimensionaler clifford - kleinische rume negativer krmmung . _ 83 ( 1931 ) , 168174 . j. seifert and d. weber . die beiden dodekaederrume , _ math . z. _ 37 ( 1933 ) , 237253 . three - dimensional hyperbolic manifolds of lbell type . _ siberian math . j. _ 28 ( 1987 ) , 5053 .

let @xmath0 be the right - angled hyperbolic dodecahedron or @xmath1-cell , and let @xmath2 be the group generated by reflections across codimension - one faces of @xmath0 . we prove that if @xmath3 is a torsion free subgroup of minimal index , then the corresponding hyperbolic manifold @xmath4 is determined up to homeomorphism by @xmath5 modulo the symmetry group of @xmath0 .

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diagonal and off - diagonal properties of 2d solid @xmath0he systems doped with a quantized vortex have been investigated via the shadow path integral ground state method using the fixed - phase approach . the chosen approximate phase induces the standard onsager - feynman flow field . in this approximation the vortex acts as a static external potential and the resulting hamiltonian can be treated exactly with quantum monte carlo methods . the vortex core is found to sit in an interstitial site and a very weak relaxation of the lattice positions away from the vortex core position has been observed . also other properties like bragg peaks in the static structure factor or the behavior of vacancies are very little affected by the presence of the vortex . we have computed also the one - body density matrix in perfect and defected @xmath0he crystals finding that the vortex has no sensible effect on the off - diagonal long range tail of the density matrix . within the assumed onsager feynman phase , we find that a quantized vortex can not auto - sustain itself unless a condensate is already present like when dislocations are present . it remains to be investigated if backflow can change this conclusion . quantized vortices are one of the most genuine manifestation of the presence of superfluidity in many body quantum systems , but from the microscopic point of view no complete understanding of them has been reached yet . recently quantized vortices have been related to the supersolidity issue @xcite ; in fact , arguments in favor of a vortex phase in low temperature solid @xmath0he , preceding the supersolid transition , are appeared in literature @xcite . with respect to this possible connection one of the fundamental questions to be answered is : what does a quantum vortex look like in solid helium from a microscopic point of view ? dealing with vortices is a really hard task for microscopic methods , and it calls for some approximations or assumptions [ 5 - 10 ] . in fact , the wave function has to be an eigenstate of the angular momentum , so it needs a phase . following the well established routine for the ground state , once chosen a variational ansatz for the wave function , one could be tempted to correct it by means of exact zero temperature quantum monte carlo ( qmc ) techniques . unfortunately this is actually not possible because of the sign problem that affects qmc methods . the most followed recipe is to improve the variational description via qmc , but releasing the exactness of the methods in favor of approximations that allow to avoid the sign problem , like for example fixed phase @xcite or fixed nodes @xcite . we study here the properties of a single vortex in solid @xmath0he via the shadow - pigs ( spigs ) method with fixed phase approximation . the many - body wave function can be written as @xmath1 , where @xmath2 is a many - body phase , @xmath3 is the modulus of the wave function and @xmath4 are the coordinates of the @xmath5 particles . @xmath6 describes a quantum state of the system if it is a solution of the time independent schrdinger equation : from @xmath7 it is possible to obtain two coupled differential equations for @xmath2 and for @xmath3 . the fixed phase approximation consists in assuming the functional form of @xmath8 as given and to solve the equation @xmath9\psi_0(r ) = e\psi_0(r).\ ] ] for @xmath3 . solving ( [ schreff ] ) is equivalent to solve the original time independent schrdinger equation for the @xmath5-particle with an extra potential term @xmath10 . the simplest choice for the phase is the well known onsager - feynman ( of ) phase @xcite : @xmath11 ( where @xmath12 is the angular polar coordinate of the @xmath13-th particle ) . @xmath6 is an eigenstate of the @xmath14 component of the angular momentum operator @xmath15 with eigenvalue @xmath16 , being @xmath17 the quantum of circulation . this choice for @xmath2 gives rise to the standard of flow field : in fact the extra - potential in ( [ schreff ] ) reads @xmath18 where @xmath19 is the radial polar coordinate of the @xmath13-th particle . in order to sustain a quantized vortex , the system should display a macroscopic phase coherence , and at @xmath20 k this means that solid @xmath0he should house a bose - einstein condensate ( bec ) . it is known from qmc results that no bec is present in the perfect crystal [ 12 - 15 ] , but if the vortex turns out to be able to induce a bec it could be a self - sustaining excitation . on the other hand , it is largely accepted that defects are able to induce bec @xcite , and then a defected crystal can safely sustain a quantized vortex . here we report on the study of a two dimensional ( 2d ) @xmath0he crystal with and without dislocations . in fact , dislocations can be included in the 2d crystal without imposing boundary constraints @xcite . moreover the 2d system allows to reach large distances keeping the number of particle in the simulation at a tractable level , and this is a desirable feature when interested in off - diagonal properties of the system . we face the task of solving ( [ schreff ] ) with the extra - potential given by ( [ pot ] ) when @xmath21 with the spigs method @xcite , which allows to obtain the lowest eigenstate of a given hamiltonian by projecting in imaginary time a swf @xcite taken as trial wave function . the spigs method is unbiased by the choice of the trial wave function and the only inputs are the interparticle potential and the approximation for the imaginary time propagator @xcite . as he - he interatomic potential we have considered the hfdhe2 aziz potential @xcite and we have employed the pair - suzuki approximation @xcite for the imaginary time propagator with time step @xmath22 k@xmath23 . one difficulty with ( [ pot ] ) is that the potential is long range so that either one puts the system in a bucket @xcite or one should consider a vortex lattice @xcite . such complications can be avoided by multiplying @xmath24 in ( [ pot ] ) by a smoothing function @xmath25 ( @xmath26 being the side of the simulation box ) so that standard periodic boundary conditions can be applied . with this choice , the extra - potential is equivalent to the of one only for @xmath27 , the provided @xmath28 is no more an exact eigenstate of @xmath15 but it is close to it in the interesting region of the vortex core if @xmath29 is large enough . here we have used @xmath30 . we have performed simulations at @xmath31@xmath32 in a nearly squared box designed to house a perfect triangular crystal with @xmath33 lattice sites , and a crystal with 10 vacancies ( @xmath34 ) ; such vacancies in the initial configuration transform themselves in dislocations @xcite . integrated vortex energy @xmath35 with error bars as a function of the distance from the core in perfect 2d solid @xmath0he at @xmath31@xmath32.,width=264 ] in fig . [ fig : e2d ] we report our results for the integrated vortex energies @xmath36 ( @xmath37 and @xmath38 are , respectively , the energy of the particles that lie inside the disk of radius @xmath39 in the system with and without vortex ) as a function of the distance from the core in the perfect crystal . the center of mass of the system is not fixed , and we find that , independently on the starting configuration of the crystal , the vortex core sits in an interstitial site . we also find a very small relaxation of the surrounding lattice around the vortex core . comparison between the radial defect distribution @xmath40 as a function of the distance from the origin ( vortex core when vortex is present ) for a 2d solid @xmath0he at @xmath31@xmath32 with ( filled symbols ) and without ( open symbols ) a vortex in the perfect ( circles ) and in defected ( squares ) crystal.,width=264 ] in order to study the effects of the vortex on the crystal properties we have monitored the static structure factor , the pair distribution function and the radial defect distribution @xmath40 ( i.e. the distribution of the particle whose coordination is different from 6 as a function of the distance from the origin where the vortex core is located ) . in fig . [ fig : x2d ] we plot our results for @xmath40 both for the perfect and for the defected crystal . we find that in the defected case the radial defect distribution is about an order of magnitude larger than in the perfect one ; however , the results of the system with and without vortex are very close each other so that we conclude that the of vortex does not affect in a sensible way the disorder which behaves as in the system without vortex . we come at a similar conclusion for the crystalline structure , since both the static structure factor and the pair distribution function show no appreciable differences for the system with and without the vortex . one body density matrix @xmath41 in 2d solid @xmath0he without ( open symbols ) and with ( filled symbols ) a vortex for the perfect crystal ( circles ) and for the crystal with dislocations ( squares).,width=264 ] we have computed also the one - body density matrix @xmath41 for the perfect and for the defected crystal with and without the vortex in order to investigate the vortex effects on the off - diagonal properties . our results are reported in fig . [ fig : c2d ] . @xmath41 for the crystal with vortex are indistinguishable within the error bars from the ones obtained without vortex . we conclude that the of vortex is not able to induce a bose - einstein condensation ( bec ) in the perfect crystal , or to increase the already present condensate fraction in the defected one . since no bec is present in the perfect crystal and the of vortex is not able to induce it by itself , we can conclude that perfect 2d solid @xmath0he can not sustain vortices of the of type . thus the of wave function is a possible representation of a vortex in solid @xmath0helium only when bec is already present , like in a defected crystal . preliminary results in three dimensional solid @xmath0he seem to confirm such conclusions for the of vortex . in the liquid phase , the of phase @xcite has been improved with the inclusion of back - flow ( bf ) correlations @xcite . the effect of backflow increases @xcite at higher densities and it might well become dominant in the solid phase . computations along this line are in progress . since the bf terms acts mainly near the vortex core , we might expect that bf will not modify much the off - diagonal properties in 2d , where the vortex core is a point defect , while it could become relevant in 3d where the vortex core is an extended defect . this work was supported by regione lombardia and cilea consortium through a lisa initiative ( laboratory for interdisciplinary advanced simulation ) 2010 grant [ link : http://lisa.cilea.it ] .

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constraints on possible lorentz symmetry violation ( lv ) of order @xmath0 for electrons and photons in the framework of effective field theory ( eft ) are discussed . using ( i ) the report of polarized mev emission from grb021206 and ( ii ) the absence of vacuum erenkov radiation from synchrotron electrons in the crab nebula , we improve previous bounds by @xmath1 and @xmath2 respectively . we also show that the lv parameters for positrons and electrons are different , discuss electron helicity decay , and investigate investigate how prior constraints are modified by the relations between lv parameters implied by eft . the past few years have witnessed a rapid development of powerful constraints on some types of lorentz symmetry violation ( lv ) that have been suggested by quantum gravity scenarios . while no current suggestion of lv is firm enough to be considered a prediction , there is nevertheless great interest in the possibility of lv induced by planck scale physics since it offers the hope of an observational window into quantum gravity . to date no lv phenomena have been observed ( although the ultra high energy cosmic ray events detected by the akeno giant air shower array ( agasa ) , could possibly turn out to be harbingers of lv physics @xcite ) . the absence of lv provides important constraints on viable quantum gravity theories . moreover , these constraints are interesting in their own right as they extend the domain where relativity has been tested far beyond its previous frontiers . the primary purpose of this paper is to further strengthen the bounds on lv of order @xmath3 for photons and electrons , where @xmath4 gev is the planck energy , the presumed energy scale of quantum gravity . we use the reported observation @xcite of polarized gamma rays from the gamma ray burst grb021206 to improve the birefringence constraint by ten orders of magnitude . [ the results of @xcite have been challenged @xcite and defended @xcite . if the polarization turns out to be weaker , then the birefringence constraint from grb021206 is weakened ( or eliminated ) . ] by consideration of the vacuum erenkov process for the electrons producing the highest frequency synchrotron radiation from the crab nebula we improve on the old birefringence constraint by two orders of magnitude . a secondary purpose is to revisit previous constraints in light of the effective field theory ( eft ) analysis of @xcite , some of which are strengthened and some weakened or limited in applicability . we show that eft implies that the lv parameters for positrons are opposite ( in two senses ) compared to electrons , and we discuss a new lv process of helicity decay " , in which an electron of one helicity decays to a state with the opposite helicity . finally we pull together the strongest constraints to date and present them in a logarithmic plot that allows their nature and relative strength to be easily compared to previous work . we adopt the framework of effective field theory as developed e.g. in @xcite , focusing on the electron - photon sector since this involves no other particles and there are many observations allowing a number of independent constraints to be combined . we assume rotational symmetry is preserved in a preferred frame , which is taken to coincide with that of the cosmic microwave background radiation , and consider only lv suppressed by one power of the ratio @xmath0 , which arises from mass dimension five operators in the lagrangian . ( we thus assume that lower mass dimension lv operators are suppressed by a symmetry or other mechanism , otherwise they would be expected to dominate @xcite . ) under these assumptions the most general photon and electron dispersion relations are @xcite e^2 & = & p^2 p^3/m photons [ eq : disp - ph ] + e^2 & = & m^2 + p^2 + _ r , l p^3/m [ electrons ] where @xmath5 , @xmath6 , and @xmath7 are independent dimensionless parameters , and @xmath8 gev is factored out rather than the planck mass @xmath9 for computational convenience . we adopt units with @xmath10 and the low energy speed of light @xmath11 . the sign in the photon dispersion relation ( [ eq : disp - ph ] ) corresponds to the helicity ( i.e. right or left circular polarization ) , while the labels @xmath12 and @xmath13 in the electron dispersion relation ( [ electrons ] ) apply for positive and negative electron helicity respectively ( see below for more details ) . the bound @xmath14 @xcite is provided by measurements of spin - polarized torsion pendulum frequency @xcite . _ new birefringence constraint_. the dispersion relation ( [ eq : disp - ph ] ) implies that electromagnetic waves of opposite helicity have different phase velocities , which leads to a rotation of linear polarization direction through the angle ( t)=t/2=k^2 t/2 m [ rotation ] for a plane wave with wave - vector @xmath15 . observations of polarized radiation from distant sources can hence be used to place an upper bound on @xmath5 . the best previous bound , @xmath16 , was obtained by gleiser and kozameh @xcite , using the observed 10% polarization of ultraviolet light from a distant galaxy . ( see also @xcite for similar birefringence bounds in the context of different types of lorentz symmetry breaking . ) recently the prompt emission from the gamma ray burst grb021206 was observed using the rhessi detector @xcite . a linear polarization of 80% @xmath17 20% was reported @xcite . [ this claim has been challenged @xcite and defended @xcite . ] during the five seconds of emission the intensity varied strongly on a timescale of small fractions of a second consistently across the spectral window 0.15 - 2 mev . the data @xcite indicate a major contribution to the flux comes from photons significantly distributed over at least the energy range 0.1 - 0.5 mev . the constraint arises from the fact that if the angle of polarization rotation ( [ rotation ] ) were to differ by more than @xmath18 over the range 0.1 - 0.3 mev ( and hence by more than @xmath19 over the range 0.1 - 0.5 mev ) , the instantaneous polarization at the detector would fluctuate sufficiently for the net polarization of the signal to be suppressed well below the observed value . ( a stronger constraint could clearly be obtained by taking into account more precisely the spectral characteristics of the signal and detector . ) the difference in rotation angles for wave - vectors @xmath20 and @xmath21 is = ( k_2 ^ 2-k_1 ^ 2 ) d/2 m , [ diffrotation ] where we have replaced the time @xmath22 by the distance @xmath23 from the source to the detector ( divided by the speed of light ) . while the distance to grb021206 is unknown , it is well known that most cosmological bursts have redshifts in the range 1 - 2 corresponding to distances of greater than a gpc . using the distance distribution derived in ref . @xcite we conservatively take the minimum distance to this burst as 0.5 gpc , corresponding to a redshift of @xmath24 . this then yields the constraint ||<5.010 ^ -15/d_0.5 . where @xmath25 is the distance to the burst in units of 0.5 gpc . _ new erenkov - synchrotron constraint_. in a region of the lv parameter space there is an energy threshold for a free electron to emit a photon in a process called vacuum erenkov radiation . the threshold can occur either with emission of a soft photon or a hard photon depending on the parameters @xcite . in the soft photon case the threshold is @xmath26 , from which it follows that the strength of the constraint on @xmath27 scales as the inverse cube of the electron energy , and that energies of order 10 tev for the electron are required in order to put constraints of order unity on the lv parameters @xcite . electrons of energy up to 50 tev are inferred via the observation of 50 tev gamma rays from the crab nebula which are explained by inverse compton ( ic ) scattering . since the erenkov rate is orders of magnitude higher than the ic scattering rate , the erenkov process must not occur for these electrons @xcite . this yields a constraint on @xmath27 of order @xmath28 . neither photon helicity should be emitted , so the absolute value @xmath29 is bounded , which strengthens the ic erenkov constraint . on the other hand , it could be that only one electron helicity produces the ic photons and the other loses energy by vacuum erenkov radiation . hence we can infer only that at least one of @xmath6 and @xmath7 satisfies the bound . a complementary constraint was derived in @xcite by making use of the very high energy electrons that produce the highest frequency synchrotron radiation in the crab nebula . for negative values of @xmath27 the electron has a maximal group velocity less than the speed of light , hence there is a maximal synchrotron frequency that can be produced regardless of the electron energy @xcite . observations of the crab nebula reveal synchrotron radiation at least out to 100 mev ( requiring electrons of energy 1500 tev in the lorentz invariant case ) , which implies that at least one of the two parameters @xmath30 must be greater than @xmath31 ( this constraint is independent of the value of @xmath5 ) . we can not constrain both @xmath27 parameters in this way since it could be that all the crab synchrotron radiation is produced by electrons of one helicity . hence for the rest of this discussion let @xmath27 stand for whichever of the two @xmath27 s satisfies the synchrotron constraint . this must be the same @xmath27 as satisfies the ic erenkov constraint discussed above , since otherwise the energy of these synchrotron electrons would be below 50 tev rather than the lorentz invariant value of 1500 tev . the crab spectrum is well accounted for with a single population of electrons responsible for both the synchrotron radiation and the ic @xmath32-rays . if there were enough extra electrons to produce the observed synchrotron flux with thirty times less energy per electron , then the electrons of the other helicity which would be producing the ic @xmath32-rays would be too numerous . we now use the existence of these synchrotron producing electrons to improve on the vacuum erenkov constraint . for a given @xmath33 , some definite electron energy @xmath34 must be present to produce the observed synchrotron radiation . ( this is higher for negative @xmath27 and lower for positive @xmath27 than the lorentz invariant value @xcite . ) values of @xmath29 for which the vacuum erenkov threshold is lower than @xmath35 for either photon helicity can therefore be excluded . ( this is always a hard photon threshold , since the soft photon threshold occurs when the electron group velocity reaches the low energy speed of light , whereas the velocity required to produce any finite synchrotron frequency is smaller than this . ) for negative @xmath27 , the erenkov process occurs only when @xmath36 @xcite , so the excluded parameters lie in the region @xmath37 . _ implications of eft for prior constraints_. photon time of flight.the lorentz violating dispersion relation ( [ eq : disp - ph ] ) implies that the group velocity of photons , @xmath38 , is energy dependent . this leads to an energy dependent dispersion in the arrival time at earth for photons originating in a distant event @xcite , which was previously exploited for constraints @xcite . the dispersion of the two polarizations is larger since the difference in group velocity is then @xmath39 rather than @xmath40 , but the time of flight constraint remains many orders of magnitude weaker than the birefringence one from polarization rotation . in fig . [ fig : all ] we use the eft improvement of the constraint of @xcite which yields @xmath41 . photon decay and photon absorption.the constraints from photon decay @xmath42 and absorption @xmath43 must be reanalyzed to take into account the different dispersion for the two photon helicities , and the different parameters for the two electron helicities , but there is a further complication : both these processes involve positrons in addition to electrons . previous constraint derivations have assumed that these have the same dispersion , but that need not be the case @xcite . we show below for the @xmath44 corrections that it is indeed not so . taking into account the above factors could not significantly improve the strength of the constraints ( which is mainly determined by the energy of the photons ) . we indicate here only what the helicity dependence of the photon dispersion implies , neglecting the important role of differing parameters for electrons and positrons and their helicity states . the strongest limit on photon decay came from the highest energy photons known to propagate , which at the moment are the 50 tev photons observed from the crab nebula @xcite . since their helicity is not measured , only those values of @xmath29 for which _ both _ helicities decay could be ruled out . the photon absorption constraint came from the fact that lv can shift the standard qed threshold for annihilation of multi - tev @xmath32-rays from nearby blazars such as mkn 501 with the ambient infrared extragalactic photons @xcite . lv depresses the rate of absorption of one photon helicity and increases it for the other . although the polarization of the @xmath32-rays is not measured , the possibility that one of the polarizations is essentially unabsorbed appears to be ruled out by the observations which show the predicted attenuation@xcite . _ electron and positron dispersion_. the dirac equation in the lorentz violating eft including the dimension five operators can be written @xcite as = 0 , [ dirac ] where @xmath45 is the unit timelike 4-vector that specifies the preferred frame . if we choose coordinates aligned with @xmath45 , so that @xmath46 , an electron or positron mode of energy @xmath47 and momentum @xmath48 in the @xmath49 direction contributes to the field operator via @xmath50 , where the upper sign here and below is for an electron and the lower for a positron , and @xmath51 is the spinor . inserting this in the deformed dirac equation ( [ dirac ] ) yields = 0 . [ diracep ] the helicity operator acting on @xmath51 is @xmath52 @xcite , where @xmath53 . this is hermitian and commutes with @xmath54 times the operator in ( [ diracep ] ) , which is also hermitian . hence helicity remains a good quantum number in the presence of this lorentz violation . assuming without loss of generality that @xmath55 , a spinor for helicity @xmath56 therefore satisfies @xmath57 , or equivalently @xmath58 . for helicity eigenstates therefore = 0 . this has the form of the standard dirac equation , with @xmath47 replaced by @xmath59 and @xmath48 replaced by @xmath60 . hence the dispersion relation is given by @xmath61 . for @xmath62 this yields e^2 = p^2 + m^2 + 2(_1 + h_2)e^3/m . with the definitions @xmath63 and @xmath64 , the parameters in the dispersion relations for positive and negative helicity states respectively are thus @xmath6 and @xmath7 for electrons , and @xmath65 and @xmath66 for positrons . _ possible new constraints from helicity decay_. if @xmath6 and @xmath67 are unequal , say @xmath68 , then a positive helicity electron can decay into a negative helicity electron and a photon , even when the lv parameters do not permit the vacuum erenkov effect . in this process , the large @xmath12 or small ( @xmath69 ) @xmath13 component of a positive helicity electron transitions to the small @xmath12 or large @xmath13 component of a negative helicity electron respectively . such helicity decay " has no threshold energy , so whether this process can be used to set constraints on @xmath30 is solely a matter of the decay rate . it can be shown ( assuming @xmath70 ) that for electrons of energy less than the transition energy @xmath71 , the lifetime of an electron susceptible to helicity decay is greater than @xmath72 . at the limit of the best current bound @xmath73 , the transition energy is approximately 10 tev and the lifetime for electrons below this energy is greater than @xmath74 seconds . this is long enough to preclude any terrestrial experiments from seeing the effect . the lifetime above the transition energy is instead bounded below by @xmath75 , which is @xmath76 seconds for energies just above 10 tev . the lifetime might therefore be short enough to provide new constraints . such a constraint might come from the crab nebula . suppose that @xmath7 is below the synchrotron constraint ( i.e. @xmath77 ) , so that @xmath6 must satisfy both the synchrotron and erenkov constraints as explained above . then positive helicity electrons must have an energy of at least 50 tev to produce the observed synchrotron radiation . these must not decay to negative helicity electrons ( since those are unable to produce the synchrotron emission ) , which would require that the transition energy be greater than 50 tev if the decay rate is fast enough . this would yield the constraint @xmath78 . _ combined constraints_. the combined constraints are shown logarithmically in figure [ fig : all ] . ) and electron ( @xmath27 ) lv parameters . the birefringence constraint uses the observed polarization of mev photons from grb021206 . the synchrotron and ic cerenkov constraints use the observation of 0.1 gev synchrotron and 50 tev inverse compton radiation from the crab nebula , respectively . for the origin of other constraints see text . for negative parameters the negative of the logarithm of the absolute value is plotted , and a region of width @xmath79 is excised around each axis . the synchrotron and erenkov constraints are known to apply only for at least one @xmath30 . the ic and synchrotron erenkov lines are truncated where they cross . prior photon decay and absorption constraints are shown in dashed lines since they do not account for the eft relations between the lv parameters.,title="fig : " ] the vast improvement in the birefringence constraint overwhelms the new synchrotron erenkov constraint , while the latter improves the previous birefringence constraint @xcite by @xmath80 . the allowed region is defined above and below by the birefringence bound @xmath81 , on the left by the synchrotron bound @xmath82 , and on the right by the ic erenkov bound @xmath83 . if the polarization of grb021206 proves incorrect , the allowed region will expand vertically to the synchrotron erenkov lines . the combined constraints severely limit first order planck suppressed lv , making any theory that predicts this type of lv very unlikely . the most useful improvements at this stage would be to strengthen the positive @xmath27 and @xmath84 bounds .

the time - varying nature of the underlying channel is one of the most significant design challenges in wireless communication systems . in particular , real - time media traffic typically has a stringent delay constraint , so the exploitation of long blocklength frames is infeasible and the entire frame may fall into deep fading channel states . furthermore , the receiver may have limited resources to feed the estimated channel state information back to the transmitter , which precludes adaptive transmission and forces the transmitter to use a stationary coding strategy . the above described situation is modeled as a slowly fading channel with receiver side information only , which is an example of a non - ergodic _ composite channel_. a composite channel is a collection of component channels @xmath0 parameterized by @xmath1 , where the random variable @xmath1 is chosen according to some distribution @xmath2 at the beginning of transmission and then held fixed . we assume the channel realization is revealed to the receiver but not the transmitter . this class of channel is also referred to as the _ mixed channel _ @xcite or the _ averaged channel _ @xcite in literature . the shannon capacity of a composite channel is given by the verd - han generalized capacity formula @xcite @xmath3 where @xmath4 is the liminf in probability of the normalized information density . this formula highlights the pessimistic nature of the shannon capacity definition , which is dominated by the performance of the `` worst '' channel , no matter how small its probability . to provide more flexibility in capacity definitions for composite channels , in @xcite we relax the constraint that all transmitted information has to be correctly decoded and derive alternate definitions including the _ capacity versus outage _ and the _ expected capacity_. the capacity versus outage approach allows certain data loss in some channel states in exchange for higher rates in other states . it was previously examined in @xcite for single - antenna cellular systems , and later became a common criterion for multiple - antenna wireless fading channels @xcite . see ( * ? ? ? * ch . 4 ) and references therein for more details . the expected capacity approach also requires the transmitter to use a single encoder but allows the receiver to choose from a collection of decoders based on channel states . it was derived for a gaussian slow - fading channel in @xcite , and for a composite binary symmetric channel ( bsc ) in @xcite . channel capacity theorems deal with data transmission in a communication system . when extending the system to include the source of the data , we also need to consider the data compression problem which deals with source representation and reconstruction . for the overall system , the end - to - end distortion is a well - accepted performance metric . when both the source and channel are stationary and ergodic , codes are usually designed to achieve the same end - to - end distortion level for any source sequence and channel realization . nevertheless , practical systems do not always impose this constraint . if the channel model is generalized to such scenarios as the composite channel above , it is natural to relax the constraint that a single distortion level has to be maintained for all channel states . in parallel with the development of alternative capacity definitions , we introduce generalized end - to - end distortion metrics including the _ distortion versus outage _ and the _ expected distortion_. the distortion versus outage is characterized by a pair @xmath5 , where the distortion level @xmath6 is guaranteed in receiver - recognized non - outage states of probability no less than @xmath7 . this definition requires csir based on which the outage can be declared . the expected distortion is defined as @xmath8 , i.e. the achievable distortion @xmath9 in channel state @xmath1 averaged over the underlying distribution @xmath2 . these alternative distortion metrics are also considered in prior works . in @xcite the average distortion @xmath10 , obtained by averaging over outage and non - outage states , was adopted as a fidelity criterion to analyze a two - hop fading channel . here @xmath11 is the variance of the source symbols . the expected distortion was analyzed for the mimo block fading channel in the high snr regime @xcite and in the finite snr regime @xcite . various coding schemes for expected distortion were also studied in a slightly different but closely related broadcast scenario @xcite . data compression ( source coding ) and data transmission ( channel coding ) are two fundamental topics in shannon theory . for transmission of a discrete memoryless source ( dms ) over a discrete memoryless channel ( dmc ) , the renowned source - channel separation theorem ( * ? ? ? * theorem 2.4 ) asserts that a target distortion level @xmath12 is achievable if and only if the channel capacity @xmath13 exceeds the source rate distortion function @xmath14 , and a two - stage separate source - channel code suffices to meet the requirement . this theorem enables separate designs of source and channel codes with guaranteed optimal performance . it also extends to stationary and ergodic source and channel models @xcite @xcite . separate source - channel coding schemes provide flexibility through modularized design . from the source s point of view , the source can be transmitted over any channel with capacity greater than @xmath14 and be recovered at the receiver subject to a certain fidelity criterion ( the distortion @xmath12 ) . the source is indifferent to the statistics of each individual channel and consequently focuses on source code design independent of channel statistics . despite their flexibility and optimality for certain systems , separation schemes also have their disadvantages . first of all , the source encoder needs to observe a long - blocklength source sequence in order to determine the output , which causes infinite delay . second , separation schemes may increase complexity in encoders and decoders because the two processes of source and channel coding are acting in opposition to some extent . source coding is essentially a data compression process , which aims at removing redundancy from source sequences to achieve the most concise representation . on the other hand , channel coding deals with data transmission , which tries to add some redundancy to the transmitted sequence for robustness against the channel noise . if the source redundancy can be exploited by the channel code , then a joint source - channel coding scheme may avoid this overhead . in particular , transmission of a gaussian source over a gaussian channel , and a binary symmetric source over a bsc , are both examples where optimal performance can be achieved without any coding @xcite . this is because the source and channel are matched " to each other in the sense that the transition probabilities of the channel solve the variational problem defining the source rate - distortion function @xmath14 and the letter probabilities of the source drive the channel at capacity @xcite . a careful inspection of the shannon separation theorem reveals some important underlying assumptions : a single - user channel , a stationary and ergodic source and channel , and a single distortion level maintained for all transmissions . violation of any of these assumptions will likely prompt reexamination of the separation theorem . for example , cover et . al . showed that for a multiple access channel with correlated sources , the separation theorem fails @xcite . in @xcite vembu et al . gave an example of a non - stationary system where the source is transmissible through the channel with zero error , yet its minimum achievable source coding rate is twice the channel capacity . in this work , we illustrate that different end - to - end distortion metrics lead to different conclusions about separability even for the same source and channel model . in fact , source - channel separation holds under the distortion versus outage metric but fails under the expected distortion metric . in @xcite we proved the direct part of source - channel separation under the distortion versus outage metric and established the converse for a system of gaussian source and slow - fading gaussian channels . here we extend the converse to more general systems of stationary sources and composite channels . source - channel separation implies that the operation of source and channel coding does not depend on the statistics of the counterpart . however , the source and channel do need to communicate with each other through a _ negotiation interface _ even before the actual transmission starts . in the classical view of shannon separation for stationary ergodic sources and channels , the source requires a rate @xmath14 based on the target distortion @xmath12 and the channel decides if it can support the rate based on its capacity @xmath13 . for generalized source / channel models and distortion metrics , the interface is not necessarily a single rate and may allow multiple parameters to be agreed upon between the source and channel . after communication through the appropriate negotiation interface , the source and channel codes may be designed separately and still achieve the optimal performance . vembu et al . studied the transmission of non - stationary sources over non - stationary channels and observed that the notion of ( strict ) domination ( * ? ? ? * theorem 7 ) dictates whether a source is transmissible over a channel , instead of the simple comparison between the minimum source coding rate and the channel capacity . the notion of ( strict ) domination requires the source to provide the distribution of the _ entropy density _ and the channel to provide the distribution of the _ information density _ as the appropriate interface . the source - channel interface concept also applies after the actual transmission starts . at the transmitter end , we see examples where the source sequence is directly supplied to the channel , such as the uncoded transmission of a gaussian source over a gaussian channel . but more generally there is certain processing on the source side , and the processed output , instead of the original source sequence , is supplied to the channel . the _ transmitter interface _ contains what the source actually delivers to the channel . for example , in separation schemes the interface is the source encoder output ; in hybrid digital - analog schemes @xcite the interface is a combination of vector quantizer output and quantization residue . similarly we can introduce the concept of a _ receiver interface_. instead of directly delivering the channel output sequence to the destination , the receiver may implement certain decoding and choose the channel decoder output as the interface . the interfaces at the transmitter and the receiver are the same in classical shannon separation schemes , since the channel code requires all transmitted information to be correctly decoded with vanishing error , but in general the two interfaces can be different . for example , the receiver interface may include an outage indicator or partial decoding when considering generalized capacity definitions . different transmission schemes can be compared by their end - to - end performance . nevertheless , the concept of source - channel interface opens a new dimension for comparison . ideally the interface complexity should be measured by some quantified metrics . transmission schemes with low interface complexity are also appealing in view of simplified system design . we expect a performance enhancement when the source and channel exchange more information through a more sophisticated interface , and illustrate the tradeoff between interface complexity and end - to - end performance through some examples in this work . the rest of the paper is organized as follows . we review alternative channel capacity definitions and define corresponding end - to - end distortion metrics in section [ sec : performance ] . in section [ sec : source - channel ] we provide a new perspective of source - channel separation generalized from shannon s classical view and also introduce the concept of source - channel interface . in section [ sec : outagedistortion ] we establish the separation optimality for transmission of stationary ergodic sources over composite channels under the distortion versus outage metric . in section [ sec : interfacebsc ] we consider various schemes to transmit a binary symmetric source ( bss ) over a composite bsc and show the tradeoff between achievable expected distortion and interface complexity . conclusions are given in section [ sec : con ] . we first review alternate channel capacity definitions derived in @xcite to provide some background information . we then define alternate end - to - end performance metrics for the entire communication system , including the source and the destination . the channel @xmath15 is statistically modeled as a sequence of @xmath16-dimensional conditional distributions @xmath17 . for any integer @xmath16 , @xmath18 is the conditional distribution from the input space @xmath19 to the output space @xmath20 . let @xmath21 and @xmath22 denote the input and output processes , respectively . each process is specified by a sequence of finite - dimensional distributions , e.g. @xmath23 . in a composite channel , when the channel side information is available at the receiver , we represent it as an additional channel output . specifically , we let @xmath24 , where @xmath1 is the channel side information and @xmath25 is the output of the channel described by parameter @xmath1 . throughout , we assume the random variable @xmath1 is independent of @xmath21 and unknown to the encoder . thus for each @xmath16 @xmath26 the information density is defined similarly as in @xcite @xmath27 consider a sequence of @xmath28 codes . let @xmath29 be the probability that the receiver declares an outage , and @xmath30 be the decoding error probability given that no outage is declared . we say that a rate @xmath31 is outage-@xmath32 achievable if there exists a sequence of @xmath28 channel codes such that @xmath33 and @xmath34 . the _ capacity versus outage _ @xmath35 is defined to be the supremum over all outage-@xmath32 achievable rates , and is shown to be @xcite @xmath36 \leq q \right\}. \label{eqn : outage}\ ] ] the operational implication of this definition is that the encoder uses a single codebook and sends information at a fixed rate @xmath35 . assuming repeated channel use and independent channel state at each use , the receiver can correctly decode the information a proportion @xmath7 of the time and turn itself off a proportion @xmath32 of the time . we further define the _ outage capacity _ @xmath37 as the long - term average rate , which is a meaningful metric if we are only interested in the fraction of correctly received packets and approximate the unreliable packets by surrounding samples , or if there is some repetition mechanism where the receiver requests retransmission of lost information from the sender . the value @xmath32 can be chosen to maximize the long - term average throughput @xmath38 . this notion provides another strategy for increasing reliably - received rate . although the transmitter is forced to use a single encoder at a rate @xmath39 without channel state information , the receiver can choose from a collection of decoders , each parameterized by @xmath40 and decoding at a rate @xmath41 , based on csir . denote by @xmath42 the error probability associated with channel state @xmath40 . the expected capacity @xmath43 is the supremum of all achievable rates @xmath44 of any code sequence that has @xmath45 approaching zero . in a composite channel , different channel states can be viewed as virtual receivers , and therefore the expected capacity is closely related to the capacity region of a broadcast channel ( bc ) . in the broadcast system the channel from the input to the output of receiver @xmath40 is @xmath46 under certain conditions , it is shown that the expected capacity of a composite channel equals to the maximum weighted sum - rate over the capacity region of the corresponding broadcast channel , where the weight coefficient is the state probability @xmath47 ( * ? ? ? * theorem 1 ) . using broadcast channel codes , the expected capacity is derived in @xcite for a gaussian slow - fading channel and in @xcite for a composite bsc . the expected capacity is a meaningful metric if _ partial _ received information is useful . for example , consider sending an image using a multi - resolution ( mr ) source code over a composite channel . decoding all transmitted information leads to reconstructions with the highest fidelity . however , in the case of inferior channel quality , it still helps to decode partial information and get a coarse reconstruction . next we introduce alternative end - to - end distortion metrics as performance measures for transmission of a stationary ergodic source over a composite channel . we denote by @xmath48 the source alphabet and the source symbols @xmath49 are generated according to a sequence of finite - dimensional distributions @xmath50 , and then transmitted over a composite channel @xmath51 with conditional output distribution @xmath52 it is possible that the source generates symbols at a rate different from the rate at which the channel transmits symbols , i.e. a length-@xmath16 source sequence may be transmitted in @xmath53 channel uses with @xmath54 . the channel _ bandwidth expansion ratio _ is defined to be @xmath55 . for simplicity we assume @xmath56 in this and the next two sections , but the discussions can be easily extended to general cases with @xmath57 . the numerical examples in section [ sec : interfacebsc ] will explicitly address this issue . here we design an encoder @xmath58 that maps the source sequence to the channel input . note that the source and channel encoders , whether joint or separate , do not have access to channel state information @xmath1 . however , the receiver can declare an outage with probability @xmath59 based on csir . in non - outage states , we design a decoder @xmath60 that maps the channel output to a source reconstruction . we say a distortion level @xmath12 is outage-@xmath32 achievable if @xmath61 and @xmath62 where @xmath63 is the distortion measure between the source sequence @xmath64 and its reconstruction @xmath65 . the _ distortion versus outage @xmath6 is the infimum over all outage-@xmath32 achievable distortions . in order to evaluate we need the conditional distribution @xmath66 . assuming the encoder @xmath67 and the decoder @xmath68 are deterministic , this distribution is given by @xmath69 here @xmath70 is the indicator function . note that the channel statistics @xmath18 and the source statistics @xmath50 are fixed , so the code design is essentially the appropriate choice of the outage states and the encoder - decoder pair @xmath71 . we denote by @xmath9 the achievable average distortion when the channel is in state @xmath1 , and it is given by @xmath72 where the summation is over all @xmath73 such that @xmath74 and @xmath75 . notice that the transmitter can not access channel state information so the encoder @xmath76 is independent of @xmath1 ; nevertheless the receiver can choose different decoders @xmath77 based on csir . in a composite channel , each channel state is assumed to be stationary and ergodic , so for a fixed channel state @xmath1 we can design source - channel codes such that @xmath78 approaches a constant limit @xmath9 for large @xmath16 ; however , it is possible that @xmath78 approaches different limits for different channel states . the expected distortion metric captures the distortion averaged over various channel states . using the conditional distribution @xmath66 in and the definition of @xmath9 in , the average distortion can be written as @xmath79 the expected distortion @xmath80 is the infimum of all achievable average distortions @xmath81 . for transmission of a source over a channel , the system consists of three concatenated blocks : the encoder @xmath67 that maps the source sequence @xmath64 to the channel input @xmath82 ; the channel @xmath18 that maps the channel input @xmath82 to channel output @xmath83 , and the decoder @xmath68 that maps the channel output @xmath83 to a reconstruction of the source sequence @xmath65 . in contrast , a separate source - channel coding scheme consists of five blocks . the encoder @xmath67 is separated into a source encoder @xmath84 and a channel encoder @xmath85 where the index set @xmath86 of size @xmath87 serves as both the source encoder output and the channel encoder input . equivalently , each index in @xmath86 can be viewed as a block of @xmath88 bits ( * ? ? ? * defn . 5 ) . the decoder @xmath68 is also separated into a channel decoder @xmath89 and a source decoder @xmath90 . the difference between a general system and a separate source - channel coding system is summarized in fig . [ fig : system3block ] . separation does not imply isolation - the source and channel encoders and decoders still need to agree on certain aspects of their respective designs . there are three interfaces through which they exchange information , the negotiation interface , the transmitter interface and the receiver interface . for classical shannon separation schemes with an end - to - end distortion target @xmath12 , these interfaces are summarized in table [ table : shannoninterface ] . the negotiation interface is a single rate comparison between @xmath14 and @xmath13 . since the shannon capacity definition requires that all transmitted information be correctly decoded , the transmission rate @xmath39 is the same as the receiving rate @xmath91 . assuming stationary and ergodic systems , these rates do not depend on the blocklength @xmath16 . however , these constraints can be relaxed to include more source - channel transmission strategies as separation schemes . .interface for shannon separation schemes [ cols="<,<",options="header " , ] [ table : interfaceresiduesplitting ] we provide some numerical examples to compare different schemes in this section . we assume the two states of the composite bsc have crossover probabilities @xmath92 and @xmath93 , and the bandwidth expansion ratio @xmath94 . for various schemes.,width=288 ] in fig . [ fig : d1d2 ] we plot the achievable distortion pair @xmath95 for each scheme . for the broadcast coding scheme , by varying the auxiliary variable @xmath96 from @xmath97 and @xmath98 , we change the rate allocation between the base layer @xmath99 and the refinement layer @xmath100 . the separation schemes using the shannon capacity code and the capacity versus outage code are the special cases of @xmath101 and @xmath98 , respectively . they are marked by the two end - points of the broadcast distortion region boundary . for the quantization residue splitting scheme , we calculate the distortion pairs @xmath95 for different parameters @xmath102 and @xmath103 . the plotted curve is the convex hull of all achievable distortion pairs . note that the broadcast scheme is a special case of the residue splitting scheme with @xmath104 , so the broadcast distortion region lies strictly within the residue splitting distortion region . there are two systematic codes , one targeting at each channel state . they are represented by two points , both out of the residue splitting distortion region . in fig . [ fig : edcompare ] we plot the expected distortion of various schemes for different channel state distributions . each systematic code achieves a single distortion pair , so the expected distortion is simply the weighted average and increases linearly with the bad channel state probability @xmath105 . for broadcast and residue splitting schemes , we need to choose the optimal point on the distortion region boundary at each channel state probability . since the broadcast scheme is a special case of the residue splitting scheme , its expected distortion is no less , and sometimes strictly larger , than that of the residue splitting scheme . for different ranges of @xmath105 , the scheme that achieves the lowest expected distortion is also different . for @xmath106 or @xmath107 it is the residue splitting scheme , for @xmath108 it is the systematic code for the good channel state , and for @xmath109 it is the systematic code for the bad channel state . expected distortion alone does not provide the complete picture for comparison of the schemes . in fig . [ fig : interfaceenc ] and [ fig : interfacedec ] we assume the channel state probability @xmath110 and illustrate the tradeoff between the expected distortion and the transmitter / receiver interface complexity for different schemes , where the complexity is measured by bits per source symbol delivered through the interface . for the broadcast scheme , we can reduce the expected distortion by increasing @xmath96 , which reduces the base layer rate but increases the refinement layer rate and the total rate , hence a higher interface complexity . however , the distortion - complexity curve is not strictly decreasing . after we reach the minimum expected distortion , it does not provide any more benefit to further increase the interface complexity . the same trend is also observed in the residue splitting scheme . at channel state probability @xmath111 , the systematic code targeting the good state has the lowest expected distortion , nevertheless it also has the highest interface complexity . the choice about the appropriate scheme and operating points ( parameters ) depends on the system designer s view about this distortion - complexity tradeoff . we consider transmission of a stationary ergodic source over non - ergodic composite channels with channel state information at the receiver ( csir ) . to study the source - channel coding problem for the entire system , we include a broader class of transmission schemes as separation schemes by relaxing the constraint of shannon separation , i.e. a single - number comparison between source coding rate and channel capacity , and introducing the concept of a source - channel interface which allows the source and channel to agree on multiple parameters . we show that different end - to - end distortion metrics lead to different conclusions about separation optimality , even for the same source and channel models . specifically , one such generalized scheme guarantees the separation optimality under the distortion versus outage metric . separation schemes are in general suboptimal under the expected distortion metric . we study the performance enhancement when the source and channel coders exchange more information through a more sophisticated interface , and illustrate the tradeoff between interface complexity and end - to - end performance through the example of transmission of a binary symmetric source over a composite binary symmetric channel . in fig . [ fig : expected_enc ] , the multi - resolution source code can be constructed as follows . consider three independent auxiliary random variables @xmath112@xmath113bernoulli@xmath114 , @xmath115@xmath113bernoulli@xmath116 , and @xmath117@xmath113bernoulli@xmath118 , where @xmath119 and @xmath120 , @xmath121 are given by . also define @xmath122 which has a bernoulli distribution with parameter @xmath123 . these variables are related to the source symbol through the relationship @xmath124 _ random codebook generation _ : generate @xmath125 sequences @xmath126 , @xmath127 , by uniform and independent sampling over the strong typical set @xmath128 . similarly , generate @xmath129 sequences @xmath130 , @xmath131 , drawn uniformly and independently over @xmath132 . _ decoding _ : if only the index @xmath139 is received , the decoder declares the estimate of the source sequence as @xmath140 . if both indices are received , the source is reconstructed as @xmath141 . following the procedures in @xcite and ( * theorem 1 ) we can easily verify the following distortion targets are achievable : @xmath142 , @xmath143 . in practice the mr source code can be implemented as a multi - stage vector quantization , which has an _ additive _ successive refinement structure @xcite . as shown in fig . [ fig : expected_enc ] , in channel state 2 only the base layer description is received and source dec 2 determines the base reconstruction @xmath144 . when both layers are received , source dec 1 determines a refinement sequence @xmath145 based on the refinement layer encoding index only , and add it to the base reconstruction @xmath144 to obtain the overall reconstruction @xmath65 . on the contrary , for general mr source codes the overall reconstruction may require a joint decoding of indices from both layers . the additive refinement structure reduces coding complexity , provides scalability , and does not incur any performance loss under certain conditions ( * ? ? ? * theorem 3 ) , which are all satisfied in this example . the broadcast channel code design , for a chosen @xmath146 , is summarized as follows . _ random codebook generation _ : generate @xmath147 independent codewords @xmath148 , @xmath149 , by i.i.d . sampling of a bernoulli@xmath116 distribution . generate @xmath150 independent codewords @xmath151 , @xmath152 , by i.i.d . sampling of a bernoulli@xmath153 distribution . _ decoding _ : given channel output @xmath155 , in state 2 we determine the unique @xmath156 such that @xmath157 in state 1 we look for the unique indices @xmath158 such that @xmath159 following the analysis of ( * ? ? ? * theorem 14.6.2 ) , we can show that the channel decoding error probability approaches zero as long as the encoding rates satisfy . roughly speaking , in channel state 2 , we observe @xmath160 where the channel noise @xmath161 is a bernoulli@xmath162 sequence . we want to decode the @xmath163 sequence subject to the overall interference - plus - noise @xmath164 , which is a bernoulli sequence with parameter @xmath165 , hence the achievable rate @xmath166 . in channel state 1 , we observe @xmath167 since @xmath168 , the sequence @xmath163 can be decoded and then subtracted off . we then decode @xmath169 subject to the noise @xmath170 , and the rate @xmath171 is achievable . m. effros and a. goldsmith . capacity definitions and coding strategies for general channels with receiver side information . in _ proc . inform . theory ( isit ) _ , page 39 , cambridge ma , august 1998 . m. effros , a. goldsmith , and y. liang . capacity definitions of general channels with receiver side information . in _ proc . inform . theory ( isit ) _ , pages 921925 , nice , france , june 2007 . c. t.k . ng , d. gndz , a. goldsmith , and e. erkip . recursive power allocation in gaussian layered broadcast coding with successive refinement . in _ communications _ , glasgow , scotland , june 2007 . to appear . c. t.k . ng , d. gndz , a. goldsmith , and e. erkip . minimum expected distortion in gaussian layered broadcast coding with successive refinement . in _ proc . inform . theory ( isit ) _ , pages 22262230 , nice , france , june 2007 . g. d. hu . on shannon theorem and its converse for sequence of communication schemes in the case of abstract random variables . in _ proc . 3rd prague conf . on inform . theory , stat . decision functions , random processes _ , pages 285333 , czechoslovak academy of sciences , prague , 1964 . c. tian , a. steiner , s. shamai(shitz ) , and s. diggavi . expected distortion for gaussian source with a broadcast transmission strategy over a fading channel . in _ proc . ieee inform . theory workshop on wireless networks _ , pages 15 , bergen norway , july 2007 .

we consider transmission of stationary and ergodic sources over non - ergodic composite channels with channel state information at the receiver ( csir ) . previously we introduced alternate capacity definitions to shannon capacity , including the capacity versus outage and the expected capacity . these generalized definitions relax the constraint of shannon capacity that all transmitted information must be decoded at the receiver . in this work alternate end - to - end distortion metrics such as the distortion versus outage and the expected distortion are introduced to relax the constraint that a single distortion level has to be maintained for all channel states . for transmission of stationary and ergodic sources over stationary and ergodic channels , the classical shannon separation theorem enables separate design of source and channel codes and guarantees optimal performance . for generalized communication systems , we show that different end - to - end distortion metrics lead to different conclusions about separation optimality even for the same source and channel models . separation does not imply isolation - the source and channel still need to communicate with each other through some interfaces . for shannon separation schemes , the interface is a single - number comparison between the source coding rate and the channel capacity . here we include a broader class of transmission schemes as separation schemes by relaxing the constraint of a single - number interface . we show that one such generalized scheme guarantees the separation optimality under the distortion versus outage metric . under the expected distortion metric , separation schemes are no longer optimal . we expect a performance enhancement when the source and channel coders exchange more information through more sophisticated interfaces , and illustrate the tradeoff between interface complexity and end - to - end performance through the example of transmitting a binary symmetric source over a composite binary symmetric channel .

the famous @xmath2-queens problem asks for the number of arrangements of @xmath2 nonattacking queens the largest possible number on an @xmath1[d : n ] chessboard . ( see , for instance , @xcite . ) there is no known general formula , other than the very abstract , that is to say impractical one we obtained in part ii . solutions have been found only by individual analyses for small @xmath2 . in this series of five papers @xcite we treat the problem by separating the board size , @xmath2 , from the number of queens , @xmath0,[d : q ] and rephrasing the whole problem in geometry . we also generalize to every piece @xmath3 whose moves are , like those of the queen , rook , and bishop , unlimited in length . such pieces are known as `` riders '' in fairy chess ( chess with modified rules , moves , or boards ) ; an example is the nightrider , whose moves are those of the knight extended to any distance . the problem then , given a fixed rider @xmath3 , is : [ pr : formula ] find an explicit , easily evaluated formula for @xmath4[d : indistattacks ] , the number of nonattacking configurations of @xmath0 unlabelled pieces @xmath3 on an @xmath5 board . two kinds of piece were previously solved : the rook , which is elementary , and the bishop , for which arshon and kotovec found a single formula . aside from those two , formulas in terms of @xmath2 have been found for only a few riders and only for small numbers of pieces for instance , up to 6 queens or 3 nightriders and mostly heuristically , without rigorous proof . ( given the power of computers , proving a formula is more difficult than making an educated guess , which itself is by no means easy even for an expert like vclav kotovec @xcite . ) finding a single comprehensive formula , for all numbers of pieces @xmath0 and all board sizes @xmath2 , for any piece other than the rook and bishop especially for the queen , the original problem of this type looks impossible . one difficulty is not being certain what such a formula should look like . [ pr : format ] describe the nature of a formula for @xmath4 for an arbitrary piece . one would wish there to be one style of formula that applies to all riders . this ideal is realized to an extent . we proved in part i that for each rider @xmath3 , @xmath4 is a quasipolynomial function of @xmath2 of degree @xmath6 and that the coefficients of powers of @xmath2 are given by polynomials in @xmath0 , up to a simple normalization ; for instance , the leading term is @xmath7 . being a quasipolynomial means that for each fixed @xmath0 , @xmath4 is given by a cyclically repeating sequence of polynomials in @xmath2 ( called the _ constituents _ of the quasipolynomial ) ; the shortest length of such a cycle is the period of @xmath4 . that raises a fundamental question . [ pr : period ] what is the period @xmath8[d : p ] of the quasipolynomial formula for @xmath4 ? ( the period , which pertains to the variable @xmath2 , may depend on @xmath0 . ) the period tells us how much data is needed to rigorously determine the complete formula ; @xmath9 values of the counting function determine it completely , since the degree is @xmath6 and the leading coefficient is known . the difficulty with this computational approach is that , in general , @xmath8 is hard to determine and seems usually to explode with increasing @xmath0 . ( indeed , we have reason to believe the period increases at least exponentially for any rider with at least three moves ; see theorem [ t : exponential ] . ) a better way would be to find information about the @xmath4 that is valid for all @xmath0 . [ pr : coeffs ] for a given piece @xmath3 , find explicit , easily evaluated formulas for the coefficients of powers of @xmath2 in the quasipolynomials @xmath4 , valid for all values of @xmath0 . a complete solution to problem [ pr : coeffs ] would solve problem [ pr : formula ] . we think that is unrealistic but we have achieved some results . in part i we took a first step : in each constituent polynomial , the coefficient @xmath10 of @xmath11 is ( neglecting a denominator of @xmath12 ) itself a polynomial in @xmath0 of degree @xmath13 , that varies with the residue class of @xmath2 modulo a period @xmath14 that is independent of @xmath0 . in other words , if we count down from the leading term there is a general formula for the @xmath15th coefficient as a function of @xmath0 that has its own intrinsic period ; the coefficient is independent of the overall period @xmath8 . this opens the way to explicit formulas and in part ii we found such a formula for the second leading coefficient as well as the complete quasipolynomial for an arbitrary rider with only one move an unrealistic game piece but mathematically informative . in part iii we found the third and fourth coefficients by concentrating on _ partial queens _ , whose moves are a subset of the queen s . still , that is only scratching at the surface ; we want to go further . in this part we present our current state of knowledge about specific pieces : the queen , rook , bishop , and nightrider and the pieces that have a subset of their moves . our goal is to prove exact quasipolynomial formulas for a fixed number @xmath0 of each piece , where @xmath0 is ( unavoidably ) small , and along the way to see how many complete formulas we can prove for coefficients of high powers of @xmath2 . how large a number @xmath0 and how many coefficients we can handle depends on the piece . for the rook , naturally , we get well - known formulas for all @xmath0 . at the other extreme we have only very partially solved three nightriders , for which the formula was previously found heuristically , without proof , by kotovec . no formula for four nightriders has even been guessed ; it is conceivable , but judging by the 11-digit denominator we computed ( see table [ tb : nightriders ] ) not probable , that it could be obtained by a painstaking analysis using our method . we offer no hope for five . one reason we want a quasipolynomial is to substitute @xmath16 to get the number of combinatorially distinct types of configuration , as explained in section i.5 . the arshon kotovec formula for bishops is not a quasipolynomial and does not help us find the bishops counting quasipolynomials . for that reason we consider the bishops problem only partially solved and we give it close attention in part v. we summarize our geometrical approach : we create a configuration of points and lines to represent attacks of specified slopes . the boundaries of the square determine a hypercube in @xmath17[d : configsp ] and the attack lines determine hyperplanes whose @xmath18-fractional points within the hypercube represent attacking configurations , which must be excluded ; the nonattacking configurations are the integral points inside the hypercube and outside every hyperplane , so it is they we want to count . the combination of the hypercube and the hyperplanes is an inside - out polytope @xcite . the ehrhart theory of inside - out polytopes implies quasipolynomiality of the counting function and that the period divides the _ denominator _ @xmath19,[d : d ] defined as the least common multiple of the denominators of all coordinates of vertices of the inside - out polytope . then we apply the inside - out adaptation of ehrhart lattice - point counting theory , in which we combine by mbius inversion the numbers of lattice points in the polytope that are in each intersection subspace of the hyperplanes . we also investigate the denominators of individual vertices , which provide a better understanding of the period because the overall denominator bounds it . in general in ehrhart theory the period and the denominator need not be equal and often are not , so it is surprising that in all our examples , and for any rider with only one move , they are . we can not prove that is always true for the inside - out polytopes arising from the problem of nonattacking riders , but this observation suggests that our approach to that problem may be a good test case for understanding the relationship between denominators and periods . we find an exact formula for the denominator of a one - move rider in proposition [ p:1moveiv ] , and we introduce a notion of trajectories to give intuition about finding the denominator for a piece with two moves ( section [ sec : config2 m ] ) . we show that when a piece has three or more moves , by letting the number @xmath0 of pieces increase we obtain a sequence of inside - out polytope vertices with denominators that increase exponentially , and the polytope denominators may increase even faster . these vertices arise from geometrical constructions related to fibonacci numbers . a summary of this paper : section [ parti ] recalls some essential notation and formulas from parts i iii . section [ sec : period ] describes the concepts we use to analyze the periods . we turn to the theory of attacking configurations of pieces with small numbers of moves in sections [ sec : fewer ] and [ sec : moremoves ] , partly to establish formulas and conjectural bounds for the denominators of their inside - out polytopes , especially for partial queens and nightriders , and partly to support the exponential lower bound on periods and our many conjectures . after connecting to our theory the known results on rooks in section [ r ] , we discuss the current state of knowledge and ignorance about bishops and semi - bishops in section [ b ] . section [ q ] treats the queen as well as the partial queens that are not the rook , bishop , and semi - bishop . section [ n ] concerns the nightrider and sub - nightriders , whose nonattacking placements have not been the topic of any previous theoretical discussion that we are aware of . we conclude with questions related to these ideas and with proposals for research . for example , since counting nonattacking rider placements is an accessible topic in ehrhart theory , we suggest in section [ simplified ] fairy chess pieces with relatively simple behavior that might provide insight into the central open problem of a good general bound on the period of the counting quasipolynomial in terms of @xmath0 and the set of moves . in future work we will prove stronger properties , such as that every positive integer @xmath20 appears as a denominator given enough pieces ; specifically , about @xmath21 pieces where @xmath22 depends on the piece s set of moves . we append a dictionary of notation for the benefit of the authors and readers . we must mention kotovec s book @xcite , replete with formulas , mostly generated by himself , for all kinds of nonattacking chess problems . we learned of this work after beginning our research ; then properties of kotovec s bishops and queens formulas inspired much of our detailed results . for instance , we saw that the coefficients of highest degree are constant ( independent of @xmath2 ) ; then we proved most of the observed constancies . kotovec conjectured some formulas for high - degree coefficients ; we prove some of those conjectures . we saw that the bishops quasipolynomials ( for @xmath23 ) all have period 2 ; in part v we prove that is true for every number ( at least 3 ) of bishops . kotovec conjectured that the period of a queens quasipolynomial is a product of fibonacci numbers ; we take a step toward a proof . anyone who wants to know the actual number of nonattacking placements of @xmath0 of our four principal pieces will find answers in the online encyclopedia of integer sequences @xcite . table [ tb : oeis ] gives sequence numbers in the oeis . the first sequence in each column is the sequence of square numbers . after that it gets interesting . ' '' '' @xmath0 & rooks & bishops & queens & nightriders + ' '' '' 1 & a000290 & a000290 & a000290 & a000290 + ' '' '' 2 & a163102 * & a172123 & a036464 & a172141 + ' '' '' 3 & a179058 & a172124 & a047659 & a173429 + ' '' '' 4 & a179059 & a172127 & a061994 & + ' '' '' 5 & a179060 & a172129 & a108792 & + ' '' '' 6 & a179061 & a176886 & a176186 & + ' '' '' 7 & a179062 & a187239 & a178721 & + ' '' '' 8 & a179063 & a187240 & & + ' '' '' 9 & a179064 & a187241 & & + ' '' '' 10 & a179065 & a187242 & & + each configuration - counting problem arises from making two choices : a chess piece , and the number of pieces . ( the size of the board is considered a variable within the problem . ) the pieces are placed on the integral points , @xmath24 for @xmath25 : = \{1,\ldots , n\}$ ] , in the interior of an integral dilation @xmath26 ^ 2 $ ] of the unit square . we call the set @xmath27 ^ 2 = ( n+1)(0,1)^2 \cap { \mathbb{z}}^2 , \label{d:[n]2}\ ] ] whose dilation factor is @xmath28 , the _ board _ , in full the _ integral square board_. we also call the open or closed unit square the `` ( square ) board '' ; it will always be clear which board we mean . we sometimes consider a general board @xmath29,[d : cb ] which is any rational convex polygon , i.e. , it has rational corners . ( we call the vertices of @xmath29 its _ corners _ to avoid confusion with other points called vertices . ) when we do not mention @xmath29 or a polygonal board , our board will always be square . the piece @xmath3[d : p ] has _ moves _ defined as all integral multiples of a finite set @xmath30[d : moveset ] of non - zero , non - parallel integral vectors @xmath31,[d : mr ] which we call the _ basic moves_. each one must be reduced to lowest terms ; that is , its two coordinates need to be relatively prime ; and no basic move may be a scalar multiple of any other . thus , the slope of @xmath32 contains all necessary information and can be specified instead of @xmath32 itself . we say two distinct pieces _ attack _ each other if the difference of their locations is a move . in other words , if a piece is in position @xmath33 , it attacks any other piece in the lines @xmath34 for @xmath35 and @xmath36 . attacks are not blocked by a piece in between , and they include the case where two pieces occupy the same location . ( the set @xmath30 is @xmath37 for the bishop , @xmath38 for the queen , @xmath39 for the nightrider , and of course @xmath40 for the rook . ) the number @xmath0 is the number of pieces that are to occupy places on the board ; we assume @xmath41 . a _ configuration _ @xmath42[d : config ] is any choice of locations for the @xmath0 pieces , including on the board s boundary , where @xmath43 denotes the position of the @xmath15th piece @xmath44 . ( the boundary , while not part of the board proper , is necessary in our counting method . ) therefore , @xmath45 is an integral point in the @xmath46-fold dilation of the @xmath6-dimensional closed , convex polytope @xmath47.[d : cp ] . if we are considering the undilated board , @xmath45 is a fractional point in @xmath48 . we consider these two points of view equivalent ; it will always be clear which kind of board or configuration we are dealing with . any integral point @xmath45 in the dilated polytope , or its fractional representative @xmath49 in the undilated board , represents a placement of pieces on the board , and vice versa ; thus we use the same term `` configuration '' for the point and the placement . in this part @xmath29 is usually the square board ; then the closed and open polytopes are @xmath50^{2q}}$ ] and @xmath51 . the constraint for a _ nonattacking configuration _ is that the pieces must be in the board proper ( so @xmath52 or its dilation ) and that no two pieces may attack each other . in other words , if there are pieces at positions @xmath53 and @xmath54 , then @xmath55 is not a multiple of any @xmath56 ; equivalently , @xmath57 for each @xmath56 , where @xmath58[d : mrperp ] . for counting we treat nonattacking configurations as interior integral lattice points in the dilation of an inside - out polytope @xmath59 , where @xmath47 and @xmath60 is the _ move arrangement_[d : ap ] , whose members are the _ move hyperplanes _ ( or _ attack hyperplanes _ ) @xmath61 for @xmath62 ; the equations of these hyperplanes are called the _ move equations _ ( or _ attack equations _ ) of @xmath3 . thus ( by the definition of `` interior '' of an inside - out polytope @xcite ) , a configuration @xmath63 is nonattacking if and only if it is in @xmath64 and not in any of the hyperplanes @xmath65 . the _ intersection lattice _ @xmath66[d : l ] is the lattice of all intersections of subsets of the move arrangement , ordered by reverse inclusion . the mbius function of @xmath66 is denoted by @xmath67.[d : mu ] a _ vertex _ of @xmath59 is any point in @xmath68 that is the intersection of facets of @xmath68 and hyperplanes of @xmath60 . for instance , it may be a vertex of @xmath68 , or it may be the intersection point of hyperplanes if that point is in @xmath68 , or it may be the intersection of some facets and some hyperplanes . each subspace @xmath69[d : u ] is the intersection of hyperplanes involving a set @xmath70 consisting of @xmath71[d : kappa ] of the @xmath0 pieces . the _ essential part _ of @xmath72 is the subspace @xmath73[d : tu ] of @xmath74 that satisfies the same attack equations as @xmath72 . define @xmath75[d : alphau ] to be the number of integral points in the dilation of @xmath76 , i.e. , @xmath77 ( the utility of this quantity is that it is independent of @xmath0 , because @xmath73 is independent of the value of @xmath0 used to construct it from @xmath72 . ) by ehrhart theory @xmath75 is a quasipolynomial of degree @xmath78 . since @xmath79 , the number of lattice points in @xmath80 is @xmath81 . the parity theorem ( theorem ii.4.1 ) tells us that @xmath75 is an even or odd function of @xmath2 ( depending on the codimension of @xmath72 ) . what it does not say is how that affects the number of undetermined coefficients in computing @xmath75 , which is , in particular , the number of values of the function we need to interpolate all the coefficients . in general , an ehrhart quasipolynomial of degree @xmath82 with period @xmath8 has @xmath83 coefficients that have to be computed . ( the leading coefficient is the same for all constituents ; it is the volume of @xmath84 . ) the full theorem , then , should be this : [ t : strongparity ] for a subspace @xmath85 whose equations involve @xmath71 pieces , for which @xmath75 has period @xmath8 , the number of values of @xmath75 that are sufficient to determine all the coefficients in all constituents is @xmath86 , where @xmath87 if @xmath88 is even and @xmath89 if it is odd . let @xmath90 and @xmath91 . thus , @xmath92 has degree @xmath93 . let the constituents of @xmath92 be @xmath94 ; that means @xmath95 . we take subscripts modulo @xmath8 so that , e.g. , @xmath96 . write @xmath97 since @xmath98 ( corollary ii.4.1 ) , @xmath99 subtracting , @xmath100 n^j = 0 , $ ] which implies that @xmath101 for @xmath102 . it follows that only the coefficients for @xmath103 need to be computed . there are @xmath104 coefficients with @xmath102 for @xmath105 . for @xmath106 , corollary ii.4.1 says that @xmath107 is an even or odd polynomial ( depending on @xmath82 ) and so is @xmath108 if the period is even . the number of coefficients to determine , other than @xmath109 , is therefore @xmath110 for @xmath107 and the same for @xmath108 if it exists . summing these up , there are @xmath111 independent coefficients to be computed in @xmath92 . the quasipolynomial for the number of nonattacking configurations of @xmath0 unlabelled pieces on an @xmath1 board expands in powers of @xmath2 in the form @xmath112 with labelled pieces the number is @xmath113[d : distattacks ] , which equals @xmath114 . our task is to find the coefficients @xmath115 , or in practice @xmath116 , which we know to be polynomials in @xmath0 that may differ for each residue class of @xmath2 modulo the period @xmath8 ( theorem i.4.2 ) . ehrhart theory says that the leading coefficient of @xmath113 is the volume of the polytope @xmath50^{2q}}$ ] , i.e. , 1 ; so @xmath117 for every piece . in the proofs we assume acquaintance with the counting theory of part ii for the square board , @xmath118 ^ 2 $ ] in the notation of part ii . for a basic move @xmath119 , we define @xmath120 , @xmath121,[d : cdhat ] and @xmath122.[d : barn ] in part ii we defined @xmath123 the number of ordered pairs of positions that attack each other along slope @xmath124 ( they may occupy the same position ; that is considered attacking ) . similarly , @xmath125 the number of ordered triples that are collinear along slope @xmath124 . proposition ii.3.1 gives general formulas for @xmath92 and @xmath126 . we need only a few examples for later use : @xmath127 \frac{5}{12}n^3+\frac{7}{12}n & \text{for $ n$ odd } \end{cases } \\ & = \frac{5}{12}n^3+\frac{11}{24}n + ( -1)^n \left\ { \frac{1}{8}n \right\ } , \\ \beta^{\pm2/1}(n ) = \beta^{\pm1/2}(n ) & = \begin{cases } \frac{3}{16}n^4+\frac{1}{4}n^2 & \text{for $ n$ even , } \\[3pt ] \frac{3}{16}n^4+\frac{5}{8}n^2+\frac{3}{16 } & \text{for $ n$ odd . } \end{cases } \\ & = \frac{3}{16}n^4+\frac{7}{16}n^2+\frac{3}{32 } - ( -1)^n\left\ { \frac{3}{16}n^2+\frac{3}{32 } \right\}. \end{aligned } \label{e : attacklines}\ ] ] a point @xmath42 of an inside - out polytope @xmath128 , associated with @xmath0 copies of a piece @xmath3 on a board @xmath29 , represents a configuration of @xmath0 pieces on the board , with piece @xmath44 located at position @xmath129 . a vertex @xmath45 of @xmath128 is determined by @xmath6 equations that are either _ move equations _ , associated to hyperplanes @xmath130 , or _ boundary equations _ , also called _ fixations _ in this part , of the form @xmath131 an edge line @xmath132[d : ce ] of @xmath29 ; for the square board a fixation is one of @xmath133 , @xmath134 , @xmath135 , and @xmath136 . ( for a configuration in an @xmath137-fold dilation @xmath138 , a fixation has the form @xmath139 . ) the vertex @xmath45 represents a configuration with @xmath44 on the boundary of the board ( if it has a fixation ) or attacking one or more other pieces ( if in a hyperplane ) . we call the configuration of pieces that corresponds to a vertex @xmath45 a _ vertex configuration_. let us write @xmath140[d : deltabz ] for the least common denominator of a fractional point @xmath141 and call it the _ denominator of @xmath45_. the denominator @xmath142 of the inside - out polytope is the least common multiple of the denominators @xmath140 of the individual vertices . one way to find @xmath140 for a vertex @xmath45 is to find its coordinates by intersecting move hyperplanes of @xmath3 and facet hyperplanes of @xmath48 . there is an equivalent method to find @xmath140 . for a set of move equations and fixations producing a vertex configuration @xmath45 , notice that the @xmath140-multiple of @xmath45 has integer coordinates and no smaller multiple of @xmath45 does . this proves : [ l : deltan ] for a vertex @xmath45 of @xmath143 , @xmath140 equals the smallest integer @xmath137 such that a configuration @xmath144 satisfying the move equations and fixations @xmath145 for edge lines @xmath132 ( @xmath133 , @xmath134 , @xmath146 , or @xmath147 on the square board ) has integral coordinates . we define @xmath148 [ d : deltaq ] to be the maximum value of the denominator @xmath140 over all vertices @xmath45 of the inside - out polytope for @xmath0 . ( it is not to be confused with @xmath149 , which means the least common multiple of vertex denominators . ) we expect the period to be weakly increasing with @xmath0 and also with the set of moves ; that is , if @xmath150 , the period for @xmath151 pieces should be a multiple of that for @xmath0 ; and if @xmath152 has move set containing that of @xmath3 , then the period of @xmath152 should be a multiple of that of @xmath3 . ( the pieces need not be partial queens . ) we can not prove either property , but they are obvious for denominators , and we see in examples that the period equals the denominator . we write @xmath153 for the denominator of the inside - out polytope @xmath154^{2q}},{\mathscr{a}}_{\mathbb p}^q)$ ] . ( the optional superscript in @xmath155 shows the number of pieces . ) [ p : dincrease ] let @xmath29 be any board , let @xmath156 , and suppose @xmath3 and @xmath152 are pieces such that every basic move of @xmath3 is also a basic move of @xmath152 . then the denominators satisfy @xmath157 and @xmath158 . the first part is clear if we embed @xmath17 into @xmath159 as the subspace of the first @xmath6 coordinates , so the polytope @xmath48 is a face of @xmath160 and the move arrangement @xmath155 in @xmath17 is a subarrangement of the arrangement induced in @xmath17 by @xmath161 . the second part is obvious since @xmath162 . although in ehrhart theory periods often are less than denominators , we observe that not to be true for our solved chess problems . we believe that will some day become a theorem . [ cj : p = d ] for every rider piece @xmath3 and every number of pieces @xmath163 , the period of the counting quasipolynomial @xmath4 equals the denominator @xmath164^{2q}},{\mathscr{a}}_{\mathbb p})$ ] of the inside - out polytope for @xmath0 pieces @xmath3 . a _ partial queen _ @xmath165[d : partq ] is a piece with @xmath166 basic moves that are horizontal or vertical ( obviously , @xmath167 ) and @xmath168 basic moves at @xmath169 to the horizontal ( also , @xmath170 ) . we studied partial queens in part iii . table [ tb : pqueensperiods ] contains a list of the partial queens with their names and what we know or believe about the periods of their counting quasipolynomials . in particular , we think four of the partial queens are uniquely special . ' '' '' & + & & @xmath171 & @xmath172 & @xmath173 & @xmath174 & @xmath175 & @xmath176 + semirook & @xmath177 ' '' '' & 1 & 1 & 1 & 1 & 1 & 1 + rook & @xmath178 ' '' '' & 1 & 1 & 1 & 1 & 1 & 1 + semibishop & @xmath179 ' '' '' & 1 & 1 & 1 & 1 & 1 & 1 + subqueen & @xmath180 ' '' '' & 1 & 1 & 1 & 1 & 1 & 1 * ( @xmath181 ) + semiqueen & @xmath182 ' '' '' & 1 & 1 & 2 * & 2 * & 6 * & 12@xmath183 ( @xmath184 ) + bishop & @xmath185 ' '' '' & 1 & 2 & 2 & 2 & 2 & 2 + frontal queen & @xmath186 ' '' '' & 1 & 2 & 6 * & 12@xmath183 & 60@xmath183 & 420@xmath183 ( @xmath184 ) + queen & @xmath187 ' '' '' & 1 & 2 & 6 * & 60 * & @xmath188 & 360360 * ( @xmath184 ) + [ 2 pt ] [ cj : period1 ] the rook and semirook , the semibishop , and the subqueen are the only four pieces that have period 1that is , whose counting functions are polynomials in @xmath2 . conjecture [ cj : period1 ] is supported by the fact that , by theorem [ t : maxcd ] , the only one - move pieces with period 1 are the semirook and semibishop . the rook is obvious . if conjecture [ cj : trajectories ] is true corollary [ cor : subqueen ] establishes the same for the subqueen . any other one - move piece has larger period and denominator . conjecture [ cj : period1 ] follows from proposition [ p : dincrease ] and conjecture [ cj : p = d ] , if the latter is true . the denominator of the inside - out polytope of a one - move rider can be explicitly determined for arbitrary boards @xmath29 . given a move @xmath119 , the line parallel to @xmath32 through a corner @xmath189 of @xmath29 may pass through another point on the boundary of @xmath29 . call that point the _ antipode _ of @xmath189 . the antipode may be another corner of @xmath29 . when @xmath32 is parallel to an edge @xmath190 of @xmath29 , we consider @xmath53 and @xmath54 to be each other s antipodes . [ p:1moveiv ] for a one - move rider @xmath3 with move @xmath191 , the denominator of the inside - out polytope @xmath192 equals the least common denominator of the corners of @xmath29 when @xmath193 , and when @xmath194 it equals the least common denominator of the corners of @xmath29 and their antipodes . a vertex of @xmath192 is generated by some set of hyperplanes , possibly empty , and a set of fixations . the total number of hyperplanes and fixations required is @xmath6 . when @xmath193 , because there are no move equations involved , a vertex of the inside - out polytope is a corner of @xmath29 . when @xmath194 , a vertex is determined by its fixations and the intersection @xmath72 of the move hyperplanes it lies in . let @xmath195 be the partition of @xmath196 $ ] into blocks for which @xmath15 and @xmath197 are in the same block if @xmath65 is one of the hyperplanes containing @xmath72 . the number of hyperplanes necessary to determine @xmath72 is @xmath0 minus the number of blocks of @xmath195 . ( @xmath72 will be contained in additional , unnecessary hyperplanes if a block of @xmath195 has three or more members ; we do not count those . ) consider a particular block of @xmath195 , which we may suppose to be @xmath198 $ ] for some @xmath199 . we need @xmath200 fixations on the @xmath168 pieces to specify a vertex , so there must be two fixations that apply to the same @xmath201 $ ] , anchoring @xmath53 to a corner of @xmath29 . the remaining @xmath202 fixations fix the other values @xmath54 for @xmath203 $ ] to either @xmath53 s corner or its antipode . it follows that all vertices @xmath204 of the inside - out polytope satisfy that each @xmath53 is either a corner or a corner s antipode for all @xmath15 . furthermore , with at least two pieces and for every corner @xmath189 , it is possible to create a vertex containing @xmath189 and its antipode as components , from which the proposition follows . for the square board , the corners are @xmath205 , @xmath177 , @xmath179 , and @xmath180 , and the antipodes have denominator @xmath206 . alongside proposition ii.6.2 , this proves conjecture ii.6.1 . [ t : maxcd ] on the square board with @xmath207 copies of a one - move rider with basic move @xmath191 , the period of @xmath4 is @xmath206 . this is an example where conjecture [ cj : p = d ] is true : the period agrees with the denominator . we continue with a general board @xmath29 . one fruitful kind of configuration of two - move riders with moves @xmath208 and @xmath209 involves a _ trajectory _ , composed of a sequence of distinct pieces in which the first piece @xmath210 is anchored at a corner of @xmath29 by two fixations , and the position of each subsequent piece @xmath211 is determined by one move hyperplane @xmath212 involving @xmath213 and by one fixation . the _ basic moves _ of a trajectory are the moves @xmath214 , each of which equals either @xmath208 or @xmath209 . we can regard a trajectory as a sequence of points in the board instead of pieces ; the position of @xmath211 is denoted by @xmath215 . a trajectory is _ primitive _ if any piece after @xmath210 that is placed at a corner of @xmath29 is its last . any trajectory can be decomposed into primitive trajectories in the following manner . suppose @xmath216 is a trajectory , perhaps accompanied by an additional set of trajectories in order to completely determine a vertex @xmath45 of the inside - out polytope , in which @xmath217 and @xmath218 is a corner . we break @xmath219 into @xmath220 and @xmath221 with @xmath222 anchored at its corner by two fixations . we have replaced the move hyperplane @xmath223 by a new fixation that had been implied by this move hyperplane because it had forced @xmath222 to be in a corner . conversely , any equation , whether from a move hyperplane or a fixation , that fixes @xmath222 in its corner is sufficient to give the same vertex @xmath45 as we get from @xmath219 and the other accompanying trajectories . therefore @xmath219 can be replaced by @xmath224 and @xmath225 in determining @xmath45 . a _ simple trajectory _ is a primitive trajectory in which no two consecutive slopes are equal and no segment @xmath226 lies on the boundary of the board . figure [ fig : twomove ] shows that primitive trajectories can involve complex dynamics . the pattern of piece placements depends on the range into which the slopes fall ( less than @xmath227 , between @xmath227 and @xmath89 , between @xmath89 and @xmath228 , and greater than @xmath228 ) . in most cases , the piece positions and the denominator of the corresponding vertex follow a pattern that is difficult to describe completely . , @xmath229 , and @xmath230 . , title="fig:",width=182 ] , @xmath229 , and @xmath230 . , title="fig:",width=182 ] , @xmath229 , and @xmath230 . , title="fig:",width=182 ] we believe that for two - move riders simple trajectories encompass the full scope of possible denominators , as conjectured here . [ cj : trajectories ] the points @xmath231 that occur as components of a vertex @xmath45 of an inside - out polytope of @xmath0 two - move riders are determined by simple trajectories . such points can only occur as points along a @xmath168-point simple trajectory where @xmath232 , as intersection points of two simple trajectories of @xmath15 and @xmath197 points where @xmath233 , or as self - intersection point of a @xmath168-point simple trajectory where @xmath232 . as a consequence , denominators of @xmath45 only arise from such points . intersection points have to be included in the conjecture . if @xmath189 is the intersection of segments @xmath234 and @xmath235 of the trajectory or trajectories , segments parallel respectively to moves @xmath32 and @xmath236 , @xmath45 may also have a piece @xmath237 located at @xmath189 ( so @xmath238 ) , which is fixed in place by the move equations connecting it to @xmath44 by move @xmath32 and @xmath239 by move @xmath236 . the length limits on the trajectories come from the fact that there are only @xmath0 pieces . for an intersection point , one piece @xmath237 has to be reserved to place on the intersection in order to generate that point s denominator . with @xmath237 on the intersection it is no longer necessary to have a piece on the final points of the intersecting trajectory or trajectories . example [ ex : threemove ] demonstrates that the conjecture is false for three or more moves . [ p : denom2move ] let @xmath29 be the square board and let @xmath240 and @xmath82 be relatively prime positive integers . if conjecture [ cj : trajectories ] is true , then the denominator of the inside - out polytope for @xmath0 two - move riders with moves @xmath177 and @xmath241 is @xmath242 we consider the two - move rider with move @xmath191 ; the argument is similar for the other signs . we assume @xmath243 since otherwise the period is 1 and the problem is trivial . assuming conjecture [ cj : trajectories ] , we obtain all vertices of the inside - out polytope by combining simple trajectories . construct a trajectory @xmath219 by fixing a piece @xmath244 at the corner @xmath205 and following the two moves alternately , with slope first @xmath124 , then @xmath89 , etc . at each step , stop when the move hits an edge of the square , place the next piece there , and then begin the next move . so @xmath245 , @xmath246 , @xmath247 have coordinates @xmath248 , @xmath249 , @xmath250 , and so forth . if @xmath0 is sufficiently large , this generates a trajectory that continues until it reaches @xmath251 , where it stops . evidently , @xmath219 is simple and has no self - intersections . the only other simple trajectory is @xmath252 , a rotation of @xmath219 by 180@xmath183 around the center of the square . @xmath252 begins at @xmath180 and is centrally symmetric to @xmath219 . if @xmath253 , then @xmath219 ends at @xmath245 with coordinates @xmath254 . @xmath219 and @xmath252 do not intersect , so @xmath82 is the only denominator . if @xmath255 , then @xmath245 in trajectory @xmath219 has coordinates @xmath248 and @xmath246 has coordinates @xmath249 . this zigzag pattern continues up to @xmath256 at @xmath257 , where @xmath258 , and then @xmath259 on the line @xmath251 with @xmath260-coordinate @xmath261 . ( see the illustration in figure [ fig : pfdenom2move ] . ) by central symmetry , the points along @xmath219 and @xmath252 have the same denominators and neither @xmath219 nor @xmath252 has a self - intersection . thus , if @xmath262 , there is a denominator @xmath82 as well as @xmath240 and the overall denominator of the configuration is @xmath263 . until a piece has @xmath264-coordinate @xmath228 . in this example @xmath265 and the coordinates of the even pieces are @xmath266 , @xmath267 , @xmath268 , and @xmath269.,width=192 ] it remains to calculate the intersections of @xmath219 and @xmath252 if they are not the same trajectory and if @xmath0 is big enough for them to intersect . they are the same when @xmath270 is an integer ( that means @xmath271 ) and @xmath272 ( @xmath273 has coordinates @xmath180 in that case ) ; then the denominator is @xmath240 , which is also the denominator when there are @xmath0 points in @xmath219 alone if @xmath274 . now assume @xmath219 and @xmath252 are not the same ; that is , @xmath270 is fractional ( @xmath275 ) . intersections occur when a sloped edge of one trajectory ( say @xmath219 ) intersects a horizontal edge of the other trajectory ( say @xmath252 ) . the horizontal edges of @xmath252 occur at @xmath264-coordinates @xmath276 , @xmath277 . the sloped edge joins , say , @xmath278 to @xmath279 . the @xmath260-coordinate of the intersection point is @xmath280 , whose denominator is @xmath82 . thus , points on the edges and points of intersection have denominators @xmath240 and @xmath82 , implying that the least common denominator of all points of the configuration is @xmath263 . for there to be an intersection , however , @xmath279 must have greater @xmath264-coordinate than a horizontal edge of @xmath252 ; thus , @xmath281 , or @xmath282 . as the horizontal edge in @xmath252 is @xmath283 , the number of pieces necessary for an intersection to occur is @xmath284 , the denominator @xmath82 appears only if @xmath285 . otherwise , the overall denominator in the trajectories is @xmath240 . proposition [ p : denom2move ] applies to the subqueen , which has one diagonal move and one horizontal or vertical move . therefore : [ cor : subqueen ] if conjecture [ cj : trajectories ] is true , the denominator and period of the subqueen are @xmath228 . a piece with three or more moves enters a new domain of complexity that begins when we have just three copies of it . ( in this section and from now on we assume the square board . ) for such a piece , the denominator @xmath149 grows exponentially , or super - exponentially , with @xmath0 ( theorem [ t : exponential ] ) . conjecture [ cj : p = d ] would imply that the period of its counting quasipolynomial also grows at least exponentially . two fruitful constructions that yield the largest known vertex denominators @xmath140 combine @xmath286 move equations and three fixations . these constructions produce pieces arranged in a golden parallelogram configuration when the piece has at least three moves ( see example [ ex : parallelograms ] ) or in a twisted fibonacci spiral when the piece has at least four moves ( see example [ ex : twisted ] ) . with three or more moves , a new key configuration appears . it is a triangle of pairwise attacking pieces . we can calculate the corresponding denominator @xmath20 . consider a piece with the three moves @xmath287 , @xmath288 , and @xmath289 . since no move is a multiple of another , there exist nonzero integers @xmath290 , @xmath291 , and @xmath292 [ d : w ] such that @xmath293 with @xmath294 . the @xmath295 are unique up to negating all of them . [ p : triangular ] for @xmath172 , a triangular configuration of three pieces on the square board , attacking pairwise along three distinct move directions @xmath287 , @xmath288 , and @xmath289 , together with three fixations that fix its position in the square @xmath118 ^ 2 $ ] , gives a vertex @xmath45 of the inside - out polytope . its denominator is @xmath296 the pieces may be at corners , and there may be two pieces on the same edge . the three fixations may be choosable in more than one way but they will give the same denominator . there is a unique similarity class of triangles with edge directions @xmath297 , @xmath298 , and @xmath299 , if we define triangles with opposite orientations to be similar . we can assume the the pieces are located at coordinates @xmath300 with @xmath301 ( by diagonal reflection ) , with @xmath302 ( by suitably numbering the pieces ) , with @xmath303 ( by horizontal reflection ) , and with @xmath304 below the line @xmath305 ( by a half - circle rotation ) . the reflections change the move vectors @xmath306 by negating or interchanging components ; that makes no change in equation . we number the slopes so that @xmath297 , @xmath298 , and @xmath299 are , respectively , the directions of @xmath307 , @xmath305 , and @xmath308 . given these assumptions the triangle must have width @xmath309 , since otherwise it will be possible to enlarge it by a similarity transformation while keeping it in the square @xmath118 ^ 2 $ ] ; consequently @xmath310 and @xmath311 . furthermore , the slopes satisfy @xmath312 . ( if @xmath313 we say the slope @xmath314 and treat it as greater than all real numbers . if @xmath315 we say @xmath316 and treat it as less than all real numbers . @xmath317 can not be 0 . ) our configuration has @xmath318 so two slopes are nonnegative but @xmath319 may be negative . that gives two cases . if @xmath320 , we choose fixations @xmath310 , @xmath321 , and @xmath311 . ( a different choice of fixations is possible if @xmath307 is horizontal or vertical , if @xmath308 is horizontal or vertical , or if @xmath305 is horizontal , not to mention combinations of those cases . note that the denominator computation depends on the differences of coordinates rather than their values . in each horizontal or vertical case the choice of fixations affects only the triangle s location in the square , not its size or orientation . ) if @xmath322 , we choose fixations @xmath323 and @xmath311 . the rest of the proof is the same for both cases . first we prove that the configuration is a vertex . that means the locations of the three pieces are completely determined by the fixations and the fact that @xmath324 . we know the similarity class of @xmath325 and its orientation . the fixations of @xmath244 and @xmath246 determine the length of the segment @xmath305 . that determines the congruence class of @xmath325 , and the fixations determine its position . thus , @xmath45 is a vertex . we now aim to find the smallest integer @xmath137 such that @xmath326 has integral coordinates , i.e. , it embeds in the integral lattice @xmath327\times[0,n]$ ] . by the definition of @xmath290 , @xmath291 , and @xmath292 , we know that @xmath328 , @xmath329 , and @xmath330 gives an integral triangle that is similar to @xmath325 and similarly or oppositely oriented , because its sides have the same slopes . if @xmath331 is oppositely oriented to @xmath325 ( that means @xmath332 is above the line @xmath333 ) , we can make the orientations the same by negating all @xmath295 . given these restrictions @xmath331 is as compact as possible , for if some multiple @xmath334 were smaller ( @xmath335 ) and integral , then @xmath336 with integers @xmath337 , so @xmath338 would be a proper divisor of 1 . by lemma [ l : deltan ] , . we can now translate @xmath331 to the box @xmath327\times[0,n]$ ] where @xmath340 [ ex : threemove ] for the three - move partial nightrider with move set @xmath341 , since @xmath342 the denominator is @xmath343 as shown in figure [ fig : threemove ] . , @xmath344 , @xmath345 , @xmath310 , @xmath321 , and @xmath346 . the coordinates are @xmath347 , @xmath348 , and @xmath349 . this illustrates proposition [ p : triangular ] when @xmath350 . the value of @xmath137 is @xmath351.,height=163 ] , @xmath352 , @xmath345 , @xmath310 , @xmath321 , and @xmath346 . the coordinates are @xmath205 , @xmath353 , and @xmath354 . this illustrates proposition [ p : triangular ] when @xmath322 . the value of @xmath137 is @xmath355.,height=163 ] now , for pieces with three or more moves , we explore vertex configurations that use only three moves . we motivate the general case by studying the semiqueen @xmath356 which has a horizontal , vertical , and diagonal move : @xmath357 . to supplement the standard move - hyperplane notation we define @xmath358 and @xmath359[d : xy ] for the hyperplanes that express an attack along a file ( column ) or rank ( row ) . in this subsection and the next we use the terminology `` golden rectangle configuration '' and `` discrete fibonacci spiral '' , which is inspired by the following two concepts . ( we index the fibonacci numbers @xmath360[d : fib ] so that @xmath361 . ) a _ golden rectangle _ is a rectangle whose sides are in the ratio @xmath362 , @xmath363 being the golden ratio @xmath364.[d : phi ] the rectangle that has side lengths @xmath360 and @xmath365 is a close approximation to such a rectangle . the _ fibonacci spiral _ is an approximation to the _ golden spiral _ ( the logarithmic spiral with growth factor @xmath363 ) where squares of fibonacci side length are arranged in an outwardly spiraling manner and each has a quarter circle inscribed , as shown in figure [ fig : fibspiral ] . there are multiple vertex configurations of @xmath0 semiqueens that have denominator @xmath366 . in this analysis we take the diagonal move to be @xmath367 . the _ golden rectangle configuration _ with @xmath0 semiqueens is defined by the move equations @xmath368 for all @xmath15 such that both indices fall between @xmath228 and @xmath0 , inclusive , and fixations @xmath369 , @xmath370 , and either @xmath371 if @xmath372 is even or @xmath373 if @xmath372 is odd . these fixations define the smallest square box that contains all pieces in the configuration . they also serve to locate the configuration in the unit - square board , by giving the unique positive integer @xmath137 such that dividing by @xmath137 fits the shrunken configuration @xmath45 into the square board with three queens fixed on its boundary ; thus @xmath45 is a vertex ( and the shrinkage justifies the use of the term _ fixation _ for equations like @xmath371 ) . the denominator @xmath140 is that integer @xmath137 . ( see lemma [ l : deltan ] . ) the denominator of this configuration for every value of @xmath0 is therefore @xmath374 figure [ fig : q21fib1](a ) shows the golden rectangle configuration of @xmath375 semiqueens ; the configuration fits in an @xmath376 rectangle . we note that figure [ fig : q21fib1](b ) is a related expanding discrete fibonacci spiral that has the same denominator ; a similar spiral will figure more prominently in configurations with four moves . it is straightforward to find the coordinates of @xmath44 , which we present without proof . we assume coordinates with origin in the lower left corner of figure [ fig : q21fib1](a ) . for the semiqueen @xmath377 , when the pieces are arranged in the golden rectangle configuration , @xmath244 is in position @xmath177 and for @xmath378 , @xmath379 is in position @xmath380 the step from @xmath381 to @xmath44 is @xmath382 a key idea is that we can apply a linear transformation to the golden rectangle configuration to create * six * _ golden parallelogram configurations _ for any piece with three moves , some of which may coincide if there is symmetry in the move set . to define the golden parallelogram , in the golden rectangle configuration consider the semiqueens @xmath383 at position @xmath177 , @xmath384 at @xmath179 , and @xmath385 at @xmath180 . they form the smallest possible triangle ; they and the construction rule determine the positions of the remaining pieces . for an arbitrary piece @xmath3 with moves @xmath297 , @xmath298 , and @xmath299 , we consider the smallest integral triangle involving three copies of @xmath3 , which we discussed in proposition [ p : triangular ] . we apply to the golden rectangle configuration a linear transformation that takes vectors @xmath386 and @xmath387 to any ordered choice of two of the vectors @xmath388 , @xmath389 , and @xmath390 , with a minus sign on one of them if needed to ensure that the third side of the triangle has the correct orientation ; that transforms the golden rectangle with the @xmath391 in their locations to a golden parallelogram with pieces @xmath44 in the transformed locations and with @xmath392 forming the aforementioned smallest triangle ; hence , there are six possible golden parallelograms . [ ex : parallelograms ] for the three - move partial nightrider ( defined in section [ n ] ) the vectors are @xmath393 , @xmath394 , and @xmath395 . the corresponding six distinct golden parallelogram configurations are presented visually in figure [ fig : parallelograms ] . the precise linear transformations are given in table [ tab : parallelograms ] . we see that of these six parallelograms , the one yielding the largest denominator is that in the upper left . .,title="fig:",height=182 ] .,title="fig:",height=182 ] .,title="fig:",height=182 ] .,title="fig:",height=182 ] .,title="fig:",height=182 ] .,title="fig:",height=182 ] .the linear transformations corresponding to the golden parallelogram configurations of 13 pieces in figure [ fig : parallelograms ] , along with the denominator @xmath20 for each configuration . [ cols="^ " , ] + @xmath20 & 152 & 125 & 139 + [ tab : parallelograms ] these golden parallelograms appear to maximize the denominator , from which we infer formulas for the largest denominators for various pieces of interest . for a piece with exactly three moves , the vertex configuration giving the largest denominator is one of the golden parallelogram configurations . [ ex : q21parallelogram ] the semiqueen has ( up to symmetry ) only one other golden parallelogram besides the golden rectangle ; it is shown in figure [ fig : q21fib2 ] . it has a larger denominator than the golden rectangle configuration when @xmath0 is odd and @xmath396 . semiqueens has the largest denominator when @xmath0 is odd.,height=240 ] here , if we consider @xmath245 to be in position @xmath205 , then @xmath244 is in position @xmath397 and for @xmath378 , @xmath379 is in position @xmath398 [ conj : semiqueen ] the largest denominator of a vertex configuration for @xmath0 semiqueens @xmath356 is @xmath366 if @xmath0 is even and is @xmath399 if @xmath0 is odd . [ ex : q12parallelograms ] the frontal queen @xmath400 gives the three distinct golden parallelogram configurations shown in figure [ fig : q12fib1 ] . . for twelve pieces , the denominators are 25 , 21 , and 20 , respectively.,title="fig:",height=192 ] . for twelve pieces , the denominators are 25 , 21 , and 20 , respectively.,title="fig:",height=192 ] . for twelve pieces , the denominators are 25 , 21 , and 20 , respectively.,title="fig:",height=192 ] once again , the largest denominator depends on @xmath0 . because the piece positions for @xmath378 for the configuration in figure [ fig : q12fib1](a ) are @xmath401 the largest denominator for such a configuration with @xmath0 pieces is @xmath402 on the other hand , in the configuration in figure [ fig : q12fib1](c ) , the piece positions for @xmath378 are @xmath403 which yields a largest denominator of such a configuration with @xmath0 pieces of @xmath404 [ conj : frontal ] the largest denominator of a vertex configuration for @xmath0 frontal queens @xmath400 is @xmath405 if @xmath0 is even and @xmath406 if @xmath0 is odd . the symmetry in the piece positions for the two configurations is remarkable . suppose the linear transformation that creates a golden parallelogram carries @xmath407 and @xmath408 . it is possible to write an explicit formula for the denominator of the resulting golden parallelogram configuration . we have not computed the entire formula . it has not more than @xmath409 cases , with one case for each value of @xmath410 and one subcase for each of the @xmath411 sign patterns of the components of @xmath388 and @xmath389 ( sign @xmath89 can be combined with sign @xmath412 ) , and in some of those subcases one further subcase for each of the @xmath413 magnitude relations between @xmath414 and @xmath415 or between @xmath416 and @xmath417 . [ t : exponential ] the denominators @xmath153 of any piece that has three or more moves increase at least exponentially ; specifically , they are bounded below by @xmath418 when @xmath419 , where @xmath363 is the golden ratio . to prove the theorem it suffices to produce a vertex of @xmath154^{2q } } , { \mathscr{a}}_{\mathbb p}^q)$ ] with denominator exceeding @xmath420 . first consider the semiqueen . the points @xmath383 and @xmath421 of the golden rectangle have coordinates @xmath177 and @xmath422 . letting @xmath423 or @xmath424 gives an @xmath260-difference of @xmath425 for a golden rectangle of @xmath0 pieces . similarly , letting @xmath426 or @xmath427 gives a @xmath264-difference of @xmath428 . the golden rectangle is a vertex configuration so it follows by lemma [ l : deltan ] that the vertex @xmath45 has @xmath429 . a calculation shows that @xmath430 for @xmath431 . the theorem for @xmath356 follows . an arbitrary piece with three ( or more ) moves has a golden parallelogram configuration formed from the golden rectangle by the linear transformation @xmath432 and @xmath433 . we may choose these moves from at least three , so we can select @xmath297 to have @xmath434 and @xmath298 to have @xmath435 . the displacement from @xmath383 to @xmath421 becomes that from @xmath244 at @xmath388 to @xmath436 at @xmath437 . this displacement is @xmath438 . since @xmath439 , the @xmath260-displacement is at least that for @xmath356 ; therefore the denominator of the corresponding vertex for @xmath3 is bounded below by @xmath425 , just as it is for @xmath356 . similarly , the @xmath264-displacement for @xmath426 is bounded below by @xmath428 . this reduces the problem to the semiqueen , which is solved . we know that @xmath149 is weakly increasing , by proposition [ p : dincrease ] . if , as we believe , the period equals @xmath149 , then the period increases at least exponentially for any piece with more than two moves . we think any board has a similar lower bound , say @xmath440 where @xmath441 is a constant depending upon @xmath29 , but we ran into technical difficulties trying to prove it . when a piece has four or more moves , the diversity of vertex configurations increases dramatically and the denominators grow much more quickly . again we start with the piece with the simplest four moves , the queen . the _ discrete fibonacci spiral _ with @xmath0 queens is defined by the move hyperplanes @xmath442 for all @xmath15 such that both indices fall between @xmath228 and @xmath0 , inclusive , and fixations for pieces @xmath443 , @xmath444 , and @xmath445 . the fixations are @xmath446 figure [ fig : qfib1 ] shows the discrete fibonacci spiral of @xmath447 queens . the bounding rectangle of the discrete fibonacci spiral with @xmath0 queens has dimensions @xmath448 by @xmath449 so the vertex s denominator is @xmath448 . the largest denominator that appears in any vertex configuration for @xmath0 queens is @xmath448 . the queen appears to satisfy an extremely special property that is not shared with three - piece riders nor with other four - piece riders . the initial data ( for @xmath450 ) seems to indicate that it is possible to construct vertex configurations that generate _ all _ denominators up to @xmath448 . for every integer @xmath451 between @xmath228 and @xmath448 inclusive , there exists a vertex configuration of @xmath0 queens with denominator @xmath451 . [ ex : q8configs ] the eighth fibonacci number is @xmath452 . the spiral in figure [ fig : qfib1 ] exhibits a denominator of @xmath453 . for each @xmath454 there is a vertex configuration of seven or fewer queens with denominator @xmath451 ( we do not show them ) . figure [ fig : q8configs ] provides seven vertex configurations of eight queens in which the denominator ranges from 14 to 20 , as one can tell from the size of the smallest enclosing square and lemma [ l : deltan ] . [ denomq ] the denominator of the inside - out polytope for @xmath0 queens is @xmath455 $ ] , where @xmath456=\{1,2,\ldots , f_q\}$ ] . the appearance of fibonacci numbers in kotovec s conjecture [ kotfib ] was one of the main motivations for this line of study . unlike in the case of three - move pieces , a simple transformation of the queen s fibonacci spiral to general four - move pieces does not suffice ; the configuration also experiences an extra expansion as pieces are added . consider the nightrider @xmath457 . example [ ex : threemove ] shows that the numbers @xmath290 , @xmath291 , and @xmath292 are rearrangements of the triple @xmath458 . when we fit four nightriders in the next fibonacci spiral , the smallest triangle must be dilated by an additional factor of four . we define a _ twisted fibonacci spiral _ of @xmath0 pieces @xmath3 with moves @xmath459 to be defined by the move equations @xmath460 for all @xmath15 such that both indices fall between @xmath228 and @xmath0 , inclusive . in addition , choose the three fixations to ensure that the square box bounding all the pieces is as small as possible and so that all coordinates are integral . by varying the choice of @xmath297 , @xmath298 , @xmath299 , and @xmath461 we get different vertex configurations . consider nightriders in the following example . [ ex : twisted ] the most obvious analog of the queens discrete fibonacci spiral for the nightriders is given in figure [ fig : n41 ] , for which @xmath462 , @xmath463 , @xmath464 , and @xmath465 . there is an alternate vertex configuration with larger denominator , the `` expanding kite '' shown in figure [ fig : n42 ] , that is a twisted fibonacci spiral in which @xmath466 , @xmath467 , @xmath464 , and @xmath465 . for any piece @xmath3 , there is a vertex configuration that maximizes the denominator and is a twisted fibonacci spiral . unlike for queens , the maximum denominator @xmath148 of a vertex configuration of @xmath0 pieces @xmath3 is difficult to compute . furthermore , it can not be expected that for all integers @xmath137 between @xmath228 and @xmath148 inclusive , there will exist a vertex configuration of @xmath0 pieces @xmath3 with denominator @xmath137 . as an example , with three nightriders the only possible denominators are @xmath468 . rooks illustrate our approach nicely because they are well understood . the well - known elementary formula is @xmath469 where @xmath470 denotes the falling factorial . thus , @xmath471 is a quasipolynomial of period 1 ( that is , a polynomial ) and degree @xmath6 , in accordance with our general theory . we want to study its coefficients . the coefficient of @xmath11 is @xmath472 where @xmath473[d : s ] denotes the stirling number of the first kind , defined as @xmath89 if @xmath474 or @xmath475 . for instance , @xmath476 these formulas , derived from equation , agree with the general partial queens formulas in theorem iii.3.1 . ( recall that the rook is the partial queen @xmath477 . ) the sign of each term in the summations in is @xmath478 , so that is the sign of @xmath10 for @xmath479 . for @xmath480 , @xmath481 because @xmath482 the rook is one of few pieces for which we know the sign of every term in @xmath113 . [ p : rookform ] the coefficient @xmath483 is a polynomial in @xmath0 of degree @xmath13 . it has a factor @xmath484 . the coefficient of @xmath485 is @xmath486 whose sign is @xmath478 . schlmilch s formula @xcite @xmath487 ( which involves the stirling numbers @xmath488[d : s ] of the second kind ) tells us that @xmath489 is a polynomial in @xmath0 of degree @xmath490 with leading term @xmath491 this term , as the leading term , must have the same sign as @xmath489 for large @xmath0 . so the leading coefficient of @xmath483 is as in equation and the sign of this coefficient is @xmath478 . it is easy to infer from that the polynomial equals 0 if @xmath492 , i.e. , @xmath493 ; therefore @xmath494 is a factor . the number of combinatorial types of nonattacking configuration of @xmath0 ( unlabelled ) rooks is @xmath12 . to prove it we may substitute @xmath16 into equation ( by theorem i.5.3 ) or apply theorem i.5.8 , which says that every piece with two basic moves has @xmath12 combinatorial configuration types . the semirook has only one of the rook s moves and is consequently the least interesting of all riders . we mention it because it is a second example with period 1 ; also because it has no diagonal move and , as such , exemplifies corollary iii.3.2 , that partial queens with at most one diagonal move have a coefficient @xmath495 that is independent of @xmath2 . counting formulas for @xmath496 are given in proposition ii.6.1 ( where one should take @xmath497 and , for @xmath498 , explicitly in tables iii.4.1 and iii.4.2 . the leading coefficients of those polynomials are in theorem iii.3.1 and tables iii.3.1 and iii.3.2 . the number of combinatorial types with @xmath0 of any one - move rider equals 1 ( theorem i.5.8 ) . here we treat the bishop and its scion the semibishop . the bishop s basic move set is @xmath499 . the quasipolynomial formulas for up to 6 bishops , published by kotovec in early editions of @xcite most of which were found by him are : @xmath500 & u_{\mathbb b}(2;n ) = \frac{n^4}{2}-\frac{2 n^3}{3}+\frac{n^2}{2}-\frac{n}{3}. \\[5pt ] & u_{\mathbb b}(3;n ) = \left\{\frac{n^6}{6}-\frac{2 n^5}{3}+\frac{5 n^4}{4}-\frac{5 n^3}{3}+\frac{4 n^2}{3}-\frac{2 n}{3}+\frac{1}{8}\right\ } - ( -1)^n\dfrac{1}{8}. \\[5pt ] & u_{\mathbb b}(4;n ) = \left\{\frac{n^8}{24}-\frac{n^7}{3}+\frac{11 n^6}{9}-\frac{29 n^5}{10}+\frac{355 n^4}{72}-\frac{35 n^3}{6}+\frac{337 n^2}{72}-\frac{73 n}{30}+\frac{1}{2}\right\ } \\ & \qquad\qquad\ - ( -1)^n\left\{\frac{n^2}{8}-\frac{n}{2}+\frac{1}{2}\right\}.\\[5pt ] & u_{\mathbb b}(5;n ) = \left\{\frac{n^{10}}{120}-\frac{n^9}{9}+\frac{49 n^8}{72}-\frac{118 n^7}{45}+\frac{523 n^6}{72}-\frac{2731 n^5}{180}+\frac{3413 n^4}{144}-\frac{4853 n^3}{180 } \right . \\ & \qquad\qquad\quad \left.+\frac{2599 n^2}{120}-\frac{1321 n}{120}+\frac{9}{4}\right\ } - ( -1)^n\left\{\frac{n^4}{16}-\frac{7 n^3}{12}+\frac{17 n^2}{8}-\frac{85 n}{24}+\frac{9}{4}\right\}.\\[5pt ] & u_{\mathbb b}(6;n ) = \left\{\frac{n^{12}}{720}-\frac{n^{11}}{36}+\frac{37n^{10}}{144}-\frac{4813n^9}{3240}+\frac{8819n^8}{1440}-\frac{72991n^7}{3780}+\frac{2873n^6}{60 } \right . & \\ & \qquad\qquad\quad \left . -\frac{100459n^5}{1080}+\frac{199519n^4}{1440}-\frac{498557n^3}{3240}+\frac{14579n^2}{120}-\frac{7517n}{126}+\frac{765}{64}\right\ } & \\ & \qquad\qquad\ - ( -1)^n\left\{\frac{n^6}{48}-\frac{n^5}{3}+\frac{221n^4}{96}-\frac{211n^3}{24}+\frac{467n^2}{24}-\frac{47n}{2}+\frac{765}{64}\right\}. \end{aligned}\ ] ] the formula for @xmath171 is due to dudeney ( * ? ? ? * problem 318lion - hunting solution ) and those for @xmath501 to fabel @xcite ( kotovec @xcite supplies these attributions ) . the formulas for @xmath502 are special cases of our theorems iii.4.1and iii.4.2 , thereby supporting the correctness of those theorems . kotovec found the formulas for @xmath503 by calculating the values @xmath504 for many values of @xmath2 , looking for an empirical recurrence relation , deducing a generating function , and from that getting the quasipolynomial . ( reference @xcite has details of his method of calculation applied to queens . ) his approach , while excellent for finding formulas , does not prove their validity because it does not bound the period though intuitively period 2 is plausible since one could guess that odd and even board sizes would have separate polynomials . in fact , 2 is the complete story on the period . in theorem v.1.1 we provide the missing upper bound of 2 that rigorously establishes period 2 for every @xmath505 and hence the correctness of kotovec s quasipolynomial formulas . together with the fact that we know the degree @xmath6 and the leading coefficient @xmath506 of the constituent polynomials , this implies that , if the first @xmath507 values of a candidate quasipolynomial are correct , then we have @xmath504 . since kotovec did check those values for @xmath508 @xcite , his formulas are proved correct . despite the overall period 2 , in kotovec s formulas the six leading coefficients do not vary with the parity of @xmath2 . kotovec conjectured expressions for @xmath509 , @xmath510 , and @xmath511 in terms of @xmath0 alone , which we proved in theorem iii.3.1 ( and see tables iii.3.1 and iii.3.2 ) since the bishop is the partial queen @xmath512 . that theorem also gives the periods of @xmath513 , @xmath514 , and @xmath495 . [ c : btopcoeffs ] in @xmath504 the coefficients @xmath10 for @xmath515 are constant as functions of @xmath2 , while @xmath495 has period @xmath516 ( if @xmath517 ) . the number of combinatorial types of nonattacking configuration of @xmath0 unlabelled bishops is @xmath12 , by theorem i.5.8 . this is therefore the value of @xmath518 , which we know even though we do not know the general formula for @xmath504 . a surprising development during our work on this project was kotovec s discovery that , in 1936 , arshon had solved the @xmath2-bishops problem , the number of ways to place @xmath2 nonattacking bishops on an @xmath1 board @xcite . his method was to count independently the number of ways to place @xmath15 nonattacking bishops on the black squares , @xmath519[arshon ] ( arshon s @xmath14 ) , and on the white squares , @xmath520 ( arshon s @xmath521 ) , of the @xmath1 chessboard . when @xmath2 is even , @xmath522 when @xmath2 is odd , @xmath523 and @xmath524 with these formulas arshon solved the @xmath2-bishops problem . this work was forgotten until kotovec rediscovered it . it was an easy step for him to write down an explicit formula for the number of placements of @xmath0 nonattacking bishops ( * ? ? ? * fourth ed . , p. 140 ) . kotovec then restated the arshon equations with no signed terms by using stirling numbers of the second kind ( * ? ? ? * fourth ed . , p. 142 ) . his formula for @xmath0 bishops is @xmath525 since the number of terms depends on @xmath2 , these formulas do not tell us the quasipolynomial form of @xmath504 . neither do they allow us to substitute @xmath16 to obtain the number of combinatorial types of nonattacking configuration ( though for the bishop this number is known , obtained from theorem i.5.8 ) . we consequently take the point of view that bishops formulas , like those for other pieces , call for a quasipolynomial analysis via ehrhart theory . the _ semibishop _ @xmath526 has just one of the bishop s moves , say @xmath527 . thus , it is an example of a one - move rider ( section ii.6 ) . as such it has counting functions @xmath528 from proposition ii.6.1 ( in which we have @xmath529 ) . all of these are polynomials in @xmath2 . ( letting @xmath0 vary gives a power series in two variables that is truncated , for each @xmath0 , above the @xmath530 term . we have not investigated that power series . ) [ p : semibishop ] the counting function for nonattacking unlabelled semibishops on the square board is @xmath531 which is a polynomial function of @xmath2 of degree @xmath6 . this is an immediate consequence of proposition [ p : triangleboard ] below . alternatively , it can be proved similarly to that proposition . explicit formulas for the coefficients @xmath10 for @xmath532 are in theorem iii.3.1 and tables iii.3.1 and iii.3.2 . we prepare for the proof of theorem [ p : semibishop ] by changing the board . right triangle board _ ( _ triangular board _ for short ) has legs parallel to the axes and hypotenuse in the direction of the semibishop s move ; thus , it is the set @xmath533[d : ct ] . the _ @xmath1 triangular board _ is the set of integral points in the interior of the dilation by @xmath534 , i.e. , @xmath535 write @xmath536 for the counting function of nonattacking placements of @xmath0 unlabelled semibishops on an @xmath1 triangular board . most of our theory for the square board applies equally well to the triangular board ; we omit details . [ p : stirling ] the stirling number of the first kind , @xmath537 , is a polynomial function of @xmath2 of degree @xmath0 . the coefficient of @xmath11 in @xmath538 is a polynomial function of @xmath0 of degree @xmath13 . the fact that @xmath537 is a polynomial in @xmath2 of degree @xmath6 is well known ; see e.g. @xcite . we give a proof here which we believe to be new , using ehrhart theory in the spirit of our chess series . we do not know a prior reference for the fact that the coefficients are polynomials . [ p : triangleboard ] the counting function for nonattacking unlabelled semibishops on the triangular board is @xmath539 we prove both propositions together . the @xmath1 integral right triangle board has @xmath2 diagonals ( parallel to the hypotenuse ) of lengths @xmath540 , each of which can have at most one semibishop . the number of ways to place @xmath0 labelled semibishops is the sum of all products of @xmath0 of these @xmath2 values , i.e. , the elementary symmetric function @xmath541 , which equals @xmath542 . that proves the first part of the proposition . proposition [ p:1moveiv ] applies because the semibishop is a one - move rider . the corners of the triangular board @xmath543 are @xmath205 , @xmath180 , and @xmath179 . for the move @xmath180 , the corner @xmath179 has no antipode while the corners @xmath205 and @xmath180 serve as each other s antipodes . thus , every vertex is integral and the denominator @xmath544 , so @xmath545 is a polynomial in @xmath2 . theorem i.4.2 says that the coefficients are polynomials in @xmath0 with the stated degrees . there are four pieces in this section : the queen , and three others that are like defective queens without being similar to bishops or rooks , which we call the semiqueen , the frontal queen , and the subqueen . as they are all partial queens , counting formulas for @xmath502 are special cases of our theorems iii.4.1 and iii.4.2 and are presented in tables iii.4.1 and iii.4.2 . the four leading coefficients of the general counting polynomial are implied by theorems iii.3.1 , iii.4.2 , and iii.4.2 ; see tables iii.3.1 and iii.3.2 . all our formulas that were also calculated by kotovec in @xcite agree with his . let @xmath546[d : zetar ] be a primitive @xmath547-th root of unity . the basic move set for the queen @xmath548 is @xmath549 . the quasipolynomial formulas for up to four queens are : @xmath550 u_{\mathbb q}(2;n ) & = \frac{n^4}{2}-\frac{5 n^3}{3}+\frac{3 n^2}{2}-\frac{n}{3 } = \frac{n(n-1)(3n^2 - 7n+2)}{6 } \\[5pt ] u_{\mathbb q}(3;n ) & = \left\{\frac{n^6}{6}-\frac{5 n^5}{3}+\frac{79 n^4}{12}-\frac{25 n^3}{2}+11 n^2-\frac{43 n}{12}+\frac{1}{8}\right\ } + ( -1)^n \left\{\frac{n}{4}-\frac{1}{8}\right\}. \\[5pt ] u_{\mathbb q}(4;n ) & = \left\{\frac{n^8}{24}-\frac{5n^7}{6}+\frac{65n^6}{9}-\frac{1051n^5}{30}+\frac{817n^4}{8}-\frac{19103n^3}{108}+\frac{3989n^2}{24}-\frac{18131n}{270}+\frac{253}{54}\right\ } \\ & \quad + ( -1)^n \left\{\frac{n^3}{4}-\frac{21n^2}{8}+7n-\frac{7}{2}\right\ } + \operatorname{re}(\zeta_3^n ) \frac{32(n-1)}{27 } + \operatorname{im}(\zeta_3^n ) \frac{40\sqrt3}{81}. \end{aligned}\ ] ] the formula for two queens is given by theorem iii.4.1 and is originally due to lucas ( * ? ? ? * , exemple ii ) . the formula for three queens is implied by theorem iii.4.2 and is originally due to landau @xcite . kotovec gives formulas for up to six queens , calculated by him for @xmath551 and calculated for six queens by karavaev @xcite and ( * ? ? ? * sequence a176186 ) . the number of combinatorial types of nonattacking configuration for @xmath0 pieces is @xmath552 . the numbers for @xmath553 are in table [ tb : queens ] . for two queens the number is what one expects from two pieces with four basic moves ( proposition i.5.6 ) . kotovec conjectured formulas for @xmath509 and @xmath510 based on the known and heuristically derived formulas ( mostly by him ) for @xmath554 for small @xmath0 . theorem iii.3.1 proves his conjectures along with a formula for @xmath511 and further proves that @xmath555 is constant as a function of @xmath2 , but that the next two coefficients are not . [ c : qtopcoeffs ] in @xmath554 the coefficients @xmath10 for @xmath556 are constant as functions of @xmath2 ; but @xmath514 has period @xmath516 if @xmath23 and @xmath495 has period @xmath516 if @xmath517 . exact formulas are @xmath557 the periodic parts are @xmath558 for @xmath514 and @xmath559 for @xmath495 . unlike the case of bishops and semibishops , the period of @xmath554 is not simple , although kotovec ( * ? ? ? * ed . , p. 31 ) makes the following remarkable conjecture . [ kotfib ] the counting quasipolynomial for @xmath0 queens has period @xmath455 $ ] , the least common multiple of all positive integers up through the @xmath0th fibonacci number @xmath448 . the observed periods up to @xmath184 ( see ( * ? ? * ed . , pp . 19 , 2728 ) for @xmath184 ) agree with this proposal , and the theory of section [ 4move ] lends credence to its veracity . kotovec conjectures , yet more strongly , the exact form of the denominator of the generating function @xmath560 : it is a product of specific cyclotomic polynomials raised to specific powers ; see ( * ? ? ? , p. 22 ) . the conjecture implies that , when written in standard ehrhart form with denominator @xmath561 , the generating function has many cancellable factors . this , too , is not predicted by ehrhart theory ; but as it is too systematic and elegant to be accidental , it presents another tantalizing question . kotovec s evidence , indeed , suggests that @xmath554 has a recurrence relation of length far less than its period . a proof of these conjectures seems to require a new theoretical leap forward . we summarize the known numerical results for queens in table [ tb : queens ] . unlike in the case of bishops and semibishops , the period of @xmath554 is not simple and we have no general formula in terms of @xmath0 . ' '' '' & types & period & denom + ' '' '' @xmath562 1 & 1 & 1 & 1 + ' '' '' 2 & 4 & 1 & 1 + ' '' '' 3 & 36 & 2 & 2 + ' '' '' 4 & 574 * & 6 * & 6 + ' '' '' 5 & 14206 * & 60 * & + ' '' '' 6 & 501552@xmath563 & 840@xmath563 & + ' '' '' 7 & & 360360 * & + the _ semiqueen _ @xmath356 is the queen without one of its diagonal moves ( think of it as having lost the left ( or right ) arm in battle ) . conjecture [ conj : semiqueen ] gives a conjectural upper bound for the quasipolynomial period of @xmath564 $ ] when @xmath0 is even and @xmath565 $ ] when @xmath0 is odd . since we do not expect all denominators in @xmath566 $ ] or @xmath567 $ ] to appear , we do not expect this bound to be tight for large @xmath0 , although it agrees with kotovec s formulas for @xmath568 in ( * ? ? ? * ed . , pp . 732733 ) . the _ frontal queen _ is the partial queen @xmath400 that can advance and retreat but can not move sideways . our counting formulas for @xmath569 are the same as kotovec s heuristic ones @xcite . conjecture [ conj : frontal ] implies a conjectural upper bound for the period @xmath570 of the counting quasipolynomial for @xmath0 frontal queens of @xmath571 $ ] when @xmath0 is even and @xmath572 $ ] when @xmath0 is odd . again , this should not be considered tight , because we do not expect that all denominators from @xmath228 to the maximum will appear . however , kotovec s heuristic approach did find that the period when @xmath173 is 6 , which is our bound . the _ subqueen _ ( kotovec s `` semi - rook + semi - bishop '' ) is the partial queen @xmath573 , with one horizontal or vertical and one diagonal move . it is the fourth and final piece whose counting function @xmath574 is a polynomial for all @xmath0 , if our conjecture [ cj : trajectories ] is correct . ( there are two subqueens that differ in chirality : the right - handed subqueen has , say , a vertical move @xmath179 and a diagonal move @xmath180 to the right ; the left - handed subqueen has the vertical move and a diagonal move @xmath367 to the left . mixing the two types is outside our competence since they will behave like two different pieces . in our work all subqueens are right - handed . ) kotovec noticed that @xmath470 is a factor of @xmath574 for @xmath0 up to @xmath447 . for instance , @xmath575 we would like to have an explanation for this . kotovec also presents a formula for the number of ways to place @xmath2 nonattacking subqueens on an @xmath1 board : @xmath576 we hope that our method will be able to prove this . the basic move set for a nightrider @xmath457 is @xmath577 . as always , it is easy to see that @xmath579 , @xmath580 , @xmath581 , and not quite so easy to find @xmath582 by hand . many more values of @xmath583 are in the oeis ( see table [ tb : oeis ] ) . in theorem ii.3.1 , all @xmath584 and the period is 2 , so @xmath585 . therefore , @xmath586 this formula was found independently by kotovec . his ( * ? ? ? , p. 312 ) has an enormous formula for three nightriders ( undoubtedly correct , though unproved ) that is too complicated to reproduce here . a proof may be accessible using our techniques . we summarize the known numerical results for nightriders in table [ tb : nightriders ] . we calculated the denominator for four nightriders by using mathematica to find all vertices of the inside - out polytope and then the least common multiple of their denominators . ' '' '' & types & period & denom + ' '' '' @xmath562 1 & 1 & 1 & 1 + ' '' '' 2 & 7 & 2 & 2 + ' '' '' 3 & 36 * & 60 * & 60 + ' '' '' 4 & & & 14559745200 + from theorem ii.3.1 we get a generalization . define a _ partial nightrider _ @xmath587 to have any @xmath168 ( @xmath588 ) of the nightrider s moves . ( there are three different pieces @xmath589 ; see section [ simplified ] . all have the same formula for two pieces . ) letting @xmath590 , @xmath591 & = \left\ { \frac{1}{2}n^4 - \frac{5k}{24}n^3 + \frac{k-1}{2}n^2 - \frac{11k}{48}n \right\ } + ( -1)^n \frac{3k}{48}n . \end{aligned } \label{e : u2nk}\ ] ] a direct consequence of theorem ii.5.1 is that we know the second coefficient of the counting quasipolynomial of @xmath587 : @xmath592 this formula for @xmath457 was conjectured by kotovec . @xmath593 is its own leading coefficient . theorem ii.5.1 gives the leading coefficient of every @xmath10 for all partial nightriders . [ t : ngammaleading ] ( i ) for a partial nightrider @xmath587 , the coefficient @xmath483 of @xmath11 in @xmath594 is a polynomial in @xmath0 , periodic in @xmath2 , with leading term @xmath595 specializing to the complete nightrider @xmath457 , computer algebra gives us a general formula for the third coefficient , @xmath510 . it and the periods of @xmath511 and @xmath555 are new . both agree with kotovec s data . [ thm : n ] the third coefficient of the nightrider counting quasipolynomial is independent of @xmath2 ; it is @xmath596 the next coefficient , @xmath511 , is periodic in @xmath2 with period @xmath516 and periodic part @xmath597 the coefficient @xmath555 has period @xmath516 and periodic part @xmath598 just as in theorem iii.3.1 , we calculate @xmath510 by determining the contribution from all subspaces @xmath72 defined by two move equations , each of the form @xmath599 for @xmath600 and @xmath601 $ ] . this is done in lemma [ l : ncodim2 ] . there is no contribution to @xmath510 from subspaces of codimension @xmath89 or @xmath228 . the coefficient @xmath511 may have contributions from subspaces of codimensions @xmath228 to @xmath602 . since the contribution of a subspace of codimension @xmath602 comes from the leading coefficient , it is independent of @xmath2 . we did not compute these leading coefficients . a subspace of codimension @xmath516 contributes zero by theorem ii.4.2 , or simply by observing the formula in equation ( ii.2.5 ) . because by equation @xmath603 provides a periodic contribution of @xmath604 to @xmath605 , the periodic part of @xmath606 is @xmath607 , consisting of one contribution from each hyperplane in @xmath608 . the calculations for hyperplanes and subspaces of type @xmath609 imply that @xmath555 is periodic with period @xmath516 . that is because a periodic contribution can come only from a subspace of codimension 1 , 2 , or 3 . equation shows that hyperplanes make no contribution to @xmath555 . subspaces of codimension 2 with periodic part all have period 2 with values that are the periodic coefficients of @xmath610 in the formulas for types @xmath611 , @xmath609 , and @xmath612 . subspaces of codimension 3 contribute zero by theorem ii.4.2 . thus , the periodic part of @xmath613 is the sum of @xmath614 from type @xmath611 , @xmath615 from type @xmath609 , and @xmath616 from type @xmath612 , giving a total periodic part of @xmath513 of @xmath617 [ l : ncodim2 ] the total contribution to @xmath618 of all subspaces with codimension @xmath516 is @xmath619 n^{2q-2 } + \left [ \frac{7}{12}(q)_3+\frac{227}{144}(q)_3+\frac{55}{72}(q)_4 \right ] n^{2q-4 } \right . \\&\quad \left . + \left [ \frac{1}{8}(q)_3+\frac{65}{144}(q)_4 \right ] n^{2q-6 } \right\ } \\ & - ( -1)^n \left\ { \left [ \frac{1}{4}(q)_3+\frac{3}{16}(q)_3+\frac{5}{24}(q)_4 \right ] n^{2q-4 } + \left [ \frac{1}{8}(q)_3+\frac{11}{48}(q)_4 \right ] n^{2q-6 } \right\ } \\ & + \bigg[\bigg ( \frac{527}{1728 } -\frac{1}{8 } \zeta_{12}^{3n } + \frac{2}{27 } \zeta_{12}^{4n } -\frac{13}{64 } \zeta_{12}^{6n } + \frac{2}{27 } \zeta_{12}^{8n } -\frac{1}{8 } \zeta_{12}^{9n } \\ & \qquad+ \frac{599}{1600 } -\frac{4}{25 } \zeta_{20}^{4 n } + \frac{1}{8 } \zeta_{20}^{5 n } -\frac{4}{25 } \zeta_{20}^{8 n } + \frac{1}{64 } \zeta_{20}^{10 n } -\frac{4}{25 } \zeta_{20}^{12 n } + \frac{1}{8 } \zeta_{20}^{15 n } -\frac{4}{25 } \zeta_{20}^{16 n } \\ & \qquad+\frac{51}{256}-\frac{19}{256}\zeta_4^n-\frac{13}{256 } \zeta_4^{2 n}-\frac{19}{256 } \zeta_4^{3 n}\bigg)2(q)_3\bigg ] n^{2q-6}.\end{aligned}\ ] ] [ e : ncodim2 ] the striking symmetry in the coefficients of powers of each @xmath620 in the last @xmath621-term is due to the coefficients being real numbers . there are four types of subspace , of which only type @xmath609 involves calculations that are substantially different from those in lemma iii.3.4 . from section [ parti ] , for a subspace determined by equations involving @xmath71 pieces , @xmath75 is a quasipolynomial of degree @xmath78 . * type @xmath622 * : : the subspace @xmath72 is defined by two move equations involving the same two pieces . the contribution to @xmath618 is @xmath623 . since @xmath72 lies in four hyperplanes , the mbius function is @xmath624 and the contribution to @xmath625 is @xmath626 . * type @xmath611 * : : the subspace @xmath72 is defined by two move equations of the same slope involving three pieces . there is one subspace of this type for each of the four slopes . the number of points in each subspace is @xmath627 from equation . there are @xmath628 ways to choose three nightriders , and the mbius function is @xmath516 . thus we multiply @xmath627 by @xmath629 to find that the contribution to @xmath618 is @xmath630 \right\ } , \ ] ] so that to @xmath625 is @xmath631 and that to @xmath613 is @xmath632(q)_3 $ ] . * type @xmath609 * : : the subspace @xmath72 is defined by two move equations of different slopes involving three pieces , say @xmath633 . it suffices to find the contributions when @xmath634 and @xmath635 ; the other three combinations of slopes are symmetric to these three , generating a multiplicative factor of @xmath516 . we write @xmath636 when we need to mention the slope . + the mbius function is @xmath637 . + we choose @xmath638 in @xmath0 ways , @xmath639 in @xmath640 ways , and @xmath641 in @xmath642 ways . + for each value of @xmath643 we calculated the denominators of all vertex coordinates of @xmath50^{2q}}\cap { \mathcal{u}}$ ] using mathematica . that gave us the denominator @xmath164^{2q}}\cap { \mathcal{u}})$ ] and hence an upper bound on the period of @xmath75 in each case . using mathematica again we found quasipolynomial formulas for the number of placements of the three nightriders , @xmath75 . these formulas were calculated by varying the position of @xmath638 in the @xmath1 grid as @xmath2 varied in a residue class modulo @xmath164^{2q}}\cap { \mathcal{u}})$ ] . the calculations were carried out for @xmath644 , which covers at least five periods in every case . by theorem [ t : strongparity ] and the fact that @xmath75 has degree @xmath645 , there are @xmath646 coefficients to determine in @xmath75 ; as the periods are bounded by 20 in every case , that is enough data to infer them all with redundancy . we found that the period always equals the denominator . + _ case @xmath647 . _ the vertex denominators here are @xmath648 so @xmath164^{2q}}\cap { \mathcal{u } } ) = { \operatorname{lcm}}(2,3,4 ) = 12 $ ] . the number of placements is @xmath649 \frac{53}{288}n^4+\frac{7}{36}n^2-\frac{2}{9 } & \text{for $ n\equiv \pm4{\ , \operatorname{mod}\ , } 12 $ , } \\[3pt ] \frac{53}{288}n^4+\frac{7}{36}n^2+\frac{1}{2 } & \text{for $ n\equiv 6{\ , \operatorname{mod}\ , } 12 $ , } \\[3pt ] \frac{53}{288}n^4+\frac{7}{36}n^2+\frac{5}{18 } & \text{for $ n\equiv \pm2{\ , \operatorname{mod}\ , } 12 $ , } \\[3pt ] \frac{53}{288}n^4+\frac{55}{144}n^2+\frac{21}{32 } & \text{for $ n\equiv 3{\ , \operatorname{mod}\ , } 6 $ , } \\[3pt ] \frac{53}{288}n^4+\frac{55}{144}n^2+\frac{125}{288 } & \text{for $ n\equiv \pm1{\ , \operatorname{mod}\ , } 6$. } \end{cases}\ ] ] note that the coefficient of @xmath650 , which becomes a contribution to @xmath555 , has period @xmath516 . + _ case @xmath651 . _ the vertex coordinate denominators here are @xmath516 , @xmath355 , and @xmath652 so @xmath164^{2q}}\cap { \mathcal{u } } ) = 20 $ ] . the number of placements is @xmath653 \frac{27}{160}n^4+\frac{1}{4}n^2+\frac{4}{5 } & \text{for $ n\equiv \pm4,\pm8 { \ , \operatorname{mod}\ , } 20 $ , } \\[3pt ] \frac{27}{160}n^4+\frac{1}{4}n^2-\frac{1}{2 } & \text{for $ n\equiv 10{\ , \operatorname{mod}\ , } 20 $ , } \\[3pt ] \frac{27}{160}n^4+\frac{1}{4}n^2+\frac{3}{10 } & \text{for $ n\equiv \pm2 , \pm6 { \ , \operatorname{mod}\ , } 20 $ , } \\[3pt ] \frac{27}{160}n^4+\frac{1}{4}n^2-\frac{9}{32 } & \text{for $ n\equiv \pm5{\ , \operatorname{mod}\ , } 20 $ , } \\[3pt ] \frac{27}{160}n^4+\frac{1}{4}n^2+\frac{83}{160 } & \text{for odd $ n\not\equiv \pm5{\ , \operatorname{mod}\ , } 20$. } \end{cases}\ ] ] + _ case @xmath654 . _ the vertex denominators here are @xmath516 and @xmath355 so @xmath164^{2q}}\cap { \mathcal{u}})=4 $ ] . the number of placements is @xmath655 \frac{11}{64}n^4+\frac{1}{4}n^2+\frac{1}{4 } & \text{for $ n\equiv 2{\ , \operatorname{mod}\ , } 4 $ , } \\[3pt ] \frac{11}{64}n^4+\frac{1}{4}n^2+\frac{19}{64 } & \text{for $ n$ odd . } \end{cases}\ ] ] the contribution to @xmath510 is therefore @xmath656 . that to @xmath555 is @xmath657(q)_3.$ ] last , the contribution to @xmath658 has period @xmath659 and is @xmath660 times @xmath661 * type @xmath662 * : : the subspace @xmath72 is defined by two move equations involving four distinct pieces . for every pair of hyperplanes , the number of attacking configurations is @xmath663 , whose value is given in equation . we must also multiply by the number of ways in which we can choose this pair of hyperplanes , which is @xmath664 . the mbius function is @xmath228 . we conclude that the contribution to @xmath618 is @xmath665 \right\},\ ] ] that to @xmath510 is @xmath666 , and that to @xmath555 is @xmath667 . curiously , not only is the quasipolynomial for every subspace as a whole an even function , so is each constituent ; equivalently , opposite constituents @xmath668 and @xmath669 are equal , for every @xmath15 . we do not know why . type @xmath609 contributes period @xmath670 to @xmath658 , as one can see from lemma [ l : ncodim2 ] . we therefore expect @xmath658 to have period that is a multiple of 60 ; however , we are far from proving this . this computational method can be applied to larger numbers of any piece , limited only by human effort and computing power . it should be feasible to deduce , at the least , nightrider formulas for @xmath511 , @xmath555 , and @xmath671 . work on nonattacking chess placements raises many questions , some of which have general interest . we can not reach satisfactorily strong conclusions about the queen and nightrider in part because their periods grow too rapidly as @xmath0 increases , which we now understand by way of the twisted fibonacci spirals in section [ 4move ] . it would be desirable to study simplified analogs , hoping not only for hints to solve those pieces but to find general patterns in the period and coefficients . as having four move directions is complicated , we propose handicapping the pieces by eliminating some of their moves . as we saw in part iii , partial queens @xmath165 are approachable because the queen s moves are individually simple . we suggest further study of the following variants , some of which have been investigated by kotovec . a. a generalization of the subqueen is a rider with two moves , @xmath177 and @xmath191 . the denominator of this piece was determined in proposition [ p : denom2move ] . this piece , and especially its period , would facilitate analysis of the effect of non - unit slopes . the nightrider s main complication comes from the non - unit slopes . we propose as worthy subjects the partial nightriders with only two moves : a. the _ lateral nightrider _ , which moves along slope @xmath672 ( or equivalently @xmath673 ) . we conjecture a period of @xmath355 for @xmath23 . we verified this for @xmath501 . b. the _ inclined nightrider _ , which moves along slopes @xmath674 and @xmath516 . we propose a period of @xmath675 for @xmath23 . we only know this for @xmath172 but we have evidence for @xmath173 from analyzing trajectories ( with help from arvind mahankali ) and it is certainly correct if conjecture [ cj : trajectories ] is true . c. the _ orthonightrider _ , whose directions have the orthogonal slopes @xmath674 and @xmath676 . we propose the period is @xmath677 for @xmath23 . this is correct for @xmath172 and evidence suggests it for higher @xmath0 . these should exhibit some of the complexity of the nightrider without being so opaque . it appears that the lateral nightrider should behave more nicely than the other two . for one , from our analysis of subspaces of type @xmath609 in theorem [ thm : n ] , it is expected to have a smaller period . furthermore , the denominators generated by configurations similar to those in figure [ fig : twomove ] behave much more nicely than the others . a. a simple three - move rider would have moves @xmath678 . this should be investigated . b. the partial nightrider @xmath679 . we discussed it briefly in section [ n ] , finding the counting formula and the period @xmath516 for @xmath171 . the period for three pieces appears to be @xmath670 . these periods are the same as for the complete nightrider but we expect @xmath679 to have a smaller period than @xmath457 when @xmath517 . it would be valuable to produce a conjectural expression for the number of combinatorial types of nonattacking configuration for the queen , a partial queen with three moves , or any other piece with more than two moves ( one or two moves being easy ; see proposition i.5.6 ) . we found that periods equal denominators . that is not a general truth about ehrhart quasipolynomials . is there always equality in nonattacking rider problems , and if so , is there an interesting reason ? in section [ q ] we saw that denominators arising from vertex configurations of queens can be determined by understanding configurations of points and lines . indeed , the fibonacci numbers seem to arise from optimal configurations of points and lines in some imprecise notion of optimality . understanding which denominators appear is a fundamental problem . the configurations in section [ sec : config2 m ] may be related to the theory of billiards . if there are only two moves , with a linear transformation one can ensure that the angle of incidence equals the angle of reflection and make it equal to any desired angle less than a right angle . the square board becomes a parallelogram , but a theory of two - move riders on polygonal boards would be able to handle that . @xmath191 coordinates of move vector ( pp . ) 2 . @xmath680 @xmath681 of @xmath682 ( p. ) 3 . @xmath124 slope of line or move ( p. ) 4 . @xmath448 fibonacci numbers ( p. ) 5 . @xmath166 # of horizontal , vertical moves of partial queen ( p. ) 6 . @xmath168 # of diagonal moves of partial queen ( p. ) 7 . @xmath683 basic move ( p. ) 8 . @xmath684 orthogonal vector to move @xmath32 ( p. ) 9 . @xmath2 size of square board ( p. ) 10 . @xmath685 ( p. ) 11 . @xmath686 ^ 2 $ ] square board ( p. ) 12 . # of nonattacking labelled configurations ( p. ) 13 . period of quasipolynomial ( p. ) 14 . # of pieces on a board ( p. ) 15 . stirling number of the first kind ( p. ) 16 . stirling number of the second kind ( p. ) 17 . # of nonattacking unlabelled configurations ( p. ) 18 . arshon s bishops numbers ( p. ) 19 . @xmath689 , @xmath690 piece position ( p. ) 20 . @xmath42 configuration ( p. ) 1 . @xmath691 # of 2-piece attacks on slope @xmath124 ( p. ) 2 . @xmath75 # of attacking configurations in essential part of subspace @xmath72 ( p. ) 3 . @xmath692 # of 3-piece attacks on slope @xmath124 ( p. ) 4 . @xmath693 coefficient of @xmath11 in @xmath694 ( p. ) 5 . @xmath695 primitive @xmath547-th root of unity ( p. ) 6 . @xmath71 # of of pieces in equations of @xmath72 7 . @xmath67 mbius function of intersection lattice ( p. ) 8 . @xmath363 golden ratio @xmath696 ( p. ) 1 . @xmath60 move arrangement of piece @xmath3 ( p. ) 2 . @xmath29 closed board : usually the square @xmath118 ^ 2 $ ] ( p. ) 3 . @xmath132 edge line of the board ( p. ) 4 . @xmath599 hyperplane for move @xmath191 ( p. ) 5 . @xmath66 intersection lattice ( p. ) 6 . @xmath697 closed , open polytope ( p. ) 7 . @xmath50^{2q } } , { ( 0,1)^{2q}}$ ] closed , open hypercube ( p. ) 8 . @xmath698^{2q}},{\mathscr{a}}_{\mathbb p})$ ] inside - out polytope ( p. ) 9 . @xmath699 open inside - out polytope ( p. ) 10 . triangular board @xmath700 ( p. 11 . subspace in intersection lattice ( p. ) 12 . essential part of subspace @xmath72 ( p. ) 13 . hyperplane of equal @xmath260 coordinates ( p. ) 14 . hyperplane of equal @xmath264 coordinates ( p. ) seth chaiken , christopher r. h. hanusa , and thomas zaslavsky , a @xmath0-queens problem . i. general theory . _ electronic j. combin . _ , 21 ( 2014 ) , no . 3 , paper # p3.33 , 28 pp . mr 3262270 , zbl 1298.05021 , arxiv:1303.1879 . ii . the square board . _ j. algebraic combin . _ , 41 ( 2015 ) , no . 3 , 619642 . mr 3328174 , zbl 1314.05008 , arxiv:1402.4880 . henry ernest dudeney , _ amusements in mathematics_. 1917 . reprinted by dover publications , 1958 . project gutenberg edition : + https://www.gutenberg.org/files/16713/16713-h/16713-h.htm#x_318_lion-huntinga karl fabel , anordnung gleichartiger schachfiguren . [ arrangement of identical chess pieces . ] in eero bonsdorff , karl fabel , and olavi riihimaa , _ schach und zahl _ [ _ chess and number _ ] , pp . walter rau verlag , dsseldorf , 1966 . , _ non - attacking chess pieces ( chess and mathematics ) _ [ _ ach a matematika - poty rozmstn neohroujcch se kamen _ ] . [ self - published online book ] , apr . 2010 ; second ed . 2010 , third ed . , 2011 , fourth ed . 2011 , fifth ed . , 2012 ; 6th ed . , 2013 , 795 pp . + http://www.kotesovec.cz/math.htm and https://oeis.org/wiki/user:vaclav_kotesovec

parts i iii showed that the number of ways to place @xmath0 nonattacking queens or similar chess pieces on an @xmath1 chessboard is a quasipolynomial function of @xmath2 whose coefficients are essentially polynomials in @xmath0 and , for pieces with some of the queen s moves , proved formulas for these counting quasipolynomials for small numbers of pieces and high - order coefficients of the general counting quasipolynomials . in this part , we focus on the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside - out polytope . we find an exact formula for the denominator when a piece has one move , give intuition for the denominator when a piece has two moves , and show that when a piece has three or more moves , geometrical constructions related to the fibonacci numbers show that the denominators grow at least exponentially with the number of pieces . furthermore , we provide the current state of knowledge about the counting quasipolynomials for queens , bishops , rooks , and pieces with some of their moves . we extend these results to the nightrider and its subpieces , and we compare our results with the empirical formulas of kotovec .

low - mass x - ray binaries ( lmxbs ) are mass - exchange binaries that contain an accreting black hole or neutron star primary and a low - mass secondary star . accretion takes place through an accretion disc which encircles the compact object and regulates the flow of material onto it making these objects the brightest x - ray sources in the sky . lmxbs provide an ideal playground for exploring the physics of compact objects yielding the confirmation of the existence of stellar mass black holes . optical observations are crucial to prove this . by observing the radial velocity curve of the companion star , one can determine the mass function of the system , which represents a minimum mass for the accreting compact object . this experiment can be best applied to a subclass of lmxbs , the so called x - ray transients ( xts ) , in which x - ray activity occurs only during well - defined outburst episodes . between outbursts , the emission from the accretion flow fades to the point that the companion star is clearly visible and it is nearly undisturbed by irradiation ; hence it can be used to derive a dynamical mass for the compact object ( eg . + the distribution of black hole masses can only be determined from the study of x - ray binaries ( eg . , * ? ? ? * ) and it is intricately related to the population and evolution of massive stars , the energetics and dynamics of supernova explosions , and the critical mass dividing neutron stars and black holes . several attempts to extract statistical information from the observed mass distribution have been made @xcite , however , the small number of black holes prevents us from extracting very compelling conclusions . about @xmath510@xmath14 - 10@xmath15 stellar mass black holes are believed to exist in the galaxy @xcite while @xmath510@xmath1610@xmath17 are expected to be members of xts @xcite . unfortunately only @xmath520 black hole candidates have reliable dynamical mass determinations to date . in summary , it becomes necessary to increase the sample of black hole masses . + the best place to look for black holes are xts , which are detected in outburst by x - ray all - sky monitors . this was the case of ky tra , a _ historical _ x - ray transient discovered in 1974 by the ariel v instruments @xcite . after a short precursor , ky tra reached an outburst peak flux of 0.9crab in the 3 - 6kev band , before decaying with an e - folding time of about 2 months @xcite . the source also showed a low intensity outburst in 1990 which was significantly fainter than the discovery 1974 outburst @xcite . six months later , an upper limit to the quiescent x - ray luminosity of @xmath52 - 10@xmath1810@xmath19ergs@xmath9 was derived . the ultrasoft x - ray spectrum seen by ariel v and the hard tail observed by sigma @xcite strongly suggests that ky tra is a black hole candidate . the optical counterpart was identified 12 days after the x - ray maximum of the 1974 outburst at @xmath20=17.5 @xcite . surprisingly , no further optical studies have been done since . + in this paper we present an optical study of the counterpart of the x - ray transient ky tra in quiescence to test its identification and obtain information about the donor star . the field of ky tra was observed on ut 2004 may 16 with the eso vlt u4 telescope at paranal and the focal reducer and low dispersion spectrograph ( fors2 ) . seven 300s exposures were obtained in johnson @xmath2 , spinning over 0.6hours . we also obtained series of 600s images in @xmath2 on ut 2007 jun 16 and 17 on the 3.6 m telescope at la silla observatory , equipped with the eso faint object spectrograph and camera ( efosc ) . in the first night of this run the sky was partially covered by clouds and no calibrations or images in other filters than @xmath2 were obtained . fortunately weather improved on jun 17 and we could take also @xmath4 and @xmath3 images of ky tra with 1500s and 1000s respectively . furthermore @xmath3 , @xmath2 and @xmath7 images were taken on ut 2010 june 03 on the eso vlt u1 telescope at paranal and fors2 using 80s , 45s and 400s integration respectively in each band . + for calibration , we observed the field of ky tra on ut 2015 jan 18 with the 1.3 m smarts telescope at cerro tololo ( chile ) . we took @xmath4 , @xmath3 and @xmath2 images with , respectively , 800 , 500 and 500 seconds of exposure time . we observed also the landolt field sa107 with 30 seconds integration in each of the three bands . an observing log is presented in table 1 . + lcccc * photometry : * & & & & + date & exp . time ( s ) & @xmath21 t ( hr ) & filter & telescope / + & & & & instrument + 2004/05/16 & 300 & 0.6 & @xmath2 & eso vlt u4 / fors2 + 2007/06/16 & 600 & 6.3 & @xmath2 & eso 3.6 m / efosc + 2007/06/17 & 600 & 6.0 & @xmath2 & + & 1000 & - & @xmath3 & + & 1500 & - & @xmath4 & + 2010/06/03 & 80 & - & @xmath2 & eso vlt u1 / fors2 + & 45 & - & @xmath3 & + & 300 & - & @xmath7 & + 2015/01/18 & 500 & - & @xmath2 & smarts 1.3 m / andicam + & 500 & - & @xmath3 & + & 800 & - & @xmath4 & + + * spectroscopy : * & & & & + date & exp . time ( s ) & & grism & telescope / + & & & & instrument + 2004/05/16 & 2000 & & gris-600ri & eso vlt u4 / fors2 + all images were corrected for bias and flat - fielded in the standard way using iraf . in particular the 2007 @xmath2-band frames were affected by fringing so these images were corrected using a _ master flat _ , i.e. the result of the median of a set of dithered exposures of our field . point spread function psf photometry was obtained with daophot ii @xcite whereas differential photometry of the 2004 and 2007 series were performed creating a photometric reference level , through an _ ensemble _ of isolated stars in the ky tra field , following the technique described in @xcite . + we finally calibrated a set of stars in the 2015 field that we then used as secondary standards to calibrate the 2004 and 2007 images , taken under non photometric conditions . to do so , we used the observatory extinction coefficients together with the zero points extracted from the landolt standards . a single 2000s spectrum was obtained on the night of 2004 may 16 using fors2 at the eso vlt u4 telescope at paranal . we used the gris-600ri grism , and a 2@xmath182 binning in both the spatial and spectral direction , which provides a wavelength coverage of @xmath2252008400 at 1.68pix@xmath9 dispersion and 600 km s@xmath9 resolution . standard procedures were used to de - bias and flat - field the spectra . the one - dimensional spectra were extracted using optimal extraction routines which maximize the final signal - to - noise ratio . a hg cd ar ne arc was obtained to provide the wavelength calibration scale . the optical counterpart of ky tra was identified at @xmath0=15:28:16.59 and @xmath1=-61:52:58.1 ( 2000 ) by @xcite 12 days after the outburst . these authors showed a schmidt plate of the proposed counterpart but no deep image with the target in quiescence has been published to date . in figure [ figure : field ] we show an improved finding chart of ky tra ( @xmath2-band , 1800s exposure and 0@xmath239 seeing , taken on 2007 june 17 ) . the field is 2x2arcmin and the star at the position proposed by @xcite is marked in the center . for comparison we have also marked the star labeled as s in that paper . in this figure we also show a zoom of the central region of 30x30arcsecs where an elongation of the source profile along the nw direction is clearly visible . we measure an elongation coefficient ( the iraf `` ellipsoidal '' parameter ) of 0.68 , significantly larger than the typical values obtained for nearby field stars ( 0.32 ) . although this requires confirmation through better seeing quality images , it strongly suggests that the counterpart is double and the transient is blended with an interloper . + to obtain a precise astrometric solution , we used the positions of the astrometric standards selected from the usno - b1 astrometric catalog with a nominal 0@xmath232 uncertainty . hundreds of reference objects can be identified in our field from which we selected 392 , discarding the stars with significant proper motions . the iraf tasks ccmap / cctran were applied for the astrometric transformation of the images . formal rms uncertainties of the astrometric fit for our images are @xmath240@xmath2325 in both right ascension and declination , which is compatible with the maximum catalog position uncertainty of the selected standards . the star within the @xcite error box has coordinates @xmath0=15:28:16.97 and @xmath1=-61:52:58.2 with a conservative estimate of our 3@xmath25 astrometric uncertainty of @xmath240 @xmath233 in both ra and dec . + we test this identification by cross - matching the @xmath3 , @xmath2 , and @xmath7 photometry of the field of ky tra taken on 2010 to build the ( @xmath3-@xmath2)(@xmath3-@xmath7 ) diagram of all the objects detected in the three photometric bands . our proposed target ( marked with a circle in figure [ figure : diagram ] ) shows a clear @xmath7 excess above the main stellar locus , confirming it is the true quiescent counterpart of ky tra . note that very close to the kt tra counterpart there is another source of similar colour and h@xmath0 excess . this source is at 58arcsec from the xt and its nature is unknown . however , ky tra should be located even higher in this diagram since the presence of the interloper dilutes its actual h@xmath0 excess . + we can refine the location of the counterpart by choosing which of the components of the blend is actually the xt counterpart . to do so , we first aligned the @xmath3 , @xmath2 , and @xmath7 images and then calculated the centroids of the profile targeted as ky tra and of a set of 40 stars around it . in figure [ figure : centroids ] ( bottom ) we show the modulus of the shifts between @xmath3 , @xmath2 and @xmath7 , defined as @xmath26^{(1/2)}$ ] and @xmath27^{(1/2)}$ ] where @xmath28 are the positions of the centroids in pixels . the centroid of ky tra measured in h@xmath0 is clearly shifted with respect to its position in the @xmath3 and @xmath2 images . the top panel in figure [ figure : centroids ] shows a zoom of the h@xmath0 image centered on our target , with the white cross marking its centroid and the black cross the centroid measured in the @xmath2-band image . this indicates that the xt is the component of the blend located at the nw . assuming that the @xmath7 centroid is the actual position of the target , we derived an improved source position of @xmath0=15:28:16.97 and @xmath1=-61:52:57.8 with an uncertainty of @xmath240 @xmath233 in both ra and dec . -@xmath2)(@xmath3-@xmath7 ) diagram of all the objects detected in the three photometric bands in our field of view . our proposed target , marked with a circle , shows a clear @xmath7 excess above the main stellar locus . under the hypothesis that the light of ky tra is contaminated by an interloper , it should be located even higher in this diagram , with a larger @xmath7 excess . the source closest to ky tra is unknown.,width=340 ] profile of ky tra with a white cross marking its centroid . the black cross marks the centroid measured in the @xmath2 image . ( lower ) modulus of the shifts ( in pixels ) between the image centroid positions for ky tra and 40 surrounding stars as measured in our @xmath3 , @xmath2 and @xmath7 images ( for details see text).,width=340 ] the colours of ky tra in quiescence were obtained on ut 2007 june 17 . we calculated @xmath2=20.88@xmath60.01 , @xmath3=21.75@xmath60.01 and v=22.83@xmath60.12 . photometric error estimates on the magnitudes are based on a combination of poisson statistics and the error contribution of the stars used for calibration . these magnitudes , however , correspond to the transient blended with the interloper ( see section [ astrometry ] ) so they need to be corrected taking this into account . to do so , we cleaned the contribution of the contaminant component by subtracting its best psf fit . the initial centers for fitting the profiles of the components were determined by visual inspection of the @xmath2 image , where they are more clear . the two components of the blend have approximately equal brightness : we calculated @xmath2=21.47@xmath60.09 , @xmath3=22.3@xmath60.1 and v= 23.6@xmath60.1 for the top component , which is the xt counterpart as suggested by the @xmath7 image . the @xmath29 and @xmath30 colours are typical of a m0 star , although note that this is not corrected for interstellar reddening . @xcite report a reddening e(b - v)@xmath310.5 . an estimation of the interstellar reddening for any sky region can be obtained from the nasa / ipac infrared science archive . the reddening quoted for the field of ky tra , e(b - v)=0.7 , implies a corrected @xmath32=0.77 and @xmath33=1.15 consistent with a @xmath5k0 - 2v companion . nevertheless , it should be noted that this is likely an upper limit to the true spectral type of the donor star because we have neglected any contribution from a residual accretion disc into the observed colour . the limited seeing conditions prevents us from deblending the flux of the two stars in every single image . therefore , we studied the variability of ky tra by integrating the total flux of the blend from the series of images in the @xmath2-band in the two different epochs : in 2004 may with less than one hour of observations and in 2007 june 16 and 17 with about 6 hours coverage . the 2004 images were obtained during twilight so photometry accuracy was dominated by a bright sky level . unfortunately on 2007 june 16 we were affected by poor weather conditions with variable transparency caused by clouds . on 2007 june 17 no clouds were present although sky transparency was not ideal . in this night we get @xmath57% photometry for a 21.5 magnitude star . in summary , we found that the target is not variable above the error levels in any of our nights ( i.e. , @xmath34=0.07 , 0.14 and 0.07mags for 2004 may and 2007 june 16 and 17 respectively ) . this is clearly seen in fig.[figure : variability ] where we plot the scatter in the observed magnitudes . the scatter around the mean magnitude of ky tra ( marked with a circle in fig.[figure : variability ] ) is consistent with that displayed by the field stars of similar mean brightness . note that the presence of the interloper dilutes the orbital modulation of ky tra , which would explain the lack of photometric variability on our images . because the interloper contributes around half of the total flux , we can only conclude that the target is not variable by @xmath350.15mags . + after re - scaling the zero - point of each night we found that the total flux remains stable at @xmath2=20.88@xmath60.01 ( @xmath2=21.47@xmath60.09 for ky tra , i.e. the top component of the blend ) in our entire dataset , from 2004 until 2007 . + band series of images of each night . the blend ( i.e , ky tra plus the interloper ) is marked with a circle.,width=340 ] profile.,width=340 ] although very noisy , our spectrum shows the h@xmath0 emission line characteristic of x - ray transients in quiescence . we obtained its full - width - half - maximum ( fwhm ) from a gaussian fit of the continuum rectified spectrum within the range @xmath610000km / s , centered on the h@xmath0 line . this is shown in fig . [ figure : ha_gaussian ] . after subtracting quadratically the instrumental resolution we find fwhm=2700@xmath6280km / s where the error is the formal 1-@xmath34 on the fitted parameter as derived through @xmath36 minimization . we note that the quoted error is within the typical 10% standard deviation caused by intrinsic line variability and therefore we take this as realistic @xcite . we also extracted the equivalent width ( ew ) by integrating the h@xmath0 flux after continuum normalization and find @xmath37 . note , however , that this value is diluted by the extra continuum of the interloper and hence the true ew of ky tra is underestimated by a factor 2 . although ky tra is a very promising black hole candidate , it had not been studied in the optical band since its discovery in 1974 . we have observed ky tra in quiescence and confirmed its identification . the finding chart we present will certainly be helpful in performing observations with eelt - class telescopes in order to obtain dynamical information on the mass of the compact object . + a rough estimate of the period can be made by combining the @xcite expression for the averaged radius of a roche lobe with kepler s third law to get the well - known relationship between the secondary s mean density and the orbital period : @xmath38g@xmath39 , where @xmath40 is the mean density and @xmath41 the orbital period in hours . under the hypothesis that the light of ky tra is contaminated by the interloper , we calculate an orbital period of about 8h assuming a k0v star . it should be noted that this is likely an upper limit since , as we pointed out in sect . 4.1 , we have neglected any residual contribution from an accretion disc to the colour of ky tra . an independent estimation can be made using the empirical relation @xmath42 which predicts the orbital period of xts with orbital periods less than 1day given only its visual outburst amplitude @xmath43 @xcite . during outburst , optical emission is dominated by the reprocessing of the x - rays in the accretion disc where most of the reprocessed energy is radiated in the ultraviolet . according to the irradiated model predictions @xmath440 @xcite resulting in @xmath4517.5 at the outburst peak @xcite . taking @xmath4=23.6 in quiescence , the total outburst amplitude is @xmath56.1mag so this would lead to an orbital period of about 12hours . we can place a robust upper limit for the period of about 15h corresponding to a minimum outburst amplitude if the source were not contaminated by any interloper . + after analyzing the h@xmath0 emission profiles of 12 dynamically confirmed black holes and 2 neutron star x - ray transients ( xts ) in quiescence , @xcite has found a tight correlation between the fwhm of the h@xmath0 line and the velocity semi - amplitude of the donor star , where k@xmath10=0.233(13)@xmath18fwhm . we have applied this relation to ky tra and predict k@xmath10=630@xmath674km this can be combined with our rough estimates of the orbital period to infer the mass function @xmath46 of the binary . our upper limit @xmath47h implies @xmath48 m@xmath12 while @xmath49h would lead to @xmath50 m@xmath12 . more accurate constraints require an accurate determination of the orbital period . + despite the foregoing , no variability has been found above the error levels , i.e. @xmath50.07mags , indicating that we may be looking at the binary at very low inclination . however , given the contaminating flux from an interloper , the variability would be diluted to the extent that any intrinsic variability above @xmath50.15mag would not be detectable given our error levels , and so the inclination might not be as low . interestingly , ky tra has one of the broadest h@xmath0 lines among sxts ( a summary of the parameters of the xts can be found in @xcite table 3 ) , suggesting that , if ky tra is viewed at low inclination , it must have a very short orbital period . some clues about the inclination are provided by the ew of the h@xmath0 line , since it depends on the binary geometry . the ew tends to increase with inclination because , when the disc is seen at large inclinations , its continuum brightness decreases . an interesting exercise is to locate ky tra in the ew fwhm diagram shown in figure 5 of @xcite . regions of constant inclination and @xmath51 , where @xmath52 is the mass of the compact object and @xmath53 is the orbital period , are defined in the diagram . both the relatively large ew and the high @xmath51 factor are reminiscent of xte j1118 + 480 and suggest that ky tra also hosts a black hole seen at moderately high inclination . furthermore , we point out that given the contaminating flux from an interloper , the observed @xmath54 would just be a lower limit to the true @xmath54 since the latter would be diluted by the excess continuum . clearly , more higher quality photometry is necessary to resolve these issues and draw further firm conclusions . jmc - s acknowledges financial support * from * conicyt through the fondecyt project no . 3140310 and basal - cata pfb-06/2007 , and jc to the spanish ministerio de educacin , cultura y deportes under grants aya201018080 and aya2013 - 42627 . we specially want to thank jose l. prieto for his help in obtaining the smarts observations . we also acknowledge the referee ( phil charles ) whose comments greatly improved the manuscript . bailyn , c. d. , jain , . k. , coppi , p. , orosz , j. a. , 1998 , apj , 499 , 367 barret d. , bouchet , l. , m , rou , p. , roques j. p. , cordier b. , laurent p. , lebrun f. , paul j. , sunyaev , r. , churazov e. , gilfanov m. , diachkov a. , khavenson n. , novikov b. , chulkov , i. , kuznetsov , a. , 1992 , apj , 394 , 615 barret d. , mandrou p. , roques j. p. , denis m. , lebrun f. , claret a. , goldwurm a. , laurent p. , churazov e. , gilfanov m. , sunyaev r. a. , bogomolov a. , khavenson n. , kuleshova n. , tserenin i. , sukhanov k. , 1993 , a&a , 97 , 241 barret d. , motch c. , pietsch w. , voges w. , 1995 , a&a , 296 , 459 brown , g. e. , & bethe , h. a. 1994 , apj , 423 , 657 casares j. , 2007 , in karasv . , eds , proc . 238 , black holes : from stars to galaxies - across the range of masses . cambridge univ . press , cambridge casares j. , 2015 , astro-ph.sr , arxiv:1506.00639 charles p.a . , & coe m. 2006 , in lewin w.h.g . , van der klis m. , eds , compact stellar x - ray sources . cambridge univ . press , cambridge , 215 , 265 , p. 3 farr , w. m. , sravan , n. , cantrell , a. , kreidberg , l. , bailyn , c. d. , mandel , i. , kalogera , v. 2011 , apj , 499 , 367 honeycutt r. k. , 1992 , pasp , 104 , 435 kaluzienski l. j. , holt s. s. , boldt e. a. , serlemitsos p. j. , eadie g. , pounds k. a. , ricketts m. j. , watson m. , 1975 , apjl , 201 , 121 kreidberg l. , bailyn c.d . , farr w.m . , kalogera v. , 2012 , apj , 757 , 36 murdin , p. , griffiths , r. e. , pounds , k. a. , watson , m. g. , longmore , a. j. , 1977 , mnras , 178 , 27 zel f. , psaltis , d. , narayan , r. , mcclintock , j. e. 2010 , apj , 725,1818 pounds k. a. , holt s. s. , kaluzienski l. j. , boldt e. a. , serlemitsos p. j. , 1974 , iau circ . , 2729 paczyski , b. , 1971 , ara&a , 9 , 183 romani , r. w. 1998 , a&a , 333 , 583 shahbaz t. , & kuulkers e. , 1998 , mnras , 295 , l1 stetson , p.b . , 1987 , pasp , 99 , 191 van paradijs , j. , & mcclintock , j. , 1995 , in x rays binaries . cambridge univ . press , cambridge white , n. e. , & van paradijs , j. 1996 , apj , 473 , 25 yungelson l. r. , lasota j .- p . , nelemans g. , dubus g. , van den heuvel e. p. j. , dewi j. , portegies zwart s. , 2006 , a&a , 454 , 559

we present deep optical images of the _ historical _ x - ray transient ky tra in quiescence from which we confirm the identification of the counterpart reported by @xcite and derive an improved position of @xmath0=15:28:16.97 and @xmath1=-61:52:57.8 . in 2007 june we obtained @xmath2 , @xmath3 and @xmath4 images , where the counterpart seems to be double indicating the presence of an interloper at @xmath51.4arcsec nw . after separating the contribution of ky tra we calculate @xmath2=21.47@xmath60.09 , @xmath3=22.3@xmath60.1 and @xmath4=23.6@xmath60.1 . similar brightness in the @xmath2 band was measured in may 2004 and june 2010 . variability was analyzed from series of images taken in 2004 , spanning 0.6h , and in two blocks of 6h during 2007 . we find that the target is not variable in any dataset above the error levels @xmath50.07mags . the presence of the interloper might explain the non - detection of the classic ellipsoidal modulation ; our data indicates that it contributes around half of the total flux , which would make a variation < 0.15 mags not detectable . a single spectrum obtained in 2004 may shows the @xmath7 emission characteristic of x - ray transients in quiescence with a full - width - half - maximum @xmath8 280 km s@xmath9 . if the system follows the fwhm k@xmath10 correlation found by @xcite , this would correspond to a velocity semi - amplitude of the donor star of k@xmath10=630@xmath674 km s@xmath9 . based on the outburst amplitude and colours of the optical counterpart in quiescence we derive a crude estimate of the orbital period of 8 h and an upper limit of 15 h which would lead to mass function estimates of @xmath11 9m@xmath12 and @xmath1316m@xmath12 respectively . = 1 [ firstpage ] keyword1 keyword2 keyword3 binaries : close x - rays : binaries stars : individual : ky tra

if the rate of gravitational lenses among known quasars is roughly 0.1% of all quasars ( turner , ostriker & gott 1984 ) , the expected number of new gravitationally lensed quasars in the sloan digital sky survey s ( sdss ; york et al . 2000 ) final spectroscopic quasar sample will be approximately @xmath5 . furthermore , the entire sdss photometric sample may contain an order of magnitude more lensed systems ( since the photometric observations reliably probe @xmath6 mag fainter than the spectroscopic sample ) . the sdss is a project to conduct parallel photometric and spectroscopic surveys of 10,000 deg@xmath7 of the sky centered approximately on the north galactic pole , using a dedicated wide - field 2.5-m telescope at apache point observatory ( apo ) in new mexico , usa . photometric observations are done in five broad optical bands ( @xmath8 , @xmath9 , @xmath10 , @xmath11 , and @xmath12 , centered at 3561 , 4676 , 6176 , 7494 , and 8873 respectively ( fukugita et al . 1996 ; stoughton et al . the imaging camera consists of a 5@xmath136 array of large photometric ccds ( 2048@xmath132048 pixels ) and 24 astrometric ccds ( 2048@xmath13128 pixels ) ( gunn et al . the imaging data are reduced by the photometric pipeline ( lupton et al . 2001 ) using information from the astrometric pipeline ( pier et al . 2002 ) and the 0.5-m photometric calibration telescope ( hogg et al . 2001 ; smith et al . 2002 ) . spectroscopic observations are done with a multi - fiber spectrograph covering 3800 to 9200 with the resolution r@xmath141800 . we are interested in using the sdss to find new lensed quasars since they have proven to be useful for cosmological tests , especially for the measurement of the cosmological constant by the number counts of lensed quasars ( fukugita & turner 1991 ) as well as the measurement of the hubble constant by observations of the time delay between multiple components ( refsdal 1964 ) . finding new lensed quasars in a large , homogeneous survey , such as the sdss will contribute greatly to the statistics of lensed quasars ( useful for measurements of the cosmological constant ) and the determination of their time delays ( useful for constraining the hubble constant ) . to date the sdss has yielded one new two - image lensed quasar , sdss 1226@xmath00006a , b ( inada et al . 2002 ) and here we report on the second lensed quasar system discovered among the sdss quasars . to maximize the likelihood of discovering additional lens systems , we are studying the sdss parameters of previously known lensed quasars . in particular , we studied the parameters of sdss j1226@xmath00006 , and using these parameters we developed an algorithm to select sdss 1226-like objects from the sdss database . this algorithm should be sensitive to lensed quasars with separations on the order of @xmath15 to @xmath16 . applying this algorithm to the approximately 10,000 sdss quasars discovered prior to 2001 december 1 , we identified five lensed quasar candidates . by definition one of them is sdss 1226@xmath00006 . another candidate is sdss j0924 + 0219 ( 09@xmath17 24@xmath18 55@xmath1987 , + 02@xmath20 19@xmath21 @xmath22 , j2000 ) which is identified as a @xmath1 quasar in the sdss database . we obtained photometric follow - up observations of sdss j0924 + 0219 using the magellan consortium s walter baade 6.5-m ( wb6.5 m ) telescope at las campanas observatory under good seeing conditions ( @xmath23 ) . additional spectroscopic observations of the components that were resolved by the wb6.5 m images were obtained with the keck ii telescope at the w. m. keck observatory on mauna kea in @xmath24 seeing . observations of the three other candidates from this sample have been taken with the wb6.5 m telescope and indicate that these systems are also likely to be lensed quasars ; these systems will be discussed in future papers . section 2 of this paper briefly describes the new algorithm which led to the discovery of sdss0924 + 0219 . section 3 describes the follow - up observations and show the results of them . in 4 , we discuss some of the interesting aspects of this system . finally , we present a summary of this paper in 5 . we now describe the manner in which sdss j0924 + 0219 was selected as a gravitational lens candidate based upon the object parameters in the sdss object catalog ( lupton et al . first , we selected objects that were confirmed to be quasars in the sdss spectroscopic survey ( see richards et al . 2002 for details of the sdss quasar target selection algorithm ) . we rejected quasars whose redshifts are less than 0.6 since many low - redshift quasars are extended objects ( schneider et al . 2002 ) , making it hard to distinguish them from unresolved lensed quasars . next , we restricted our lensed quasar candidate sample using some sdss catalogued parameters , specifically the galaxy profile fitting likelihood . as the sdss data are passed through the data reduction pipelines , each extended object is fitted with a set of possible galaxy profiles ( lupton et al . 2001 ) and labeled with the likelihood that each profile explains the data . these likelihoods are useful for searching for extended quasars , so we optimized our search criteria to use these values , based on our study of the first sdss lensed quasar , sdss j1226@xmath00006 . this algorithm targets lensed quasars whose separations are approximately @xmath25 , because we empirically confirmed that lensed quasars whose separations are less than 10 do not have large `` extended '' parameters in the sdss object catalog ; we can not distinguish these small separation lensed quasars from single , unresolved quasars . for lensed quasars that have separations of more than 25 , each lensed component should appear as a separate entity in the sdss catalog . as a result of applying this algorithm to the approximately 10,000 sdss quasars ( in @xmath26 deg@xmath27 ) , we selected five lensed quasar candidates . one of these five candidates is the first sdss lensed quasar , sdss 1226@xmath00006 , and another of these candidates is sdss j0924 + 0219 . if the total lensing rate is roughly 0.1% of all quasars , then on the order of @xmath28 lensed quasars are expected from the approximately @xmath29 sdss quasars that were known at the start of this work . about half of the lensed quasars should have @xmath25 separations ( chiba & yoshii 1999 ) ; therefore , the result of this algorithm is consistent with the theoretical estimate . whether or not this selection algorithm is the optimal way to select moderate separation lens candidates from the sdss imaging data remains to be seen , since it could be biased towards sdss j1226@xmath00006-like lens systems . however , the successful discovery of two lensed quasars suggests that it is a reasonable method to use for our initial lens search . photometric follow - up observations of sdss j0924 + 0219 were obtained using the wb6.5 m telescope . the data were taken on 2001 december 15 with @xmath8 , @xmath9 , @xmath10 , and @xmath11 filters using the magellan instant camera ( magic , a 2048@xmath132048 ccd camera ) ; the seeing was 055@xmath30075 fwhm . the pixel size was 0069 . the exposure time was 300 sec in each band . each ccd frame was bias - subtracted and flat - field corrected . additional spectroscopic data were taken on 2002 january 12 with the keck ii echellette spectrograph and imager ( esi ; sutin 1997 ) , using the mit - ll 2048@xmath134096 ccd camera and a 175 line mm@xmath2 grating . we used the echellette mode . the resolution of the echellette mode of this spectrograph is 11.4 km sec@xmath2 pixel@xmath2 . the spatial resolution scale of this spectrograph is 0153 pixel@xmath2 . the spectral range covers 3900 to 11,000 . the exposure time was 1200 sec . we set the slit direction so that two of the three components ( components a and b , see below ) were on the slit at the same time . the two components are separated by 178 and the seeing was less than 10 fwhm ; the two components are clearly distinct in the 2-dimensional esi image . the two spectra were extracted separately using the usual method of summing the flux in a window around each object and subtracting sky from neighboring windows on either side of the trace . the only difference from a simple single spectrum extraction was that we were careful to exclude the other object from the sky windows . there was no need ( and no attempt ) to fit the two spectra simultaneously because we can not see any overlap along the slit in the 2-dimensional esi image . however , flux of component d ( see below ) actually affect the component a spectrum ( we set the slit width 10 ) . we could not see it in the 2-dimensional esi image , because component d is close to the component a , and is more than two magnitudes fainter than component a. the estimated contaminations are 6.0% of the flux of component a in @xmath9 band ( 4000@xmath315500 ) , 6.8% in @xmath10 band ( 5500@xmath317000 ) , and 7.4% in @xmath11 band ( 7000@xmath318500 ) , respectively . we used a single slit position that included components a and b , which puts component g ( see below ) slightly off to one side . since it is much fainter ( about 2 mag fainter than component a ) , it is not noticeable in the spectrum . the sdss images ( sky and bias subtracted and flat - field corrected ) are shown in all bands in figure 1 . total magnitudes of the five components are 18.68@xmath320.02 , 18.43@xmath320.01 , 18.34@xmath320.01 , 18.09@xmath320.01 , and 17.98@xmath320.03 in @xmath33 , @xmath34 , @xmath35 , @xmath36 , and @xmath37 , respectively , while the still preliminary 2.5-m filter - based photometry will be called @xmath38 ( stoughton et al . 2002 ) ] . these errors in the magnitudes are statistical errors . we show the full follow - up magic @xmath10 image including nearby stars ( star a , star b , and star c ) as well as sdss j0924 + 0219 in figure 2 . the non - psfs subtracted ( hereafter `` original '' ) @xmath8 , @xmath9 , @xmath10 , and @xmath11 images of sdss j0924 + 0219 and the psf - subtracted images of each band are shown in figure 3 . the upper panels are the original images and the lower panels are the psf - subtracted images . we subtracted psfs using stars from the original images in all bands . we used star a for the @xmath8 image , star b for the @xmath10 and the @xmath11 images and star c for the @xmath9 image , respectively . the `` peak '' flux and the center coordinates of stars a , b , and c and components a , b , and c were calculated by the single gaussian fit ( these results agree with the results obtained by the `` imexamine '' task in iraf ) . we named the three stellar components `` components a c '' according to their magnitudes , the center extended object `` component g '' and the unknown component which remains after subtracting psfs `` component d '' . the flux ratios between components a and b are 0.47 , 0.44 , 0.43 , and 0.43 in @xmath8 , @xmath9 , @xmath10 , and @xmath11 , respectively , and those between components a and c are 0.44 , 0.43 , 0.41 , and 0.40 . in the original @xmath11 image one can see components g and d , both are much more prominent in the psf - subtracted images ( except for @xmath8 ) . we show the reduced @xmath39 from the psf subtraction in table 1 . the reduced @xmath39 of the @xmath9 , @xmath10 , and @xmath11 images of component a and component c are large , because component a has contamination from components g and d , and component c has contamination from component g. we can not see components d and g in the @xmath8 psf - subtracted image . this fact suggests that component d is not a quasar but rather a galaxy . we give our estimated magnitudes and colors of the three stellar components , component g and component d in table 2 . we used stars a , b , and c as the photometric standard stars using their sdss catalog magnitudes and positions . spectra of components a and b taken with esi on keck ii are shown in figure 4 . , , , and emission lines are seen clearly ; both components are quasars at the same redshift ( @xmath1 ) . the velocity difference between the two components is less than 100 km sec@xmath2 , calculated using the emission lines . the widths of the emission lines of both components are also in good agreement . we do not have a spectrum of component c , but the colors of this component indicate that this component is also a quasar . a photometric redshift can be computed for component c following richards et al . ( 2001 ) , assuming an @xmath40 color of @xmath41 . the resulting photometric redshift is 1.33@xmath320.20 , which is consistent with the redshifts of components a and c , @xmath421.524 . the redshifts and the widths of the emission lines are summarized in table 3 . we calculated the celestial coordinates of components a , b , and c based on the sdss celestial coordinates of the three stars ( star a , star b , and star c , figure 2 ) common to the magic @xmath10 image and the sdss data . the separations between components a and b , b and c , and a and c are calculated to be 178@xmath32004 , 150@xmath32003 , and 114@xmath32004 , respectively . we also calculated the celestial coordinates of component g and component d after subtracting psfs . the results are summarized in table 4 . sdss j0924 + 0219 is certainly a lensed quasar because of its morphology ( there is a galaxy between three almost same color quasars ) and the small velocity difference and the small line width differences between components a and b. however , even though there are no emission line profile differences between components a and b , the ratio of the spectra of the two quasars is not constant with @xmath43 ; it steadily decreases from 0.7 at 4000 to 0.2 at 9000 as shown in figure 5 . the flux of component d would affect the spectrum of component a , but the estimated contaminations are not large ( see 3.1 ) and it would not give a dramatically change in the spectrum of component a. the difference in the flux density ratio between the two components indicates that the two quasars might not be from the same physical source . it is possible that the slit used for the keck spectroscopic observation was not precisely aligned along the line between components a and b ; combined with differential refraction , this could have produced a spectroscopic flux density ratio different from the photometric flux ratio . however , this can not be the full explanation , since it can not explain why the spectroscopic flux ratio is not smooth ( figure 5 ) . another possibility for the differences in the spectra of the two components is that continuum variations of the source quasar combined with the differential time delay causes the wavelength dependence in the flux density ratio . continuum variations are often seen in quasars ( trvese , kron , & bunone 2001 ) , and differences in flux ratios between lensed component continua are seen in some catalogued lensed quasar systems , e.g. he 1104@xmath01805 ( wisotzki et al . 1995 ) and fbq 0951 + 2635 ( schechter et al . we investigate whether the large differences between the photometric flux ratios ( see 4.1 ) and the spectroscopic flux density ratio could be caused by the continuum variations combined with the differential time delay . we estimate the time delay between components a and b using the sis model ( peacock 1999 ) : @xmath44 where @xmath45 , @xmath46 , @xmath47 , @xmath48 , and @xmath49 are the angular size distances from observer to the source quasar , from observer to the lensing galaxy , from the lensing galaxy to the source quasar , the einstein radius in arcseconds and the redshift of the lensing galaxy , respectively . the observed separations from the center of the lensing galaxy are represented as @xmath50 and @xmath51 . we suppose the redshift of the lensing galaxy is 0.4 ( see below ) , the velocity dispersion of the lensing galaxy is 230 km sec@xmath2 ( see below ) , @xmath5270 km sec@xmath2 mpc@xmath2 , @xmath53 , and @xmath54 . using these parameters , we determine a time delay of about 15 days . quasars generally do not experience large variations , such as shown in figure 4 ( or figure 5 ) , in less than 15 days , therefore , there is a significant possibility that other additional phenomena cause the differences between the two components of sdss j0924 + 0219 . one such effect might be microlensing . we can not see clear evidence of microlensing in both spectra ( figure 4 ) , but microlensing should be universal in quadruple lens ( witt , mao , & schechter 1995 ) , therefore , it is natural that microlensing events happen to both components or to one of the two components and cause the difference between the two components ( if we suppose the same situation of q2237 + 0305 ( walsh , carswell , & weymann 1979 ) , the microlensing optical depth ( schmidt , webster , & lewis 1998 ) supports this fact ) . furthermore , if there is another object which darkens component d or component d is a second lensing galaxy ( see below discussion ) , it might have an effect on the observed spectrum of component a , i.e. the reddening of component a could be larger than that of component b. however , this extinction should be time invariant ( effectively ) , and therefore , we should see the same extinction in the the photometric flux ratios , but we can not see it . confirming spectra and direct imaging are needed to determine whether the differences between the spectroscopic and direct imaging flux ratios are real or an observational artifact . although the sis model with an external shear ( kochanek 1991 ) predicts the existence of a fourth lensed component whose amplification is as bright as the brightest of the other three lensed components , there are no stellar components except components a , b , and c in all the images ( figure 3 ) . using the positions of components a c and of lensing galaxy g ( which of course might not be the only lensing galaxy ) we can fit the sis model with an external shear , with a projected potential : @xmath55 where @xmath56 is the einstein radius of the sis model in arcseconds , @xmath10 and @xmath57 are the radial and the angular parts , respectively , of the angular position on the sky , and @xmath58 is the position angle of the shear , measured east of north . fitting this model to the positions ( but not the fluxes ) of components a c , we get @xmath59 , @xmath60 , and @xmath61 , with a source position ( @xmath62r.a . , @xmath62dec . ) = ( @xmath63 , @xmath64 ) relative to component g. this value of @xmath56 corresponds to a velocity dispersion 230 km sec@xmath2 with the estimated redshift of the lensing galaxy ( see final paragraph of this section ) . the observed positions of components a c , the observed position of component d and the predicted positions of the lensed components are given in table 5 below . with six constraints and five free parameters it is no surprise that we obtain small residuals ( @xmath65 ) from the observed positions of components a c . although the flux ratios of observed components were _ not _ taken as constraints , the predicted flux ratios between components a and b and between components a and c are in agreement with the observations . the predicted position of the fourth lensed component ( hereafter , component d@xmath66 ) is shown in figure 6a and figure 6b . figure 6a is the image plane of this model , and figure 6b is the enlarged part of the @xmath11 subtracted image of figure 3 with the measured positions of components a c , component d and the predicted position of component d@xmath66 . component d is separated by only about 015 from the predicted position of component d@xmath66 ( the filled circle on figure 6a or the open circle on figure 6b ) . also , the predicted position of component d@xmath66 is within the region occupied by component d ( figure 6b ) . these results suggest that it is possible that either 1 ) component d really is the `` missing '' fourth lensed component , or 2 ) that component d is a object which is mixed component d@xmath66 ( predicted fourth lensed component ) with a foreground object that is obscuring the light from component d@xmath66 . according to schechter & wambsganss 2002 , microlensing causes demagnificating ( or vanishing ) the fourth lensed image at `` saddlepoint '' . the difference between the expected magnitude and the observed magnitude is large ( @xmath67 ) , but this marginally consistent with figure 3 of schechter & wambsganss 2002 . futhermore , some literatures reported that milli - lensing and/or micro - lensing produce anomalous flux ratios of four - image gravitationally lensed systems ( subramanian , chitre , & narasimha 1985 , metcalf & madau 2001 , keeton 2003 , and kochanek & dalal 2003 ) . the second case is also likely . if there are a foreground dusty galaxy superposed upon component d@xmath66 , it could be obscuring and reddening the light from component d@xmath66 . this might also explain why component d is bluer than component g ; component d may have some contribution from component d@xmath66 , which is expected to be relatively blue . the reduced @xmath39 of component d from the psf subtraction in the @xmath11 image is 20.91 and it is comparable with the other stellar component ( table 1 ) , therefore , there is a possibility that component d includes a stellar component . in addition to reddening by the foreground object , the demagnifying effect of microlensing might also darken component d@xmath66 , and therefore , component d might be much fainter than component d@xmath66 . two additional minor possibilities are that component d is a second lensing galaxy and contributes significantly to the lensing potential , which would change the lensing model such that the predicted position of component d@xmath66 is incorrect , and that this lensing system has a very interesting lensing potential which produces only three lensed components . according to keeton , kochanek , & seljak ( 1997 ) ; kassiola & kovner ( 1993 ) ; and wallington & narayan ( 1993 ) , there are some cases that non - singular lensing potentials with large shears , large ellipticities , or large core radii produce only three lensed component . however , sdss j0924 + 0219 can not be explained by these standard lensing models , such as non - singular lensing potentials with large shears because components of a `` standard '' three component lensing system are expected to be on the same side of the lensing galaxy in the non - singular lens models ( kassiola & kovner 1993 ; wallington & narayan 1993 ) while the three components of sdss j0924 + 0219 are not on the same side of the presumed lensing galaxy ( component g ) . spectroscopy of image d is just one way for the question to be resolved . higher resolution imaging , say with hst , may resolve it without any need for spectra . according to fukugita , shimasaku , & ichikawa ( 1995 ) , a typical elliptical galaxy at @xmath68 has @xmath69 and @xmath70 , which are close to the estimated colors of component g ( see table 2 ) . the estimated colors and the spherical appearance of component g , therefore , indicate that it may be an elliptical galaxy at @xmath68 . this would be the primary lensing galaxy of this lensing system . if the redshift of the lensing galaxy is 0.4 , the @xmath71 separation of components a and b requires the velocity dispersion of the lensing galaxy to be about 230 km sec@xmath2 in the sis model , while the faber - jackson law predicts 225 km sec@xmath2 from the @xmath11 magnitude of this galaxy ( @xmath7220.8 ) . here we assume that @xmath73 , @xmath74 km sec@xmath2 ( blanton et al . 1999 and kochanek 1996 ) , @xmath75 , @xmath76 , @xmath77 km sec@xmath2 mpc@xmath2 , and the @xmath78-corrections of @xmath11 are 0.2 for a @xmath79 elliptical galaxy and 0.4 for a @xmath80 ( inada 2001 ) . this result favors a redshift of 0.4 for the lensing galaxy , and a velocity dispersion of about 230 km sec@xmath2 . using a selection algorithm tuned to recover a previously discovered lensed quasar ( sdss 1226@xmath00006 ) , we have identified several additional lensed quasar candidates from the sdss data . we confirmed that one of them , sdss j0924 + 0219 , is a lensed quasar with follow - up observations using the walter baade 6.5-m and the keck ii telescopes . the redshift of the source quasar is @xmath1 . the maximum separation is @xmath3 . the velocity difference between component a and component b is very small , less than 100 km sec@xmath2 . we can directly see the lensing galaxy in the original magic @xmath11 image , and we can see it more clearly in the psf - subtracted images . the estimated colors and the magnitudes of the lensing galaxy are consistent with those of a typical elliptical galaxy at @xmath80 with a velocity dispersion of 230 km sec@xmath2 . we can see only a faint red component near the predicted position of the fourth lensed image ( using the sis model with an external shear ) . we consider that this faint red component is a fourth lensed component darkened and reddened by a foreground objects and microlensing . to settle the issue of what causes the lack of the fourth quasar component , we need to obtain deeper and higher resolution images and fainter spectroscopy of component d. funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , princeton university , the united states naval observatory , and the university of washington . g. t. r. and d. p. s. acknowledge support from national science foundation grant ast99 - 00703 and d. j. e. acknowledges support from national science foundation grant ast00 - 98577 and an alfred p. sloan research fellowship . finally , we thank the staffs of the keck and las campanas observatories for their excellent assistance . m. r. , et al . 2001 , , 121 , 2358 chiba , m. & yoshii , y. 1999 , , 510 , 42 falco , e. e. , et al . 1999 , , 523 , 617 fukugita , m. , ichikawa , t. , gunn , j. e. , doi , m. , shimasaku , k. , & schneider , d. p. 1996 , , 111 , 1748 fukugita , m. , shimasaku , k. , & ichikawa , t. 1995 , pasp , 107 , 945 fukugita , m. & turner , e. l. 1991 , , 253 , 99 gunn , j. e. , et al . 1998 , , 116 , 3040 hogg , d. w. , finkbeiner , d. p. , schlegel , d . j. , & gunn , j. e. 2001 , , 122 , 2129 inada , n. 2001 , master s thesis of university of tokyo , department of physics . inada , n. , et al . 2003 , in preparation kassiola , a. & kovner , i. 1993 , , 417 , 450 keeton , c. r. , 2003 , , 584 , 664 keeton , c. r. , kochanek , c. s. , & seljak , u. 1997 , , 482 , 604 kochanek , c. s. , & dalal , n. 2003 , , submitted ( astro - ph/0302036 ) kochanek , c. s. , 1996 , , 466 , 638 kochanek , c. s. , 1991 , , 373 , 354 koopmans , l. v. e. , de bruyn , a. g. , xanthopoulos , e. , & fassnacht , c. d. 2000 , a&a , 356 , 391 lupton , r. h. , gunn , j. e. , ivezi , . , knapp , g. r. , kent , s. , & yasuda , n. 2001 , adass x , 2001adass-10 , 269 metcalf , b. r. , madau , p , 2001 , , 563 , 9 peacock , j. a. , 1999 , cambridge university press , `` cosmological physics '' pier , j. r. , munn , j. a. , hindsley , r. b. , hennessy , g. s. , kent , s. m. , lupton , r. h. , & ivezi , . 2002 , , submitted refsdal , s. 1964 , , 128 , 307 richards , g. t. , et al . 2002 , , 123 , 2945 richards , g. t. , et al . 2001 , , 122 , 1151 schechter , p. l. , gregg , m. d. , becker , r. h. , helfand , d. j. , & white , r. l. 1998 , , 115 , 1371 schechter , l. p. & wambsganss . 2002 , , submitted ( astro - ph/0204425 ) schneider , d. p. , et al . 2002 , , 123 , 567 smith , a. , et al . 2002 , , 123 , 2121 stoughton , c. , et al . 2002 , , 123 , 485 subramanian , k. , chitre , s. m. , & narasimha , d. 1985 , , 289 , 37 sutin , b. m. 1997 , proc . sipe , 2871 , 1116 wallington , s. & narayan , r. 1993 , , 403 , 517 trvese , d. , kron , r. g. , & bunone , a. 2001 , , 551 , 103 turner , e. l. , ostriker , j. p. , & gott , j. r. , iii 1984 , , 284 , 1 wisotzki , l. , koehler , t. , ikonomou , m. , & reimers , d. 1995 , a&a , 297 , l59 witt , h. j. , mao , s. , & schechter , l. p. 1995 , , 443 , 18 york , d. g. , et al . 2000 , , 120 , 1579 cccccccc component a & 19.66@xmath320.02 & 19.46@xmath320.01 & 18.97@xmath320.02 & 18.87@xmath320.02 & 0.20 & 0.49 & 0.10 + component b & 20.49@xmath320.05 & 20.34@xmath320.04 & 19.89@xmath320.04 & 19.79@xmath320.03 & 0.15 & 0.45 & 0.10 + component c & 20.55@xmath320.05 & 20.38@xmath320.05 & 19.94@xmath320.04 & 19.91@xmath320.03 & 0.17 & 0.44 & 0.03 + component d & @xmath8122.30 & 22.45@xmath320.12 & 21.82@xmath320.06 & 21.61@xmath320.05 & & 0.63 & 0.21 + component g & @xmath8122.30 & 22.73@xmath320.13 & 21.25@xmath320.05 & 20.78@xmath320.05 & & 1.48 & 0.47 + star a & 20.80@xmath320.01 & 18.06@xmath320.01 & 16.68@xmath320.02 & 16.03@xmath320.01 & 2.74 & 1.38 & 0.65 + star b & 20.72@xmath320.09 & 18.13@xmath320.01 & 16.72@xmath320.02 & 15.68@xmath320.01 & 2.59 & 1.41 & 1.04 + star c & 23.15@xmath320.06 & 21.01@xmath320.04 & 19.47@xmath320.02 & 18.57@xmath320.02 & 2.14 & 1.54 & 0.90 + cccccccc ( 1857.40 ) & 4684.75 & 38.9 & 1.5222@xmath320.003 & & 4690.14 & 42.4 & 1.5251@xmath320.003 + ( 1892.03 ) & 4775.94 & 36.0 & 1.5242@xmath320.002 & & 4779.39 & 43.6 & 1.5261@xmath320.003 + ( 1908.73 ) & 4815.97 & 48.5 & 1.5231@xmath320.001 & & 4818.40 & 45.4 & 1.5244@xmath320.002 + ( 2798.75 ) & 7063.51 & 61.2 & 1.5238@xmath320.001 & & 7063.01 & 60.8 & 1.5236@xmath320.001 + crrrr component a & 09 24 55.8293 & + 02 19 25.356 & + 0.0108 & + 0.847 + component b & 09 24 55.8327 & + 02 19 23.565 & + 0.0142 & @xmath00.944 + component c & 09 24 55.7659 & + 02 19 24.691 & @xmath00.0549 & + 0.182 + component d & 09 24 55.8653 & + 02 19 24.897 & + 0.0468 & + 0.388 + component g & 09 24 55.8185 & + 02 19 24.509 & 0.0000 & 0.000 + lccccccc component a & + 0.162 & + 0.847 & 1.00 & & + 0.1688 & + 0.8446 & 1.00 + component b & + 0.213 & @xmath00.944 & 0.44 & & + 0.2209 & @xmath00.9289 & 0.38 + component c & @xmath00.789 & + 0.182 & 0.42 & & @xmath00.7548 & + 0.1868 & 0.41 + component d & + 0.702 & + 0.388 & 0.09 & & & & + component d@xmath66 & & & & & + 0.6532 & + 0.4962 & 0.85 +

we report the discovery of a new gravitationally lensed quasar from the sloan digital sky survey , sdss j092455.87 + 021924.9 ( sdss j0924 + 0219 ) . this object was selected from among known sdss quasars by an algorithm that was designed to select another known sdss lensed quasar ( sdss 1226@xmath00006a , b ) . five separate components , three of which are unresolved , are identified in photometric follow - up observations obtained with the magellan consortium s 6.5 m walter baade telescope at las campanas observatory . two of the unresolved components ( designated a and b ) are confirmed to be quasars with @xmath1 ; the velocity difference is less than 100 km sec@xmath2 according to spectra taken with the w. m. keck observatory s keck ii telescope on mauna kea . a third stellar component , designated c , has the colors of a quasar with redshift similar to components a and b. the maximum separation of the point sources is @xmath3 . the other two sources , designated g and d , are resolved . component g appears to be the best candidate for the lensing galaxy . although component d is near the expected position of the fourth lensed component in a four image lens system , its properties are not consistent with being the image of a quasar at @xmath4 . nevertheless , the identical redshifts of components a and b and the presence of component c strongly suggest that this object is a gravitational lens . our observations support the idea that a foreground object reddens the fourth lensed component and that another unmodeled effect ( such as micro- or milli - lensing ) demagnificates it , but we can not rule out the possibility that sdss0924 + 0219 is an example of the relatively rare class of `` three component '' lens systems .

it is implicitely assumed that the independent degrees of freedom of any mechanical system satisfy , in general , second order differential equations , as it happens for the newton equations of the centre of mass of the mechanical system . this is reflected in the general assumption that the lagrangian of any mechanical system is an explicit function @xmath0 of the time @xmath1 , the @xmath2 degrees of freedom @xmath3 and their time derivatives @xmath4 . lagrangians depending on higher order derivatives are scarcely used , and their usefulness is left to some especific problems . in this work we are going to give three different kinds of arguments to justify the use of lagrangians depending up to the acceleration of the centre of charge for describing charged elementary spinning particles . the centre of charge of an elementary particle thus satisfies fourth order differential equations . in section [ rigid ] we analyze the motion of the centre of charge of a charged rigid body , considered as a model of a classical elementary particle . in section [ invariance ] we suggest that the order of the invariant differential equation satisfied by the evolution of a point depends on the number of parameters of the kinematical group of the theory . since the galilei and poincar groups are ten - parameter groups , and if the point represents the position of the centre of charge of an elementary particle , the corresponding differential equation must be of fourth order . section [ geometrical ] analyzes the most general differential equation satisfied by a point in three dimensional space , which is of fourth order . finally , section [ dirac ] describes the model of a dirac particle obtained from a general formalism for describing classical spinning particles and compares its structure with the previous predicted motions . let us consider that an elementary particle is described as a nonrelativistic rigid body . a rigid body is a mechanical system of six degrees of freedom . three represent the position of a point and the other three the orientation of a body frame attached to that point . usually , it is described by the location of the centre of mass , which is represented by the point @xmath5 , and the orientation by the principal axis of inertia located around @xmath5 . the centre of mass satisfies second order dynamical equations and moves like a point of mass @xmath6 , the total mass of the system , under the total external force . the torques of the external forces produce a change in the orientation of the body . in this way a rigid body moves and rotates . and and another arbitrary point @xmath7 of a rigid body.,width=264 ] if we describe the evolution of a different point @xmath7 , it will follow , in general , a helical trajectory around the centre of mass , like the one depicted in the figure [ fig:1 ] . if an elementary particle is a charged rigid body , it is clear that we also need to know its electromagnetic structure , which can be reduced to the knowledge of the centre of charge and the different multipoles . if assumed some spherical symmetry for the electric field produced by the particle , we are left with the location of the centre of charge and no further multipoles . the position of this point will be used to determine there the actions of the external fields and to compute from there the fields generated by the particle . in general , depending how the mass and charge are distributed , these two points will be different points as we shall assume here . therefore , if we try to describe the evolution of the centre of mass , we have to determine also at any time the location of the centre of charge to compute the external forces . dynamical equations for the centre of mass will be written as @xmath8 the electromagnetic force @xmath9 depends , in general , on the electric and magnetic external fields defined at the charge position @xmath7 and on the velocity of the charge @xmath10 which appears in the magnetic term . the relative motion between @xmath7 and @xmath5 when analyzed by an observer who sees the centre of mass at rest like in figure [ fig:2 ] is a circular motion with constant velocity , in particular in the free case . around the centre of mass at rest @xmath5 . , width=264 ] if we define a unit vector @xmath11 in the direction of the normal acceleration @xmath12 of point @xmath7 , @xmath13 where @xmath14 is the radius of the circular motion , then , the centre of mass position can be written as @xmath15 for the centre of mass observer , the angular velocity is also orthogonal to the plane subtended by the velocity @xmath16 and acceleration @xmath17 of point @xmath7 , and given by @xmath18 in this way , we have also solved the problem of the rotation of the charged rigid body by analyzing the evolution of just the centre of charge . therefore , it will be simpler , from a theoretical point of view , just to describe the evolution of a single point , the centre of charge @xmath7 , instead of the orientation of the rigid body and its centre of mass @xmath5 , which will be in some average position of point @xmath7 , and obtained from ( [ eq : q ] ) once the trajectory of @xmath7 is computed . the elimination of the @xmath19 among equations ( [ eq : cm ] ) and ( [ eq : q ] ) will give us , in general , a fourth order differential equation for the variable @xmath7 . if the rigid body is no longer free , we shall assume that its stiffness means that the relationship between @xmath5 , @xmath20 and @xmath7 and their derivatives given in ( [ eq : q ] ) and ( [ eq : w ] ) still remain valid , even in the presence of an external interaction . therefore , if the centre of mass and centre of charge of a rigid body are different points , it is sufficient to describe the evolution of the centre of charge . then , the rigid body is reduced to a system of only three degrees of freedom which satisfy fourth - order differential equations . we see that a lagrangian depending on the acceleration of point @xmath7 , could reproduce such equations . the following nonrelativistic free lagrangian @xmath21 produces the dynamical equations for the point @xmath7 @xmath22 which can be factorized in terms of a point @xmath5 , defined as in ( [ eq : q ] ) in the form @xmath23 a free motion for the point @xmath5 which can be interpreted as the centre of mass , and a harmonic motion of point @xmath7 around @xmath5 with a constant frequency @xmath24 . it is the presence of the acceleration term in the lagrangian which produces the definition of a point @xmath5 , different from @xmath7 , moving freely . this lagrangian describes a particle with spin @xcite , where its detailed analysis can be found in this reference . let us consider the trajectory @xmath25 , @xmath26 $ ] followed by a point of a mechanical system for an arbitrary inertial observer @xmath27 . any other inertial observer @xmath28 is related to the previous one by a transformation of the spacetime kinematical group such that their relative spacetime measurements of any spacetime event are given by @xmath29 where the functions @xmath30 and @xmath31 define the corresponding spacetime transformation of the kinematical group @xmath32 , of parameters @xmath33 , among any two observers . then the description of the trajectory of that point for observer @xmath28 is obtained from @xmath34.\ ] ] if we eliminate @xmath1 as a function of @xmath35 from the first equation and substitute into the second we shall get @xmath36 since observer @xmath28 is arbitrary , equation ( [ eq : robs ] ) represents the complete set of trajectories of the point for all inertial observers . elimination of the @xmath37 group parameters among the function @xmath38 and their time derivatives will give us the differential equation satisfied by all the trajectories of the point . let us assume that the trajectory is unrestricted in such a way that the above group parameters are essential in the sense that no smaller number of them gives the same family of trajectories . this differential equation is invariant by construction because it is independent of the group parameters and therefore independent of any inertial observer . if @xmath32 is either the galilei or poincar group , it is a ten - parameter group so that we have to work out in general up to the fourth derivative to obtain sufficient equations to eliminate the group parameters . therefore the order of the differential equation is dictated by the number of parameters and the structure of the kinematical group . if the point @xmath7 represents the position of the centre of charge of an elementary particle we get again that it satisfies , in general , a fourth order differential equation . but at the same time it is telling us that to obtain the invariant differential equation satisfied by the centre of charge of an elementary particle , it is sufficient to obtain its trajectory in an arbitrary reference frame , and to follow the above procedure of elimination of the group parameters . a continuous and differentiable curve in three - dimensional space , @xmath39 , is a regular curve if at any point @xmath40 has a tangent vector @xmath41 . it has associated three orthogonal unit vectors , @xmath42 , @xmath11 and @xmath43 , called respectively the tangent , normal and binormal . if using the arc length @xmath40 as the curve parameter , they satisfy the frenet - serret equations @xmath44 where the overdot means the derivative with respect to @xmath40 . the knowledge of the curvature @xmath45 and torsion @xmath46 , together the boundary values @xmath47 , @xmath48 , @xmath49 and @xmath50 , completely determine the curve , because the above equations are integrable . in terms of the vector @xmath51 , known as darboux vector , the frenet - serret equations can be rewritten as @xmath52 so that , in units of arc length , darboux vector represents the instantaneous angular velocity undergone by the local triad frame . by changing the notation @xmath53 , we have @xmath54 and this allows us to eliminate the three unit vectors @xmath42 , @xmath11 and @xmath43 , in terms of the three derivatives @xmath55 , @xmath56 . if we replace them in the next order derivative @xmath57 , one obtains the most general differential equation satisfied by the point @xmath7 , i.e. , the fourth order differential system @xmath58 let us consider that an elementary particle , instead of being a rigid body , is just a localized mechanical system . by localized we mean that , at least , it is described by the evolution of a single point @xmath7 . this point could be the centre of mass , but , as mentioned before , in order to determine the external forces , we also need to know the location of the centre of charge to compute the actions of the external fields . let us assume that the elementary particle is charged . by the previous arguments , if its electric field is spherically symmetric , we are reduced to know the evolution just of the centre of charge . the particle will have a centre of mass but we make the assumption that the centre of mass and the centre of charge are not necessarily the same point . let us consider that the geometrical regular curve @xmath39 in three - dimensional space represents the trajectory of the centre of charge of an elementary particle . when the corresponding inertial observer uses its time as the evolution parameter , kinematics enters into the scene . let us assume now that the motion of the particle is free . this means that we can not distinguish one point of the evolution from another , so that the motion has to be at a constant velocity such that the arc length @xmath59 , where @xmath16 is the velocity of the charge , must be independent of the time @xmath1 . otherwise , if @xmath60 is not the same we can distinguish one instant of the evolution from another , as far as the displacement of the charge is concerned . at the same time , darboux vector has to be also independent of time . the frenet - serret triad moves and rotates in a free motion with constant linear and angular velocities . the curvature and torsion are necessarily constants of the motion . thus @xmath61 , and , in the free case , the equations ( [ eq : masgeneral ] ) are reduced to @xmath62 if the curvature and torsion are constant the curve is a helix , which can be factorized in terms of a central point @xmath63 which is moving in a straight trajectory , while the point @xmath7 satisfies @xmath64 an isotropic harmonic motion of frequency @xmath24 , around point @xmath5 . the point @xmath5 clearly represents the centre of mass position of the free particle and its expression in terms of the centre of charge position is exactly the same as in the case for the rigid body ( [ eq : q ] ) . the centre of charge of a free elementary particle is describing a helix at a constant velocity for any inertial observer . if we make a nonrelativistic analysis , the relationship of the velocity measurements among two arbitrary inertial observers @xmath27 and @xmath28 , is given by @xmath65 , where @xmath66 is the constant velocity of @xmath27 as measured by @xmath28 and the constant rotation matrix @xmath14 is their relative orientation . now , @xmath67 in a relativistic analysis @xmath68 where @xmath66 is also the velocity of observer @xmath27 measured by @xmath28 and @xmath14 represents their relative orientation , and thus @xmath69 taking the time derivative of both expressions ( [ una ] ) and ( [ dos ] ) if @xmath70 has to be also constant for observer @xmath28 , we get that @xmath71 , irrespective of @xmath66 and of the rotation matrix @xmath14 . this means that the vector @xmath72 must be a constant vector . the centre of charge necessarily moves along a straight trajectory at a constant velocity , for every inertial observer , and the above general helix degenerates into a straight line and because the point is not accelerated @xmath73 . this is the usual description of the spinless or pointlike free elementary charged particle , whose centre of charge and centre of mass are represented by the same point . however , in this relativistic analysis , there is one alternative not included in the nonrelativistic approach . the possibility that the charge of an elementary particle will be moving at the speed of light and , in that case , @xmath74 , for any inertial observer . this means that the centre of the helix is always moving at a velocity @xmath75 , and , if it represents the centre of mass , this particle is a massive particle . in a variational description of this system the lagrangian should depend up to the acceleration of the point @xmath7 in order to obtain fourth order differential equations . this dependence on the acceleration will give a contribution to the spin of the particle and there is also another contribution from the rotation of the system , because the body frame rotates with angular velocity @xmath20 . the motion of the charge around the centre of mass produces the magnetic moment of the particle . when we introduce the time as the curve parameter if the trajectory is necessarily a helix at a constant velocity @xmath76 , it is equivalent the use of the arc length @xmath77 or the time as the curve parameter . but if the curve is regular it must have at any point a tangent vector @xmath78 . this implies that the velocity of the charge can never be reached by any inertial observer . otherwise , the velocity of the charge will be zero at a certain time but different from zero for subsequent times , because the point is accelerated , which is contradictory with the requirement that the point describes a trajectory with a constant velocity . this kinematical argument requires that the kinematical group of spacetime symmetries must contain a limit velocity which can not be reached by any inertial observer . among the possible ten - dimensional kinematical groups found by bacry and levy - leblond @xcite , only the de sitter groups @xmath79 and @xmath80 and the poincar group @xmath81 contain such a limit velocity @xmath82 . if the spacetime is flat then only the poincar kinematical description is singled out by this requirement , so that the charge , necessarily , must be moving at this limit velocity . in summary , there are only two possibilities for a free motion of the centre of charge of an elementary particle . one , the charge is moving along a straight line at any constant velocity , the centre of mass is the same point , the system has no magnetic moment and it can be described either in a relativistic or a non - relativistic framework . in the other , the particle has spin and magnetic moment , and the charge moves along a helix at the speed of light . because all known elementary particles , quarks and leptons , are spin @xmath83 particles , and the charged ones have magnetic moment , we are left only with this last possibility if we want to give a more improved description of an elementary particle . here , only the relativistic description is allowed . this is consistent with dirac s theory of the electron , because the eigenvalues of the components of dirac s velocity operator are @xmath84 . this means that dirac s spinor @xmath85 is expressed in terms of the position of the charge @xmath7 , because the external fields @xmath86 are defined and computed at this point . this last possiblity is the description of the centre of charge of a relativistic spinning elementary particle obtained in the kinematical formalism @xcite , and which satisfies dirac s equation when quantized . the classical structure of this dirac particle is analyzed in the next section . in this formalism dirac particles are localized and also orientable mechanical systems . by orientable we mean that we have to attach to the above point @xmath7 , a local cartesian frame to describe its spatial orientation . this frame could be the frenet - serret triad . the rotation of the frame will also contribute to the total spin of the particle . when quantizing the system , the spin @xmath83 is coming from the presence of the orientation variables . otherwise , if there are no orientation variables , no spin @xmath83 structure is described when quantizing the system . this twofold structure of the classical spin has produced a pure kinematical interpretation of the gyromagnetic ratio @xcite . the dependence of the lagrangian on the acceleration is necessary for the particle to have magnetic moment and for the separation between the centre of mass and centre of charge . in the kinematical formalism @xcite , an elementary particle is , by definition , a mechanical system which in addition to being indivisible , as a consequence of the atomic hypothesis @xcite , it can never be deformed so that all allowed states are only kinematical modifications of any one of them . this means that when the state of an elementary particle changes it is possible to find an inertial observer who measures the particle in the same state as before . an electron , if not annihilated , always remains an electron under any external force . this means that in a variational approach the initial @xmath87 and final @xmath88 states of the evolution are related by a transformation of the kinematical group @xmath89 . therefore , the boundary variables of the variational approach , necessarily span a homogeneous space of the kinematical group of space - time symmetries . when quantizing all classical systems characterized by such homogeneous spaces , their hilbert space of pure states carries a projective unitary irreducible representation of the kinematical group @xcite . it thus satisfies wigner s definition of a quantum elementary particle . in this way , the parameters of the kinematical group become the classical variables we need to consider , as the boundary values of the variational formalism for describing an elementary particle . in the relativistic and non - relativistic approach , these variables are reduced to the ten variables @xmath90 and @xmath91 , interpreted respectively as the time , position of the charge , velocity of the charge and orientation . in the relativistic case we have three disjoint , maximal homogeneous spaces of the poincar group spanned by these variables with the constraint either @xmath92 , @xmath93 or @xmath94 . it is the manifold with @xmath93 , as suggested by the kinematical arguments of the previous section , which leads to dirac s equation when quantizing the system . because the lagrangian depends on the next order derivative of the boundary variables , it thus depends also on the acceleration of the point @xmath7 and on the angular velocity . it is thus clear that the point @xmath7 can not be the centre of mass because satisfies fourth - order differential equations . because the external interaction is defined at this point @xmath7 , is why we consider it represents the position of the charge . then , for a dirac particle , the charge located at point @xmath7 , is moving at the speed of light @xmath93 . the classical expression which gives rise to dirac s equation is @xmath95 where the energy @xmath96 is expressed as the sum of two terms , @xmath97 , or translational energy and the other , which depends on the spin of the system , or rotational energy . the spin comes from the dependence of the lagrangian @xmath98 of both , the acceleration @xmath99 , and the angular velocity @xmath100 , and if we define @xmath101 it takes the form @xmath102 the first part @xmath103 , or _ zitterbewegung _ part , is related to the separation between the centre of charge from the centre of mass and takes into account this relative orbital motion . it quantizes with integer values . the second part @xmath104 is the rotational part of the body frame and quantizes with both integer and half - integer values . the total angular momentum with respect to the origin of observer s frame is @xmath105 so that the spin @xmath106 is the angular momentum of the system with respect to the centre of charge @xmath7 , and not with respect to the centre of mass @xmath5 . by this reason , it is not a conserved quantity for a free particle , but satisfies the dynamical equation @xmath107 this is exactly the same dynamical equation satisfied by dirac s spin operator in the quantum case . this has to be taken into account when comparing the analysis of this spin with other approaches , for instance , with bargmann - michel - telegdi spin observable @xcite , which clearly represents the angular momentum with respect to the centre of mass of the system . once a mechanical system has two distinguished points , the centre of charge and centre of mass , we must clarify with respect to which of these points is defined the angular momentum of the system . this is important , for instance , in the so called _ proton spin crisis_. if the spin of the proton is the angular momentum with respect to its centre of mass , and we add the three dirac spin operators of the three quarks we can not obtain the spin of the proton . what we have to add are the three spins of the quarks with respect to their corresponding centre of mass , if assumed that the motion of the quarks is in an @xmath108 orbital angular momentum state . when expressed dirac s spin and the centre of mass position in terms of the velocity and acceleration of the charge they take , respectively , the form @xmath109 dirac s spin is always orthogonal to the osculator plane of the trajectory of the charge @xmath7 , in the direction opposite to the binormal for a positive energy particle , and in the opposite direction for the antiparticle this implies a difference in chirality between matter and antimatter . the acceleration of the charge is pointing from @xmath7 to the centre of mass , as it corresponds to a helix . it is shown that the dynamical equation of point @xmath7 for the free particle and in the centre of mass frame is given by @xmath110 and where the spin vector @xmath106 is constant in this frame , as depicted in fig . [ fig:3 ] . the radius of the zitterbewegung motion is @xmath111 , and the angular velocity @xmath112 . when considered in the centre - of mass frame it is a system of three degrees of freedom ; two are the @xmath113 and @xmath114 components of the position of the charge on the zitterbewegung plane and the third is the phase of the rotation of the body frame . this phase is the same as the phase of the orbital motion and because the velocity @xmath93 is constant , we are just left with a single and independent degree of freedom , for instance , the @xmath113 coordinate . the dirac particle , when considered in the centre of mass frame , is equivalent to a one - dimensional harmonic oscillator of frequency @xmath24 . if we quantize this harmonic oscillator , because the centre of mass is at rest the oscillator is in its ground state of energy @xmath115 . no further excited states are allowed according to the atomic principle @xcite . the classical spin parameter @xmath116 thus becomes @xmath117 , when quantized . motion of the charge of the electron in the centre of mass frame . the magnetic moment of the particle is produced by the motion of the charge . the total spin @xmath106 is half the value of the zitterbewegung part @xmath118 when quantizing the sytem , so that when expressing the magnetic moment in terms of the total spin we get a @xmath119 gyromagnetic ratio @xcite . the body frame attached to the point @xmath7 , which could be frenet - serret triad , rotates with angular velocity @xmath20 , has not been depicted.,width=264 ] when seen from an arbitrary observer ( see figure [ fig:4 ] ) , the motion of the charge is a helix , so that according to ( [ eq : dynspin ] ) dirac s spin precess around the direction of the conserved linear momentum @xmath120 . the spin with respect to the centre of mass is defined as @xmath121 it is a conserved quantity for a free particle . the centre of mass velocity is @xmath122 , and the linear momentum is written as usual as @xmath123 . this means that the transversal motion of the charge is at the velocity @xmath124 . a moving electron takes a time @xmath125 times longer than for an electron at rest to complete a turn , as a result of the time dilation measurement . the faster the centre of mass of the electron moves the slower is the rotation frequency of the centre of charge around the centre of mass . the internal clock of a fast electron is running slower . precession of dirac s spin @xmath106 along the linear momentum @xmath120 . the tranversal motion of the charge takes a time @xmath125 longer than when the centre of mass at rest , to complete a turn . the three vectors @xmath72 , @xmath17 and @xmath126 , properly normalized , form the frenet - serret triad of the motion of the charge . the spin with respect to the centre of mass @xmath127 , is a constant of the motion for the free particle.,width=264 ] 99 rivas m 2001 _ kinematical theory of spinning particles _ ( dordrecht : kluwer ) + see also some other works related to this formalism through the webpage bacry h and levy - leblond j m 1968 _ j. math . phys . _ * 9 * 1605 rivas m , aguirregabiria j m and hernndez a 1999 _ phys . lett . a _ * 257 * 21 rivas m 2008 _ j. phys . a : math . theor . _ * 41 * 304022 ( _ preprint _ arxiv:0709.0192v2 ) rivas m 1994 _ j. math . phys . _ * 35 * 3380 bargmann v , michel l and telegdi v l 1959 _ phys . rev . * 2 * 435 aguirregabiria jm 2006 _ dynamics solver _ , computer program for solving different kinds of dynamical systems , which is available from his author through the web site at the server of the theoretical physics dept . of the university of the basque country , bilbao , spain

in particle physics , most of the classical models consider that the centre of mass and centre of charge of an elementary particle , are the same point . this presumes some particular relationship between the charge and mass distribution , a feature which can not be checked experimentally . in this paper we give three different kinds of arguments suggesting that , if assumed different points , the centre of charge of an elementary spinning particle moves in a helical motion at the speed of light , and it thus satisfies , in general , a fourth order differential equation . if assumed a kind of rigid body structure , it is sufficient the description of the centre of charge to describe also the evolution of the centre of mass and the rotation of the body . this assumption of a separation betwen the centre of mass and centre of charge gives a contribution to the spin of the system and also justifies the existence of a magnetic moment produced by the relative motion of the centre of charge . this corresponds to an improved model of a charged elementary particle , than the point particle case . this means that a lagrangian formalism for describing elementary spinning particles has to depend , at least , up to the acceleration of the position of the charge , to properly obtain fourth order dynamical equations . this result is compared with the description of a classical dirac particle obtained from a general lagrangian formalism for describing spinning particles .

in @xcite , ozsvth , stipsicz , and szab use the heegaard floer knot complex @xmath2 of a knot @xmath3 to define a piecewise linear function @xmath4 with domain @xmath1 $ ] . the function @xmath5 induces a homomorphism from the smooth knot concordance group to the group of functions on the interval @xmath1 $ ] . among its properties , @xmath4 provides bounds on the four - genus , @xmath6 , the three - genus , @xmath7 , and , consequently , the concordance genus , @xmath8 . this note describes one approach to defining @xmath4 and proving its basic properties . thanks go to matt hedden , jen hom , slaven jabuka , swatee naik , shida wang , and c .- m . michael wong for their assistance . the heegaard floer knot complex of a knot @xmath9 , denoted @xmath2 , is a free , finitely generated @xmath10$]module , where @xmath11 is the field of two elements . it has two filtrations , called the _ algebraic _ and _ alexander _ filtrations . it is also graded ; the grading is called the _ maslov _ or _ homological _ grading . the filtrations are compatible with the boundary map , which lowers the maslov grading by one . the action of @xmath12 lowers filtration levels by one and lowers the maslov grading by two . the homology of @xmath2 is isomorphic to @xmath13 $ ] as a module , with @xmath14 $ ] at grading @xmath15 . it follows that @xmath16 is at grading @xmath17 . the minimal algebraic filtration levels of a cycle representing the grading 0 generator is 0 , as is the minimal alexander filtration level . the complex is well - defined up to bifiltered chain homotopy equivalence . since @xmath2 is , as an unfiltered complex , the heegaard floer complex of @xmath18 , we write its homology group as @xmath19 . figure [ fig37 ] is a schematic diagram of the complex for the torus knot @xmath20 . the complex has nine filtered generators , with algebraic and alexander filtration levels indicated by the first and second coordinate , respectively . five of the generators , indicated with black dots , have maslov grading 0 ; the four white dots represent generators of maslov grading one . the boundary map is indicated by the arrows . the rest of @xmath2 consists of the @xmath16 translates of this finite complex ; for instance , applying @xmath12 shifts the diagram one down and to the left . , title="fig : " ] a _ real - valued ( discrete ) filtration _ on a vector space @xmath21 is a collection of subspaces @xmath22 indexed by the real numbers . this collection must satisfy the following properties : 1 . @xmath23 if @xmath24 .. 05 in 2 . @xmath25 .. 05 in 3 . @xmath26 .. 05 in 4 . ( _ discreteness _ ) @xmath27 is finite dimensional when @xmath24 . .1 in given a discrete filtration @xmath22 on @xmath21 , we can define an associated function on @xmath21 , which we temporarily also denote by @xmath28 , given by @xmath29 . notice that @xmath30 ) = c_s$ ] . given an arbitrary real - valued function @xmath31 on @xmath21 , one can define an associated filtration with @xmath32))$ ] . the resulting filtration need not be discrete . .05 in * notation * in cases in which more than one filtration might be under consideration , we will write @xmath33 rather than @xmath34 . .05 in a set of vectors @xmath35 in the real filtered vector space @xmath21 is called a _ filtered basis _ if it is linearly independent and every @xmath34 has some subset of @xmath35 as a basis . for any @xmath38 $ ] , the convex combination of alexander and algebraic filtrations , @xmath39 , defines a real - valued function on @xmath40 , to which we associate a filtration denoted @xmath36 . that is , for all @xmath41 , @xmath42 is spanned by all vectors @xmath43 such that @xmath44 . if @xmath45 , the filtration @xmath36 on @xmath37 is a filtration by subcomplexes and is discrete . to see that these are subcomplexes , suppose that @xmath46 . write @xmath47 where @xmath48 for all @xmath49 . since @xmath50 , we only need to check that for each @xmath49 , @xmath51 . let @xmath52 have @xmath53 and @xmath54 . then @xmath55 and @xmath56 since both @xmath57 and @xmath58 are nonnegative , @xmath59 , as desired . the discreteness of the filtration depends on two properties of @xmath37 . first , for the three - genus @xmath60 , one has @xmath61 for all @xmath62 . from this it follows that for given @xmath63 , there are @xmath64 and @xmath65 such that @xmath66 ( with care , one can show that @xmath67 and @xmath68 . ) second , the alexander filtration is discrete , so the quotient ( @xmath69 is finite dimensional . let @xmath28 be any discrete real filtration on @xmath70 satisfying the additional property that @xmath71 is subcomplex for all @xmath72 . let @xmath73 contains a nontrivial element of grading 0@xmath74 . @xmath75 . consider the knot @xmath76 with @xmath2 as illustrated in figure [ fig37 ] . the portion of the complex shown has homology @xmath11 , at maslov grading 0 . we use the notation @xmath77 to denote the complex @xmath2 with filtration @xmath36 . the set @xmath78 is generated by the bifiltered generators with alexander and algebraic filtration levels satisfying @xmath79 .05 in * observation * the lattice points which contain a filtered generator at filtration level @xmath80 all lie on a line of slope @xmath81 with lattice points parametrized by the pair @xmath82 . alternatively , if a line of slope @xmath83 contain distinct lattice points representing bifiltration levels of generators at the same @xmath36 filtration level , then @xmath84.05 in in the diagram for @xmath20 , the illustrated line in the plane corresponds to @xmath85 and @xmath86 . since the lower half plane bounded by this line contains a generator of @xmath87 of grading level 0 , while no half plane bounded by a parallel line with smaller value of @xmath72 contains such a generator , we have @xmath88 . continuing with @xmath76 , it is now clear that for @xmath89 ( that is , for @xmath90 ) , the least @xmath72 for which @xmath78 contains a generator of @xmath91 corresponds to the line through @xmath92 , which has filtration level @xmath93 . for @xmath94 ( that is , for @xmath95 ) , the least @xmath72 for which @xmath78 contains a generator of @xmath96 corresponds to the line through @xmath97 , which has filtration level @xmath98 . multiplying by @xmath99 and checking the value @xmath100 yields @xmath101 if @xmath102 and @xmath103 are two filtered complexes , there is a natural filtration @xmath104 on @xmath105 , defined via : @xmath106 notice that the direct sum is infinite . the following is basically from @xcite , in which the @xmath107-invariant is shown to be additive . [ addthm ] @xmath108 . the image of the map @xmath109 contains a nontrivial element of grading 0 if and only if for some @xmath110 and @xmath111 with @xmath112 the image of the map @xmath113 contains such an element . by the kunneth formula , this will be the case if and only if the image of @xmath114 \otimes_{\f[u , u^{-1 } ] } \f[u , u^{-1 } ] \cong \f[u , u^{-1 } ] \ ] ] contains such an element . for this to be the case , there must be an @xmath115 such that the image of @xmath116 contains an element of grading @xmath115 and the image of @xmath117 contains an element of grading @xmath118 . if this is the case , then the images of @xmath119 and @xmath120 both contain elements of grading 0 . note that @xmath121 . let @xmath122 and @xmath123 . since the images of @xmath124 and @xmath125 each contain elements of grading 0 , @xmath126 does also , and thus @xmath127 . if @xmath128 , then for some @xmath129 and @xmath111 with @xmath130 , @xmath131 contains an element of grading 0 . but clearly either @xmath132 , in which case @xmath133 would not contain an element of grading 0 , or @xmath134 fails to contain such an element , yielding a contradiction . for each @xmath135 $ ] , @xmath136 according to @xcite , the complex @xmath137 is bifiltered chain homotopy equivalent to @xmath138 . suppose that @xmath139 and @xmath140 are bifiltered generating sets for each . if @xmath141 has bifiltration level @xmath142 and @xmath143 has bifiltration level @xmath144 , then @xmath145 is a bifiltered generator of the tensor product with bifiltration level @xmath146 . the @xmath36 filtration levels of @xmath141 and @xmath143 are @xmath147 and @xmath148 , respectively . the @xmath36 filtration level of the tensor product is the sum of these two . stated in another way , the previous argument shows that @xmath149 is filtered chain homotopy equivalent to @xmath150 . for an arbitrary knot @xmath9 , @xmath151 . the complex @xmath2 with filtration @xmath36 has a dual complex , @xmath152 with ( decreasing ) filtration @xmath153 . the set of dual vectors @xmath154 with @xmath155 are those that vanish on @xmath156 for all @xmath157 . one first proves that @xmath158 can be defined as the maximal filtration level of a class in @xmath152 which represents a nontrivial generator of ( co)homology in grading 0 . the proof is completed by applying the result of @xcite that @xmath159 , where @xmath28 is either the algebraic or alexander filtration . we now present some basic results concerning @xmath4 and its derivative . an initial observation is that @xmath161 and , since @xmath2 is finitely generated , @xmath4 is continuous at 0 . thus , we focus on @xmath162 . we continue to abbreviate @xmath163 . [ mainthm1 ] @xmath164.0 in 1 . for every knot @xmath9 , @xmath165 is a continuous piecewise linear function . .05 in 2 . at a nonsingular point of @xmath160 , the value of @xmath166 is @xmath167 , where @xmath168 is the bifiltration level of some filtered generator of @xmath21 with homological grading 0 . singularities in @xmath160 can occur only at values of @xmath80 such that some line of slope @xmath169 contains at least two lattice points , @xmath168 and @xmath170 , each of which represents the algebraic and alexander gradings of filtered generators of @xmath171 of homological grading @xmath15 . .05 in 4 . if @xmath160 has a singularity at @xmath80 , then the jump in @xmath160 at @xmath80 , denoted @xmath172 , satisfies @xmath173 for some pair @xmath174 for which there are lattice points @xmath168 and @xmath170 as in the previous item . the proof is discussed in terms of the diagram of the complex , as illustrated for the knot @xmath20 in the previous section . suppose @xmath175 and there is precisely one lattice point @xmath168 with @xmath176 which represents the bifiltration level of a filtered generator of @xmath40 . ( this will be the case for all but a finite number of values of @xmath80 . ) for nearby @xmath80 , say @xmath177 , the value of @xmath178 will be such that the same vertex ( at @xmath168 ) lies on the line @xmath179 . that is , for all nearby values of @xmath80 , the value of @xmath72 is given by @xmath180 . written differently , @xmath181 in particular , we see that @xmath4 is piecewise linear off a finite set . now consider a singular value of @xmath80 , at which @xmath182 and there are two or more pairs @xmath168 for which @xmath183 . notice that this line in the @xmath168plane has slope @xmath184 . for @xmath177 close to @xmath80 and @xmath185 , we have @xmath186 for one of those pairs @xmath168 . if @xmath177 is near @xmath80 and @xmath187 , then @xmath188 for another ( or possibly the same ) of these pairs , @xmath170 . notice that these are equal at @xmath80 , giving the continuity of @xmath4 . we now see that a singularity of @xmath4 occurs if @xmath189 . with these observations , the proofs of ( 1 ) , ( 2 ) , and ( 3 ) are complete . for ( 4 ) , our computations have shown that the change in @xmath160 , denoted @xmath190 , is given by @xmath191 for some appropriate @xmath168 and @xmath170 . since both are assumed to lie on a line of slope @xmath192 , we have @xmath193 , so @xmath194 this completes the proof of the theorem . for any knot @xmath9 and for @xmath195 with @xmath196 , @xmath197 where @xmath198 is some integer if @xmath199 is odd , or half - integer if @xmath199 even . by theorem [ mainthm1 ] ( 4 ) , @xmath200 for some pair of integers @xmath49 and @xmath201 , where there are two lattice points on a line of slope @xmath202 thus , we want to constrain the possible differences between the first coordinates of such lattice points . for @xmath203 , @xmath204 . since @xmath196 , in reduced terms , this is either @xmath205 or @xmath206 if @xmath199 is odd or even , respectively . two lattice points on such a line have first coordinates differing by a multiple of @xmath199 or of @xmath207 , if @xmath199 is odd or even , respectively . the completes the proof . for nonsingular points of @xmath160 , @xmath208 . according to @xcite , if @xmath9 is of genus @xmath209 , then all elements of @xmath2 have filtration level @xmath168 where @xmath210 it follows immediately from the second statement of theorem [ mainthm1 ] that @xmath211 . we also observe that the genus of @xmath9 constrains the possible points of singularity of @xmath160 . [ gbound ] suppose that @xmath212 has a singularity at @xmath195 , with @xmath196 . then : * if @xmath199 is odd , @xmath213 .. 05 in * if @xmath199 is even , @xmath214 . suppose that a line of slope @xmath215 , where @xmath216 contains two distinct points of the form @xmath168 with @xmath217 . it follows quickly that the genus bound implies @xmath218 to express this in terms of @xmath80 , suppose @xmath195 with @xmath196 . then @xmath219 if @xmath199 is odd , then @xmath220 . if @xmath199 is even , say @xmath221 , then @xmath222 and @xmath223 , with @xmath224 and @xmath198 relatively prime . in the first case , with @xmath199 odd , we have @xmath225 , so @xmath226 . in the second case , with @xmath199 even , we have @xmath227 , so @xmath228 . if knots @xmath229 and @xmath230 are concordant , then there is an equality of @xmath231invariants : @xmath232 for all @xmath233 and @xmath234 , @xmath235 . here @xmath236 denotes @xmath237 surgery on @xmath9 , @xmath231 is the heegaard floer correction term , and @xmath238 is a spin@xmath239 structure , with @xmath83 given by a specific enumeration of spin@xmath239 structures ; all are described in @xcite . ( in the case that @xmath237 is odd , this range of @xmath83 includes all possible spin@xmath239 structures . ) if @xmath237 is large , then @xmath240 , where @xmath241 is the largest grading of a class @xmath154 in the homology of @xmath242 for which @xmath243 is nontrivial for all @xmath244 , and @xmath245 is some rational function defined on the integers , independent of @xmath9 . in the case that @xmath9 is slice , we see that the maximal grading @xmath246 , where @xmath247 is the unknot . this implies that for a slice knot @xmath9 , @xmath248 . we have a nesting of complexes @xmath249 since @xmath250 is at @xmath36 filtration level @xmath15 , it follows that @xmath251 ; thus @xmath252 . however , @xmath253 is also slice , so @xmath254 . it follows that @xmath255 . an additive invariant of knots that vanishes on slice knots is a concordance invariant . the concordance - genus @xmath8 of a knot @xmath9 , defined in @xcite , is the minimal genus among all knots concordant to @xmath9 . since @xmath4 is a concordance invariant , the genus bounds in section [ secgenus ] apply to the concordance genus . for all nonsingular points of @xmath4 , @xmath256 . the jumps in @xmath257 occur at rational numbers @xmath258 . for @xmath199 odd , @xmath259 . if @xmath199 is even , @xmath260 . let @xmath261 denote the bifiltered subcomplex @xmath262 . we let @xmath263 denote the minimum value of @xmath83 such that the homology of @xmath261 contains a nontrivial grading 0 element of the homology of @xmath264 ( that is , in @xmath265 ) , which we recall is isomorphic to @xmath13 $ ] with 1 at grading level 0 . there is the following result of hom and wu @xcite , built from work of rasmussen @xcite . ( in @xcite the invariant @xmath266 is described ; the equivalence with @xmath267 is presented in @xcite . ) @xmath268 . based on this , we show that @xmath4 provides a bound on @xmath6 . for all @xmath135 $ ] , @xmath269 . since @xmath270 is at @xmath36 filtration level @xmath271 , we have the containment @xmath272 since @xmath273 contains an element of grading 0 in the homology of @xmath2 , so does @xmath274 . thus , @xmath275 . by the previous proposition , @xmath276 . considering @xmath253 , we have @xmath277 ; it follows that @xmath278 . combining these yield @xmath279 multiplying by @xmath99 yields the desired conclusion . here we sketch a proof of proposition 1.10 of @xcite . the argument is essentially the same as used in @xcite to prove the corresponding fact about @xmath280 . let @xmath281 and @xmath282 be knots with identical diagrams , except at one crossing which is either negative or positive , respectively . then for @xmath283 $ ] , @xmath284 first note that @xmath285 can be changed into the slice knot @xmath286 by changing a negative crossing to positive . thus , @xmath287 . it follows that @xmath288 next note that @xmath289 can be changed into the slice knot @xmath290 by changing one negative crossing to positive and one positive crossing to negative . thus , it too has four - genus at most 1 : it bounds a singular disk with two singularities of opposite sign , and these can be tubed together . a simple computation for @xmath291 yields @xmath292 for @xmath293 . thus , @xmath294 which we rewrite as @xmath295 combining equations [ eq1 ] and [ eq2 ] , @xmath296 adding @xmath297 to all terms yields the desired conclusion , @xmath284 * note * this argument can be easily modified to show that if there is a singular concordance from @xmath9 to @xmath298 with a single positive double point , then @xmath299 for small @xmath80 , @xmath4 is determined by the @xmath107 invariant defined in @xcite . we review the definition below . here is the statement of the result . [ tauthm ] for @xmath80 small , @xmath300 . the quotient complex @xmath301 is denoted @xmath302 . it is filtered by the alexander filtration and has homology @xmath11 , supported in grading 0 . the invariant @xmath280 is defined to be the least integer @xmath49 such that the map on homology @xmath303 is surjective . we wish to relate @xmath304 to an invariant of @xmath2 . the needed technical result is the following . [ lemmatau ] if @xmath304 , then there is a cycle @xmath305 representing an element in @xmath306 of grading 0 . consider the following commutative diagram joining short exact sequences . the vertical maps are obtained by quotienting by @xmath307 . @xmath308 .1 in the complex at the top right corner is naturally isomorphic to the cokernel of @xmath309 . considering the associated long exact sequences and using notation that is chosen to reflect that used on the chain level , we have the following commutative diagram . @xmath310 .1 in from the definition of @xmath107 , there is an element @xmath311 that maps under @xmath312 to a generator @xmath313 of @xmath314 . the map @xmath315 is the quotient map of @xmath316$]-modules , @xmath316 \to f[u]/u\f[u ] \cong \f$ ] , and thus is surjective . choose a @xmath317 such that @xmath318 . then @xmath319 , so there is a @xmath320 such that @xmath321 . for @xmath80 small we consider the filtration @xmath36 and the filtration level @xmath322 . then one has @xmath323 . by lemma [ lemmatau ] , this subcomplex contains a cycle that represents an element of grading 0 in @xmath306 . thus , for this @xmath36 filtration , @xmath324 . on the other hand , suppose that @xmath325 . then there would exist a cycle @xmath326 representing a generator of @xmath306 of grading 0 . however , the image of @xmath154 in @xmath327 would be an element in @xmath328 that represents a generator of @xmath327 . but @xmath107 is by definition the lowest level at which this can occur . thus , we see that @xmath329 . to conclude , recall that @xmath330 , so @xmath331 , as desired . * note . * with care , one can check that in this argument , the condition that @xmath80 be small can be made precise by requiring that @xmath332 . of course , once the result is established for some set of small @xmath80 , then theorem [ gbound ] provides the bound @xmath333 . to conclude this note , we explain why @xmath4 as defined here agrees with that of @xcite . in section 3 of @xcite , for @xmath334 , @xmath4 in @xcite is defined as follows . the construction begins with the @xmath335$]module @xmath336 } \f[v^{1/n}]$ ] , where @xmath12 acts on @xmath337 $ ] via multiplication by @xmath338 . observe that there are ( rational ) filtrations @xmath339 and @xmath340 on @xmath341 which are consistent with those on the @xmath316$]submodule @xmath2 . the action of @xmath342 lowers filtration levels by @xmath343 . thus , @xmath344 lowers filtration levels by 2 , as it should . there is a rational grading on @xmath2 defined via the maslov grading , @xmath345 , and alexander filtration . if @xmath62 is an element at filtration level @xmath168 , then : @xmath346 ( in @xcite , only generators at algebraic filtration level 0 are used to define gr@xmath347 , so @xmath348 and the formula @xmath349 is presented . ) one checks that @xmath12 continues to lower gradings by 2 , so on the extension to @xmath341 , @xmath350 lowers gradings by @xmath351 and @xmath342 lowers gradings by @xmath352 . continuing to follow @xcite , if @xmath62 is a filtered generator of @xmath2 with @xmath353 , then the boundary @xmath354 is explicitly defined so that @xmath355 , with the values of @xmath356 given explicitly . this extends naturally to a boundary operator on all of @xmath341 . given that the operator @xmath354 is well - defined , it is a simple matter to determine its value . suppose that @xmath62 is a filtered generator of @xmath2 at filtration level @xmath168 , maslov grading @xmath209 , and suppose also that @xmath353 . let @xmath313 denote one of the terms in this sum , at filtration level @xmath170 , necessarily of grading level @xmath357 . then viewed as an element of @xmath341 , @xmath62 is of grading @xmath358 , and @xmath313 has grading level @xmath359 . in @xmath360 , the term @xmath361 appears , and @xmath362 is such that gr@xmath363 . rewriting this , we have @xmath364 . that is , @xmath365 as two examples , figure [ fig37a ] illustrates the complexes @xmath366 for @xmath76 , with @xmath367 and @xmath368 . the construction is straightforward using equation [ eqn1b ] and the fact that @xmath350 shifts along the diagonal a distance of @xmath369 down and to the left . the portion of the complex illustrated was chosen because its homology is @xmath11 in grading 0 and represents the generator of the homology of @xmath370 in grading 0 . it is apparent from these examples that the alexander filtration is not a filtration of the chain complex , as some arrows increase the alexander filtration level . however , as is easily verified , the algebraic filtration is a filtration on the chain complex . the value of @xmath4 as just defined is equal to @xmath374 , where @xmath72 is the least number for which the homology of @xmath375 contains an element of grading 0 which represents a nontrivial element of the homology of @xmath370 . suppose that using this definition of @xmath4 , we have @xmath182 . this implies that @xmath376 contains a cycle @xmath154 representing a nontrivial generator of grading 0 in the homology of @xmath370 . write @xmath377 , where the @xmath378 are filtered generators . some @xmath378 has filtration level @xmath379 , and none of the @xmath378 has algebraic filtration level greater than @xmath72 . from the refiltration formula given in equation [ eqn1b ] , @xmath380 , we see that generators of @xmath2 at filtration level @xmath168 and grading 0 yield generators of grading 0 in @xmath370 at filtration level @xmath381 . ( recall that shifting down to the left by @xmath80 units decreases the grading level by @xmath382 . ) we are thus led to consider the transformation @xmath383 its inverse is given by @xmath384 under this transformation , for a fixed value of @xmath72 , the vertical line @xmath385 , is carried to the line ( in the @xmath2plane ) @xmath386 relabeling the coordinate system @xmath387 , this is the line @xmath388 comparing with equation [ eqn0 ] , we see that the homology of the filtered complex @xmath389 contains a generator of grading 0 that is nontrivial in the homology of @xmath2 , and that this is not the case for @xmath390 for any @xmath391 . thus , the value of @xmath4 as defined in section [ sectionu ] is @xmath374 , and the definitions agree .

ozsvth , stipsicz , and szab have defined a knot concordance invariant @xmath0 taking values in the group of piecewise linear functions on the closed interval @xmath1 $ ] . this paper presents a description of one approach to defining @xmath0 and of proving its basic properties .

in einstein s theory of general relativity , linearization of the field equations shows that small perturbations of the metric obey a wave equation ( misner , thorne & wheeler 1973 ) . these small disturbances , referred to as gravitational waves , travel at the speed of light . however , some other gravity theories predict a dispersive propagation ( see @xcite for references ) . the most commonly considered form of dispersion supposes that the waves obey a klein gordan type equation : @xmath0 \psi = 0.\ ] ] physically , the dispersive term is ascribed to the quantum of gravitation having a non - zero rest mass @xmath1 , or equivalently a non - infinite compton wavelength @xmath2 . the group velocity of propagation for a wave of frequency @xmath3 is then @xmath4,\ ] ] valid for @xmath5 ; only in the infinite frequency limit is general relativity recovered , with waves traveling at the speed of light @xcite . over the past few decades a number of different _ dynamic _ tests of this dispersive hypothesis have been described , i.e. tests making use of direct observations of gravitational waves or their radiation reaction effects @xcite . in this paper we add another method to this list ; we consider gravitational radiation from _ eccentric _ binary systems . such binaries emit gravitational radiation at ( infinitely many ) harmonics of the orbital frequency @xcite . our idea lies simply in measuring the phase of arrival of these harmonics . dispersion of the form described by equation ( [ eq : vg ] ) would be signaled by the higher harmonics arriving slightly earlier than the lower harmonics , as compared to the general relativistic waveform . we present a rough estimate of the bounds that might be obtained , deferring a more accurate calculation to a future study ( barack & jones , in preparation ) . the plan of this paper is as follows . in [ sect : dotb ] we derive formulae to make a simple estimate of the bounds that might be obtained using our method . in [ sect : r ] we estimate bounds obtainable on @xmath6 for lisa observations of two sorts of binary systems . finally in [ sect : c ] we summarize our findings and compare with those of other authors . to derive a rigorous estimate of the bound one should add the graviton mass to the list of unknown source parameters to be extracted from the measured signal , as was done by will in the case of circular orbits @xcite . the @xmath1dependent waveform can then be computed , allowing calculation of the fisher information matrix @xmath7 , which could then be inverted , the @xmath8 component , evaluated at @xmath9 , giving the best bound obtainable @xcite . for the case of eccentric binaries such a calculation is not easy , and so in this paper we make a preliminary estimate of the possible bounds , without going to the trouble of calculating @xmath10 . we will begin by deriving a general formula for estimating the bound on @xmath6 that could be obtained from a system which produces gravitational waves at two different frequencies , say @xmath11 and @xmath12 . the two gravitational waves will travel with ( different ) speeds @xmath13 and @xmath14 , and so their journey times to the detector a distance @xmath15 away will differ by a time interval @xmath16 given by @xmath17.\ ] ] multiplying this by @xmath18 , where @xmath3 is a characteristic frequency in the problem , gives the accumulated difference in phase of arrival of the two signals caused by the dispersion , measured in terms of radians of phase of @xmath3 : @xmath19.\ ] ] this is to be compared with the accuracy with which the phase of arrival of the waves can be extracted from the noisy gravitational wave data stream . in the high signal to noise ratio regime the error in extracting the phase of a continuous signal can be written as @xmath20 where we follow the notation of @xcite . in this formula @xmath21 is the signal to noise ratio of the measurement and @xmath22 is a dimensionless factor that depends upon how many unknown parameters ( including the phase ) need be extracted from the signal . the lower bound that can be placed on @xmath6 comes from equating @xmath23 and @xmath24 to give : @xmath25.\ ] ] this shows that the best bounds will come from high mass ( i.e. high @xmath26 ) , high eccentricity , low orbital frequency systems . we will now apply this method of estimation to eccentric binary systems . in general many more than two harmonics will contribute significantly to @xmath21 , so equation ( [ eq : lambdageneral ] ) is not directly applicable . in order to take advantage of this spread we will make the following identifications . we will set @xmath21 equal to the total signal to noise of the observation . to identify appropriate frequencies , consider a plot of the signal to noise of the n - th harmonic , @xmath27 , verses gravitational wave frequency @xmath28 . we will set @xmath3 to be the frequency at which this curve peaks , and @xmath29 as the frequencies corresponding to the lower and upper full - width - at - half - maximum . in reality only discrete harmonic frequencies exist , but for the purpose of defining @xmath30 , @xmath31 and @xmath3 , we will treat the curve as continuous , interpolating to find the necessary frequencies . ( a formalism using only discrete frequencies would have introduced spurious step - wise changes in our bounds on @xmath6 as a function of eccentricity ) . identification of a suitable @xmath22 value , which quantifies the error in phase measurement , is more problematic . @xcite consider errors in measuring the phase of a single monochromatic signal of known sky location ; they find that @xmath32 for a large fraction of the possible binary orientations . @xcite examine extreme mass ratio inspirals . they find phase measurement errors which again yield @xmath33 ( see the @xmath34 parameter of their table iii ) . however , even in the dispersionless case of general relativity , the relative phasing of the detected harmonics is non - trivially determined by the source s sky location and the relative orientation of the detector and binary system @xcite . the phase differences we are considering here are _ additional _ delays caused by dispersive propagation . clearly , then , the results of @xcite and @xcite do not directly apply to our problem . only a full fisher matrix calculation will accurately show how we can disentangle the phase differences contributed by dispersion , measurement error and those intrinsic to the binary . we expect that in those situations where the system parameters , including its sky location and orientation relative to lisa , are measured accurately , the dispersion - induced phase delays will be measured accurately too . in the absence of a full fisher matrix calculation to evaluate the correct measurement errors we will set @xmath35 , but note that this is the weakest link in our estimate . in particular , if the various geometric factors that enter the problem conspire such that a dispersionless signal from a certain binary is very similar to the dispersed signal from a binary with slightly different parameters ( e.g. a slightly different sky location ) , then the errors in @xmath36 could be very much larger than estimated here . also , @xmath22 will depend upon the type of system being studied . it will generally be smaller for systems where information in addition to the gravitational wave signal is available , e.g. galactic binary systems where optical measurements give accurate sky locations . note , however , that @xmath22 enters the bound on @xmath6 only rather weakly , as @xmath37 , and so we hope that our ignorance of this factor will not change our qualitative conclusions . it is expected that gravitational radiation reaction will result in most binary systems detectable by ground based interferometers being nearly perfectly circular @xcite and so will be unusable for deriving a bound of the sort described here . we will therefore concentrate exclusively on ( two sorts of ) binaries in the lisa band . in equation ( [ eq : lambdageneral ] ) we will set @xmath35 , as discussed above . when calculating @xmath21 we will assume an integration time of one year . we computed the lisa noise using the online sensitivity curve generator @xcite , which included a fit to the galactic white dwarf background @xcite . we consider here the inspiral of a solar - mass type black hole into a massive one . these are excellent systems from our point of view , as they are expected to dominate the lisa inspiral event rate and , crucially , many will have very large eccentricities @xcite . to see if such systems can indeed be used to obtain a bound on @xmath6 , in figure [ fig1 ] we plot the eccentricity - orbital frequency phase space for a @xmath38 binary at a distance of @xmath39gpc . the upper curve describes the innermost stable orbit ( iso ) @xcite ; binary systems in nature only exist _ below _ this curve . the lower curve gives the minimum eccentricity required for the system to be detectable , with multiple harmonics contributing significantly to @xmath21 . [ our exact criterion is to see if @xmath21 exceeds some detection threshold @xmath40 when the single strongest harmonic is removed from the sum . we have set @xmath41 , as would be reasonable if computational power does not limit the search @xcite ] . our methods are only applicable for systems _ above _ this curve . it follows that we can use binary systems which lie _ between _ these two curves to bound @xmath6 . fortunately we see that this means that binaries in a significant portion of the @xmath42 plane are of use to us . to illustrate this , a trajectory of a plausible lisa source is shown between the two curves , with an eccentricity at the iso of about @xmath43 . this system spends about @xmath44 years between the two curves . in figure [ fig2 ] we show the actual bounds on @xmath6 that could be obtained from observations of extreme mass ratio systems . the distance is still fixed at @xmath39gpc , but now we fix the orbital frequency at @xmath45hz and leave the eccentricity as a free parameter . results for binary systems with @xmath46 and several different values of @xmath47 are given , as indicated . the following features are of note : ( i ) each curve terminates at a minimum eccentricity below which the system is undetectable and/or fewer than two harmonics contribute significantly to @xmath21 , and at a maximum eccentricity above which the system is dynamically unstable . ( ii ) for a system of given masses , the bound increases slightly ( i.e. becomes stronger ) the larger the eccentricity . ( iii ) stronger bounds are obtained from more massive systems , and can be obtained for wider ranges of the eccentricity . lisa will be able to detect gravitational waves from a large number of low mass galactic binaries , consisting of white dwarfs and/or neutron stars @xcite . to investigate the suitability of these systems for bounding @xmath6 , in figure [ fig3 ] we plot the eccentricity frequency phase space for a galactic @xmath48 binary at a distance @xmath39kpc . we set @xmath49 , although a lower value could be used for electromagnetically studied binaries @xcite . we do not show the iso curve as for all plausible eccentricities such a binary would go dynamically unstable in the much higher ligo frequency band . clearly , binaries in a large portion of the phase space are of use for bounding @xmath6 . however , unlike the case of the extreme mass ratio inspiral , there is no compelling reason to expect the eccentricities of these systems to be large . many of them will have gone through a period of mass transfer in the past , which is believed to be an efficient circularizer . nevertheless , as we require merely _ one or more of them _ to have a sufficiently large eccentricity , greater than about @xmath50 , a bound on @xmath6 may well be obtained . in figure [ fig4 ] we present the bounds on @xmath6 that would be obtained from observations of various equal - mass binaries at a distance of @xmath39kpc and with an orbital frequency @xmath45hz . the qualitative form is the same as in figure [ fig2 ] , except we terminate the curves at the high eccentricity end at @xmath51 as such extreme eccentricities seem unlikely . in table [ table : lambdag_dynamic ] we collect together reported and proposed dynamic bounds on @xmath6 that have appeared in the literature , and add two proposed bounds from this work . as is clear from perusal of the table and figures [ fig2 ] and [ fig4 ] , the bounds presented here for low mass galactic systems are comparable to those of @xcite , while our bounds from extreme mass ratio inspirals are comparable to those of @xcite for massive black hole coalescence . it should be remembered that our numbers can only be regarded as estimates , particularly given our rough guess as to the accuracy with which dispersion - induced phase delays can be measured . however , even if @xmath22 , the parameter which quantifies this error , were four orders of magnitude larger than assumed here , our results for extreme mass ratio inspirals would still beat both solar system and galactic low mass binary tests . the method presented here has several advantages over other methods . the analysis of @xcite requires knowledge of the initial relative phases of the x - ray and gravitational wave signals from an accreting white dwarf system ; it is not clear if the accretion process will be sufficiently well understood to allow this . less problematically , the method of @xcite requires knowledge of the phasing of the binary inspiral waveform in the strongly chirping regime , as it is this frequency variation that allows the dispersion test . in contrast , the method described here is very simple , requiring only that multiple harmonics can be detected . it is not necessary for the binary to be chirping significantly , and correlation with other ( i.e. non - gravitational ) radiation is not required . returning to equation ( [ eq : kg ] ) , in the static regime the solution is of the form of a yukawa - type potential , i.e. a newtonian one suppressed by an exponential @xmath52 . this offers the possibility of bounding @xmath6 by looking for departures from newtonian gravity in the non - radiative regime . such results are given in table [ table : lambdag_static ] . @xcite used planetary ephemeris data to obtain their bound , while @xcite cited evidence of gravitational binding of galaxy clusters , suggesting that the exponential suppression is not important over length scales of the order of a mpc . the bounds that could be obtained by using the methods described in this paper would be better than the solar system bounds by around @xmath53 orders of magnitude . however , they are weaker than those from galaxy clusters by @xmath54 orders of magnitude . therefore , if equation ( [ eq : kg ] ) is the correct linearization of the true theory of gravity , and if galaxy clusters are indeed gravitationally bound , the bounds on @xmath55 from the dynamic sector are much weaker than those from the static sector . however , the possibility remains that equation ( [ eq : kg ] ) is not the correct linearization , the static potential is not suppressed , but the wave propagation is nevertheless dispersive , i.e. equation ( [ eq : vg ] ) holds but is not derived from an equation of the form of equation ( [ eq : kg ] ) . the only way of settling this is to use the methods proposed in the dynamic regime . it could even be the case that neither equations ( [ eq : kg ] ) nor ( [ eq : vg ] ) are correct , but that gravitational waves have some other form of dispersion . the method considered here ( or any of the methods referred to in table [ table : lambdag_dynamic ] ) could be used to identify this . having used simple estimates to establish the competitiveness of the method presented here with other dynamic tests , we are currently working to improve the accuracy of our calculation by using the fisher information matrix to calculate the bound ( rather than the methods of [ sect : dotb ] ; barack & jones , in preparation ) . we also aim to extend the scope of the investigation by considering the full range of anticipated gravitational wave sources for both ground and space - based detectors . it is a pleasure to thank leor barack , shane larson , ben owen , steinn sigurdsson and nico yunes for useful discussions during this investigation , and the anonymous referee for providing comments which improved the manuscript . the center for gravitational wave physics is supported by the national science foundation under cooperative agreement phy 01 - 14375 . lll 1 & radio pulsars & @xmath56 + 2 & 4u1820 - 30 & @xmath57 + 3 & @xmath58 , @xmath59 & @xmath57 + 2 & ideal low mass binary & @xmath60 + 4 & ( @xmath61@xmath62 & @xmath63 + 3 & @xmath64 , @xmath65 & @xmath66 + 4 & ( @xmath67@xmath68 & @xmath69

we describe a method by which gravitational wave observations of eccentric binary systems could be used to test general relativity s prediction that gravitational waves are dispersionless . we present our results in terms of the graviton having a non - zero rest mass , or equivalently a non - infinite compton wavelength . we make a rough estimate of the bounds that might be obtained following gravitational wave detections by the space - based lisa interferometer . the bounds we find are comparable to those obtainable from a method proposed by will , and several orders of magnitude stronger than other dynamic ( i.e. gravitational wave based ) tests that have been proposed . the method described here has the advantage over those proposed previously of being simple to apply , as it does not require the inspiral to be in the strong field regime nor correlation with electromagnetic signals . we compare our results with those obtained from static ( i.e. non - gravitational wave based ) tests .

consider , for example , a record @xmath15 , where the index @xmath16 counts the days in the record , @xmath17,2, ... ,@xmath18 . this record @xmath15 may represent the maximum daily temperature or the daily amount of precipitation , measured at a certain meteorological station . for eliminating the periodic seasonal trends , we concentrate on the departures of the @xmath15 , @xmath19 , from their mean daily value @xmath20 for each calendar date @xmath16 , say 1st of april , which has been obtained by averaging over all years in the record . quantitatively , correlations between two @xmath21 values separated by @xmath22 days are defined by the ( auto)-correlation function @xmath23 if the @xmath21 are uncorrelated , @xmath24 is zero for @xmath22 positive . if correlations exist up to a certain number of days @xmath25 , the correlation function will be positive up to @xmath25 and vanish above @xmath25 . a direct calculation of @xmath24 is hindered by the level of noise present in the finite records , and by possible nonstationarities in the data . to reduce the noise we do not calculate @xmath24 directly , but instead study the `` profile '' @xmath26 we can consider the profile @xmath27 as the position of a random walker on a linear chain after @xmath28 steps . the random walker starts at the origin and performs , in the @xmath16th step , a jump of length @xmath21 to the right if @xmath21 is positive , and to the left if @xmath21 is negative . the fluctuations @xmath29 of the profile , in a given time window of size @xmath1 , are related to the correlation function @xmath0 . for the relevant case ( 1 ) of long - term power - law correlations , @xmath30 the mean - square fluctuations @xmath31 , obtained by averaging over many time windows of size @xmath1 ( see below ) asymptotically increase by a power law @xcite,@xmath32 for uncorrelated data ( as well as for correlations decaying faster than @xmath33 ) , we have @xmath34 . for the analysis of the fluctuations , we employ a hierarchy of methods that differ in the way the fluctuations are measured and possible trends are eliminated ( for a detailed description of the methods we refer to kantelhardt et al . @xcite ) . 1 . ) in the simplest type of fluctuation analysis ( fa ) ( where trends are not going to be eliminated ) , we determine the difference of the profile at both ends of each window . the square of this difference represents the square of the fluctuations in each window . 2 . ) in the _ first - order _ detrended fluctuation analysis ( dfa1 ) , we determine in each window the best linear fit of the profile . the variance of the profile from this straight line represents the square of the fluctuations in each window . 3 . ) in general , in the @xmath22th order dfa ( dfan ) we determine in each window the best @xmath22th order polynomial fit of the profile . the variance of the profile from these best @xmath22th - order polynomials represents the square of the fluctuations in each window . by definition , fa does not eliminate trends similar to the hurst method and the conventional power spectral methods @xcite . in contrast , dfan eliminates trends of order @xmath22 in the profile and @xmath35 in the original time series . thus , from the comparison of fluctuation functions @xmath36 obtained from different methods , one can learn about long - term correlations and types of trends , which can not be achieved by the conventional techniques . figure 1 shows the results of the fa and dfa analysis of the maximum daily temperatures @xmath15 of the following weather stations ( the length of the records is written within the parentheses ) : ( a ) cheyenne ( usa , 123 y ) , ( b ) edinburgh ( uk , 102 y ) , ( c ) campbell island ( new zealand , 57 y ) , and ( d ) sonnblick ( austria , 108 y ) . the results are typical for a large number of records that we have analyzed so far ( see eichner et al.@xcite and koscielny - bunde et al.@xcite ) . cheyenne has a continental climate , edinburgh is on a coastline , campbell island is a small island in the pacific ocean , and the weather station of sonnblick is on top of a mountain . in the log - log plots , all curves are ( except at small @xmath1 values ) approximately straight lines . for both the stations inside the continents and along coastlines , the slope is @xmath37 . there exists a natural crossover ( above the dfa crossover ) that can be best estimated from fa and dfa1 . as can be verified easily , the crossover occurs roughly at @xmath38 , which is the order of magnitude for a typical grosswetterlage . above @xmath39 , there exists long - range persistence expressed by the power - law decay of the correlation function with an exponent @xmath40 . these results are representative for the large number of records we have analyzed . they indicate that the exponent is `` universal '' , i.e. , does not depend on the location and the climatic zone of the weather station . below @xmath39 , the fluctuation functions do not show universal behavior and reflect the different climatic zones . however , there are exceptions from the universal behavior , and these occur for locations on small islands and on top of large mountains . in the first case , the exponent can be considerably larger , @xmath41 , corresponding to @xmath4 . in the second case , on top of a mountain , the exponent can be smaller , @xmath42 , corresponding to @xmath43 . next we consider precipitation records . figure 2 shows the results of the fa and dfa analysis of the daily precipitation @xmath44 of the following weather stations ( the length of the records is written within the parentheses ) : ( a ) cheyenne ( usa , 117 y ) , ( b ) edinburgh ( uk , 102 y ) , ( c ) campbell island ( new zealand , 57 y ) , and ( d ) sonnblick ( austria , 108 y ) . the results are typical and represent a large number of records that we have analyzed so far @xcite ) . in the log - log plots , all curves are ( except at small @xmath1 values ) approximately straight lines at large times , with a slope close to 0.5 . if there exist long - term correlations , then they are very small . some exceptions are again stations on top of a mountain , where the exponent might be around 0.6 , but this happens only very rarely . in most cases , the exponent is between 0.5 and 0.55 , pointing to uncorrelated or weakly correlated behavior at large time spans . unlike the temperature records , the exponents actually do not depend on specific climatic or geographic conditions . figure 3 summarizes the results for exponents @xmath45 for ( a ) temperature records and ( b ) precipitation records . different climatological conditions are marked in the histograms . first we concentrate on the temperature records ( fig . one can see clearly that for stations that are neither on islands nor on summits , the average exponent is close 0.65 , with a variance of 0.03 . for the islands ( where only few records are available ) the average value of @xmath45 is 0.78 , with quite a large variance of 0.08 . the variance is large , since stations on larger islands , like wrangelija , behave more like continental stations , with an exponent close to 0.65 . for the precipitation records ( fig . 3b ) , the average exponent @xmath45 is close to 0.54 , with a variance close to 0.05 , and does not depend significantly on the climatic conditions around a weather station . since for the temperature records the exponent for continental and coastline stations does not depend on the location of the meteorological station and its local environment , the power - law behavior can serve as an ideal test for climate models where regional details can not be incorporated and therefore regional phenomena like urban warming can not be accounted for . the power - law behavior seems to be a global phenomenon and therefore should also show up in the simulated data of the global climate models ( gcm ) . the state - of - the - art climate models that are used to estimate future climate are coupled atmosphere - ocean general circulation models ( aogcms ) @xcite . the models provide numerical solutions of the navier stokes equations devised for simulating meso - scale to large - scale atmospheric and oceanic dynamics . in addition to the explicitly resolved scales of motions , the models also contain parameterization schemes representing the so - called subgrid - scale processes , such as radiative transfer , turbulent mixing , boundary layer processes , cumulus convection , precipitation , and gravity wave drag . a radiative transfer scheme , for example , is necessary for simulating the role of various greenhouse gases such as co@xmath46 and the effect of aerosol particles . the differences among the models usually lie in the selection of the numerical methods employed , the choice of the spatial resolution@xmath47 , and the subgrid - scale parameters . three scenarios have been studied by the models , and the results are available , for four models , from the ipcc data distribution center @xcite . the first scenario represents a control run where the co@xmath46 content is kept fixed . in the second scenario , one considers only the effect of greenhouse gas forcing ( ghg ) . the amount of greenhouse gases is taken from the observations until 1990 and then increased at a rate of 1% per year . in the third scenario , the effect of aerosols ( mainly sulfates ) in the atmosphere is taken into account . only direct sulfate forcing is considered ; until 1990 , the sulfate concentrations are taken from historical measurements , and are increased linearly afterwards . the effect of sulfates is to mitigate and partially offset the greenhouse gas warming . although this scenario represents an important step towards comprehensive climate simulation , it introduces new uncertainties - regarding the distributions of natural and anthropogenic aerosols and , in particular , regarding indirect effects on the radiation balance through cloud - cover modification etc . @xcite . ) , greenhouse gas forcing only ( @xmath48 ) and greenhouse gas plus aerosols ( @xmath49 ) . all curves are obtained by applying dfa3 to the monthly mean of the daily maximum temperatures generated by the four aogcms . the lines with slopes 0.65 and 0.5 are shown as a guide to the eye . for details of the records , we refer to @xcite.,height=672 ] for the test , we consider the monthly temperature records from those four aogcms . data for these three scenarios are available from the internet : csiro - mk2 ( melbourne ) , ccsr / nies ( tokyo ) , echam4/opyc3 ( hamburg ) , and cgcm1 ( victoria , canada ) . we extracted the data for six representative sites around the globe ( prague , kasan , seoul , luling [ texas ] , vancouver , and melbourne ) . for each model and each of the three scenarios , we selected the temperature records of the four grid points closest to each site , and bilinearly interpolated the data to the location of the site . figure 4 shows representative results of the fluctuation functions , calculated using dfa3 , for two sites ( kasan [ russia ] and luling [ texas ] ) for the four models and the three scenarios . as seen in figure 4 most of the dfa curves approach the slope of 0.5 . however , the control runs seem to show a somewhat better performance , i.e. , many of them have a slope close to 0.65 ( e.g. , luling ( csiro - mk2 ) ) , and the greenhouse gas only scenario show the worst performance . the actual long - term exponents @xmath45 for the three scenarios of the 4 models for the 6 cities are summarized in figure 5(a)-(c ) . each histogram consists of 24 blocks and every block is specified by the model and the city . ) obtained from the simulations of the four aogcms ( listed in ( a ) ) , for six sites ( listed in ( b ) ) . the three panels are for the three scenarios : ( a ) control run , ( b ) greenhouse gas forcing only , and ( c ) greenhouse gas plus aerosol forcing . the entries in each box represent `` model - site''.,scaledwidth=100.0% ] for the control run ( fig . 5(a ) ) there is a peak at @xmath37 but more than half of the exponents are below @xmath50 . for the greenhouse gas only scenario ( fig . 5(b ) ) , the histogram shows a pronounced maximum at @xmath51 . for best performance , all models should have exponents @xmath45 close to 0.65 , corresponding to a peak of height 24 in the window between 0.62 and 0.68 . actually , more than half of the exponents are close to 0.5 , while only 3 exponents are in the proper window between 0.62 and 0.68 . figure 5(c ) shows the histogram for the greenhouse gas plus aerosol scenario , where , in addition to the greenhouse gas forcing , also the effects of aerosols are taken into account . for this case , there is a pronounced maximum in the @xmath45 window between 0.56 and 0.62 ( more than half of the exponents are in this window ) , while again only 3 exponents are in the proper range between 0.62 and 0.68 . this shows that although the greenhouse gas plus aerosol scenario is also far from reproducing the scaling behavior of the real data , its overall performance is better than the performance of the greenhouse gas scenario . the best performance is observed for the control run , which points to remarkable deficiencies in the way the forcings are introduced into the models . the long - term correlations in a record @xmath52 effect strongly the statistics of the extreme values in the record , as has been shown recently in [ 2 ] . the central quantity in extreme value statistics ( evs ) is the return time @xmath53 between two events of size greater or equal to a certain threshold @xmath54 . the basic assumption in conventional evs is , that the events are uncorrelated ( at least when the time lag between them is sufficiently large ) . in this case , one can obtain the mean return time @xmath55 simply from the probability @xmath56 that an event greater or equal to @xmath54 occurs , @xmath57 . since the events are uncorrelated , also the return intervals are uncorrelated and follow the poisson - statistics ; i.e. their distribution function @xmath58 is a simple exponential , @xmath59 . + for long - term correlations , it has been shown in [ 2 ] that the distribution function @xmath58 changes into a _ stretched _ exponential function , @xmath60 , \eqno(4)\ ] ] for @xmath3 between zero and one . for @xmath3 above one , in the case of short - term correlations , @xmath61 reduces to the poisson distribution . for the record of return intervals obtained from two artificial long - term correlated records @xmath62 with @xmath63 ( upper curves ) and @xmath64 ( lower curves ) . the distribution of the @xmath15 values has been chosen as gaussian , with zero mean and variance one . for the return intervals , the thresholds @xmath65 and @xmath66 have been considered . the straight lines in the figure have slopes @xmath67 , suggesting that the return intervals are long - term correlated in the same way as the @xmath15,height=288 ] in addition , the return intervals become long - term correlated , with an exponent that is approximately identical to @xmath3 [ 2 ] . this is seen in fig . 6 , where the fluctuation functions @xmath36 of the return intervals ( obtained by dfa2 ) are shown for two artificial long - term correlated records with @xmath63 and 0.7 . two values of thresholds @xmath65 and @xmath66 have been considered for each value of @xmath3 . the distribution of @xmath15 values has been chosen as gaussian , with zero mean and variance one . in the double logarithmic plot , all the curves approach straight lines with slopes @xmath67 , suggesting that the return intervals are long - term correlated in the same way as the @xmath15 . + as a consequence , small return intervals are more likely to be followed by small intervals and large intervals are more likely to be followed by large intervals . accordingly , for long - term correlated records it is more likely than for uncorrelated records that a sequence of large return times is followed by a sequence of short return times . + this fact may be relevant for the occurence of floods . it is well known that river flows are long - term correlated with exponents @xmath3 between 0.3 and 0.9 , in most cases close to 0.4 . in the last decades , the frequency of large floods in europe has increased . it is possible , that this increase is due to global warming , but it is also possible that it has been triggered by the long - term correlations . we are grateful to prof . schellnhuber , dr . j.w . kantelhardt , and prof . s. brenner for very useful discussions . we like to acknowledge financial support by the deutsche forschungsgemeinschaft and the israel science foundation . eichner , j. , bunde , a. , havlin , s. , koscielny - bunde , e. , schellnhuber , h.j . `` power - law persistence and trends in the atmosphere : a detailed study of long temperature records '' . e ( 2003 ) submitted . govindan , r. b. , vjushin , d. , brenner , s. , bunde , a. , havlin , s. , schellnhuber , h .- j . `` global climate models violate scaling of the observed atmospheric variability '' . ( 2002 ) : 028501 . govindan , r. b. , vjushin , d. , brenner , s. , bunde , a. , havlin s. , h .- j . schellnhuber h .- j . `` long - range correlations and trends in global climate models : comparison with real data '' . physica a 294 ( 2001 ) : 239 ; vjushin , d. , govindan , r. b. , brenner , s. , bunde , a. , havlin , s. , schellnhuber , h .- j . `` lack of scaling in global climate models '' . c 14 ( 2002 ) : 2275 . hasselmann , k. `` multi - pattern fingerprint method for detection and attribution of climate change '' , multi - fingerprint detection and attribution analysis of greenhouse gas , greenhouse gas - plus - aerosol and solar forced climate change . climate dynamics 13 ( 1997 ) : 601 - 634 and references therein . houghton , j.t . ( editor ) . `` climate change 2001 : the scientific basis , contribution of working group i to the third assessment report of the intergovernmental panel on climate change ( ipcc ) '' . cambridge : cambridge university press , 2001 . intergovernmental panel on climate change . _ the regional impacts of climate change . an assessment of vulnerability _ , edited by r. t. watson , m. c. zinyowera , and r. h. moss . cambridge : cambridge university press , 1998 . koscielny - bunde , e. , bunde , a. , havlin , s. , roman , h. e. , goldreich , y. , schellnhuber , h .- j . `` indication of a universal persistence law governing atmospheric variability '' . ( 1998 ) : 729 - 732 . pelletier , j.d . , turcotte , d.l . `` long - range persistence in climatological and hydrological time series : analysis , modeling and application to drought hazard assessment '' . j. hydrol . 203 ( 1997 ) : 198 - 208 .

we review recent results on the appearance of long - term persistence in climatic records and their relevance for the evaluation of global climate models and rare events . the persistence can be characterized , for example , by the correlation @xmath0 of temperature variations separated by @xmath1 days . we show that , contrary to previous expectations , @xmath0 decays for large @xmath1 as a power law , @xmath2 . for continental stations , the exponent @xmath3 is always close to 0.7 , while for stations on islands @xmath4 . in contrast to the temperature fluctuations , the fluctuations of the rainfall usually can not be characterized by long - term power - law correlations but rather by pronounced short - term correlations . the universal persistence law for the temperature fluctuations on continental stations represents an ideal ( and uncomfortable ) test - bed for the state of - the - art global climate models and allows us to evaluate their performance . in addition , the presence of long - term correlations leads to a novel approach for evaluating the statistics of rare events . in : `` nonextensive entropy - interdisciplinary applications '' , edited by m. gell - mann and c. tsallis , new york oxford university press , 2003 . power - law persistence in the atmosphere : + analysis and applications armin bunde@xmath5 , jan eichner@xmath5 , rathinaswamy govindan@xmath6 , shlomo havlin@xmath7 , + eva koscielny - bunde@xmath8 , diego rybski@xmath6 and dmitry vjushin@xmath7 _ @xmath5institut fr theoretische physik iii , universitt giessen , d-35392 giessen , germany _ _ @xmath7minerva center and department of physics , bar - ilan university , israel _ _ @xmath9potsdam institute for climate research , d-14412 potsdam , germany _ ( submitted : 20 december 2002 , revised : 31 january 2003 ) [ [ section ] ] the persistence of weather states on short terms is a well - known phenomenon : a warm day is more likely to be followed by a warm day than by a cold day and vice versa . the trivial forecast that the weather of tomorrow is the same as the weather of today was , in previous times , often used as a `` minimum skill '' forecast for assessing the usefulness of short - term weather forecasts . the typical time scale for weather changes is about one week , a time period which corresponds to the average duration of so - called general weather regimes " or `` grosswetterlagen '' , so this type of short - term persistence usually stops after about one week . on larger scales , other types of persistence occur , one of them is related to circulation patterns associated with blocking @xcite . a blocking situation occurs when a very stable high - pressure system is established over a particular region and remains in place for several weeks . as a result , the weather in the region of the high remains fairly persistent throughout this period . furthermore , transient low - pressure systems are deflected around the blocking high so that the region downstream of the high experiences a larger than usual number of storms . on even longer terms , a source for weather persistence might be slowly varying external ( boundary ) , forcing such as sea surface temperatures and anomaly patterns for example . on the scale of months to seasons , one of the most pronounced phenomenon is the el ni@xmath10o southern oscillation ( enso ) event , which occurs every three to five years and which strongly affects the weather over the tropical pacific as well as over north america @xcite . the question is , _ how _ the persistence , that might be generated by very different mechanisms on different time scales , decays with time @xmath1 . the answer to this question is not simple . correlations , and in particular long - term correlations , can be masked by trends that are generated , for instance , by the well - known urban warming phenomenon . even uncorrelated data in the presence of long - term trends may look like correlated data , and , on the other hand , long - term correlated data may look like uncorrelated data influenced by a trend . therefore , in order to distinguish between trends and correlations , one needs methods that can systematically eliminate trends . those methods are available now : both wavelet techniques ( wt)@xmath11 and detrended fluctuation analysis ( dfa)@xmath12 can systematically eliminate trends in the data and thus reveal intrinsic dynamical properties such as distributions , scaling and long - range correlations that are often masked by nonstationarities . in recent studies @xcite we have used dfa and wt to study temperature and precipitation correlations in different climatic zones on the globe . the results indicate that the temperature variations are long - range power - law correlated above some crossover time that is of the order of 10 days . above 10 d , the persistence , characterized by the autocorrelation @xmath0 of temperature variations separated by @xmath1 days , decays as @xmath13 where , most interestingly , the exponent @xmath3 has roughly the same value @xmath14 for all continental records . for small islands the correlations are more pronounced , with @xmath3 around 0.4 . this value is close to the value obtained recently for correlations of sea - surface temperatures @xcite . in marked contrast , for most stations the precipitation records do not show indications of long - range temporal correlations on scales above 6 months . our results are supported by independent analysis by several groups @xcite . the fact that the correlation exponent varies only very little for the continental atmospheric temperatures , presents an ideal test - bed for the performance of the global climate models , as we will show below . we present an analysis of the two standard scenarios ( greenhouse gas forcing only and greenhouse gas plus aerosols forcing ) together with the analysis of a control run . our analysis points to clear deficiencies of the models . for further discussions we refer to govindan et al . @xcite . finally , we review a recent approach to determine the statistics of rare events in the presence of long - term correlations . the chapter is organized in five sections . in section 2 , we describe one of the detrending analysis methods , the detrended fluctuation analysis ( dfa ) . in section 3 , we review the application of this method to both atmospheric temperature and precipitation records . in section 4 , we describe how the `` universal '' persistence law for the atmospheric temperature fluctuations on continental stations can be used to test the three scenarios of the state - of - the - art climate models . in section 5 , finally , we describe how the common extreme value statistics is modified in the presence of long - term correlations .

the decays @xmath14 and @xmath15 provide interesting tests @xcite of chiral perturbation theory , chpt . the dominant contribution to the @xmath14 decay is @xmath16 and can therefore be computed with reasonable accuracy in chpt . the @xmath16 term vanishes for @xmath15 in the su(3 ) limit . however large @xmath17 contributions mediated by pseudoscalar mesons @xcite are expected for @xmath18 with values depending on the amount of singlet - octet mixing @xcite . a precise measurement of the @xmath15 decay rate is also of interest in connection with the @xmath19 decay . in fact the absorptive part of the decay rate , @xmath20 , is proportional to @xmath21 . this constrains the dispersive part , @xmath22 and eventually the possibility of determining the v@xmath23 parameter of the ckm matrix @xcite . measurements of @xmath24 and @xmath25 have been recently published by the na48 collaboration @xcite . we describe a new measurement of @xmath26 obtained with @xmath27-mesons from @xmath28 decays at da@xmath3ne , the frascati @xmath4factory . in da@xmath3ne the electron and positron beams have energy @xmath29 where @xmath30 = 12.5 mrad is half of the beam crossing angle . @xmath2-mesons are produced with a cross section of @xmath31 3 @xmath32b and a momentum of 12.5 mev / c toward the center of the rings . the center of mass energy , @xmath33 , the position of the beam crossing point ( @xmath34 ) and the @xmath2 momentum are determined by measuring bhabha scattering events . in a typical run of integrated luminosity @xmath35dt @xmath31 100 nb@xmath36 , lasting about 30 minutes , we have @xmath37 kev , @xmath38 kev / c , @xmath39 m , and @xmath40 m . the detector consists of a large cylindrical drift chamber , dc @xcite , surrounded by a lead - scintillating fiber sampling calorimeter , emc @xcite , both immersed in a solenoidal magnetic field of 0.52 t with the axis parallel to the beams . the dc tracking volume extends from 28.5 to 190.5 cm in radius and is 340 cm in length . for charged particles the transverse momentum resolution is @xmath41 and vertices are reconstructed with a spatial resolution of @xmath31 3 mm . the calorimeter is divided into a barrel and two endcaps and covers 98@xmath42 of the solid angle . photon energies and arrival times are measured with resolutions @xmath43 and @xmath44 respectively . the photon entry points are determined with an accuracy @xmath45 along the fibers , and @xmath46 cm in the transverse directions . a photon is defined as a calorimeter cluster not associated to a charged particle , by requiring that the distance along the fibers between the cluster centroid and the impact point of the nearest extrapolated track be greater than 3@xmath47 . two small calorimeters , qcal @xcite , made with lead and scintillating tiles are wrapped around the low - beta quadrupoles to complete the hermeticity . the trigger @xcite uses information from both the calorimeter and the drift chamber . the emc trigger requires two local energy deposits above threshold ( @xmath48 mev in the barrel , @xmath49 mev in the endcaps ) . recognition and rejection of cosmic - ray events is also performed at the trigger level , checking for the presence of two energy deposits above 30 mev in the outermost calorimeter plane . the dc trigger is based on the multiplicity and topology of the hits in the drift cells . the trigger has a large time spread with respect to the beam crossing time . it is however synchronized with the machine radio frequency divided by four , @xmath50 = 10.85 ns , with an accuracy of 50 ps . during the period of data taking the bunch crossing period at da@xmath3ne was @xmath51 = 5.43 ns . the @xmath52 of the bunch crossing producing an event is determined offline during the event reconstruction . the @xmath2-meson decays into @xmath53 @xmath54 of the time . the production of a @xmath27 is tagged by the observation of a @xmath55 decay . @xmath15 and @xmath56 decay vertices are reconstructed along the direction opposite to the @xmath57 in the @xmath2 rest frame and required to be inside a given fiducial volume , @xmath58 . we call @xmath59 the ratio of interest . the numerators and denominators are found from : @xmath60 where @xmath61 and @xmath62 are the numbers of observed events and estimated background , @xmath63 , @xmath64 , @xmath65 and @xmath66 are respectively the trigger efficiency , the tagging efficiency , the acceptance in the fiducial volume and the selection efficiency for the two decays . the efficiencies @xmath64 and @xmath63 are equal at the few per mil level and cancel in the ratio @xmath67 . background and selection efficiencies must be separately determined . for this analysis the drift chamber is used to measure the @xmath68 decay and to determine the direction of the @xmath27 , the calorimeter is used to measure the photon energies and impact points and to reconstruct the @xmath27 decay vertex by time of flight . the data sample was collected during 2001 and 2002 for an integrated luminosity of @xmath31 362 pb@xmath36 corresponding to the production of @xmath69 @xmath2 . details of the analysis can be found in reference @xcite . @xmath15 events have a very clear signature , being the only source of @xmath31 250 mev photon pairs that balance the momentum of the observed @xmath57 . this allows the use of very loose selection criteria . on the other hand @xmath56 , the dominant neutral decay , is characterised by a large multiplicity of lower energy photons . the final error on @xmath67 is dominated by the error on the number of @xmath15 events . before full event reconstruction , the data are passed through a filter to reject machine background and cosmic ray events . as discussed later , this filter has a modest impact on the events of interest for this analysis . @xmath55 decays are selected with the following requirements : * two tracks with opposite charge that form a vertex with cylindrical coordinates @xmath70 cm , @xmath71 cm , and no other tracks connected to the vertex ; * @xmath72 momentum 100 mev / c @xmath73 120 mev in the @xmath2 rest system , and @xmath74 invariant mass 490 mev @xmath75 505 mev . the @xmath7 decay provides an unbiased tag for the @xmath27 when it decays into neutral particles and a good measurement of the @xmath27 momentum , @xmath76 , where @xmath77 is the central value of the @xmath78 momentum determined with bhabha scattering events . the angular resolution on the @xmath27 direction is determined from @xmath79 events by measuring the angle between @xmath80 and the line joining the @xmath2 vertex and the @xmath74 reconstructed vertex . the widths of the angular distributions are @xmath81 , @xmath82 . the position of the @xmath6 vertex for @xmath15 and @xmath56 decays is measured using the photon arrival times on the emc . each photon defines a time of flight triangle shown in fig . the three sides are the @xmath27 decay length , @xmath83 ; the distance from the decay vertex to the calorimeter cluster centroid , @xmath84 ; and the distance from the cluster to the @xmath2 vertex , @xmath85 . the equations to determine the unknowns @xmath86 and @xmath84 are : @xmath87 where @xmath88 is the photon arrival time on the emc , @xmath89 is the @xmath27 velocity and @xmath30 is the angle between @xmath90 and @xmath91 . only one of the two solutions is kinematically correct . the value of @xmath83 is obtained from the energy weighted average , @xmath92 where _ i _ is the photon index . the accuracy of this method is checked by comparing in the @xmath79 decays the position of the @xmath27 decay vertex measured both by timing with the calorimeter and , with a much better precision , by tracking with the drift chamber . the resolution function , @xmath93 , is determined with @xmath56 and @xmath9 events by the distribution of the residuals @xmath94 where @xmath83 is the average obtained with all the photons but the @xmath95 . we measure for the @xmath56 sample @xmath96 ( cm ) and for the @xmath15 sample @xmath97 ( cm ) . the @xmath27 fv is defined in cylindrical coordinates as 30 cm @xmath98 170 cm , @xmath99 140 cm . the fraction of @xmath6-mesons decaying in the fv is @xmath100 . the identification of the bunch crossing that originated the event is crucial to locate the vertex in space . an error by one bunch crossing period results in a displacement of the @xmath6 vertex of about 33 cm and decreases the probability of correctly associating the photon clusters . the bunch crossing is determined by identifying one of the two pions of the @xmath72 decay and by measuring its track length , momentum and time of flight . thus an error of one ( or more ) crossing periods can occur if there is an incorrect track - to - cluster association or the track parameters are poorly measured . to minimise the number of events with an incorrect bunch - crossing assignment , we perform a consistency check of the time of flight of the pions along their trajectory @xmath101 measured with the dc , @xmath102 with the corresponding cluster time measured by the calorimeter , @xmath103 . requiring @xmath104 ns for at least one pion , the probability of correctly identifying the bunch crossing is @xmath105 . this additional cut retains @xmath106 of the original @xmath107 event sample . the probability of identifying the correct bunch crossing was measured with a sample of @xmath79 decays where the position of the @xmath108 vertex , @xmath109 , is reconstructed by tracking in the dc and the position of the two - photon vertex , @xmath110 , by timing with the emc . the difference @xmath111 is used to isolate the events in which the bunch crossing is incorrectly determined . the @xmath56 decay has a large branching fraction , 21% , and thus has very small background . given the large statistics we retain only 1 out of 10 decays . the selection of @xmath56 events requires at least three calorimeter clusters with the following properties the main sources of inefficiencies are : 1 ) geometrical acceptance ; 2 ) cluster energy threshold ; 3 ) merging of clusters ; 4 ) accidental association to a charged track ; 5 ) dalitz decay of one or more @xmath113 s . the effect of these inefficiencies is to modify the relative population for events with 3 , 4 , 5 , 6 , 7 and @xmath114 , clusters without significant loss of efficiency . monte carlo simulation shows that the selection efficiency is @xmath115 . a comparison between data and monte carlo of the relative populations and of the distribution of the total energy , @xmath116 , shows that only events with 3 and 4 clusters are contaminated by background . this is due to @xmath79 decays where one or two charged pions produce a cluster not associated to a track and neither track is associated to the @xmath6 vertex or to @xmath117 decays , possibly in coincidence with machine background particles ( @xmath118 or @xmath119 ) that shower in the qcal and generate soft neutral particles . to reduce this background , for the 3-cluster population we further require at least two clusters in the barrel with at least one of them with energy @xmath120 mev and for the 4-cluster population at least one cluster in the barrel with energy @xmath48 mev . the probability to have a cluster with @xmath121 mev has been evaluated using the 6-cluster events . the probability of having a given number of clusters in the barrel depends only on geometry and has been evaluated by monte carlo simulation . we obtain @xmath122 and @xmath123 . additionally , an event with 3 clusters is accepted only if an additional cluster is found in qcal within a time window of 10 ns with respect to the @xmath27 decay time . the probability of such an occurrence is @xmath124 . the @xmath79 background is rejected by imposing a veto ( _ track veto _ ) on the events with charged tracks not associated to the @xmath57 decay and with the first hit in the drift chamber at a distance of less than 30 cm from the position of the @xmath27 vertex . the track veto also rejects about @xmath125 of the @xmath56 events with dalitz decays [ fig2 ] shows the distribution of the total energy for events with different numbers of clusters together with the results of the monte carlo simulation . the relative fraction of events is shown in table [ table1 ] . the difference between data and monte carlo simulation for events with @xmath126 5 clusters is due to split clusters . the contamination from accidental clusters originated by machine background is negligible . the residual background contamination in events with 3 and 4 clusters is evaluated by monte carlo simulation and amounts to @xmath127 and @xmath128 respectively . + using the known value for the @xmath56 branching fraction , we obtain @xmath12 where the first error represents the statistical and systematic error on r combined in quadrature and the second is due to the uncertainty in the @xmath13 branching fraction . a decay width of @xmath132 ev is in agreement with @xmath17 predictions of chpt provided the value of the pseudoscalar mixing angle is close to our recent measurement of @xmath133 @xcite . we want to thank our technical staff : g.f.fortugno for his dedicated work to ensure an efficient operation of the kloe computing center ; m.anelli for his continuous support to the gas system and the safety of the detector ; a.balla , m.gatta , g.corradi and g.papalino for the maintenance of the electronics ; m.santoni , g.paoluzzi and r.rosellini for the general support to the detector ; c.pinto ( bari ) , c.pinto ( lecce ) , c.piscitelli and a.rossi for their help during shutdown periods . this work was supported in part by doe grant de - fg-02 - 97er41027 ; by eurodaphne , contract fmrx - ct98 - 0169 ; by the german federal ministry of education and research ( bmbf ) contract 06-ka-957 ; by graduiertenkolleg ` h.e . phys . and part . astrophys . ' of deutsche forschungsgemeinschaft , contract no . gk 742 ; by intas , contracts 96 - 624 , 99 - 37 ; and by tari , contract hpri - ct-1999 - 00088 . 9 g. dambrosio , g. ecker , g. isidori and h. neufeld , `` radiative non - leptonic kaon decays '' , in the second da@xmath3ne physics handbook , eds . l. maiani , g. pancheri and n. paver , vol.i , p.265 , 1995 . e. ma and b. r. holstein , phys . d 24 ( 1984 ) 2346 ; j. f. donoghue , b. r. holstein and y - c . r. lin , nucl . b277 ( 1986 ) 651 . g. dambrosio and d. espriu , phys . b175 ( 1986 ) 237 . a. lai et al . , phys . b551 ( 2003 ) 7 . k. hagiwara et al . d66 ( 2002 ) 010001 - 1 . the kloe collaboration , m. adinolfi et al . a488 ( 2002 ) 51 . the kloe collaboration , m. adinolfi et al . , nucl . instr . and meth . a482 ( 2002 ) 363 . the kloe collaboration , m. adinolfi et al . instr . and meth . a483 ( 2002 ) 649 . the kloe collaboration , m. adinolfi et al . instr . and meth . a492 ( 2002 ) 134 . s. di falco , g. lanfranchi and e. santovetti , _ measurement of the ratio @xmath134 _ , kloe note n 182 , january 2003 . g. d. barr et al . , phys . b358 ( 1995 ) 399 . the kloe collaboration , a. aloisio et al . b541 ( 2002 ) 45 .

we have measured the ratio @xmath0 using the kloe detector . from a sample of @xmath1 @xmath2-mesons produced at da@xmath3ne , the frascati @xmath4factory , we select @xmath5 @xmath6-mesons tagged by observing @xmath7 following the reaction @xmath8 . from this sample we select 27,375 @xmath9 events and obtain @xmath10 . using the world average value for @xmath11 , we obtain @xmath12 where the second error is due to the uncertainty on the @xmath13 branching fraction . pacs : keywords : -1 cm , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

perturbative gauge theory and gravity in four dimensions are quite dissimilar from a dynamical viewpoint . gauge theory ( _ e.g. _ pure yang - mills theory ) is a renormalisable theory that is strongly coupled in the infrared and asymptotically free in the ultraviolet . gravity on the other hand is a weakly coupled theory in the infrared but strongly coupled in the ultraviolet . by power counting , gravity in four dimensions is potentially a non - renormalisable theory . pure gravity scattering amplitudes are finite at one - loop with the first divergence occurring at two - loops@xcite . supersymmetry generally softens the uv behaviour in a quantum field theory . for example , maximally supersymmetric yang - mills is a finite theory@xcite and supergravity theories have a finite s - matrix until at least three loops@xcite . although four - dimensional power counting and counter - term arguments suggest that supergravity theories are non - renormalisable@xcite this has , so far , not been tested by direct computations . arguments based on power counting within unitary cuts suggest that the first counter term in maximal supergravity@xcite is expected at five loops@xcite . recently , initiated by the duality between gauge theories and a twister string theory@xcite , there has been much progress in the computation of amplitudes in gauge theory . in this talk we discuss how these ideas may be applied to gravity calculations and the results thereof . we will first review the recent progress in computing physical on - shell tree amplitudes for gravity theories particularly focusing on the on - shell recursion relations@xcite and the mhv - vertex construction@xcite . later we will discuss one - loop amplitudes . a surprising result is that the one - loop amplitudes of @xmath4 sym and @xmath5 supergravity@xcite occur to be expressible in terms of scalar box integral functions - despite the expectation from power counting . supergravity multi - loop amplitudes are not directly addressed , however , the structure of amplitudes at tree - level and one - loop have , through factorisation and unitarity , important consequences on the structure of higher loop amplitudes . -0.5 truecm graviton scattering amplitudes are extremely difficult to evaluate using conventional feynman diagram techniques . in this section we review alternative methods : 1 ) the kawai - lewellen - tye relations , 2 ) on shell recursion relations and 3 ) mhv vertex constructions . * 1 ) * gravity amplitudes can be constructed through the kawai , lewellen and tye ( klt)-relations@xcite as squares of gauge theory amplitudes . the klt relations are inspired by the nave string theory relation @xmath6 and have the explicit form , up to five points , @xmath7 where @xmath8 are the tree - level colour - ordered gauge theory partial amplitudes . we suppress factors of @xmath9 in the @xmath8 and @xmath10 in the @xmath11 . the klt relations are helpful in the calculation of gravity tree amplitudes , however they have some undesirable features . the factorisation structure is not manifest and the expressions do not tend to be compact , as the permutation sums grow rather quickly with @xmath12 . in fact , the berends , giele and kuijf ( bgk ) form of the mhv gravity amplitude@xcite , @xmath13\ , , \cr } \]]is rather more compact than that of the klt sum ( as is the expression in @xcite . ) in the above we use the definitions , @xmath14 \;\equiv\ ; \sum_{a = i}^j\spb k.a\spa a.l\,,$ ] and @xmath15 . in terms of the above weyl spinors we often use twistor variables @xmath16 and @xmath17 . the mhv amplitudes for graviton scattering display a feature not shared by the yang - mills expressions , they depend not only on the holomorphic variables @xmath18 , but also on the anti - holomorphic @xmath19 variables ( within the @xmath20 for the klt expression ) . * 2 ) * in a recent computational approach for amplitudes , britto , cachazo , feng and witten@xcite obtained on - shell recursion relations for trees . the recursion relations are based on factorisation properties of amplitudes and are thus applicable to a wide range of theories and in particular to gravity@xcite . the technique is based on analytically shifting a pair of external legs , @xmath21 and on determining the physical amplitude , @xmath22 , from the poles in the shifted amplitude , @xmath23.this leads to a recursion relation of the form , @xmath24 where the factorisation is only on these poles , @xmath25 , where legs @xmath26 and @xmath27 are connected to different sub - amplitudes . an essential condition for the recursion relations is that the shifted amplitude @xmath23 vanishes for large @xmath28 . whereas proven for gauge theory amplitudes a general proof ( for arbitrary helicities@xcite ) in gravity is an open problem . recursion relations based on the analyticity in the complex plane can also be used at loop level both to calculate rational terms@xcite and the coefficients of integral functions@xcite . * 3 ) * finally , we would like to mention the csw construction@xcite of amplitudes and its generalisation to gravity@xcite . in this approach mhv - amplitudes are treated as fundamental vertices and generic scattering amplitudes are expanded in terms of these mhv - vertices . considering the n@xmath29mhv amplitude with @xmath12 external legs . one would begin by drawing all diagrams which may be constructed using mhv vertices . ( 100,75)(80,-40 ) ( 30,30)(0,30)(30,30)(10,50)(30,30)(40,50)(30,30)(10,10 ) ( 30,30)(50,30)(120,30)(100,50)(120,30)(140,50 ) ( 120,30)(100,30)(120,30)(140,30)(120,30)(120,10 ) ( 240,30)(220,30)(240,30)(260,50)(240,30)(260,10 ) ( 240,30)(270,30 ) ( 30,30)2(120,30)2(240,30)2 ( 7,1)[c]@xmath30(-10,32)[c]@xmath31 ( 7,59)[c]@xmath32(39,55)[c]@xmath33 ( 58,30)[c]@xmath34(93,30)[c]@xmath34 ( 75,30)[c]@xmath35 ( 160,30)[c]@xmath36(212,30)[c]@xmath34 the contribution from each diagram is a product of @xmath37 mhv vertices and @xmath38 propagators as indicated above . the contribution of a given diagram to the total amplitude can be calculated by evaluating the product of mhv amplitudes and propagators , @xmath39 where the propagators are computed on the set of momenta @xmath40 and @xmath41 , and the mhv vertices are evaluated at shifted momenta @xmath42 and @xmath43 . the momenta @xmath40 are external and the momenta @xmath44 internal , and given by momentum conservation at each mhv - vertex . a key feature is the interpretation of the mhv amplitudes for internal legs . for yang - mills where the mhv vertices only depend on @xmath18 the correct interpretation is@xcite @xmath45 for an arbitrary reference spinor @xmath46 . for gravity amplitudes we must also solve for @xmath47 which is less obvious@xcite and will be a function of the momentum of the negative helicity legs . it turns out that all spinors are uniquely defined in terms of the shifted momenta @xmath48 and @xmath49 if we demand that they are null vectors obeying momentum conservation at each vertex . explicitly they are given by shifting the negative helicity legs @xmath50 by @xmath51 and leaving @xmath52 of the positive helicity legs @xmath53 untouched . the @xmath54 parameters , @xmath55 are uniquely fixed@xcite by demanding a ) overall momentum conservation , b ) momentum conservation at each vertex and finally c ) that the internal momenta , @xmath49 , are massless @xmath56 . as an example of how the mhv - vertex constructions works for gravity , we can consider the @xmath12-point nmhv amplitude with three negative helicity legs @xmath57 . the mhv - vertex approach gives the amplitude in the form , @xmath58 where the sum runs over diagrams involving all choices of @xmath59 and all permutations of the negative and positive helicity legs . to illustrate the correct continuation , the three negative helicity @xmath19 must be shifted @xmath60 . imposing the momentum constraints leaves us with a shift , @xmath61 together with the cyclic shifts of the other two legs . momentum is conserved for any value of the parameter @xmath28 . requiring @xmath62 then fixes @xmath28 uniquely as @xmath63 and the mhv vertex expansion is completely determined . in a yang - mills theory , the loop momentum polynomial in a one - loop @xmath12-point diagram will generically be of degree @xmath65 . @xmath0 one - loop amplitudes exhibits considerable simplification and the loop momentum integral will be of degree @xmath66@xcite . consequently , from a passarino - veltman reduction@xcite , the amplitudes can be expressed as a sum of scalar box integrals with rational coefficients , @xmath67 determining the amplitude then reduces to determining the rational coefficients @xmath68 . inspired by the duality in @xcite , considerable progress has recently been made in determining such coefficients , @xmath68 , using a variety of methods based on unitarity@xcite . for @xmath1 supergravity the equivalent power counting arguments@xcite give a loop momentum polynomial of degree @xmath69 which is consistent with eq . ( [ stringrelation ] ) . reduction for @xmath70 leads to a sum of tensor box integrals with integrands of degree @xmath66 which would then reduce to scalar boxes _ and _ triangle , bubble and rational functions , @xmath71 where the @xmath72 are present for @xmath73 , @xmath74 for @xmath75 and @xmath76 for @xmath77 . there is evidence that all one - loop amplitudes of @xmath1 , like the @xmath0 amplitudes eq . ( [ onlyboxeseq ] ) , can be expressed as a sum over scalar box integrals , the so called `` no - triangle hypothesis''@xcite . firstly , in the few definite computations at one - loop level , triangle or bubble functions do not appear . the first computation was of the four - point amplitude@xcite where only box functions appear ( although this is consistent with power counting ) . beyond this computations of the five and six point mhv - amplitudes yielded only scalar box - functions@xcite . secondly , the factorisation properties of physical amplitudes do not demand the presence of these functions . since the four and five point amplitudes are triangle - free then in any factorisation limit of a higher point function the triangles must vanish . in this spirit an ansatz for the @xmath12-point mhv amplitude was constructed@xcite entirely of box functions consistent in all soft limits . thirdly , one can check whether the amplitudes composed purely from box - functions precisely give the expected soft divergence in a @xmath12 graviton amplitude@xcite , @xmath78 } = { i c_\gamma \kappa^2 } \bigg [ { \sum_{i < j } s_{ij } \ln[-s_{ij } ] \over 2\epsilon } \bigg ] \!\!\times\ ! m^{\rm tree}_{[1,2,\ldots , n]}. \cr } \label{iramplitudeeq } \vspace{-0.15cm}\ ] ] in @xcite the box coefficients were explicitly computed for the six - point nmhv amplitudes confirming the above claims . the `` no - triangle '' hypothesis applies to one - loop amplitudes . however , by factorisation it should have implications beyond one - loop suggesting the uv behaviour of maximal supergravity may be significantly milder than expected from power counting . 99 g. t hooft and m. j. g. veltman , annales poincare phys . theor . a * 20 * ( 1974 ) 69 ; m. h. goroff and a. sagnotti , nucl . b * 266 * ( 1986 ) 709 . s.mandelstam , nucl.phys . b * 213 * , 149 ( 1983 ) . m. t. grisaru , p. van nieuwenhuizen and j. a. m. vermaseren , phys . * 37 * , 1662 ( 1976 ) ; m. t. grisaru , phys . b * 66 * ( 1977 ) 75 ; e. tomboulis , phys . b * 67 * ( 1977 ) 417 . s. deser , j. h. kay and k. s. stelle , phys . lett . * 38 * , 527 ( 1977 ) ; p. s. howe and u. lindstrom , nucl . b * 181 * , 487 ( 1981 ) . e. cremmer , b. julia and j. scherk , phys . b * 76 * , 409 ( 1978 ) ; e. cremmer and b. julia , phys . b * 80 * , 48 ( 1978 ) . z. bern , l.j . dixon , d.c . dunbar , m. perelstein and j.s . rozowsky , ; . p. s. howe and k. s. stelle , phys . b * 554 * , 190 ( 2003 ) . e. witten , commun . phys . * 252 * , 189 ( 2004 ) . r. britto , f. cachazo and b. feng , nucl . b * 715 * ( 2005 ) 499 ; r. britto , f. cachazo , b. feng and e. witten , phys . * 94 * ( 2005 ) 181602 . j. bedford , a. brandhuber , b. spence and g. travaglini ; nucl . b * 721 * ( 2005 ) 98 ; f. cachazo and p. svrcek . hep - th/0502160 . f. cachazo , p. svrcek and e. witten , jhep * 0409 * , 006 ( 2004 ) . a. brandhuber , b. spence and g. travaglini , nucl . b * 706 * , 150 ( 2005 ) ; c. quigley and m. rozali , jhep * 0501 * , 053 ( 2005 ) ; j. bedford , a. brandhuber , b. spence and g. travaglini , nucl . b * 706 * , 100 ( 2005 ) ; a. brandhuber , b. spence and g. travaglini , jhep * 0601 * , 142 ( 2006 ) . g.georgiou and v.v.khoze , jhep * 0405 * , 070 ( 2004 ) ; j.b.wu and c.j.zhu jhep * 0409 * , 063 ( 2004 ) ; x.su and j.b.wu . mod . phys . a * 20 * ( 2005 ) 1065 l. j. dixon , e. w. n. glover and v. v. khoze , jhep * 0412 * , 015 ( 2004 ) ; z. bern , d. forde , d. a. kosower and p. mastrolia , hep - ph/0412167 ; s. d. badger , e. w. n. glover and v. v. khoze . jhep * 0503 * ( 2005 ) 023 n. e. j. bjerrum - bohr , d. c. dunbar , h. ita , w. b. perkins and k. risager , jhep * 0601 * , 009 ( 2006 ) . z. bern , n. e. j. bjerrum - bohr and d. c. dunbar , jhep * 0505 * ( 2005 ) 056 . bjerrum - bohr , d.c . dunbar and h. ita , phys . lett b * 621 * ( 2005 ) 183 ; hep - th/0606268 . h. kawai , d. c. lewellen and s. h. h. tye , nucl . b * 269 * , 1 ( 1986 ) . f. a. berends , w. t. giele and h. kuijf , phys . b * 211 * , 91 ( 1988 ) . z. bern , l. j. dixon and d. a. kosower , phys . d * 73 * , 065013 ( 2006 ) ; c. f. berger et al hep - ph/0604195 ; hep - ph/0607014 . z. bern , n. e. j. bjerrum - bohr , d. c. dunbar and h. ita , jhep * 0511 * , 027 ( 2005 ) ; hep - ph/0603187 . k. risager , jhep * 0512 * , 003 ( 2005 ) . s. giombi , r. ricci , d. robles - llana and d. trancanelli , jhep * 0407 * , 059 ( 2004 ) ; j. b. wu and c. j. zhu , jhep * 0407 * , 032 ( 2004 ) . z. bern and d. a. kosower , phys . lett . * 66 * , 1669 ( 1991 ) ; nucl . b * 379 * , 451 ( 1992 ) ; z. bern and d. c. dunbar , nucl . b * 379 * , 562 ( 1992 ) . z. bern , l.j . dixon , d.c . dunbar and d.a . kosower , . g. passarino and m. veltman , nucl . b * 160 * , 151 , ( 1979 ) . z. bern , l.j . dixon , d.c . dunbar and d.a . kosower , . z. bern , v. del duca , l.j . dixon and d. a. kosower , phys . d * 71 * , 045006 ( 2005 ) ; z. bern , l. j. dixon and d. a. kosower . d * 72 * ( 2005 ) 045014 ; r. britto , f. cachazo and b. feng , phys . d * 71 * ( 2005 ) 025012 ; r. roiban , m. spradlin and a. volovich . * 94 * ( 2005 ) 102002 ; r. britto , f. cachazo and b. feng . b * 725 * ( 2005 ) 275 ; s. j. bidder , d. c. dunbar and w. b. perkins , jhep * 0508 * ( 2005 ) 055 . s. j. bidder , n. e. j. bjerrum - bohr , l. j. dixon and d. c. dunbar , phys . b * 606 * , 189 ( 2005 ) ; s. j. bidder , n. e. j. bjerrum - bohr , d. c. dunbar and w. b. perkins , phys . b * 608 * ( 2005 ) 151 ; phys . b * 612 * ( 2005 ) 75 ; r. britto , e. buchbinder , f. cachazo and b. feng , phys . d * 72 * ( 2005 ) 065012 . z. bern , d.c . dunbar and t. shimada , phys . b * 312 * , 277 , ( 1993 ) ; d.c . dunbar and p.s . norridge , . . z. bern , l. j. dixon , m. perelstein and j. s. rozowsky , nucl . b * 546 * , 423 ( 1999 ) . d. c. dunbar and p. s. norridge , class . * 14 * ( 1997 ) 351 .

we review recent progress in computations of amplitudes in gauge theory and gravity . we compare the perturbative expansion of amplitudes in @xmath0 super yang - mills and @xmath1 supergravity and discuss surprising similarities . # 1_#1 _ # 1#1 = 0.1 cm = -2.7 cm = -1.5 cm = 23 . truecm = 15.7truecm 40 pt plus 1000pt minus 100pt # 1 # 1= to [ section ] # 1#2#3nucl . phys . b * # 1 * , # 3 ( # 2 ) # 1#2#3phys . lett . b * # 1 * , # 3 ( # 2 ) # 1#2#3phys . rev . d * # 1 * , # 3 ( # 2 ) # 1#2#3class . and quant . grav . * # 1 * , # 3 ( # 2 ) # 1#2#3phys . rev . lett . * # 1 * , # 3 ( # 2 ) # 1#2#3jhep * # 1 * , # 3 ( # 2 ) # 1#2#3_j . high ener . phys . _ * # 1*:#3 ( # 2 ) # 1#2#3int . j. mod . phys . * a # 1 * , # 3 ( # 2 ) # 1#2#3j . math . phys . * # 1 * , # 3 ( # 2 ) # 1[hep - th/#1 ] # 1[hep - ph/#1 ] # 1eq . ( [ # 1 ] ) # 1eq . ( [ # 1 ] ) # 1#2eqs . ( [ # 1 ] ) and ( [ # 2 ] ) # 1#2eqs . ( [ # 1 ] ) and ( [ # 2 ] ) # 1fig . # 1 # 1#2figs . [ # 1 ] and [ # 2 ] # 1#2figs . [ # 1 ] and [ # 2 ] # 1section [ # 1 ] # 1section [ # 1 ] # 1#2sections [ # 1 ] and [ # 2 ] # 1appendix [ # 1 ] # 1appendix [ # 1 ] # 1#2appendices [ # 1 ] and [ # 2 ] # 1#2appendices [ # 1 ] and [ # 2 ] # 1table [ # 1 ] # 1table [ # 1 ] # 1#2f ^#1_4:#2 # 1#2f^#1_:#2 # 1 = = = by - by - by by 2 -to # 1.#2#1#2 # 1.#2 # 1.#2(#1#2 ) @=11 # 10.05 cm /-0.22 cm # 1 # 1a_(#1 ) # 1(#1 ) # 1#2#3#4(#1#2#3#4 ) # 1#2x_#1 # 2 # 1@xmath2 # 1.#2#1#2 # 1.#2 # 1.#2(#1#2 ) # 1.#2.#3 |#2 | # 1.#2.#3 ^-|#2 |^+ # 1.#2.#3 ^+|#2 |^+ # 1.#2.#3 ^-|#2 |^- # 1.#2.#3 ^+|#2 |^- # 1.#2.#3 ^-|#2 |^+ # 1a_#1 # 1#2 _ # 2 k_#1 # 1#2 k_#1 k_#2 ( # 1#2_#1k_#2 # 1(_i _ # 1 k_i _ # 1,i ) # 1(_i k_#1 k_i _ # 1 , i ) # 1#2 ( _ i ( k_#1 k_i _ # 1 , i + k_#2 k_i _ # 2 , i ) ) # 1 ( _ i _ # 1 k_i _ # 1,i ) # 1 ( _ i k _ # 1 k_i _ # 1,i ) # 1#2t^[#1]_#2 ( # 1#2#1|k_abc|#2 # 1#2#1^+|_abc|#2^+ # 1#2#1||#2 # 1#1 # 1#2#3#1|#2|#3 # 1#2#3 [ # 1|#2|#3 # 1#2#3#1^+||#3^+ swat-06/481 + 2.cm * similarities of gauge and gravity amplitudes * 2.cm 2.cm n. e. j. bjerrum - bohr@xmath3 , david c. dunbar@xmath3 and harald ita@xmath3 0.5 cm 0.5 cm _ department of physics + university of wales swansea + swansea , sa2 8pp , uk _ .3 cm

a stripe is a complicated object . its degrees of freedom may include charge and spin of the stripe particles , as well as transverse fluctuations of its position . one potentially promising route to finding the right 1d model is to study simple solvable models of an antiferromagnet in 2d in the presence of a domain wall . a single domain wall can be created artificially , e.g. , by wrapping a system with an odd number of rows on a cylinder . one may then study an isolated stripe at any filling @xmath4 by adding or removing the right number of electrons . a 2d model may look unrealistic , but with universality and a bit of luck the resulting 1d theory may have just the right symmetry of the vacuum and the correct quantum numbers of elementary excitations . consider a 2d model , a variant of the @xmath0@xmath1 model in the ising limit @xmath5 @xcite : @xmath6,\end{aligned}\ ] ] with the usual exclusion of doubly occupied sites . the dominating @xmath7 term minimizes the number of frustrated ( ferromagnetic ) bonds and thus sets the ground state . the energy of an af bond @xmath8 controls interactions of charged quasiparticles on the domain wall and in the bulk . making @xmath9 prevents phase separation in the bulk , as well as on a stripe . for @xmath13 , the stripe will contain @xmath14 quasiparticles with the quantum numbers of electrons , @xmath15 , @xmath16 ( fig . [ fig : electron ] ) . because of the zigzag geometry of the stripe , the electron quasiparticles feel a strong ( of order @xmath7 ) staggered magnetic field . it confines electrons with opposite spins to different rows of the stripe . within its row , an electron can delocalize to reduce kinetic energy . in the tight - binding model ( [ t - j - ham ] ) , such hopping is a multistage process , which involves pushing a neighbor spin out of the way . precisely at half - filling ( @xmath17 ) , the domain wall is bond - centered . a single doped hole ends up at the domain wall ( fig . [ fig : holon ] , left ) . if it starts to move alng the wall , two remarkable things happen . first , the hole leaves its spin @xmath18 behind in the form of a spinon and then propagates freely as a spinless object ( a holon , fig . [ fig : holon ] , right ) . no additional frustrated bonds are produced afterwards . furthermore , no costly spinons would be left behind had we started with 2 holes . second , the holon ( as well as the spinon ) resides on a transverse kink of the domain wall . these elementary excitations are thus maximally strongly coupled to transverse fluctuations of the stripe . kink direction can be specified by assigning a holon ( or a spinon ) a transverse flavor @xmath19 @xcite .

we argue that effective 1d models of stripes in the cuprate superconductors can be constructed by studying ground states and elementary excitations of domain walls in 2d model antiferromagnets . this method , applied to the @xmath0@xmath1 model with ising anisotropy , yields two such limiting cases : an ordinary 1d electron gas and a 1d gas of holons strongly coupled to transversal fluctuations of the stripe . formation of charge stripes in some of the cuprate superconductors @xcite is a peculiar phenomenon in its own right . a possible connection between the stripes and high - temperature superconductivity makes them even more attractive to a theorist . the puzzle is nevertheless quite hard : to date there is no microscopic theory describing the physics of stripes in the cuprates . some progress has been made . hartree - fock studies of the hubbard model near half - filling @xcite have revealed that doped charges segregate in the form of narrow stripes . the stripes are domain walls separating antiferromagnetic ( af ) domains with opposite orientations of the neel vector , in agreement with experiments . however , the predicted linear density of charge on a stripe , @xmath2 hole per unit cell , indicates that there are no charges able to carry current . experimentally , @xmath3 for non - overlapping stripes @xcite . in the absence of a microscopic theory , it is reasonable to look for a model description in which a stripe is a 1d object interacting with the surrounding antiferromagnet . a stripe is characterized by a ground state , by its elementary excitations , and by the interactions of the excitations among themselves and with the environment .

since their discovery , luminous submm galaxies ( smgs ) have been proposed as candidates for the progenitors of the most massive spheroids in the local universe ( smail , ivison & blain 1997 ; hughes et al . 1998 ; lilly et al . 1999 ; blain et al . 2002 ) . the recent measurement of the redshift distribution , space densities and clustering of this population provides strong support for this proposed relationship ( chapman et al . 2003a , 2004 ; blain et al . these galaxies have large bolometric luminosities , @xmath610@xmath1810@xmath19 , characteristic of ultraluminous infrared galaxies ( ulirgs , sanders & mirabel 1996 ) . if their intense restframe far - infrared ( far - ir ) emission arise from dust - obscured star formation , then the estimated rates are @xmath20m@xmath10yr@xmath11 , sufficient to form the stellar population of a massive elliptical galaxy in only a few dynamical times , given a sufficient gas reservoir . alternatively , a substantial fraction of the submm emission in these galaxies could arise from an obscured agn ( e.g. almaini et al . it has proved difficult to distinguish whether agn or starburst activity powers the dust heating and associated far - ir radiation in these luminous submm galaxies ( frayer et al . 1998 ; alexander et al.2003 ; chapman et al . optical and near - infrared spectroscopy or x - ray observations have frequently been used to search for the signatures of agn , in both local ulirgs and those at high - redshifts ( sanders & mirabel 1996 ; fabian et al . 2000 ; ivison et al . 2000 ; barger et al . 2001 ; frayer et al . 2003 ; chapman et al . 2003a ; alexander et al . 2003 , 2004 ; swinbank et al . however , merely identifying the presence of an agn within a ulirg does not immediately mean that it must be the dominant source of far - ir radiation . energetic arguments must be used to estimate what fraction of a ulirg s luminosity arises from the agn . a much simpler test is available if the far - ir emission is resolved the geometry of the emission from an agn means it is not a natural source to heat dust over an extended region hence any extended far - ir emission is very likely to arise from star formation . unfortunately , the coarse resolution of most far - ir and submm instruments , e.g. @xmath21 fwhm at 850@xmath8 m with scuba on the jcmt , means that the emission is rarely resolved except in the most local galaxies ( le floch et al . there have been recent claims for the detection of submm emission on @xmath22kpc scales around some powerful high - redshift agn ( ivison et al . 2000 ; stevens et al.2003 , 2004 ) , however , these are rare and extreme objects whose characteristics may have little bearing on those of typical submm galaxies . to disentangle the mechanisms responsible for the far - ir emission in the population of submm galaxies at @xmath233 ( chapman et al . 2003a ) , will likely require sub - arcsecond resolution to map emission on kpc scales well beyond the capabilities of current far - ir / submm facilities . one way to circumvent the limited spatial resolution of far - ir / submm instruments is by exploiting the tight far - ir radio correlation observed for infrared galaxies ( e.g. helou et al . 1985 ; condon 1992 ) and the high angular resolution capabilites of long - baseline radio inteferometers , such as the multi - element microwave linked interferometer ( merlin ) or the very large array ( vla ) , to infer the sub - arcsecond distribution of far - ir emission within submm galaxies . one caveat of this approach is that the far - ir radio correlation has only been demonstrated locally on relatively large scales , @xmath24kpc ( yun , reddy & condon 2001 ) , and it the precise correlation may break down on the smallest scales ( m. yun , in preparation ) . nevertheless , if a significant extended component of the continuum radio emission from the submm population is seen then this would provide strong support for the far - ir emission being similarly extended . the deepest 1.4-ghz observations from the vla can detect the synchrotron emitting disks and nuclear starbursts ( formed from the coalescence of radio supernovae and their remnants ) of a ulirg such as arp220 out to redshifts of @xmath254 . indeed , a large fraction of the bright submm population at high redshifts are detected as @xmath8jy radio sources ( smail et al . 2000 ; barger , cowie & richards 2000 ; chapman et al . 2001 ; ivison et al . 2002 ) , as expected given their submm luminosities and the local far - ir radio correlation ( chapman et al.2004 ) . the highest spatial resolution available from the vla at 1.4ghz is 1.5 , but by combining deep 1.4-ghz observations from merlin and the vla it is possible to produce datasets which combine both high sensitivity and high spatial resolution , @xmath26scales , sufficient to map @xmath8jy radio sources and test the extent of the far - ir activity in these galaxies . these sub - arcsecond radio maps , which indirectly trace the far - ir morphologies , of the smg population can also be compared and contrasted with the restframe uv structures visible in _ hst _ imaging on similar scales . such comparisons may help to constrain the extent of obscuration in smgs relative to other samples of high - redshift sources selected in the restframe uv ( e.g. adelberger & steidel 2000 ) . in this paper , we present sensitive merlin / vla radio and _ hst _ restframe uv observations of smgs at comparable , sub - arcsecond resolution . we discuss the sample and observations in 2 , describe our main results in 3 and discuss our conclusions in 4 . we assume a @xmath27-cdm cosmology with @xmath28 , @xmath29 and @xmath30kms@xmath11mpc@xmath11 , so that 1arcsec corresponds to 8.4kpc physical size at @xmath31 . our primary observational dataset is the combined merlin and vla radio map of a @xmath32 region centered on the hubble deep field ( hdf ) region , which has sufficient sensitivity and resolution to attempt a radio morphological analysis of the submm galaxies in this area . the smg sample in this region comes from the parent catalog of submm - detected , optically - faint @xmath8jy radio galaxies used in the spectroscopic survey of chapman et al . ( 2003a , 2004 ) . we identify 14 submm - detected galaxies lying within a @xmath33-diameter field centered on 12 36 48.0 + 62 15 40 ( j2000 ) which have 850@xmath8 m fluxes brighter than 4mjy and are detected at 1.4ghz in the vla a - array observations of this region with a flux of @xmath34jy . this radio flux limit should ensure a useful constraint on the source morphology from the merlin observations . the deep merlin observation of the hdf region ( muxlow et al . 1999 ; muxlow et al . 2004 ) comprise @xmath35hrs integration at 1.4ghz of a @xmath32 field centered on the hdf and including the hubble flanking fields ( hff ) . these data were acquired in february 1996 and april 1997 . the merlin data were supplemented with 42hrs of 1.4-ghz vla a - array observations ( richards 2000 ) , and combined and deconvolved in the sky - plane due to computational limitations . we use a map with a restored 0.3-@xmath36 beam which has an rms noise level of 3.3@xmath8jybeam@xmath11 . to register the radio and optical data , radio sources associated with compact galaxies have been used to align the radio map with panoramic ground - based imaging ( see below ) . for the hdf , this matching involves 128 @xmath37 optical sources with radio counterparts and yields an rms of 0.3@xmath36 ( capak et al . 2004 ) . we next search the _ hst _ database for deep imaging observations of smgs that lie within merlin field . as our primary goal is a comparison of the coarse morphologies of the sources in the radio and optical wavebands , the choice of camera used for the _ hst _ observations is less critical than it would be for a detailed morphological analysis of smgs ( chapman et al . hence , we search for any observations of smgs within the hdf merlin field using the stis and acs cameras . due to the intensive study of this field we identify _ hst _ imaging of 13 smgs from our sample which lie in the merlin field and list these in table 1 . four of these galaxies come from the _ hst _ targeted survey of 13 smgs observed with stis by chapman et al . ( 2003b ) , see table 1 . in addition to these , a further 9 smgs serendipitously fall in the hdf / hff region which is covered by acs imaging from the goods project ( giavalisco et al . 2004 ) . however , in one of these 13 sources ( smmj123651.76 + 621221.3 ) , the smg is not uniquely identified , and may lie behind a foreground elliptical galaxy ( dunlop et al . 2004 ) , and an accurate comparison of the morphology using the merlin / vla radio and _ hst _ in the optical is impossible . we exclude this object from further comparison . the cycle 10 stis imaging of galaxies in our sample uses the open filter , clear50ccd ( central wavelength 5733a ) , and have durations of two or three orbits ( 5.07.5ks ) . the reduction and analysis of these images is described in chapman et al . ( 2003b ) . the resolution of these images is 0.06@xmath36 fwhm and they have a typical point - source sensitivity limit of 27.4 mag . ( ab ) . the acs observations of the smgs lying within the goods - n field were obtained from the stsci v1.0 release ( august , 2003 ) , in the @xmath38 and @xmath39 bands . the reduction of these data is described by giavalisco et al . ( 2004 ) and the typical point - source sensitivity is 28 mag . ( ab ) and resolution of 0.07@xmath36 . we register the _ hst _ images to the radio coordinate frame by aligning them with the deep suprimecam images from capak et al . ( 2004 ) which are tied to the radio frame . to achieve this we first smooth the _ hst _ images to the ground - based seeing , then match all @xmath40-@xmath41 sources ( except the smg ) , and transform the coordinate grids using the iraf task , geotran . the morphological characteristics of this sample have been classified by eye from the _ hst _ imaging in fig . 1 and are presented in table 1 . these rough morphological classes are subject to uncertain structured dust extinction ( smail et al . 1999 ) , and may not represent the true physical morphology of the system . optical photometry for our smg sample in the @xmath42 and @xmath43-bands ( table 1 ) was measured from the subaru suprimecam imaging published by capak et al . we use a 3 diameter aperture centered on the radio source and the limiting magnitudes are @xmath44 and @xmath45 ( 5@xmath41 ) . 11 of these 12 smgs in our sample have secure spectroscopic redshifts from the keck survey of chapman et al . ( 2004 table 1 ) . these enable us to measure physical sizes and luminosities for these galaxies and also calculate k - corrections between galaxies in the sample to directly compare their restframe properties . @xmath46 photometry exists for the remaining source ( smmj123646.1 + 621449 ) , allowing us to estimate a photometric redshift . using the hyper - z software ( bolzonella et al . 2000 ) , we derive a photometric redshift of @xmath47 . the median redshift of the sample is @xmath48 , representative of that measured for a larger spectroscopic samples of submm galaxies chapman et al . ( 2003a , 2004 ) . using the registered radio and _ hst _ optical imaging we show in fig . 1 the 1.4-ghz merlin / vla contours overlayed on the _ hst _ images for the 12 smgs in the joint sample . we note that at the median redshift of our sample , the _ hst _ imaging corresponds to restframe wavelengths of 17001800a . the median diameter of the radio emission from the 12 galaxies is @xmath49 or @xmath50kpc ( measured above the 3-@xmath41 contour , fig . 1 ) , showing that the typical source in our sample is well - resolved at the resolution of our merlin / vla map . the radio morphologies for these submm galaxies split into two broad classes : those dominated by an unresolved component , often centered on a sub - component of the _ hst_-optical emission ( 4/12 or 33% ) , and extended structures on scales @xmath60.51(8/12 or 67% ) . remarkably , the extended radio morphology in these latter sources often appears to trace the same uv - bright , large - scale structures seen in the _ hst _ optical images ( fig . 1 ) . with sensitive , high - resolution imagery in the radio and optical , we can compare the radio emission ( as a tracer of the dust emission ) to restframe - uv emission on kpc - scales within the systems . in fig . 2 we show the radio and restframe uv surface brightness profiles along the major axes of three representative resolved sources . in some cases the uv emission shows broad similarities to the radio emission . however , fig . 2 demonstrates that the uv / radio flux ratios can vary significantly over the extent of the galaxy , and the sfrs derived from the respective wavelengths will differ accordingly . for example , in smmj123621.3 + 621708 the extended restframe uv emission does not trace the more compact radio emission , with the peak of the radio emission actually coinciding with a clear deficit in the uv emission . such anti - correlations in the radio and uv are similar to those seen in many local ulirgs ( chamandaris et al . 2002 ) . to test whether there is any evidence for a correlation between internal reddening and the distribution of obscured star formation traced by the radio emission , we construct @xmath51 color maps of the 12 smgs using the acs imaging of the goods - n region . only smmj123712.0 + 621325 and smmj123707.2 + 621408 show redder internal color structure which corresponds to the radio morphology . in all other cases , the radio emission does not correspond to any regions with unusually red colors in the _ hst _ @xmath51 maps . smmj123707.2 + 621408 and smmj123712.0 + 621325 actually show bluer colors in the vicinity of the peaks in the radio emission . this suggests that the uv emission may not always probe the true site of far - ir emission ; and indeed the uv - inferred bolometric luminosities in smgs on average underpredict the true luminosities by factors @xmath5210 ( chapman et al . 2004 ) . a more quantitative test of the degree of obscuration within these galaxies is gleaned through comparing the variation in the total radio / uv flux ratio measurements of each galaxy with that derived locally for the regions of intense far - ir emission pinpointed by the merlin / vla morphology . the monochromatic rest - frame 2000aluminosities ( l@xmath53 ) of our smgs are estimated from linear interpolation between the @xmath42- and @xmath43-band magnitudes ( table 1 ) . we estimate l@xmath54 using the measured radio flux ( table 1 ) , k - corrected using synchrotron slope of @xmath55 ( @xmath56 ) to restframe 1.4ghz at the observed redshift , and then transformed to the far - ir using the local far - ir radio correlation from helou et al . we take this route to estimate the far - ir emission on arcsecond - scales as our submm data lack both the spatial resolution and full spectral information needed to estimate l@xmath57 more directly . we note that garrett ( 2002 ) and kovacs et al.(2004 ) have shown that the far - ir radion relation does not seem to be change substantially out to @xmath582.5 , and so the approach we have adopted should be reliable . in this manner we predict a median total , far - ir luminosity of @xmath59l@xmath10 for the 12 smgs , equivalent to star formation rates of @xmath60m@xmath10yr@xmath11 for stars more massive than 0.1m@xmath10 based on a salpeter imf . to investigate the variation of far - ir / uv ratios within the smgs we first calculate the ratios for the total emission from the galaxies . we then measure the same ratio in a region encompassing the peak of the radio emission using a fixed circular aperature enclosing all radio emission above the 3@xmath41 contour shown in fig . 1 . we compare the total and locally derived l@xmath54/l@xmath53 ratios for the 12 sources in fig . 3 . we see that when the regions pinpointed as the sites of strong activity by the merlin / vla radio morphology are considered , the obscuration levels increase by factors of 28 over the ratios derived for the whole galaxy . even measured over the entire extent of the systems , the obscurations we derive are considerably higher than those seen in high redshift , restframe uv - selected populations which are typically l@xmath54/l@xmath610.1100 ( e.g. the lyman - break galaxies , adelberger & steidel 2000 ; or the bx / bm galaxies , steidel et al . 2004 ) . having discovered that the radio emission in over half the smgs in our sample is extended on 1 scales we now wish to test whether this emission extends to even larger scales , @xmath62 or @xmath63kpc . we can constrain the fraction of radio flux arising on @xmath64 scales by a simple comparison of the 1.4-ghz fluxes of smgs in maps from the vla in its b - array ( 5 synthesized beam ) and a - array ( 1.5 beam ) configurations . the higher resolution a - array observations may resolve out a portion of the total emitted flux density of any source which has significant extended structure on scales larger than the 1.5 vla a - array beam . as no b - array coverage is available for the hdf - n we have assembled vla b - array and a - array fluxes for a sample of 50 @xmath8jy radio sources , including 5 submm galaxies , from the our ssa22 survey field ( see chapman et al.2004 ) . we calculate the median and bootstrap errors on the flux ratios for the radio sources from the data taken at the two resolutions . we start by confirming that sources show either similar or higher fluxes in the b - array map compared to the a - array data , as expected considering the median radio angular sizes of @xmath8jy sources ( windhorst et al . 1993 ) we find median ratios of @xmath65 and that for 30% of the sample agree within 2-@xmath41 , suggesting there is no calibration offset between the fluxes from the two maps . for the five submm - detected sources we obtain a median flux ratio of @xmath66 , compared to @xmath67 for an @xmath43-band matched sample of 8 radio sources undetected or unobserved in the submm waveband . this suggests that the submm - luminous section of the @xmath8jy radio populations is no more extended than the general population , after we have removed low redshift galaxies ( with bright apparent magnitudes ) . moreover , we see that there is only weak evidence that the radio emission in the smgs is extended on @xmath68kpc scales . the far - ir emission from local ulirgs generally arises from the nuclear regions of the systems , while the uv light is more extended , although it contributes a negligible fraction of the bolometric emission ( goldader et al . 2002 ; surace & sanders 2000 ) . we have been able to trace the distribution of these two components in similarly luminous galaxies at high redshifts through the combination of sub - arcsecond imagery in the radio and restframe uv from our merlin / vla and _ hst _ observations . these maps allowed us to investigate the relative distribution of obscured and unobscured star formation on kpc - scales within luminous submm galaxies at @xmath69 , when the universe was only a fifth of its current age . as expected the uv morphologies of this sample are similar to the ( overlapping ) sample of submm galaxies imaged with _ hst_/stis and analysed by chapman et al . ( 2003b ) , as well as the acs imaging in smail et al . the sample exhibits irregular and frequently highly complex morphologies in their restframe @xmath70a emission compared to optically - selected galaxies at similar redshifts , and have scale lengths far in excess of comparably luminous local galaxies . turning to the radio morphologies , we find that in @xmath71% ( 8/12 ) the merlin / vla radio exhibits resolved radio emission on @xmath30.5@xmath72 scales ( @xmath73kpc ) which mirrors the general form of the restframe uv morphology seen by _ we interpret this as strong support for the radio emission tracing spatially - extended , massive star formation within these galaxies . this situation is very unlike local ulirgs , where the high surface brightness far - ir / radio emission is restricted to a compact nuclear region with an extent of less than @xmath5kpc ( charmandaris et al . 2002 ) . a more detailed analysis of the distribution of uv and radio emission within these galaxies ( fig . 2 & 3 ) shows that the correspondence is rarely one - to - one , with variations of the uv / radio flux ratio of factors of a few on kpc - scales within galaxies . it is likely that comparisons using even higher resolution data would show even stronger variations , as are seen in local ulirgs ( e.g. bushouse et al . 2002 ) , but which are diluted at the current resolution . similarly , there is only weak evidence for correlations between the restframe uv colors and the positions of the radio emitting regions within these galaxies . here , longer wavelength near - ir observations with _ hst_/nicmos , or from the ground , may reveal variations in the uv spectral slope that would show a better correlation of reddening with radio intensity ( smail et al . 2004 ) . in the remaining @xmath630% ( 4/12 ) of smgs , the radio emission is much more compact and is essentially unresolved suggesting it arises in a region with a scale size of order @xmath5kpc or less ( fig . 1 ) . in two of these cases , the compact radio emission is centered on a bright uv source ( in one case clearly the nucleus of a face - on spiral galaxy , which is a strong x - ray source and is also the only one of the sources in our sample which shows agn signatures in its uv spectrum ) , while in the other two systems the compact radio component is spatially offset by several kpc from the uv source . these configurations reflect either compact , nuclear starbursts and/or a dominant contribution from an agn to the radio emission . in half of these cases the agn / nuclear starburst is also strongly obscured at restframe wavelengths of @xmath74a . stevens et al . ( 2003 , 2004 ) have recently presented evidence for submm emission resolved on @xmath75kpc scales ( including apparent filaments ) in the rare and extreme environments around powerful radio galaxies and absorbed qsos at @xmath7624 . our results demonstrate that in many cases ( @xmath470% ) the far - ir emission ( as seen in our merlin / vla radio maps ) of the general smg population is extended on scales @xmath77kpc . however , the interferometric measurements are not suited to measuring larger scale , diffuse emission . our vla b - array versus a - array comparison ( 3.2 ) , however , suggests that the typical field smg does not have submm emission ( as traced by the radio ) extending on scales much larger than @xmath781(@xmath79kpc ) . our high resolution radio and optical imaging allows us to address the relative obscuration of the galaxies . adelberger & steidel ( 2000 ) have suggested that high - luminosity galaxies at high redshift have much stonger obscuration than lower - luminosity galaxies , as measured by the ratio of far - ir to restframe - uv luminosity . this is certainly true on large - scales in the submm galaxies . on smaller scales within the smgs , we have seen that the obscuration is roughly @xmath80 higher over the region of intense radio ( and by implication far - ir ) emission , compared to the average over the whole galaxy . this suggests that there is highly structured reddening within the submm galaxies , such anisotropic obscuration would be a natural consequence of channels being blown through the dust around the star - formation regions by vigorous winds . it is worth considering that few star - burst galaxies with l@xmath81l@xmath10 exist locally ; most galaxies in our neighborhood with these luminosities have strong and obvious agn components . however , at @xmath82 , the median redshift of the radio - selected smgs , the most active galaxies were evidently forming stars at rates of @xmath61700m@xmath10yr@xmath11 in regions extending over @xmath640kpc@xmath13 . the large physical extent of this activity contrasts markedly with the compact , nuclear starbursts typical of local redshift ulirgs . this suggests that the some of the observational properties of the star formation activity in these galaxies ( e.g. mix of dust temperatures , ease of superwind generation , etc . ) may differ markedly from that seen in local `` analogs '' . however , we also note that the star formation surface density inferred from our radio observations is @xmath645m@xmath10yr@xmath11kpc@xmath14 , comparable to the upper - limit estimated for such activity in local starburst galaxies by meurer et al . this argues that the small - scale physical mechanisms which limit the star formation process within these galaxies are similar to those operating in the most vigorous systems locally . while evidence for massive amounts of molecular gas in submm galaxies has now been established ( frayer et al . 1998 ; neri et al . 2003 ; greve et al . in preparation ) , and x - ray luminosities are consistent with a dominant role for star formation in the energetics of smgs ( alexander et al . 2004 ) , our discovery of spatially extended radio morphologies is perhaps the strongest piece of evidence that star formation dominates the bolometric output of the majority of the submm galaxy population . in summary , we have compared the restframe uv and radio morphologies on sub - arcsecond scales of a small sample of highly luminous , dusty galaxies for which precise redshifts are available . this analysis shows that the radio emission , which we adopt as a proxy for the far - ir emission , in these galaxies is resolved in the majority of galaxies implying that dust heating ( and by implication , massive star formation ) is occuring on @xmath73kpc scales within these systems . currently , this represents our only constraint on the likely submm morphology of these galaxies , and one which will not be further testable until alma comes on - line . the overall structure of the radio emission matches that seen in the restframe uv although there are strong variations in the relative emission on kpc - scales which we interpret as resulting from highly structured dust obscuration within the galaxies . this structured obscuration may reflect from anisotropic dispersal of the dust as superwinds driven by the star formation activity blows channels through the intergalactic medium . such channels would provide the opportunity for ly@xmath15 photons to escape from these otherwise highly - obscured systems , explaining the unexpected strength of this line in their spectra ( chapman et al . 2003a , 2004 ) . = 0.2 cm smmj123606.9 + 621021 & 2.509 & 25.6 & 25.2 & 11.6@xmath833.5 & 74.4@xmath834.1 & e & disturbed , merger + smmj123616.2 + 621514@xmath84 & 2.578 & 26.8 & 25.7 & 5.8@xmath831.1 & 53.9@xmath838.4 & e & 3 components + smmj123622.7 + 621630@xmath84 & 2.466 & 25.6 & 25.4 & 7.7@xmath831.3 & 70.9@xmath838.7 & e & merging disks + smmj123629.1 + 621046@xmath85 & 1.013 & 26.1 & 24.6 & 5.0@xmath831.3 & 81.4@xmath838.7 & e & disturbed + smmj123655.8 + 621200@xmath86 & 2.743 & 25.4 & 25.3 & 8.0@xmath831.8 & 21.0@xmath836.2 & e & disturbed , merger + smmj123707.2 + 621408 & 2.484 & 26.9 & 26.0 & 6.3@xmath831.3 & 45.3@xmath837.9 & e & blue & red pair + smmj123712.0 + 621325 & 1.992 & 26.0 & 25.8 & 4.1@xmath831.3 & 53.9@xmath838.1 & e & disturbed with dusty component + smmj123712.1 + 621212@xmath87 & 2.914 & 27.0 & 25.5 & 8.0@xmath831.8 & 21.0@xmath834.0 & e & disturbed , double source + smmj123618.3 + 621551@xmath84 & 1.865 & 26.0 & 25.9 & 7.3@xmath831.1 & 150.5@xmath8311.2 & c & small group + smmj123621.3 + 621708@xmath88 & 1.988 & 25.1 & 24.9 & 7.8@xmath831.9 & 148.1@xmath8311.2 & c & linear + smmj123635.6 + 621424 & 2.005 & 24.2 & 24.2 & 5.5@xmath831.4 & 87.8@xmath838.8 & c & disturbed , face - 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we compare high - resolution optical and radio imaging of 12 luminous submillimeter ( submm ) galaxies at a median @xmath0 observed with _ hubble space telescope _ ( _ hst _ ) and the merlin and vla radio interferometers at comparable spatial resolution , @xmath1 ( @xmath2kpc ) . the radio emission is used as a tracer of the likely far - infrared morphology of these dusty , luminous galaxies . in @xmath330% of the sample the radio emission appears unresolved at this spatial scale , suggesting that the power source is compact and may either be an obscured agn or a compact nuclear starburst . however , in the majority of the galaxies , @xmath470% ( 8/12 ) , we find that the radio emission is resolved by merlin / vla on scales of @xmath5(@xmath610kpc ) . for these galaxies we also find that the radio morphologies are often broadly similar to their restframe uv emission traced by our _ hst _ imaging . to assess whether the radio emission may be extended on even larger scales , @xmath7 , resolved out by the merlin+vla synthesized images , we compare vla b - array ( 5beam ) to vla a - array ( 1.5 beam ) fluxes for a sample of 50 @xmath8jy radio sources , including 5 submm galaxies . the submm galaxies have comparable fluxes at these resolutions and we conclude that the typical radio emitting region in these galaxies are unlikely to be much larger than @xmath5(@xmath610kpc ) . we discuss the probable mechanisms for the extended emission in these galaxies and conclude that their luminous radio and submm emission arises from a large , spatially - extended starburst . the median star formation rates for these galaxies are @xmath9m@xmath10yr@xmath11 ( m@xmath12m@xmath10 ) occuring within regions with typical sizes of @xmath640kpc@xmath13 , giving a star formation density of 45m@xmath10yr@xmath11kpc@xmath14 . such vigorous and extended starburst appear to be uniquely associated with the submm population . a more detailed comparison of the distribution of uv and radio emission in these systems shows that the broad similarities on large scales are not carried through to smaller scales , where there is rarely a one - to - one correspondance between the structures seen in the two wavebands . we interpret these differences as resulting from highly structured internal obscuration within the submm galaxies , suggesting that their vigorous activity is producing wind - blown channels through their obscuring dust clouds . if correct this underlines the difficulty of using uv morphologies to understand structural properties of this population and also may explain the surprising frequency of ly@xmath15 emission in the spectra of these very dusty galaxies . # 1 * # 1 * @xmath16 i@xmath17

what is the astrophysical object producing r - process elements and how has r - process enrichment proceeded through cosmic time ? clear answers to these questions are still veiled . recent updates from both observational and theoretical studies of the r - process site , in particular by the detection of a near - infrared light bump in the afterglow of a short - duration @xmath6-ray burst ( sgrb)(e.g . * ) and successful nucleosynthesis calculations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , have indicated neutron star ( ns ) mergers as the most promising astronomical site . core - collapse supernovae ( cc - sne ) driven by neutrino heating are another candidate for the r - process site , buy they are unable to synthesize heavy r - process elements such as eu and ba . the required conditions for r - process nucleosynthesis are not achieved in the proto - ns winds due to the strong effect of neutrino absorption ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this fact has turned out to strengthen the ns merger origin scenario . new research on r - process synthesis has renewed interest in the galactic chemical evolution ( gce ) of r - process elements in terms of the enrichment taking place through ns mergers . though previous researchers have pointed out the fatal problems with the notion of enrichment by ns mergers @xcite , @xcite successfully reproduced an observed eu feature in the galaxy using a gce model that considers a unique propagation of the ejecta of ns mergers in the scheme of hierarchical galaxy formation . they also found that an early release of eu from ( at least some fraction of ) ns mergers with a short timescale of @xmath7 year is an inevitable condition for explaining the presence of very low - metallicity ( [ fe / h]@xmath8 ) stars enriched by r - process elements . such a prompt enrichment by ns mergers is also claimed by other gce studies @xcite , and can be a possible channel in population synthesis models @xcite . however , the coalescence time for an ns merger depends critically on the binary evolution during a common envelope phase which is poorly understood . in addition , the above requirement is in contrast with a typical ns merger time of @xmath9@xmath10 gyr deduced from the analysis of _ swift _ sgrb samples @xcite . @xcite have extended the discussion of the site of r - process elements to nearby dwarf spheroidal ( dsph ) galaxies . they have found compelling evidence for ns mergers as the origin of r - process elements , in the observed chemical feature of eu in the draco , sculptor , and carina dsphs ; these dsphs exhibit a constant [ eu / h ] of @xmath11 with no apparent increase in the eu abundance over a metallicity range of @xmath12@xmath13 . this implies no eu production events as more than @xmath14 cc - sne increase the galactic fe abundance . this level of rarity of eu production is compatible with the frequency expected for ns mergers @xcite . the next question is how these dsphs which do not undergo any eu production events for [ fe / h]@xmath12 are enriched up to [ eu / h]@xmath11 until [ fe / h]@xmath15 . @xcite examined eu abundances for [ fe / h]@xmath16 in the draco and sculptor dsphs from a purely observational viewpoint and concluded that [ eu / h ] remarkably increases from @xmath17 to @xmath11 inside these dsphs , including a conspicuous intermittent increase in the draco dsph . these revealed eu features suggest that the contributor of eu in the early dsphs is active only in the low - metallicity environment with an occurrence rate that is not as frequent as in cc - sne . deduced from [ ba / h ] for stars with upper limits on eu data or no detection of eu together with new data . the eu floor is assigned at [ eu / h]@xmath18 , which is implied by the sculptor dsph . the data are originally from @xcite , @xcite , @xcite , and @xcite for the draco dsph ; @xcite , @xcite , @xcite , @xcite , and @xcite for the sculptor dsph ; and @xcite , @xcite , and @xcite for the carina dsph . ] besides the mechanism of cc - sne by neutrino heating in the standard scenario , another proposed option for r - process enrichment is an explosion triggered by fast rotations and high magnetic fields ( e.g. , * ? ? ? * ) . these magnetorotational cc - sne ( mr - sne ) can produce heavy r - process elements and thus may contribute to their enrichment during early star formation @xcite . in addition , it is possible to presume that the emergence of mr - sne is inclined toward very low - metallicity stars in which the rotational velocity is expected to be high ( e.g. , * ? ? ? * ) . in this letter , we provide compelling evidence for heavy r - process enrichment by mr - sne at an early stage of star formation inside dsphs . this letter is organized as follows . theoretical arguments on the chemical feature of eu abundance in the draco and sculptor dsphs are first presented in section 2 . then , the interpretation of these arguments is incorporated into the models of chemical evolution , and the results are shown in section 3 . subsequently we discuss the astrophysical site of early eu production in terms of nucleosynthesis in mr - sne ( section 4 ) and finally conclude with section 5 . chemical features of eu abundance for faint ( i.e. , less massive ) dsphs in the luminosity range of @xmath19 , i.e. , the draco , carina , and sculptor dsphs , is divided into two discrete trends : ( i ) a remarkably increasing [ eu / h ] for [ fe / h]@xmath20 , and ( ii ) a broadly constant [ eu / h ] for [ fe / h]@xmath21 . the first feature is also supported by the trend of [ ba / h ] increasing versus [ fe / h ] @xcite . it suggests that eu production episodes occur only in the low - metallicity regime . here we attempt to unravel how the early eu enrichment proceeds in the draco and sculptor dsphs ( note that the carina dsph has only one datum for [ fe / h]@xmath16 ) . the observed correlation of [ eu / h ] with [ fe / h ] is shown in figure 1 . in this figure , for the stars ( [ fe / h]@xmath16 ) measured with only upper limits of eu abundance or without the detection of eu , we deduce [ eu / h ] from their ba abundances with an assumed pure r - process ratio of [ ba / eu]=@xmath22 @xcite . this procedure is validated by the fact that the stars with [ fe / h]@xmath12 exhibit a [ ba / eu ] ratio compatible with the value assumed in the translation ( i.e. , [ ba / eu]=@xmath22 ) and thus are considered to be of a pure r - process origin before s - process nucleosynthesis can occur @xcite . first we see the enrichment path in the draco . it begins with a jump of [ eu / h ] from @xmath23 to @xmath24 between metallicities [ fe / h]@xmath25 and @xmath26 . to pin down the metallicity of the first eu production episode more precisely , acquisition of more [ eu / h ] data between @xmath27[fe / h]@xmath28 are awaited . this includes verification of the possibility of intrinsic scatter in [ eu / h ] at [ fe / h]@xmath29 implied by two data points . though the subsequent path leading to a plateau of [ eu / h]@xmath11 is hard to detect , it is possible to predict the presence of a few jump - like increases in [ eu / h ] as indicated by dotted arrows if we consider the errors in [ fe / h ] and [ eu / h ] of individual stars . in fact , such a small number of eu production events is strongly supported by a relatively high metallicity where the first eu jump happens . the corresponding metallicity is determined by how many cc - sne producing fe occur before the first event , which is directly connected to the frequency of eu production event . for the draco dsph with metallicities of [ fe / h]@xmath29 to @xmath26 , this gives about three to six events in total as shown by our calculations in the following section . here we will deduce the actual rate of events from this total number of @xmath9@xmath30 . the stellar metallicity distribution function ( mdf ) for the draco @xcite indicates that @xmath31% of stars populate the metallicity range of [ fe / h]@xmath32 . using the kroupa imf @xcite , we obtain the corresponding stellar mass for this metallicity range of @xmath33@xcite and find that this stellar population presumably hosted @xmath34 total cc - sne in total . accordingly , we conclude that r - process production events happen at a rate of one per @xmath35 cc - sne in the early evolution of draco . in addition , we can estimate the average eu yield of a single event . the average metallicity of [ fe / h]@xmath36 @xcite is expected to result from the conversion mass fraction of the initial gas to stars of @xmath37% , which is calculated by models of dwarf galaxies ( * ? ? ? * ; * ? ? ? * see also 3 ) . then , the hypothesis that an initial gas cloud with a mass of @xmath38was enriched to [ eu / h]=@xmath39 by three to six events leads to a eu yield of @xmath40 . in the sculptor dsph , a rise from the eu floor happens at [ fe / h]@xmath41 , a lower metallicity than draco s , i.e. , [ fe / h]@xmath8 . then , this eu jump is followed by a continuous increase in [ eu / h ] if we ignore an outlier showing a very low eu abundance ( [ eu / h]=@xmath42 ) at [ fe / h]@xmath43 . we note that three unpublished data sets of [ ba / h ] at [ fe / h]@xmath44 @xcite suggest @xmath45 } \rangle=-2.2 $ ] . such a relatively continuous feature is naturally understood if we take into consideration that the observed stellar mass of sculptor is about eight times more than that of the draco . we expect that the occurrence frequency of early eu production events in the sculptor will increase by the same factor , and as a result , more than 20 occurrences of eu production are likely to build the smooth enrichment path as observed . irrespective of its higher event rate , the final enrichment level should be the same as for draco since the mass ratio of initial gas between the two dsphs is broadly equivalent to the ratio of the eu event rate between the two . therefore , different mass - scaled dsphs , i.e. , draco , carina , and sculptor , share the same plateau of [ eu / h]@xmath11 . however , the evolutionary path to the plateau varies in accordance with the number of eu production episodes . to validate the theoretical interpretation presented in the previous section , we calculate the evolution of eu abundance for the draco and sculptor dsphs , adopting the following properties of eu producers . we assume two production sites for eu : ns mergers and one that emerges selectively in a low - metallicity regime , likely mr - sne ( see section 4 ) . due to the severe rarity of ns mergers ( one per thousands of cc - sne , * ? ? ? * ) ) , the two faint dsphs did not host any ns mergers over their whole lifetimes . mr - sne , however , contribute to eu enrichment in the early phase with a moderately rare event rate . here we assume that mr - sne operate with a rate of one per 200 cc - sne in the metallicity range of @xmath5}<-2 $ ] for the reference case . this rate gives three occurrences in draco as already discussed , while 21 total events for sculptor are deduced from its stellar mass and mdf @xcite . in our calculations , each mr - sn is assumed to occur after every 200 cc - sne with a progenitor metallicity lower than [ fe / h]=@xmath12 . for the eu mass ejected from a single mr - sn , we adopt the yield of @xmath46 . by contrast , for the first mr - sn for both dsphs , a smaller amount of eu mass is found to give a better fit to the observed jumps from an eu floor , and thus @xmath47is assumed as the eu mass of the first mr - sn . as discussed in 4 , both eu yields are within the predictions from hydrodynamical simulations of mr - sne . in addition to the reference case , for draco we calculate another case in which the mr - sn rate is assumed to be one per 120 ccsne with a eu yield of @xmath48(@xmath49for the first mr - sn ) . the chemical evolution models are modified from the gce model to account for the star formation rate ( sfr ) and the imf as done by @xcite . to reproduce the observed mdfs with a mean [ fe / h ] of @xmath50 and @xmath51 for the draco and sculptor dsphs , respectively @xcite , we adjust the sfr together with a steep imf ( @xmath52=@xmath53 ) as in the large magellanic cloud . here we assume that the period of star formation continues for 4 gyr for draco ( cf . * ) and 6 gyr for sculptor @xcite , respectively . the initial eu abundance ( i.e. , the eu floor ) is set to be [ eu / h]=@xmath54 . the results from our models are shown in figure 2 . } < -2 $ ] while staying dormant for @xmath5}>-2 $ ] . a resultant smaller event rate ( three : solid curve ; five : dashed curve ) in the draco is attributed to its lower stellar mass compared with the sculptor . the pre - enriched level is set at @xmath4 } = -5 $ ] . ] the observed values of [ fe / h ] corresponding to the jump from a pre - enriched level of [ eu / h ] for the two dsphs are well reproduced by our models . the higher jump of [ eu / h ] at higher [ fe / h ] in draco can be interpreted as a result of a lower frequency of mr - sne owing to a smaller total stellar mass than sculptor . the subsequent ladder - like increasing feature of [ eu / h ] in draco is also in good agreement with observations . on the other hand , a gradually increasing trend after the first jump until the plateau - like abundance ( i.e. , [ eu / h]@xmath11 ) is predicted by the model for sculptor . three data points giving the low upper limits of [ eu / h ] between @xmath55[fe / h]@xmath56 suggest a more delayed first jump . accordingly , the result of another model in which 12 total events are assumed to occur with a rate of one per 350 ccsne is shown by the dashed blue curve . the agreement of our models with observations for both draco and sculptor suggest that eu production episodes occur more frequently than ns mergers , but much less frequently than cc - sne and with a high eu yield . as discussed in the previous sections , the early chemical evolution of the draco and sculptor dsphs demands a large amount of eu production such as @xmath57 per single event . this value far exceeds the assumed amount ( @xmath58 ) from the ejecta of canonical cc - sne with neutrino - driven winds in gce models ( e.g. , * ? ? ? * ) , while it is smaller than the expectation from an ns merger ( @xmath59 ) . here we claim that the implied eu yields agree well with the prediction from mr - sne , based on the recent calculations by @xcite . in their models , a typical case ( i.e. , b11@xmath601.00 ) results in a eu mass @xmath61 in the jet - like ejecta and a total mass of r - process nuclei of @xmath62 . inside the ejecta , the values of @xmath63 vary between 0.15 and 0.5 and an average in the region where r - process operates ( i.e. , @xmath64 ) is @xmath65 . the resultant nucleosynthesis pattern is shown by the red curve in figure 3 , compared with the observed solar abundance pattern ( filled circles : * ? ? ? * ) . in this calculation , we used reaction rates based on the finite - range droplet mass model which is available in the reaction rate library reaclib @xcite , leading to a relatively low ba / eu ratio of [ ba / eu]@xmath66 . however , a recent improvement @xcite indicates that the production of isotopes in the region including the rare earth peak has increased by a few factors @xcite , which may lead to a match with the solar pattern , giving @xmath67 and a [ ba / eu ] ratio close to [ ba / eu]=@xmath22 . in addition , @xcite pointed out that the total mass of ejecta can be larger by a few factors in more energetic jet - like explosion models . this strong jet model gives @xmath68}=-0.55 $ ] ( the blue curve ) . accordingly , the eu mass of @xmath57 is indeed within the prediction by mr - sn models . besides the above jet - like explosions , another scenario that would eject a significant amount of eu is from mr - sne , which release a massive amount of moderate neutron - rich ejecta . according to recent updated magnetohydrodynamic simulations that resolve the magnetorotational instability @xcite , the ejecta due to neutrino heating enhanced by fast rotation and large magnetic fields have much lower values of @xmath63 for the cases of explosion without jet - like outflows . as a result , the ejecta are a moderate neutron - rich state . to obtain the nucleosynthesis results for the corresponding case , we set higher @xmath63 values inside the ejecta in the jet model ( see section 5.2 of nishimura et al . 2012 ) and obtain results for @xmath69 , as shown by the dashed curve in figure 3 . the ejecta mass including the total r - process nuclei of @xmath70 @xcite gives @xmath71 , which is broadly equivalent to the mass predicted by the jet - like explosion models . we show that the properties of r - process production events that are important for early star formation , i.e. , the frequency and yield , can be assessed from the recently revealed eu feature of two less massive dsphs . the frequency is estimated to be about one per @xmath1@xmath2 cc - sne in a low - metallicity regime , while this frequency decreases to a level similar to or less than that of ns mergers for @xmath5}>-2 $ ] . we identify high frequency events at low - metallicity together with the implied high eu yield as mr - sne . accordingly , we have the following picture of r - process enrichment among galaxies . in dwarf galaxies such as the draco and sculptor dsphs , mr - sne at an early phase are the only contributors of the nucleosynthesis of heavy r - process elements . however , in more massive galaxies , including our own , in addition to mr - sne with their metallicity - dependent frequencies , ns mergers enrich the ism with heavy r - process elements over the whole lifetime of a galaxy at a rate of one per thousands of cc - sne . since enrichment by mr - sne is expected to start much earlier than ns mergers , the presence of extremely metal - poor stars enriched by r - process elements as observed in the galactic halo is naturally explained by our scheme . we predict that the frequency of mr - sne is fairly small in the relatively metal - rich galaxies . thus their occurrences should be detected with a high rarity in the present - day universe . mr - sne are likely to be associated with magnetars , the rate of which is estimated to be @xmath72 in the galaxy @xcite . we accordingly anticipate that a few percent or less of magnetars are the end result of mr - 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one of the hottest open issues involving the evolution of r - process elements is fast enrichment in the early universe . clear evidence for the chemical enrichment of r - process elements is seen in the stellar abundances of extremely metal poor stars in the galactic halo . however , small - mass galaxies are the ideal testbed for studying the evolutionary features of r - process enrichment given the potential rarity of production events yielding heavy r - process elements . their occurrences become countable and thus an enrichment path due to each event can be found in the stellar abundances . we examine the chemical feature of eu abundance at an early stage of @xmath0 } \lesssim -2 $ ] in the draco and sculptor dwarf spheroidal ( dsph ) galaxies . accordingly , we constrain the properties of the eu production in the early dsphs . we find that the draco dsph experienced a few eu production events , whereas eu enrichment took place more continuously in the sculptor dsph due to its larger stellar mass . the event rate of eu production is estimated to be about one per @xmath1@xmath2 core - collapse supernovae , and a eu mass of @xmath3per single event is deduced by associating this frequency with the observed plateau value of @xmath4 } \sim -1.3 $ ] for @xmath5 } \gtrsim -2 $ ] . the observed plateau implies that early eu enrichment ceases at @xmath5 } \approx -2 $ ] . such a selective operation only in low - metallicity stars supports magnetorotational supernovae , which require very fast rotation , as the site of early eu production . we show that the eu yields deduced from chemical evolution agree well with the nucleosynthesis results from corresponding supernovae models .

the @xmath0-state potts model on a graph @xmath12 with vertices @xmath13 and edges @xmath14 can be defined geometrically through the cluster expansion of the partition function @xcite @xmath15 where @xmath16 and @xmath17 are respectively the number of connected components ( clusters ) and the cardinality ( number of links ) of the edge subsets @xmath18 . we are interested in the case where @xmath19 is a finite regular two - dimensional lattice of width @xmath20 and length @xmath9 , so that @xmath21 can be constructed by a transfer matrix @xmath3 propagating in the @xmath9-direction . in @xcite , we studied the case of cyclic boundary conditions ( periodic in the @xmath9-direction and non - periodic in the @xmath20-direction ) . we decomposed @xmath21 into linear combinations of certain restricted partition functions ( characters ) @xmath22 ( with @xmath8 ) in which @xmath23 _ bridges _ ( that is , marked non - contractible clusters ) wound around the transfer ( @xmath9 ) direction . we shall often refer to @xmath23 as the _ level_. unlike @xmath21 itself , the @xmath22 could be written as ( restricted ) traces of the transfer matrix , and hence be directly related to its eigenvalues . it was thus straightforward to deduce from this decomposition the amplitudes in @xmath21 of the eigenvalues of @xmath3 . the goal of this work is to repeat this procedure in the case of toroidal boundary conditions . note that as in the cyclic case some other procedures exist . first , read and saleur have given in @xcite a general formula for the amplitudes , based on the earlier coulomb gas analysis of di francesco , saleur , and zuber @xcite . they obtained that the amplitudes of the eigenvalues are simply @xmath24 at the level @xmath25 and @xmath26 at @xmath27 . for @xmath28 they obtained that , contrary to the cyclic case , there are several differents amplitudes at each level @xmath23 . their number is equal to @xmath29 , the number of divisors of @xmath23 . they are given by : @xmath30 where @xmath23 is the level considered , and @xmath31 is a divisor of @xmath23 which labels the different amplitudes for a given level . @xmath32 is defined as : @xmath33 here , @xmath34 and @xmath35 are respectively the mbius and euler s totient function @xcite . the mbius function @xmath34 is defined by @xmath36 , if @xmath37 is an integer that is a product @xmath38 of @xmath39 _ distinct _ primes , @xmath40 , and @xmath41 otherwise or if @xmath42 is not an integer . similarly , euler s totient function @xmath35 is defined for positive integers @xmath37 as the number of integers @xmath43 such that @xmath44 and @xmath45 . the value of @xmath46 depends on @xmath0 and is given by : @xmath47 note that in eq . ( [ deflambda ] ) we may write @xmath48 , where @xmath49 is the @xmath37th order chebyshev polynomial of the first kind . the term @xmath50 in eq . ( [ eqsal ] ) is due to configurations containing a cluster with `` cross - topology '' @xcite ( see later ) . the drawback of the derivation in ref . @xcite is that since it relies ultimately on free - field techniques it is _ a priori _ valid only at the usual ferromagnetic critical point ( @xmath51 ) and in the continuum limit ( @xmath52 ) . but one may suspect , in analogy with the cyclic case , that these amplitudes would be valid for any finite lattice and for any inhomogeneous ( i.e. , edge - dependent ) values of the coupling constants @xmath53 . to our knowledge , no algebraic study proving this statement does exist in the literature . indeed , when the boundary conditions are toroidal , the transfer matrix ( of the related six - vertex model , to be precise ) does no longer commute with the generators of the quantum group @xmath54 . therefore , there is no simple algebraic way of obtaining the amplitudes of eigenvalues , although some progress has been made by considering representations of the periodic temperley - lieb algebra . a good review is given by nichols @xcite . chang and shrock have studied the potts model with toroidal conditions from a combinatorial point of view @xcite . using a diagrammatic approach they obtained some general results on the eigenvalue amplitudes . in particular , they showed that the sum of all amplitudes at level @xmath23 equals @xmath55 they also argued that it was because @xmath3 enables permutations among the bridges , due to the periodic boundary conditions in the transverse ( @xmath20 ) direction , that there were different amplitudes for a given level @xmath23 . without them , all the amplitudes at level @xmath23 would be equal ( to a global factor ) to @xmath56 . finally , they computed explicitly the amplitudes at levels @xmath57 and @xmath58 ; one may check that those results are in agreement with eq . ( [ eqsal ] ) . using the combinatorial approach we developed in @xcite , we will make the statements of chang and shrock more precise , and we will give in particular a new interpretation of the amplitudes using the characters of the cyclic group @xmath11 . then , by calculating sums of characters of irreducible representations ( irreps ) of this group , we will reobtain eq . ( [ eqsal ] ) and thus prove its validity for an arbitrary finite @xmath59 lattice . as will become clear below , the argument relies exclusively on counting correctly the number of clusters with non - trivial homotopy , and so the conclusion will hold true for any edge - dependent choice of the coupling constants @xmath53 as well . our approach will have to deal with several complications due to the boundary conditions , the first of which is that the bridges can now be permuted ( by exploiting the periodic @xmath20-direction ) . in the following this leads us to consider decomposition of @xmath21 into more elementary quantities than @xmath22 , namely characters @xmath60 labeled by @xmath23 _ and _ a permutation of the cyclic group @xmath11 . however , @xmath60 is not simply linked to the eigenvalues of @xmath61 , and thus we will further consider its expansion over related quantities @xmath62 , where @xmath10 labels an irreducible representation ( irrep ) of @xmath11 . it is @xmath62 which are the elementary quantities in the case of toroidal boundary conditions .. the present approach , using only the cyclic group @xmath11 , is far simpler and for the first time allows us to prove eq . ( [ eqsal ] ) . note also that some misprints had cropped up in ref . @xcite , giving in particular wrong results for the amplitudes at level @xmath63 . ] the structure of the article is as follows . in section [ sec2 ] , we define appropriate generalisations of the quantities we used in the cyclic case @xcite and we expose all the mathematical background we will need . then , in section [ sec3 ] , we decompose restricted partition functions and as a byproduct the total partition function into characters @xmath22 and @xmath60 . finally , in section [ sec4 ] , we obtain a general expression of the amplitudes of eigenvalues which involves characters of irreps of @xmath11 . using number theoretic results ( ramanujan sums ) we then proceed to prove its equivalence with the formula ( [ eqsal ] ) of read and saleur . as in the cyclic case , the existence of a periodic boundary condition allows for non - trivial clusters ( henceforth abbreviated ntc ) , i.e. , clusters which are not homotopic to a point . however , the fact that the torus has _ two _ periodic directions means that the topology of the ntc is more complicated that in the cyclic case . indeed , each ntc belongs to a given homotopy class , which can be characterised by two coprime numbers @xmath65 , where @xmath66 ( resp . @xmath67 ) denotes the number of times the cluster percolates horizontally ( resp . vertically ) @xcite . the fact that all clusters ( non - trivial or not ) are still constrained by planarity to be non - intersecting induces a convenient simplification : all ntc in a given configuration belong to the same homotopy class . for comparison , we recall that in the cyclic case the only possible homotopy class for a ntc was @xmath68 . it is a well - known fact @xcite that the difficulty in decomposing the potts model partition function or relating it to partition functions of locally equivalent models ( of the six - vertex or rsos type)is due solely to the weighing of the ntc . although a typical cluster configuration will of course contain trivial clusters ( i.e. , clusters that are homotopic to a point ) with seemingly complicated topologies ( e.g. , trivial clusters can surround other trivial clusters , or be surrounded by trivial clusters or by ntc ) , we shall therefore tacitly disregard such clusters in most of the arguments that follow . note also that a ntc that span both lattice directions in the present context corresponds to @xmath69 . non - trivial clusters ( ntc ) , here represented in red and blue colours . each ntc is characterised by its number of branches , @xmath70 , and by the permutation it realises , @xmath71 . within a given configuration , all ntc have the same topology . ] consider therefore first the case of a configuration having a single ntc . for the purpose of studying its topology , we can imagine that is has been shrunk to a line that winds the two periodic directions @xmath65 times . in our approach we focus on the the properties of the ntc along the direction of propagation of the transfer matrix @xmath3 , henceforth taken as the horizontal direction . if we imagine cutting the lattice along a vertical line , the ntc will be cut into @xmath66 horizontally percolating parts , which we shall call the @xmath66 _ branches _ of the ntc . seen horizontally , a given ntc realises a permutation @xmath72 between the vertical coordinates of its @xmath66 branches , as shown in fig . [ fig1 ] . up to a trivial relabelling of the vertical coordinate , the permutation @xmath72 is independent of the horizontal coordinate of the ( imaginary ) vertical cut , and so , forms part of the topological description of the ntc . we thus describe totally the topology along the horizontal direction of a ntc by @xmath66 and the permutation @xmath73 . note that there are restrictions on the admissible permutations @xmath72 . firstly , @xmath72 can not have any proper invariant subspace , or else the corresponding ntc would in fact correspond to several distinct ntc , each having a smaller value of @xmath66 . for example , the case @xmath74 and @xmath75 is not admissible , as @xmath72 corresponds in fact to two distinct ntc with @xmath70 . in general , therefore , the admissible permutations @xmath72 for a given @xmath66 are simply cyclic permutations of @xmath66 coordinates . secondly , planarity implies that the different branches of a ntc can not intersect , and so not all cyclic permutations are admissible @xmath72 . for example , the case @xmath74 and @xmath76 is not admissible . in general the admissible cyclic permutations are characterised by having a constant coordinate difference between two consecutive branches , i.e. , they are of the form @xmath77 for some constant @xmath78 , with all coordinates considered modulo @xmath66 . for example , for @xmath74 , the only admissible permutations are then finally @xmath79 and @xmath80 . ) when we discuss in detail the attribution of `` black points '' to one or more different ntc . it will then be shown that the admissible permutations at level @xmath23 correspond to the cyclic group @xmath11 . for example , the admissible permutations at level @xmath63 are @xmath81 , @xmath79 , @xmath82 and @xmath80 . ] consider now the case of a configuration with several ntc . recalling that all ntc belong to the same homotopy class , they must all be characterised by the same @xmath66 and @xmath72 . alternatively one can say that the branches of the different ntc are entangled . henceforth we denote by @xmath83 the number of ntc with @xmath84 in a given configuration . note in particular that , seen along the horizontal direction , configurations with no ntc and configurations with one or more ntc percolating only vertically are topologically equivalent . this is an important limitation of our approach . let us denote by @xmath85 the partition function of the potts model on an @xmath59 torus , restricted to configurations with exactly @xmath83 ntc characterised by the index @xmath84 and the permutation @xmath86 ; if @xmath72 is not admissible , or if @xmath87 , we set @xmath88 . further , let @xmath89 be the partition function restricted to configurations with @xmath83 ntc of index @xmath66 , let @xmath90 be the partition function restricted to configurations with @xmath83 ntc _ percolating horizontally _ , and let @xmath21 be the total partition function . obviously , we have @xmath91 , and @xmath92 , and @xmath93 . in particular , @xmath94 corresponds to the partition function restricted to configurations with no ntc , or with ntc percolating only vertically . in the case of a generic lattice all the @xmath85 are non - zero , provided that @xmath72 is an admissible cyclic permutation of length @xmath66 , and that @xmath95 . the triangular lattice is a simple example of a generic lattice . note however that other regular lattices may be unable to realise certain admissible @xmath72 . for example , in the case of a square lattice or a honeycomb lattice , all @xmath85 with @xmath96 and @xmath97 are zero , since there is not enough `` space '' on the lattice to permit all ntc branches to percolate horizontally while realising a non - trivial permutation . such non - generic lattices introduce additional difficulties in the analysis which have to be considered on a case - to - case basis . in the following , we consider therefore the case of a generic lattice . the construction and structure of the transfer matrix @xmath98 can be taken over from the cyclic case @xcite . in particular , we recall that @xmath99 acts towards the right on states of connectivities between two time slices ( left and right ) and has a block - trigonal structure with respect to the number of _ bridges _ ( connectivity components linking left and right ) and a block - diagonal structure with respect to the residual connectivity among the non - bridged points on the left time slice . as before , we denote by @xmath100 the diagonal block with a fixed number of bridges @xmath23 and a trivial residual connectivity . each eigenvalue of @xmath98 is also an eigenvalue of one or more @xmath100 . in analogy with @xcite we shall sometimes call @xmath101 the transfer matrix at level @xmath23 . it acts on connectivity states which can be represented graphically as a partition of the @xmath20 points in the right time slice with a special marking ( represented as a _ black point _ ) of precisely @xmath23 distinct components of the partition ( i.e. , the components that are linked to the left time slice via a bridge ) . a crucial difference with the cyclic case is that for a given partition of the right time slice , there are more possibilities for attributing the black points ( for @xmath102 ) . considering for the moment the black points to be indistinguishable , we denote the corresponding dimension as @xmath103 . it can be shown @xcite that n_tor(l , l ) = ll & l=0 + 2l-1 l-1 & l=1 + 2l l - l & 2 [ defntor ] and clearly @xmath104 for @xmath105 . suppose now that a connectivity state at level @xmath23 is time evolved by a cluster configuration of index @xmath66 and corresponding to a permutation @xmath72 . this can be represented graphically by adjoining the initial connectivity state to the left rim of the cluster configuration , as represented in fig . [ fig1 ] , and reading off the final connectivity state as seen from the right rim of the cluster configuration . evidently , the positions of the black points in the final state will be permuted with respect to their positions in the intial state , according to the permutation @xmath72 . as we have seen , not all @xmath72 are admissible . we will show in the subsection [ sec : decklcl ] that the possible permutations at a given level @xmath23 ( taking into account all the ways of attributing @xmath23 black points to cluster configurations ) are the elements of the cyclic group @xmath11 . of all permutations at level @xmath23 , not just the admissible permutations . therefore the dimension of @xmath100 they obtained was @xmath106 . although this approach is permissible ( since in any case @xmath100 will have zero matrix elements between states which are related by a non - admissible permutation ) it is more complicated @xcite than the one we present here.]the number of possible connectivity states without taking into account the possible permutations between black points , the dimension of @xmath107 is @xmath108 , as @xmath11 has @xmath23 distinct elements . let us denote by @xmath109 ( where @xmath110 ) the @xmath111 standard connectivity states at level @xmath23 . the full space of connectivities at level @xmath23 , i.e. , with @xmath23 distinguishable black points , can then be obtained by subjecting the @xmath109 to permutations of the black points . it is obvious that @xmath100 commutes with the permutations between black points ( the physical reason being that @xmath101 can not `` see '' to which positions on the left time slice each bridge is attached ) . therefore @xmath100 itself has a block structure in a appropriate basis . indeed , @xmath100 can be decomposed into @xmath112 where @xmath113 is the restriction of @xmath100 to the states transforming according to the irreducible representation ( irrep ) @xmath114 of @xmath11 . note that as @xmath11 is a abelian group of @xmath23 elements , it has @xmath23 irreps of dimension @xmath115 . one can obtain the corresponding basis by applying the projectors @xmath116 on all the connectivity states at level @xmath23 , where @xmath116 is given by @xmath117 here @xmath118 is the character of @xmath72 in the irrep @xmath114 and @xmath119 is its complex conjugate . the application of all permutations of @xmath11 on any given standard vector @xmath109 generates a regular representation of @xmath11 , which contains therefore once each representation @xmath114 ( of dimension @xmath115 ) . as there are @xmath111 standard vectors , the dimension of @xmath112 is thus simply @xmath103 . instead of @xmath11 we would have had algebraic degeneracies , which would have complicated considerably the determination of the amplitudes of eigenvalues . in fact , it turns out that even by considering @xmath11 there are degeneracies between eigenvalues of different levels , as noticed by chang and shrock @xcite . but these degeneracies depend of the width @xmath20 , and have no simple algebraic interpretation . ] we now define , as in the cyclic case @xcite , @xmath22 as the trace of @xmath121 . since @xmath100 commutes with @xmath11 , we can write @xmath122 in distinction with the cyclic case , we can not decompose the partition function @xmath21 over @xmath22 because of the possible permutations of black points ( see below ) . we shall therefore resort to more elementary quantities , the @xmath120 , which we define as the trace of @xmath123 . since both @xmath100 and the projectors @xmath116 commute with @xmath11 , we have @xmath124 obviously one has @xmath125 the sum being over all the @xmath23 irreps @xmath114 of @xmath11 . recall that in the cyclic case the amplitudes of the eigenvalues at level @xmath23 are all identical . this is no longer the case , since the amplitudes depend on @xmath114 as well . indeed @xmath126 in order to decompose @xmath21 over @xmath120 we will first use auxiliary quantities , the @xmath127 defined as : @xmath128 @xmath129 being an element of the cyclic group @xmath11 . so @xmath127 can be thought of as modified traces in which the final state differs from the initial state by the application of @xmath129 . note that @xmath130 is simply equal to @xmath131 . because of the possible permutations of the black points , the decomposition of @xmath21 will contain not only the @xmath132 but also all the other @xmath127 , with @xmath133 . we will show that the coefficients before @xmath127 coincide for all @xmath133 that belong to the same class _ with respect to the symmetric group @xmath134_. is an abelian group , each of its elements defines a class of its own , if the notion of class is taken with respect to @xmath11 itself . what we need here is the non - trivial classes defined with respect to @xmath134 . ] we will note these classes @xmath135 ( corresponding to a level @xmath136 ) and it is thus natural to define @xmath137 as : @xmath138 the sum being over elements @xmath133 belonging to the class @xmath135 . this definition will enable us to simplify some formulas , but ultimately we will come back to the @xmath127 . once we will obtain the decomposition of @xmath21 into @xmath127 , we will need to express the @xmath127 in terms of the @xmath120 to obtain the decomposition of @xmath21 into @xmath120 , which are the quantities directly linked to the eigenvalues . ( [ defkldtor ] ) and ( [ projpd ] ) yield a relation between @xmath120 and @xmath127 : @xmath139 these relations can be inverted so as to obtain @xmath127 in terms of @xmath120 , since the number of elements of @xmath11 equals the number of irreps @xmath114 of @xmath11 . multiplying eq . ( [ kldfc ] ) by @xmath140 and summing over @xmath114 , and using the orthogonality relation @xmath141 one easily deduces that : @xmath142 note that @xmath143 in the following we will obtain an expression of the amplitudes at the level @xmath23 which involves sums of characters of the irreps @xmath114 of @xmath11 . in order to reobtain eq . ( [ eqsal ] ) , we will have to calculate these sums . we give here the results we shall need . @xmath11 is the group generated by the permutation @xmath144 . it is abelian and consists of the @xmath23 elements @xmath145 , with @xmath146 .. ] the cycle structure of these elements is given by a simple rule . we denote by @xmath147 ( with @xmath148 ) the integer divisors of @xmath23 ( in particular @xmath149 and @xmath150 ) , and by @xmath151 the set of integers which are a product of @xmath147 by an integer @xmath37 such that @xmath152 and @xmath153 , is @xmath154 . ] if @xmath155 then @xmath156 consists of @xmath147 entangled cycles of the same length @xmath157 . we denote the corresponding class @xmath158 . the number of elements of @xmath159 , and so the number of such @xmath156 , is equal to @xmath160 , where @xmath35 is euler s totient function whose definition has been recalled in the introduction .. ] consider @xmath161 as an example . the elements of @xmath161 in the class @xmath162 are @xmath163 and @xmath164 . the elements in @xmath165 are @xmath166 and @xmath167 . is not an element of @xmath161 since it is not entangled . ] there is only one element @xmath168 in @xmath169 , and only @xmath170 in @xmath171 . indeed , the integer divisors of @xmath172 are @xmath115 , @xmath173 , @xmath174 , @xmath172 , and we have @xmath175 , @xmath176 , @xmath177 , @xmath178 . @xmath11 has @xmath23 irreps denoted @xmath10 , with @xmath179 . the corresponding characters are given by @xmath180 .. ] we will have to calculate in the following the sums given by : @xmath181 these sums are slight generalizations of ramanujan s sums . corresponds exactly to a ramanujan s sum . ] using theorem @xmath182 of ref . @xcite , we obtain that : @xmath183 where @xmath78 is supposed to be in @xmath184 and @xmath31 is given by @xmath185 . the mbius function @xmath34 has been defined in the introduction . note that all @xmath78 which are in the same @xmath184 lead to the same sum ; we can therefore restrain ourselves to @xmath78 equal to an integer divisor of @xmath23 in order to have the different values of these sums . indeed , we will label the different amplitudes at level @xmath23 by @xmath31 . by generalising the working for the cyclic case , we can now obtain a decomposition of the @xmath22 in terms of the @xmath89 . to that end , we first determine the number of states @xmath186 which are _ compatible _ with a given configuration of @xmath89 , i.e. , the number of initial states @xmath186 which are thus that the action by the given configuration produces an identical final state . the notion of compatibility is illustrated in fig . [ fig2 ] . which are compatible with a given cluster configuration contributing to @xmath187 . ] we consider first the case @xmath69 and suppose that the @xmath78th ntc connects onto the points @xmath188 . the rules for constructing the compatible @xmath186 are identical to those of the cyclic case : 1 . the points @xmath189 must be connected in the same way in @xmath186 as in the cluster configuration . 2 . the points @xmath188 within the same bridge must be connected in @xmath186 . 3 . one can independently choose to associate or not a black point to each of the sets @xmath188 . one is free to connect or not two distinct sets @xmath188 and @xmath190 . the choices mentioned in rule 3 leave @xmath191 possibilities for constructing a compatible @xmath186 . the coefficient of @xmath192 in the decomposition of @xmath22 is therefore @xmath193 , since the allowed permutation of black points in a standard vector @xmath186 allows for the construction of @xmath23 distinct states , and since the weight of the @xmath83 ntc in @xmath22 is @xmath115 instead of @xmath194 . it follows that k_l = _ j = l^l l n_tor(j , l ) n_1=1 . which are compatible with a given cluster configuration contributing to @xmath195 . ] we next consider the case @xmath196 . let us denote by @xmath197 the points that connect onto the @xmath31th branch of the @xmath78th ntc ( with @xmath198 and @xmath199 ) , and by @xmath200 all the points that connect onto the @xmath78th ntc . as shown in fig . [ fig3 ] , the @xmath186 which are compatible with this configuration are such that 1 . the connectivities of the points @xmath201 are identical to those appearing in the cluster configuration . 2 . all points @xmath197 corresponding to the branch of a ntc must be connected . 3 . we must now count the number of ways we can link the branches of the @xmath78 ntc and attribute @xmath23 black points so that the connection _ and the position of the black points _ are unchanged after action of the cluster configuration . for @xmath28 , there are no compatible states ( indeed it is not possible to respect planarity and to leave the position of the black points unchanged ) . for @xmath27 and @xmath25 there are respectively @xmath202 and @xmath203 compatible states . note that these results do not depend on the precise value of @xmath66 ( for @xmath196 ) . the rule @xmath174 implies that the decomposition of @xmath22 with @xmath28 does not contain any of the @xmath89 with @xmath196 . we therefore have simply @xmath204 the decomposition of @xmath205 and @xmath206 are given by : @xmath207 @xmath208 note that the coefficients in front of @xmath89 do not depend on the precise value of @xmath66 when @xmath196 . to simplify the notation we have defined @xmath209 . since the coefficients in front of @xmath192 and @xmath210 in eqs . ( [ expk1tor])([expk0tor ] ) are different , we can not invert the system of relations ( [ expkltor])([expk0tor ] ) so as to obtain @xmath211 in terms of the @xmath22 . it is thus precisely because of ntc with several branches contributing to @xmath210 that the problem is more complicated than in the cyclic case . in order to appreciate this effect , and compare with the precise results that we shall find later , let us for a moment assume that eq . ( [ expkltor ] ) were valid also for @xmath212 . we would then obtain @xmath213 where the coefficients @xmath214 have already been defined in eq . ( [ defbltori ] ) . the coefficients @xmath56 play a role analogous to those denoted @xmath215 in the cyclic case @xcite ; note also that @xmath216 for @xmath217 . chang and schrock have developed a diagrammatic technique for obtaining the @xmath56 @xcite . supposing still the unconditional validity of eq . ( [ expkltor ] ) , one would obtain for the full partition function @xmath218 this relation will be modified due to the terms @xmath210 realising permutations of the black points , which we have here disregarded . to get things right we shall introduce irrep dependent coefficients @xmath219 and write @xmath220 . neglecting @xmath210 terms would lead , according to eq . ( [ devztorosim ] ) , to @xmath221 independently of @xmath114 . we shall see that the @xmath210 will lift this degeneracy of amplitudes in a particular way , since there exist certain relations between the @xmath219 and the @xmath56 . in order to simplify the formulas we will obtain later , we define the coefficients @xmath222 for @xmath223 by : @xmath224 for @xmath28 , @xmath222 is simply equal to @xmath56 , they are different only for @xmath27 , as we have @xmath26 but @xmath225 . in order to reobtain the expression ( [ eqsal ] ) of read and saleur for the amplitudes we will use that : @xmath226 where @xmath46 has been defined in eq . ( [ defe0 ] ) . the relations ( [ expkltor])([expk0tor ] ) were not invertible due to an insufficient number of elementary quantities @xmath22 . let us now show how to produce a development in terms of @xmath127 , i.e. , taking into account the possible permutations of black points . this development turns out to be invertible . a standard connectivity state with @xmath23 black points is said to be _ @xmath129-compatible _ with a given cluster configuration if the action of that cluster configuration on the connectivity state produces a final state that differs from the initial one just by a permutation @xmath129 of the black points . this generalises the notion of compatibility used in sec . [ sec : charkl ] to take into account the permutations of black points . let us first count the number of standard connectivities @xmath186 which are @xmath129-compatible with a cluster configuration contributing to @xmath85 . for @xmath69 , @xmath227 contains only the identity element @xmath228 , and so the results of sec . [ sec : charkl ] apply : the @xmath192 contribute only to @xmath130 . we consider next a configuration contributing to @xmath85 with @xmath196 . the @xmath186 which are @xmath129-compatible with this configuration satisfy the same three rules as given in sec . [ sec : charkl ] for the case @xmath196 , with the slight modification of rule 3 that the black points must be attributed in such a way that _ the final state differs from the initial one by a permutation @xmath129_. this modification makes the attribution of black points considerably more involved than was the case in sec . [ sec : charkl ] . first note that not all the @xmath129 are admissible . to be precise , the cycle decomposition of the allowed permutations can only contain @xmath72 , as @xmath72 is the permutation between the branches realised by a single ntc . therefore the admissible permutations contain only @xmath72 and are such that @xmath229 , denoting by @xmath147 the number of times @xmath72 is contained . we note @xmath135 the corresponding classes of permutations and @xmath137 the corresponding @xmath230 , see eq . ( [ defklctor ] ) . note that the number of classes of admissible permutations at a given level @xmath23 is equal to the number of integers @xmath147 dividing @xmath23 , i.e. @xmath29 . furthermore , inside these classes , not all permutations are admissible . indeed , the entanglement of the ntc imply the entanglement of the structure of the allowed permutations . we deduce from all this rules that , as announced , the admissible permutations at level @xmath23 are simply the elements of the cyclic group @xmath11 . which are @xmath231-compatible with a given cluster configuration contributing to @xmath195 . the action of the cluster configuration on these connectivity states permutes the positions of the two black points . ] let us now consider the decomposition of @xmath127 , which is depicted in fig . [ fig4 ] , @xmath129 being an authorized permutation different from identity and containing @xmath147 times the permutation @xmath72 of length @xmath66 . then , only the @xmath85 , with @xmath232 , contribute to the decomposition of @xmath127 . we find that the number of @xmath186 which are @xmath129-compatible with a given clusters configuration of @xmath85 is @xmath233 . is simply @xmath234 for @xmath235 but is different for @xmath236 , see eq . ( [ defntor ] ) . ] therefore we have : @xmath237 from this we infer the decomposition of @xmath137 : @xmath238 we will use the decomposition of @xmath137 in the following as it is simplier to work with @xmath89 than with @xmath85 ( but one could consider the @xmath85 too ) . it remains to study the special case of @xmath239 . this is in fact trivial . indeed , in that case , the value of @xmath66 in @xmath89 is no longer fixed , and one must sum over all possible values of @xmath66 , taking into account that the case of @xmath69 is particular . since @xmath240 , one obtains simply eqs . ( [ expkltor])([expk0tor ] ) of sec . [ sec : charkl ] up to a global factor . to obtain the decomposition of @xmath89 in terms of the @xmath127 , we invert eq . ( [ knpn1 ] ) for varying @xmath147 and fixed @xmath196 and we obtain : @xmath241 since the coefficients in this sum do not depend on @xmath66 ( provided that @xmath196 ) , we can sum this relation over @xmath66 and write it as @xmath242 where we recall the notations @xmath243 and @xmath244 , corresponding to permutations consisting of @xmath147 cycles of the same length @xmath245 . consider next the case @xmath69 . for @xmath246 one has simply @xmath247 recalling eq . ( [ expzj1i ] ) and the fact that for @xmath248 the @xmath210 do not appear in the decomposition of @xmath22 . however , according to eqs . ( [ expk1tor])([expk0tor ] ) , the @xmath210 do appear for @xmath25 and @xmath27 , and one obtains @xmath249 inserting the decomposition ( [ expzjn1>1 ] ) of @xmath210 into eq . ( [ expz11i ] ) one obtains the decomposition of @xmath250 over @xmath251 and @xmath137 : @xmath252 we proceed in the same fashion for the decomposition of @xmath253 , finding @xmath254 upon insertion of the decomposition ( [ expzjn1>1 ] ) of @xmath210 , one arrives at @xmath255 since @xmath256 , we conclude from eqs . ( [ expzj12])([expzjn1>1 ] ) and from eq . ( [ defbltil ] ) that , for any @xmath83 , @xmath257 the decomposition of @xmath258 is therefore @xmath259 the culmination of the preceeding section was the decomposition ( [ expzj ] ) of @xmath90 in terms of @xmath127 ( as @xmath137 is the sum of the @xmath127 with @xmath129 being an element of @xmath11 belonging to the class @xmath135 ) . however , it is the @xmath120 which are directly related to the eigenvalues of the transfer matrix @xmath98 . for that reason , we now use the relation ( [ klcfd ] ) between the @xmath127 and the @xmath62 to obtain the decomposition of @xmath90 in terms of @xmath62 . the result is : @xmath260 where the coefficients @xmath261 are given by @xmath262 indeed , @xmath263 , and since @xmath137 corresponds to the level @xmath136 , we have @xmath264 . ( recall that @xmath135 is the class of permutations consisting of @xmath147 cycles of the same length @xmath265 . ) as explained in sec . [ sec : coef_bl ] , the @xmath261 are not simply equal to @xmath266 because of the @xmath196 terms . using eq . ( [ relcdc ] ) we find that they nevertheless obey the following relation @xmath267 but from eq . ( [ defbldj ] ) the @xmath261 with @xmath268 are trivial , i.e. , equal to @xmath266 independently of @xmath114 . this could have been shown directly by considering the decomposition ( [ expkltor ] ) of @xmath22 . the decomposition of @xmath21 over @xmath62 is obviously given by @xmath269 where @xmath270 i.e. @xmath271 this is the central result of our article : we have obtained a rather simple expression of the amplitudes @xmath219 in terms of the characters of the irrep @xmath114 . a priori , for a given level @xmath23 , there should be @xmath23 distinct amplitudes @xmath219 because @xmath11 has @xmath23 distinct irreps @xmath114 . however , because of the fact that two different permutations in the same class @xmath272 correspond to the same coefficient @xmath273 , there are less distinct amplitudes : some @xmath219 are the same . indeed , the eq . ( [ defbld ] ) giving the amplitudes of the eigenvalues contains generalized ramanujan s sum , so using the subsection [ clprop ] , the @xmath10 whose @xmath78 are in the same @xmath274 correspond to the same amplitude @xmath275 . for example , at level @xmath172 , there are only four distinct amplitudes : @xmath276 , @xmath277 , @xmath278 and @xmath279 , since we have @xmath280 and @xmath281 . an important consequence of the expression of the @xmath6 is that they satisfy @xmath282 i.e. , the sum of the @xmath23 ( not necessarily distinct ) amplitudes @xmath6 at level @xmath23 is equal to @xmath56 . this has been previously noted by chang and shrock @xcite , except that they stated it was the sum of @xmath283 amplitudes , not @xmath23 , as they did not notice that only permutations in the cyclic group @xmath11 were admissible . note also that for @xmath28 , eq . ( [ defbld ] ) can be written more simply as : @xmath284 since @xmath285 for @xmath28 . we now restrict ourselves to this case , as the amplitudes at levels @xmath286 and @xmath115 are simply @xmath24 and @xmath26 . we now calculate the ramanujan s sums appearing in eq . ( [ def2bld ] ) . using eq . ( [ passage ] ) , we obtain : @xmath287 remember that @xmath31 is given by @xmath185 for @xmath78 in the set @xmath184 , and so is an integer divisor of @xmath23 . using the expression of the @xmath288 given in eq . ( [ def2bltil ] ) , we finally recover the formula ( [ eqsal ] ) of read and saleur . in particular , the term @xmath289 in the definition ( [ defbltil ] ) of @xmath222 corresponds to degenerate cluster configurations . note that the number of different amplitudes at level @xmath23 is simply equal to the number of integer divisors of @xmath23 . in particular , when @xmath23 is prime , there are only two different amplitudes : @xmath290 which corresponds to @xmath291 ( @xmath292 is the identity representation ) and @xmath293 which corresponds to the @xmath294 other @xmath6 ( as they are all equal ) . using that @xmath295 , we find : @xmath296 this could have been simply directly showed using eq . ( [ def2bld ] ) . indeed , for @xmath23 prime , @xmath11 contains @xmath81 and @xmath294 cycles of length @xmath23 . as @xmath295 , we deduce that @xmath297 . for @xmath293 , one needs just use that @xmath298 . to summarise , we have generalised the combinatorial approach developed in ref . @xcite for cyclic boundary conditions to the case of toroidal boundary conditions . in particular , we have obtained the decomposition of the partition function for the potts model on finite tori in terms of the generalised characters @xmath120 . we proved that the formula ( [ eqsal ] ) of read and saleur is valid for any finite lattice , and for any inhomogeneous choice of the coupling constants . furthermore , our physical interpretation of this formula is new and is based on the cyclic group @xmath11 . the eigenvalue amplitudes are instrumental in determining the physics of the potts model , in particular in the antiferromagnetic regime @xcite . generically , this regime belongs to a so - called berker - kadanoff ( bk ) phase in which the temperature variable is irrelevant in the renormalisation group sense , and whose properties can be obtained by analytic continuation of the well - known ferromagnetic phase transition @xcite . due to the beraha - kahane - weiss ( bkw ) theorem @xcite , partition function zeros accumulate at the values of @xmath0 where either the amplitude of the dominant eigenvalue vanishes , or where the two dominant eigenvalues become equimodular . when this happens , the bk phase disappears , and the system undergoes a phase transition with control parameter @xmath0 . determining analytically the eigenvalue amplitudes is thus directly relevant for the first of the hypotheses in the bkw theorem . for the cyclic geometry , the amplitudes are very simple , and the real values of @xmath0 satisfying the hypothesis of the bkw theorem are simply the so - called beraha numbers , @xmath299 with @xmath300 , independently of the width @xmath20 . for the toroidal case , the formula is more complicated , and there can be degeneracies of eigenvalues between different levels which depend on the width @xmath20 of the lattice , as shown by chang and shrock @xcite . the role of the beraha numbers will therefore be considered in a future work .

we consider the @xmath0-state potts model in the random - cluster formulation , defined on _ finite _ two - dimensional lattices of size @xmath1 with toroidal boundary conditions . due to the non - locality of the clusters , the partition function @xmath2 can not be written simply as a trace of the transfer matrix @xmath3 . using a combinatorial method , we establish the decomposition @xmath4 , where the characters @xmath5 are simple traces . in this decomposition , the amplitudes @xmath6 of the eigenvalues @xmath7 of @xmath3 are labelled by the number @xmath8 of clusters which are non - contractible with respect to the transfer ( @xmath9 ) direction , and a representation @xmath10 of the cyclic group @xmath11 . we obtain rigorously a general expression for @xmath6 in terms of the characters of @xmath11 , and , using number theoretic results , show that it coincides with an expression previously obtained in the continuum limit by read and saleur .

the system simulated . ( a ) : schematic of the bec collision in real space in the lab frame . ( b ) : slice of the velocity distribution @xmath98 in the center - of - mass frame at @xmath99 and @xmath100s calculated using the positive - p method . this is about a third of the collision time , and the maximum time achievable with that method . the condensates are located around @xmath101mm / s . the halo of scattered atoms is clearly seen , as are the coherent frequency doubling peaks at @xmath102mm / s . the collision is along the @xmath103 axis . [ [ the - relationship - of - the - hybrid - mathcalh_a - to - s - ordered - operators ] ] the relationship of the hybrid @xmath31 to @xmath74-ordered operators ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ first , a brief exposition of the standard formalism used in deriving phase - space quantum dynamics will be necessary . writing the state of the system as a density matrix @xmath1 , it can also be expressed as a distribution @xmath104 over a family of basis operators @xmath105 parameterised by variables in the set @xmath4 . if the distribution @xmath106 is real and non - negative , this corresponds , in turn , to an ensemble of @xmath107 sets of random variables @xmath4 ( `` configurations '' ) chosen according to the distribution @xmath108 , in the limit when @xmath109 . in practice one computes a finite but large ensemble ( @xmath110 ) and knows properties of @xmath1 to within a statistical uncertainty that can be confidently estimated from the properties of the finite ensemble . the dynamics of the system is described by the master equation @xmath111,\ ] ] while expectation values of observables are @xmath112.\ ] ] these are most readily related to the computational ensemble of random variables through the use of the `` operator identities '' , that are specific to each formulation . for example , in the positive - p method one chooses @xmath3 to be an off - diagonal coherent - state operator . letting @xmath46 label discrete points in the computational lattice with @xmath113 volume per point , defining @xmath114 one has @xmath115 where @xmath116 , @xmath117 is a coherent state on the @xmath46 lattice point with the complex amplitude @xmath118 and anihilation operator @xmath119 . then , one finds ( omitting ubiquitous local @xmath46 dependence ) the operator identities : @xmath120{\widehat{\lambda}}_{pp}\\ { \widehat{\lambda}}_{pp}{\widehat{\psi}^{\dagger}}=\psi_2^*{\widehat{\lambda}}_{pp } \quad&;&\quad { \widehat{\lambda}}_{pp}{\widehat{\psi}}=\left[\psi_1+{\frac{\partial}{\partial\psi_2^*}}\right]{\widehat{\lambda}}_{pp},\end{aligned}\ ] ] which are the source of the positive - p identities in the main text . combined with ( [ rhodef ] ) and ( [ master ] ) these allow one to obtain a partial differential equation for @xmath121 that is equivalent to the full quantum evolution of @xmath122 . for the positive - p representation , this is a fokker - planck equation , and it corresponds exactly to the langevin equations given in ( 5 ) of the main text combining the identities with ( [ obs ] ) and @xmath123=1 $ ] one finds @xmath124 with a function @xmath125 that is obtained from @xmath44 via the operator identities , so that in the calculation it corresponds to an ensemble average of @xmath125 . for example , for @xmath126 , the function is can also be obtained , but gives the same value of @xmath41 in the @xmath109 limit . ] the initial coherent state corresponds to @xmath128 . it has been shown that the glauber - sudarshan p distribution described by a coherent state operator basis @xmath129 ( similar to the positive - p but diagonal ) can be described as the limit of a representation over @xmath74-ordered basis states @xmath130 where @xmath74 can take on continuous values from -1 to 1 , and @xmath131}.\ ] ] here @xmath132 is a kernel operator that becomes the vacuum @xmath133 in the limit of @xmath134 and the local displacement operator is @xmath135 so that coherent states are @xmath136 . it was also shown there that the wigner distribution corresponds to @xmath137 , hence a variation of @xmath74 from 0 to 1 looks like a good candidate to create the @xmath31 hybrid formulation between truncated wigner and positive - p . the `` truncation '' refers to ad - hoc removal of third order partial derivatives of the wigner distribution @xmath108 in its evolution equation to make it interpretable as langevin stochastic equations of the samples . this removal is the reason why truncated wigner treatments do not reproduce the full quantum dynamics . first , though , one must take into account the off - diagonality that is responsible for the difference between the glauber - sudarshan p and positive - p : @xmath138 . notably one of the bases such as e.g. @xmath139 $ ] can also reproduce the positive - p formulation but are not useful for generalisation to @xmath140 , and do not reproduce the same itermediate operator identities . ] that reproduces the positive - p is @xmath141}\nonumber\\ & = & \prod_{{{\mathbf{x}}}}{\widehat{d}}(\vec{v})_{{{\mathbf{x}}}}{\widehat{t}}(0,-1)_{{{\mathbf{x}}}}{\widehat{d}}^{-1}(\vec{v})_{{{\mathbf{x}}}}\label{lambdapp}\end{aligned}\ ] ] where the `` displacement - like '' operator @xmath142 is obtained by the replacement @xmath143 in @xmath144 , and the second line follows because the trace in the denominator evaluates to one . the reason for this particular replacement is that for the positive - p distribution one requires @xmath3 to depend _ analytically _ on two separate complex variables , hence their complex conjugates must be removed . here these analytic variables are @xmath145 and @xmath146 . the extension of this @xmath3 onto a family of @xmath74-ordered bases is @xmath147}\nonumber\\ & = & \prod_{{{\mathbf{x}}}}{\widehat{d}}(\vec{v})_{{{\mathbf{x}}}}{\widehat{t}}(0,-s)_{{{\mathbf{x}}}}{\widehat{d}}^{-1}(\vec{v})_{{{\mathbf{x}}}}.\label{habase}\end{aligned}\ ] ] this then interpolates towards the wigner representation . note that since the truncated wigner evolution is deterministic , then if one takes the formally off - diagonal basis set with @xmath137 but imposes @xmath148 in the initial conditions , it will remain exactly equivalent to the normal truncated wigner formulation of ( [ opt ] ) with @xmath137 . one obtains the identities is expanded in number states . ] @xmath149{\widehat{\lambda}}^{{{\mathcal{a}}}}_s\\ { \widehat{\psi}^{\dagger}}{\widehat{\lambda}}^{{{\mathcal{a}}}}_s&=&\left[\psi_2^*+\frac{1+s}{2}{\frac{\partial}{\partial\psi_1}}\right]{\widehat{\lambda}}^{{{\mathcal{a}}}}_s \\ { \widehat{\lambda}}^{{{\mathcal{a}}}}_s{\widehat{\psi}^{\dagger}}&=&\left[\psi_2^ * -\frac{1-s}{2}{\frac{\partial}{\partial\psi_1}}\right]{\widehat{\lambda}}^{{{\mathcal{a}}}}_s\\ { \widehat{\lambda}}^{{{\mathcal{a}}}}_s{\widehat{\psi}}&=&\left[\psi_1+\frac{1+s}{2}{\frac{\partial}{\partial\psi_2^*}}\right]{\widehat{\lambda}}^{{{\mathcal{a}}}}_s\end{aligned}\ ] ] which are exactly the same as was obtained by a naive blending of the operator identities in the main text provided we identify @xmath150 . regarding initial conditions , the diagonal @xmath74-ordered representation ( [ opt ] ) for a coherent state @xmath151 was found by cahill and glauber to be gaussian @xmath152 when one additionally imposes @xmath153 as is done in the main text , this is equivalent to ( [ habase ] ) , justifying the initial conditions given in the main text that contain complex gaussian noise of variance @xmath154 . @xmath9-dependent predictions of halo density ( at @xmath32 , @xmath33mm / s in velocity space ) for several times ( circles ) with uncertainty shown as vertical bars at the same location . the corresponding fits ( dashed ) are quadratic for the @xmath78 hybrid , and constant - value for @xmath31 . fitting is via minimisation of rms deviation in units of @xmath83 data uncertainty . linear or quadratic fits to the @xmath31 hybrid data are not more statistically significant than the constant - value fit , and hence would be poorly conditioned . predictions of halo density ( at @xmath32 , @xmath33mm / s in velocity space ) from hybrids @xmath31 and @xmath78 compared with short - time full quantum dynamics and approximate methods . triple lines , where visible , are @xmath83 uncertainty . prediction data based on @xmath95 values of @xmath9 , each with @xmath84 trajectories , and quadratic / constant - value fitting for @xmath31 / @xmath78 hybrids , respectively . note the agreement with truncated wigner to within statistical uncertainty . times detailed in the previous figure ( above ) are highlighted . predictions of the number of scattered atoms at several times , as a function of the @xmath9 segment @xmath155 $ ] used for extrapolation from a quadratic fit to @xmath31 results . triple lines , where visible , are @xmath83 uncertainty . dashed lines indicate the final predictions using all the available @xmath9 values . data used was from the same simulations as in fig . 2 of the main text . there is no statistically significant trend with @xmath156 visible , suggesting that the fitting function that is a quadratic polynomial in @xmath9 is appropriate within statistical precision .

a controlled hybridization between full quantum dynamics and semiclassical approaches ( mean - field and truncated wigner ) is implemented for interacting many - boson systems . it is then demonstrated how simulating the resulting hybrid evolution equations allows one to obtain the full quantum dynamics for much longer times than is possible using an exact treatment directly . a collision of sodium becs with @xmath0 atoms is simulated , in a regime that is difficult to describe semiclassically . the uncertainty of physical quantities depends on the statistics of the full quantum prediction . cutoffs are minimised to a discretization of the hamiltonian . the technique presented is quite general and extension to other systems is considered . the calculation of the full quantum dynamics of a many - body interacting system from the microscopic description is a long - standing `` difficult '' problem with potential applications in many fields of physics if only one could make it numerically tractable . the difficulty is that the size of the hilbert space grows exponentially with the number of particles or orbitals , while path integral monte carlo is foiled by the rapid appearance of random phases . how new headway against this problem can be made will be demonstrated below . outside of fully integrable systems or 1d , where mps / dmrg - based methods are successful , simplified descriptions are used , e.g. mean - field theory , bogoliubov diagonalization , long - wavelength or strong interaction expansions , and wigner - distribution based `` c - field '' methods@xcite . however , some interesting problems fall outside the regimes of validity of these , typically where several competing effects are important or there is a transition between regimes that require different approximations . in quantum gases this occurs with rising density when interactions between the coherent component and incoherent particles already become of essence during the evolution , but the gas is not yet dense enough for the c - field descriptions to describe it with only highly occupied modes . ( see @xcite for a comprehensive review of c - field methods and their validity ) . this may occur e.g. in quenches of the gas@xcite , colliding becs@xcite , dynamics of the cooling and trapping , shock waves and the effects of obstacles@xcite or disorder@xcite . this kind of dynamics is often amenable to phase - space approaches that randomly sample the full quantum dynamics , such as positive - p@xcite , stochastic wavefunctions@xcite , and stochastic gauges@xcite . they are successful when collective behaviour is important , but interactions between individual particles are not too strong . the density matrix @xmath1 of the system is re - described in terms of a probability distribution @xmath2 of basis operators @xmath3 that is subsequently randomly sampled . these samples @xmath4 are then evolved according to stochastic evolution equations that are chosen to keep the entire quantum dynamics of the microscopic description . a serious limitation is the `` noise catastrophe '' : after some finite time , an exponential ( or faster ) growth of the noise variance occurs , imposing a maximum feasible simulation time @xmath5@xcite . while some phenomena can be simulated@xcite , an extension of @xmath5 is much sought - after , and will be demonstrated here . the underlying reasons why phase - space methods can overcome the hilbert space complexity , are that quantities of physical interest usually involve contributions from many particles , and that limited precision is sufficient if it is well controlled . as in monte - carlo methods , there is no need to follow the amplitudes of all possible configurations as long as one can predict physical quantities with a _ well - controlled uncertainty_. however and now we come to the central idea to be demonstrated here this can be taken further : there is also no true need to actually follow the troublesome exact quantum evolution equations provided that one can still predict what they would give _ with a well - controlled uncertainty_. how can such a roundabout prediction be achieved ? if one has at one s disposal two , or more , independent approximate methods that produce evolution equations `` @xmath6 '' and `` @xmath7 '' without a noise catastrophe , but which bear sufficient resemblance to the full quantum dynamics equations `` @xmath8 '' , then hybrid equations can be constructed ( possibly ad - hoc ) with a continuous blending parameter @xmath9 in a scheme resembling @xmath10 whose details will be non - universal . here @xmath11 gives full quantum dynamics , and @xmath12 the original approximate methods . the hybrids will still contain a noise catastrophe , _ but at a later time _ than the full quantum treatment @xmath8 . therefore , long times @xmath13 that are not accessible by @xmath8 will be accessible by some range of @xmath14 $ ] . if a physical quantity varies smoothly , preferably monotonically , as a function of @xmath9 for hybrid @xmath15 , then an extrapolation can be made to @xmath11 , based on several calculations in the accessible range @xmath16 $ ] . one extrapolation is not yet very convincing , however , it can be checked using the other independent hybrids @xmath17 . when they all agree , one has an `` interpolation between extrapolations '' that is robust and much more reliable . conceptually this step is similar to comparing results obtained using different summation techniques in diagrammatic monte - carlo calculations@xcite . the remainder of this letter will demonstrate this procedure on a system of colliding becs ( schematic shown in @xcite ) . the parameters are chosen to be close to an early experiment at mit@xcite , but deliberately with fewer atoms , to put the system in the dilute yet bose - stimulated regime where truncated wigner and simple quasiparticle methods fail : an @xmath18 atom bec of @xmath19 is prepared in an elongated magnetic trap with frequencies @xmath20 hz , at a temperature low enough to discount the thermal component ( not unusual in experiments ) . a brief bragg laser pulse coherently imparts a velocity kick of @xmath21 to half the atoms along the long ( x ) condensate axis . the speed of the kicked atoms is supersonic ( sound velocity in the cloud is @xmath22 mm / s ) . the trap is simultaneously turned off so that the wave - packets collide freely , producing a halo of scattered atom pairs moving at speeds @xmath23 relative to the overall centre of mass . this scattered halo exhibits a rich behaviour , which has been the repeated focus of experiments@xcite and theory@xcite . the high - density regime of a similar system has been treated in detail with c - field methods in @xcite . bogoliubov expansions and/or a pair - creation simplification treat the spontaneous regime , or special cases when bec evolution is negligible or speed is highly supersonic@xcite ( a stochastic bogoliubov treatment gives promising results in broader cases@xcite ) . however , major discrepancies between predictions for halo density and correlations arise when bec evolution or bose stimulation is appreciable . correlations depend on the sizes of phase grains@xcite , which develop a complicated and poorly understood shape@xcite and dynamics@xcite in this case . parallels to unresolved questions in other fields of physics have been noted , such as the `` hbt puzzle '' in heavy ion collisions@xcite . trustworthy calculations that reach the end of the collision ( observed in experiments@xcite but not reached by positive - p@xcite ) could shed light on all these issues . fig . [ fig - scat - nhalo ] includes predictions from gross - pitaevskii ( gp ) mean field , truncated wigner , and positive - p calculations . the time reachable by positive - p ( @xmath24 ) is less than a half of the collision time @xmath25s , and both gp and wigner give an error . the first does not treat scattering , while for a lattice fine enough to encompass all physics the second becomes valid only for @xmath26 atoms ( one needs @xmath27 atoms per lattice site@xcite ) . n.b . the @xmath28-dependent difference between @xmath29 and its effective lattice value@xcite is @xmath30 here , so it has not been corrected for . wigner ( purple ) , positive - p ( red ) , gp ( dashed ) and hybrid @xmath31 calculations at various blending parameters @xmath9 . ( a ) : total number of scattered atoms , from integration of k - space density ( excluding the narrow bec region ) . ( b ) : peak density of the halo ( at @xmath32 , @xmath33mm / s in velocity space ) . triple lines show 1@xmath34 uncertainty . , title="fig : " ] wigner ( purple ) , positive - p ( red ) , gp ( dashed ) and hybrid @xmath31 calculations at various blending parameters @xmath9 . ( a ) : total number of scattered atoms , from integration of k - space density ( excluding the narrow bec region ) . ( b ) : peak density of the halo ( at @xmath32 , @xmath33mm / s in velocity space ) . triple lines show 1@xmath34 uncertainty . , title="fig : " ] now let us turn to obtaining the full quantum dynamics for times longer than with the positive - p . the dynamics equations in the truncated wigner , gp , and positive - p treatments share the gp kernel with certain additions , and turn out similar enough to play the role of the @xmath6 , @xmath7 , and @xmath8 . the dynamical gp equation for the complex field @xmath35 corresponding to the cold atom hamiltonian @xmath36 $ ] is @xmath37\psi({{\mathbf{x}}})$ ] . an initial condensate wavefunction @xmath38 normalised to @xmath39 leads to initial conditions @xmath40 . expectation values of observables @xmath41 are calculated by making the replacements @xmath42 and @xmath43 in @xmath44 . for example , the density is @xmath45 . in the truncated wigner method , the dynamics is obtained by standard methods ( e.g.@xcite ) based on the basis operator identities ( @xmath46 dependence implied ) @xmath47{\widehat{\lambda}}\ \ ; \ \ { \widehat{\psi}^{\dagger}}{\widehat{\lambda}}=\left[\psi^*+\frac{1}{2}{\frac{\partial}{\partial\psi}}\right]{\widehat{\lambda}}\ ] ] whose importance for us will be seen below . the equation of motion is as for gp but with the replacement @xmath48 on the rhs . however , in the initial conditions the condensate field is admixed with half a virtual particle per mode as @xmath49 , where @xmath50 is a local complex gaussian noise with the ensemble averages @xmath51 and @xmath52 . to calculate observables one ensemble averages a modified expression @xmath53 $ ] that is obtained via @xmath54 = \int\!d\vec{v}p(\vec{v})\operatorname{tr}\left[{\widehat{o}}{\widehat{\lambda}}\right]$ ] and subsequent replacements ( [ wigop ] ) , which give @xmath55 . e.g. @xmath56 . the positive - p method uses two independent fields @xmath57 and @xmath58 and the identities @xmath59{\widehat{\lambda } } , \vspace*{1pt}\\ { \widehat{\lambda}}{\widehat{\psi}^{\dagger}}&\psi_2^*{\widehat{\lambda } } & { \widehat{\lambda}}{\widehat{\psi}}&\left[\psi_1+{\frac{\partial}{\partial\psi_2^*}}\right]{\widehat{\lambda}}. \end{array}\ ] ] the @xmath60 obey the ito stochastic equations @xmath61&\!\!\!\psi_1({{\mathbf{x}}})\\ i\hbar\dot{\psi}_2({{\mathbf{x}}})&\!\!=&\!\!\left [ h_{\rm sp}({{\mathbf{x } } } ) + g\rho({{\mathbf{x}}})^*-i\sqrt{ig}\,\xi_2({{\mathbf{x}}},t ) \right]&\!\!\!\psi_2({{\mathbf{x } } } ) \end{array}\hspace*{-0.1cm}\ ] ] with `` complex density '' @xmath62 . here the @xmath63 are delta - correlated real gaussian noise fields with the ensemble averages @xmath64 and @xmath65 . initial conditions are @xmath66 and observables are obtained with the replacements @xmath67 and @xmath68 . the next step will be to hybridize the truncated wigner with the positive - p into treatment @xmath31 . it is most straightforward to proceed from hybrid operator identities for an off - diagonal expansion @xmath69{\widehat{\lambda } } & { \widehat{\psi}^{\dagger}}{\widehat{\lambda}}&\left[\psi_2^*+\frac{1+\lambda}{2}{\frac{\partial}{\partial\psi_1}}\right]{\widehat{\lambda } } \vspace*{1pt}\\ { \widehat{\lambda}}{\widehat{\psi}^{\dagger}}&\left[\psi_2^ * -\frac{1-\lambda}{2}{\frac{\partial}{\partial\psi_1}}\right]{\widehat{\lambda } } & { \widehat{\lambda}}{\widehat{\psi}}&\left[\psi_1+\frac{1+\lambda}{2}{\frac{\partial}{\partial\psi_2^*}}\right]{\widehat{\lambda } } \end{array}\ ] ] one obtains : @xmath70 and initial @xmath71 . the usual truncated - wigner - like discarding of high - order derivatives in the relevant fokker - planck equations , gives dynamics @xmath72&\!\!\!\psi_1({{\mathbf{x}}})\\ i\hbar\dot{\psi}_2({{\mathbf{x}}})&\!\!=&\!\!\left[h_{\rm sp}({{\mathbf{x}}})+g\rho'({{\mathbf{x}}})^*-i\sqrt{ig\lambda}\,\xi_2({{\mathbf{x}}},t)\right]&\!\!\!\psi_2({{\mathbf{x } } } ) \end{array}\ ] ] with @xmath73 . as an aside , this corresponds to a representation based on an off - diagonal operator basis using @xmath74-ordered@xcite coherent - like states with @xmath75 ( see @xcite for details ) . fig . [ fig - scat - nhalo ] shows the performance of this hybrid for several values of @xmath9 for two halo quantities of interest . as desired , @xmath76 calculations last for longer than the full quantum dynamics . here the simulation time scales as @xmath77 , but this is not universal . hybridization of the gp and positive - p methods into treatment @xmath78 simply entails replacing @xmath79 by @xmath80 in the equations ( [ ppeq ] ) and following the positive - p prescription from then on . here @xmath81 . with hybrids in hand , extrapolations of the total number of scattered atoms to the full qd limit @xmath11 are shown in fig . [ fig - extrap ] for several times @xmath82 . halo peak density is in@xcite . @xmath9-dependent predictions for several times @xmath82 ( symbols ) and corresponding quadratic fits ( dashed line ) . fitting is via minimisation of rms deviation in units of @xmath83 data uncertainty . data points use @xmath84 trajectories . ] an issue here is deciding upon a fitting function linear , quadratic , otherwise ? firstly , an acceptable fit must not have any statistically significant mismatch with the data . secondly , to exclude spurious ill - conditioned parameters , one should choose a fit that minimises the uncertainty in the extrapolated value at @xmath11 ( see below ) . one must also beware of possible stiffness in the unseen @xmath9 , and sensitivity to this is the primary reason why several independent hybrids are needed . details of fig . [ fig - extrap ] are consistent with a lack of stiffness in the unsimulated large @xmath9 region : firstly , for @xmath85 at which the whole @xmath9 sequence is seen , there are no inflections . secondly , the two hybrids approach the @xmath11 value from different sides but agree . also , extrapolations from only a low-@xmath9 portion of the available data should agree with ones that use the whole sequence . this is confirmed in @xcite . agreement between the @xmath31 and @xmath78 extrapolations in fig . [ fig - extrap ] is rather good at long times , but it remains to provide a well - defined uncertainty for the final prediction . methods to obtain the statistical uncertainty of the @xmath11 extrapolation are known@xcite . in this endeavour it is very helpful to know the underlying distribution of the data points @xmath86 , which are ensemble averaged observables . conveniently , it is known to be gaussian by the central limit theorem , and the shown 1@xmath34 uncertainty @xmath87 is its standard deviation . one rather simple way to proceed is to generate a number @xmath88 of `` synthetic '' data sets , where in the @xmath89th set one generates @xmath90 , with @xmath63 being gaussian random variables of variance 1 , mean zero . the synthetic data @xmath91 are distributed with the same mean as the original @xmath92 but double the variance . now one calculates an extrapolated qd prediction @xmath93 for @xmath11 for each synthetic set @xmath89 , and uses the distribution of these @xmath93 to obtain the final uncertainty @xmath94 . predictions from @xmath31 and @xmath78 that match within statistical uncertainty are trustworthy to this accuracy . the final predictions from both hybrid methods for the number of scattered atoms are shown in fig . [ fig - predict ] , and for halo density in @xcite . predictions of from hybrids @xmath31 and @xmath78 compared with short - time full quantum dynamics and approximate methods . triple lines , where visible , are @xmath83 uncertainty . uses @xmath95 values of @xmath9 , as per fig . [ fig - extrap ] . ] one sees that the useful simulation time has been extended several - fold , allows one to reach the end of the collision here , and determine the total scattered atoms to be @xmath96 ( at @xmath85=1.7ms ) . the much worse precision of the @xmath31 result stems from the inherent vacuum noise in wigner calculations and shorter segment of @xmath9 values . however , for halo density , it is @xmath78 that is more noisy . regarding limits of applicability , at very long times the uncertainty becomes excessive for all hybrids since the short @xmath9 intervals give badly conditioned extrapolations . hence the bare simulation time in the @xmath8 treatment must not be too small to ensure a sufficiently long @xmath9 interval . it is also crucial that the blending @xmath9 enter the dynamics in a global way : artificial boundaries@xcite could make observables depend stiffly on the boundary position . for cold gases low densities can be treated perturbatively , while at high enough densities c - field treatments are valid , so that one expects that the blending method will be most useful at intermediate densities that `` fall through the cracks '' between these two methods . the relative simplicity of not requiring a projection onto low - energy modes may also make blending appealing in other regimes . finally , while the emphasis has been on cold boson dynamics , the general equation - blending approach should be broadly applicable . for hard - core boson or fermion systems other approximations would have to be hybridised with a different complete phase - space description @xmath8 . one can also hybridise `` imaginary - time '' evolution for thermal equilibrium states , or monte - carlo path - integrals with the aim of predicting the ab - initio result for longer @xmath97 than is normally allowed by the fermion sign problem . _ concluding , _ it has been demonstrated how the full quantum dynamics of a macroscopic interacting 3d system can be calculated for much longer times than was possible with the previously most effective method , the positive - p representation . quantitative predictions for bec collisions in the dilute stimulated regime were obtained . the hybrid dynamical equations used , while not actually simulating complete quantum dynamics _ per se _ , can be used to confidently predict the full quantum dynamics ( within a given accuracy ) when several families of hybrids are available . i am grateful to scott hoffmann , peter drummond , georgy shlyapnikov , boris svistunov , joel corney , anatoli polkovnikov , and evgeny burovskiy for stimulating discussions . this research was supported by the european community under the contract meif - ct-2006 - 041390 . lptms is a mixed research unit no . 8626 of cnrs and universit paris - sud . 99 a. sinatra _ et al . _ , phys . rev . lett . * 87 * , 210404 ( 2001 ) . p. b. blakie , m. j. davis , phys . rev . a * 72 * , 063608 ( 2005 ) . p. b. blakie _ et al . _ , adv . phys . * 57 * , 363 ( 2008 ) . l. e. sadler _ et al . _ , nature * 443 * 312 ( 2006 ) . a. p. chikkatur _ et al._phys . rev . lett * 85 * , 483 ( 2000 ) . a. perrin _ et al . _ , phys . rev . lett . * 99 * , 150405 ( 2007 ) . j. m. vogels _ et al . _ , phys . rev . lett . * 89 * 020401 ( 2002 ) . z. dutton _ et al . _ , science * 293 * , 663 ( 2001 ) ; 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many environments have been considered regarding the origin of deuterium @xcite . they include pregalactic cosmic rays ( crs ) from quasars and collapsing objects , shock waves , and neutron stars . in general , the crs induce nuclear reactions producing d , @xmath5he , li , be , and b nuclides . pregalactic crs or cosmological crs generated before the galaxy formation also produce @xmath6li ( via the @xmath7 fusion @xcite ) and @xmath5he ( via @xmath2he+@xmath8 nuclear spallation @xcite ) . @xmath3li productions have been calculated for the crs in specific environments : the crs accelerated in structure formation shocks at the galaxy formation epoch @xcite and the crs from supernova remnants at the pregalactic epoch @xcite . since a metal pollution proceeds along with a stellar activity in the universe , the crs would come to contain metals such as c , n , and o. therefore , the pregalactic cr nucleosynthesis would also produce be and b through reactions of ( c , n , or o)+(@xmath8 or @xmath9 ) @xcite and ( @xmath5he or @xmath9)+@xmath10he or @xmath6li)+@xmath11 followed by ( @xmath3he or @xmath6li)+@xmath12be+@xmath13 with byproducts @xmath11 and @xmath13 @xcite . another possible source of the cr is an energy injection at decay and annihilation of exotic long - lived particles @xcite . a constraint on the mass of a hypothetical stable heavy neutrino has been derived through calculation of its present cosmological energy density @xcite . an unstable heavy neutrino was then considered , and constraints on its mass and lifetime were derived @xcite . the electromagnetic decay of the unstable particle is constrained through distortions in the energy spectrum of cosmic microwave background radiation @xcite . the constraints on hypothetical heavy neutrino @xcite and primordial black holes @xcite were then derived from the effect on light element abundances through energy densities in detailed calculations of big bang nucleosynthesis ( bbn ) . the decay of unstable heavy neutrinos also affects nuclear abundances through nonthermal photodissociation of nuclei @xcite . the radiative decay induces electromagnetic cascades of energetic photons , electrons , and positrons during the propagation of the nonthermal photon emitted at the decay @xcite . effects of hadronic injections at the decay were studied @xcite . _ investigated hadronic cascades of proton and antiproton and dissociations of @xmath2he @xcite . et al . _ @xcite extensively studied the effects on abundances of nuclei up to @xmath0li and @xmath0be . they considered the reaction , i.e. , @xmath14h(@xmath15)@xmath16h , for d production , and the reaction , i.e. , @xmath0be(@xmath17)@xmath0li , for @xmath0be destruction , where 1(2@xmath183)4 stands for a reaction @xmath19 . antiprotons injected at decays of exotic long - lived particles could dissociate @xmath2he and produce d and @xmath5he @xcite . the cross sections of @xmath20he annihilation have been measured @xcite , and the yields of d , @xmath5h , and @xmath5he at the annihilation were calculated as a function of energy of antiproton @xcite . effects of exotic particles on nuclear abundances through hadronic showers have been extensively studied with realistic initial spectra of injected hadrons @xcite . the standard bbn ( sbbn ) model explains primordial light element abundances inferred from astronomical observations well @xcite . modifications of the bbn model are then constrained from the consistency between theoretical predictions and observations of abundances . among light elements produced during the bbn , however , the lithium has an unexplained discrepancy between sbbn prediction and observational determinations of its primordial abundances @xcite . spectroscopic observations of metal - poor stars ( mpss ) indicate an abundance measured by number relative to hydrogen , i.e. , @xmath0li / h@xmath21 @xcite.23 lower than sbbn prediction.]37934 , among 91 stars of the globular cluster m4 has a high lithium abundance ( @xmath0li / h=@xmath22 ) consistent with the abundance of the sbbn model . ] this abundance is a factor of 24 higher than the sbbn prediction when we adopt the baryon - to - photon ratio determined from the observation of the cosmic microwave background radiation with wilkinson microwave anisotropy probe ( wmap ) @xcite . after the lithium problem was recognized , the neutron injection during the bbn was suggested to be a solution since it can reduce @xmath0be abundance via @xmath0be(@xmath17)@xmath0li(@xmath23)@xmath2he , although it increases d abundance via @xmath14h(@xmath15)@xmath16h simultaneously @xcite . such a neutron injection is realized in the hadronic decay of exotic long - lived massive particles @xcite . important reactions caused by injected nonthermal hadrons have been identified in a statistical study , which are shown to be closely associated with resulting elemental abundances @xcite . a wide parameter region of the lifetime and the abundance of a long - lived particle was studied , and a parameter region for @xmath0li reduction has been found @xcite . and @xmath24 are generated by the particle decays , they can convert protons to neutrons , and a reduction of @xmath0li abundance realizes along with an enhancement of d abundance . when the decays do not generate any mesons , and muons and neutrinos are generated , on the other hand , induced electron antineutrinos convert protons to neutrons . in this case , the dissociation of once enhanced d by nonthermal photons can reduce d abundance to the level consistent with observations . ] in this paper , we focus solely on the parameter region for @xmath0li reduction , and derive a model independent constraint on a relation between abundances of d and @xmath0li , by using recent d abundance data . in sec . [ sec2 ] , we describe input physics and assumptions adopted in this paper . we prove that the assumption of thermal neutron injection ( tni ) leads to a conservative lower limit on the ratio of the increase of d abundance to the decrease of @xmath0li abundance . in sec . [ sec3 ] , we describe the tni model and the bbn model , as well as adopted observational constraint on primordial nuclear abundances . the tni is assumed to occur instantaneously , and the injection time and the abundance of injected neutron are used as parameters in this model . in sec . [ sec4 ] , results of the bbn calculations are shown , and a relation between abundances of d and @xmath0li is derived . [ sec5 ] , we estimate an effect of antinucleon annihilation with @xmath2he on the abundance relation . in sec . [ sec6 ] , we estimate amounts of @xmath3li production induced by the antinucleon+@xmath2he annihilation . in sec . [ sec7 ] , conclusions are done finally . in appendix [ app1 ] , we list important nuclear reactions which work in a parameter region for the reduction of primordial @xmath0li abundance . in appendix [ app2 ] , approximate analytic estimates of d and @xmath0li abundances are shown . in this paper , we adopt notation of @xmath25 with a real number @xmath11 and an integer @xmath26 , and @xmath27 with a parameter @xmath28 and a real number @xmath13 . the boltzmann s constant ( @xmath29 ) , the reduced planck s constant ( @xmath30 ) , and the light speed ( @xmath31 ) are normalized to be unity . in this paper , we concentrate on a production of d and a reduction of @xmath0be and @xmath0li induced by hadronic energy injection at temperature @xmath32 ( or cosmic time @xmath33 this injection epoch corresponds to that of the solution to the @xmath0li problem by the hadronic energy injection model @xcite . the injection produces energetic nucleons , antinucleons , and mesons . such hadrons can scatter background nuclei so that many energetic hadrons are generated and hadronic showers composed of energetic hadrons are developed @xcite . main reactions changing abundances of d and ( @xmath0li+@xmath0be ) li and @xmath0be abundances gives primordial @xmath0li abundance before the start of the early stellar activity . this is because @xmath0be produced at bbn epoch is transformed to @xmath0li by electron capture after the recombination of @xmath0be@xmath34 ion . ] in this parameter range @xcite are @xmath35 @xmath36 respectively , where @xmath37h , @xmath38 , or @xmath39 is a byproduct . if the neutron injection time is @xmath40 s as considered here , effects of long - lived mesons are negligible @xcite . in sec . [ sec21 ] , we comment that energetic proton , antiproton , and nuclei quickly thermalize while an energetic neutron can induce inelastic scatterings off background proton . in sec . [ sec22 ] , we present that the bbn calculation for the case of the tni provides a lower limit on the ratio , i.e. , @xmath41d/@xmath42li@xmath43 , where @xmath44 is a difference between final number densities of nuclide @xmath45 in this model ( @xmath46 ) and the sbbn model ( @xmath47 ) . a more precise estimation of the ratio @xmath41d/@xmath42li@xmath43 should include the annihilation of antineutron with @xmath2he . nuclear data on the annihilation , however , contains a large uncertainty . it is shown that effects of hadronic showers composed of energetic neutron and antineutron always enhance the ratio @xmath41d/@xmath42li@xmath43 above that of the tni model . in sec . [ sec23 ] , we see that the assumption of instantaneous thermalization of nonthermal neutron leads to a lower limit on the ratio . we assume instantaneous thermalizations of nonthermal @xmath8 , @xmath48 , and nuclei for the following reason . an inelastic scattering of two nucleons can be triggered by incident nucleons with energies of @xmath49 gev ( fig . 1 in ref . ) . such incident nucleons are thus relativistic to a certain degree . because of the coulomb interaction via electric charge , a relativistic proton undergoes coulomb energy loss . the loss rate for @xmath50 is given ( eq . [ a.18 ] in ref . @xcite ) by @xmath51 where @xmath52 and @xmath53 is the kinetic energy and the charge number of proton , respectively , @xmath9 is the fine - structure constant , and @xmath54 is the electron mass . @xmath55 is a parameter associated with coulomb divergence ( eqs . [ 13.13 ] and [ 13.43 ] in ref . @xcite ) . here , @xmath56 is the velocity of the proton , and @xmath57 is the lorentz factor . @xmath58{\mathstrut 4\pi\alpha n_e / m_e}$ ] is the plasma frequency of background plasma composed of electron and positron @xcite . @xmath59 and @xmath60 are the total number density and the energy density , respectively , of electron and positron plasma . the total number density is given by @xmath61^{3/2}\exp(-m_e / t)$ ] for @xmath62 and @xmath63 for @xmath64 with the mass fraction @xmath65 of @xmath2he to total baryon , the baryon - to - photon number ratio @xmath66 , and the number density @xmath67 of background photon @xcite . cross sections for inelastic scattering of two nucleons are @xmath68 mb . the reaction rate is then given by @xmath69 the rate of energy degradation via coulomb scattering is , on the other hand , given by @xmath70~e_{,{\rm gev}}^{-1},\nonumber\\ \label{eq5}\end{aligned}\ ] ] where the numerical factor in the second line corresponds to the case of @xmath71 . nonthermal protons generated at hadron injections hardly trigger an inelastic collision before they lose energies because of quick thermalization , i.e. , @xmath72 for @xmath73 . the same holds true for antiprotons , and nuclei with larger charge numbers . nonthermal protons effectively stop without inducing hadronic scatterings . hadronic showers then contain only neutrons and antineutrons as mediator particles which can interact with background nuclei nonthermally by the energy injected at the particle decay . main reactions between a nonthermal neutron and a background proton , which is much more abundant than background neutron at @xmath74 , are @xmath75 where @xmath76 and @xmath77 are nonnegative integers . the elastic scattering corresponds to @xmath78 in eq . ( [ eq6 ] ) . for a same set of @xmath76 and @xmath77 values , reaction thresholds of the second reaction are higher than those of the third by @xmath79 , where @xmath80 and @xmath81 are the masses of neutron and proton , respectively . the first reaction does not change the combination of nucleon isospins so that the number of energetic particle , i.e. , neutron , is not changed . the second reaction could increase the number of energetic neutron , while the third decreases it both by the unit of one . two protons from the third reaction stop instantaneously . if the sum of rates for the second reaction over @xmath76 and @xmath77 is larger than that for the third , nonthermal neutron abundance goes up from the abundance of originally injected neutron . if the total rate for the second is smaller than that for the third , however , the nonthermal neutron abundance goes down . if the both rates balance approximately , the nonthermal neutron abundance does not change during developments of hadronic showers . cross sections of the second and third reactions have been measured , and they equate within the statistical errors @xcite . although an isospin symmetry in the two reactions seems to exist , it is not yet verified experimentally . uncertainties in reaction rates affect a net number of neutrons which are generated in the universe . the net abundance of nonthermal neutron is the most important quantity determining abundances of d and @xmath0li . then , one should be cautious about the uncertainties in reaction rates when a parameter space for @xmath0li reduction is searched . recent previous bbn calculations including hadronic particle injection were based on biased network codes in which either reaction of the second and third types is included for some sets of @xmath76 and @xmath77 @xcite . the present study escapes from these uncertainties , and obtains a conservative lower limit on @xmath41d/@xmath42li@xmath43 . in this subsection , we focus on the processes occurring at the time of neutron injection , @xmath82 , and omit the index for the time @xmath82 on physical quantities for simplicity . firstly we describe changes in d and @xmath0li abundances caused by injections of neutrons and antineutrons by the following two equations . the amount of @xmath0li reduction is approximately proportional to the total abundances of injected nonthermal neutron , i.e. , @xmath83 since @xmath0be is destroyed by neutron [ eq . ( [ eq2 ] ) ] . the equation for @xmath84 is @xmath85\right\}\nonumber\\ & & + \bar{n}_1 p^\prime_{1\rightarrow 2}(n)\left[1+p_{2\rightarrow 3}(n)(1+\cdots)\right ] , \label{eq9}\end{aligned}\ ] ] where @xmath86 and @xmath87 are the abundances of primary neutron and antineutron , respectively , injected at the considered event , @xmath88 and @xmath89 are the probabilities that the @xmath76-th generation neutrons and antineutrons , respectively , generate the @xmath90-th generation species for @xmath91 or @xmath92 . we note that @xmath93 is a sum of components for multiple reactions ( @xmath94 ) . if no neutron is emitted at a reaction induced by a @xmath76-th neutron , the net number of neutron changes by @xmath95 . the @xmath96 value is then @xmath95 for this reaction @xmath94 . the first and second terms of the right hand side ( rhs ) correspond to neutrons originating from primary neutrons and antineutrons , respectively . we neglect effects of the @xmath97 scattering off background @xmath8 and @xmath2he . since annihilation cross sections of @xmath98 and @xmath99he reactions are significant in comparison with total cross sections @xcite , generated @xmath97 are typically lost after at most a few reactions unaccompanied with annihilations . the change in d abundance is described as @xmath100\right\}\nonumber\\ & \hspace{-5pt}&\hspace{-5pt}+\bar{n}_1 \left\{p^\prime_{1\rightarrow 2}(d)+p^\prime_{1\rightarrow 2}(n ) \left[p_{2\rightarrow 3}(d)+p_{2\rightarrow 3}(n ) \cdots\right]\right\},\nonumber\\ \label{eq10}\end{aligned}\ ] ] the first term of the rhs is for deuterons produced via @xmath14h(@xmath15)@xmath16h . note that the injected neutrons are mostly captured by proton , and converted to d for @xmath40 s. the second term is for the sum of the @xmath90-th deuterons produced mainly via @xmath2he spallation by the @xmath101-th neutrons which originate from primary neutrons . the third term includes deuterons produced at annihilations with @xmath2he , and the sum of the @xmath90-th deuterons produced mainly via @xmath2he spallation by the @xmath102-th generation neutrons originating from primary antineutrons . the present model is constrained by an overproduction of d as described below . we then conserve the model by keeping d abundances low while reducing @xmath0li abundances . when instantaneous thermalizations of energetic @xmath26 and @xmath97 are assumed , no secondary or higher order energetic particles would be generated . then , an equation , i.e. , @xmath103 , holds . accordingly , one obtains @xmath104 , and @xmath105 , where subscript 1 in @xmath106 and @xmath107 indicates that the amounts count only particles originating from primary neutrons and not higher order neutrons . the @xmath108 ratio is estimated as follows : first , we assume the symmetry in injected amounts of neutron and antineutron ( @xmath109 ) . the following relation then holds : @xmath110 the @xmath111 value is given by @xmath112 where @xmath113 and @xmath114 are number densities of @xmath14h and @xmath2he , respectively . in the epoch after the @xmath2he production , the ratio is @xmath115 . @xmath116 and @xmath117 are cross sections for annihilation by hydrogen and @xmath9 particle , respectively . the ratio in the parenthesis with subscript @xmath97 indicates the value for annihilation of @xmath97 . @xmath118 is the fraction of the @xmath119he annihilation into exit channels including species @xmath120 . although an estimation of @xmath111 [ eq . ( [ eq12 ] ) ] is associated with uncertainties , an example estimation is shown as follows : nuclear data on @xmath20he annihilation at low energies indicate fractions for the production of @xmath92 and @xmath26 , i.e. , @xmath121@xmath122 and @xmath123<0.4 $ ] @xcite . we then assume the similarity of the fractions for @xmath48 and @xmath97 , and take values of @xmath124 and @xmath125 . in addition , we assume the simple scaling of @xmath126 with the mass number @xmath45 , and @xmath127 @xcite . in this case , the equation , @xmath128 , holds , and eq . ( [ eq11 ] ) becomes @xmath129 here the assumption of instantaneous thermalization is removed , i.e. , @xmath130 . a relation between yields of the @xmath102-th generation neutron and deuteron derives from eqs . ( [ eq9 ] ) and ( [ eq10 ] ) as @xmath131 the quantity @xmath132 is described by an integration of a distribution function in energy of the @xmath133-th generation neutron multiplied by a rate for production of species @xmath120 . a lower limit on @xmath134 is estimated utilizing experimental data on cross sections as @xmath135 where @xmath136 for @xmath137 and @xmath9 , and @xmath138 and @xmath26 represents an effective cross section for production of @xmath94 at the reaction with @xmath120 , as explained below . we defined @xmath139 which is a sum of cross sections @xmath140 for final states @xmath141 weighted according to the net increase in deuteron number . similarly we defined @xmath142 and @xmath143 as the sums of cross sections weighted according to the net increase in neutron number . the value in the second line of eq . ( [ eq15 ] ) was estimated as follows : we adopt values of @xmath144 mb from the mirror reaction , i.e. , @xmath145 ( fig . 7 of ref . ) , and @xmath146 mb ( fig . 6 of ref . ) . in addition , an asymmetry in cross sections of @xmath147 [ eq.([eq7 ] ) ] and @xmath148 [ eq.([eq8 ] ) ] was allowed conservatively by 20 % of the total inelastic cross section at maximum , i.e. , @xmath149 mb . by comparing eq . ( [ eq13 ] ) with eqs . ( [ eq14 ] ) and ( [ eq15 ] ) , it is found that an addition of contribution from the @xmath102-th generation neutron always enhances the @xmath108 ratio . in this model , in addition to the sbbn , we consider an extra production of d and a destruction followed by some degrees of reproduction of @xmath0be ( sec . [ sec4 ] ) . we write the ratio between changes of ( @xmath0li+@xmath0be ) and d as a function of the kinetic energy of neutron , i.e. , @xmath52 , and @xmath150 . it is given by @xmath151 where @xmath152 is the change of @xmath45 abundance caused by neutrons with energy @xmath52 in the universe of temperature @xmath150 . @xmath153 is the cross section for the reaction @xmath154 as a function of @xmath52 . @xmath155 is the destruction fraction of @xmath0li , which is produced via the reaction @xmath0be(@xmath17)@xmath0li , during its propagation in the cooling universe . @xmath156 is the survival fraction of d , which is produced via the reaction @xmath14h(@xmath15)@xmath16h , during its propagation . if energetic @xmath0li and @xmath92 nuclei are produced by the respective reactions , they instantaneously lose their energies through the coulomb scattering , and are thermalized soon after the productions @xcite . the quantities @xmath155 and @xmath156 should then be taken as values for thermal maxwell - boltzmann distribution of @xmath0li and d ( see appendix [ app2 ] ) . note that although the quantities , @xmath155 and @xmath156 , depend on @xmath150 , we omit to express the argument . the ratio of @xmath157 is roughly speaking smaller at higher energies as seen hereinbelow while the ratio of @xmath158 is larger at higher energies ( see fig . [ fig6 ] in appendix [ app2 ] ) . figure [ fig1 ] shows the ratio of thermonuclear reaction rates estimated with recommended rates given by descouvemont _ et al . _ @xcite [ for @xmath0be(@xmath17)@xmath0li ] and ando _ et al . _ @xcite [ for @xmath14h(@xmath15)@xmath16h ] . because of a decrease in the @xmath0be(@xmath17)@xmath0li rate at high energies , the ratio decreases at high temperatures . at low temperatures ( @xmath159 ) , @xmath0li is not destroyed , i.e. , @xmath160 , although @xmath0be is transformed to @xmath0li via @xmath0be(@xmath17)@xmath0li . the amount of @xmath0li reduction is , therefore , small [ eq . ( [ eq16 ] ) ] . an efficient destruction of @xmath0li then prefers an operation of @xmath0be(@xmath17)@xmath0li at higher temperature . at high temperatures ( @xmath161 ) , on the other hand , the @xmath0be production in the sbbn is not yet completed . although @xmath0be nuclei are converted to @xmath0li , the same nuclei are produced via the reaction @xmath5he(@xmath162)@xmath0be later in lower temperatures until the reaction stops ( appendix [ app2 ] ) . in a white region at @xmath163 , therefore , the reduction of @xmath0li is most efficient . be(@xmath17)@xmath0li and @xmath14h(@xmath15)@xmath16h as a function of the temperature @xmath164 . in a right shaded region at @xmath161 , @xmath0be is reproduced through the reaction @xmath5he(@xmath162)@xmath0be after its destruction by neutron , while in a left shaded region at @xmath159 , the destruction of @xmath0li by the proton capture is inefficient . the white region bounded by the shaded region , i.e. , @xmath163 , is the best temperature region in which extra neutrons efficiently reduce a final @xmath0li abundance.[fig1],width=302 ] when energetic neutrons are injected , they experience an energy loss , especially the coulomb scattering off the background electrons and positrons through interaction via their magnetic moments @xcite . nonthermal neutrons are then quickly thermalized . nevertheless , a small abundance of energetic neutrons can react with background h and @xmath0be before they could be thermalized . at high neutron energies , the ratio of cross sections for @xmath0be(@xmath17)@xmath0li and @xmath14h(@xmath15)@xmath16h is small . although the ratio of rates averaged over maxwell - boltzmann distribution is shown in fig . [ fig1 ] , the trend in reaction rate as a function of temperature roughly traces that in cross section as a function of energy . neutrons with higher energies thus relatively prefer the production of d over the destruction of @xmath0be . in order to obtain a conservative lower limit on @xmath165li@xmath43 , we assume that nonthermal neutrons instantaneously thermalize , and cause a preferential reduction of @xmath0li . even if energetic hadrons induced by a hadronic energy injection were instantaneously thermalized , thermalized antinucleon can destroy background @xmath2he nuclei through annihilation processes ( sec . [ sec5 ] ) . we assume that the tni occurs at time @xmath82 instantaneously with a number density of injected neutron @xmath84 .. the abundance is measured as the number density relative to that of total baryons , i.e. , @xmath166 . we use the bbn code by kawano @xcite with the sarkar s correction @xcite to @xmath2he abundance . reaction rates relating to light nuclei of mass number @xmath167 are updated with the jina reaclib database v1.0 @xcite . we adopt the neutron lifetime of @xmath168 s @xcite . we adopt an upper limit on the abundance ratio @xmath0li / h from a recent observation of mpss , i.e. , log(@xmath0li / h)@xmath169 derived with the 3d nonlocal thermal equilibrium model @xcite . taking the two @xmath4 ( standard deviation ) uncertainty , we assume the primordial abundance of @xmath170 . a consistency between a theoretical prediction and observations of mpss requires a reduction of ( @xmath0li+@xmath0be ) during the bbn in amounts of at least @xmath171 . the sbbn prediction of deuterium abundance is ( d / h)@xmath172 . the final value of d / h@xmath173 after the d production caused by the neutron injection should not deviate from primordial abundance inferred from observations of lyman-@xmath9 absorption system in the foreground of quasi - stellar objects ( qso ) . recent measurement of a damped lyman @xmath9 system qso sloan digital sky survey ( sdss ) j1419 + 0829 was performed most precisely of all qso absorption systems ever found @xcite . we adopt the best measured abundance , log(d / h)=@xmath174 ( best ) , and a mean value of ten qso absorption line systems including j1419 + 0829 , log(d / h)=@xmath175 ( mean ) @xcite .. li , and suggested a solution to the @xmath0li problem by the @xmath0be destruction at the hadronic decay of a long - lived exotic particle followed by a depletion of d in the cosmic chemical evolution . ] figure [ fig2 ] shows calculated abundances of h and @xmath2he , i.e. , @xmath176 and @xmath65 , respectively , ( mass fractions ) , and other nuclides ( number ratios relative to h ) as a function of the temperature @xmath177 . solid lines correspond to cases of different injection times of @xmath178 , @xmath179 , @xmath180 , and @xmath181 s , for the same injected abundance @xmath182 . dashed lines show fiducial abundances of the sbbn model . in all cases , final values of baryon - to - photon ratios are the wmap9 value @xmath183 ( model @xmath184cdm ; wmap data only ) @xcite . in appendix [ app1 ] , we describe important reactions through which nuclear abundances are affected . it is seen that abundances of t , @xmath0li , and @xmath0be are changed much by the tni , and that increases in abundances of t and @xmath0li depend significantly on @xmath82 . at a large value of @xmath82 , the destruction reaction of t , i.e. , @xmath5h(@xmath92 , @xmath26)@xmath2he , is ineffective because of a low temperature . the final t abundance is then large . t nuclei produced during the bbn epoch decay to @xmath5he with the half life of @xmath185 yr @xcite . the final @xmath5he abundance is , therefore , given by a sum of abundances of t and @xmath5he at bbn . since the @xmath5he abundance is much larger than the t abundance in the bbn epoch , increases of t abundance change the final @xmath5he abundance by only negligible amounts . he , i.e. , @xmath176 and @xmath65 , respectively , ( mass fractions ) , and other nuclides ( number ratios relative to h ) as a function of the temperature @xmath164 . solid lines are for cases of neutron injection by @xmath182 at @xmath178 , @xmath179 , @xmath180 , and @xmath181 s. dashed lines are for standard bbn model . in all cases , final values of baryon - to - photon ratios are fixed to the wmap value @xmath183 @xcite.[fig2],width=302 ] figure [ fig3 ] shows contours for final abundances of d ( solid lines ) and @xmath0li ( dashed lines ) in the ( @xmath82 , @xmath166 ) plane . in a narrow region indicated at @xmath186 , @xmath187 s , @xmath188 ) by points , the primordial abundance inferred from observations of @xmath0li @xcite is reproduced within the two @xmath4 uncertainty keeping the d abundances close to the observed value @xcite . this region is , therefore , the most preferred region . dark ( black ) points correspond to calculated d abundances in the 12 @xmath4 range of the best observed value , while light ( green ) points correspond to those in the 5 @xmath4 range of the mean value . it is found that d abundances in the 11 @xmath4 range of the best value and the 4 @xmath4 range of the mean value are never accompanied with li abundances in the observational 2 @xmath4 range in this model . the recent precise determination of d abundance in the qso absorption line systems thus completely excludes the solution to the li problem in this model . this calculation itself should be similar to a recent calculation for neutron injection which concluded that this model can provide a solution to the li problem @xcite . our different conclusion results from the use of the new observational constraints on primordial d abundance . effects on abundances of @xmath26 , d , @xmath5h , @xmath5he , @xmath3li , @xmath0li , and @xmath0be are different in different parameter cases . reasons for that are described in appendixes [ app1 ] and [ app2 ] . li ( dashed lines ) in the ( @xmath82 , @xmath166 ) plane . analytical estimates with eqs . ( [ eqb4 ] ) , ( [ eqb7 ] ) , ( [ eqb9 ] ) , and ( [ eqb11 ] ) are also shown by thin dotted lines . points at @xmath186 , @xmath187 s , @xmath188 ) indicate parameter sets which reproduce the observed @xmath0li abundance in the 2 @xmath4 range @xcite while keeping d abundance in the 12 @xmath4 range of the best value [ dark ( black ) points ] , and the 5 @xmath4 range of the mean value [ light ( green ) points ] , respectively @xcite . [ fig3],width=302 ] figure [ fig4 ] shows a region on the parameter plane of ( @xmath189li / h , @xmath41d / h ) which can be occupied in this model . the lines with arrows indicate the regions which satisfy observational constraints on abundances of d ( 12 @xmath4 for the best value , and 5 @xmath4 for the mean value ) @xcite and @xmath0li ( 2 @xmath4 ) @xcite . a lower limit on @xmath190 as a function of @xmath189li / h can be read from this figure . the points at ( @xmath189li / h , @xmath41d / h ) @xmath191 , @xmath192 ) satisfy the constraints . abundances in this parameter region give close agreement with those found in a recent detailed study on effects of hadronic decay @xcite as their most favorable results of abundances . in our preferred parameter region , the abundance of d is related to that of @xmath0li for the adopted nuclear reaction rates ( sec . [ sec31 ] ) and baryon - to - photon ratio @xcite as described by @xmath193\times 10^{-5}. \label{eq17}\ ] ] this constraint is free of many uncertainties related to nuclear and electromagnetic reactions for nonthermal particles produced by the neutron injection . although primary antinucleons , and secondary and higher order neutrons always increase the ratio @xmath41d/@xmath194li@xmath43 ( see secs . [ sec22 ] and [ sec5 ] ) , their effects depend [ eqs . ( [ eq15 ] ) and ( [ eq18 ] ) ) ] on information of relative injected amounts of @xmath26 , @xmath8 , @xmath97 and @xmath48 , and their injected energy spectra . the information itself depends on the decay property of the long - lived exotic particle such as its mass and decay modes . equation ( [ eq17 ] ) then corresponds to the most conservative model independent lower limit on d / h as a function of @xmath0li / h . li ( horizontal ) . the lines with arrows indicate the parameter regions which satisfy the 12@xmath4 constraint ( best ) and the 5 @xmath4 constraint ( mean ) of d abundance and the 2@xmath4 constraint of @xmath0li abundance . the dark ( black ) and light ( green ) points inside the region bounded by two lines correspond to parameter sets satisfying those constraint regions shown in fig . [ fig4],width=302 ] in the case of earliest neutron injection at @xmath178 s ( fig . [ fig2 ] ) , effects of additional neutrons are removed by efficient nuclear reactions . especially , although the @xmath0be abundance reduces right after the neutron injection , the reaction @xmath5he(@xmath162)@xmath0be enhances @xmath0be again . in the best case of injection , i.e. , @xmath195 s , the @xmath0be abundance decreases and the d abundance increases a little less efficiently . in the case of later injection , i.e. , @xmath196 s , the d abundance increases via @xmath14h(@xmath15)@xmath16h , and is not affected by already inefficient d destruction reactions . the resulting d abundance is thus larger than in the best case . the @xmath0be conversion to @xmath0li by neutron capture efficiently proceeds . however , the reaction @xmath0li(@xmath23)@xmath2he is no longer operative . the resulting decrease in the mass - number - seven ( @xmath0li+@xmath0be ) abundance is , therefore , very small . in the case of the latest injection , i.e. , @xmath197 s , some portion of injected neutrons decay with the lifetime @xmath198 s @xcite before they could trigger the d production via @xmath14h(@xmath15)@xmath16h . this leads to a suppressed d production . the reduction of ( @xmath0li+@xmath0be ) is not operative as in the previous case . if the neutron injection has a duration , deviations in final abundances would be approximately given by weighted average over time of deviations obtained in this instantaneous injection model . when amounts of neutron injection are small , i.e. , @xmath199 , both of the @xmath0be destruction and the d production are efficient . if the injection is strong , i.e , @xmath200 , however , the efficiency of @xmath0be destruction plateaus since it gets difficult for neutrons to find @xmath0be nuclei with an already small abundance [ cf . eq . ( [ eqb7 ] ) ] . the efficiency in the d production , on the other hand , is not suppressed since the target of neutrons at the reaction @xmath14h(@xmath15)@xmath16h is proton whose abundance is very large , and dose not change significantly in this model for parameter values of @xmath166 and @xmath82 considered here . the antinucleon ( @xmath201)+@xmath2he annihilation ( as considered in ref . @xcite ) is an important process which always operates when @xmath201 s are produced . the annihilation of ( thermalized ) @xmath201 and @xmath2he affects the elemental abundances even when productions of secondary particles via @xmath2he spallations by energetic hadrons can be neglected . the annihilations produce light mesons , @xmath26 , @xmath8 , @xmath92 , @xmath202 , and @xmath5he . generated neutrons of abundance @xmath84 are almost completely captured by protons , and produce deuterons if the time of neutron injection is @xmath203 s. the final abundance of d produced through the @xmath201+@xmath2he annihilation is then given by @xmath204 where @xmath205h is the number densities of primary @xmath201 injected simultaneously at the injection of neutrons [ cf . ( [ eq9 ] ) ] relative to that of background hydrogen . the ratio in the parenthesis with subscript @xmath201 is the value for annihilation of species @xmath201 [ cf . ( [ eq12 ] ) ] . @xmath206 is the fraction of annihilation into final states including a deuteron to that for total annihilation . equation ( [ eq18 ] ) is transformed to an equation : @xmath207p_{\rm sur}({\rm d } ) , \nonumber\\ \label{eq19}\end{aligned}\ ] ] where @xmath208 is the number abundance of generated neutron relative to that of @xmath14h . @xmath209 is the effective number of primary antinucleons per primary neutron , and @xmath210 is the number ratio between the neutron produced secondarily by the annihilation of @xmath201 plus @xmath2he , and the primary neutron . the square bracket in the second line is the quantity averaged over @xmath211 and @xmath48 with weights of @xmath212 . the equation , i.e. , @xmath213 , is satisfied . we try an example estimation . we assume that abundances of nonthermal primary antinucleons are twice as large as those of primary neutron . this leads to @xmath214 . we assume @xmath215 , @xmath216 @xcite for both @xmath97 and @xmath48 , @xmath127 @xcite , and @xmath115 , as done in deriving eq . ( [ eq13 ] ) . the following equation is then derived : @xmath217 using eqs . ( [ eq19 ] ) and ( [ eq20 ] ) , we obtain @xmath218 this component should add to the production of d in the present model in which only effects of neutron were taken into account . the total change of d abundance is , therefore , given by @xmath219p_{\rm sur}({\rm d})+\delta { \rm d_1^{\rm ann}}/{\rm h}>1.03 [ \delta n_{\rm inj}/(x n_{\rm b})]p_{\rm sur}({\rm d})$ ] . in this case , the abundance of d in the preferred parameter region ( sec . [ sec4 ] ) is @xmath220\times 10^{-5}. \label{eq22}\ ] ] figure [ fig5 ] shows contours for final abundances of d ( solid lines ) and @xmath0li ( dashed lines ) on the ( @xmath82 , @xmath166 ) plane in the case that the additional d production from the annihilation is taken into account by the lower limit , i.e. , eq . ( [ eq21 ] ) . we find that this small fraction of additional d production narrows the best parameter region in figs . [ fig3 ] and [ fig4 ] without moving contours in fig . [ fig3 ] significantly . for the case that effects of the antinucleon+@xmath2he annihilation are taken into account as described in sec . parameter regions reproducing observed abundances of d @xcite and @xmath0li @xcite are narrower than those in fig . dotted lines are contours for final abundance of @xmath3li for the case that @xmath3li production through secondary reactions triggered by antinucleon is taken into account ( see sec . [ sec6 ] ) . [ fig5],width=302 ] the @xmath221he annihilation produces nonthermal @xmath5h and @xmath5he . the nuclides with mass number three can react with background @xmath2he , and produce @xmath3li . the decay of @xmath5h can be neglected since its half life , i.e. , @xmath222 y @xcite , is much longer than time scales of related processes [ e.g. , inverse of eq . ( [ eq4 ] ) ] in the relevant temperature range . the abundance from this @xmath201-induced @xmath3li production is then estimated as @xmath223 \nonumber\\ & \hspace{-5pt}&\hspace{-5pt}\times p_{\rm sur}(^6{\rm li } ) , \label{eq23}\end{aligned}\ ] ] where @xmath224 is the ratio of the cross section for annihilation into final states including a nuclide @xmath45 to that for total annihilation . @xmath225 and @xmath226 are the cross section and the threshold energy for the reaction @xmath45(@xmath227)@xmath3li , @xmath228 and @xmath229 are the kinetic energy and the velocity of @xmath45 , and @xmath230 is the distribution function of secondary @xmath45 produced at the annihilation of @xmath201+@xmath2he as a function of @xmath228 . @xmath231 is the total reaction rate of nuclide @xmath45 as a function of @xmath228 , and @xmath232 is the survival fraction of @xmath3li , which is produced via the secondary reaction @xmath45(@xmath227)@xmath3li , during its propagation . we try an example estimation for this component of nonthermal @xmath3li production . we assume @xmath214 , @xmath216 , @xmath233 and @xmath234 @xcite for both @xmath97 and @xmath48 , @xmath127 @xcite , and @xmath115 , as done in sec . the energy spectra of mass - three - nuclides , i.e. , @xmath230 were assumed to be given by an result of experiment measuring the spectrum for @xmath5he at @xmath20he annihilation @xcite . in the experiment , no dependence of the spectrum on the initial @xmath48 energy has been observed , and the nuclide @xmath5he in the final state can be identified without being confused with other hadronic species . the reaction cross sections @xmath225 are taken from ref . the total rate @xmath235 is assumed to be the coulomb loss rate since the coulomb loss dominates as long as the energy is not too high . at temperature @xmath236 , the coulomb loss rate of relativistic charged particles is given by eq . ( [ eq3 ] ) . the rate of non - relativistic charged particles is given @xcite by @xmath237 . \label{eq24}\end{aligned}\ ] ] the @xmath3li survival fraction @xmath232 is calculated in our bbn code . in fig . [ fig5 ] , dotted lines correspond to abundance ratios , i.e. , @xmath3li / h@xmath238li / h@xmath239li / h@xmath240 , @xmath241 , @xmath242 , and @xmath243 ( from bottom to top ) . in high temperature environments , @xmath3li nuclei produced in the reaction @xmath45(@xmath227)@xmath3li are effectively destroyed via proton burning , i.e. , @xmath244 . a significant production of @xmath3li then occurs at relatively low temperature when the @xmath3li destruction is ineffective and the energy loss rate of secondary nuclides @xmath5h and @xmath5he through coulomb scattering off background @xmath245 is diminished because of the reduced abundances of @xmath245 through their pair annihilation . the injections of energetic hadrons could have occurred in the early universe by hypothetical events of decays or annihilations of long - lived exotic particles , or evaporations of exotic objects . the injections cause scattering of thermal nuclei by energetic hadrons , and showers of nonthermal nucleons , antinucleons , and nuclei can develop . neutrons generated at the exotic events can react with @xmath0be and reduce final abundances of @xmath0li ( which are mainly produced via the electron capture of @xmath0be ) . it has been suggested that the @xmath0be reduction can be a solution to a discrepancy between theoretical @xmath0li abundances of the sbbn model and that inferred from observations of galactic metal - poor stars . the theoretical abundance is about a factor of three larger than the observational one . based on an analysis of related physical processes , we prove that the assumption of instantaneous thermalization of injected neutron provides the way to derive a conservative limit on the relation between abundances of d and @xmath0li in the hadronic energy injection model , which is independent of uncertainties in generations and reactions of nonthermal hadrons originating from the injections ( sec . [ sec2 ] ) . furthermore , two important points are stressed : 1 ) an uncertainty in cross sections of inelastic @xmath1 scattering [ eqs . ( [ eq6 ] ) , ( [ eq7 ] ) , and ( [ eq8 ] ) ] affects the total number of neutrons generated from the primary neutron injection , which is critical for resulting abundances of d and @xmath0li . 2 ) one must include effects of annihilations of antinucleons with @xmath2he on a primordial d abundance even if antinucleons generated with neutron were instantaneously thermalized . we then consider a simple model in which extra thermal neutrons are injected in a late epoch of the bbn . we estimate the probability that primordial abundances of @xmath0li in this model can be consistent with observed abundances . relations between primordial abundances of d and @xmath0li are obtained in a manner to conserve the probability securely . we perform a bbn calculation , and find a very small parameter region of the neutron injection time ( @xmath82 ) and the number density ( @xmath84 ) of injected neutron in which @xmath0li abundances are within the 2 @xmath4 uncertainty range determined from observation and changes in d abundance are minimum . in the preferred parameter region , the injection time is @xmath246 s , and its number density is @xmath188 times as large as that of total baryonic matter . a typical pattern of nucleosynthesis in the parameter region is analyzed ( appendix [ app1 ] ) . situations of d production and @xmath0li reduction are observed especially ( appendix [ app2 ] ) . we derive a model - independent result [ eq . ( [ eq17 ] ) ] that a reduction of @xmath0li abundance from the sbbn value down to the observational two @xmath4 upper limit is necessarily accompanied by an undesirable increase of d abundance up to at least the 12 @xmath4 upper limit ( best observed value ) and the 5 @xmath4 upper limit ( mean observed value ) . when effects of antinucleons+@xmath2he annihilations are considered utilizing a possible example case , the preferred parameter regions become narrower in the present model . bbn models involving any injections of extra neutron are , therefore , not likely to accommodate alone a reduction of primordial @xmath0li abundance to the observed level . we analyzed nucleosynthesis with a bbn code , and found important reactions operating in the case of extra neutron injection of @xmath247 at @xmath248 s corresponding to @xmath71 ( see fig . [ fig2 ] ) . we list rates of dominant reactions for productions ( @xmath249 ) and destructions ( @xmath250 ) of respective nuclides . in what follows , @xmath254 denotes the rate , i.e. , @xmath255 for a reaction 1(2,3)4 with the cross section @xmath4 , and the relative velocity @xmath56 . rates are measured in the unit of cm@xmath5 s@xmath257 mol@xmath257 . an instantaneous production of extra neutron has been assumed : @xmath258 [ [ d ] ] d ~ @xmath259 @xmath260 where @xmath261 or @xmath8 , and @xmath262he or @xmath202 . rates for final states of @xmath263he and @xmath264h are @xmath265 and @xmath266 cm@xmath5s@xmath257mol@xmath257 , respectively . @xmath267 @xmath268 the rate for @xmath5he(@xmath269)@xmath2he is about 30 times smaller than that for @xmath5h(@xmath270)@xmath2he . @xmath271 @xmath272 @xmath273 @xmath274 the destruction and production of @xmath3li still operate efficiently in this low temperature environment . the abundance of @xmath3li is , therefore , the steady state abundance determined from @xmath275 . @xmath276 @xmath277 @xmath0li also experiences destruction and production efficiently . the @xmath0li abundance is the steady state abundance . @xmath278 @xmath279 @xmath0be is only transformed into @xmath0li , and the @xmath0li abundance instantaneously relaxes to the steady state abundance . the evolution of extra neutron abundance , i.e. , @xmath280 , is described simply by @xmath281 where the first , second , and third terms of the rhs correspond to the dilution by cosmic expansion , the reduction by the radiative proton capture , and the reduction by @xmath282-decay , respectively . @xmath283 denotes the reaction rate , i.e. , @xmath255 , for a reaction of species @xmath120 and @xmath94 . for a simplistic understanding , we assume that the destruction by the radiative capture reduces the neutron abundance instantaneously compared to the time scale of hubble expansion , and that the neutron gradually decreases via the @xmath282-decay thereafter ( for a result of precise calculation , see fig . [ fig2 ] ) . the extra neutron abundance at @xmath284 is then approximately solved to be @xmath285 ^ 3 \nonumber\\ & & \times \exp\left\{-\left[(n_{\rm h } \langle { \rm h}+n \rangle)_{t_{\rm inj } } + \tau_n^{-1 } \right ] ( t - t_{\rm inj})\right\ } , \nonumber\\ \label{eqb2}\end{aligned}\ ] ] where @xmath286 is the scale factor of universe with the redshift @xmath287 , and quantities with subscript @xmath82 represent values at time @xmath82 . the abundance of extra d is given by an integration of production rate via @xmath14h(@xmath15)@xmath16h . assuming an instantaneous production of extra d , its abundance at @xmath284 is given by @xmath288 the change in d abundance at the neutron injection is then given by @xmath289 when an abundance of extra neutron is much larger than that of thermal background neutron , the evolution of @xmath0be abundance is described by @xmath290 using eq . ( [ eqb2 ] ) , an approximate solution is obtained : @xmath291 ^ 3 \exp\left[-\int_{t_{\rm inj}}^t \delta n \langle ^7{\rm be}+n \rangle~dt\right]\nonumber\\ & \hspace{-5pt}\approx&\hspace{-5pt } n_{^7{\rm be}}(t_{\rm inj } ) \left[\frac{a(t_{\rm inj})}{a(t)}\right]^3\nonumber\\ & \hspace{-5pt}&\hspace{-5pt}\times \exp\left[-\frac{\langle ^7{\rm be}+n \rangle_{t_{\rm inj}}}{(n_{\rm h } \langle { \rm h}+n \rangle)_{t_{\rm inj}}+\tau_n^{-1}}\delta n_{\rm inj}\right ] . \label{eqb6}\end{aligned}\ ] ] the survival fraction of d produced by the extra neutrons is estimated using a simplified rate equation for d , i.e. , @xmath293 where the factors of two in numerator and denominator in the rhs are for the number of d lost in one reaction , and for avoiding a double counting of initial state d nuclei , respectively . we approximately take the hydrogen number density to be constant . the survival fraction is then given by @xmath294^{-1}.~~~~~ \label{eqb9}\end{aligned}\ ] ] in the above equation , @xmath295d / h@xmath296 is given by the sum of value in the sbbn model , i.e. , @xmath295d / h@xmath297 , plus @xmath298d / h@xmath296 [ eq . ( [ eqb4 ] ) ] . the destruction fraction of @xmath0li produced via the conversion of @xmath0be by neutron capture is roughly estimated taking account of only the instantaneous proton burning of @xmath0li ( see appendix [ app1 ] ) . the evolution of @xmath0li abundance is described by @xmath299 the destruction fraction of @xmath0li is then given by @xmath300.~~~ \label{eqb11}\end{aligned}\ ] ] figure [ fig6 ] shows approximate values of @xmath156 and @xmath155 calculated with eqs . ( [ eqb9 ] ) and ( [ eqb11 ] ) . the increase of d abundance relative to that in the sbbn is not taken into account in the curve for @xmath156 . when the amount of extra neutron injection is larger , ( d / h)@xmath301 is larger , and resultingly the @xmath156 value decreases by the self destruction of d [ eq . ( [ eqb9 ] ) ] . a neutron injection at lower temperature triggers a production of d with a higher survival probability . the @xmath0li nuclei produced via the conversion of @xmath0be has a smaller destruction probability at lower temperature . ] , and fraction of @xmath0li which is destroyed by proton capture [ @xmath155 ] after their propagations through the universe from the time ( @xmath82 ) or the temperature [ @xmath164 ] of their productions . in drawing curves , approximate eqs . ( [ eqb9 ] ) and ( [ eqb11 ] ) are used.[fig6],width=302 ] in fig . [ fig3 ] , thin dotted lines correspond to abundance ratios , i.e. , d / h@xmath304d / h@xmath305d / h@xmath306 [ with eqs . ( [ eqb4 ] ) and ( [ eqb9 ] ) ] and @xmath0li / h@xmath307li / h@xmath308h@xmath309 [ with eqs . ( [ eqb7 ] ) and ( [ eqb11 ] ) ] , respectively . in calculating the ratios , we read abundance evolution profiles in the sbbn model and used them . the analytical result ( dotted lines ) are rather consistent with the results of full calculation ( solid and dashed lines ) . , * * , ( ) . , , , * * , ( ) . , * * , ( ) . , , , * * , ( ) . , , , , * * , ( ) . , , , * * , ( ) . , , , * * , ( ) . , * * , ( ) . , * * , ( ) . , , , , * * , ( ) . , , , * * , ( ) . , , , * * , ( ) . , , , , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . , , , , * * , ( ) . , * * , ( ) . , * * , ( ) . , * * , ( ) . m. pospelov and j. pradler , phys . lett . * 106 * , 121305 ( 2011 ) . , * , , , , , * * , ( ) . m. -m . kang , y. hu , h. -b . hu and s. -h . zhu , jcap * 1205 * , 011 ( 2012 ) . , , , , * * , ( ) . m. kusakabe , a. b. balantekin , t. kajino and y. pehlivan , phys . d * 87 * , 085045 ( 2013 ) . h. ishida , m. kusakabe and h. okada , arxiv:1403.5995 [ astro-ph.co ] . , * * , ( ) . , * * , ( ) . , * * , ( ) . , , , , , , , , , , , * * , ( ) . , , , , , * * , ( ) . , , , , , , * * , ( ) . , , , * * , ( ) . g. hinshaw _ et al . _ [ wmap collaboration ] , astrophys . j. suppl . * 208 * , 19 ( 2013 ) . , , , , , * * , ( ) . , * * , ( ) .

the injections of energetic hadrons could have occurred in the early universe by decays of hypothetical long - lived exotic particles . the injections induce the showers of nonthermal hadrons via nuclear scattering . neutrons generated at these events can react with @xmath0be nuclei and reduce @xmath0be abundance solving a problem of the primordial @xmath0li abundance . we suggest that thermal neutron injection is a way to derive a model independent conservative limit on the relation between abundances of d and @xmath0li in a hadronic energy injection model . we emphasize that an uncertainty in cross sections of inelastic @xmath1 scattering affects the total number of induced neutrons , which determines final abundances of d and @xmath0li . in addition , the annihilations of antinucleons with @xmath2he result in higher d abundance and trigger nonthermal @xmath3li production . it is concluded that a reduction of @xmath0li abundance from a value in the standard big bang nucleosynthesis ( bbn ) model down to an observational two @xmath4 upper limit is necessarily accompanied by an undesirable increase of d abundance up to at least an observational 12 @xmath4 upper limit from observations of quasi - stellar object absorption line systems . the effects of antinucleons and secondary particles produced in the hadronic showers always lead to a severer constraint . the bbn models involving any injections of extra neutrons are thus unlikely to reproduce a small @xmath0li abundance consistent with observations .

the evolution of the universe is determined by the complex interplay of a large number of processes that involve the exchange of mass and energy . a global view of these mass and energy flows has been taken to investigate the universal content of baryonic mass @xcite , energy @xcite , metals @xcite and star formation @xcite . these quantities give us useful insights in to the evolution of the physical universe as a whole and they help us understand how individual processes contribute to the evolution . in this work , we aim to investigate a key part of this global energy interchange : radiative cooling of diffuse gas on cosmological scales . in particular , we aim to identify the cooling channels that dominate the energy budget of the diffuse gas that traces the large - scale structure of the universe and that feeds galaxies and fuels star formation through accretion processes . for our purposes , we define this diffuse matter as the sum of three main components : i ) the intergalactic medium ( igm ) , defined as the low - density gas that is not gravitationally bound to galaxies and their haloes ; ii ) the circum - galactic medium ( cgm ) , which is denser than the igm , resides in the immediate surroundings of galaxies , and is bound to galaxy haloes ; and iii ) the intra - cluster medium ( icm ) , which is the hot , shock - heated gas that accumulates in the gravitational potentials of groups and clusters . both the cgm and the icm are gravitationally bound to haloes and represent the densest fraction of diffuse gas . while the distinctions between these components are often not clear in observations , they can be more cleanly defined in simulations . in the rest of the paper , we will consider gas in simulations as diffuse " if its hydrogen number density is @xmath5 @xmath1 . thermal and kinetic energy are injected in to the diffuse gas by gravity , by galactic winds powered by stellar winds and supernova explosions , and by feedback due to black hole accretion ( agn feedback ) , cosmic rays and other processes . the gas cools via a variety of mechanisms and emits radiation in a broad range of wavelengths . the cooling radiation can in turn heat up gas locally or far away , can be absorbed and re - emitted at lower frequencies by dust , or ( if the mean free path of photons is very large ) can contribute to the background radiation field . cooling processes within galaxies are thought to contribute substantially to the diffuse infrared , optical , uv and x - ray backgrounds and are an essential ingredient in galaxy formation . these are still not well understood , but can be somewhat constrained and quantified by observation . the radiation produced by the cooling of diffuse gas , on the other hand , is very difficult to observe directly because of its very low surface brightness ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? currently , the igm can therefore be more effectively studied in absorption rather than in emission . in this work , we aim to provide a global view of how cooling occurs in diffuse gas . to this end , we use a subset of runs from the overwhelmingly large simulations ( owls hereafter ) project @xcite to address a number of questions concerning the emission and cooling processes of diffuse gas , including : i ) which transitions , ions , and gas phases are responsible for most of the cooling radiation , and should thus be the focus of theoretical modelling efforts ? ii ) which wavebands and emission lines carry most of the energy emitted by diffuse gas through cooling and fluorescence radiation , and should thus be primary targets for observational studies ? iii ) what is the relative importance of continuum emission and line emission ? iv ) what is the global emission of diffuse gas , and how does it change with time ? how much energy is stored in the igm ? since our focus is diffuse gas , we will ignore emission by dust and by gas with densities exceeding @xmath6 @xmath1 . the paper is organized as follows . section [ tables ] describes the emissivity tables used to calculate the gas emission and shows the dependence of the emissivity on temperature for a large sample of ions , individual emission lines and the continuum . section [ owlproj ] discusses the simulations and the methodology used to calculate the gas emission . section [ volume ] presents results for the fraction of energy emitted by different ions , elements and the continuum in the reference simulation set ( _ ref _ in the following ) at @xmath2 and @xmath3 . in section [ sims ] we compare results from different simulations to understand the effect of varying the physical prescriptions . finally , we draw our conclusions in section [ conclusion ] and present a series of numerical convergence tests in the appendix . .adopted solar abundances , from @xcite , @xcite and @xcite . [ cols="<,^,<,^",options="header " , ] in this appendix we provide the full lists of the individual emission lines that contribute most of the cooling energy . the fractions of the mean energy emitted per unit volume for individual lines as a function of temperature are shown in figs . [ lines_vs_temp_z=0 ] and [ lines_vs_temp_z=2 ] . only lines that contribute at least 1 per cent of the total emitted energy in at least one temperature bin are listed . table [ linelist1 ] lists line for h , he , c , n , o , ne and mg . table [ linelist2 ] lists lines for si , s , ca and fe .

in this work we investigate the cooling channels of diffuse gas ( i.e. @xmath0 @xmath1 ) in cosmology . we aim to identify the wavelengths where most of the energy is radiated in the form of emission lines or continuum radiation , and the main elements and ions responsible for the emission . we use a subset of cosmological , hydrodynamical runs from the owls project to calculate the emission of diffuse gas and its evolution with time . we find that at @xmath2 ( @xmath3 ) about 70 ( 80 ) per cent of the energy emitted by diffuse gas is carried by emission lines , with the continuum radiation contributing the remainder . hydrogen lines in the lyman series are the primary contributors to the line emission , with a share of 16 ( 20 ) per cent . oxygen lines are the main metal contributors at high redshift , while silicon , carbon and iron lines are strongest at low redshift , when the contributions of agb stars and supernova ia explosions to the metal budget become important and when there is more hot gas . the ionic species carrying the most energy are o , c , c , si , si , fe and s. the great majority of energy is emitted in the uv band ( @xmath4 ) , both as continuum radiation and line emission . with almost no exception , all the strongest lines fall in this band . at high energies , continuum radiation is dominant ( e.g. , 80 per cent in the x - ray band ) , while lines contribute progressively more at lower energies . while the results do depend on the details of the numerical implementation of the physical processes modeled in the simulations , the comparison of results from different simulations demonstrates that the variations are overall small , and that the conclusions are fairly robust . given the overwhelming importance of uv emission for the cooling of diffuse gas , it is desirable to build instruments dedicated to the detection and characterisation of diffuse uv emission . = -0.8 in [ firstpage ] intergalactic medium diffuse radiation radiation mechanisms : thermal method : numerical

in the fourier decomposition of the azimuthal distribution of particles , the first and second coefficients correspond to the directed , @xmath30 , and elliptic , @xmath31 , flow , respectively @xcite . the elliptic flow is expected to be sensitive to the system evolution at the time of maximum compression @xcite . ollitrault @xcite showed that in a hydro model the elliptic flow is proportional to the initial space elliptic anisotropy of the overlapping region weighted by the number of nucleon collisions in the beam direction . this initial space elliptic anisotropy , which we will call @xmath32 , has been calculated for a woods - saxon density distribution and shown to be almost insensitive to the nucleon - nucleon cross section @xcite . it is enlightening to plot @xmath33 versus centrality @xcite in order to look for changes in the reaction mechanism or properties of the nuclear matter . thus the motivation for this work is to find a signature ( elliptic flow ) , scan this signature as a function of a control parameter ( centrality ) , and , after first dividing out the geometry of the initial state , look for a change in the physics ( unexpected behavior ) . na49 has published directed and elliptic flow results from the na49 main time projection chambers for a set of data taken with a medium impact parameter trigger @xcite . we now have a new set of data taken with a minimum bias trigger so that we can study the flow centrality dependence . also , the tracks from the main and vertex tpcs are combined resulting in full coverage of the forward hemisphere . the data in the graphs below presenting flow as a function of rapidity have been reflected about mid - rapidity . the data have been integrated over @xmath34 and in some cases also over @xmath35 using as weights the measured double differential cross sections @xcite . the data have been sorted into six centrality bins using the zero degree calorimeter , with `` cen1 '' being the most central and `` cen6 '' the most peripheral . the impact parameter values for these bins have been estimated from the number of participants which were obtained by integrating the yields @xcite . slightly higher values of @xmath36 , used in the oral presentation of this paper , were determined from the fraction of the total cross section corresponding to each bin . only some of the available data has been analyzed so far . thus these data are preliminary and no systematic errors have been included yet . the rapidity dependence of directed and elliptic flow integrated over the whole range of measured impact parameters up to about 11 fm is shown in fig . [ fig : v - y ] . the pion @xmath30 values hug the axis near mid - rapidity and the @xmath31 values for both pions and protons appear to slightly peak somewhat away from mid - rapidity . for pions the @xmath30 and @xmath31 values are shown for different centrality bins in fig . [ fig : cen ] . both sets of flow values increase continuously as the reaction becomes more peripheral . the elliptic flow values for pions have been integrated over rapidity up to @xmath37 and are shown in fig . [ fig : b ] , together with simulations from rqmd v2.3 @xcite . the flow from rqmd peaks at a medium impact parameter whereas the flow from experiment continues to rise . in fig . [ fig : b - ep ] the @xmath31 values have been divided by the initial space elliptic anisotropy @xcite . in addition , results from rqmd v3.0 @xcite which includes a phase transition are shown . typical hydro results @xcite are also shown . the data are below hydro indicating a lack of complete equilibration in the reaction @xcite . the data are above the rqmd resonance gas and tantalizingly close to the rqmd phase transition calculation . clearly , it is important to process the full set of na49 data and obtain final results . 9 a.m. poskanzer and s.a . voloshin , phys . c * 58 * , 1671 ( 1998 ) . h. sorge , phys . letters * 78 * , 2309 ( 1997 ) . ollitrault , phys . d * 46 * , 229 ( 1992 ) . p. jacobs and g. cooper , star note sn0402 ( 1999 ) . h. sorge , phys . * 82 * , 2048 ( 1999 ) ; h. sorge , this conference ( 1999 ) . h. heiselberg and a .- m . levy , phys . c * 59 * , 2716 ( 1999 ) . h. appelshuser _ et al . _ , na49 , phys . letters * 80 * , 4136 ( 1998 ) . a.m. poskanzer _ et al . _ , na49 , nucl . phys . * a638 * , 463c ( 1998 ) . g. cooper , na49 , this conference . f. sikler , na49 , this conference . h. sorge , phys . c * 52 * , 3291 ( 1995 ) . p. kolb , j. sollfrank , u. heinz , preprint nucl - th/9906003 . voloshin and a.m. poskanzer , preprint nucl - th/9906075 .

new data with a minimum bias trigger for 158 gev / nucleon pb + pb have been analyzed . directed and elliptic flow as a function of rapidity of the particles and centrality of the collision are presented . the centrality dependence of the ratio of elliptic flow to the initial space elliptic anisotropy is compared to models . j. bchler@xmath0 , d. barna@xmath1 , l.s . barnby@xmath2 , j. bartke@xmath3 , r.a . barton@xmath2 , l. betev@xmath4 , h. biakowska@xmath5 , a. billmeier@xmath6 , c. blume@xmath7 , c.o . blyth@xmath2 , b. boimska@xmath5 , j. bracinik@xmath8 , f.p . brady@xmath9 , r. brockmann@xmath10 , r. brun@xmath0 , p. buni@xmath11 , l. carr@xmath12 , d. cebra@xmath9 , g.e . cooper@xmath13 , j.g . cramer@xmath12 , p. csat@xmath1 , v. eckardt@xmath14 , f. eckhardt@xmath15 , d. ferenc@xmath9 , h.g . fischer@xmath0 , z. fodor@xmath1 , p. foka@xmath6 , p. freund@xmath14 , v. friese@xmath15 , j. ftacnik@xmath8 , j. gl@xmath1 , r. ganz@xmath14 , m. gadzicki@xmath6 , e. gadysz@xmath3 , j. grebieszkow@xmath16 , j.w . harris@xmath17 , s. hegyi@xmath1 , v. hlinka@xmath8 , c. hhne@xmath15 , g. igo@xmath4 , m. ivanov@xmath8 , p. jacobs@xmath13 , r. janik@xmath8 , p.g . jones@xmath2 , k. kadija@xmath18 , v.i . kolesnikov@xmath19 , m. kowalski@xmath3 , b. lasiuk@xmath17 , p. lvai@xmath1 , a.i . malakhov@xmath19 , s. margetis@xmath20 , c. markert@xmath7 , b.w . mayes@xmath21 , g.l . melkumov@xmath19 , j. molnr@xmath1 , j.m . nelson@xmath2 , g. odyniec@xmath13 , m.d . oldenburg@xmath6 , g. plla@xmath1 , a.d . panagiotou@xmath22 , a. petridis@xmath22 , m. pikna@xmath8 , l. pinsky@xmath21 , a.m. poskanzer@xmath13 , d.j . prindle@xmath12 , f. phlhofer@xmath15 , j.g . reid@xmath12 , r. renfordt@xmath6 , w. retyk@xmath16 , h.g . ritter@xmath13 , d. rhrich@xmath23 , c. roland@xmath7 , g. roland@xmath6 , a. rybicki@xmath3 , t. sammer@xmath14 , a. sandoval@xmath7 , h. sann@xmath7 , a.yu . semenov@xmath19 , e. schfer@xmath14 , n. schmitz@xmath14 , p. seyboth@xmath14 , f. siklr@xmath24 , b. sitar@xmath8 , e. skrzypczak@xmath16 , r. snellings@xmath13 , g.t.a . squier@xmath2 , r. stock@xmath6 , p. strmen@xmath8 , h. strbele@xmath6 , t. susa@xmath25 , i. szarka@xmath8 , i. szentptery@xmath1 , j. sziklai@xmath1 , m. toy@xmath26 , t.a . trainor@xmath12 , s. trentalange@xmath4 , t. ullrich@xmath17 , d. varga@xmath1 , m. vassiliou@xmath22 , g.i . veres@xmath1 , g. vesztergombi@xmath1 , s. voloshin@xmath13 , d. vrani@xmath27 , f. wang@xmath13 , d.d . weerasundara@xmath12 , s. wenig@xmath0 , c. whitten@xmath4 , n. xu@xmath13 , t.a . yates@xmath2 , i.k . yoo@xmath15 , j. zimnyi@xmath1 @xmath22department of physics , university of athens , athens , greece . + @xmath13lawrence berkeley national laboratory , university of california , berkeley , usa . + @xmath2birmingham university , birmingham , england . + @xmath8institute of physics , bratislava , slovakia . + @xmath1kfki research institute for particle and nuclear physics , budapest , hungary . + @xmath0cern , geneva , switzerland . + @xmath3institute of nuclear physics , cracow , poland . + @xmath7gesellschaft fr schwerionenforschung ( gsi ) , darmstadt , germany . + @xmath9university of california at davis , davis , usa . + @xmath19joint institute for nuclear research , dubna , russia . + @xmath6fachbereich physik der universitt , frankfurt , germany . + @xmath21university of houston , houston , tx , usa . + @xmath20kent state university , kent , oh , usa . + @xmath4university of california at los angeles , los angeles , usa . + @xmath15fachbereich physik der universitt , marburg , germany . + @xmath14max - planck - institut fr physik , munich , germany . + @xmath5institute for nuclear studies , warsaw , poland . + @xmath16institute for experimental physics , university of warsaw , warsaw , poland . + @xmath12nuclear physics laboratory , university of washington , seattle , wa , usa . + @xmath17yale university , new haven , ct , usa . + @xmath25rudjer boskovic institute , zagreb , croatia . + @xmath28present address : university of bergen , norway . + @xmath29deceased . +

type ia supernovae ( sne ia ) play important roles in astrophysics as a standard candle for measuring cosmological distances and as main production sites of iron group elements . it is commonly agreed that the exploding star is a mass - accreting carbon - oxygen ( c+o ) white dwarf ( wd ) . however , it is not clarified yet whether the wd accretes h / he - rich matter from its binary companion [ single degenerate ( sd ) scenario ] , or two c+o wds merge [ double degenerate ( dd ) scenario ] ( e.g. , * ? ? ? * ; * ? ? ? observations have provided the following constraints on the nature of companion stars . some evidences support the sd model , such as the presence of circumstellar matter ( csm ) @xcite and detections of hydrogen in the circumstellar - interaction type sne ( ia / iin ) like sn 2002ic @xcite and ptf11kx @xcite . on the other hand , there has been no direct indication of the presence of companions , e.g. , ( 1 ) the lack of companion stars in the images of sn 2011fe @xcite , some sn ia remnants ( snrs ) @xcite , sn 1572 ( tycho ) @xcite and sn 1006 @xcite , ( 2 ) the lack of ultraviolet ( uv ) excesses of early - time light curves @xcite , and ( 3 ) the lack of hydrogen features in the spectra @xcite . both ( 2 ) and ( 3 ) are expected from the collision between ejecta and a companion . in particular , detailed observations of sn 2011fe in m101 require stringent constraints on the progenitor , i.e. , @xcite excluded the presence of a red - giant ( rg ) , a helium star , or a main - sequence ( ms ) star of @xmath3 . @xcite further excluded a solar mass ms companion from an early uv observation with _ swift _ because of no signature of shock interaction between ejecta and a companion . the tightest constraints come from no signature of shock interaction with the companion . however , if the binary separation , @xmath4 , is much larger than the companion radius , @xmath5 , i.e. , @xmath6 , the solid angle subtended by the companion would be much smaller , and so would be the effect of shock interaction . in their spin - down scenario , @xcite and @xcite argued that the donor star in the sd model might shrink rapidly before the wd explosion , because it exhausts its hydrogen - rich envelope before the sn ia explosion during a long spin - down phase of the rapidly rotating , super - chandrasekhar mass wd . in such a case , the companion star is much smaller than its roche lobe , which reduces the shock signature . this also explains the lack of hydrogen in the spectra of sne ia and possibly the unseen companion in the snr @xcite . @xcite presented possible evolutionary routes to super - chandrasekhar mass wds in the wd+ms channel of the sd model . they also compared the spin - down time of wds with the companion star s ms lifetime and discussed their final states at explosions , although their main - focus was to explain the observed extremely luminous sne ia . in this letter , we apply our method in @xcite to wd+rg binaries , and calculate the wd mass distribution beyond the chandrasekhar mass limit for both the wd+ms and wd+rg systems . we then estimate the brightness distribution of sne ia , assuming that the brightness depends on the wd mass . we further confirm that , in most of the wd+rg systems , the companion has evolved off from a rg to a helium ( or c+o ) wd before the sn ia explosion and such a compact companion does not show any prominent shock signatures nor indications of hydrogen . in section [ binary_evolution ] , we describe our basic assumptions and methods . section [ numerical_results ] presents our numerical results . section [ discussion ] discusses various perspectives on the progenitors of sne ia . based on the sd model , we followed binary evolutions in which a wd accretes hydrogen - rich matter from its companion . there are two well studied evolutionary paths to sne ia , the wd+ms and wd+rg channels . our basic assumptions in binary evolutions are essentially the same as those in @xcite . mass - accreting wds blow optically thick winds if the mass transfer rate exceeds the critical rate , @xmath7 @xcite . the wd winds collide with the secondary s surface and strip off its surface layer . if the mass - stripping is efficient enough , the mass transfer rate is attenuated and the binary avoid the formation of a common envelope even for a rather massive secondary of @xmath8@xmath9 . thus , the mass - stripping effect widens the donor mass range of sn ia progenitors . we have incorporated the mass - stripping effect in the same way as in @xcite , i.e. , the mass stripping rate @xmath10 is proportional to the wind mass - loss rate @xmath11 as @xmath12 . in this study , we assume @xmath13 as a lower representative value for the wd+ms system , because @xcite found that @xmath14 is between a few to 10 to reproduce the optical and x - ray light curve behaviors of some supersoft x - ray sources . if we adopt a larger value , we could have a more massive secondary . for the wd+rg systems , @xmath14 is calculated from equation ( 21 ) of @xcite . mass - accreting wds spin up because of angular momentum gain from the accreted matter @xcite . if the wd rotates rigidly , its mass can only slightly exceed the chandrasekhar mass of no rotation , @xmath15 with electron mole number @xmath16 . if the wd rotates differentially , however , its mass can significantly exceed @xmath17 ( e.g. , * ? ? ? * ) . @xcite concluded that the wd increases its mass beyond @xmath17 when the accretion rate to the wd is as high as @xmath18 yr@xmath19 . they showed that the gradient of angular velocity is kept around the critical value for the dynamical shear instability and that this differential rotation law is strong enough to support wds whose masses significantly exceed @xmath17 @xcite . @xcite showed that highly differential rotation may not be realized due to baroclinic instability . however , his stability condition is not a sufficient condition but just a necessary condition for instability , so that his conclusion is premature ( see * ? ? ? * ) . in the present study , we simply assume that mass - accreting wds are supported by the above differential rotation law and the mass can increase without carbon being fused at the center as long as @xmath20 yr@xmath21 . we assume that , when @xmath22 , hydrogen shell - burning occurs intermittently on the wd and the resultant nova outbursts eject a large part of the envelope mass @xcite . as a result , the net growth rate of the wd mass is significantly reduced ( @xmath23 yr@xmath19 ) . then the timescale of angular momentum deposition to the wd core would become much longer than @xmath24 yr and comparable to the timescale for the eddington - sweet meridional circulation ( @xmath25 yr ) to redistribute angular momentum . this redistribution causes a contraction of the wd core and its central density increases high enough to trigger an sn ia explosion before the wd mass significantly increases . thus , if the wd mass increases beyond @xmath26 , we define the wd mass at the sn ia explosion as the wd mass when the mass transfer rate drops to @xmath27 . rrrrr 1.381.5 & 48.2 & 41.1 & 19.8 & 44.8 + 1.51.6 & 13.7 & 22.7 & 6.7 & 17.9 + 1.61.7 & 13.6 & 14.0 & 2.0 & 13.8 + 1.71.8 & 9.6 & 10.0 & 0.5 & 9.8 + 1.81.9 & 8.2 & 6.2 & 0.0 & 7.2 + 1.92.0 & 2.9 & 3.7 & 0.0 & 3.3 + 2.02.1 & 1.8 & 1.6 & 0.0 & 1.7 + 2.12.2 & 1.3 & 0.8 & 0.0 & 1.1 + 2.22.3 & 0.7 & 0.0 & 0.0 & 0.4 rrrr 1.381.6 & 62.7 & 1.12.1 & 67.4 + 1.61.8 & 23.6 & 1.01.1 & 17.3 + 1.82.0 & 10.5 & 0.91.0 & 10.2 + 2.02.3 & 3.2 & 0.70.9 & 5.1 figures [ zregevl_10_11_strip_ms_rg_sc15_2012 ] and [ zregevl_08_09_strip_ms_rg_sc15_2012 ] show the parameter regions that produce sne ia in the @xmath28 ( orbital period companion mass ) plane for the metallicity of @xmath29 . we plot results for four initial wd masses of @xmath30 , 1.0 , 0.9 , and @xmath31 because ( 1 ) no c+o wds of @xmath32 are expected for normal metallicity @xcite and ( 2 ) the @xmath33 region is too small for the wd+ms systems and none for the wd+rg systems . the wds inside these sn ia regions ( labeled `` initial '' ) will increase their masses from ( a ) @xmath34 , ( b ) 1.0 , ( c ) 0.9 , and ( d ) 0.8 @xmath35 to @xmath36 , 1.5 , 1.6 , ... , @xmath37 ( @xmath38 step from outside to inside contours ) and reach the regions labeled `` final . '' we stop the binary evolution when the mass transfer rate decreases to @xmath39 . both the wd+ms and wd+rg systems can produce a super - chandrasekhar mass wd up to @xmath40 . the sn ia regions for different initial wd masses , @xmath41 , 0.8 , 0.9 , 1.0 , and @xmath42 are calculated both for the wd+ms and wd+rg systems . we then estimate the sn ia birth rate in our galaxy as @xmath43 yr@xmath19 and @xmath44 yr@xmath19 for the constant star formation rate . here we assume the initial distribution of binaries given by equation ( 1 ) of @xcite , i.e. , @xmath45 and the distribution of mass ratio @xmath46 . we also estimate the delay time distribution ( dtd ) of sne ia as shown in figure [ wd_mass_distribution_c3]a , where the delay time ( @xmath47 ) is the elapsed time from binary birth to explosion . the spin - down time of wds is not included . the computational method is the same as that in @xcite . these values are normalized to fit the dtd at 11 gyr @xcite . our dtd shows a featureless power law ( @xmath48 ) from 0.1 to 12 gyr , which is consistent with totani et al.s ( 2008 ) observation . the present results are essentially the same as our previous results by @xcite in which we assume that the wd explodes as an sn ia at @xmath49 . in general , the wd+ms systems consist of a young population of sne ia corresponding to short delay times ( @xmath50 gyr ) and the wd+rg systems an old population of long delay times ( @xmath51 gyr ) . we further calculate the distribution of the wd masses at sn ia explosions as in table [ wd_mass_distribution_sup_ch ] and in figure [ wd_mass_distribution_c3]b d . we also plot the number ratio of the wds for the wd+rg systems with the initial companion masses of @xmath52 , which are expected to occur in elliptical galaxies ( see * ? ? ? it is clear that the wd mass distribution in ellipticals is confined into a narrower range of 1.381.6@xmath53 than in late type galaxies . after the mass transfer rate drops to @xmath39 , the wd stops growing in mass . there are three characteristic mass ranges for the final evolution of wds toward an sn ia explosion ( see * ? ? ? * for detail ) . \(1 ) in the extremely massive case , the differentially rotating wd explodes as an sn ia soon after the wd mass exceeds @xmath54 owing to a secular instability . this is not the present case , because it happens for @xmath55 in only low metallicity environments . \(2 ) for the mid - mass range of @xmath56@xmath54 , the wd is differentially rotating and its mass exceeds the maximum mass for rigid rotation . as angular momentum in the wd core is lost or redistributed toward rigid rotation , the wd core contracts until its central density and temperature become high enough to ignite carbon . thus the timescale of contraction until the sn explosion is @xmath57 yr due to angular momentum transport by the eddington - sweet meridional circulation . as for the other angular momentum transport mechanisms , @xcite showed that magneto - dipole radiation leads to spin - down in a typical timescale of @xmath25@xmath58 yr for @xmath59 when the magnetic field of the wd is as weak as @xmath60 g. they also showed that the @xmath61-mode instability is not significant in spinning - down wds . thus , we here assume that the eddington - sweet meridional circulation is the most effective process for spin - down . \(3 ) for the lower mass range of @xmath36@xmath62 , the wd can be supported by rigid rotation while it exceeds the critical mass of non - rotating wds for carbon ignition . thus , the wd contracts with the spin - down timescale , which is determined by angular momentum loss from the wd and thus depends on the strength of the magnetic field of the wd . the final fate of the wd depends on the spin - down time as we discuss below . it may take more than @xmath63 yr for weak magnetic fields of @xmath60 g @xcite . if the spin - down time is not much longer than @xmath63 yr , the compressional heating due to the spin - down would dominate the radiative cooling of the wd ( see equation ( 7 ) of * ? ? ? because the spin - down time is not unique , its variation causes a variety of thermal state of wd cores when carbon ignites at the center . this would lead to a variation of the carbon ignition density and thus a variation of @xmath64ni mass and brightness of sne ia even for the same wd mass . if the spin - down time is much longer than @xmath65 yr , on the other hand , the central density at the carbon ignition could become high enough to induce collapse @xcite . this collapse produces a quite little amount of @xmath64ni as @xmath66 @xcite , which might correspond to a faint transient . in most of the wd+ms systems , the companion remains to be an ms ( central hydrogen burning ) star until the `` final '' stage of evolution . as shown in figures [ zregevl_10_11_strip_ms_rg_sc15_2012 ] and [ zregevl_08_09_strip_ms_rg_sc15_2012 ] , these systems have an orbital period shorter than @xmath67 day and a companion mass smaller than @xmath40 . this type of pre - supernova binaries satisfy the constraint on sn 2011fe posed by @xcite , @xmath68 . if @xmath56@xmath54 and the spin - down time ( @xmath25 yr ) is shorter than the ms lifetime of the companion , the wd explodes before the companion evolves off the main - sequence . in most of these cases , the companion s mass further satisfies the condition of @xmath69 posed by @xcite on the snr 2011fe , because the companion s mass further decreases from the `` final '' mass in figures [ zregevl_10_11_strip_ms_rg_sc15_2012 ] and [ zregevl_08_09_strip_ms_rg_sc15_2012 ] by the amount of roughly @xmath70 yr@xmath71 yr@xmath72 , which is transferred to the wd and ejected by nova outbursts . on the other hand , if @xmath36@xmath62 and the spin - down time ( @xmath63 yr ) is longer than the ms lifetime of the companion , the companion becomes a helium wd and csm has disappeared . such a case might correspond to the case of no csm like sn 2011fe @xcite . as already discussed in our previous paper @xcite , the mass - stripping effect produces a large amount of csm , say , a few to several solar masses ( @xmath73 initial mass minus final mass , as seen in figures [ zregevl_10_11_strip_ms_rg_sc15_2012 ] and [ zregevl_08_09_strip_ms_rg_sc15_2012 ] ) . if the spin - down time is short enough , or if the wd is forced to be rigidly rotating during accretion due to strong magnetic fields , the wd may explode in the csm . then we may observe the interaction between the ejecta and the csm like in sne ia / iin . in the wd+rg systems , after the mass transfer rate drops to @xmath39 , the companion rg further evolves and finally becomes a helium ( or c+o ) wd in a timescale of @xmath74 yr . this timescale is shorter than the spin - down time in both the mid - mass range ( @xmath75 : @xmath25 yr for the eddington - sweet circulation ) and lower mass range ( @xmath76 : @xmath63 yr for the magneto - dipole radiation ) wds . the companion rg has already evolved to a wd when the primary wd explodes as an sn ia . therefore , the immediate progenitor is a wide binary consisting of wd+wd for all the cases . these immediate progenitors satisfy all the constraints mentioned in section [ intro ] . it might correspond to sn 2011fe , which shows no csm . it also explains the lack of hydrogen in the spectra of sne ia and possibly the unseen companions of sn 1572 ( tycho ) @xcite , sn 1006 @xcite , and snr 0509 - 67.5 @xcite . in the above discussion , we assumed that the spin - down time is @xmath77@xmath58 yr . however , if the spin - down time is short enough , or if the wd is forced to be rigidly rotating during accretion , the wd may explode before the companion evolves off the red - giant or asymptotic giant branch . then we may observe the interaction between the ejecta and the csm like in kepler s snr @xcite and in ptf11kx @xcite . in our progenitor models , various types of sne ia can be explained as follows : normal sne ia correspond to the lower mass range of wds , @xmath78 ( or @xmath79 ) . the brightness variation can be explained as a variety of spin - down time . the brighter group of sne ia such as sn 1991 t correspond to the mid - mass range of wds , @xmath80 ( or @xmath81 ) . we think that for these brighter sne ia , the brightness variation stems mainly from a variation of wd mass . here we set the border between the normal and brighter sne at about @xmath62 . however , it depends on the timescale of the eddington - sweet circulation which may become longer near rigid rotation . therefore , this border could be as massive as @xmath82 . now we explain the luminosity distribution of sne ia with our model . in the observation , peak brightnesses of sne ia depend monotonically on the @xmath83 , where @xmath83 is the @xmath84-magnitude decay from the maximum in 15 days . table [ wd_mass_vs_luminosity_sup_ch ] compares the wd mass distribution in our model with the observational @xmath83 distribution @xcite . for the brighter group of sne ia ( @xmath59 ) , the distribution of @xmath83 is in good agreement with the wd mass distribution . this may be a support for our theoretical expectation that the brightness of sne ia is determined mainly by the wd mass because the thermal state is similar among various wd cores due to its relatively short spin - down time . for the fainter group of sne ia ( @xmath85 ) , however , the brightness of sne ia depends not only on the wd mass but also on the spin - down time as mentioned in the previous subsection . to summarize , we examined the final fate of the two progenitor models of sne ia , the wd+ms and wd+rg systems . a major part of the wd+ms systems reasonably satisfy the stringent constraints on sn 2011fe in m101 . most cases of the wd+rg systems satisfy even more stringent constraints on snr 0509 - 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taking into account the rotation of mass - accreting white dwarfs ( wds ) whose masses exceed the chandrasekhar mass , we extend our new single degenerate model for the progenitors of type ia supernovae ( sne ia ) , accounting for two types of binary systems , those with a main sequence companion and those with a red - giant ( rg ) companion . we present a mass distribution of wds exploding as sne ia , where the wd mass ranges from 1.38 to @xmath0 . these progenitor models are assigned to various types of sne ia . a lower mass range of wds ( @xmath1 ) , which are supported by rigid rotation , correspond to normal sne ia . a variety of spin - down time may lead to a variation of brightness . a higher mass range of wds ( @xmath2 ) , which are supported by differential rotation , correspond to brighter sne ia such as sn 1991 t . in this case , a variety of the wd mass may lead to a variation of brightness . we also show the evolutionary states of the companion stars at sn ia explosions and pose constraints on the unseen companions . in the wd+rg systems , in particular , most of the rg companions have evolved to helium / carbon - oxygen wds in the spin - down phase before the sn ia explosions . in such a case , we do not expect any prominent signature of the companion immediately before and after the explosion . we also compare our new models with the recent stringent constraints on the unseen progenitors of sne ia such as sn 2011fe .

data on the production of strange b mesons is rapidly accumulating . at lep the delphi collaboration has measured the probability for a @xmath3 antiquark to hadronize into a weakly decaying strange b meson @xcite . assuming the hadronization is dominated by the fragmentation of the heavy @xmath3 antiquark , we determined the fragmentation functions @xmath4 and @xmath5 from the measured total probability for a @xmath3 antiquark to hadronize into the lowest strange - beauty states @xmath6 and @xmath7 @xcite . here @xmath8 is the fraction of the @xmath3 antiquark momentum carried by the @xmath0 or @xmath1 at the scale @xmath9 . the momentum distributions of the @xmath0 and @xmath1 mesons produced in @xmath10 annihilation at the energy of the @xmath11 mass were then predicted @xcite . here we extend our calculations to the production of the @xmath0 and @xmath1 states in @xmath2 annihilation at the tevatron , using the fragmentation functions previously determined from the @xmath10 data taken at lep . ii presents the cross section for the process in terms of the parton distribution functions and subprocesses and the fragmentation functions , which we discuss in secs . iii and iv , respectively . in sec . v we present numerical results for the transverse momentum @xmath12 distributions of the strange @xmath0 and @xmath1 mesons , as well as the total cross section for @xmath13 . finally , we discuss these results and summarize our conclusions . in the parton model the cross section for the production by fragmentation of strange b mesons in proton - antiproton annihilation involves three main factors : the structure functions of the initial partons in the proton and antiproton , the subprocess in which the initial partons produce a particular parton in the final state , and the fragmentation of this final parton into the strange b meson . the transverse momentum distribution of the @xmath0 meson is then of the form @xmath14 here @xmath15 and @xmath16 are the structure functions of the initial partons @xmath17 and @xmath18 carrying fractions of the total momenta @xmath19 and @xmath20 in the proton and antiproton , respectively . the production of parton @xmath21 with momentum @xmath22 is described by the hard subprocess cross section @xmath23 at the scale @xmath9 . and @xmath24 is the fragmentation function for the parton @xmath21 to yield the meson @xmath0 with momentum fraction @xmath8 at the scale @xmath9 . the factorization scale @xmath9 is of the order of the transverse momentum of the fragmenting parton . the soft physics is contained in the structure functions , while the parton subprocess is a hard process and can be reliably calculated in perturbation theory . we shall also assume the fragmentation of @xmath3 into @xmath0 is also a hard process , an assumption that will require further discussion below , since the @xmath25 quark is not very heavy . clearly , there is a formula very similar to eq . ( [ bs ] ) for the production of the @xmath1 , which we need not write explicitly . because the radiative decay @xmath26 is so fast and the @xmath27 mass difference is so small , the transverse momentum of @xmath1 is hardly affected in the decay @xmath28 , and thus contributes to the inclusive production of the @xmath0 . we will , therefore , present each individually in the figures showing @xmath29 and @xmath30 and also include their sum in a table , for convenience . in the fragmentation process @xmath31 we have included gluons as well as @xmath3 antiquarks . although the direct @xmath32 fragmentation process does not occur until order @xmath33 there is a significant contribution coming from the altarelli - parisi evolution of the @xmath3 antiquark fragmentation functions from the heavy quark mass to the collider energy scale @xmath34 , which is of order @xmath35 due to the splitting @xmath36 . this has been shown to be significant for the production of @xmath37 mesons @xcite and we have also included it here . in eq . ( [ bs ] ) we have assumed a factorization scale @xmath9 , with the parton distribution functions being due to soft processes below this scale , while the parton cross sections for the subprocesses are due to hard contributions above this scale . for the parton distribution functions we used the most recent cteq version 3 @xcite , in which we chose the leading - order fit , since our calculation is also a leading - order one . for the subprocesses we used the tree - level cross sections , to be consistent , since the parton fragmentation functions , which are discussed in the following section , were calculated only to leading order . for the production of @xmath3 antiquarks we included the processes @xmath38 , @xmath39 , and @xmath40 , while for the production of gluons @xmath41 we included the processes @xmath42 , @xmath43 , and @xmath44 , where @xmath45 denotes any of the quarks , @xmath46 . for the running strong coupling constant @xmath47 we used the simple one - loop result evolved from its value at @xmath48 : @xmath49 here @xmath50 is the number of active flavors at the scale @xmath9 and we chose @xmath51 @xcite . from the experimental results on @xmath52-quark production cross sections at the tevatron we know that there are discrepancies between the calculations and the experiments about a factor of two . therefore , adopting a more complicated form for the strong coupling constant , or using the next - to - leading order subprocess cross sections , or taking the next - to - leading order structure functions is not critical . at this point , we are primarily interested in how important the fragmentation process is for @xmath0 and @xmath1 production and how the shapes of the calculated momentum distributions compare with forthcoming data . the factorization scale @xmath9 in eq . ( [ bs ] ) deserves further discussion before proceeding . the scale @xmath9 superficially appears to have been invented only to separate the production of @xmath0 mesons into structure functions , subprocess cross sections , and fragmentation functions . in principle , this is so and the production is independent of the choice of @xmath9 . but in practice , the results do depend on @xmath9 , since only if all the factors are calculated to all orders in @xmath47 will the dependence on @xmath9 cancel . however , for example , the fragmentation functions were calculated only to leading order and , consequently , the results will depend on @xmath9 . we shall investigate the sensitivity of the results to the choice of @xmath9 by varying @xmath9 from @xmath53 to @xmath54 , where @xmath55 is our central choice of the scale @xmath9 . to obtain the fragmentation functions at the scale @xmath9 we have numerically integrated the altarelli - parisi evolution equations @xcite from the scale @xmath56 , which is of the order of the @xmath52-quark mass . the evolution equations for the fragmentation functions @xmath57 and @xmath58 are @xmath59 and @xmath60 with similar , coupled equations for the @xmath1 fragmentation functions . the splitting functions @xmath61 , to leading order in @xmath62 , are explicitly given in ref . @xcite . as boundary conditions in these calculations we have used the fragmentation functions at the scale @xmath56 calculated in ref . @xcite : @xmath63\ , . \label{dz1}\end{aligned}\ ] ] here @xmath8 is the fraction of the @xmath3 antiquark momentum carried by the @xmath0 meson , @xmath64 , and @xmath65 is the @xmath0 meson @xmath66-wave radial wavefunction at the origin . the corresponding fragmentation function for @xmath67 is @xcite @xmath68\ , , % \label{dz2}\end{aligned}\ ] ] the induced gluon fragmentation functions @xmath69 are of order @xmath70 and become important at large values of the scale @xmath9 relative to the @xmath3 antiquark fragmentation functions @xmath71 , which are of order @xmath72 . the boundary conditions for the gluon fragmentation functions are @xmath73 for @xmath74 , the threshold for producing @xmath0 or @xmath1 mesons from a gluon . the value of @xmath75 can be determined from the decay constant @xmath76 of the @xmath0 meson using the relation @xcite @xmath77 with @xmath78 mev @xcite and @xmath79 gev @xcite . if we choose the @xmath52 quark mass to be @xmath80 gev the only remaining parameter in these fragmentation functions is @xmath81 . this strange - quark mass parameter @xmath81 has been determined previously @xcite using the same initial fragmentation functions from the experimental value @xcite of the probability @xmath82 for a @xmath3 antiquark to fragment into weakly decaying strange - beauty mesons : @xmath83 \;. \label{y}\ ] ] since the total probability for the @xmath3 antiquark to fragment into a @xmath84 meson is independent of scale , the initial scale @xmath56 in eq . ( [ y ] ) , which is of the order of the @xmath52-quark mass , was chosen to be @xmath85 as in ref . we previously found @xcite @xmath86@xmath87 mev@xmath88 based on the delphi measurement @xmath89 and assuming @xmath90 . prior to the delphi measurement there were older data from lep @xcite which were consistent with this measurement . there is now also a new measurement from cdf @xcite of the quantity @xmath91 since the total fragmentation probability @xmath92% , the corresponding value of @xmath93 can be estimated assuming @xmath94 . using the newer cdf data , eq . ( [ x ] ) , we obtain @xmath95 for @xmath96% , which is consistent with the delphi measurement . in the present calculations , therefore , we used the same value @xmath97 @xcite but a slightly different value of @xmath98 @xcite . for the strange quark mass parameter we then obtain @xmath99 using the scale - independent relation eq . ( [ y ] ) . this value of @xmath81 is not significantly different from the previous determination and we shall use it in our fragmentation functions eqs . ( [ dz1 ] ) ( [ dz2 ] ) . the strong coupling constant in eqs . ( [ dz1 ] ) ( [ dz2 ] ) then has the value @xmath100 . while this value of @xmath47 is uncomfortably large we note that it only enters , together with @xmath81 and @xmath65 , in the normalization of the fragmentation functions , which has been determined empirically through eq . ( [ y ] ) . the evolution equations were numerically integrated and the fragmentation functions at the scale @xmath101 were combined with the structure functions and parton cross sections for the subprocesses to obtain the cross section , eq . ( [ bs ] ) . specifically , we calculated the transverse momentum distributions of @xmath0 and @xmath1 mesons produced in @xmath2 collisions at the tevatron energy @xmath102 tev . we assumed a cut - off on the transverse momentum of @xmath103 gev@xmath88 and considered only the rapidity range @xmath104 . in fig . [ fig1 ] we show the cross section @xmath29 for both @xmath0 and @xmath1 . to investigate the sensitivity of these results to the scale @xmath9 we have also included the results for @xmath105 and @xmath106 . when the scale @xmath9 is less than @xmath107 , which only happens for the case of @xmath106 , we chose the larger of @xmath108 . from fig . [ fig1 ] it is clear that the choice of scale @xmath9 is not critical for the transverse momentum distribution ; in fact , the variation over the range @xmath109 is comparable to the current discrepancies between the measured and calculated @xmath52 production cross sections . as one might expect , the @xmath110 cross section is larger than the @xmath111 cross section at all @xmath112 , however , their ratio is not precisely the naive prediction @xmath113 , but is about 20% smaller . in fig . [ fig2 ] we show the total cross section for the production of @xmath0 and @xmath1 mesons with transverse momentum above a minimum value @xmath114 . as in fig . [ fig1 ] only the range @xmath103 gev and @xmath104 was considered . the sensitivity to the scale @xmath9 was investigated , as before , by considering @xmath106 and @xmath105 , and the choice of this scale is clearly not critical . figure [ fig2 ] provides useful estimates of the production rates of @xmath0 and @xmath1 mesons at the tevatron due to the fragmentation process . for convenience , we have also presented these cross section estimates in table [ table ] , including the sum @xmath115 . the production rates and transverse momentum distributions for @xmath0 and @xmath1 meson production in @xmath2 collisions presented here assume that fragmentation is the dominant process and , therefore , are probably underestimates . at large enough transverse momentum the fragmentation process should dominate , as it falls off more slowly , even though it is only a part of the full order @xmath116 contribution . we also note that in our calculation the light quark fragmentation contribution is not included , since it is of even higher order in @xmath47 than the induced gluon fragmentation . at the initial heavy quark mass scale , the light quark fragmentation function @xmath117 is of order @xmath118 for @xmath119 . but the @xmath120 has been shown @xcite to be suppressed by @xmath121 relative to @xmath4 . therefore , at the initial scale none of these light quark fragmentation functions are important . the light quark fragmentation functions might acquire a logarithmic enhancement from the evolution of the altarelli - parisi equations , but the splitting kernels @xmath122 and @xmath123 are of order @xmath47 , while @xmath124 and @xmath125 are of order @xmath126 . therefore , the most important sources of induced light quark fragmentation come from ( i ) @xmath127 , and ( ii ) @xmath128 . hence , the order of the induced light - quark fragmentation functions at the scale @xmath9 can be at most of order @xmath129 , which is suppressed by @xmath47 relative to the induced gluon fragmentation . since the induced gluon fragmentation only contributes at about the 1520% level , which is mostly due to the large gluon luminosity at low @xmath130 , the induced light - quark fragmentation contribution is expected to be very small , at the level of a few % . and thus , the light quark contribution can be safely ignored in compare with other uncertainties in the calculation . our fragmentation functions , which we used for the boundary conditions for the altarelli - parisi equations at the scale of the heavy @xmath52 quark mass , are rigorously correct in perturbative qcd if the constituent quarks are much heavier than @xmath131 ; e.g , @xmath37 mesons . also higher order qcd corrections can , in principle , be systematically calculated . in the case of the @xmath0 meson , higher order corrections in the fragmentation functions might be large due to the fact that the strong coupling constant is evaluated at @xmath132 and is equal to about 0.75 . but we fitted the parameter @xmath81 to the experimental measurement of the total probability for @xmath133 and @xmath81 turned out to be about 300 mev@xmath88 . we can , therefore , be confident of the overall normalization , which is proportional to @xmath134 . the shapes of the fragmentation functions are only moderately sensitive to the value of @xmath81 and comparing our results for the momentum distribution will , therefore , primarily test the assumption that their _ shapes _ are adequately given by perturbative qcd at the scale of the @xmath52 quark mass . our approach also appears somewhat similar to the approach used in some monte carlo programs , e.g. , pythia @xcite and herwig @xcite . however , the main difference lies in the fragmentation of the partons . the fragmentation models used in pythia or herwig are rather different in spirit from our perturbative qcd fragmentation , and is based on string fragmentation , though the actual fragmentation models used in these two monte carlo programs are somewhat different from each other . in the string fragmentation picture , given an initial @xmath135 , it is assumed that a new @xmath136 pair may be created according to some pre - assigned probabilities such that a meson @xmath137 is formed and a @xmath138 is left behind . this @xmath138 may at a later stage pair off with a @xmath139 , and so on . the relative probabilities used in pythia to create @xmath140 pairs are set at @xmath141 , where @xmath142 are default values . this choice is based on a quantum mechanical effect and is only approximate . for example , in a paper by lusignoli et al . @xcite , in order to reproduce the @xmath37 meson production using herwig to within a reasonable range , they had to increase the probabilities of creating @xmath143 pairs in the cluster splitting . thus , this ad hoc feature of string fragmentation in these monte carlo programs makes it less predictive than our perturbative qcd fragmentation functions , because in our perturbative qcd fragmentation model the probability for @xmath144 is determined reliably once @xmath145 , @xmath81 , as well as the nonperturbative parameter of the bound - state , e.g. , the decay constant @xmath76 , are given . the string fragmentation picture is rather phenomenological since the parameter @xmath146 can be adjusted freely to fit the experimental measurements . another weakness of such a string fragmentation model is that it must assume the @xmath147 ratio of @xmath148 , from naive spin counting , while our perturbative qcd fragmentation model can reliably predict this ratio once @xmath81 is given . in fact , the ratio is slightly larger ( 20% ) than @xmath148 , as expected for a non - zero @xmath81 , in the hqet @xcite in which @xmath149 is the leading heavy quark - spin symmetry breaking effect . the comparison with forthcoming data from the fermilab tevatron should be instructive . this work was supported by the u. s. department of energy , division of high energy physics , under grant de - fg02 - 91-er40684 and de - fg03 - 93er40757 . 99 p. abrau _ ( delphi collaboration ) , z. phys * c61 * , 407 ( 1994 ) . k. cheung and r.j . oakes , phys . b337 * , 181 ( 1994 ) . k. cheung , phys . * 71 * , 3413,(1993 ) ; 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[ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ] 1 . [ fig1 ] the transverse momentum distributions of @xmath0 and @xmath1 mesons for @xmath103 gev , @xmath104 , and @xmath150 , and @xmath151 at @xmath152 tev . [ fig2 ] the @xmath0 and @xmath1 total cross sections for @xmath13 with @xmath103 gev , @xmath104 , and @xmath150 , and @xmath151 at @xmath152 tev .

the production rates and transverse momentum distributions of the strange - beauty mesons @xmath0 and @xmath1 at @xmath2 colliders are calculated assuming fragmentation is the dominant process . results are given for the tevatron in the large transverse momentum region , where fragmentation is expected to be most important . epsf + * strange - beauty meson production at @xmath2 colliders * + kingman cheung _ center for particle physics , university of texas , austin , tx 78712 _ robert j. oakes _ department of physics and astronomy , northwestern university , evanston , il 60208 _

quantum discord ( qd ) is a measure of quantum correlation defined by ollivier and zurek almost ten years ago @xcite and , yet , a subject of increasing interest today @xcite . it is well known that , for a bipartite pure state , the definition of qd coincides with that of the entanglement of formation ( eof ) . but it has remained an open question how those two quantities would be related for general mixed states . here , we present this desired relation for arbitrarily mixed states and show that the eof and the qd obey a monogamic relation . surprisingly , this necessarily requires an extension of the bipartite mixed system to its tripartite purified version . nonetheless , we obtain a conservation relation for the distribution of eof and qd in the system - the sum of all possible bipartite entanglement shared with a particular subsystem , as given by the eof , can not be increased without increasing , _ by the same amount _ , the sum of all qd shared with this same subsystem . when extended to the case of a tripartite mixed state , this relation results in a new proof of the strong subadditivity of entropy , with stronger bounds depending on the balance between the sum of eof and the sum of qd shared with a particular subsystem . as an example of the importance of this conservation relation , we explore the distribution of entanglement in the deterministic quantum computation with one single pure qubit and a collection of @xmath0 mixed states ( dqc1 ) . the algorithm , developed by knill and laflamme @xcite , is able to perform exponentially faster computation of important tasks , @xcite when compared with well - known classical algorithms , without any entanglement between the pure qubit and the mixed ones @xcite . arguably , the power of the quantum computer is supposed to be related to qd , rather than entanglement @xcite . here , using the conservation relation , we have shown that even in the supposedly entanglement - free quantum computation there is a certain amount of multipartite entanglement between the qubits and the environment , which is responsible for the non - zero qd ( see fig . 1 ) . represents the pure qubit , @xmath1 the maximally mixed state , obtained through maximal entanglement with the environment @xmath2 . from the left to the right , @xmath1 is initially entangled with @xmath2 ( purple bar ) . the protocol is then executed and @xmath3 , although not directly entangled with @xmath1 , gets entangled with the pair @xmath4 as the qd between @xmath3 and @xmath1 increase . ] let us first consider an arbitrary system represented by a density matrix @xmath5 with @xmath3 and @xmath1 representing two subsystems and @xmath2 representing the environment . it is important to emphasize that the environment , here , is constituted by the universe minus the subsystems @xmath3 and @xmath1 , since , in this case , @xmath5 is a pure density matrix . there is an important monogamic relation between the entanglement of formation ( eof ) @xcite and the classical correlation ( cc ) @xcite between the two subsystems developed by koashi and winter @xcite , that we employ to understand the distribution of entanglement . it is given by @xmath6 where @xmath7 is the eof between @xmath3 and @xmath1 , @xmath8 is the cc between @xmath3 and @xmath2 , and @xmath9 is the usual shannon entropy @xcite of @xmath3 . further , @xmath10 and analogously for @xmath11 and @xmath12 . explicitly , cc reads @xmath13 $ ] where the maximum is taken over all positive operator valued measurements @xmath14 performed on subsystem @xmath2 , with probability of @xmath15 as an outcome , @xmath16 and @xmath17 . one can easily understand eq . ( [ koashi ] ) . the entropy @xmath18 measures the amount of correlation ( classical and/or quantum ) between @xmath3 with the external world . if we divide the external world into two parts , @xmath1 and @xmath2 , the amount of quantum correlation between @xmath3 and @xmath1 , plus the amount of classical correlation between @xmath3 and the complementary part @xmath2 , must be equal to @xmath19 . in this sense , eq . ( [ koashi ] ) poses constraints on the ability that system @xmath3 has to share correlations with other systems . for this reason it is called a monogamous relation . we can show a different aspect of eq . ( [ koashi ] ) by adding to both of its sides the mutual information between @xmath3 and @xmath2 , @xmath20 . after some manipulation we obtain @xmath21 where @xmath22 is the conditional entropy and @xmath23 is the qd between subsystem @xmath3 and the environment @xmath2 . ( [ uni1 ] ) tells us that the entanglement between two arbitrary subsystems , a and b , is related to the quantum discord between one of the subsystems ( a ) and the environment e. it is important to note that , although in eq . ( [ uni1 ] ) the eof is written as a function of the qd between @xmath3 and @xmath2 , it is straightforward to write it as a function of the discord between @xmath1 and @xmath2 . in that case , @xmath24 . in the same way , we can evaluate the qd between the subsystems @xmath3 and @xmath1 , @xmath25 which gives the quantum discord between a and b as a function of the entanglement between a and e. remarking that , since the global state is pure , @xmath26 which is in fact the eof of the partition @xmath2 with @xmath27 , @xmath28 , minus the eof of the partition @xmath1 with @xmath29 , @xmath30 . ( [ uni2 ] ) can be rewritten as @xmath31 this result shows that the eof and qd obey a very special monogamic relation , involving bipartite and tripartite entanglement . we now derive a very simple but powerful result regarding the distribution of bipartite entanglement . noting that @xmath32 since @xmath5 is a pure state and summing eq . ( [ uni1 ] ) and eq . ( [ uni2 ] ) , we obtain @xmath33 this important monogamic distribution of eof and qd can also be viewed as a quantum conservation law : _ given an arbitrary tripartite pure system , the sum of all possible bipartite entanglement shared with a particular subsystem , as given by the eof , can not be increased without increasing , by the same amount , the sum of all qd shared with this same subsystem . _ this last fundamental result has remarkable implications in the way that entanglement can be distributed among many parties . for example , we are now able to analyze the power of the quantum computer `` without '' entanglement in view of this last statement . in this sense , let us consider the dqc1 protocol , where the power of one pure qubit was firstly revealed . it is well - known that any quantum computation executed over @xmath0 maximally mixed states does not give rise to exponential speedup when compared with the classical computation . however , knill and laflamme @xcite demonstrated that if just one single pure qubit is added to this set , the situation changes dramatically @xcite - for instance , the dqc1 gives an exponential speedup for the computation of the normalized trace of an unitary operator , @xmath34 . the dqc1 consists of a pure qubit that is represented here by the subsystem @xmath3 and a completely mixed state of @xmath35 qubits , @xmath36 that is represented by @xmath1 . as illustrated in fig . ( [ fig1 ] ) , we observe that initially the subsystem @xmath3 is pure and has zero entanglement and zero discord with respect to @xmath1 and @xmath2 . on the other hand , the subsystem @xmath1 is given by the maximally mixed state . it is important to emphasize here that even a completely mixed state manifests its _ quantumness _ by the fact that it is impossible to distinguish the infinitely many ensembles that can realize it . an alternative way to look at this property is to consider it as an entangled state with an external environment which has as many degrees of freedom as necessary to purify the whole system . thus , we consider here the degree of mixture of @xmath1 as due to the entanglement between an environment @xmath2 , which does not interact with @xmath1 , and @xmath3 . however , it has interacted with @xmath1 in the past , being thus responsible for its _ mixedness_. this approach has been fundamental for the understanding of important tasks in quantum information like schumacher compression , quantum state merging , and entanglement theory @xcite . given this initial situation , we consider a circuit as that exposed in fig . ( [ fig1 ] ) . we suppose that the subsystem @xmath3 is a qubit in the initial state @xmath37 ( an eigenvector of the pauli matrix @xmath38 ) and apply a hadamard quantum gate , followed by a control unitary on the remaining @xmath35 mixed qubit state . thus , after this process , the state of the subsystem @xmath3 and @xmath1 is given by @xmath39 since it gives a separated state with respect to @xmath3 and @xmath1 @xcite . expanding the state on the eigenstate basis @xmath40 of the unitary operator @xmath41 with eigenvalues @xmath42 and considering the purifying system eigenbasis @xmath43 , the joint @xmath44 state can be written as @xmath45 thus the expectation values of @xmath46 and @xmath47 on @xmath3 provide the normalized trace of @xmath41 : @xmath48 and @xmath49 . at the end of the process , just before the measurement that determines @xmath50 and @xmath51 , we will have finite qd between @xmath3 and @xmath1 , @xmath52 @xcite , but no entanglement between them , @xmath53@xcite . using the results given in eq . ( [ uni2 ] ) and eq . ( [ unix ] ) , we meticulously examine the eof and the qd distribution . for this purpose , we examine how the initial entanglement between @xmath1 and @xmath2 is affected during the computation . according to eq . ( [ uni2 ] ) , @xmath54 , but from eq . ( [ rhodq ] ) , @xmath55 . similarly , we see that @xmath56 , an so there really is no bipartite entanglement between @xmath3 with @xmath1 or @xmath2 . but in a similar fashion to eq . ( [ uni21 ] ) we can write @xmath57 allowing us to analyze the eof and the qd distribution in the qdc1 . prior the computation , @xmath58 , and @xmath59 . however , @xmath60 increases after the controlled unitary operation , implying necessarily in a redistribution of the entanglement between the parties . in order to @xmath60 to increase some multipartite entanglement @xmath61 must exist . this entanglement is indeed signaled by the mixed state of @xmath3 alone after the computation . furthermore , using eq . ( [ unix ] ) , it is straightforward to show that for the dqc1 this entanglement unbalance can be measured by the qd between @xmath1 and @xmath2 , since @xmath62 and finally @xmath63 nevertheless , it is important to emphasize that the power of the quantum computer does not come only from the entanglement present between @xmath1 and @xmath2 , or even between @xmath3 and @xmath4 . in the dqc1 , it is clear that it comes from the protocol ability to redistribute entanglement and quantum discord . this property is intrinsic of the protocol and does not rely on the particularities of the environment . the dqc1 protocol ability to transfer entanglement and its efficiency against classical algorithms for special tasks can be tested in the light of the subsystem @xmath1 initial entanglement with @xmath2 . had we started with a non - maximally mixed state for @xmath1 , meaning a non - maximally entanglement with @xmath2 , instead of eq . ( [ rhodq ] ) , one would have ended up with @xmath64 in this case @xmath65 or @xmath66 gives @xmath67 $ ] @xcite containing , thus , less information about the trace of @xmath41 when @xmath1 is initially less entangled with @xmath2 . the worst case is when @xmath1 is in a definite state @xmath68 ( no entanglement with e ) , when we have access to only one eigenvalue of @xmath41 . curiously this corresponds to the situation where a maximal entanglement between @xmath3 and @xmath1 would be available at the end , which certainly does not contribute to any speedup for this special purpose . therefore , we suggest that one should look carefully at the redistribution of entanglement during any quantum computation , and its implication for the speedup of certain protocols . in the present situation , we see that this ability for entanglement redistribution is a necessary ( but not sufficient ) ingredient for efficient quantum computation . at this point , one could imagine what would be the implications of such a relation when some information is lacking for the description of the global state , i. e. , when the tripartite state involving systems @xmath3 , @xmath1 , and @xmath2 is mixed . in that case eq . ( [ koashi ] ) becomes an inequality @xcite and , therefore , eq . ( [ uni1 ] ) turns into @xmath69 similarly , by changing @xmath1 for @xmath2 in the equation above , it now reads @xmath70 , which when added to eq . ( [ u1 ] ) gives @xmath71 with @xmath72 being the balance between the entanglement and the quantum discord in the system . the inequality ( [ ineq ] ) can be stronger than the strong subadditivity ( ss ) @xcite , @xmath73 depending on @xmath74 . for @xmath75 it gives a remarkable lower bound for @xmath76 , which is more restrictive than ( [ sub ] ) and must be fulfilled by any quantum system . thus , we can define a more restrictive inequality than the ss , @xmath77 with @xmath78 where @xmath74 is given by the balance between eof and qd , eq . ( [ delta ] ) . it is important to emphasize that the ss , despite of being more difficult to prove , is essentially derived through extensions of its classical counterpart , but correlations play a different role in quantum systems . so , it is not surprising that a more restrictive bound may occur . to exemplify this let us suppose we have a convex mixed state @xmath79 , where @xmath80 : @xmath81 + \alpha|000\rangle$ ] , with @xmath82 , and @xmath83 is the identity operator over the joint hilbert space of @xmath84 . let us define two quantities @xmath85 and @xmath86 in fig . ( [ fig2 ] ) we plot @xmath87 and @xmath88 as a function of @xmath89 , and , in the inset , we plot @xmath74 for a fixed @xmath90 . it is easy to see that in this situation @xmath74 can be positive or negative . when @xmath91 the inequality given by eq . ( [ ineq ] ) is weaker than the ss given by eq . ( [ sub ] ) . however , when @xmath75 , meaning that the eof of all bipartions is larger than their qd , eq . ( [ ineq ] ) is stronger than eq . ( [ sub ] ) , limiting the lower bound for @xmath76 . this is a strikingly different bound imposed on the entropies of quantum systems , which is not shared by their classical counterpart . the inequality above recovers the ss only when @xmath92 is null , meaning that the distribution of bipartite entanglement is equal to the amount of distributed quantum discord or smaller than that . it is important to emphasize here the essential role that ss plays in classical and quantum information theories @xcite . many fundamental inequalities , as nonnegativity of entropy and subadditivity , can be derived from that . to the best of our knowledge , the only inequality known to be independent is the one proposed in ref . @xcite , which is valid when the ss saturates on some particular subsystems configuration . it is straightforward to show that the inequality in ref . @xcite is independent of ( [ newstrsub ] ) as well , when @xmath75 , since in this case ss can not be saturated . however something else can be learned from this saturation . given a quadripartite quantum system @xmath93 such that ss is saturated for the three triples @xmath94 , @xmath95 , and @xmath96 then , @xmath97 @xcite . substituting @xmath98 by @xmath99 and using the monogamic relation , eq . ( [ koashi ] ) , and the conservation law , eq . ( [ unix ] ) , it is straightforward to show that when @xmath100 we have @xmath101 . so , as in eq . ( [ ineq ] ) , the difference between the eof and the qd is of fundamental importance . ) , @xmath87 , in red ( solid line ) and the difference between the right and left hand sides of eq . ( [ ineq ] ) , @xmath88 , in blue ( dotted line ) . combining these two quantities the stronger inequality eq . ( [ newstrsub ] ) is obtained . the difference between its right and left hand sides is given by the shaded area . the inset shows @xmath102 . ] to summarize , we have given a monogamic relation between the eof and the qd . for that , we have derived a general interrelation on how those quantities are distributed in a general tripartite system . we applied this relation to show that in the dqc1 the entanglement present between one of the subsystems and the environment is responsible for the non - zero quantum discord . since the maximally mixed state is entangled with the environment , we show that the circuit described by the dqc1 distributes this initial entanglement between the pure qubit and the mixed state . our results suggest that the protocol ability to redistribute entanglement is a necessary condition for the speedup of the quantum computer . in addition , we have extended the discussion for an arbitrary tripartite mixed system showing the existence of an inequality for the subsystems entropies which is stronger than the usual ss . 99 h. ollivier and w. h. zurek , phys . lett . * 88 * , 017901 ( 2001 ) . b. p. lanyon , et al . rev . lett . * 101 * , 200501 ( 2008 ) , a. shabani and d. a. lidar , phys . rev . 102 * , 100402 ( 2009 ) , t. werlang , et al . a * 80 * , 024103 ( 2009 ) , k. modi , et al . * 104 * , 080501 ( 2010 ) . e. knill and r. laflamme , phys . 81 * , 5672 ( 1998 ) . d. poulin , r. blume - kohout , r. laflamme , and h. ollivier , phys . . lett . * 92 * , 177906 ( 2004 ) . p. w. shor and s. p. jordan , arxiv:0707.2831 ( 2007 ) . a. datta , a. shaji , and c. m. caves , phys . lett . * 100 * , 050502 ( 2008 ) . c. e. lpez , g. romero , f. lastra , e. solano , and j. c. retamal , phys . * 101 * , 080503 ( 2008 ) ; j. maziero , t. werlang , f. f. fanchini , l. c. cleri , and r. m. serra , phys . rev . a * 81 * , 022116 ( 2010 ) . c. h. bennett , d. p. divincenzo , j. a. smolin , and w. k. wootters , phys . a * 54 * , 3824 ( 1996 ) . l. henderson and v. vedral , j. phys . a * 34 * , 6899 ( 2001 ) ; v. vedral , phys . lett * 90 * , 050401 ( 2003 ) . m. koashi and a. winter , phys . a * 69 * , 022309 ( 2004 ) . m. a. nielsen and i. l. chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , england , 2000 ) . b. schumacher , phys . a * 51 * , 2738 ( 1995 ) . m. horodecki , j. oppenheim , and a. winter , nature * 436 * , 673 ( 2005 ) ; m. horodecki , j. oppenheim , and a. winter , nature * 436 * , 673 ( 2005 ) ; m. f. cornelio , m. c. de oliveira , f. f. fanchini arxiv:1007.0228 ( 2010 ) . e.h lieb and m.b . ruskai , j. math . phys . * 14 * , 1938 ( 1973 ) . n. linden and a. winter , commun . math . phys . * 259 * , 129 ( 2005 ) .

we present a direct relation , based upon a monogamic principle , between entanglement of formation ( eof ) and quantum discord ( qd ) , showing how they are distributed in an arbitrary tripartite pure system . by extending it to a paradigmatic situation of a bipartite system coupled to an environment , we demonstrate that the eof and the qd obey a conservation relation . by means of this relation we show that in the deterministic quantum computer with one pure qubit the protocol has the ability to rearrange the eof and the qd , which implies that quantum computation can be understood on a different basis as a coherent dynamics where quantum correlations are distributed between the qubits of the computer . furthermore , for a tripartite mixed state we show that the balance between distributed eof and qd results in a stronger version of the strong subadditivity of entropy .

information on the diffusion of ions and pretransitional increase of copper defect numbers is mainly obtained from line shape and spin - lattice relaxation measurements on copper . the total relaxation rate is a sum of the phonon contribution and the two defect parts : @xmath80 quadruploar spin 3/2 copper nuclei couple to electric field gradients due to phonons and defects ; the phonon part is well approximated by @xmath81 above the debye temperature ( with @xmath82 a constant ) @xcite . in the simplest approximation of point charges and exponential autocorrelation , the relaxation rate due to one kind of defects is given by eq . ( 1 ) in the paper . hopping times and defect numbers are temperature - dependent and in principle arrhenius - like , except copper defects close to the order - disorder transition . a linear scale plot of spin - lattice relaxation vs. temperature below @xmath8 ( fig . [ t1_lin ] ) clearly demonstrates that two distinct defect contributions are neccesary to reproduce the temperature dependence of the relaxation . we must therefore fit a superposition of the form of ( [ t1 ] ) , with many adjustable parameters . to constrain the fit , we used several additional results . we easily determined the number of copper defects using measured integrals of the nmr line , as described in the paper . an additional field - dependent relaxation measurement was performed at 280 k to increase the reliability of the remaining fitting parameters two activation energies for mercury ( defect formation and jump ) and one for copper ( jump ) , and three multiplicative constants ; field and temperature dependecies were fitted simultaneously . attempt frequencies were constrained to one order of magnitude using raman spectroscopy data ( where anomalous low - frequency modes are identified with the attempt frequencies @xcite ) , and agree with typical values for similar compounds ( e.g. cui @xcite ) . the activation energies resulting from the fit are given in the paper . the approximation of independent defects is most probably adequate for temperatures not very close to @xmath8 , but the reliability of our fitting model can be questioned near @xmath8 , as we suppose that only the number of copper defects grows dramatically . however , the line shape measurements indicate that indeed copper defects are much more prevalent than mercury intensity of the @xmath3cu line drops sharply , while @xmath21hg shows no appreciable change ( within the relatively large error margin ) . above @xmath8 the relaxation rate is significantly higher than in the low temperature phase , due to the large number of fast mercury ions . if we assume that the relaxation model of eq . ( 1 ) in the paper is at least approximately valid above @xmath8 , clearly the contribution of mercury ion diffusion will dominate the relaxation rate ( as we know from the absence of motional narrowing in cu that copper motion is much slower , and @xmath83 ) . we can thus use mercury hopping times obtained directly from nmr line shapes to make a comparison with the @xmath3cu relaxation . in this limit @xmath84 is roughly proportional to @xmath85 ( with @xmath39 the number of mobile ions , and @xmath45 the conductivity as defined in eq . ( 2 ) in the paper ) . when @xmath84 , @xmath86 and @xmath46 are plotted on top of each other , we see that the relaxation rate follows the conductivities within the error margin ( fig . [ t1 ] insert ) , confirming our qualitative analysis . @xmath3cu spin - lattice relaxation rate below @xmath8 , on a linear scale . raman phonon process and two defect diffusion contributions are plotted separately . insert is relaxation rate above @xmath8 ( circles ) , which is seen to roughly fall between @xmath45 ( squares ) and @xmath46 ( dotted line).,width=340 ] motional narrowing enables us to make selective nmr measurements on moving mercury ions . the spin - spin relaxation time of stationary ions , @xmath87 , is of the order of @xmath88 @xmath48s , while the fast ions have some ten times longer relaxation times ( fig . [ stimecho]a ) . if we use a stimulated spin echo sequence@xcite ( fig . [ stimecho]b ) we can exploit this difference . the first two pulses excite only spins which are moving at the moment of excitation ( if @xmath89 and @xmath90 ) and thus tag them magnetically . after a diffusion time @xmath91 the third pulse is applied , causing a stimulated echo . a ) spin - spin relaxation for mercury at 380 k , obtained using a conventional spin echo sequence . b ) the stimulated spin echo ( sse ) sequence . radio - frequency pulses and spin echo signal are represented schematically.,width=291 ] however , tagged spins which have been trapped during the time @xmath91 and are stationary at the moment of the application of the third pulse , have shorter @xmath92 and do not contribute to the echo signal . thus the echo amplitude should be anomalously small for diffusion times comparable to and larger than the dynamic correlation time @xmath54 . the size of this dip in diffusion time dependence provides an estimate of the absolute number of correlated ions . such behavior is observed in simulation as well even the same functional dependence ( a stretched exponential ) can be fitted to experimental and simulation results . it is possible to obtain a good agreement between experiment and simulation in both correlation time and fraction of correlated ions ( for a vacancy fraction of 0.4 and effective copper diffusion coefficient @xmath93 ) , but the simplifying assumption of single copper and mercury jump probabilities makes the stretching parameter of the simulation curve somewhat closer to 1 . this could be quickly amended by introducing a jump probability distribution by hand , but was deemed physically untransparent . the agreement between simulation and experiment is already quite impressive in the simple version , and more sophisticated models are needed to account for the finer effects . in absence of trapping , the echo decay should be a simple exponential in @xmath91 , with the decay time equal to the spin - lattice relaxation time @xmath78 . the spin - spin relaxation is single exponential ( fig . [ stimecho]a ) , the line shape does nt change noticeably for any @xmath91 , and @xmath21hg is a spin 1/2 nucleus . hence the deviations from an exponential decay in the sse experiment can only be due to dynamical trapping .

we present the observation of glasslike dynamic correlations of mobile mercury ions in the ionic conductor cu@xmath0hgi@xmath1 , detected in both nmr and nonlinear conductivity experiments . the results show that dynamic cooperativity appears in systems seemingly unrelated to glassy and soft arrested materials . a simple kinetic two - component model is proposed , which seems to provide a good description of the cooperative ionic dynamics . in glass - forming materials particles increasingly move together as the glass transition is approached @xcite . such cooperativity is also found in other arrested systems @xcite and seems to be intimately connected to the slow dynamics . here we report on the observation of large - scale dynamic correlations in a distinctly nonglassy system the conductive phase of the ionic conductor cu@xmath0hgi@xmath1 . using carefully designed nuclear magnetic resonance experiments we prove that mercury ions are the main contributors to conduction ( establishing cu@xmath0hgi@xmath1 as the first known mercury conductor ) , and show that mercury diffusion is anomalous . these results urge for a more detailed examination of ionic motion . therefore nonlinear conductivity measurements are used as a probe for dynamical heterogeneity , revealing a characteristic correlation time scale . to explain the cooperativity we propose a simple model related to previous work on glasses @xcite , with two essential ingredients disorder and existence of two kinds of particles , slow ( copper ) and fast ( mercury ) . we compare the results with recent studies of arrested and glass - forming materials @xcite , thus establishing an unexpected connection between seemingly different fields . cu@xmath0hgi@xmath1 used in experiments was in powder form , synthetized according to standard procedure @xcite . x - ray diffraction at 300 k showed no appreciable contamination with iodides , and all applied techniques ( nmr , dsc , conductivity ) saw a sharp transition at @xmath2 k , providing further evidence of phase purity . free induction decays were used to record nmr line shapes , while a recovery sequence was employed in the @xmath3cu relaxation measurements . conductivity was measured in a two - contact cylindrical cell with graphite electrodes , using a low distortion voltage source and lock - in amplifier . at all temperatures , sample resistance was above 1 m@xmath4 . low frequency ( 7 hz ) conductivity agrees quantitatively with previously published values @xcite . the third harmonic current @xmath5 provided nonlinear conductivity ; the heating contribution to @xmath5 was estimated to be small due to large sample resistance , and more importantly , uniform over the employed frequency range ( linear conductivity is not peaked ) . instrumental harmonic distortion effects were also negligible between @xmath6 hz and @xmath7 khz . all measurements were reproducible after several temperature cycles across @xmath8 . before studying cooperative ionic motion in cu@xmath0hgi@xmath1 , we must identify the charge carriers and nature of the insulator - conductor transition at @xmath8 . ever since the discovery of ionic conduction in cu@xmath0hgi@xmath1 @xcite , it has remained unclear which ion species predominantly carries current in the conducting phase above @xmath8 @xcite . here we obtain direct proof of mercury motion from nmr experiments a substantial motional narrowing of the mercury line in the conductive phase [ fig . [ nmr](a ) ] . in contrast , the copper line is broadened , indicating quasistatic disorder . this suggests that the transition is not melting of the copper sublattice ( unlike the related cui @xcite ) , but rather an order - disorder transition @xcite with the copper ions remaining virtually static . mercury motion is then enhanced in the changed energy landscape above @xmath8 . slow copper diffusion and disorder are essential for explaining mercury dynamic cooperativity , so we perform nmr line shape and relaxation measurements on copper to provide a microscopic picture of the transition . the structure of cu@xmath0hgi@xmath1 in the ordered phase below @xmath8 contains 8 tetrahedral positions per unit cell , with only 3 occupied by copper or mercury @xcite . thus one expects that a relatively small activation energy is neccessary to create point defects by moving ions from `` regular '' to normally empty tetrahedral positions . nmr enables us to follow these motions in an ion - specific way , and observe how they lead to a transition to the disordered phase . naturally abundant copper nuclei are quadrupolar and thus sensitive to electric field gradients ( efgs ) present in the material . this is already obvious in the nmr line shapes the local environment of copper ions does nt have cubic symmetry , leading to a quadrupolar splitting of the line . while all cu@xmath9 which are on regular sites have roughly the same local environment , ions on normally vacant sites ( defects ) experience much larger efgs and their nmr lines are substantially shifted and broadened @xcite . the signal from defects thus contributes as a low , broad background easily separable from the narrow line of `` regular '' cu@xmath9 . the integral of the narrow line can be used to obtain the relative number of defects . this number is expected to grow anomalously fast close to an order - disorder transition , as is indeed observed . one can even see power - law behavior close to @xmath8 , described reasonably well with a mean - field approach . we define an order parameter @xmath10 , with @xmath11 and @xmath12 the number of regular copper atoms and copper defects , respectively . the transition being first order , @xmath13 drops abruptly to zero above @xmath8 . a landau expansion yields @xmath14 close to @xmath8 , with @xmath15 the order parameter at @xmath8 , @xmath16 the reduced temperature and @xmath17 the critical exponent : @xmath18 , comparing well with the experimental value of 0.58 ( fig . [ nmr]b right insert ) . at @xmath8 the lattice reorganizes and the distinction between `` defects '' and `` regular '' copper ions is lost , as all cu@xmath9 positions are equally probable @xcite . however , no motional narrowing of the copper line is observed above @xmath8 . instead , the line broadens due to larger efgs caused by electrostatic disorder . ( color online ) a ) nmr lines of copper and mercury , below and above @xmath19 . frequency is relative to larmor frequencies , 145.6 mhz for @xmath20cu and 91.2 mhz for @xmath21hg . pulse excitation width is @xmath22 khz . copper lines are normalized to have the same integrals , while the low - temperature mercury line is multiplied by 10 after normalization . significant motional narrowing is observed for mercury above @xmath8 . b ) spin - lattice relaxation measurements for @xmath3cu . left insert is frequency dependence at 280 k. lines are fits obtained from a superposition of phonon and two defect diffusion processes ( mercury and copper ) . temperature ranges where each of the processes comes into play are indicated . right insert is order parameter in dependence on reduced temperature . , width=325 ] to learn more about defect dynamics we measure copper spin - lattice relaxation [ fig . [ nmr](b ) ] . diffusing defects create fluctuating efgs , which influence the relaxation . in addition to a conventional raman phonon mechanism @xcite , we observe effects of both mercury and copper defect diffusion on the relaxation rate below @xmath8 . the contribution from one defect type is @xcite @xmath23^{2}}\ ] ] with @xmath24 a temperature - independent quadrupolar coupling constant , @xmath25 hopping time and @xmath26 larmor frequency . the hopping process is thermally activated @xcite , with @xmath27 , where @xmath28 is the attempt frequency and @xmath29 the hopping activation energy . the numbers of defects also follow arrhenius - type laws except close to @xmath8 . combining the temperature and frequency dependences of the relaxation rate , with attempt frequencies estimated from raman spectroscopy @xcite , we obtain @xmath30 for both copper and mercury defects @xcite . the values are @xmath31 k and @xmath32 k for copper and mercury , respectively . thus already below @xmath8 mercury has a significantly lower hopping activation energy than copper . microscopic reasons are as of yet unclear . ( color online ) a ) dc conductivity ( circles ) is smaller than conductivity predicted from mercury nmr ( squares ) , indicating anomalous behavior . the discrepancy between conductivities closely follows the number of correlated ions ( diamonds , estimated from nonlinear conductivity and nmr ) b ) nonlinear conductivity in dependence on frequency . peaks clearly show the existence of a cooperativity time scale . nonlinear response below @xmath19 is negligible . lines are guides to the eye . , width=328 ] above @xmath8 the mercury diffusion rate increases for an order of magnitude and the @xmath21hg line becomes motionally narrowed . the line shape is well fitted by a lorentzian curve and the hopping time can be extracted from the linewidth using @xmath33 , where @xmath34 is the static linewidth ( below @xmath8 ) . the simple formula is valid for @xmath35 , so we have taken into account corrections for finite @xmath25 where neccessary @xcite . employing the einstein relation for mobility , we can try to calculate the conductivity from extracted hopping times : @xmath36 where @xmath37 is the charge of hg@xmath38 ions , @xmath39 their number density ( @xmath40 @xmath41 ) and @xmath42 a hopping distance of the order of the interatomic spacing ( @xmath43 ) . if we now take this conductivity and compare it to the measured dc values , we observe the first sign of anomalous behavior : in a region @xmath44 k above @xmath8 , @xmath45 is significantly larger than @xmath46 ( fig . [ corr_panel]a ) . thus relation ( [ sigma1 ] ) , valid for simple stochastic motion of ions , does nt correctly predict the long - time transport . one may ask if this is due to the existence of some new , intermediate timescale above @xmath47 @xmath48s where motional correlations arise , or just well - known short range correlation effects quantified with the haven ratio @xcite . to resolve the question , we measure the nonlinear conductivity @xmath49 , defined by @xmath50 , in dependence on frequency @xmath51 ( fig . [ corr_panel]b ) . although dynamical correlation effects often bear small influence on linear response , they are intimately related to nonlinear susceptibilities . a quantitative measure is the four - point correlation function @xcite , @xmath52 ( with @xmath53 a suitable dynamic parameter , e.g. intermediate scattering function ) , representing the correlation of time changes at different points in space . thus if many ions move synchronously on a characteristic timescale @xmath54 , @xmath55 will have a peak at @xmath54 . generalized fluctuation - dissipation theorems connect @xmath56 , the spatial integral of @xmath57 , with the corresponding nonlinear susceptibility @xcite , making dynamical correlations measurable . in contrast to the case of a dielectric ( or magnetic ) material , where one measures the response of dipoles to an external field , we detect the response of mobile charges , and the natural response function is @xmath58 instead of the susceptibility @xmath59 . a lot of activity is currently aimed at modeling dynamical heterogeinity in soft and glassy systems @xcite , but experimental data are still scarce the first report on nonlinear susceptibility of a glass - former ( glycerol ) only appeared recently @xcite . here we see similar effects , but in a rather unexpected material . characteristic correlation timescales are revealed through peaks in @xmath49 at frequencies @xmath60 , and the relative number of correlated ions , @xmath61 , can be estimated from integrals of the peaks [ fig . [ corr_panel](a ) ] . in our work we focused on @xmath58 as it is sensitive to correlated motion ; we note , however , that the new correlation timescale also affects the linear conductivity @xmath62 . a shoulderlike feature is visible at the frequency @xmath63 in the frequency spectrum of @xmath64 : this strengthens the analogy with supercooled liquids , where , similarly , the peak of the nonlinear response occurs close to the characteristic relaxation frequency visible in linear response . ( color online ) stimulated spin echo nmr measurements at 380 k , giving evidence of mercury ion trapping at the characteristic timescale @xmath54 . insert is raw measurement , the square denoting the zoomed - in segment where dynamic trapping effects are visible . main graph is compensated for spin - lattice relaxation , showing only the correlation contribution . full line is from simulation , and the dotted line a stretched exponential fit.,width=340 ] except very close to the transition , @xmath54 is substantially longer than the mercury hopping time @xmath25 . thus a simplistic conduction model can be used to explain the discrepancy between @xmath45 and @xmath46 . we assume that mercury ions move vigorously most of the time , but sometimes get constrained to small volumes . nmr lines of these ions are broad and do not contribute to the principal narrow line . occasionally several trapped ions arrange favourably , and leave the `` trap '' together . thus the effective number of charge carriers is diminished and the characteristic correlation timescale appears . this is essentially a `` cooperatively rearranging regions '' ( crr ) scenario , well known in glass science @xcite . a similar mechanism was also proposed for colloidal gel relaxations @xcite , and seems to offer a good phenomenological explanation of our data . contrary to glass - forming liquids , where @xmath61 has no effect on @xmath46 , here the correlations influence it . we obtain direct experimental evidence for this model from a different nmr experiment on mercury stimulated spin echo ( sse ) measurements ( fig . [ se ] ) . moving spins experience much smaller average local fields than trapped ones , leading to a difference in spin decoherence times . this can be exploited to selectively excite and detect only ions moving at a given moment . the sse sequence @xcite is perfectly suited for such an experiment @xcite . after correcting for spin - lattice relaxation , we can directly observe how , of all ions moving at the time of excitation , a sizeable fraction becomes trapped after a time @xmath65 ( fig . [ se ] ) . this experiment provides an absolute scale for @xmath61 , and we can make a comparison with the difference between @xmath45 and @xmath46 [ fig . [ corr_panel](a ) ] . the agreement is gratifying , both in absolute scale and temperature dependence , implying that the difference can be attributed to a diminished effective number of carriers , confirming the phenomenological model . however , microscopic questions remain : what causes confinement , and how are correlated jumps performed ? to answer them , we propose a very simple mechanism , related to investigations of spin glasses ( essentialy a limiting case of the edwards - anderson hamiltonian with diffusion @xcite ) . aside from disorder , the basic requirement is the existence of two kinds of atoms in the material , with different diffusion coefficients , and short - range interactions . in cu@xmath0hgi@xmath1 this is realized with copper and mercury , on a fixed iodine background . if we take low - temperature activation energies to be representative , we can conclude that mercury diffuses 10@xmath66 to 10@xmath67 times faster than copper in the interesting temperature range where correlations appear . as slow copper ions move around , they occasionally form compartments with several trapped mercury ions inside . the compartments can then `` open '' due to cation rearrangement once a path is open , many fast mercury ions use it sequentially to empty the compartment . such behavior is indeed observed in a two - dimensional random walk simulation . simulations were run on a square lattice with periodic boundary conditions and initially randomly placed ions . in every step the mercury ions moved in random ( allowed ) directions , while the copper ions moved with a certain probability @xmath68 ( which is essentially the ratio of copper and mercury diffusion coefficients ) . in the course of simulation large mercury `` islands '' form and dissolve ( fig . [ siml ] , insert ) . to calculate the four - point correlation function @xmath69 , we used the persistence function @xcite , defined as @xmath70 if nothing has happened on site @xmath71 for @xmath72 , and @xmath73 otherwise . @xmath56 is calculated as the variance of the autocorrelation of @xmath74 , evaluated at all mercury sites @xcite . a characteristic correlation timescale is revealed ( fig . [ siml ] ) , and the curves qualitatively follow the crr prediction @xcite . the only parameters we have to set are @xmath68 and effective particle concentration , taking care that the number of vacant sites is above the percolation threshold @xcite . for realistic concentrations and @xmath68 , correlation times become about @xmath75 , in fair agreement with experiment . the sse decay curve ( fig . [ se ] ) can also be predicted surprisingly well . ( color online ) some results of the 2d simulation . inserted frame at 2000 steps shows formed mercury islands ( @xmath76 , simulation box 300x300 cells ) . a correlation timescale is nicely visible in the time dependence of the four - point correlation @xmath56 , for several @xmath77 . , width=321 ] the model is `` minimal '' , in the sense that we obtain dynamical heterogeinity with the minimum number of assumptions . this hints at a considerable universality of such correlations . important effects have , however , been neglected : the iodine lattice potential , short - range electrostatic correlations , electronic dynamics and phonons . correct temperature behavior can not be obtained without taking them into account . in contradistinction to glass - forming materials , no slowing down of the correlated dynamics with decreasing temperature is observed in cu@xmath0hgi@xmath1 ; we suspect that this is due to the additional periodic potential , which tends to restore an ordered state ( and succeeds at @xmath8 ) . also , we believe that a full three - dimensional simulation would show similar cooperative behavior ( with modified vacancy numbers due to a lower percolation threshold ) , but this needs to be proven . more elaborate simulations are needed to better understand these issues . from our results we can conclude that ingredients needed for large - scale motional correlations are quite ubiquitous , so it is reasonable to believe that ionic cooperativity is important for many other systems as well . in disordered ionic conductors it might offer a more convincing explanation of nonlinear response than standard hopping models @xcite , opening up new perspectives for studying ion dynamics . even more important is the connection with arrested materials , which shows that dynamical correlations are more universal than previously thought . the observed interplay between lattice potential and dynamical heterogeneity is very interesting in itself and could provide a unique possibility for exploring the emergence of glasslike correlations . we thank d. cini and v. stilinovi for dsc and x - ray measurements and a. duli , h. buljan , s. marion and m. s. grbi for helpful discussions and comments . the research leading to these results was supported by equipment financed from the european community s seventh framework programme ( fp7/2007 - 2013 ) under grant agreement no . 229390 solenemar and by funding from the croatian ministry of science , education and sports through grant no . 119 - 1191458 - 1022 . 1 _ dynamical heterogeneities in glasses , colloids , and granular media _ , edited by l. berthier , g. biroli , j .- p . bouchaud , l. cipeletti , and w. van saarloos ( oxford university press , new york , 2011 ) c. crauste - thibierge , c. brun , f. ladieu , d. lhote , g. biroli , and j .- p . bouchaud , _ phys . rev . lett . _ * 104 * , 165703 ( 2010 ) a. duri and l. cipelletti , _ europhys . lett . _ * 76 * , 972 ( 2006 ) . o. dauchot , g. marty , g. biroli , _ phys . rev . lett . _ * 95 * , 265701 ( 2005 ) . a. s. keys , a. r. abate , s. c. glotzer and d. j. durian , _ nature phys . _ * 3 * , 260 ( 2007 ) . p. mayer _ et al . _ , _ phys . rev . lett . _ * 93 * , 115701 ( 2004 ) . d. apeta and d. k. sunko , _ phys . rev . b _ * 74 * , 220201(r ) ( 2006 ) . g. adam and j. h. gibbs , _ j. chem . phys . _ * 43 * , 139 ( 1965 ) . precipitation from an aqueous solution , similar to l. suchow , p. h. keck , _ j. am . chem . soc . _ * 75 * , 518 ( 1953 ) l. suchow and g. r. pond , _ j. am . chem . soc . _ * 75 * , 5242 ( 1953 ) s. hull and d. a. keen , _ j. phys . : cond . matt . _ * 12 * , 3751 ( 2000 ) . j. a. a. ketelaar , _ z. kristallogr . _ * 80 * , 190 ( 1931 ) l. eriksson , p. wang and p. werner , _ z. kristallog . _ * 197 * , 235 ( 1991 ) j. b. boyce and b. a. huberman , _ solid state commun . _ * 21 * , 31 ( 1977 ) m. lumsden , m. steinitz and e. j. mcalduff , _ j. appl . phys . _ * 77 * , 6039 ( 1995 ) f. reif , _ phys . rev . _ * 100 * , 1597 ( 1955 ) . this is nicely seen in x - ray diffraction experiments , ref . [ 13 ] . a. abragam , _ the principles of nuclear magnetism_. ( oxford university press , oxford , 2002 ) . n. w. ashcroft and n. d. mermin , _ solid state physics_. ( holt , rinehart & winston , new york , 1976 ) j. i. mcomber , d. f. shrivera and m. a. ratner , _ j. phys . chem . solids _ * 43 * , 895 ( 1982 ) . for details on the rather intricate fitting process and comments on the behavior of copper @xmath78 above @xmath8 , see supplemental material . a. abragam , ( ref . [ 18 ] ) . even at the lowest temperature ( 345 k ) the correction to a lorentzian result is less than 20@xmath79 . g. e. murch , _ solid state ionics _ * 7 * , 177 ( 1982 ) . j .- p . bouchaud and g. biroli , _ phys . rev . b _ * 72 * , 064204 ( 2005 ) . n. laevi , f. w. starr , t. b. schrder and s. c. glotzer , _ j. chem . phys . _ * 119 * , 7372 ( 2003 ) . j. n. fry and p. j. e. peebles , _ astrophys . j. _ * 221 * , 19 ( 1978 ) . l. berthier and g. biroli , _ rev . mod . phys . _ * 83 * , 587 ( 2011 ) . t. r. kirkpatrick , d. thirumalai and p. g. wolynes , _ phys . rev . a. _ * 40 * , 1045 ( 1989 ) . e. l. hahn , _ phys . rev . _ * 80 * , 580 ( 1950 ) . for a detailed description of the stimulated echo measurement technique , see supplemental material . s. f. edwards and p. w. anderson , _ j. phys . f _ * 5 * , 965 ( 1975 ) . c. toninelli , m. wyart , l. berthier , g. biroli and j .- p . bouchaud , _ phys . rev . e _ * 71 * , 041505 ( 2005 ) . the system is given an equilibration time of several ( typically 10 ) @xmath54 , after which the calculation of @xmath56 begins . the effective number of available vacancies is possibly smaller than the geometric ratio obtained from the crystal structure , due to coulomb repulsion between ions ( some evidence for this is presented for the similar compound ag@xmath0hgi@xmath1 in t. hibma , h. u. beyeler and h. r. zeller , _ j. phys . c : solid state phys . _ * 9 * , 1691 ( 1976 ) ) . a. heuer , s. murugavel , and b. roling , _ phys . rev . b _ * 72 * , 174304 ( 2005 ) .

the preparation of an intense beam of polarized antiprotons is _ the _ crucial point for the physics program proposed by the pax collaboration @xcite at the future fair facility in darmstadt . a possibility to overcome this experimental challenge is seen in elastic scattering of antiprotons off a polarized @xmath5h target @xcite . this conjecture is motivated by the result of the filtex experiment @xcite , where a sizeable effect of polarization buildup was achieved in a storage ring by scattering of unpolarized protons off polarized hydrogen atoms at low beam energies of 23 mev . recent theoretical analyses @xcite suggest that the polarization effect observed in ref . @xcite is solely due to the spin dependence of the hadronic ( proton - proton ) interaction , which gives rise to the so - called spin - filtering mechanism , i.e. leads to different rates of removal of beam protons from the ring for different polarization states of the hydrogen target . contrary to what was assumed before @xcite , proton scattering on the polarized electrons of hydrogen atoms does not provide sizeable effects for the polarization buildup @xcite . accordingly , only the hadronic interaction of antiprotons with nucleons or nuclei can be used to produce polarized antiprotons on the basis of the spin - filtering mechanism @xcite . since the spin - dependent part of the @xmath6 interaction is still poorly known experimentally , the polarization buildup mechanism in elastic scattering of stored antiprotons off a polarized @xmath5h target is planned to be studied in a new experiment at cern @xcite at intermediate energies . some theoretical estimations of the expected polarization effects were already presented , based on the amplitudes of the paris @xcite and jlich @xcite @xmath7 potential models and the nijmegen @xcite @xmath7 partial - wave analysis . in this context , it is important to explore other antiproton nucleus interactions as possible source for the antiproton polarization buildup too . therefore , we present here results of a study of polarization effects in antiproton - deuteron ( @xmath8 ) scattering for beam energies up to 300 mev @xcite . besides the issue of polarization buildup for antiprotons , @xmath9 scattering on a polarized deuteron , if it will be studied experimentally , can be also used as a test for our present knowledge of the @xmath10 and @xmath4 interactions . our investigation is based on the glauber - sitenko theory for @xmath0 scattering and it utilizes the @xmath11 interaction models developed by the jlich group @xcite as input for the elementary amplitudes . considering the full spin dependence of the forward @xmath2 elastic scattering amplitude @xcite and using the optical theorem , one can show that the total polarized @xmath8 cross section can be written as @xmath12 where @xmath13 is the polarization of the antiproton beam and @xmath14 ( @xmath15 ) is the vector ( tensor ) polarization of the deuterium target , @xmath16 is the unpolarized and @xmath17 ( @xmath18 ) are the polarized total cross sections . the unit vector @xmath19 is fixed by the direction of the beam momentum . one can find from eq . ( [ totalspin ] ) that only the cross sections @xmath20 and @xmath21 are connected with the spin - filtering mechanism @xcite and , thus , determine the rate of the polarization buildup in the scattering of unpolarized antiprotons off polarized deuterons . the tensor cross section @xmath22 is not related to the polarization of the beam and , therefore , is not relevant for the spin - filtering . however , this cross section , as well as the unpolarized cross section @xmath16 , determine the lifetime of the beam . we utilize the glauber theory of multiple scattering @xcite for investigating the @xmath2 scattering process . for the elementary @xmath23 amplitudes we use those of the jlich models a and d @xcite . details on the applied formalism can be found in ref . @xcite . in order to check the reliability of this approach we calculate the unpolarized total and differential @xmath2 cross sections where we can compare our results with available experimental information . in the evaluation of the polarized cross sections @xmath17 ( @xmath24 , we take into account the coulomb - nuclear interference terms , which are added to the corresponding purely hadronic cross sections . the pure coulomb amplitude does not contribute to @xmath25 , but it gives an important contribution to @xmath16 . in order to calculate the contribution of the coulomb - nuclear interference terms one can not use the optical theorem because of the coulomb singularity at the scattering angle @xmath26 , and therefore we use here the method of ref . @xcite , adapted for the case of @xmath27 scattering @xcite . for the polarized cross sections we use only the single - scattering approximation . as was shown in refs . @xcite , in forward elastic scattering of antiprotons off nuclei the glauber theory of diffractive multiple scattering , though in principle a high - energy approach , works rather well even at fairly low antiproton beam energies . the reason for this is that due to strong annihilation effects , the @xmath23 elastic differential cross section is peaked in forward direction already at rather low energies and , therefore , suitable for application of the eikonal approximation , which is the basis of the glauber theory . the elastic spin - averaged @xmath23 scattering amplitude can be parameterized as @xmath28 where @xmath29 is the total unpolarized @xmath23 cross section , @xmath30 is the ratio of the real to imaginary part of the forward amplitude @xmath31 , @xmath32 is the slope of the diffraction cone , @xmath33 is the transferred 3-momentum , and @xmath34 is the @xmath23 cms momentum . we use eq . ( [ fpn ] ) to represent the scattering amplitudes of the jlich @xmath7 models in analytical form . when performing the fit we found that even at beam energies as low as 10 - 25 mev the parameter @xmath35 is large , i.e. @xmath36 ( gev / c)@xmath37 , reflecting the fact that the @xmath23 amplitude is indeed peaked in forward direction . results for the total unpolarized @xmath2 cross section are displayed in fig . [ totpd ] together with experimental information @xcite . one can see that the single - scattering approximation ( shown here for model d only ) overestimates the total unpolarized cross section by roughly 15% , cf . the dotted line . but the shadowing effect generated by @xmath23 double scattering reduces the cross section ( solid line ) and leads to a good agreement with the experiment . the results for model a ( including also double scattering ) are very similar ( dashed line ) and also in agreement with the data . predictions for differential cross sections are presented in fig . [ pbard179 ] . also here the single - scattering mechanism as well as the double - scattering terms were included in the corresponding calculation . the abb form factor @xcite is used for the deuteron . at @xmath38 = 179.3 mev data for the elastic differential cross section are available @xcite . these data ( squares in fig . [ pbard179 ] ) are nicely reproduced by our model calculation for forward angles . also the differential cross sections for elastic ( @xmath39 ) plus inelastic ( @xmath40 ) scattering events , measured at the neighboring energy of @xmath38 = 170 mev as well as at some lower energies @xcite ( circles ) , are well described . results for the spin - dependent @xmath2 cross sections @xmath20 and @xmath21 are obtained in the single - scattering approximation and presented in fig . [ totpdx ] ( right - hand side ) . we show predictions based on the purely hadronic part , @xmath41 , as well as full results , including the coulomb - nuclear interference term , i.e. @xmath42 for @xmath43 . the @xmath7 model d predicts large values for @xmath20 around 40 mev and for @xmath21 around 25 mev . in case of model a the most pronounced spin dependence is seen at considerably higher energies . compared with the results for the @xmath44 reaction , shown on the left - hand side of fig . [ totpdx ] , the spin - dependent @xmath3 cross sections @xmath20 and @xmath21 are of similar magnitude or even larger . with regard to the purely hadronic contribution , @xmath41 ( i=1,2 ) , the cross sections for @xmath45 and @xmath2 are , in general , of opposite sign @xcite . it should be said , however , that the sign does not affect the spin - filtering mechanism . the effect of the coulomb - nuclear interference is somewhat smaller for @xmath27 scattering than for the @xmath44 case . this difference comes from the additional @xmath46 amplitudes entering the expression for @xmath47 in case of the @xmath2 reaction , cf . @xcite for details . 10.5 cm one can see from fig . [ totpdx ] that the largest values for the polarized @xmath2 cross sections ( and also those for @xmath45 ) are expected at very low energies , i.e. for @xmath38 less than 10 mev , where the cross sections are dominated by the coulomb - nuclear interference term . however , as was already mentioned above , at these energies the pure coulomb cross section becomes rather large , so that the method of spin - filtering for the polarization buildup can not be applied due to the decrease of the beam lifetime . acording to the analysis of the kinetics of polarization @xcite , the polarization buildup is determined mainly by the ratio of the polarized total cross sections to the unpolarized one ( @xmath16 ) @xcite . let as define the unit vector @xmath48 , where @xmath49 is the target polarization vector , which in the case of @xmath2 scattering enters eq . ( [ totalspin ] ) . the non - zero antiproton beam polarization vector @xmath13 , produced by the polarization buildup , is collinear to the vector @xmath50 for any directions of @xmath51 and can be calculated from consideration of the kinetics of polarization . the general solution for the kinetic equation for @xmath45 scattering is given in ref . here we assume that this solution is valid for the @xmath2 scattering also . therefore , for the spin - filtering mechanism of the polarization buildup the polarization degree at the time @xmath52 is given by @xcite @xmath53 , \label{pdeg}\ ] ] where @xmath54\right \}. \label{omega}\ ] ] here @xmath55 is the areal density of the target and @xmath56 is the beam revolving frequency . one should note that the tensor cross section @xmath22 from eq . ( [ totalspin ] ) does not contribute to @xmath57 . assuming the condition @xmath58 ) , which was found in refs . @xcite for the @xmath45 scattering in rings at @xmath59 @xmath60 and @xmath61 c@xmath62 , one can simplify eq . ( [ pdeg ] ) . if one denotes the number of antiprotons in the beam at the time moment @xmath52 as @xmath63 , then the figure of merit ( fom ) is @xmath64 . this value is maximal at the moment @xmath65 , where @xmath66 is the beam life time , which is determined by the total cross section @xmath16 of the interaction of the antiprotons with the deuteron target as @xmath67 to estimate the efficiency of the polarization buildup mechanism it is instructive to calculate the polarization degree @xmath68 at the time @xmath69 @xcite . in our definition for @xmath20 and @xmath21 , which differ from that in refs . @xcite , we find @xmath70 the polarization degree @xmath71 for @xmath72 ( @xmath73 ) at @xmath74 is shown in fig . [ poldeg2 ] versus the beam energy . for the ease of comparison the polarization degree for the @xmath45 and @xmath75 cases are shown too . the results for @xmath76 ( @xmath77 ) are shown in fig . [ poldeg1 ] . 10.5 cm 10.5 cm one can see that , except for the @xmath78 case , at energies below 100 mev the polarization degree is small due to large total coulomb cross section . however , @xmath71 increases with increasing energy . for longitudinal polarization maximal values of about 10 - 15% are predicted above 150 mev . the transversal polarization degree is smaller than the longitudinal one for both models a and d. for the @xmath2 case the transversal polarization is expected to be larger than for @xmath45 , having a maximum of around @xmath79% at 150 -250 mev ( see fig . [ poldeg1 ] ) . the obtained values for the polarization degree are somewhat smaller than those presented in @xcite , based on the amplitudes of the nijmegen @xmath7 analysis @xcite . experiments for determining the spin - dependent part of the cross sections of the @xmath80 and @xmath2 scattering are planned for the near future @xcite . such data should allow one to discriminate between the different @xmath7 amplitudes @xcite . in this work we have used two @xmath7 potential models developed by the jlich group for a calculation of @xmath2 scattering within the glauber theory and found that this approach allows one to describe the experimental information on ( unpolarized ) differential and total @xmath2 cross sections , available at @xmath81 mev , quantitatively . for those spin - independent observables the difference in the predictions based on those two models turned out to be rather small . the double - scattering corrections to the unpolarized cross section were found to be in the order of 15% in the energy range where the data are available . but we found that even at such low energies as 10 - 25 mev they are not larger than 20 - 25% . this means that , most likely , the glauber approximation does work reasonably well for @xmath2 scattering down to fairly small energies . the predictions for the spin - dependent cross sections for @xmath2 scattering , presented in this work , exhibit a fairly strong model dependence , which is due to uncertainties in the spin dependence of the elementary @xmath45 and @xmath46 interactions . still , for both considered models we find that the magnitude of the spin - dependent cross sections is comparable or even larger than those for @xmath45 . thus , our results suggest that @xmath2 elastic scattering can be used for the polarization buildup of antiprotons at beam energies of 100 - 300 mev with similar and possibly even higher efficiency than @xmath45 scattering . however , it is obvious , that only concrete experimental data on the spin - dependent part of the cross sections of @xmath80 and @xmath2 scattering will allow one to confirm or disprove the feasibility of the spin filtering mechanism for the polarization buildup . this work was supported in part by the heisenberg - landau program . 99 barone v _ et al _ [ pax collaboration ] 2005 arxiv:0505054[hep - ex ] rathmann f _ et al _ 2005 _ phys . lett . _ * 94 * 014801 rathmann f _ et al _ 1993 _ phys . lett . _ * 71 * 1379 milstein a i and strakhovenko v m 2005 _ phys . _ e * 72 * 066503 nikolaev n n and pavlov f 2005 arxiv:0512051[hep - ph ] nikolaev n n and pavlov f 2007 _ aip conf . proc . _ * 915 * 932 meyer h o 1994 _ phys . _ e * 50 * 1485 lenisa p and rathmann f [ pax collaboration ] 2005 arxiv:0512021[nucl - ex ] . barschel c _ et al _ [ pax collaboration ] 2009 arxiv:0904.2325[nucl - ex ] dmitriev v f , milstein a i and strakhovenko v m 2008 _ nucl . instrum . meth . _ b * 266 * 1122 dmitriev v f , milstein a i and salnikov s g 2010 _ phys . _ b * 690 * 427 ; salnikov s g , contribution to this conference . hippchen t , haidenbauer j , holinde k and mull v 1991 _ phys . c _ * 44 * 1323 ; mull v , haidenbauer j , hippchen t and holinde k 1991 _ phys . _ c * 44 * 1337 mull v and holinde k 1995 _ phys . c _ * 51 * 2360 rekalo m p , piskunov n m and sitnik i m 1998 _ few - body syst . _ * 23 * 187 franco v and glauber r j 1966 _ phys . rev . _ * 142 * 1195 kondratyuk l a , shmatikov m zh and bizzarri r 1981 _ yad . fiz . _ * 33 * 795 [ 1981 _ sov . phys . _ * 33 * 413 ] dalkarov o d and karmanov v a 1985 _ nucl . phys . _ a * 445 * 579 bizzarri r _ et al _ 1974 _ nuovo cim . _ * 22 a * 225 kalogeropoulos t and tzankos g s 1980 _ phys . d * 22 * 2585 burrows r d _ et al _ 1970 _ austr . j. phys . _ * 23 * 919 carroll a s _ et al _ 1974 _ phys .

antiproton - deuteron ( @xmath0 ) scattering is calculated at beam energies below 300 mev within the glauber approach , utilizing the amplitudes of the jlich @xmath1 models . a good agreement is obtained with available experimental data on upolarized differential and integrated @xmath2 cross sections . predictions for polarized total @xmath0 cross sections are presented , obtained within the single scattering approximation including coulomb - nuclear interference effects . it is found that the total longitudinal and transversal @xmath3 cross sections are comparable in absolute value to those for @xmath4 scattering . the kinetics of polarization buildup is considered . # 1

confinement effects play an important role in the thermodynamics of several materials such as polymers , liquid crystals , and magnets . for example , capillary condensation stands as a well known example of how phase equilibrium is affected by the confluence of surface and finite - size effects . in particular , due to the wall - particle interaction , a fluid between two plates undergoes a gas - liquid transition at a lower pressure than it does in the bulk . these effects of confinement are due to the additional contributions to the thermodynamic potential of the solvation force ( finite - size effect ) and the wall - fluid interfacial tension ( surface effect).@xcite a more complicated physical situation arises in the case of thin films of polymer mixtures on selective substrates.@xcite an @xmath4 polymer mixture which undergoes a phase separation below a bulk critical temperature @xmath5 , develops , when cast into a thin film over a substrate , an interface between the two phases which runs parallel to the substrate . this interface appears provided there is a substrate affinity for one of the components the confinement is established between the polymer - air and polymer - substrate boundaries . a model fluid confined between two parallel walls that exert opposite surface fields , has been often considered in order to investigate the underlying physics in systems with competing boundaries . in this case , the interplay between wetting and phase separation is very important , unlike the case of capillary condensation in which wetting plays a small role . the competition between surface effects leads to an interesting and unusual behavior : phase coexistence is restricted to temperatures below the wetting temperature @xmath6 even in the limit of infinite separation between the plates . the wetting temperature depends on the surface field and it can be far from the bulk critical temperature.@xcite the aforementioned scenario , first predicted using a mean - field approximation , has been confirmed subsequently via monte carlo simulations and transfer - matrix calculations in two dimensions.@xcite however , when the effect of gravity is considered phase coexistence is restored up to the bulk critical temperature.@xcite the confinement studies described above deal with phase separating systems , in which the phases coexisting along a line of first - order transitions have the same symmetry , e.g. , ferromagnetic thin films . surface effects in systems with ordering ( antiferromagnetic ) interactions have been investigated mostly within the context of binary alloys undergoing a first - order phase transition , with particular emphasis on the surface - induced order and surface - induced disorder phenomena,@xcite although some investigations have been done in the context of multilayer adsorption . more recently , attention has turned to the surface critical behavior of binary alloys displaying continuous ordering reactions and , in particular , to the dependence of the universality class on surface orientation.@xcite @c@c@l@ tag&structure & + 1 & @xmath7 & @xmath8 + 2 & @xmath9 & @xmath10 + 3 & @xmath11 & @xmath12 + 4 & @xmath13 & @xmath14 + 5 & @xmath15 & @xmath16 + @xmath17&@xmath18 & @xmath19 + 6 & @xmath20 & @xmath21 + 7 & @xmath22 & @xmath23 + 8 & @xmath24 & @xmath25 + 9 & @xmath26 & @xmath27 + [ t1 ] in this paper we investigate the interplay between finite - size and surface effects in ising antiferromagnets in the presence of an external field . in particular , we are interested in systems with surfaces that preserve the symmetry of the order parameter . in other words , we shall study thin films which develop antiferromagnetic ( afm ) ordering in each plane parallel to the surfaces . our layered system can be described by the following hamiltonian : @xmath28 where the spin variable @xmath29 takes the value of @xmath30 or @xmath31 depending if the spin at site @xmath32 is pointing up or down , respectively . we have assumed that surface sites , in layers 1 and @xmath33 for an @xmath33-layer film , experience a surface field @xmath1 in addition to the external magnetic field @xmath2 . on physical grounds , it is natural to expect that the pair interactions at and near the surfaces differ from those in the bulk . we approximate the position dependence of the pair couplings , by allowing the nearest - neighbor intralayer surface coupling ( @xmath34 ) to differ from the bulk one ( @xmath35 ) . here we restrict ourselves to case of @xmath36 ( antiferromagnetic ) , but we allow @xmath34 to assume any real value . also , we specialize ourselves in the case of localized symmetric surface fields , i.e. , the field at each surface is the same and acts only at the surface sites . in the remaining of the paper , the effective pair interactions , the surface field , and the external magnetic field ( @xmath2 ) shall be expressed in terms of the bulk afm coupling ( @xmath36 ) . the ratio of surface to bulk coupling is then denoted by @xmath0 . confinement effects in the order - disorder transitions for the particular case of @xmath37 and @xmath38 have been reported previously.@xcite in this paper , we give a full description of the surface and finite - size effects in terms of the variables @xmath1 and @xmath0 . the ground - state properties of the hamiltonian ( [ our - ham ] ) are derived in sec . this zero - temperature analysis is used to identify the different sequences of ground states displayed by the film as a function of the external field @xmath2 . moreover , it is shown that for antiferromagnetic systems with symmetry - preserving surface orientations and nearest - neighbor interactions , a zero - temperature phase diagram can be drawn as a function of @xmath0 and @xmath1 , for any value of the number of layers @xmath33 and external field @xmath2 . in sec.[ft ] , we use a cluster variation free energy@xcite to describe the finite - temperature behavior of the system as a function of surface variables @xmath0 , @xmath1 and the number of layers @xmath33 . particular attention is devoted to the analysis of the critical curve ( in the @xmath2-@xmath3 plane ) for each one of the different regions of the zero - temperature phase diagram . we close with a summary of the important results ( sec . in the absence of surface and finite - size contributions , that is in the bulk , the hamiltonian ( [ our - ham ] ) reduces to @xmath39 for a two - sublattice antiferromagnet such as body - centered or simple cubic , the hamiltonian ( [ bulk - ham ] ) has three different ground states as a function of the external field @xmath2:ferromagnetic ( @xmath40 ) for @xmath41 ; antiferromagnetic ( @xmath42 ) for @xmath43 and again ferromagnetic ( @xmath44 ) for @xmath45 . the critical field @xmath46 , equal to the coordination number @xmath47 [ recall that all quantities in eq.([our - ham ] ) as well as in eq . ( [ bulk - ham ] ) are normalized to @xmath35 ] , determines the point where the critical curve @xmath49 meets the field axis . for the afm thin films studied here [ see hamiltonian ( [ our - ham ] ) ] , the possible ground - state ( gs ) structures are listed in table [ t1 ] along with their corresponding energy . we considered only the case @xmath38 since the results for @xmath50 can obtained straightforwardly from the symmetry properties of hamiltonian ( [ our - ham ] ) . the nomenclature in table [ t1 ] is as follows : structure number 4 corresponds to @xmath51 , which means that both surfaces are ferromagnetic ( @xmath40 ) and that the remaining @xmath52 inner layers are antiferromagnetically ordered . structure @xmath17 , a special case to be discussed later in the paper , has both surfaces in a ferromagnetic state ( @xmath44 ) , the subsurface layers are ferromagnetic but with magnetization in the opposite direction ( @xmath40 ) , and the remaining @xmath53 layers are antiferromagnetic . we arrived at the set of gs in table [ t1 ] as follows . since only nearest - neighbor interactions are included in the hamiltonian and the ( uniform ) surface field acts locally at surface sites , the presence of long - period superstructures can be ruled out . a possible set of ground states for hamiltonian ( [ our - ham ] ) was then constructed by combining all possible surface and bulk ground states . for the sake of definiteness , let us consider a body - centered cubic film with surfaces in the ( 110 ) direction . the bulk ground states consist of two ferromagnetic structures ( with opposite magnetization ) plus an ordered cscl - type afm structure . the ( 110 ) surfaces constitute face - centered rectangular lattices , for which the possible ground states are a checkerboard afm structure and two ferromagnetic states of opposite magnetization . the nine ground - state structures obtained by combining the surface and bulk ground states are listed in table [ t1 ] . these structures are ground states of hamiltonian ( [ our - ham ] ) in the limit of weak coupling between the surface and the subsurface layers . for strong coupling between the surfaces and the bulk , we found only one additional ground - state structure gs @xmath17 in table [ t1].@xcite the hamiltonian in eq . ( [ our - ham ] ) distinguishes between the pair interactions in the surface layers from the rest , thus allowing us to define the surface coordination number @xmath54 as a function of the surface coupling parameter @xmath0 @xmath55 where the intralayer and interlayer coordination are denoted by @xmath56 and @xmath57 , respectively . recall that all quantities in eq . ( [ our - ham ] ) are given in terms of @xmath35 and , therefore , @xmath54 in ( [ zs ] ) actually accounts for the surface energy . for a bcc(110 ) film , @xmath58 and @xmath59 , and the bulk coordination number is @xmath60 . even in the absence of an applied surface field @xmath1 , the surfaces are under the influence of a `` missing neighbors '' field @xmath61 that arises from the disruption of the translational symmetry perpendicular to the surfaces . this missing neighbors field produces an inhomogeneous magnetization profile . thus , with increasing external field , the surfaces may turn into a ferromagnetic state before the bulk does . application of a surface field @xmath62 can restore the magnetization profile to the homogeneous condition . note that the missing neighbors field depends on @xmath0 , since @xmath63 is a measure of the difference between the environment at the surfaces and in the bulk [ see eq . ( [ zs ] ) ] . the missing neighbors field can be written as @xmath64 we derive this value for the missing neighbors field later in paper , by considering the stability of the different gs structures as a function of @xmath0 and @xmath1 . a direct comparison between the energies @xmath65 for each structure @xmath32 gives the ground state for every set of values of the thermodynamic variables ( see table [ t1 ] ) . however , it is more useful and less tedious to consider a physical sequence of gs structures ( as a function of the applied field ) and examine its domain of stability as we vary the surface variables @xmath0 and @xmath1 . as a starting point , consider the following case : upon the application of an external field @xmath2 ( in either direction ) , a film with @xmath66 and @xmath67 will pass from an afm state in all layers ( small @xmath68 ) to a state with ferromagnetic surfaces and an afm bulk and , finally , for large @xmath68 , to a ferromagnetic state in all planes . this case is represented by the sequence 1 - 4 - 5 - 6 - 9 of gs structures [ a schematic view is presented in fig.[f1](b ) . see also table [ t1 ] for the nomenclature ] . the characteristic value of the external field at the transition between different gs structures is indicated in fig . [ f1 ] . in general , the transition between gs structures @xmath69 and @xmath70 occurs at @xmath71 , which is determined by equating the corresponding energies . ground - state sequence 1 - 4 - 5 - 6 - 9 ( hereafter referred as i ) in fig.[f1](b ) , provides some useful insight on confinement versus finite - size effects . at the beginning of this section we considered the ground states of an infinite antiferromagnet , which in the nomenclature of table [ t1 ] , correspond to gs sequence 1 - 5 - 9 in the limit of @xmath72 . thus , ground - state structures 4 and 6 are due the confinement effects . when either gs 4 or gs 6 become unstable in favor of gs 5 , surface effects are lost and the film is subject only to the finite - size effects . an external surface field will produce an asymmetry in the gs sequence since the hamiltonian is not invariant under the transformation @xmath73 , @xmath74 . applying a surface field @xmath38 reduces the surface ferromagnetism ( @xmath40 ) in gs 4 and enhances it in gs 6 ( @xmath44 ) . the domain of stability of gs 4 shrinks to zero when @xmath75 becomes @xmath47 . this particular value of @xmath1 defines the missing - neighbors field [ eq . ( [ hm ] ) ] . applying an external surface field is not the only way to eliminate surface effects in afm thin films . a homogeneous condition can also be attained in the film by setting neutral boundary conditions ( @xmath76 ) and increasing the pair interactions at the surfaces to a given value @xmath77 . the characteristic value of the surface coupling that compensates for the missing neighbors effect is given by : @xmath78 for this value of the surface coupling gs 4 and gs 6 become unstable simultaneously [ eq . ( [ vm ] ) is equivalent to the condition @xmath79 . for values of the surface coupling larger than @xmath77 , keeping the neutrality at the boundaries , ordering becomes stronger at the surfaces than in the inner layers . this situation is represented in fig . [ f1](c ) . it is worth noting that the gs sequence 1 - 2 - 5 - 8 - 9 , hereafter referred as ii , is stable not only in the case of @xmath76 but for a range of values of @xmath80 and @xmath1 . we will return to this point later in the paper . reducing the surface coupling make surface ordering less stable , until @xmath0 reaches the characteristic value @xmath81 for which gs 5 becomes unstable [ @xmath82 in ( [ vps ] ) corresponds to the condition @xmath83 . the remaining ground - state sequence 1 - 4 - 6 - 9 ( hereafter vii ) are depicted in fig . [ f1](a ) . sequence vii remains unaltered for @xmath84 regardless the strength of @xmath0 : the phase coexistence between spin - up and spin - down is regulated by the surface field @xmath1 . large , negative values of @xmath0 increase the critical - point temperature . in the alloy terminology , sequence vii represents the situation of a binary - alloy thin film with an ordered bulk coexisting with a surface miscibility gap . this will become apparent in sec . [ ft ] where we discuss the finite - temperature properties of hamiltonian ( [ our - ham ] ) . ( c ) becomes sequence iii in ( a ) as the surface field is increased . the transition @xmath85 occurs at @xmath86 when gs 8 becomes unstable ( see table [ t2 ] ) . a further increase of the surface field @xmath1 establishes 1 - 2 - 3-@xmath17 - 6 - 9 in ( b ) as the stable gs sequence ( iv ) . observe the appearance of gs @xmath17 and the disordered gap between gs 3 and gs @xmath17 . the characteristic field between gs 2 and gs 3 is @xmath87 . see the text.,width=172 ] 2 ( c ) becomes sequence iii in ( a ) as the surface field is increased . the transition @xmath85 occurs at @xmath86 when gs 8 becomes unstable ( see table [ t2 ] ) . a further increase of the surface field @xmath1 establishes 1 - 2 - 3-@xmath17 - 6 - 9 in ( b ) as the stable gs sequence ( iv ) . observe the appearance of gs @xmath17 and the disordered gap between gs 3 and gs @xmath17 . the characteristic field between gs 2 and gs 3 is @xmath87 . see the text.,width=172 ] sequences i , ii , and vii ( fig . [ f1 ] ) were obtained by analyzing the stability of the corresponding gs sequences upon variations of the surface coupling @xmath0 for neutral boundary conditions . as expected , a similar variation of gs sequences will appear as we increase the surface field . consider for example sequence ii in fig . [ f1](c ) : setting higher values for the surface field eventually overcome the ordering tendencies at the surfaces . ground - state structure 8 then becomes unstable and sequence ii turns into the new 1 - 2 - 5 - 6 - 9 gs sequence ( iii ) depicted in fig.[f2](a ) . the asymmetry of sequence iii is interesting . for very negative values of @xmath2 sequence iii looks like sequence ii ( in the same range of @xmath2 ) , with long - range order dictated by the surfaces . on the other hand , for large positive values of @xmath2 , sequence iii looks like sequence i , for which the bulk is responsible for the afm ordering . this similarity is due to the fact that sequence i evolves into iii when the surface field increases beyond @xmath88 ( missing neighbors field ) for @xmath89 . the homogeneous antiferromagnetic thin film ( gs 5 ) , with constant energy for given @xmath0 and @xmath33 , becomes rapidly unstable with increasing @xmath1 . for sufficiently large @xmath1 , gs 5 is replaced by another zero - magnetization structure , gs @xmath17 in table [ t1 ] , with energy given by @xmath90 for a given value of the number of layers @xmath33 and the coordination at the surfaces , @xmath91 is constant while @xmath92 depends only on @xmath1 . when the surface field reaches the value of @xmath93 structures 5 and @xmath17 have the same energy . a unique feature of gs 5 and gs @xmath17 is that they remain degenerate over a _ finite _ range of the external field @xmath2 . from eq . ( [ h14 ] ) and fig.[f2](a ) we can see that the ground state for an afm film at a surface field value given by ( [ h14 ] ) is a _ mixture _ of gs 5 and gs @xmath17 for @xmath94 . on average , a scan in @xmath2 will show a layer magnetization of @xmath95 at the surfaces together with subsurface magnetization of @xmath96 and afm bulk ( zero magnetization ) . thermal excitations destroy this degeneracy between gs 5 and gs @xmath17 in most of the interval @xmath97 in favor of gs 5 , except near the ends , i.e.@xmath98 and @xmath99 , where gs @xmath17 is pinned by the onset of stability of gs 3 and the presence of gs 6 . traces of the ground - state degeneracy between gs 5 and gs @xmath17 are observable at low temperatures . for the other structures listed in table [ t1 ] , a transition similar to @xmath100 does not occur , mainly due to the symmetry in the boundary conditions . ground - state sequence iii evolves into 1 - 2 - 3-@xmath17 - 6 - 9 sequence ( iv hereafter ) at @xmath101 . the situation is shown schematically in fig . [ f2](b ) for @xmath102 . observe that the appearance of gs 3 has established a disorder gap between gs 2 ( afm surfaces ) and gs @xmath17 ( afm bulk ) . this behavior is unique in the sense that in all previous cases the ordered domain was a compact interval in @xmath2 . this characteristic brings some interesting features into the @xmath2-@xmath3 phase diagram , such as the splitting of the film s critical @xmath49 curve into two distinct critical curves.@xcite . sequence v in ( a ) is obtained from iv in fig.[f2](b ) , when the surface afm - phase in gs 2 becomes unstable upon the reduction the pair interaction at surfaces . a coexistence between spin - up ( gs 3 ) and spin - down ( gs 1 ) magnetizations is established at the surfaces [ _ cf_. fig . [ f1](a ) ] . for sufficient negative values of @xmath0 and large @xmath1 , a line of first - order transitions , ending at a critical point , occurs outside the antiferromagnetic critical curve . sequence i in fig.[f1](c ) turns into sequence vi in ( b ) in the same way as iii becomes iv ( fig . [ f2 ] ) , that is , replacing gs 5 by gs @xmath17 . a difference arises , however , since in this case gs 4 is adjacent to gs @xmath17 , and a coexistence line will appear _ inside _ the afm region . see the text for details.,width=172 ] 2 . sequence v in ( a ) is obtained from iv in fig.[f2](b ) , when the surface afm - phase in gs 2 becomes unstable upon the reduction the pair interaction at surfaces . a coexistence between spin - up ( gs 3 ) and spin - down ( gs 1 ) magnetizations is established at the surfaces [ _ cf_. fig . [ f1](a ) ] . for sufficient negative values of @xmath0 and large @xmath1 , a line of first - order transitions , ending at a critical point , occurs outside the antiferromagnetic critical curve . sequence i in fig.[f1](c ) turns into sequence vi in ( b ) in the same way as iii becomes iv ( fig . [ f2 ] ) , that is , replacing gs 5 by gs @xmath17 . a difference arises , however , since in this case gs 4 is adjacent to gs @xmath17 , and a coexistence line will appear _ inside _ the afm region . see the text for details.,width=172 ] increasing the surface field does not change sequence iv into another gs sequence . however , for a large value of @xmath1 , antiferromagnetic order at the surfaces becomes unstable upon reduction of the surface coupling , and iv changes into sequence v composed of gs sequence 1 - 3-@xmath17 - 6 - 9 for @xmath103 [ fig.[f3](a ) ] . observe in fig . [ f3](a ) that gs 1 is now adjacent to gs 3 . the difference between gs 1 and gs 3 resides at the surfaces , which have opposite magnetization . this situation is reminiscent to the one found in sequence vii [ fig . [ f1](a ) ] where the surface field regulates the surface phase coexistence between up- and down - magnetizations . in sequence v , however , the line of first - order transitions is located outside of the well defined afm region composed by gs @xmath17 and gs 6 . ground - state sequence v is stable for large @xmath1 and negative @xmath0 . previously , we found that vii is the stable gs sequence for @xmath84 and low @xmath1 . a transition between v and vii certainly occurs , although it is mediated by the gs sequence 1 - 4-@xmath17 - 6 - 9 ( vi ) [ see fig . [ f3](b ) ] . finally , sequences i and vi are separated by the gs 5 to gs @xmath17 transition , which in this case occurs at lower values of @xmath1 since @xmath103 . we have discussed the several gs sequences that appear in confined antiferromagnets along particular paths , namely , we fixed @xmath76 and varied the surface coupling ( fig . [ f1 ] ) or alternatively , we fixed @xmath80 and increased @xmath1 ( fig . [ f2 ] ) . in general , however , the transition from one gs sequence to another does not occur at constant @xmath0 or @xmath1 . the relationship between the surface variables @xmath104 at the different transitions between gs sequences ( see table [ t2 ] ) , defines the domain of stability of each sequence , from i to vii , in the plane @xmath0-@xmath1 . the corresponding ground - state phase diagram is shown in fig . [ gspd ] . the phase diagram is symmetric with respect to @xmath76 , with the negative-@xmath1 region obtained by replacing spin - up with spin - down and @xmath2 with @xmath105 in fig.[gspd ] . the zero - temperature phase diagram provides a good reference frame to interpret some of the features reported in previous work on binary - alloy thin films with ordering interactions.@xcite the ground - state phase diagram is also a valuable guide for the investigation of the finite - temperature properties of hamiltonian ( [ our - ham ] ) to be carried out in the next section . the finite - temperature properties of hamiltonian ( [ our - ham ] ) were calculated using the cluster - variation method ( cvm ) in the pair approximation ( pa).@xcite for the two - sublattice antiferromagnets considered in this paper , the physical aspects of phase equilibrium under confinement are well captured by the pa - cvm.@xcite for bcc(110 ) films with neutral boundary conditions , a comparison between the pa and the tetrahedron approximation ( ta ) has shown that only the quantitative aspects are improved with the latter.@xcite for a general exposition of the cluster - variation method , we refer the interested reader to the excellent reviews available in the literature.@xcite the order - disorder transitions are described in the usual manner by subdividing the bcc or sc lattice into two interpenetrating sublattices @xmath106 and @xmath107 . the long - range order parameter in the @xmath108-layer defined as @xmath109 where @xmath110 is the @xmath111-sublattice magnetization in the @xmath108 layer . @l@ + @xmath112 + @xmath113 + @xmath114 + @xmath115 + @xmath116 + @xmath117 + [ t2 ] with reference to the gs phase diagram of fig . [ gspd ] , regions i iii display long - range order , either at the surfaces ( gs 2 and 8) or in the bulk ( gs 46 ) . with the exception of sequence i , the critical curves@xcite obtained in regions ii and iii show a distortion at high temperatures . our results for the critical curve in these regions , summarized in fig . [ ft1 ] , can be explained using the ground - state analysis discussed in sec . [ gs ] . phase diagrams in region i are virtually independent of the parameters @xmath0 and @xmath1 , as can be seen in fig . [ ft1](a ) . this behavior can be attributed to the fact that the afm ordering in region i is primarily due to the inner layers [ see fig.[f1](b ) and fig . [ gspd ] ] . thermal excitations can promote spin flip at the surfaces , resulting in a lower degree of ordering at surfaces relative to the bulk . in contrast , region ii is characterized by a strong afm ordering at the surfaces coupled with the afm bulk [ see fig . [ f1](c ) ] , thus preventing the formation of a ( separate ) surface critical curve . instead , the @xmath2-@xmath3 phase diagram shows an increase in the transition temperature and a broadening in the external field region for which the stable phase is antiferromagnetic . a relative small asymmetry in the critical curve is observed , due to the fact that the surface field favors the stability of gs 2 over gs 8 [ fig.[ft1](a ) ] . thus , the distortion in the phase diagrams associated with region ii stems from the relative stability of two ground - state configurations with the same symmetry , i.e. , gs 2 and gs 8 . a higher asymmetry in the phase diagrams is expected in region iii , since the critical - curve shape is dictated by the surface ordering for @xmath118 , and by an afm bulk ( with low surface ordering ) for @xmath119 . the difference in symmetry of the afm structures at each afm : fm boundaries [ see the gs sequence in fig . [ f2](a ) ] , allows the surfaces to drive the phase transition for fields close to the ( negative ) critical field value . one can see that the surfaces are developing their own critical curve , which unfolds as a ` shoulder ' in the phase diagram for negative applied field [ see fig . [ ft1](b ) ] . characteristics such as the maximum temperature of the shoulder or its extension in @xmath2 , are controlled by the surface variables @xmath0 and @xmath1 . the critical field between gs 1 and gs 2 [ @xmath120 makes apparent that the extension of the shoulder depends on the surface field . the maximum temperature in the shoulder is about @xmath121 , where @xmath122 is the nel temperature of the corresponding surface antiferromagnet . here , as in the rest of the paper , the relevant thermodynamic variables are expressed in units of the ( positive ) afm coupling . thus , in the pa a square lattice has a maximum critical temperature @xmath123 . and @xmath124 ) . the various symbols correspond to different values of the surface variables @xmath125 ) as shown in the inset . note that in region i the shape of the critical curve is virtually independent of @xmath0 and @xmath1 . ( b ) phase diagram for region iii ( solid line ) , showing the development of a ` shoulder ' as a signature of the incipient surface critical curve . the values for the surface variables are @xmath126 and @xmath127 . the antiferromagnetic domain is a compact region . the phase diagram for a square lattice is shown as reference ( dot - dashed line ) and to illustrate the process of separation between the bulk and surface critical curves ( _ cf_. fig . [ ft2 ] ) . both in ( a ) and ( b ) antiferromagnetic bcc(110 ) films with @xmath128 were considered and solved in the pair approximation of cvm.,width=211 ] and @xmath124 ) . the various symbols correspond to different values of the surface variables @xmath125 ) as shown in the inset . note that in region i the shape of the critical curve is virtually independent of @xmath0 and @xmath1 . ( b ) phase diagram for region iii ( solid line ) , showing the development of a ` shoulder ' as a signature of the incipient surface critical curve . the values for the surface variables are @xmath126 and @xmath127 . the antiferromagnetic domain is a compact region . the phase diagram for a square lattice is shown as reference ( dot - dashed line ) and to illustrate the process of separation between the bulk and surface critical curves ( _ cf_. fig . [ ft2 ] ) . both in ( a ) and ( b ) antiferromagnetic bcc(110 ) films with @xmath128 were considered and solved in the pair approximation of cvm.,width=220 ] - as pointed out previously , region iv is characterized by the formation of a disordered gap between two different ground states [ see fig . [ f2](b ) ] . at finite temperatures and deep inside region iv , the surfaces develop their own critical curve well separated from the bulk antiferromagnetic region [ see fig.[ft2](a ) showing the critical curves for a 14-layer film with @xmath126 , @xmath129 ( circles ) and @xmath130 ( triangles ) ] . since the surfaces are weakly coupled with the bulk , the surface critical curve scales with @xmath0 , i.e. , the zero - temperature width of the afm ordering is @xmath131 and the maximum critical temperature is @xmath132 . ) , the surfaces decouple from the inner layers ( bulk ) and develop their own critical curve . in ( a ) the critical curves of 14-layer antiferromagnetic thin films are shown for @xmath126 , @xmath129 ( circles ) and @xmath126 , @xmath130 ( triangles ) . the bulk critical curve showed no difference from @xmath129 to @xmath130 , hence only the former case is depicted . the solid lines represent , in the case of the surface critical curves , the phase diagram of a square afm , appropriately shifted . the solid line in the bulk phase is the one associated to @xmath133 in region i. the splitting between the surfaces and bulk critical curves occur at @xmath134 ( temperature of splitting ) and @xmath135 ( field of splitting ) , when the surface field reaches the value of @xmath136 . part ( b ) shows a detail of the phase diagram of 100-layer afm film at the very point of splitting.,width=220 ] ) , the surfaces decouple from the inner layers ( bulk ) and develop their own critical curve . in ( a ) the critical curves of 14-layer antiferromagnetic thin films are shown for @xmath126 , @xmath129 ( circles ) and @xmath126 , @xmath130 ( triangles ) . the bulk critical curve showed no difference from @xmath129 to @xmath130 , hence only the former case is depicted . the solid lines represent , in the case of the surface critical curves , the phase diagram of a square afm , appropriately shifted . the solid line in the bulk phase is the one associated to @xmath133 in region i. the splitting between the surfaces and bulk critical curves occur at @xmath134 ( temperature of splitting ) and @xmath135 ( field of splitting ) , when the surface field reaches the value of @xmath136 . part ( b ) shows a detail of the phase diagram of 100-layer afm film at the very point of splitting.,width=220 ] - between the situation of unconnected ordered domains and the phase diagrams observed in region iii , there is the case in which the zero - temperature disordered gap transforms , via thermal excitations , into a disordered region in the @xmath2-@xmath3 plane right inside the compact afm domain . an increment in the surface field translates into an increment in the height of the disordered region . at @xmath137 the afm region splits into the surface and the bulk critical curves [ see fig.[ft2](b ) ] . at finite temperatures , the splitting value of the surface field @xmath136 plays the role of @xmath138 : for @xmath139 the ordered region is compact whereas for @xmath140 there are two unconnected critical curves . expressing the free energy @xmath141 in terms of the long - range order parameters ( [ lro ] ) , the conditions determining the locus of the splitting point are given by : [ split ] @xmath142 @xmath143 equation([split : hess ] ) defines the critical temperature , at fixed external conditions ( @xmath3 , @xmath2 , @xmath1 and @xmath33 ) , when the second derivatives of the free energy are evaluated in the disordered state.@xcite since @xmath144 in the ordered state , at the splitting point ( @xmath145 ) @xmath146 is a concave function of the external field vanishing at the splitting value of the magnetic field @xmath135 . in a similar fashion , one can see that @xmath146 is a convex function of temperature , becoming zero at @xmath145 [ see fig . [ ft2](b ) ] . thus , the splitting point is defined as a saddle point of @xmath146 in the @xmath3 and @xmath2 variables . conditions ( [ split : hess - ht ] ) account for this . as a function of the surface coupling for afm bcc(110 ) films with @xmath128 and @xmath147 . the minimum of @xmath135 is due to the reentrance at low temperatures of the bulk critical curve . inset : splitting value of surface field @xmath136 as a function of the surface coupling @xmath0 , for the case of @xmath147 ( circles ) . a least - squares fit ( solid line ) gives @xmath148 . compare this with @xmath149 obtained in sec . [ gs ] for the boundary between regions iii and iv . see the text for further details.,width=220 ] using conditions ( [ split ] ) we determined the splitting value of the external field @xmath135 as a function of the surface coupling for thin ( @xmath128 ) and thick ( @xmath147 ) films . the results are shown in fig . [ ft3 ] for the case of bcc(110 ) films . the particular shape of @xmath150 can be understood as follows : since the height of the critical curve associated with the surfaces scales with @xmath0 [ see for example figs . [ ft1](b ) and [ ft2](a ) ] and because of the reentrance of the bulk critical curve , for small @xmath0 the point of contact ( splitting ) between the two critical curves is shifted to higher values of @xmath2 . as we increase the surface coupling , the splitting point moves ( clockwise ) along the bulk critical curve , reaching a minimum in @xmath2 and increasing again towards the saturation value . we found that within the pa the minimum in @xmath150 is not very sensitive to the total number of layers . for bcc(110 ) , @xmath151 occurs at @xmath152 while for sc(100 ) the @xmath135 is minimum at @xmath153 . again , this can be explained by considering the different nel temperature values for sc and bcc lattices . the ratio between the latter and the former is @xmath1541.4 ( pa ) , which is comparable to the ratio of the corresponding @xmath151 ( @xmath1541.45 ) . the behavior of the other quantities of interest can be inferred from fig . [ ft3 ] . the most interesting part , however , is contained in the inset of fig.[ft3 ] , which shows @xmath136 as a function of the surface coupling @xmath0 . a least - square fit gives @xmath155 which is almost parallel ( and very close ) to @xmath138 in eq.([h14 ] ) . for sc(100 ) films similar results were obtained and a linear fit for @xmath136 gives @xmath156 . thus , the process of splitting occurs within a narrow interval of @xmath1 . only the left half is shown . the calculations were done in the pa - cvm for the following values of the surface variables : @xmath157 and @xmath158 ( vii ) ; @xmath159 ( vi ) ; @xmath160 ( v).,width=220 ] due to the equivalence between the nel point and the critical point of a ferromagnet in zero field , the finite - temperature behavior of afm thin films , as a function of the surface coupling @xmath0 and @xmath161 , is equivalent to the multicritical phenomena occurring at the surface of semiinfinite ferromagnets.@xcite in our case , negative surface pair interactions give rise to a line of first - order transitions in regions v vii ( see fig . [ ft4 ] ) . in all cases the coexistence line separates surface ferromagnetic phases with opposite magnetization , that have the same symmetry . the bulk , however , may have different symmetry at each side of the coexistence line , thus modifying the shape of the first - order line at finite temperatures . this can be observed in fig . [ ft4 ] , where the surface coexistence curve is drawn @xmath157 and @xmath158 ( vii ) , @xmath159 ( vi ) , @xmath160 ( v ) . in each case , the coexistence curve ends in a critical point which is close , as expected , to the curie point associated with the ( 2d ) surface lattice , i.e. @xmath162 . in all the three regions v vii , the afm bulk remains undisturbed by the presence of the surface coexistence line . at @xmath163 and @xmath161 the critical end point reaches the second - order critical curve at the nel temperature @xmath164 . the multicritical behavior is the ( trivial ) superposition of two independent critical behaviors which do not interfere with each other.@xcite in this paper we performed an analysis of the confinement effects on antiferromagnets with symmetry - preserving surface orientations . the ground - state properties of the model , an ising hamiltonian with nn - pair interactions in the presence of external bulk and surface fields , shows an interesting structure . a zero - temperature phase diagram in the surface variables @xmath0 ( surface coupling ) and @xmath1 ( surface field ) was obtained for two - sublattice antiferromagnets . in this case there are seven different regions in the ground - state phase diagram . each region is characterized by a particular sequence of ground states as a function of the external field . an analysis of the ground - state phase diagram explains ( and sometimes even anticipates ) some of the features found in the @xmath2-@xmath3 critical curves . together with an examination of the finite - temperature behavior in each of the aforementioned regions , our analysis showed that the interplay between the surface variables @xmath0 and @xmath1 defines the thermodynamics of confinement in ordering systems . for example , the splitting of the critical curve into surface and bulk contributions results from the simultaneous application of seemingly competing contributions @xmath80 ( ordering ) and @xmath165 . at the other extreme , the development of a surface coexistence line for @xmath103 and @xmath38 represents a particular case of magnetic surface reconstruction . for reviews on wetting see : s. dietrich , in : _ phase transitions and critical phenomena _ , edited by c. domb and j. lebowitz ( academic press , new york , 1988 ) , vol . 12 ; d. e. sullivan and m. m. telo da gama , in : _ fluid interfacial phenomena _ , edited by c. a. croxton ( wiley , new york , 1986 ) ; m. schick , in : _ liquids at interfaces _ , edited by j. charvolin , j. f. joanny , and j. zinn - justin ( north - holland , amsterdam , 1990 ) . for a recent review of monte carlo studies on surface - induced order see : k. binder , in _ cohesion and structure of surfaces _ ( north - holland , amsterdam , 1995 ) . also see the contributions of k. binder , d. p. landau , f. schmid , and w. schweika in : _ stability of materials _ , edited by a. gonis , p. e. a. turchi , and j.kudrnovsk . nato asi series b , vol . 355 ( plenum , new york , 1996 ) . we confirmed these results by evaluating the free energy ( in the pair approximation ) at low temperatures , as function of the surface variables @xmath1 and @xmath0 for different values of the external field @xmath2 . we did not find any other ground - state structure additional to those listed in table [ t1 ] . at finite temperatures , the antiferromagnetic region [ nonzero value of the order parameter in ( [ lro ] ) ] is separated from the disordered region ( zero long - range order ) by a line of second - order phase transitions . equation ( [ split : hess ] ) defines the loci of the transition points in terms of a second derivative of free energy of the system with respect of the order parameters .

phase equilibrium in confined ising antiferromagnets was studied as a function of the coupling ( @xmath0 ) and a magnetic field ( @xmath1 ) at the surfaces , in the presence of an external field @xmath2 . the ground state properties were calculated exactly for symmetric boundary conditions and nearest - neighbor interactions , and a full zero - temperature phase diagram in the plane @xmath0-@xmath1 was obtained for films with symmetry - preserving surface orientations . the ground - state analysis was extended to the @xmath2-@xmath3 plane using a cluster - variation free energy . the study of the finite-@xmath3 properties ( as a function of @xmath0 and @xmath1 ) reveals the close interdependence between the surface and finite - size effects and , together with the ground - state phase diagram , provides an integral picture of the confinement in anisotropic antiferromagnets with surfaces that preserve the symmetry of the order parameter .

the decay @xmath0 is an anticipated lepton - number violating process in supersymmetric models@xcite , left - right supersymmetric models@xcite and in supersymmetric string unified models@xcite . for some ranges of model parameters , decay rates as high several parts per million are expected for this decay@xcite , even in light of the current experimental limit on the related @xmath6 decay@xcite . assuming that neutrinos have mass , the standard model branching ratio is expected at the o(@xmath7 ) level . the current best published limit is b(@xmath0 ) @xmath8 from the cleo experiment using 12.6 million @xmath9 pairs@xcite . this analysis uses @xmath2 @xmath10 events from the 1999 - 2001 data set to give a preliminary result of b(@xmath11 ) . the general signature of the @xmath0 signal is the presence of an isolated @xmath12 and @xmath13 which have an invariant mass consistent with that of the @xmath9(1.777 gev ) and with an energy in the centre - of - mass system ( cms ) consistent with the beam energy in the cms ( 5.29 gev ) and the rest of the particles in the event having properties that are consistent with being produced in a generic 1-prong @xmath9 decay . the analysis is performed using a blinded mass - energy region in the data which corresponds to a 3@xmath14 error ellipse centred on the expected peak of the distribution . a cut - based approach is used , in which the background in the signal - box is estimated from extrapolations using side - band data and verified with the monte carlo simulation . the determination of selection criteria in this analysis was based on monte carlo simulation of background and signal as well as on data in side - band regions further away from the signal - box . background sources arising from non-@xmath10 sources , the most problematic of which are radiative @xmath15 pair events , are reduced to small levels . this leaves the generic 1-prong @xmath9 decays as the major source of background . the dominant , and irreducible , background is from radiative muonic decays of the @xmath9 in which the two neutrinos have very little energy . the monte carlo simulation uses a complete description of the detector response employing the geant4@xcite software . the most important simulation samples are those involving @xmath16 decays . these include 27.6 million generic @xmath10 events , which use koralb employing the tauola decay package@xcite , and the @xmath0 signal incorporated into tauola . forty thousand signal events were generated which form the bulk of the signal monte carlo used for the analysis . a second signal sample with differing beam background contributions was produced in order to provide a cross check on the sensitivity of the selection to different background conditions . additionally , simulated @xmath15 events , which uses the afkqed generator@xcite , provide tools for @xmath12 particle identification and some guidance on @xmath17 event suppression . other potential backgrounds include @xmath18 and @xmath19 which were studied using the corresponding generic simulated event samples . as the selection is restricted to events in which both @xmath9 s decay via 1-prong modes , the @xmath19 background is completely negligible as indicated by the simulated events . the @xmath20 and light - quark background contributions are also found to be very small , with no events in the simulation surviving from these two sources after all cuts have been applied , and consequently have not been included further in the data - monte carlo comparisons . the two - photon processes are negligible backgrounds for events in which the beam energy is detected in a cms hemisphere . ( b ) @xmath21 and ( c ) @xmath21 vs @xmath22 for the monte carlo simulation of the @xmath0 signal . ] because there is no missing energy in the @xmath0 signal , the mass of the measured decay products is the @xmath9 mass and the energy is the full energy of the @xmath9 . ignoring the effects of initial - state radiation , the energy of the final state particles is equal to the full energy of the beam in the cms . this situation lends itself very well to the use of a mass of the @xmath23 system calculated from the kinematic fit employing the beam energy constraint on the energy of the system as used by argus , and denoted as @xmath21 . the energy variable used is the difference between the measured @xmath23 energy and the beam energy , denoted by the symbol @xmath22 . the distributions of these variables are shown in figure [ fig5](a)-(c ) . the resolutions of the core of these distributions , which represent those events with well reconstructed photons and tracks , are @xmath24 = 88 mev and @xmath25 = 19 mev . however , as is evident by the diagonal band of events present in figure [ fig5](c ) , when initial state radiation shifts the true energy of the @xmath9 from the beam energy , there a negative correlation between @xmath22 and @xmath21 and a substantial loss of resolution . the low - energy tail of the @xmath22 distribution for well reconstructed @xmath21 , evident in figure [ fig5](c ) , is populated by events with a photon reconstructed with the correct direction , but with significant energy loss . vs @xmath22 plane shown schematically overlaying the distribution of the simulated signal . the signal region is the assymetric ellipse within the elliptical blinded region . ] figure [ fig7 ] shows the various regions in the @xmath21 vs @xmath22 plane used for determining the selection criteria , measuring the background , and defining the signal and blinded regions . a ` grand side - band ' is defined as the region within the @xmath26 gev and @xmath27 gev bounds . it is used for evaluating the reliability of the estimation of the signal efficiency and this is also shown on the figure . the signal - box is an elliptical region centred on the peak of the two - dimensional distribution as determined by the monte carlo . for the positive side of @xmath22 , the ellipse has a 3@xmath14 half - axis for both the @xmath22 and @xmath21 axes whereas for the negative side of @xmath22 the ellipse has a 2@xmath14 half - axis in @xmath22 and a 3@xmath14 half - axis for @xmath21 . the resolutions used in defining the signal - box are those of the core gaussians obtained from the signal monte carlo . this asymmetric shape provides an optimal signal - box given the resolution in @xmath22 and the presence of increasing background in the negative @xmath22 region that is not present for positive @xmath22 . events that are ultimately selected and included in a study of the @xmath21 and @xmath22 variables must first pass a number of selection criteria . these requirements are designed primarily to reduce the number of bhabha and @xmath15 events . the requirements of the standard background filter which the signal events pass : * 2 charged tracks * zero net charge * @xmath28 gev * @xmath29 gev * @xmath30 or @xmath31 * @xmath32 * @xmath33 * charged track separation @xmath34 in the cms * at least one @xmath13 * fiducial region : @xmath35 where @xmath36 is the track momentum ; @xmath37 is the calorimeter energy associated with the track ; @xmath38 is the total transverse momentum of charged tracks in the cms ; and subscript cm indicates when the quantity is in the cms . these requirements have an efficiency of @xmath39 . these efficiency losses arise largely from the fiducial acceptance of the detector , but also from the branching ratio to 1-prong modes . at this stage of the analysis , @xmath40 events are selected in the data . monte carlo studies indicate that 83% of the signal events pass these requirements for events which would have otherwise been selected . subsequent selections employ requirements to remove the non-@xmath9 background sources and the @xmath41 and @xmath42 events in particular . cuts include a specific requirement on the observed cms energy , @xmath43 , to exclude the aforementioned background sources in general and a @xmath12 veto for reduction of di - muon backgrounds in particular . the event is tagged via an electron or @xmath44 on the opposing side . the missing mass on the tag side is restricted to reduce backgrounds from lost or difficlt to reconstruct tracks . to be consistent with a decay through an intermediate @xmath9 , the tag track is required to have less than 80% of the cms beam energy . in general , this will not be the case for the non-@xmath9 backgrounds . on the signal side , we impose the requirement of an identified @xmath12 and an associated @xmath13 with energy in excess of 400 mev . these cuts are applied in addition to the obvious @xmath21 and @xmath22 cuts for the final selection . the cut progression shows that once the non-@xmath9 background is removed by requiring an electron - vs-@xmath23 or @xmath45-vs-@xmath23 events , the @xmath9 monte carlo simulation tracks the data reasonably well . we present in figures [ fig14 ] and [ fig12 ] a sample of the distributions of variables employed in the analysis after all other requirements , apart from that using the one shown , have been applied . the data distributions are represented by points and the @xmath9 monte carlo simulation by histograms in these figures . the normalizations of the monte carlo distributions is fixed by the luminosity . after all other requirements , apart from the final @xmath21-@xmath22 , have been applied . the data and @xmath9 monte carlo are in the top plot and the @xmath0 signal monte carlo in the bottom plot . the parts of the distributions accepted in the selection are indicated.,width=264,height=264 ] after all other requirements , apart from the final @xmath21-@xmath22 , have been applied . the data and @xmath9 monte carlo are in the top plot and the @xmath0 signal monte carlo in the bottom plot . , width=264,height=264 ] these distributions are well described by the @xmath10 simulation in both shape and overall normalization . there are 604 events in the data in these distributions and the ratio of data to the monte carlo expectation is @xmath46 . the first error is statistical and the second is the systematic error associated with normalization of the monte carlo events . this normalization uncertainty is a component of the systematic error associated with the efficiency of selecting the signal . this includes uncertainties in integrated luminosity , cross - section , radiation treatment in the generator and branching ratio uncertainties . because the signal has a relatively stiff momentum spectrum , muon particle identification ( @xmath12-pid ) efficiency using a sample of radiative @xmath12 pair events ( @xmath47 ) from the same data set used for the rest of the analysis was exploited to study the @xmath12-pid efficiency . contamination from sources other than the @xmath12 pair , such as @xmath10 events , are negligible in this sample . the ratio of the efficiencies as a function of lab momentum of the data to that of the @xmath48 simulation was studied and a correction based upon the control sample was extracted . similarly obtained is the correction for the @xmath10 events in the monte carlo sample and the signal monte carlo . this method for applying the correction is particularly appropriate here as the polar - angle , azimuthal and momentum distributions and their correlations are very similar between the @xmath12 pair control sample and the signal and background . from these studies , an additional correction factor of 0.80 to 0.85 is applied to the efficiency as predicted by the monte carlo . there are two broad categories of background that require suppression : the background arising from non-@xmath10 sources , the most significant being @xmath49 , and those coming from standard @xmath9 decays . in the latter category , the @xmath50 form an irreducible background . the @xmath9 backgrounds are supressed by requiring the signal @xmath13 to be energetic , @xmath51 , and by imposing tight requirements on the @xmath12-pid of the track . the non-@xmath9 background is reduced by requiring there to be a non-@xmath12 tag on the non - signal hemi - sphere and by removing events with measured momentum and calorimeter energy characteristic of events and hemispheres with little or no undetected energy . these events are particularly problematic as they naturally possess a value of @xmath22 that overlaps the signal . after all the selection criteria have been applied , before applying the signal - box cut , there remain background sources in the grand side - band region that , from the monte carlo simulation , consist of @xmath52 ( 85.9% ) , @xmath53 and @xmath54 ( 10.6% ) and @xmath55 ( 3.5% ) . the non-@xmath9 background is expected to be very small in the final sample . the background is estimated by fitting the side - bands in @xmath21 on a sample of data selected to have @xmath56 . as seen in figure [ fig83 ] , the distribution of the background is reasonably uniform in the region of the final selection , which enables one to use a simple linear interpolation to estimate the density of background one expects in the signal - box region . a geometrical correction is applied to estimate the background contained in the elliptical signal - box . a low statistics check of the method is made by applying it to the @xmath10 monte carlo sample . in this case , the luminosity scaled number of events predicted from the side - bands is @xmath57 events , in good agreement with the @xmath58 events observed . this estimate of @xmath58 is the only background estimate based purely on counting monte carlo events . the data side - band measurements yield estimates of @xmath4 events . using the @xmath10 monte carlo , the above selections yield an absolute efficiency for the signal of @xmath59 . after the final selection of requiring the events fall within the @xmath21 - @xmath22 signal - box , the efficiency is @xmath3 . this efficency has a number of systematic uncertainties associated with it including those arising from : 1 . the trigger efficiency 2 . the tracking reconstruction efficiency 3 . the neutral cluster reconstruction efficiency 4 . the background filter and skim selection efficiency 5 . electron opposite hemisphere requirements ( electron tag ) 6 . @xmath60 requirements ( ` @xmath61 ' tag ) 7 . @xmath12-pid requirements 8 . the photon energy scale and resolution 9 . the photon direction reconstruction , scale and resolution 10 . the track momentum scale and resolution 11 . the track momentum direction scale and resolution 12 . the beam energy scale 13 . the beam energy spread evaluation of the efficiency done using the events in the grand side - band , as these have characteristics which are very similar to those of the signal . the good agreement between the data and monte carlo normalization for the background sources , evident in table 1 , and the shapes of , for example the @xmath12 momentum , indicates that the efficiencies are well understood . the effects of systematic items ( 1)-(7 ) are incorporated into the ratio of observed to expected events : @xmath46 . this yields an estimate of 7.3% on the systematic error associated with items ( 1)-(7 ) . a global estimate of tracking and calorimetry errors is provided by shifting the signal - box position and by modifying the size of the box according to the uncertainties in the resolution . half of the observed changes are used to estimate the systematic errors . the systematic errors assessed in this manner are presented in table [ table2 ] . monte carlo simulations revealed that the selection efficiency is insensitive to the uncertainties in the beam energy spread and the associated systematic error is negligible . on the signal efficiency & ( % ) + effects ( 1)-(7 ) & @xmath62 + track and ecal resolution : & + @xmath22 scale & @xmath63 + @xmath22 resolution & @xmath64 + @xmath21 scale & @xmath65 + @xmath21 resolution & @xmath66 + ecal scale & @xmath67 + momentum scale & negligible + beam energy spread & @xmath65 + total & @xmath68 + from the observed number of events a 90% upper limit is set on b(@xmath0 ) with the systematic errors included as suggested in @xcite using the technique of @xcite . a demonstration with the two sidebands can be made : the lower sideband had a predicted background rate of 6.1 @xmath69 2.2 events , whereas 6 are observed . for the data and @xmath10 monte carlo simulation for the @xmath0 simulated signal for those events within the @xmath56 region . this is after all cuts but that on the signal - box . this plot is made after unblinding.,width=264,height=264 ] vs @xmath22 for data . the signal - box is indicated as the assymetric ellipse and shows the 13 observed events.,width=264,height=264 ] when the signal - box was unblinded , we observe 13 events , shown in figure [ fig81 ] for the @xmath21 projection . of these , 9 had electrons identifed in the opposite hemisphere ; 5 were @xmath70-like and one was identified as both an electron and a @xmath70 . these tagging ratios are consistent with the expectations of @xmath9 decays . the distributions of the @xmath21 and @xmath22 are shown in figure [ fig83 ] . the background was estimated to be @xmath4 . this yields a limit of 11.5 event upper limit on the number of signal events @90% cl when the systematic errors are included . the 11.5 event upper limit translates into a limit : b(@xmath71cl . the probability of a background of @xmath4 events up to 13 observed events in the absence of a signal is 7.6% if one includes the systematic errors . the 1999 - 2001 data has been studied in a search for the forbidden decay @xmath0 . these studies reveal that a search with an efficiency of @xmath3 and an expected background rate of @xmath4 results in 13 events being observed . this leads to a preliminary limit of b@xmath72cl . 9 r. barbieri and l. j. hall , phys . b 338 , 212 ( 1994 ) [ arxiv : hep - ph/9408406 ] . j. hisano , t. moroi , k. tobe , m. yamaguchi and t. yanagida , phys . b 357 , 579 ( 1995 ) [ arxiv : hep - ph/9501407 ] . j. hisano and d. nomura , phys . rev . d 59 , 116005 ( 1999 ) [ arxiv : hep - ph/9810479 ] . k. s. babu , b. dutta and r. n. mohapatra , phys . b 458 , 93 ( 1999 ) [ arxiv : hep - ph/9904366 ] . s. f. king and m. oliveira , phys . rev . d 60 , 035003 ( 1999 ) [ arxiv : hep - ph/9804283 ] . m. l. brooks et al . [ mega collaboration ] , phys . 83 , 1521 ( 1999 ) [ arxiv : hep - ex/9905013 ] . this work currently provides the most stringent limit of @xmath73 at 90% cl . s. ahmed et al . [ cleo collaboration ] , phys . rev . d 61 , 071101 ( 2000 ) [ arxiv : hep - ex/9910060 ] . k. amako [ geant4 collaboration ] , nucl . instrum . meth . a 453 , 455 ( 2000 ) . z. was , nucl . 98 , 96 ( 2001 ) [ arxiv : hep - ph/0011305 ] . radiative processes are based on the born - level formulae in : a. b. arbuzov , g. v. fedotovich , e. a. kuraev , n. p. merenkov , v. d. rushai and l. trentadue , jhep 9710 , 001 ( 1997 ) [ arxiv : hep - ph/9702262 ] . r. d. cousins and v. l. highland , nucl . instrum . meth . a 320 , 331 ( 1992 ) . roger barlow , hep - ex/0203002 .

using data collected with the detector between 1999 and 2001 , we describe a preliminary search for the neutrinoless decay @xmath0 . this data sample includes data collected both on and off the @xmath1(4s ) resonance and corresponds to @xmath2 produced tau pair events . the search has an efficiency of @xmath3 and an expected background rate of @xmath4 events . we select 13 events in the final sample . as there is no evidence for a signal in this data , we set a preliminary upper limit of b(@xmath0 ) @xmath5 cl .

authors are thankful to changbom park , yun soo myung , and yeong gyun kim for helpful discussions . this work was partly supported by the it r&d program of mic / iita [ 2005-y-001 - 04 , development of next generation security technology ] ( j.w . lee ) , and by the korea research foundation grant ( moehrd , basic research promotion fund ) ( krf-2005 - 075-c00009;h .- c.k . , and krf-2006 - 312-c00095 ; j.j . lee ) and by the topical research program of apctp and the national e - science project of kisti .

we suggest that vacuum entanglement energy associated with the entanglement entropy of the universe is the origin of dark energy . the observed properties of dark energy can be explained by using the nature of entanglement energy without modification of gravity or exotic matter . from the number of degrees of freedom in the standard model , we obtain the equation of state parameter @xmath0 and @xmath1 for the holographic dark energy , which are consistent with current observational data at the @xmath2 confidence level . the cosmological constant problem is one of the most important unsolved puzzles in modern physics @xcite . there is strong evidence from type ia supernova ( sn ia ) observations @xcite that the universe is expanding at an accelerating rate . a simple explanation for this acceleration is the existence of negative pressure fluids , called the dark energy , whose pressure @xmath3 and density @xmath4 satisfy @xmath5 . ( see eq . ( [ friedmann ] ) ) . although , there are various dark energy models rely on materials such as quintessence @xcite , @xmath6-essence @xcite , phantom @xcite , and chaplygin gas @xcite among many , the identity of this dark energy remains a mystery . these models usually require fine tuning of potentials or unnatural characteristics of the materials . on the other hand , entanglement ( a nonlocal quantum correlation ) @xcite is now treated as an important physical quantity . the possibility of exploiting entanglement in quantum information processing applications such as quantum key distribution and quantum teleportation has led to intense study of this quantity by the quantum information community . recently , there has been renewed interest @xcite in studying black hole entropy using entanglement entropy @xcite in the context of the ads / cft correspondence @xcite . in this paper , we suggest that there is an unexpected relation between dark energy and entanglement which are the two most puzzling entities in modern physics . it is well known @xcite that a simple combination of the planck scale and ir cutoff @xmath7 ( of order of inverse of the hubble parameter @xmath8 ) gives an energy density comparable to the observed cosmological constant or dark energy . this can be understood in terms of the holographic principle proposed by t hooft and susskind @xcite , which is a conjecture claiming that all of the information in a volume can be described by the physics at the boundary of the volume and that the maximum entropy in a volume is proportional to its surface area . cohen et al @xcite proposed a relation between the uv cut - off @xmath9 of an effective theory and @xmath7 by considering that the total energy in a region of size @xmath7 can not be larger than the mass of a black hole of that size . thus , for @xmath10 , the zero - point vacuum energy density is bounded as [ holodark ] _ = l^-4 = m_p^2 h^2 . interestingly , saturating the bound gives @xmath4 comparable to the observed dark energy density @xmath11 for @xmath12 , the present hubble parameter . the success of this estimation over the naive estimate @xmath13 can be attributed to the fact that quantum field theory over - counts the independent physical degrees of freedom inside the volume . thus , dark energy models based on the holographic principle have an advantage over other models in that they do not need an @xmath14 mechanism to cancel the @xmath15 zero - point energy of the vacuum . this simple holographic dark energy model is suggestive , but not without problems of its own . hsu @xcite pointed out that for @xmath10 , the friedmann equation @xmath16 makes the dark energy behave like matter rather than a negative pressure fluid , and prohibits accelerating expansion of the universe . later , li @xcite suggested that holographic dark energy of the form [ holodark2 ] _ = , would give an accelerating universe , where the future event horizon ( @xmath17 ) is used instead of the hubble horizon as the ir cutoff @xmath7 . here @xmath18 is an @xmath19 constant . however this use of @xmath17 has yet to be adequately justified . attempts @xcite have been made to overcome this ir cutoff problem in other ways , for example , by using non - minimal coupling to a scalar field @xcite or interaction between dark energy and dark matter @xcite . despite some success , the holographic dark energy models usually lack either an explanation for the microscopic origin of the dark energy or an explanation for why @xmath18 , the constant that determines the characteristics of the dark energy , is approximately one . in this paper we propose that these problems can be overcome in a natural manner by identifying dark energy as entanglement energy associated with the entanglement entropy @xmath20 of the universe . our model also suggests a way to derive @xmath18 and @xmath21 from the standard model of particle physics . from the reeh - schlieder theorem @xcite it is known that the vacuum for general quantum fields violates bell inequality and has entanglement @xcite when there are causally disconnected regions . entanglements of bose @xcite and fermi @xcite states have been studied using a thermal green s function approach . in ref . @xcite it was suggested that the hadamard green s function representing quantum fluctuation of the vacuum is useful for the study of entanglement in a scalar field vacuum . these relations between vacuum quantum fluctuations and entanglement are reminiscent of the vacuum fluctuation model of dark energy @xcite . there are two natural physics related to the event horizon ; black hole physics and entanglement physics . however , identifying @xmath17 as a black hole horizon is problematic , because dark energy should not include ordinary matter energy , while black hole energy includes all the energy inside the horizon . in quantum information theory , the event horizon plays a role of an information barrier and this leads to modification of energy of subsystem inside the horizon , which is the entanglement energy . therefore , the vacuum entanglement energy is a remaining plausible candidate for holographic dark energy . the entanglement entropy is the von neumann entropy @xmath22 associated with the reduced density matrix @xmath23 of a bipartite system @xmath24 described by a density matrix @xmath25 @xcite . for pure states such as the quantum fields vacuum , @xmath20 is a good measure of entanglement . when there is an event horizon , a natural choice is to divide the system into two subsystems - inside and outside the event horizon - and to trace over one of these subsystems to calculate the entanglement , because the event horizon represents the global causal structure @xcite . thus , @xmath20 is intrinsically related to the event horizon rather than the particle horizon or the hubble horizon . the future event horizon is given by [ rh ] r_hr(t)_t^ , which can be used as a typical length scale of the system with the horizon . here we consider the flat ( @xmath26 ) friedmann universe which is favored by observations @xcite and inflationary theory @xcite and described by the metric ds^2=-dt^2+r^2(t)d^2 , where @xmath27 is the scale factor as usual . the entanglement entropy of the quantum field vacuum with a horizon is generally expressed in the form [ sent ] s_ent= , where @xmath28 is an @xmath19 constant that depends on the nature of the field . here , @xmath29 is the uv cut - off of quantum gravity and different from @xmath9 which is the uv cut - off of a low energy effective theory @xcite ( see below for details ) . @xmath20 has a form consistent with the holographic principle , although it is derived from quantum field theory without using the principle . entanglement entropy for a single massless scalar field in the friedmann universe is calculated in ref . @xcite . by performing numerical calculations on a sphere lattice , they obtained @xmath30 . if there are @xmath31 spin degrees of freedom of quantum fields in @xmath17 , due to the additivity of the entanglement entropy @xcite , we can add up the contributions from all of the individual fields to @xmath20 @xcite , that is , @xmath32 , where for simplicity we assume the same @xmath28 for all fields . in @xcite the entanglement energy @xmath33 is defined as disturbed vacuum energy due to the presence of a boundary . there , entanglement energy proportional to the radius of the spherical volume was derived from quantum field theory . thus , for the event horizon , the entanglement energy is generally given by [ alpha ] e_ent = r_h , where @xmath34 is a constant depending on the exact mathematical definition of @xmath33 . we suggest that this entanglement energy is the origin of dark energy . once we obtain @xmath4 from @xmath33 , the negative pressure @xmath3 can be derived from the conservation of energy momentum tensor , [ p ] p_= as usually done in holographic dark energy models ( see eq . ( 6 ) of ref . @xcite ) . recall that this equation can be derived from the freedmann equation with perfect fluid having a energy momentum tensor of the form [ tperfect ] t_=(_+p _ ) u_u_- p_g _ , where @xmath35 . eq . ( [ p ] ) indicates that perfect fluid with increasing energy as the universe expands has a negative pressure . then , it is straight forward to obtain @xmath36 ( see eq . ( [ omega ] ) ) . we will show below that our theory gives the desired form of holographic dark energy . thus , we can use the all known formalism of typical holographic dark energy models for our model . now let us determine the coefficient @xmath34 in eq . ( [ alpha ] ) . although the mathematical definition of entanglement energy is not well - established , there are several reasonable conjectures for @xmath33 in ref . @xcite . inspired by the holographic principle , we adopt the following definition among them : [ de ] de_entt_ent ds_ent . note that this is @xmath37 the first law of thermodynamics for @xmath33 which needs a pressure term @xmath38 a mere definition of @xmath33 we choose in this paper . in @xcite , it was shown that this definition for @xmath33 is good for black holes . our entanglement energy in eq . ( [ de ] ) is this modified vacuum energy and hence internal " energy which looks like some thermal energy " related to entanglement entropy . to calculate @xmath33 , the most natural choice for the temperature " related to the event horizon is the gibbons - hawking temperature @xmath39 @xcite . by integrating @xmath40 we obtain [ eent ] e_ent=. from eq . ( [ alpha ] ) and eq . ( [ eent ] ) , we see @xmath41 . then , the entanglement energy density within the event horizon is given by [ rho ] _ = = , which has the form ( eq . ( [ holodark2 ] ) ) for the holographic dark energy . from the above equation we immediately obtain a formula for the constant [ d1 ] d= for the first time . although the constant @xmath18 determines the characteristic of the dark energy and the final fate of the universe , it has been constrained only by observations so far . interestingly , our model can be easily verified by current observations . the equation of state for dark energy of the form in eq . ( [ holodark ] ) is as follows @xcite [ omega ] _ = - ( 1 + ) , where @xmath42 is the density parameter of the dark energy . now , by inserting the expression for @xmath18 in eq . ( [ d1 ] ) into the above equation , we obtain @xmath21 directly from the number of spin degrees of freedom @xmath31 in the standard model(sm ) : [ omega2 ] _ = - ( 1 + ) . since @xmath43 , @xmath44 , and @xmath45 , eqs . ( [ d1 ] ) and ( [ omega ] ) gives us @xmath46 and @xmath47 . thus , the above calculation explains why @xmath46 from a particle physics view point . more precisely , we choose natural values @xmath48 , @xmath49 , the dark energy density parameter for the present @xmath50 and the matter density parameter @xmath51 favored by recent observations @xcite . using @xmath52 for the sm , we obtain @xmath1 and @xmath0 for the present . and @xmath53 from snia+cmb+sdss joint analysis done by zhang and wu ( fig . 2 of @xcite ) . the bright region at the center corresponds to the region within the @xmath54 confidence level contour . the black dot denotes the best - fit point from the observations . the white dot represents our theoretical prediction for the sm and the star for the mssm with @xmath55 . the triangle denotes our prediction with @xmath56 for the sm . ( courtesy of f. wu ) [ fig1 ] , scaledwidth=50.0% ] remarkably , this theoretical value for @xmath57 is consistent with current observational data from sn ia , the cosmic microwave background ( cmb ) , and the sloan digital sky survey ( sdss ) @xcite ( see fig . 9 in @xcite and fig . 15 in @xcite ) at the @xmath2 confidence level . for example , the combination of 3-year wmap data and the supernova legacy survey data @xcite yields @xmath58 , which is in agreement with our prediction , although @xmath21 is assumed to be independent of time in that paper . very recently , zhang and wu @xcite perform a joint analysis of constraints on @xmath18 with the latest observational data including the gold sample of sn ia , the shift parameter of cmb and the baryon acoustic oscillation ( bao ) from the sdss . this gives @xmath59 , which contains our value @xmath1 within the @xmath54 region ( see fig . 1 ) . thus , our model well explains observed properties of dark energy . for the minimal supersymmetric standard model ( mssm ) , @xmath60 ; this value of @xmath31 gives @xmath61 and @xmath62 , which slightly violates the constraint @xmath63 from sn ia data @xcite . this result indicates that , for our model , sm degrees of freedom is good for @xmath31 and the planck length scale is good for the uv cut - off @xmath29 . the reasons behind this might be as follows . the origin of our entanglement energy is different from the energy of an low energy effective theory considered by cohen et al s proposal @xcite which motivates the usual holographic dark energy models . the entanglement energy is related to quantum information loss at the horizon and to the vacuum quantum fluctuation in quantum gravity theory , which is usually believed as the origin of dark energy or the holographic principle . thus , the natural uv cut - off of our model is the planck length as in many related literatures @xcite . what can we say about @xmath31 ? considering the planck scale uv cut - off , it is desirable to use also the degree of freedoms at the planck scale . this value depends on the model of the unification theory , which varies from @xmath64 to @xmath65 and makes the explicit value of @xmath18 vary approximately from 1 to 3 . however , sm is the only model that is verified by various experiments so far . therefore , it is still plausible that the degrees of freedom at the planck scale could be similar to that of sm and we can use sm degrees of freedom for @xmath31 . even in the case that a larger unification theory ( such as string theory ) is the true theory , contributions from non - sm fields to vacuum fluctuation might be negligible due to symmetry breaking of those sectors . although our theory still has some ambiguity to be resolved in these parameters , it is interesting that our theory predicts the observed @xmath18 value with the planck scale and sm degrees of the freedom . to obtain a more precise value of @xmath57 , it is essential to calculate the exact value of @xmath66 for every field @xmath67 in the sm or mssm . then @xmath68 and the number of degrees of freedom of the @xmath67-th field , @xmath69 , should satisfy the relation @xmath70 , derived by the same arguments as those leading to eq . ( [ d1 ] ) . an interesting question here is whether the above equation gives @xmath71 for the sm . using eq . ( 28 ) of ref . @xcite one can also obtain the time dependency of the equation of state ; [ omega3 ] _ & = & ( 1 + ) ( - + z)&= & -0.93 + 0.11 z for sm , where @xmath72 is the red shift parameter . in general , holographic dark energy models including ours tacitly assume the presence of the accelerating expansion of the universe . if not , the holographic dark energy could not be finite . since the accelerating universe is an observational fact , this assumption is plausible . alternatively , if we first assume finite @xmath20 at any finite time , then the accelerating universe is a natural consequence . from eq . ( [ sent ] ) , finite @xmath20 implies a finite @xmath17 . from eq . ( [ rh ] ) , it is easy to see that the event horizon exists only when @xmath73 converges , that is , the universe should accelerate ( @xmath74 ) as @xmath75 for @xmath17 to be finite ( unless the universe is oscillating ) @xcite . the accelerating universe satisfies [ friedmann ] = -=n(n-1)t^-2>0 , hence , @xmath76 , i.e. , @xmath77 . here the dot denotes a time derivative . thus , the finiteness of @xmath20 demands a finite @xmath17 and requires that the universe should accelerate and dark energy should dominate as @xmath78 . there are already many scenarios that explain the cosmic coincidence problem in the context of holographic dark energy . for example , in @xcite to solve the coincidence problem , an interaction between dark matter @xcite and dark energy was introduced . li suggested inflation at the gut scale with the minimal number of e - folds of expansion @xmath79 as a solution @xcite . in summary , we suggest a model in which dark energy is identified as the entanglement energy of the universe . this model could explain many observed properties of dark energy without modification of gravity , exotic fields or particles . using only standard model fields , the holographic principle , and entanglement theory , our model predicts the equation of state and the constant @xmath18 of dark energy which are well consistent with observations . our analysis also indicates that the holographic principle and the entanglement theory can play a fundamental role not only in the physics of black holes or string theory but also in cosmology @xcite .

the pipe nebula is a massive ( @xmath4 @xmath5 : @xcite ) filamentary ( @xmath015 pc long and @xmath03 pc wide ) dark cloud located in the southern sky @xmath05 apart from the galactic center . its short distance to the sun ( 145 pc : @xcite ) places this complex in the group of nearby molecular clouds which serve as good laboratories for star formation surveys . despite the large reservoir of mass , the pipe nebula molecular cloud is characterized by being apparently quiescent , with a very low star - formation efficiency ( @xmath00.06% for the entire cloud , @xcite ) . barnard 59 ( b59 ) , located at the northwestern end of the cloud , has formed a small cluster of low - mass stars @xcite . the low global star forming efficiency of the cloud contrasts with that of other nearby molecular clouds such as ophiuchus or taurus , where an important star - formation activity is observed . the pipe nebula is , hence , an excellent place to study the initial conditions of star formation at scales of a few parsecs . the first extensive survey toward the pipe nebula was done by @xcite through single dish observations of co isotopologues . these authors were the first to suggest a clumpy distribution for the dense gas by detecting compact c@xmath6o cores in the main body of the cloud . it was not until the last few years that several surveys @xcite were carried out to explore the physical properties of the cloud . @xcite use 2mass data to construct a high resolution extinction map of the pipe nebula through which they identify a large number of high extinction cores with typical masses between 0.2 and 5 @xmath3 . molecular line observations reveal that they are starless cores in a very early evolutionary stage , associated with dense ( @xmath4 @xmath2 ) , relatively cold ( @xmath7 k ) , and fairly quiescent gas ( typical line widths of 0.4 , @xcite ) . non - thermal gas motions inside the cores are sub - sonic and mass independent . therefore , thermal pressure appears to be the dominant source of internal pressure . in addition , these cores appear to be pressure confined , but gravitationally unbound @xcite . recently , @xcite performed an optical polarimetric survey toward the diffuse gas in the pipe nebula . they find a large scale magnetic field that appears to be mostly perpendicular to the cloud main axis . the magnetic field exerts a pressure ( @xmath010@xmath8 k@xmath2 ) that is likely responsible for driving the collapse of the gas and dust cloud along the field lines . the polarization properties significantly change along the pipe nebula . this fact allowed the authors to distinguish three regions in the cloud : b59 , the _ stem _ , and the _ bowl _ ( see fig . [ fig : pipe ] ) . b59 shows low polarization levels but high dispersion of the polarization position angles . moving through the _ stem _ toward the _ bowl _ , the polarization level increases and the dispersion decreases . these authors propose that these three regions might be in different evolutionary stages . b59 is the only magnetically supercritical region and the most evolved of the pipe nebula , the _ stem _ would be at an earlier evolutionary stage , with material still collapsing , and finally , the _ bowl _ would be at the earliest stage , with cloud fragmentation just started . . source list . [ cols="^,^,^,^,^ " , ] \(a ) see tables [ tab_dust_col_dens ] and [ tab_mol_col_dens ] for dust and line column densities . + ( b ) transition with no opacity mesurements available , thus optically thin emission is assumed to estimate a lower limit of the column densities and , consequently , of the abundances . + ( c ) due to the lack of c@xmath9s data we assume optically thin emission to obtain a lower limit of the column density and , as a result , also for the abundance . [ tab_abun ] crrrrrrrrrr & & & & & & & & & & + & & & & & & & & & & + core 109 & 0.063 & 4.00 & 47.60 & 36.57 & 11.0 & 3.9 & 2.20 & 8.54 & 20.6 & 40.6 + + relative values & & & & & & & & & & + core 109 & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 + core 40 & 16.5 & 6.3 & 2.3 & 1.4 & 4.2 & 22.2 & 20.8 & 49.1 & 7.5 & 16.4 + core 14 & 11.3 & 3.5 & 2.8 & 2.5 & 1.8 & 40.4 & 4.0 & 11.2 & 1.7 & 68.2 + core 48 & 20.2 & 5.2 & 1.3 & 0.6 & 1.8 & 83.8 & @xmath102.5 & @xmath102.6 & @xmath100.6 & 53.9 + \(a ) @xcite [ tab_resum ] table [ tab_dust_col_dens ] shows a variation of about a factor of @xmath04 around @xmath11 @xmath12 of the average h@xmath13 column densities derived for each of the cores with a 27@xmath14 beam , the one used to calculate the abundances for the molecular transitions at 3 mm . this represents , using the relationship @xmath15 @xcite , average values of a@xmath16@xmath04.4 to @xmath019.4 . the first case would represent a shallow core , more affected by the external radiation field , which tends to have a younger chemistry . the other extreme probably indicates a denser and more shielded core , where one would expect to find more complex and evolved molecules . however , note that this also depends on the time - scale needed to form the core @xcite . we find that cs ( see table [ tab_abun ] ) , an early - time molecule , is detected in all the cores with abundances with respect to h@xmath13 of a few times @xmath17 , similar to the ones found in other dense cores @xcite or the ones obtained in gas phase chemical models @xcite . it is worth to mention that cores 14 and 48 show high cs abundance , one order of magnitude higher than cores 109 and 40 . a similar result is found for the c@xmath9s abundances . the derived abundances for the early - time molecule hcn toward the cores in our sample is very uniform , and seems to be independent of their physical properties . the early - time molecule cn , a molecule that is also detected commonly in dense cores , has also a significantly lower abundance ( a factor @xmath184 ) toward core 48 than toward the rest of the sample . where detected , the cn abundance varies only within a factor of five . on the other hand , another early - time molecule such as c@xmath19h@xmath13 , shows differences in abundances of at least a factor of five among cores 14 and 48 with respect to cores 40 and 109 . late - time molecules , such as n@xmath13h@xmath20 or deuterated molecules , are not broadly detected in our sample : n@xmath13h@xmath20 is detected except in core 48 . on the contrary , n@xmath13d@xmath20 is only detected on cores 40 and 109 , and dco@xmath20 only in core 109 . we found a higher abundance of n@xmath13h@xmath20 toward core 40 than toward core 109 by a factor of @xmath02 , while @xcite found an abundance of nh@xmath19 toward core 109 higher than that of core 40 by a factor of @xmath03.4 . however , both cores 40 and 109 show higher abundances in n@xmath13h@xmath20 than cores 14 and 48 . despite cores 14 and 40 having a similar average column density , the former shows 5 times less abundance of n@xmath13h@xmath20 than the latter . moreover , core 14 does not show emission in any other late - time molecule while core 40 is detected in n@xmath13d@xmath20 showing an abundance only a factor of 4 lower than that of core 109 . briefly , the higher abundances in cores 109 and 40 with respect to core 14 and in particular to core 48 ( except for cs ) is an indication that cores 109 and 40 are more chemically evolved than cores 14 and 48 . however , the molecular abundances of these two late - time species are roughly an order of magnitude lower than the prototypical starless cores l1517b and l1498 @xcite , which suggests that cores 109 and 40 may be in an earlier evolutionary stage than cores in taurus . @xcite observed the emission of the nh@xmath19 ( 1,1 ) , nh@xmath19 ( 2,2 ) , ccs ( 2@xmath211@xmath22 ) , and hc@xmath23n ( 98 ) transitions towards 46 cores of the pipe nebula . cores 14 , 40 , 48 , and 109 were included in their observations . none of the lines were detected in core 48 , which is shown to be again the more chemically poor core of our sample . hc@xmath23n was not detected in core 14 , which also has the weakest ccs and nh@xmath19 lines . the four transitions were detected in cores 40 and 109 , but with some differences . the nh@xmath19 lines are much more intense in core 109 , a factor of @xmath04 for the ( 1,1 ) transition and @xmath09 for the ( 2,2 ) line , while the ccs line is more intense in core 40 , less than a factor of @xmath02 , and the hc@xmath23n lines are very similar in both cores , inside the rms . all these results are consistent with our observations : core 48 , which did not show emission of late - time molecules , is very poor chemically and shows a very young chemistry . core 14 , has some very weak emission of late - time molecules ( nh@xmath19 ) but only weak emission of ccs , an early - time molecule . core 40 is more evolved chemically and shows stronger emission of early - time molecules than of late - time molecules . finally , core 109 is the one showing more diversity of molecules and the more intense emission , in particular of late - time molecules . interestingly , the ccs abundance in core 109 is probably lower than in core 40 , which is consistent with the view that the ccs molecule is destroyed soon after the formation of a dense core , probably as a result of the contraction of the core @xcite . this would reinforce the view that this core is in a very advanced evolutionary state . in summary , core 109 seems to be the more chemically evolved core , probably because it is more dense and because it shows higher abundances of late - time molecules . core 40 , with three times lower column density , also shows large n@xmath13h@xmath20abundances . it might be in an intermediate chemical evolutionary stage . these two cores probably are in an evolutionary stage slightly younger than that of the prototypical starless cores @xcite . cores 48 and 14 show similar physical properties in terms of size , mass and h@xmath13 column density , to cores 109 and 40 . however , they appear to be very chemically poor and , therefore , they could be in an even younger stage of chemical evolution . table [ tab_resum ] shows the summary of the main properties of the cores relative to core 109 , which is the one that shows the strongest line emission . in this table we show the physical and chemical properties . additionally , we added the averaged polarimetric properties of the diffuse envelope around the cores @xcite : polarization fraction ( @xmath24 ) and dispersion of the polarization position angle ( @xmath25 ) . as shown in fig . [ fig : mambo ] , the polarization vectors calculated from optical extinction can not be derived at the more dense regions , where the visual extinction is higher . in fig . [ fig : mambo ] , except for the map of core 48 with the lowest rms , the polarization vectors lie in regions below the 3-@xmath26 noise level . however , the trend of the polarization vectors is in general rather uniform over the whole map . indeed , there are vectors up to very close to the dense parts of the cores . consequently , the derived magnetic field properties of the diffuse surrounding medium are also representative of those of the dense part of the cores . a relationship between the magnetic and the chemical properties of each core seems to exist . the two more chemically evolved cores , 109 and 40 , appear to be embedded in a strongly magnetized environment , as @xmath25 values clearly reflect ( see table [ tab_resum ] ) . the other two cores , 14 and 48 , do not show very different morphological properties with respect to the previous two ( size and mass ) . however , their chemical properties are completely opposed , and they are likely younger cores in chemical time - scale . interestingly , the magnetic properties of cores 14 and 48 are also opposed to those of cores 40 and 109 . cores 14 and 48 are surrounded by a molecular diffuse medium that is much more turbulent than that surrounding the two previous ones . core 14 is possibly affected by the star formation undergoing in the nearby region b59 . core 48 appears to be dominated by turbulence and constitutes an exception in the _ stem _ , whose cores have uniform magnetic properties among them , showing low @xmath24 and high @xmath25 @xcite . in summary , these four cores of the pipe nebula have similar masses and sizes , but they are in different stages of chemical evolution : cores 109 and 40 are much more evolved chemically than cores 48 and 14 . the different magnetic properties of the diffuse molecular environment suggest that cores 109 and 40 have grown in a more quiescent and slowly way ( probably through ambipolar diffusion ) , whereas the growth of cores 14 and 48 has occurred much faster , an indication that possibly a compression wave that generates turbulence or the turbulence itself @xcite . the longer time scale of the ambipolar diffusion process could explain the more evolved chemistry found toward the cores surrounded by a magnetized medium . these features suggest two different formation scenarios depending on the balance between turbulent and magnetic energy in the surrounding environment . the importance of these results is worth of a more detailed study of the pipe nebula cores in order to fully confirm these trends . we carried out observations of continuum and line emission toward four starless cores of the pipe nebula spread out along the whole cloud selected in base of their magnetic properties @xcite . we studied their physical and chemical properties , and the correlation with the magnetic field properties of the surrounding diffuse gas . 1 . the dust continuum emission of the observed pipe nebula cores shows quite different morphologies . in the sample there are diffuse cores , such as cores 40 and 48 , and compact and dense cores , such as core 109 . we have also mapped a clumpy filament , which contains the embedded core 14 . this filament is possibly undergoing fragmentation into smaller cores of sizes comparable to that of the others . we derived average radii of @xmath00.09 pc ( @xmath018600 au ) , densities of @xmath01.3@xmath1@xmath2 , and core masses of @xmath02.5 @xmath27 . the dust continuum peak coincides within the errors with @xmath28 peak derived from the 2mass catalog . the continuum emission is more sensitive toward the dense regions , up to @xmath1810 magnitudes for the densest cores . on the other hand , the diffuse emission is better traced by the extinction maps . the masses are in average @xmath03.4 times smaller . we have observed several early- and late - time lines of molecular emission toward the cores and derived their column densities and abundances . the starless cores of the pipe nebula are all very young , but they present different chemical properties possibly related to a different evolutionary stage . however , there does not seem to be a clear correlation between the chemical evolutionary stage of the cores and their position in the cloud . cores 109 and 40 show late - time molecular emission and seem to be more chemically evolved . core 109 shows high abundances of late - time molecules and it seems to be the more chemically evolved . core 40 has three times lower h@xmath13 column density than that of core 109 . it presents a large n@xmath13h@xmath20 abundance and the largest cn abundance , thus it might be in an intermediate chemical evolutionary stage . cores 48 and 14 show only early - time molecular emission , and core 14 presents weak n@xmath13h@xmath20 emission , and seem to be chemically younger than the other two cores . core 14 has a similar mass and size than core 40 , but the n@xmath13h@xmath20 , c@xmath19h@xmath13 , and cs abundances are about one order of magnitude lower than the core 40 abundances . our results and interpretation of the evolutionary stage of each core are consistent with the previous observations of @xcite in these same cores . there seems to be a relationship between the properties of the magnetic field in the cloud medium of the cores and the chemical evolutionary stage of the cores themselves . the two more chemically evolved cores , 109 and 40 , appear to be embedded in a strongly magnetized environment , with a turbulent to magnetic energy ratio of 0.05 and 0.27 , respectively . the two chemically younger cores , 14 and 48 , appear to be embedded in a more turbulent medium . this suggests that the magnetized cores probably grow in a more quiescent way , probably through ambipolar diffusion , in a time - scale large enough to develop the richer chemistry found . on the other hand , the less magnetized cores likely grow much faster , probably in a turbulence dominated process , in a time - scale too short to develop late - time chemistry . the pipe nebula has revealed to be an excellent laboratory for the study of the very early stages of the star formation . the studied cores show different morphologies , chemical evolutionary stages and magnetic properties . the physical and chemical properties are not directly linked as the competition between the magnetic field and turbulence at small scales seems to have an important influence in the core evolution . the importance of these results require a more detailed study of the chemistry and magnetic field properties of the cores to fully confirm these results . pf is supported by micinn fellowship fpu ( spain ) . pf , jmg , mtb , jmm , foa , gb , asm , and re are supported by micinn grant aya2008 - 06189-c03 ( spain ) . pf , jmg , mtb , om , foa , and re are also supported by agaur grant 2009sgr1172 ( catalonia ) . gapf is partially supported by cnpq ( brazil ) . the authors want to acknowledge all the iram 30-m staff for their hospitality during the observing runs , the operators and aods for their active support , guillermo quintana - 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[ eq_rad_trans ] can be simplified to @xmath59 . then , the mass can be calculated as @xmath60 which for a beam is @xmath61 , where @xmath62 is the distance to the source . all these calculations can be applied to any solid angle bigger than a beam . the column density for a @xmath63 transition of a molecule ( mol ) is @xmath65 & = & 1.67\times10^{14}\ , \frac{q_{\rm rot}}{g_k\,g_i } \left[\frac{s_{jki}}{\textrm{erg\,cm}^3\,\textrm{statc}^{-2}\,\textrm{cm}^{-2}}\right]^{-1 } \left[\frac{\mu}{\textrm{d}}\right]^{-2 } e^{eu / t_{\rm ex } } \left[\frac{\nu}{\textrm{ghz}}\right]^{-1 } \frac{j_{\nu}\left(t_{\rm ex}\right)}{j_{\nu}\left(t_{\rm ex}\right)-j_{\nu}\left(t_{\rm bg}\right ) } \frac{\tau}{1-e^{-\tau } } \left[\frac{\int_\mathrm{line}{t_{\rm mb } \ , d\texttt{v}}}{\textrm{k\,km\,s}^{-1}}\right ] . \nonumber \\ \label{eq_col_dens}\end{aligned}\ ] ] in case of single transitions , we have performed a gaussian fit to the spectrum or a statistical moment calculation , both using tasks from the class package . we obtain from either analysis the main beam temperature , @xmath68 , the line velocity , @xmath69 , and the integrated emission , @xmath70 . the opacity , @xmath71 , is calculated numerically in those molecules with more than one transition observed . in the other cases we have assumed @xmath72 . the excitation temperature , @xmath73 , can be calculated from the radiative transfer equation as in case of hyperfine transitions , we take into account all the hyperfine components of the selected transition . we have performed a hyperfine fit using class , which provides @xmath76 , where @xmath77 is beeing @xmath79 is the filling factor assumed to be @xmath01 . + to be able to use equation [ eq_col_dens ] as in the single transition case , we need @xmath73 , @xmath71 and @xmath70 . we can calculate @xmath73 as in equation [ eq_tex ] calculating @xmath68 as @xmath80 , and @xmath81 is given by class . for the integrated emission , we can use eq . [ eq_qrot ] can be approximated , in the limit of high temperatures , by an integral because generally the energy levels are close together . we are only interested in the high temperature limit because is when the transition is activated , so this limit is accurate enough . + @xmath89 * linear molecules : * the solution for the diatomic case is general for any lineal molecule , so long as the molecular moment of inertia is computed properly for more than 2 atoms . for lineal molecules @xmath90 , @xmath91 and @xmath92 . @xmath26 ( the symmetry number ) is 1 for heteronuclear diatomic ( c - o ) or asymmetric linear polyatomic ( o - n - n ) molecules , and 2 for homonuclear diatomic ( h - h ) or symmetric linear polyatomic ( o - c - o ) molecules . @xmath89 * non - linear molecules : * non - linear molecules have up to three moments of inertia and , thus , three rotational constants ( @xmath96 ) . in a similar way than before , but more complicated , the calculation of the rotational partition function at high temperatures is we can calculate the energy of the upper level ( @xmath98 ) as a function of the lower level ( @xmath99 ) plus the energy of the photon emitted ( both available at catalogues ) . this is , in units of temperature and using the units given in the catalogues , [ [ intrinsic - line - strength - times - squared - dipolar - momentum - smu2ssec_smu2 ] ] intrinsic line strength times squared dipolar momentum ( @xmath101)[ssec_smu2 ] ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ we can calculate the product of the _ intrinsic _ line strength , @xmath102 , and the squared dipolar momentum , @xmath103 , from the @xmath84 at 300 k ( @xmath104 ) , the line strength ( logint ) at 300 k and the lower state energy ( @xmath99 ) . all these parameters are available at the catalogues . @xmath105 = 24025 \times\left[\frac{10^{logint}}{\textrm{mhz\,nm}^2}\right]\;q^{300}_{\mathrm{rot } } \left[\frac{\nu}{\textrm{mhz}}\right]^{-1 } \left(\mathrm{exp } \left\ { 4.796\times10^{-3 } \left[\frac{e_l}{\textrm{cm}^{-1}}\right ] \right\ } \right ) \left(1-\mathrm{exp}\left\ { -1.6\times10^{-7 } \left[\frac{\nu}{\textrm{mhz}}\right ] \right\}\right)^{-1}. \nonumber \\\end{aligned}\ ] ]

the pipe nebula is a massive , nearby dark molecular cloud with a low star - formation efficiency which makes it a good laboratory to study the very early stages of the star formation process . the pipe nebula is largely filamentary , and appears to be threaded by a uniform magnetic field at scales of few parsecs , perpendicular to its main axis . the field is only locally perturbed in a few regions , such as the only active cluster forming core b59 . the aim of this study is to investigate primordial conditions in low - mass pre - stellar cores and how they relate to the local magnetic field in the cloud . we used the iram 30-m telescope to carry out a continuum and molecular survey at 3 and 1 mm of early- and late - time molecules toward four selected starless cores inside the pipe nebula . we found that the dust continuum emission maps trace better the densest regions than previous 2mass extinction maps , while 2mass extinction maps trace better the diffuse gas . the properties of the cores derived from dust emission show average radii of @xmath00.09 pc , densities of @xmath01.3@xmath1 @xmath2 , and core masses of @xmath02.5 @xmath3 . our results confirm that the pipe nebula starless cores studied are in a very early evolutionary stage , and present a very young chemistry with different properties that allow us to propose an evolutionary sequence . all of the cores present early - time molecular emission , with cs detections toward all the sample . two of them , cores 40 and 109 , present strong late - time molecular emission . there seems to be a correlation between the chemical evolutionary stage of the cores and the local magnetic properties that suggests that the evolution of the cores is ruled by a local competition between the magnetic energy and other mechanisms , such as turbulence .

analysis on metric spaces with no a priory smooth structure has rapidly developed the present time . this development is closely related to some generalizations of the differentiability . important examples of such generalizations and even an axiomatics of so - called `` pseudo - gradients '' can be found in @xcite and respectively in @xcite . in almost all above - mentioned books and papers the generalized differentiations involve an induced linear structure that makes possible to use the classical differentiations in the linear normed spaces . a new _ intrinsic _ approach to the introduction of the `` smooth '' structure for general metric spaces was proposed by o. martio and by the first author of the present paper in @xcite . a basic technical tool in @xcite is the notion of pretangent spaces at a point @xmath2 of an arbitrary metric space @xmath0 which were defined as factor spaces of families of sequences of points @xmath3 convergent to @xmath2 . in present paper we find and prove necessary and sufficient conditions under which the metric space with a marked point @xmath2 has a unique pretangent space at @xmath2 for every normalizing sequence @xmath4 , see definition [ 1:d1.3 ] below . for convenience we recall the main notions form @xcite , see also @xcite . let @xmath5 be a metric space and let @xmath2 be point of @xmath0 . fix a sequence @xmath4 of positive real numbers @xmath6 which tend to zero . in what follows this sequence @xmath4 be called a _ normalizing sequence_. let us denote by @xmath7 the set of all sequences of points from @xmath0 . [ 1:d1.1 ] two sequences @xmath8 , @xmath9 and @xmath10 , are mutually stable ( with respect to a normalizing sequence @xmath11 ) if there is a finite limit @xmath12 we shall say that a family @xmath13 is _ self - stable _ @xmath4 ) if every two @xmath14 are mutually stable . a family @xmath15 is _ maximal self - stable _ if @xmath16 is self - stable and for an arbitrary @xmath17 either @xmath18 or there is @xmath19 such that @xmath20 and @xmath21 are not mutually stable . a standard application of zorn s lemma leads to the following [ 1:p1.2 ] let @xmath22 be a metric space and let @xmath1 . then for every normalizing sequence @xmath23 there exists a maximal self - stable family @xmath24 such that @xmath25 . note that the condition @xmath26 implies the equality @xmath27 for every @xmath9 which belongs to @xmath28 . consider a function @xmath29 where @xmath30 is defined by . obviously , @xmath31 is symmetric and nonnegative . moreover , the triangle inequality for the original metric @xmath32 implies @xmath33 for all @xmath34 from @xmath28 . hence @xmath35 is a pseudometric space . [ 1:d1.3 ] the pretangent space to the space @xmath0 at the point @xmath2 w.r.t . a normalizing sequence @xmath4 is the metric identification of the pseudometric space @xmath36 . since the notion of pretangent space is basic for the present paper , we remaind this metric identification construction . define a relation @xmath37 on @xmath28 by @xmath38 if and only if @xmath39 . then @xmath40 is an equivalence relation . let us denote by @xmath41 the set of equivalence classes in @xmath28 under the equivalence relation @xmath37 . it follows from general properties of pseudometric spaces , see , for example , ( * ? ? ? * chapter 4 , th . 15 ) , that if @xmath42 is defined on @xmath43 by @xmath44 for @xmath45 and @xmath46 , then @xmath42 is the well - defined metric on @xmath43 . the metric identification of @xmath35 is , by definition , the metric space @xmath47 . remark that @xmath48 because the constant sequence @xmath49 belongs to @xmath50 , see proposition [ 1:p1.2 ] . let @xmath51 be an infinite , strictly increasing sequence of natural numbers . let us denote by @xmath52 the subsequence @xmath53 of the normalizing sequence @xmath11 and let @xmath54 for every @xmath55 . it is clear that if @xmath20 and @xmath56 are mutually stable w.r.t . @xmath4 , then @xmath57 and @xmath58 are mutually stable w.r.t . @xmath59 and @xmath60 if @xmath61 is a maximal self - stable ( w.r.t . @xmath4 ) family , then , by zorn s lemma , there exists a maximal self - stable ( w.r.t . @xmath59 ) family @xmath62 such that @xmath63 denote by @xmath64 the mapping from @xmath61 to @xmath65 with @xmath66 for all @xmath67 . if follows from that after the metric identifications @xmath64 pass to an isometric embedding @xmath68 : @xmath69 under which the diagram @xmath70 is commutative . here @xmath71 , @xmath72 are metric identification mappings , @xmath73 and @xmath74 . let @xmath0 and @xmath75 be two metric spaces . recall that a map @xmath76 is called an _ isometry _ if @xmath77 is distance - preserving and onto . [ 1:d1.4 ] a pretangent @xmath78 is tangent if @xmath79 : @xmath80 is an isometry for every @xmath59 . simple arguments give the following proposition . [ 1:p1.5 ] let @xmath0 be a metric space with a marked point @xmath2 , @xmath81 a normalizing sequence and @xmath82 a maximal self - stable family with correspondent pretangent space @xmath83 . the following statements are equivalent . \(i ) @xmath84 is tangent . \(ii ) for every subsequence @xmath85 of the sequence @xmath86 the family @xmath87 is maximal self - stable w.r.t . @xmath88 . \(iii ) a function @xmath89 is surjective for every @xmath90 . \(iv ) a function @xmath91 is surjective for every @xmath92 . for the proof see ( * ? ? * proposition 1.2 ) or ( * ? ? ? * proposition 1.5 ) . in this section we start from the simplest example of a metric space with unique pretangent spaces . [ 4:e2.1 ] let @xmath93 be the set of all non - negative , real numbers with the usual metric @xmath94 let @xmath95 be an arbitrary normalizing sequence and let @xmath96 be the marked point of @xmath0 . consider a maximal self - stable family @xmath97 . [ 4:p2.2 ] the following statements are true . \(i ) let @xmath98 . then @xmath99 if and only if there is @xmath100 such that @xmath101 \(ii ) for every two @xmath102 from @xmath97 the equality @xmath103 holds if and only if @xmath104 \(iii ) the pretangent space @xmath105 corresponding to @xmath106 is isometric to @xmath107 . \(iv ) the pretangent space @xmath105 is tangent . \(i ) if @xmath108 , then there is a finite limit @xmath109since we have @xmath110 @xmath111 for all @xmath112 , the limit relation holds with @xmath113 . suppose that @xmath114 , @xmath115 and there are @xmath116 such that @xmath117 it implies that @xmath118 so @xmath119 are mutually stable . it implies statement ( i ) . \(ii ) statement ( ii ) follows from statement ( i ) and . \(iii ) define a function @xmath120 by the rule : if @xmath121 and @xmath122 , then write @xmath123 . statements ( i),(ii ) and limit relation imply that @xmath77 is a well - defined isometry . \(iv ) let @xmath124 be a strictly increasing , infinite sequence of natural numbers and let @xmath125 be the corresponding subsequence of the normalizing sequence @xmath126 . if @xmath127 then , by statement ( i ) , there is @xmath128 such that @xmath129define @xmath130 by the rule @xmath131 it is clear that @xmath132 and @xmath133hence , by statement ( i ) , @xmath134 belongs to @xmath97 . using proposition [ 1:p1.5 ] we see that @xmath135 is tangent . statement ( i ) of proposition [ 4:p2.2 ] shows that the space @xmath107 possesses an interesting property : for every normalizing sequence @xmath126 there exists a unique pretangent space @xmath136 . the main theorem of this paper describes metric spaces which have this property . [ r:2.3 ] the uniqueness in the previous paragraph and in theorem [ 5:t2.5 ] below is understood in the usual set - theoretical sense . statement ( i ) of proposition [ 4:p2.2 ] implies that for @xmath137 the family (= the set ) @xmath50 is unique . hence @xmath136 , the metric identification of @xmath138 , is also unique . since @xmath139 i.e. , the set @xmath138 is the union of all equivalence classes @xmath140 , the uniqueness of the pretangent spaces @xmath136 gives the uniqueness of @xmath138 . let @xmath5 be a metric space with marked point @xmath2 . for each pair of nonvoid sets @xmath141 write @xmath142and write @xmath143and for every @xmath144 and every @xmath145 define @xmath146and for every @xmath147 0,1\right [ $ ] @xmath148where @xmath149 is the cartesian product of @xmath150 s . see fig . 1 . @xmath151 and @xmath152 with @xmath153\cup[2,3]$ ] and @xmath154 . nontengential limit is taken over the set @xmath152.,width=453 ] [ 5:t2.5 ] let @xmath5 be a metric space and let @xmath2 be a limit point of @xmath0 . then for every normalizing sequence @xmath4 there is a unique pretangent space @xmath155 if and only if the following three conditions are satisfied simultaneously . \(i ) the limit relation @xmath1560,\infty[\,,\ k\in[1,\infty[\]]holds . \(ii ) we have @xmath157 for every @xmath147 0,1\right [ \,$ ] . \(iii ) if @xmath158 is a sequence such that @xmath159 for all @xmath112 and @xmath160and there is @xmath161 , \label{5:eq2.92}\ ] ] then there exists a finite limit @xmath162 [ r:2.5 ] the annulus @xmath163 can be void in [ 5:eq2.8 ] . at that time we use the convention @xmath164 we need the following lemma . [ 5:l2.6 ] let @xmath5 be a metric space with a marked point @xmath2 . a pretangent space @xmath155 is unique for every normalizing sequence @xmath4 if and only if the implication @xmath165 @xmath166is true for every @xmath167 , @xmath134 @xmath168 suppose that is true . let @xmath169 be the set of all @xmath170 which are mutually stable with @xmath171 . it follows from that @xmath172 is self - stable . consider an arbitrary maximal self - stable @xmath82 , then , by definition of @xmath82 , we obtain the inclusion @xmath173 . since @xmath174 is maximal self - stable , we have also @xmath175 . hence the equality @xmath176holds for all @xmath82 , so all @xmath177 coincide . now suppose that @xmath50 is unique for every @xmath4 and there are @xmath178 and there is a normalizing sequence @xmath179 such that : @xmath167 and @xmath171 are mutually stable ; @xmath134 and @xmath171 are mutually stable ; @xmath167 and @xmath134 are not mutually stable . by zorn s lemma there exist maximal self - stable families @xmath180 and @xmath181 . it is clear that @xmath182 @xmath183 hence , the uniqueness of pretangent spaces , see remark [ r:2.5 ] , implies . assume that @xmath155 is unique . we need to verify the conditions ( i)(iii ) . \(i ) consider a function @xmath184 , @xmath185since @xmath186and @xmath187 the function @xmath77 is increasing . since we have @xmath188for every @xmath145 and all @xmath144 , the double inequality @xmath189 holds . consequently there is a finite , positive limit @xmath190 . it is clear that this limit coincides with the limit in . suppose that @xmath191 . let @xmath147 0,c_{0}\right [ \,$ ] . then there is @xmath192 such that the double inequality @xmath193holds for all @xmath194 1,k_{0}\right]$ ] . let @xmath195 be a strictly decreasing sequence of real numbers such that all @xmath196 1,k_{0}\right ] $ ] and @xmath197double inequality implies that there is a sequence @xmath198 , such that @xmath199 and @xmath200for all @xmath112 . it follows from that there are @xmath201 and @xmath202 from @xmath203 such that @xmath204for all @xmath112 . the definition of the annulus @xmath205 and imply that @xmath206 \label{5:eq2.15}\]]for all @xmath112 . define a sequence @xmath207 by the rule @xmath208then it follows from , and that @xmath209moreover and imply that @xmath210but@xmath211thus @xmath167 and @xmath171 are mutually stable , @xmath212 and @xmath171 are mutually stable but @xmath167 and @xmath213 are not mutually stable ( w.r.t . the normalizing sequence @xmath214 ) . hence , by lemma [ 5:l2.6 ] , pretangent spaces to @xmath0 at the point @xmath2 are not unique contrary to the assumption . \(ii ) let @xmath2150,1[\,$ ] . since @xmath216for all nonvoid sets @xmath217 we have @xmath218 where the upper and lower limits are taken over the set @xmath219 . hence , it is sufficient to show that @xmath220 in the last limit relation . let @xmath221 and @xmath222 be two sequences of positive real numbers such that @xmath223 for all @xmath112 , and @xmath224 and @xmath225when @xmath226 . without loss of generality we may suppose that@xmath227for all @xmath112 and there is the limit @xmath228first consider the case where @xmath229 . the triangle inequality implies that @xmath230hence,@xmath231from this , and we obtain@xmath232since @xmath233 we see that@xmath234assume now that @xmath235 ( note that equality @xmath236 contradicts the definition of the set @xmath237 . ) there exist sequences @xmath238 , @xmath239 and @xmath240 from @xmath203 which satisfy the following conditions:@xmath241@xmath242@xmath243@xmath244define new sequences @xmath245 and @xmath246 by the rules:@xmath247 relation and definitions of @xmath248 , @xmath249 , @xmath250 imply that@xmath251and@xmath252hence each from the sequences @xmath253 @xmath249 is mutually stable with @xmath254 consequently , by lemma [ 5:l2.6 ] , there are @xmath255 , @xmath256 and @xmath257 . moreover , , and imply that @xmath258 it follows from , and that@xmath259since @xmath260 , there is a finite limit @xmath261 in particulary it follows from the definitions of @xmath262 , @xmath263 that @xmath264moreover , using we obtain@xmath265consequently the equality @xmath220 holds also for the case where @xmath266 . \(iii ) let @xmath267 be a sequences of elements of @xmath268 such that @xmath269 and holds . if in @xmath270 or @xmath271 then it is clear that holds with @xmath272 so it is sufficient to take @xmath273consider the sequence @xmath274 as a normalizing sequence . let @xmath201 and @xmath202 belong to @xmath203 and @xmath275 and @xmath276conditions and imply that there is @xmath277hence , by lemma [ 5:l2.6 ] , there is a finite limit @xmath278moreover , since @xmath279 for all @xmath112 , we have @xmath280 . consequently , using and we obtain @xmath281 suppose that conditions ( i)(iii ) are satisfied simultaneously . we must to prove that @xmath155 is unique for every normalizing sequence @xmath4 . let @xmath221 be an arbitrary normalizing sequence and let @xmath282 and @xmath283 be two elements of @xmath203 such that @xmath284and@xmath285to prove the uniqueness of @xmath155 it is sufficient , by lemma [ 5:l2.6 ] , to show that @xmath167 and @xmath134 are mutually stable w.r.t . @xmath286 if @xmath287 , then , by the triangle inequality , @xmath288and@xmath289consequently , there is a finite limit @xmath290i.e . , @xmath167 and @xmath134 are mutually stable . the case where @xmath291 is similar . hence , without loss of generality we may assume that @xmath292consider first the case where @xmath293this assumption implies that for every @xmath294 there is @xmath295 such that the inclusion @xmath296holds for all natural @xmath297 where@xmath298it follows from that@xmath299if @xmath300 consequently @xmath301letting @xmath302 on the right - hand side of the last inequality and using we see that @xmath303hence@xmath304 it implies that @xmath167 and @xmath134 are mutually stable . it still remains to show that there exists a finite limit @xmath305 if @xmath306for convenience we write@xmath307for all @xmath112 . condition implies that there are @xmath308 and a natural number @xmath309 such that @xmath310for all @xmath311 it is clear that @xmath312where @xmath313 and @xmath314 are the spheres with the common center @xmath1 and radiuses @xmath315 , @xmath316 respectively . consequently we have the following inequalities@xmath317limit relations and imply that @xmath318 hence , using , we obtain @xmath319hence@xmath320i.e . , @xmath167 and @xmath134 are mutually stable . the initial version of theorem [ 5:t2.5 ] was published in @xcite . the following proposition will be helpful in the future . [ p:3.8 ] let @xmath5 be a metric space with a marked point @xmath2 , let @xmath321 and let @xmath322 . if pretangent space @xmath323 is unique for every normalizing sequence @xmath4 , then pretangent space @xmath324 is unique for every @xmath4 . apply lemma [ 5:l2.6 ] . using example [ 4:e2.1 ] as the simplest model we can construct some more interesting from the geometric point of view examples of metric spaces with unique tangent spaces . to this end we recall first some facts related to the structure of pretangent spaces to subspaces of metric spaces . [ 6:d5.1 ] the subspaces @xmath75 and @xmath325 are _ tangent equivalent _ at the point @xmath2 w.r.t . the normalizing sequence @xmath4 if for every @xmath328 and every @xmath329 with finite limits @xmath330 there exist @xmath331 and @xmath332 such that @xmath333 let @xmath334 . for a normalizing sequence @xmath4 we define a family @xmath335_y=[\tilde f]_{y,\tilde r}$ ] by the rule @xmath336_y)\leftrightarrow((\tilde y\in\tilde y)\&(\exists\,\tilde x\in\tilde f:\tilde d_{\tilde r}(\tilde x,\tilde y)=0)).\ ] ] the following two lemmas were proved in @xcite , see also @xcite . [ 6:p5.2 ] let @xmath75 and @xmath325 be subspaces of a metric space @xmath0 and let @xmath337 be a normalizing sequence . suppose that @xmath75 and @xmath325 are tangent equivalent ( w.r.t . @xmath4 ) at a point @xmath326 . then following statements hold for every maximal self - stable ( in @xmath338 ) family @xmath339 . * the family @xmath340_y$ ] is maximal self - stable ( in @xmath341 ) and we have the equalities @xmath342_y]_z=\tilde z_{a,\tilde r}=[\tilde z_{a,\tilde r}]_z.\ ] ] * if @xmath343 and @xmath344 are metric identifications of @xmath339 and , respectively , of @xmath345_y$ ] , then the mapping @xmath346_y\in\omega_{a,\tilde r}^y\ ] ] is an isometry . furthermore if @xmath347 is tangent , then @xmath348 also is tangent . * moreover , if for the normalizing sequence @xmath337 here exists a unique maximal self - stable ( in @xmath338 ) family @xmath349 , then @xmath350_y$ ] is a unique maximal self - stable ( in @xmath351 ) family which contains @xmath49 . let @xmath75 be a subspace of a metric space @xmath5 . for @xmath322 and @xmath352 we denote by @xmath353 the sphere ( in the subspace @xmath75 ) with the center @xmath2 and the radius @xmath354 . similarly for @xmath355 and @xmath352 define @xmath356 write @xmath357 and @xmath358 [ 6:t5.4 ] let @xmath75 and @xmath325 be subspaces of a metric space @xmath5 and let @xmath326 . then @xmath75 and @xmath325 are strongly tangent equivalent at the point @xmath2 if and only if the equality @xmath359 holds . using proposition [ 4:p2.2 ] , lemma [ 6:p5.2 ] and lemma [ 6:t5.4 ] we can easily obtain examples of subspaces of the euclidean space which have unique tangent spaces . the first example will be examined in details . [ e3.1 ] let @xmath360\to e^n,\ n\geq 2 $ ] , be a jordan curve in the euclidean space @xmath361 , i.e. , @xmath362 is continuous and @xmath363 for every two distinct points @xmath364 $ ] . we can write @xmath362 in the coordinate form @xmath365.\ ] ] suppose that all functions @xmath366 , are differentiable at the point @xmath96 and @xmath367 ( we use the one - sided derivatives here . ) we claim that each pretangent space to the subspace @xmath368)\subseteq e^n$ ] at the point @xmath369 is unique and tangent and isometric to @xmath370 for every normalizing sequence @xmath4 . indeed , by lemma [ 6:p5.2 ] and by proposition [ 4:p2.2 ] , it is sufficient to show that @xmath75 is strongly tangent equivalent to the ray @xmath371 at the point @xmath369 . the classical definition of the differentiability of real functions shows that limit relation holds with these @xmath75 and @xmath325 . hence , by lemma [ 6:t5.4 ] , @xmath75 and @xmath325 are strongly tangent equivalent at the point @xmath369 . [ e3.2 ] let @xmath372\to\mathbb r,\ i=1,\dots , n$ ] , be functions such that @xmath373 where @xmath374 is a constant . suppose all @xmath375 have a common finite right derivative @xmath376 at the point @xmath96 , @xmath377 . write @xmath378\},\ ] ] i.e. , @xmath0 is an union of the graphs of the functions @xmath375 . let us consider @xmath0 as a subspace of the euclidean plane @xmath379 . then for every normalizing sequence @xmath4 a pretangent space @xmath380 to the space @xmath0 at the point @xmath2 is unique , tangent and isometric to @xmath381 . [ e3.3 ] let @xmath382 be two functions from the precedent example . put @xmath383\},\ ] ] i.e. , @xmath0 is the set of points of the plane which lie between the graphs of the functions @xmath384 and @xmath385 . then for every nozmalazing sequence @xmath4 each pretangent space @xmath380 to @xmath0 at @xmath386 is unique , tangent and isometric to @xmath381 . [ e3.4 ] let @xmath387 be a positive real number . write @xmath388 i.e. , @xmath0 can be obtained by the rotation of the plane figure @xmath389 around the real axis . then each pretangent space @xmath380 to @xmath0 at the point @xmath390 is unique , tangent and isometric to @xmath381 . in the next our example we will describe the tangent space to the cantor set @xmath391 at the point @xmath96 w.r.t . the normalizing sequence @xmath392 . we recall the definition of the cantor set @xmath391 . let @xmath393 $ ] and expand @xmath394 as @xmath395 the cantor set @xmath391 is the set of all points from @xmath396 $ ] which have expansion using only the digits @xmath96 and @xmath397 . define a set @xmath398 as the smallest subset of @xmath399 which contains the cantor set @xmath391 and satisfies the equality @xmath400 for every integer @xmath401 where @xmath402 it follows from that a real number @xmath354 belongs to @xmath398 if and only if @xmath354 has a base @xmath403 expansion with the digits 0 and 2 only , i.e. , @xmath404 with @xmath405 and @xmath406 . let @xmath138 be a maximal self - stable family for which @xmath412 , see diagram . the uniqueness of @xmath138 and of @xmath136 follows form proposition [ p:3.8 ] . as in the proof of proposition [ 4:p2.2 ] we see that for every @xmath413 there exists a finite limit @xmath414 and that the function @xmath415 with @xmath416 is well - defined and distance - preserving . consequently @xmath410 is isometric to@xmath411 if the following two statements hold : to prove statement ( i ) note that @xmath422 and that for every @xmath352 we have the equality @xmath423\cap3^ic=[0,t]\cap3^jc\ ] ] if @xmath424 . since @xmath391 is closed , equality and imply that @xmath398 also is closed . more over , using we see that @xmath425 for all @xmath426 and all @xmath427 . hence @xmath417 belongs to @xmath398 for every @xmath428 , that is statement ( i ) follows . it still remains to prove , that @xmath410 is tangent . let @xmath433 be an infinite strictly increasing sequence of natural numbers and let @xmath434 . as in the proof of statement ( i ) we see that the equivalence @xmath435 holds for every @xmath436 . by statement ( ii ) we have @xmath437 where @xmath77 is defined in . consequently a function @xmath438 , see , is surjective . hence , by proposition , @xmath136 is tangent . let @xmath0 be the unique nonempty compact subset of @xmath399 for which equality holds , let @xmath442 , and @xmath443 be fixed points and @xmath444 be ratios of similarities @xmath445 , see , and @xmath446 . let us define the sets @xmath447 and @xmath448 by the rules @xmath449 where @xmath391 is the cantor set . then for @xmath442 the pretangent spaces @xmath450 is unique , tangent and isometric to @xmath451 . [ r:3.11 ] certainly , theorem [ t:3.10 ] is , on the whole , a reformulation of proposition [ p:3.9 ] but in this form the result admits generalizations for invariant sets @xmath456 of some other iterated function systems @xmath457 . [ e3.12 ] let @xmath327 be a sequence of strictly decreasing positive real numbers @xmath6 with @xmath458 and such that @xmath459 for all @xmath427 . let @xmath0 be a union of two countable sets @xmath460 and @xmath461 and the one - point set @xmath462 , @xmath463 consider the metric space @xmath464 . it is clear that the sequences @xmath465 and @xmath466 are mutually stable w.r.t . @xmath4 and @xmath467 let @xmath138 be a unique ( by proposition [ p:3.8 ] ) maximal self - stable family such that @xmath468 we claim that the pretangent space @xmath410 corresponding to @xmath138 is two - point . indeed , suppose that @xmath469 and @xmath470 . it is sufficient to prove that the equality @xmath471 holds . to this end , we note that and imply @xmath472 for all sufficiently large @xmath427 because in the opposite case @xmath473 since @xmath474 conditions imply that @xmath475 for sufficiently large @xmath476 , hence follows . now let @xmath477 and @xmath478 be a maximal self - stable family such that @xmath479 where @xmath480 and @xmath481 . since @xmath482 the pretangent space @xmath483 corresponding to @xmath484 contains at least three distinct points . consequently @xmath136 is not tangent . the initial version of this paper was produced during the visit of the first author to the mersin university ( turkey ) in february - april 2008 under the support of the tbitak - fellowships for visiting scientists programme .

we find necessary and sufficient conditions under which an arbitrary metric space @xmath0 has a unique pretangent space at the marked point @xmath1 . + + * key words : * metric spaces ; tangent spaces to metric spaces ; uniqueness of tangent metric spaces ; tangent space to the cantor set . + + * 2000 mathematic subject classification : * 54e35

a challenging problem in quantitative biology is to successfully model the evolutionary response of organisms to various environmental pressures . aside from its intrinsic interest , the development of models which can predict the time evolution of a population s genotype could prove useful in understanding a number of important phenomena , such as antibiotic drug resistance , cancer , viral replication dynamics , and immune response . perhaps the simplest formalism for modeling , at least phenomenologically , the evolutionary dynamics of replicating organisms is known as the quasispecies model @xcite . this model was introduced by manfred eigen in 1971 as a way to describe the _ in vitro _ evolution of single - stranded rna genomes @xcite . in the simplest formulation of the model , we consider a population of asexually replicating genomes , whose only source of variability is induced by point mutations during replication . we assume that each genome , denoted by @xmath5 , may be written as @xmath6 , where each `` base '' @xmath7 is drawn from an alphabet of size @xmath8 . with each genome is associated a first - order growth rate constant @xmath9 , which we assume to be genome - dependent , since different genomes are expected to be differently suited to the given environment . the set of all growth rate constants is termed the _ fitness landscape _ , which will generally be time - dependent . replication and mutation give rise to mutational flow between the genomes . if we let @xmath10 denote the number of organisms with genome @xmath5 , then , @xmath11 where @xmath12 denotes the first - order mutation rate constant from @xmath13 to @xmath5 . if @xmath14 denotes the probability that , after replication , @xmath13 produces the daughter genome @xmath5 , then clearly @xmath15 . to compute @xmath14 , we assume a per base replication error probability @xmath16 for genome @xmath5 ( different genomes may have different replication error probabilities , since some genomes may code for various repair mechanisms which other genomes do not ) . it is then readily shown that @xcite , @xmath17 where @xmath18 denotes the hamming distance between @xmath5 and @xmath13 . in order to model the relative competition between various genomes , it proves convenient to reexpress the dynamics in terms of population fractions . defining @xmath19 , and @xmath20 , we obtain the system of equations , @xmath21 where @xmath22 , and is therefore simply the mean fitness of the population . the above system of equations is physically realizable in a chemostat , which continuously siphons off organisms to maintain a constant population size @xcite . this ensures that growth is not resource limited , so the assumption of simple exponential growth is a good one . it should be pointed out , however , that it is possible to introduce a death term which places a cap on the population size , without changing the form of the quasispecies equations . if we introduce a second - order crowding term ( logistic growth ) , so that , @xmath23 then if @xmath24 is genome - independent , it is readily shown that when converting to the @xmath25 the quasispecies equations are unchanged . the quasispecies equations may be written in vector form as , @xmath26 where @xmath27 is the vector of population fractions , @xmath28 is the matrix of first - order mutation rate constants , and @xmath29 is the vector of first - order growth rate constants . for a static fitness landscape , eigen proved that @xmath30 evolves to the equilibrium distribution given by the eigenvector corresponding to the largest eigenvalue of @xmath31 @xcite . a considerable amount of research on quasispecies theory has focused on the simplest possible fitness landscape , known as the _ single fitness peak _ ( sfp ) landscape @xcite . in the sfp model , there exists a single , `` master '' sequence @xmath32 for which @xmath33 , while for all other sequences we have @xmath34 . the sfp model assumes a genome - independent mutation rate , so that @xmath35 for all @xmath5 . the sfp landscape is analytically solvable in the limit of infinite sequence length . the equilibrium behavior of the model exhibits two distinct regimes : a localized regime , where the genome population clusters about the master sequence ( giving rise to the term `` quasispecies '' ) , and a delocalized regime , where the genome population is distributed essentially uniformly over the entire sequence space . the transition between the two regimes is known as the _ error catastrophe _ , and can be shown to occur when @xmath36 , the probability of correctly replicating a genome , drops below @xmath37 @xcite . the error catastrophe is generally regarded as the central result of quasispecies theory , and it has been experimentally verified in both viruses @xcite and bacteria @xcite . indeed , the error catastrophe has been shown to be the basis for a number of anti - viral therapies @xcite . the structure of the quasispecies equations naturally lends itself to application to more complex systems than rna molecules . indeed , the model has been used to successfully model certain aspects of the immune response to viral infection @xcite . however , in their original form , the quasispecies equations fail to capture a number of important aspects of the evolutionary dynamics of real organisms . for example , it is implicitly assumed that each genome replicates _ conservatively _ , meaning that the original genome is preserved by the replication process . correct modeling of dna - based life must take into account the fact that dna replication is _ semiconservative _ @xcite . furthermore , the assumption of a genome - independent replication error probability is also too simple , since cells often have various repair mechanisms which may become inactivated due to mutations @xcite . in addition , eigen s model neglects the effects of recombination , transposition , insertions , deletions , and gene duplication , to name a few additional sources of variability . thus , a considerable amount of work remains to be done before a quantitative theory of evolutionary response is developed . nevertheless , some progress has been made . for example , semiconservative replication was recently incorporated into the quasispecies model @xcite . a simple model incorporating genetic repair was developed in @xcite . diploidy has been studied in @xcite , and finite size effects in @xcite . one area in which more realistic models need to be developed is in the nature of the fitness landscape . as mentioned previously , the most common landscape studied thus far has been the single fitness peak . however , genomes generally contain numerous genes ( even the simplest of bacteria , the mycoplasmas , have several hundred genes @xcite ) , which work in concert to confer viability to the organism . therefore , in this paper , we consider the behavior of the model for an arbitrary gene network . we assume conservative replication and a genome - independent error rate for simplicity , though we hypothesize at the end of the paper how our results change for the case of semiconservative replication . this paper is organized as follows : in the following section , we introduce our generalized @xmath0-gene model defining the `` gene network . '' we first give the quasispecies equations in terms of the population fractions of each of the various genomes . we proceed to the infinite sequence length equations , and then obtain a reduced system of equations which dictates the equilibrium solution of our model . we solve the model in section iii . for the sake of completeness , we include a simple example to illustrate how our solution method may be applied to specific systems . we go on in section iv to discuss the results and implications of our model , such as the relation to c.o . `` survival of the flattest '' @xcite , and also what our model says about the response of gene networks to mutagens . finally , we conclude in section v with a summary of our results and future research plans . consider a population of conservatively replicating , asexual organisms , whose genomes consist of @xmath0 genes . each genome @xmath5 may then be written as @xmath38 . let us assume , for simplicity , a `` single fitness peak '' landscape for each gene . that is , for each gene @xmath3 there is a `` master '' sequence @xmath39 for which the gene is functional , while for all @xmath40 the gene is nonfunctional . we assume that the fitness associated with a given genome @xmath5 is dictated by which genes in the genome are functional , and which are not . we let @xmath41 denote the fitness of organisms with genome @xmath5 such that @xmath42 for @xmath43 , while @xmath40 for @xmath44 ( we adopt the convention that @xmath45 when @xmath46 ) . the choice of the landscape @xmath47 is arbitrary , so that the activity of the various genes in the genome are generally correlated . thus , the @xmath0 genes may be regarded as defining a gene network . we assume that the fitnesses are all strictly positive . without loss of generality ( i.e. , by an appropriate rescaling of the time ) , we may assume that @xmath48 . the simplest quasispecies equations for this @xmath0-gene model are obtained by assuming a genome - independent per base replication error probability @xmath49 . we assume that gene @xmath3 has a sequence length @xmath50 , and we define @xmath51 . then @xmath52 , where , @xmath53 putting everything together , we obtain the system of equations , @xmath54 define the _ hamming class _ @xmath55 . also , define @xmath56 . by the symmetry of the landscape , we may assume that @xmath25 depends only on the @xmath57 corresponding to @xmath5 , and hence we may look at the total population fraction in @xmath58 and study its dynamics . the conversion of the quasispecies equations in terms of @xmath25 to @xmath59 is accomplished by a generalization of the method given in @xcite . the result is , @xmath60 we now let the @xmath61 in such a way that the @xmath62 and @xmath63 remain fixed . we assume that the @xmath64 are all strictly positive ( allowing an @xmath64 to be @xmath65 leads to certain difficulties which we choose not to address in this paper ) . because the probability of correctly replicating a genome is simply @xmath66 , fixing @xmath67 is equivalent to fixing the genome replication fidelity in the limit of infinite sequence length . in this limit , it is possible to show that , for each gene @xmath3 , the only terms in eq . ( 8) which survive the limiting process are the @xmath68 terms @xcite . this is equivalent to the statement that , in the limit of infinite sequence length , backmutations may be neglected . we also obtain that , @xmath69 and @xmath70 the final result is , @xmath71 it should be noted that the neglect of backmutations is only valid when one can group population fractions into hamming classes . in our case , by the symmetry of the fitness landscape , the equilibrium solution only depends on the hamming class , and hence , to find the equilibria , it is perfectly valid to `` pre - symmetrize '' the population distribution and study the resulting dynamics . thus , when studying dynamics , it is generally not valid to neglect backmutations . for example , consider a single fitness peak landscape , and suppose that a population of organisms is at its equilibrium , clustered about the fitness peak . if the organisms are then mutated , so that they are shifted away from the fitness peak , then eventually they will backmutate and reequilibrate on the fitness peak ( this situation has been observed with prokaryotes @xcite ) . if we imagine that the mutation shifts the organism from the master genome @xmath32 to some other genome @xmath72 , then it is clear that the landscape is not symmetric about @xmath13 , and furthermore that the population distribution is not symmetric about @xmath32 . thus , eq . ( 11 ) does not apply . to correctly model the reequilibration dynamics , it is necessary to consider the finite sequence length equations , and explicitly incorporate backmutations . because of the neglect of backmutations , eq . ( 11 ) may in principle be solved recursively to obtain the equilibrium distribution of the @xmath59 at any @xmath67 , assuming we know the equilibrium mean fitness , denoted @xmath73 . the problem , of course , is that @xmath73 needs to be computed . this may be done as follows : given any collection @xmath74 of indices , define @xmath75 via , @xmath76 where @xmath77 , @xmath78 , and so forth . thus , @xmath75 is simply the total fraction of the population in which the genes of indices @xmath79 are faulty , while the remaining genes are given by their corresponding master sequences . the dynamics of the @xmath75 is derived in appendix a. the result is given by , @xmath80 we can provide an intuitive explanation for this expression : because backmutations may be neglected in the limit of infinite sequence length , it follows that , once a gene is disabled , it remains disabled . therefore , given a set of indices @xmath81 , mutational flow can only occur from @xmath82 to @xmath83 for which @xmath84 ( in this paper , if @xmath85 , then @xmath86 is a proper subset of @xmath87 . if @xmath88 , then either @xmath89 is a proper subset of @xmath87 or @xmath90 ) . similarly , @xmath82 can only receive mutational contributions from @xmath91 for which @xmath92 . for such a @xmath93 , the probability of mutation to @xmath81 may be computed as follows : because the genes corresponding to the indices @xmath94 remain faulty , the neglect of backmutations means that it does not matter whether these genes are replicated correctly or not . all genes with indices in @xmath95 must remain equal to the corresponding master sequences after mutation . the probability that gene @xmath3 replicates correctly is given by @xmath96 , so the probability that all genes with indices in @xmath95 replicate correctly is @xmath97 . the genes which must be replicated incorrectly are those with indices in @xmath98 . since each such gene replicates incorrectly with probability @xmath99 , it follows that the probability of replicating all genes in @xmath100 incorrectly is @xmath101 . putting everything together , we obtain a mutational flow from @xmath102 to @xmath103 of @xmath104 . summing over all possible @xmath105 gives us the expression in eq . ( 13 ) . note that @xmath106 , so we need to solve eq . ( 13 ) in order to obtain the equilibrium distribution of the model . in this section , we proceed to solve the reduced system of equations given by eq . since this provides us with @xmath107 and @xmath108 , it follows that we can recursively solve for the equilibrium values of all @xmath59 . in vector notation , ( 13 ) may be expressed in the form , @xmath109 where @xmath110 is the vector of all @xmath111 , @xmath112 is the vector of all @xmath113 , and @xmath114 is the matrix of mutation rate constants . because of the neglect of backmutations in the limit of infinite sequence length , different regions of the genome space become mutationally decoupled , so that the largest eigenvalue of the mutation matrix @xmath114 will in general be degenerate . thus , the equilibrium of the reduced system of equations is not unique . however , for any initial condition , the system will evolve to an equilibrium , though of course different initial conditions will yield different equilibrium results . in this subsection , we define a variety of constructs which we will need to characterize the equilibrium behavior of our model . we begin with the definition of a _ node _ : we define a _ level n node _ to refer to any collection of `` knocked out '' genes with indices @xmath74 . the reason for this terminology is simple . we may imagine the set of all nodes to be connected via mutations . because of the neglect of backmutations , it follows that a node @xmath79 is accessible from a node @xmath115 via mutations if and only if @xmath116 . the result is that we can generate a directed graph of mutational flows between nodes , an example of which is illustrated in figure 1 . given some node @xmath117 , define @xmath118 . therefore , @xmath119 may be regarded as the subgraph of all nodes which are mutationally accessible from @xmath120 . an example of such a subgraph is illustrated in figure 2 . let @xmath121 denote any collection of nodes . then we may define @xmath122 . furthermore , define @xmath123 . thus , @xmath124 is the set of all nodes in @xmath121 such that no node in @xmath121 is contained within the mutational subgraph of any other node in @xmath124 . figure 3 gives an example showing the construction of @xmath124 from @xmath121 . given some node @xmath79 , define @xmath125 . we then define @xmath126 . finally , given some @xmath67 , define @xmath127 . with these definitions in hand , we are now ready to obtain the structure of the equilibrium solution at a given @xmath67 . we claim that @xmath128 . we prove this in two steps . first of all , we claim that @xmath129 for some node @xmath120 . clearly , because @xmath130 , it follows that at least one of the @xmath131 at equilibrium . let @xmath132 be a node of minimal @xmath133 such that @xmath134 . then it should be clear that , at equilibrium , we have , @xmath135 which , since @xmath134 , may be solved to give @xmath136 . so now suppose that @xmath137 . then @xmath138 . such an equilibrium can never be observed because it is unstable . to see this , let @xmath139 denote a node for which @xmath140 . then from eq . ( 13 ) we have , at equilibrium , that , @xmath141 and so @xmath142 . clearly , however , any perturbation on @xmath143 will push @xmath144 away from its equilibrium value . this equilibrium is therefore unstable , and hence , unobservable . note that since @xmath128 , it follows that the mean equilibrium fitness is a continuous function of @xmath67 . to find the equilibrium solution of the reduced system of equations , we first need to determine which @xmath145 at equilibrium . to this end , we begin with the claim that , for @xmath146 , @xmath147 unless @xmath148 . for suppose there exists @xmath149 such that @xmath150 at equilibrium . then out of the set of all nodes which satisfy the above two properties , we may choose @xmath120 to be of minimal level . we claim that , for any @xmath151 , we have that @xmath152 , for otherwise it is clear that @xmath153 . therefore , by the minimality of the level of @xmath120 , it follows that @xmath154 whenever @xmath155 is a proper subset of @xmath120 . but then the equilibrium equation for @xmath156 gives @xmath157 , and so @xmath158 . therefore , @xmath159 . however , by assumption , @xmath160 , which means that @xmath161 contains nodes in @xmath162 which are distinct from @xmath120 . denote one of these nodes by @xmath163 . then at equilibrium we have , from eq . ( 13 ) , that , @xmath164 which is clearly a contradiction . this establishes our claim . we now argue that our equilibrium solution may be found if we know @xmath165 for @xmath166 . we claim that for any @xmath167 , we may write , @xmath168 where the @xmath169 , and for @xmath146 a given @xmath170 is strictly positive if and only if @xmath171 . the above expression then holds for all @xmath120 , since we simply take @xmath172 for @xmath173 . we can prove the above formula via induction on the level of the nodes in @xmath174 . in doing so , we will essentially develop an algorithm for constructing the @xmath175 . so , let us start with @xmath176 , the minimal level nodes @xmath174 . then clearly @xmath166 , so that @xmath177 , hence the formula is correct for @xmath176 . so now suppose that , for some @xmath178 , the formula is correct for all @xmath179 such that @xmath180 . then for a level @xmath181 node in @xmath182 , denoted by @xmath183 , we have , at equilibrium , that , @xmath184 now , if @xmath185 , then @xmath186 . otherwise , @xmath187 , so the equilibrium equation may be solved to give , @xmath188 note that @xmath189 . furthermore , if @xmath190 , then no proper subset of @xmath191 is in @xmath192 . therefore , @xmath193 , so @xmath194 . conversely , if @xmath195 , then since @xmath196 , it follows that @xmath197 . therefore , the sum in eq . ( 20 ) is nonempty , hence , since the @xmath170 appearing in the sum are all strictly positive , it follows that @xmath198 . this implies that @xmath199 is strictly positive if and only if @xmath195 , which completes the induction step , and proves the claim . for each @xmath200 , we can define @xmath201 , and then define @xmath202 and @xmath203 . if , for each @xmath155 we also define @xmath204 , that is , the vector of all @xmath205 , and if we define @xmath206 , then we obtain , @xmath207 where @xmath208 . note that the @xmath209 form a linearly independent set of vectors . therefore , if @xmath210 contains more than one node , then the equilibrium solution of the reduced system of equations is not unique , but rather is defined by the parallelipiped @xmath211 . as mentioned earlier , the degeneracy in the equilibrium behavior follows from the neglect of backmutations in the limit of infinite sequence length . the various nodes in @xmath212 become mutationally decoupled in this limit , which can cause the largest eigenvalue of the mutation matrix @xmath114 to be degenerate . however , for _ finite _ sequence lengths , the quasispecies dynamics will always converge to a unique solution . in particular , if we start with the initial condition @xmath213 , then for finite sequence lengths we will converge to the unique equilibrium solution . because all nodes are mutationally connected in the infinite sequence length limit with this initial condition , we make the assumption that the way to find the infinite sequence length equilibrium which is the limit of the finite sequence length equilibria is to find the infinite sequence length equilibrium starting from the initial condition @xmath214 . this allows us to break the eigenstate degeneracy in a canonical manner . in the appendices , we describe a fixed - point iteration approach for finding the equilibrium solution of the model . within this algorithm , we also use the initial condition @xmath214 as the analogous approach to the one above for finding the infinite sequence length equilibrium which is the limit of the finite sequence length equilibria . finally , the treatment thus far has been finding the equilibrium solution of the reduced system of equations for @xmath146 . the equilibrium solution for @xmath215 is obtained by taking the limit of the @xmath146 solutions , so that @xmath216 . from the previous development it is clear that the nodes in @xmath217 may be regarded as `` source '' nodes which dictate the solution . to understand how the solution changes with @xmath67 , we therefore need to determine how @xmath217 depends on @xmath67 . we claim the following : that there exist a finite number of @xmath218 , which we denote by @xmath219 , where @xmath220 , for which @xmath221 contains distinct elements . in any interval @xmath222 , @xmath223 is constant , and may therefore be denoted by @xmath224 . the @xmath225 are all disjoint , and @xmath226 . we begin proving this claim by introducing one more definition . let @xmath227 denote the set of all sets of nodes , such that a collection of nodes @xmath121 is a member of @xmath228 if and only if @xmath229 contains distinct elements . note that since there are @xmath230 nodes , there are @xmath231 sets of nodes , hence @xmath227 consists of a finite number of elements . given some @xmath232 , we claim that @xmath233 for at most one @xmath234 . to show this , suppose that there exist @xmath235 for which @xmath236 . choose any two nodes @xmath79 , @xmath93 in @xmath237 , and note that @xmath238 , and similarly for @xmath239 . however , @xmath240 and @xmath241 implies that @xmath242 , so that @xmath243 and hence @xmath244 . therefore , @xmath245 and @xmath246 , so @xmath247 does not contain distinct elements . because this contradicts our assumption about @xmath237 , it follows that @xmath248 for at most one @xmath67 . so , since @xmath227 contains a finite number of elements , it follows that there are a finite number of @xmath67 for which @xmath249 satisfies the property that @xmath250 contains distinct elements . we can denote these @xmath67 by @xmath251 , where we assume that @xmath252 . note that if a collection of nodes @xmath121 has the property that @xmath253 , then @xmath121 must be a collection in @xmath227 . this is easy to see : @xmath121 contains some @xmath79 for which there exists a distinct @xmath254 where @xmath255 . therefore @xmath256 , which proves our contention . we now prove that @xmath223 is some constant , which we denote by @xmath224 , over @xmath222 . given some @xmath257 , let @xmath258 ( @xmath259 stands for `` supremum '' , which is the least upper bound of a set of real numbers . if @xmath260 is a set of real numbers with an upper bound , then @xmath261 exists , and satisfies the following properties : ( 1 ) @xmath262 is an upper bound for @xmath263 . ( 2 ) if @xmath264 is any upper bound of @xmath260 , then @xmath265 . ( 2 ) if @xmath266 , then there exists at least one element of @xmath260 which exceeds @xmath264 . ) . clearly , @xmath267 . we claim that @xmath268 . to show this , note first of all that @xmath269 for all @xmath270 , and that for any @xmath271 , there exists @xmath272 such that @xmath273 . for given any @xmath274 , we have , by definition of @xmath259 , that there exists some @xmath275 such that @xmath269 for all @xmath276 . in particular , this implies that @xmath277 . furthermore , if there exists @xmath271 for which @xmath269 for all @xmath272 , then @xmath278 for all @xmath279 , contradicting the definition of @xmath280 . now , suppose @xmath281 . then we can write @xmath282 and @xmath283 for all @xmath284 . then since @xmath285 , it follows by continuity that @xmath286 for @xmath287 in some neighborhood @xmath288 . but this implies that @xmath289 over @xmath290 . since @xmath291 over @xmath292 , we obtain that @xmath269 over @xmath293 , contradicting the definition of @xmath280 . we have just shown that @xmath294 . since @xmath295 over @xmath296 , we must have that @xmath268 . using a similar argument with @xmath297 , we can show that @xmath298 over @xmath299 , and so @xmath223 is constant on @xmath296 ( @xmath297 stands for `` infimum '' , and is defined as the greatest lower bound of a set of real numbers . it satisfies properties analogous to those of @xmath259 ) . suppose for two @xmath300 with @xmath301 , we have @xmath224 and @xmath302 are not disjoint . then they share at least one node , and so , by the nature of the two sets , we must have that @xmath303 . define @xmath304 to be @xmath305 for any node in @xmath224 , @xmath302 , and @xmath306 to be @xmath307 . now , @xmath308 contains some node @xmath309 such that @xmath310 for @xmath67 in @xmath311 . but if for @xmath312 we have that @xmath313 and @xmath314 , then @xmath315 and @xmath316 . since @xmath317 is monotone decreasing or increasing , it follows that @xmath318 on @xmath319 , or equivalently , @xmath320 . therefore , @xmath321 . the @xmath224 are thus all disjoint , as claimed . finally , since @xmath322 is continuous , we have that @xmath323 . if @xmath324 , then this gives @xmath325 . similarly , considering @xmath326 gives that @xmath327 for @xmath328 . therefore , @xmath329 , so @xmath226 , as claimed . the various @xmath224 may therefore be regarded as defining different `` phases '' in the equilibrium behavior of the model . physically , each `` phase '' is defined by a set of `` source nodes , '' which dictate which genes in the genome are knocked out , and which are not . the transition from @xmath224 to @xmath330 corresponds to certain genes in the genome becoming knocked out , and perhaps other genes becoming viable again . this transition can happen more than once , and so we refer to the series of @xmath331 transitions as an `` error cascade . '' because @xmath332 , for sufficiently large @xmath67 , @xmath333 for any @xmath334 . therefore , for sufficiently large @xmath67 , @xmath335 . since @xmath223 is constant on @xmath336 , it follows that @xmath337 on @xmath336 . thus , the final transition from @xmath338 to @xmath339 corresponds to delocalization over the entire genome space , which is simply the error catastrophe . once we have determined @xmath73 , we can in principle obtain the population fractions @xmath59 in the various hamming classes . the problem is that , if @xmath340 , then for any _ finite _ values of @xmath341 , we get that @xmath342 . to show this , suppose we can find @xmath343 such that @xmath344 at equilibrium . of the @xmath343 for which @xmath345 , choose a set of indices @xmath346 such that @xmath347 is as small as possible . note that if @xmath348 , with @xmath349 , then @xmath342 . now , let the nonzero @xmath350 be denoted by @xmath351 . then @xmath352 , and we have , from eq . ( 11 ) , that , at equilibrium , @xmath353 which gives @xmath354 . but , @xmath355 . therefore , @xmath356 , and so @xmath357 , hence @xmath46 . but then @xmath358 . this proves our claim . if @xmath359 , then the above claim does not present us with any problem . we can simply recursively solve eq . ( 11 ) at equilibrium for all the @xmath59 . but once any delocalization occurs , it is impossible to solve for the equilibrium distribution in terms of the hamming classes . however , we can recursively obtain the distribution of another class of population fractions , as follows : given some collection of indices @xmath79 , another collection of indices @xmath360 , and a collection of hamming distances @xmath343 , we define @xmath361 and @xmath362 as , @xmath363 it is possible to show that , @xmath364 and hence that , @xmath365 we may then derive the expression for @xmath366 . since the derivation uses techniques similar to those used in appendices a and b , we simply state the final result , which is , @xmath367 where @xmath368 , where @xmath369 are the indices of the nonzero hamming distances in @xmath370 . we claim that , at equilibrium , @xmath371 only if @xmath171 for some @xmath200 for which @xmath372 . for if @xmath373 , let @xmath374 be a node of minimal level for which there exists @xmath375 such that @xmath376 . note then that for any proper subset @xmath377 , we must have that @xmath378 . this implies that , at equilibrium , @xmath379 among all @xmath370 for which @xmath380 , there exists an @xmath381 such that @xmath382 is minimal . then we obtain , @xmath383 which gives @xmath384 . now , let @xmath385 denote the indices of the nonzero hamming distances in @xmath386 . then @xmath387 . but since @xmath388 , we get @xmath389 , so @xmath390 . therefore @xmath391 , so since @xmath372 , we have @xmath392 . the @xmath393 may be obtained recursively from eq . ( 27 ) , starting with the values of @xmath394 for @xmath395 . the idea is that , starting with the values of @xmath394 for @xmath396 , we may compute @xmath397 recursively . we then proceed down the levels , computing the @xmath398 using the values of @xmath399 and @xmath400 for @xmath401 . note then that instead of computing the @xmath402 , which will be @xmath65 as soon as any delocalization occurs , we first sum over a set of gene indices containing the delocalized genes as a subset , and only compute the population distribution for finite hamming distances of the localized genes . in this subsection , we compute various localization lengths associated with the population distribution . specifically , given a node @xmath79 , and some @xmath403 , we define two localization lengths , @xmath404 and @xmath405 , as follows : @xmath406 note that , @xmath407 and so , in analogy with @xmath408 and @xmath75 , we have that , @xmath409 we also define the localization length @xmath410 by , @xmath411 note that @xmath412 , and so is finite if and only if all the @xmath413 are finite . we can compute @xmath414 at equilibrium by finding the time derivative and setting it to @xmath65 . in appendix b we show that , @xmath415 therefore , at equilibrium , we get , @xmath416 we can characterize the behavior of the @xmath417 . first of all , we claim that @xmath418 if and only if @xmath419 . secondly , we claim that @xmath420 if and only if @xmath421 with @xmath422 for some @xmath423 . to show this , note first of all that , from physical considerations , @xmath418 if @xmath419 . if @xmath424 , then @xmath425 , and so , since @xmath426 , it follows that @xmath427 . this establishes the first part of our claim . so now suppose that @xmath428 , with @xmath429 for some @xmath430 . then @xmath431 , and so , @xmath432 which of course implies that @xmath433 . to prove the converse , let us suppose that @xmath434 . let us choose @xmath435 to be the minimal level subset for which @xmath436 . then if @xmath437 , it is clear from the expression for @xmath438 that @xmath439 for some @xmath440 , with @xmath441 . but this contradicts the minimality of @xmath442 , hence @xmath443 , so since @xmath429 , it follows that @xmath421 . this proves the converse , which establishes the second part of our claim . we now illustrate the theory developed above using a simple two - gene `` network '' as an example . we assume a genome containing two identical genes , so that @xmath444 , and we choose the following growth parameters : @xmath445 , @xmath446 , and @xmath447 . with these parameters , the system exhibits two localization to delocalization transitions . first , for @xmath448 we have @xmath359 . for @xmath449 we have @xmath450 . the error catastrophe occurs at @xmath451 . we determined the equilibrium behavior of the model by solving the finite sequence length equations for @xmath452 and @xmath453 . the details may be found in appendix c. figure 4 shows a plot of @xmath107 from the simulation results and from our theory . figure 5 shows plots of @xmath454 , @xmath455 , @xmath456 , and @xmath457 from the simulation results and from theory . the first point to note about the solution of the quasispecies equations for a gene network is that , unlike the single gene model , which exhibits a single `` error catastrophe , '' the multiple gene model exhibits a series of localization to delocalization transitions which we term an `` error cascade . '' the reason for this is that as the mutation rate is increased , the selective advantage for maintaining functional copies of certain genes in the genome is no longer sufficiently strong to localize the population distribution about the corresponding master sequences , and delocalization occurs in the corresponding sequence spaces . the more a given gene or set of genes contributes to the fitness of an organism , the larger @xmath67 will have to be to induce delocalization in the corresponding sequence spaces . eventually , by making @xmath67 sufficiently large , the selective advantage for maintaining any functional genes in the genome will disappear , and the result is complete delocalization over all sequence spaces , corresponding to the error catastrophe . the prediction of an error cascade suggests an approach for determining the selective advantage of maintaining certain genes in a genome . currently , the standard method for determining whether a gene is `` essential '' is by knocking it out , and then seeing if the organism survives . by knocking out each of the genes , one can construct a `` deletion set '' for a given organism , consisting of the minimal set of genes necessary for the organism to survive @xcite . while knowledge of the deletion set of an organism is important , it does not explain why the organism should maintain functional copies of other , `` nonessential '' genes . one possibility is that these `` nonessential '' do confer a fitness advantage to the organism , however , the time scale on which organisms are observed to grow during knockout experiments is simply too short to resolve these fitness differences . thus , an alternative approach to the deletion set method is to have organisms grow at various mutagen concentrations . by determining which genes get knocked out at the corresponding mutation rates , it is possible to determine the relative importance of various genes to the fitness of an organism . such an experiment is likely to be difficult to perform . nevertheless , if successful , it would provide a considerably more powerful approach than the deletion set method for determining fitness advantages of various genes . the results in this paper also shed light on a phenomenon which c.o . wilke termed the `` survival of the flattest '' @xcite . briefly , what wilke ( and others ) showed was that at low mutation rates , a population will localize in a region of sequence space which has high fitness . at higher mutation rates , a population will relocalize in a region of sequence space which may not have maximal fitness , but is mutationally robust @xcite . the error cascade is exactly a relocalization from a region of high fitness but low mutational support to a region of lower fitness but higher mutational support . the reason for this is that the fitness landscape becomes progressively flatter as more and more genes are knocked out , because the more genes are knocked out , the smaller the fraction of the genome which is involved in determining the fitness of the organism . this implies that an error cascade is necessary for the `` survival of the flattest '' principle to hold . robustness in this sense is therefore conferred by modularity in the genome . that is , robustness does not arise because an individual gene may remain functional after several point - mutations , but rather arises from the fact that the organism can remain viable even if entire regions ( e.g. `` genes '' ) of the genome are knocked out ( it should be noted that the idea that mutational robustness is conferred by modularity in the genome has been discussed before @xcite ) . to see this point more clearly , one can consider a `` robust '' landscape in which the genome consists of a single gene . however , unlike the single - fitness peak landscape , the organism is viable out to some hamming class @xmath458 . therefore , if @xmath459 , then @xmath34 if @xmath460 , otherwise @xmath461 , where @xmath462 . using techniques similar to the ones used in this paper ( neglect of backmutations and stability criterion for equilibria ) , it is possible to show that the equilibrium mean fitness is exactly @xmath463 , unchanged from the single - fitness peak results . thus , in contrast to robustness studies which consider finite sequence lengths and do not have a well - defined viability cutoff @xcite , in the limit of infinite sequence length there is no selective advantage in having a genome which can sustain a finite number of point mutations and remain viable . this paper developed an extension of the quasispecies model for arbitrary gene networks . we considered the case of conservative replication and a genome - independent replication error probability . we showed that , instead of a single error catastrophe , the model exhibits a series of localization to delocalization transitions , termed an `` error cascade . '' while the numerical example we used in this paper was relatively simple ( in order to clearly illustrate the theory developed ) , it is possible to have nontrivial delocalization behavior , depending on the choice of the landscape . for example , it is possible that certain genes which are knocked out in one phase can become reactivated again in the following phase . that is , instead of a delocalization , one can have a _ re - localization _ to source nodes not contained in the mutational subgraphs of the source nodes in the previous phase . this implies that the @xmath156 can exhibit discontinuous behavior . the types of equilibrium behaviors possible is something which will be explored in future research . future research also will involve incorporating more details to the multiple - gene model introduced in this paper . for example , one extension is to move away from the `` single - fitness peak '' assumption for each gene . another natural extension is to study the equilibrium behavior of the multiple - gene quasispecies equations for the case of semiconservative replication . while this is a subject for future work , we hypothesize that many of the semiconservative results would be essentially unchanged from the conservative ones . thus , we claim that at equilibrium , we would still have that @xmath128 , only this time @xmath464 is computed by replacing @xmath465 with @xmath466 . we also claim that we would still have that @xmath210 define the `` source '' nodes of the equilibrium solution . indeed , we hypothesize that , for semiconservative replication , eq . ( 13 ) becomes , @xmath467 finally , another subject for future work is the incorporation of repair into our network model . in @xcite it was assumed that only one gene controlled repair . by assuming that several genes control repair , then , in analogy with fitness , we hypothesize that instead of a single `` repair catastrophe '' @xcite , we obtain a series of localization to delocalization transitions over the repair gene sequence spaces , a `` repair cascade . '' this research was supported by the national institutes of health . the authors would like to thank eric j. deeds for helpful discussions . in this appendix , we derive eq . ( 13 ) from eq . ( 11 ) . to this end , define , @xmath468 we then have that , @xmath469 we now claim that , @xmath470 this can be proved by induction . for @xmath46 this statement is clearly true , since @xmath471 . suppose then , that for some @xmath472 , the statement is true for all @xmath473 . then we have , @xmath474 and so , @xmath475 now , for each set @xmath476 appearing in the sum , a given subset @xmath369 occurs only once . the @xmath442-element sets @xmath476 which contain @xmath369 as a subset must be of the form @xmath477 , where @xmath478 . therefore , there are @xmath479 distinct @xmath442-element sets which contain @xmath480 . rearranging the sum , we obtain , @xmath481 this completes the induction step , and proves the claim . we are almost ready to derive the expression for @xmath482 . before doing so , we state the following identity , which we will need in our calculation : @xmath483 we now have , @xmath484 which is exactly eq . in this section we derive the expression for @xmath485 . we have , @xmath486 we therefore have that , @xmath487 \nonumber \\ & = & \sum_{l = 0}^{n } \sum_{\{j_1 ' , \dots , j_l'\ } \subseteq \{i_1 , \dots , i_n\ } } ( -1)^{n - l } e^{-(1 - \alpha_{j_1 ' } - \dots - \alpha_{j_l } ' - \alpha_i ) \mu } \times \nonumber \\ & & ( \kappa_{\{j_1 ' , \dots , j_l ' , i\ } } \tilde{\langle l_i\rangle}_{\{j_1 ' , \dots , j_l'\ } } + \alpha_i \mu \kappa_{\{j_1 ' , \dots , j_l'\ } } \tilde{z}_{\{j_1 ' , \dots , j_l'\ } } + \alpha_i \mu \kappa_{\{j_1 ' , \dots , j_l ' , i\ } } \tilde{z}_{\{j_1 ' , \dots , j_l ' , i\ } } ) \times \nonumber \\ & & \sum_{k - l = 0}^{n - l } ( -1)^{k - l } \sum_{\{j_1 , \dots , j_{k - l}\ } \subseteq \{i_1 , \dots , i_n\}/\{j_1 ' , \dots , j_l'\ } } e^{\alpha_{j_1 } \mu } \cdot \dots \cdot e^{\alpha_{j_{k - l } } \mu } \nonumber \\ & & - \bar{\kappa}(t ) \tilde{\langle l_i\rangle}_{\{i_1 , \dots , i_n\ } } \nonumber \\ & = & e^{-(1 - \alpha_{i_1 } - \dots - \alpha_{i_n } - \alpha_i)\mu } \sum_{k = 0}^{n } \sum_{\{j_1 , \dots , j_k\ } \subseteq \{i_1 , \dots , i_n\ } } \times \nonumber \\ & & ( \kappa_{\{j_1 , \dots , j_k , i\ } } \tilde{\langle l_i\rangle}_{\{j_1 , \dots , j_k\ } } + \alpha_i \mu \kappa_{\{j_1 , \dots , j_k\ } } \tilde{z}_{\{j_1 , \dots , j_k\ } } + \alpha_i \mu \kappa_{\{j_1 , \dots , j_k , i\ } } \tilde{z}_{\{j_1 , \dots , j_k , i\ } } ) \times \nonumber \\ & & \prod_{j \in \{i_1 , \dots , i_n\}/\{j_1 , \dots , j_k\ } } ( 1 - e^{-\alpha_j \mu } ) \nonumber \\ & & - \bar{\kappa}(t ) \tilde{\langle l_i\rangle}_{\{i_1 , \dots , i_n\}}\end{aligned}\ ] ] which is exactly the expression in eq . the finite sequence length equations , given by eq . ( 11 ) , may be expressed in vector form , @xmath488 at equilibrium , we therefore have that , @xmath489 the equilibrium solution may be found using fixed - point iteration , via the equation , @xmath490 the iterations are stopped when the @xmath491 stop changing . this is determined by introducing a cutoff parameter @xmath492 , and stop iterating when the fractional change of each of the components after @xmath493 iterations is smaller than @xmath494 . @xmath493 is chosen to be sufficiently large so that , on average , each base mutates at least once after @xmath495 iterations . thus , we choose @xmath496 . what this method does is account for the fact that equilibration takes longer for smaller values of @xmath49 . this means that the smaller the value of @xmath49 , the more times it is necessary to iterate before comparing the changes in the @xmath497 . for our two - gene simulation , we took @xmath498 , and @xmath499 . we chose this initial condition to show that , even though backmutations may become small at large sequence lengths , they still strongly affect the equilibrium solution . by iterating a sufficient number of times , the cumulative effect of the backmutations becomes sufficiently large to lead to a unique equilibrium solution , independent of the initial condition .

in this paper , we study the equilibrium behavior of eigen s quasispecies equations for an arbitrary gene network . we consider a genome consisting of @xmath0 genes , so that each gene sequence @xmath1 may be written as @xmath2 . we assume a single fitness peak ( sfp ) model for each gene , so that gene @xmath3 has some `` master '' sequence @xmath4 for which it is functioning . the fitness landscape is then determined by which genes in the genome are functioning , and which are not . the equilibrium behavior of this model may be solved in the limit of infinite sequence length . the central result is that , instead of a single error catastrophe , the model exhibits a series of localization to delocalization transitions , which we term an `` error cascade . '' as the mutation rate is increased , the selective advantage for maintaining functional copies of certain genes in the network disappears , and the population distribution delocalizes over the corresponding sequence spaces . the network goes through a series of such transitions , as more and more genes become inactivated , until eventually delocalization occurs over the entire genome space , resulting in a final error catastrophe . this model provides a criterion for determining the conditions under which certain genes in a genome will lose functionality due to genetic drift . it also provides insight into the response of gene networks to mutagens . in particular , it suggests an approach for determining the relative importance of various genes to the fitness of an organism , in a more accurate manner than the standard `` deletion set '' method . the results in this paper also have implications for mutational robustness and what c.o . wilke termed `` survival of the flattest . ''

the title compounds @xmath0ti@xmath1o@xmath2 ( @xmath4 tb , dy , ho ) are three of the most well studied realizations of geometrical frustration @xcite . they support a long running experimental and theoretical quest for understanding of an apparently highly frustrated state when none is expected ( @xmath4 tb ) , and the canonical examples of the spin ice state @xcite with attendant emergent magnetic monopole excitations ( @xmath4 dy , ho ) @xcite . little is known about their lattice dynamics , though these are of potential importance for different reasons . in tb@xmath1ti@xmath1o@xmath2 , the formation of the low temperature state is accompanied by numerous anomalies in elastic properties @xcite , and , most recently , the hybridization of magnetic and lattice excitations has been advanced as a source of the fluctuations required to melt long - range magnetic order @xcite . in the spin ices , the monopole excitations must hop by reversing large single - ion magnetic moments @xcite , and mechanisms involving interaction between crystal field states and phonons could well play a role @xcite . thus far , the interaction of a crystal field level with a transverse acoustic phonon has been documented in tb@xmath1ti@xmath1o@xmath2 @xcite , but since the relatively low symmetry of the rare earth site ( @xmath5 ) splits the ground state terms of the tb@xmath6 , dy@xmath6 and ho@xmath6 into several levels that are spread over a similar total bandwidth to that typical of acoustic and optical phonons , further interactions may well be possible . a prerequisite for the understanding of such processes is to know the energy and symmetry of phonon modes which may be involved . investigations of pyrochlore - structured materials using electronic structure calculations have mainly been related to their potential applications as host materials to deposit actinides @xcite , and as thermal barrier coating materials due to their surprisingly low thermal conductivity at high temperatures @xcite . for the former , first - principles calculations were used in the study of defect formation in the pyrochlore structure , while for the latter thermodynamic properties were simulated using both the molecular dynamics method @xcite and _ ab initio _ calculations @xcite . for both applications , it was found that pyrochlore zirconates are generally favorable over titanates , hence more theoretical investigations on the lattice dynamics of zirconates have been carried out . in the pyrochlore titanates , available calculations of the phonon spectrum are limited to the @xmath7-point , where the energies and symmetries of phonons have previously been predicted @xcite . experimentally , the lattice dynamics of both titanate and zirconate pyrochlores were measured using the @xmath8 sensitive raman scattering @xcite and infrared absorption techniques @xcite , but neither could confirm the existence of the low - lying optical phonon modes that were thought to be responsible for the low thermal conductivity of pyrochlore materials at elevated temperatures @xcite . here we use density functional calculations to predict the entire phonon band structure of tb@xmath1ti@xmath1o@xmath2 and ho@xmath1ti@xmath1o@xmath2 . spectroscopic techniques with finite momentum transfers such as inelastic neutron scattering ( ins ) and inelastic x - ray scattering ( ixs ) are needed to determine the phonon dispersion relations across the brillouin zone in a single crystal , or to collect neutron - weighted powder averages of the phonon density of states ( phonon dos ) , and these techniques are then used to validate our calculations and symmetry assignments . we find good agreement with our model throughout the brillouin zone . by comparing the experimentally determined phonon dos in tb@xmath1ti@xmath1o@xmath2 , dy@xmath1ti@xmath1o@xmath2 , and ho@xmath1ti@xmath1o@xmath2 , we find that the phonon frequencies evolve only gradually across the series , and so our dispersions and symmetries may be taken as a good guide for understanding excitations in other nearby compounds such as yb@xmath1ti@xmath1o@xmath2 @xcite . after introducing our computational ( [ sec : computational_methods ] ) and experimental ( [ sec : experimental_methods ] ) methods , the paper presents detailed results concerning structural relaxation ( [ sec : relax ] ) and the calculation of the phonon band structure ( [ calc_phonons ] ) ; the verification of these predictions by inelastic neutron scattering ( [ sec : ins_cryst ] and [ sec : ins_pow ] ) , and inelastic x - ray scattering ( [ sec : ixs ] ) ; followed by discussion ( [ sec : discussion ] ) and conclusions ( [ sec : conclusion ] ) . sample parameters and calculated lattice heat capacities of tb@xmath1ti@xmath1o@xmath2 and ho@xmath1ti@xmath1o@xmath2 can be found in the appendix . readers interested only in general features of the phonon band structure of rare earth titanates will find an overview of the dispersion relations and partial phonon dos of ho@xmath1ti@xmath1o@xmath2 in fig . [ fig : hto_phonons ] and fig . [ fig : theory_pphdos ] respectively , and a tabulation of energies and symmetries of zone center phonons in table [ tab : phonon_symmetries ] . we have applied density functional theory within the perdew - burke - ernzerhof ( pbe ) @xcite parametrized generalized gradient approximation ( gga ) optimized for solids ( pbesol ) @xcite using the plane - wave basis projector augmented wave ( paw ) @xcite method as implemented in the vasp code @xcite . the energy cutoff of the plane - wave basis was checked for convergence of the structural parameters and subsequently fixed to 550ev . the electronic potentials of the ions were approximated by paw gga pseudo - potentials using the electronic valence contribution @xmath9 for the rare earths , @xmath10 for titanium and @xmath11 for oxygen . the @xmath12-electrons of the rare earth ions were frozen into the core states , an approach which was used previously @xcite and proven not to affect the results of phonon calculations @xcite . the primitive reduced unit cell containing 22 atoms ( see fig . [ fig : primitive_cell ] ) was sampled by a @xmath13 @xmath14-grid generated from the monkhorst - pack scheme @xcite . the total energy was minimized until the differences in the total forces were smaller than @xmath15ev / . the atom positions and volume of the reduced unit cell were relaxed at both ambient and applied external pressures to obtain the equilibrium structures . the phonon calculations for the two rare earth titanates tb@xmath1ti@xmath1o@xmath2 and ho@xmath1ti@xmath1o@xmath2 were carried out using the finite displacement method as implemented in the phonopy code @xcite . distorted atomic configurations in a @xmath16 supercell containing 176 atoms were generated and the induced forces were calculated by using vasp , with the same precision as employed for the structural relaxation . the atomic displacement amplitude of 0.01 was verified to give forces that depend linearly on the displacements . the static dielectric tensor and born effective charges were calculated using density functional perturbation theory ( dfpt ) as implemented in vasp . using phonopy , non - analytical term corrections were applied to the dynamical matrix at @xmath17 and interpolated to general @xmath18 according to the interpolation scheme by wang _ et al . _ the total and partial phonon densities of states were evaluated on a @xmath19 @xmath7-centered mesh ( whose size was tested for convergence ) using the parlinski - li - kawazoe fourier interpolation scheme @xcite and smeared with a gaussian of width @xmath20 mev . the lattice heat at constant volume was calculated from the total energy of the phonon bath in the harmonic approximation using a sampling mesh of size @xmath21 , which yielded a convergence of better than @xmath22 at the lowest temperatures . we investigated the phonon density of states using inelastic neutron scattering experiments on powders . the samples were prepared from stochiometric ratios of the oxides ho@xmath1o@xmath23 , dy@xmath1o@xmath23 or tb@xmath24o@xmath2 , and tio@xmath1 in a solid state reaction . the oxides , with 99.99% purity , were annealed at 850 c for 10 hours , then mixed and ground , and heated at 950 - 1300 c for 140 hours with several intermediate grindings . the structures were verified by combined neutron and x - ray diffraction experiments , which were carried out on hrpt @xcite at sinq , psi and the materials science beamline ( msb ) @xcite at the sls , psi . rietveld refinement of the structures ( results tabulated in appendix [ appendix : characterization ] ) as implemented in the fullprof @xcite software proved all samples to be of high quality and single phase . inelastic neutron time - of - flight measurements on the powder samples of ho@xmath1ti@xmath1o@xmath2 and tb@xmath1ti@xmath1o@xmath2 were performed on the merlin spectrometer at isis @xcite . the samples ( each of mass @xmath25 g ) were packed in envelopes of aluminum foil which were curled up to form an annular cylinder with diameter and height of 40 mm . subsequently , the samples were sealed into aluminum cans containing helium exchange gas , and cooled by a closed - cycle refrigerator on the instrument . different settings with incoming neutron energies of @xmath26 and @xmath27mev , and corresponding chopper frequencies of @xmath28 , and @xmath29hz were chosen to record data at @xmath30 and @xmath31k for @xmath32@xmath33amp hrs ( @xmath34 hours at isis full power ) each . the instrumental background was expected to be negligible and hence not measured . the raw data were corrected for detector efficiency by normalizing the intensities using a standard vanadium sample . dy@xmath1ti@xmath1o@xmath2 was investigated using the 4seasons spectrometer at j - parc @xcite . the sample ( mass @xmath35 g ) was packed in an aluminum foil envelop which was wrapped into a cylinder of 30 mm diameter and 50 mm height , and then sealed in an aluminum can with helium exchange gas . the thickness of the sample was carefully controlled so as not to exceed 0.5 mm , to optimize the inelastic signal despite the large absorption cross section of natural dysprosium . 4seasons was operated in repetition rate multiplication mode @xcite . using a fermi chopper frequency of 250 hz , the phases of the other choppers were configured so that for a single source pulse , spectra were recorded for @xmath36 mev . measurements were taken at @xmath37 k , for 8 hours . the detector pixel efficiency was calibrated using a vanadium standard sample . the instrumental background was measured to subtract the significant contribution from scattering due to phonons of the aluminum sample can from the raw data @xcite . for the experimental determination of the phonon dos ( @xmath38 ) in the incoherent approximation , we used data collected on merlin ( @xmath39 tb , ho ) with incoming neutron energies @xmath26 and 150mev and on 4seasons ( @xmath39dy ) with @xmath40mev . the low energy ( le ) setting was chosen to exploit the better instrumental resolution at lower incoming energies and resulting energy transfers , while the high energy ( he ) setting covers the entire spectrum of incoherent one - phonon scattering . the scattered neutron intensity is integrated over scattering angles @xmath41 ranging from 60 to 80 degrees ( he ) , or 85 to 135 degrees ( le ) , of which the latter was only accessible on merlin . by integration over the scattering angle we evaluate the neutron - weighted phonon density of states from the same @xmath42-range measured on the two instruments ( which do not have identical detector coverage ) . magnetic contamination from strong crystal field excitations is excluded by carefully limiting the integration to sufficiently large @xmath41 angles . for each setting , the ( incoherent ) elastic line was removed and replaced by a debye extrapolation below 12 mev in the he setting and 4.5 mev in the le setting @xcite . multiphonon and multiple scattering were removed from the signal up to fourth order using the iterative scheme of sears _ et al . _ @xcite , as extended by kresch _ the scaled multiple scattering contribution is found to be close to parity with the multiphonon sum . from the resulting one - phonon scattering profile , @xmath38 is obtained by correcting the thermal phonon occupation following bose statistics . eventually , scaled fractions obtained from the different @xmath43 settings were concatenated at 36mev , at which energy the phonon dos peaks . neutron measurements of the acoustic phonon dispersion relations in ho@xmath1ti@xmath1o@xmath2 and tb@xmath1ti@xmath1o@xmath2 single crystals were performed on the thermal triple - axis neutron spectrometer eiger at the swiss neutron spallation source sinq . the ho@xmath1ti@xmath1o@xmath2 sample used was a large single crystal of mass @xmath44 g , which was grown from a lead fluoride flux @xcite . it was originally used to characterize the spin ice state in ho@xmath1ti@xmath1o@xmath2 @xcite . the tb@xmath1ti@xmath1o@xmath2 sample was a large single crystal of mass @xmath45 g , grown in a floating zone furnace . it has been previously used to investigate diffuse and inelastic neutron scattering @xcite , and characterized extensively @xcite . the crystals were both aligned with the @xmath46 $ ] direction vertical , to give an @xmath47 scattering plane . this configuration allows the measurement of longitudinal and in - plane transverse acoustic phonons along the cubic high symmetry directions . individual phonon branches were measured in brillouin zones chosen to satisfy the selection rules for phonon scattering . the final neutron wave vector was usually fixed at @xmath48@xmath49 but needed to be increased to @xmath50@xmath49 to access phonon excitations in brillouin zones with large momentum transfers . for @xmath48@xmath49 a pyrolytic graphite filter was installed in the scattered beam to eliminate contamination by higher order scattering . for higher final neutron energies , the filter was removed and possible higher - order scattering , dominantly from optical phonons , was considered during the analysis . phonon excitations were measured at @xmath51k with constant energy scans for steep parts of the dispersion branches , and elsewhere with constant @xmath18-scans . we employed inelastic x - ray scattering ( ixs ) to access optical phonon branches in ho@xmath1ti@xmath1o@xmath2 . a rectangular rod was cut from a piece of the same boule which supplied the sample used in ref . [ ] , a floating - zone grown and oxygen annealed single crystal . the rod was aligned such that the @xmath46 $ ] direction was parallel to the long axis and was polished down to 70@xmath33 m thickness and 600@xmath33 m length . samples used for such measurements may be etched with hydrofluoric acid , but this was found to be unnecessary for ho@xmath1ti@xmath1o@xmath2 . the crystal was mounted on the id28 beamline at the esrf , grenoble . the spectrometer was operated with an incoming photon energy of 17.794kev selected by the si@xmath52 reflection of the backscattering monochromator . the needle - like sample , which was found to be aligned within two degrees in the @xmath47-plane , was mounted in a joule - thompson cryostat . in this configuration all nine detector positions correspond to @xmath18-points in the scattering plane , such that we could efficiently measure phonons at nine @xmath18-points in a single energy scan . .the crystallographic positions of the four independent atoms of @xmath0ti@xmath1o@xmath53o@xmath54 in conventional cubic cell with space group @xmath55 and origin at the ti site . the @xmath56 parameter of the o(48@xmath12 ) ions is approximately 0.33 for @xmath39 tb , dy , ho . 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we present a model of the lattice dynamics of the rare earth titanate pyrochlores @xmath0ti@xmath1o@xmath2 ( @xmath3 = tb , dy , ho ) , which are important materials in the study of frustrated magnetism . the phonon modes are obtained by density functional calculations , and these predictions are verified by comparison with scattering experiments . single crystal inelastic neutron scattering is used to measure acoustic phonons along high symmetry directions for @xmath3 = tb , ho ; single crystal inelastic x - ray scattering is used to measure numerous optical modes throughout the brillouin zone for @xmath3 = ho ; and powder inelastic neutron scattering is used to estimate the phonon density of states for @xmath3 = tb , dy , ho . good agreement between the calculations and all measurements is obtained , allowing confident assignment of the energies and symmetries of the phonons in these materials under ambient conditions . the knowledge of the phonon spectrum is important for understanding spin - lattice interactions , and can be expected to be transferred readily to other members of the series to guide the search for unconventional magnetic excitations .

nearly four decades ago , the quark confinement was shown by wilson at the strong bare coupling region @xcite . for weak coupling , perturbation theory clarified for the yang - mills system that bare coupling @xmath7 tends to vanish as the lattice spacing @xmath8 @xcite . the motivation of the present work is to attempt to extrapolate the large @xmath9 behavior of bare coupling to the asymptotically free behavior at weak coupling . for the purpose , we like to reformulate the strong coupling expansion by changing the primary variable from bare coupling to the lattice spacing itself . lattice serves us a suitable regularization , since in lattice field theories the lattice spacing @xmath9 explicitly appears in the action and enters into the physical quantities . for instance , the dimensionless correlation length @xmath10 represents a physical length scale divided by @xmath9 . it is given at strong coupling as a series @xmath11 in @xmath12 and it determines , in an implicit manner , the @xmath9 dependence of the bare coupling . mutual roles of @xmath7 and @xmath10 are exchanged by inverting the relation @xmath11 . thus we address the question whether the small @xmath10 series of @xmath7 allows us to confirm directly the weak coupling behavior predicted by perturbation theory . in the present paper , in non - linear @xmath0 model at two dimensions , we make an attempt to approximate the asymptotic behavior of bare coupling in the continuum limit via its large @xmath9 expansion . the model is of interest as a testing ground of our approach , since it enjoyes asymptotic freedom and dynamical mass generation for @xmath6 @xcite . in addition to the large @xmath13 limit , we also consider the case of finite @xmath13 . as the basic variable , rather than the correlation length in lattice space , we adopt mass @xmath1 in momentum space defined by the zero momentum limit of the two - point field correlation . the lattice spacing @xmath9 is included in the mass which is rescaled to be dimensionless and then @xmath1 must vanish in the continuum limit @xmath8 ( see ( [ mass ] ) ) . now the strong coupling expansion gives series of @xmath1 in @xmath14 . by inverting the series , we express @xmath3 as a power series in @xmath4 , which is equivalent to large @xmath9 expansion . as it would be , naive series fails to confirm the continuum behavior of @xmath3 . however , it is nontrivial and interesting to examine , when both pad and borel techniques are applied on the series , whether the continuum scaling emerges at finite @xmath13 or not . before proceeding to following sections , we remark the role of borel transform in our approach . we use borel transform as a device of dilation operation around the continuum limit . the response of scale transformation on @xmath15 is probed by rescaling @xmath1 to @xmath16 in @xmath17 and taking the @xmath18 limit . then , it is said @xmath17 scales with the exponent @xmath19 if @xmath20 the above criterion of scaling is implemented by introducing @xmath21 defined by @xmath22 and performing expansion to some finite orders in @xmath21 @xcite . suppose the function approaches to @xmath23 . then , expanding it to @xmath24 and setting @xmath25 , one has @xmath26 . further , if we take the limit @xmath27 with @xmath28 fixed , we obtain @xmath29 that is , the limit @xmath18 has transitioned to the limit @xmath30 with the cut off @xmath31 . then the scaling behavior with exponent @xmath19 manifests itself in the power of @xmath32 . note the universal quantity @xmath19 is left unchanged . on the other hand , when same operation is acted on @xmath17 in the series form @xmath33 valid at large @xmath1 , we have @xmath34 with a larger convergence radius , which is just the borel transform of the original series . we thus interpret the borel transform as a realization of scale transformation . we do not need integrating @xmath35 back to @xmath17 . though the information of @xmath15 over the whole range of @xmath1 is not obtained , what we need in lattice field theories is the behavior of @xmath15 in the neighborhood of @xmath36 . on the two - dimensional square lattice , the continuous spin fields @xmath37 are set on every sites . the action of the system is given by @xmath38 where @xmath39 and @xmath40 the fields are constrained to satisfy at every sites , @xmath41 . the mass variable @xmath1 defined via the zero momentum limit of the propagator @xmath42 is given by @xmath43 where susceptibility @xmath44 and second moment @xmath45 are , respectively , given by @xmath46 and @xmath47 . @xmath48 denotes the dimension of lattice space and @xmath49 in the present work . let us summarize the continuum limit of the model and large @xmath9 expansion of @xmath3 . the perturbative renormalization group predicts that , for @xmath6 , the correlation length behaves at weak coupling as @xmath50\big(\frac{2\pi n\beta}{n-2}\big)^{-1/(n-2)}\big(1+\sum_{k=1}^{\infty}\frac{a_{k}}{\beta^k}\big ) , \label{rg}\ ] ] where the multiplied constant @xmath51 is specified only non - perturbatively . hasenfratz et . al . has computed it via thermodynamic bethe ansatz @xcite , giving @xmath52 the terms @xmath53 @xmath54 in ( [ rg ] ) are contributions of @xmath55-loop levels and three- @xcite and four - loop @xcite results were computed in the literature . they are given as @xmath56 though three and higher loop contributions disappear in the continuum limit for the bare coupling , we can not take out the limit because only the series to finite order is at hand . hence , we include known three- and four - loop contributions in our analysis . it is known that @xmath57 has functional form of @xmath3 , the same as ( [ rg ] ) but with another multiplicative constant , say @xmath58 . however , monte carlo data @xcite showed that the difference is less than a percent at @xmath59 . since the two constants agree with each other in the large @xmath13 limit , the difference between @xmath51 and @xmath58 may actually be negligible for all @xmath6 . thus the estimation of the mass gap via strong coupling expansion becomes the estimation of @xmath51 and this is one of the aims of our work . . since the mass @xmath1 approaches to @xmath60 in the continuum limit , the physical mass of dimension @xmath61 is given by @xmath62 where @xmath63 is the finite mass scale given by @xmath64\big(\frac{2\pi n\beta}{n-2}\big)^{-1/(n-2)}\big(1+\sum_{k=1}^{\infty}\frac{a_{k}}{\beta^k}\big).\ ] ] from ( [ rg ] ) and @xmath65 , we have continuum @xmath3 to four - loop order as a function of @xmath1 , @xmath66 , \label{twoloop2}\end{aligned}\ ] ] where @xmath67 on the series expansion at large @xmath1 , we borrow the result in the work of butera and comi @xcite who computed strong coupling series of @xmath44 and @xmath45 to @xmath68 . using the result , we have expansion of @xmath1 in powers of @xmath3 , @xmath69 by inverting the above relation , we have @xmath70 based upon the series ( [ strong1 ] ) , we discuss the approximation of the continuum limit by the use of pad - borel approximation scheme . we attempt to recover the asymptotically free behavior ( [ twoloop2 ] ) from ( [ strong1 ] ) and then estimate @xmath51 . large @xmath13 limit serves us a good bench mark of our approach . so we consider that case first and then turn to finite @xmath13 in the next section . in the large @xmath13 limit , only the one - loop contribution to @xmath3 survives to give @xmath71 as briefly presented in the introduction , borel transform is given by a certain limit of delta expansion @xcite . explicitly , the logarithm is expanded and gives at @xmath25 that @xmath72 to the order @xmath31 . then using the asymptotic expansion @xmath73 ( @xmath74 denotes euler s constant ) , we have @xmath75 in the @xmath76 limit . let @xmath77 be small enough with @xmath78 kept finite , then the result represents borel transform of @xmath79 . denoting the operation of borel transform by @xmath80 we thus find @xmath81=\log \bar x+\gamma_{e}$ ] . using abbreviated simbol @xmath82 $ ] , we then obtain @xmath83 the large @xmath1 expansion of @xmath3 reads @xmath84 borel transform of the above series results to divide the @xmath85th order coefficient by the factorial of @xmath85 , @xmath86 then as a crucial step , we use pad method to extrapolate the above series to larger @xmath87 . the resultant pad - borel approximants enable us to capture the scaling behavior to be seen in the scaling region as we can see below . as a preliminary study , we have examined the behaviors of @xmath88 $ ] approximants of @xmath89 over almost possible pairs of @xmath90 at orders @xmath91 . on the contrary to the condensed matter models undergoing second order phase transition , critical behavior of the present model is known from perturbation theory as logarithmic and slowly varying . hence it is conceivable that good behaviors come from the cases where the difference between @xmath92 and @xmath85 is small . the numerical experiment confirmed it is indeed the case . we have also compared the approximants of three types , pad - borel , borel only and pad only improvements . the result at @xmath93th order is shown in fig . 1 . and @xmath94 at 6th order . two broken lines ( one for @xmath3 and the other for @xmath89 ) represent behaviors at continuum . horizontal axis corresponds to @xmath95 and @xmath96 ( for pad only case).,title="fig : " ] [ comparison ] as already reported in @xcite , borel only improvement is not sufficient for observing the asymptotic freedom . pad only case ( @xmath97}$ ] in fig . 1 ) is also insufficient as is clear from fig . however , pad - borel approximant shows enough improvement for quantitative approximation . though pade only approximation is found to be improved at higher orders , the best performance is achieved by pad - borel approximant at every order we analyzed . we therefore focus on pad - borel approximant hereafter . now , let us turn to the evaluation of the mass gap by estimating @xmath51 . since we know information at weak coupling , the estimation is carried out by fitting @xmath98 to @xmath88 $ ] order approximants of @xmath94 , @xmath99}$ ] , by adjusting the value of @xmath51 . in practice , we consider the difference between @xmath99}$ ] and @xmath98 and plot the difference by changing the value of @xmath51 . at just proper value of @xmath51 , the two functions touch with each other at a point @xmath100 and the difference is tiny over an interval including @xmath100 . a typical case is shown in fig . 2 and the result of estimation of @xmath51 is shown in table [ tab : estimation1 ] . for the reason previously written , we list only the results around the diagonal pad . the result in momentum space is simple modification of the propagator to the continuum limit @xmath105^{-1}$ ] . since in the large @xmath13 limit , @xmath3 is given by the gap equation written only with the propagator with mass square @xmath1 , we can easily obtain the large @xmath1 series both at first- and infinite - order improved actions ( for detailed presentation , see the first reference in @xcite ) . for example , at the first order it follows @xmath106 at infinite - order improvement , the right - hand side becomes just the integral of @xmath107 and expansion of @xmath3 in @xmath4 is straightforward . it now suffices for us to repeat the same procedure for the approximation of ( [ scaling2 ] ) and the constant @xmath51 at the first and inifinite orders of improved actions . here note that the change of action induces the change of the value of non - universal @xmath51 . @xmath108 and @xmath109 at first and infinite orders , respectively . table [ tab : estimation4 ] summarizes the result of our approximation . the improved action improves the approximation accuracy both at the first and at infinite orders . though the improved lattice action is conventionally used in the monte carlo analysis and perturbation theory , it is also useful in our approach . in the present work , we have analyzed pad - borel approximants of strong coupling expansion in non - linear @xmath110 model and have found good behaviors approximating the continuum limit . we close the paper by pointing out that , even working with the standard action , further higher order computation would improve the result for all @xmath13 including the limit @xmath111 . pad - borel approximants may become effective at larger @xmath87 ( smaller @xmath112 ) and the two unwanted effects , lattice artifacts and omitted loop contributions , would be weaker there . then , continuum scaling at smaller @xmath112 with a clearer sign of asymptotic freedom near @xmath113 would be seen , which allows us accurate evaluation of the mass gap for all @xmath6 . a. m. polyakov , phys . lett . * 59b * , 79 ( 1975 ) ; + e. brzin , j. zinn - justin , phys . rev . lett . * 36 * , 691 ( 1976 ) ; + e. brzin , j. zinn - justin and j. c. le guillou , phys . rev . * d14 * , 2615 ( 1976 ) ; + w.a . bardeen , b. w. lee and r.e . shrock , phys . rev . * d14 * , 985 ( 1976 ) .

based on the strong coupling expansion , we reinvestigate two dimensional @xmath0 sigma model by the use of pad - borel approximants . the conventional strong coupling expansion of the mass square @xmath1 in momentum space in @xmath2 is inverted to give @xmath3 expanded in @xmath4 . borel transform of @xmath3 with respect to @xmath1 is carried out and the result is improved as the rational function by pad method . we find the behavior of pad - borel transformed bare coupling at @xmath5th order is consistent for @xmath6 with that of continuum scaling to the four - loop perturbation theory . we estimate non - perturbative mass gap at @xmath6 and find the agreement with the exact result by hasenfratz et.al .

several lines of evidence suggest that the growth histories of supermassive black holes ( smbhs ) are closely linked to that of their host galaxies . these include the well - known scaling relations between the smbh mass ( ) and several properties of the ( bulge component of the ) hosts , observed in local relic systems ( see @xcite for a recent review ) . and the coincidence of intense star formation ( sf ) and smbh growth , at higher redshifts . indeed , for systems dominated by accretion onto the smbh identified as active galactic nuclei ( agn ) the luminosities associated with smbh accretion ( ) and with star formation ( ) are correlated over several orders of magnitude ( e.g. , @xcite , but see also @xcite ) . this suggests that the phase of fast smbh growth occurs in tandem with intense sf activity , reaching star formation rates ( sfrs ) of @xmath41000 yr^-1 @xmath5 for smbhs with @xmath10 ( i.e. @xmath11 ) . all this supports a general idea that both processes ( sf and agn ) are fed by a common reservoir of cold gas that collapses , forms stars , and ( eventually ) reaches the central region of the host galaxy to be accreted by the smbh . a particularly popular framework for the co - evolution of smbhs and their hosts focuses on major mergers between massive , gas - rich galaxies . theoretical studies highlight the ability of such mergers to account for both the observed properties of agn and sf galaxies , and for the smbh - host relations in relic systems ( see , e.g. , @xcite and the review by @xcite ) . in particular , many simulations of such mergers show a relatively short episode ( of order 100 myr ) of parallel intense sf and agn activity , with sfrs reaching several hundred yr^-1 @xmath5 ( or exceeding @xmath12 in some simulations ; see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? observationally , however , the relevance of mergers to fast smbh growth , and indeed to the co - evolutionary framework , is not yet well established . while some studies of low-@xmath13 luminous agn have reported a high occurrence of mergers ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , several other studies have demonstrated that interacting agn hosts are relatively rare at @xmath14 and their occurrence rate does not exceed what was found for non - active galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? a possible explanation for these apparently contradictory results , as put forward by @xcite , is that the merger - driven scenario may only be relevant for the epochs of fastest growth of the most massive bh ( that is , highest ) , and at @xmath15 when the overall rate of major mergers is expected to be higher ( e.g. , * ? ? ? * ; * ? ? ? * ) and the amount of gas in the relevant halos is considerably larger ( but see also * ? ? ? as the relevance of mergers to smbh growth is still debated , several recent studies have highlighted the importance of alternatives to the merger scenario . direct flows of cold gas from the intergalactic medium ( igm ; e.g. , * ? ? * ; * ? ? ? * ; * ? ? ? * ) , which may also trigger `` secular '' instabilities of the gas or the stars in the close environment of the smbh ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , are claimed to be most relevant for the early and fast growth of high - redshift smbhs . such models , however , usually produce sfrs of only a few hundred yr^-1 @xmath5 . all this suggests that the best way to test and understand the relevance of the merger - driven scenario for smbh growth is to focus on well - defined samples of fast - growing smbhs , preferably at early cosmic epochs ( @xmath16 ) , when the most massive bhs were growing at maximal rates @xcite . another consequence of the aforementioned scenarios is that such fast - growing smbhs would be predominantly found in significantly over - dense large - scale environments , where the rate of mergers is yet higher and where cold gas streams are expected to converge and provide ample gas supply to the smbhs ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? testing these ideas observationally is , however , extremely challenging . the agn - related emission dominates over most of the optical - nir spectral regime , significantly limiting the prospects of determining the host properties . the cosmic environments of the smbhs are often characterized by searching for nearby ( rest - frame ) uv - bright galaxies , without precise redshift determinations , and possibly missing dusty obscured sf galaxies . indeed , several studies provided ambiguous evidence for over - densities around some , but definitely not all @xmath17 quasars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . the advent of large and sensitive sub - millimeter ( sub - mm ) interferometric arrays , such as the iram plateau de bure interferometer ( now noema ) and the atacama large millimeter / sub - millimeter array ( alma ) , has enabled the direct observation of the hosts of high - redshift quasars in a spectral regime that is mostly uncontaminated by the agn emission . the early study of ( * ? ? ? * using the sma ) demonstrated the ability of such data to reveal major mergers among quasar hosts , presenting a close merger between two @xmath18 sf galaxies powering a _ pair _ of agn , one of which is a luminous quasar . the alma study of the same system @xcite showcased the revolutionary increase of spatial resolution and sensitivity provided by alma . in recent years , a growing number of @xmath17 quasars were studied with various sub - mm facilities ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , focusing on the continuum emission , which originates from sf - heated dust in the interstellar medium ( ism ) of the hosts , and the emission line , which is among the most efficient ism coolants ( e.g. , * ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . these studies have provided additional support for the coexistence of fast - growing smbhs and intense sf activity in their hosts , but mostly could not address the questions related to the merger - driven scenario , due to limited sensitivity , resolution , and/or field of view . indeed , no major mergers were identified in the aforementioned studies of high - redshift agn . in this study , we focus on a sample of quasar hosts at z4.8 @xmath19 , for which we have accumulated a wealth of multi - wavelength data . in our previous studies , we have shown that the smbhs powering these quasars trace an epoch of fast , eddington - limited growth from massive bh seeds , which is expected to form the most massive bhs known by @xmath20 ( * ? ? ? * t11 hereafter ) . our /spire campaign ( * ? ? ? * ; * ? ? ? * m12 and n14 hereafter ) showed that these fast - growing smbhs are hosted by sf galaxies , with @xmath21 of the systems exceeding @xmath22 , while a stacking analysis of the other @xmath23 of the systems suggested a typical sfr of @xmath4400yr^-1 @xmath5(see also * ? ? ? the extremely high sfrs found for the fir - bright sources were interpreted as tracing major merger activity , while the lower sfrs found for most systems were thought to be tracing the early stages of sf suppression by agn - driven `` feedback '' . the poor spatial resolution of the data ( @xmath24 , or @xmath25 ) was , however , insufficient to test these ideas , which can be now addressed directly with the alma fir continuum and emission line observations presented in this study . in we describe the sample , the alma observations , data reduction , and analysis . in we compare the ism properties and the occurrence of close ( interacting ) companions to those found in other samples of high - redshift agn . we summarize our main findings in . throughout this work , we assume a cosmological model with @xmath26 , @xmath27 , and @xmath28 , which provides an angular scale of about @xmath29 at @xmath18 - the typical redshift of our sources . we further assume the stellar initial mass function ( imf ) of @xcite . lccccccl bright & sdss j033119.67 - 074143.1 & 29 & 792 & @xmath30 & @xmath31 & 0.06 & ... + & sdss j134134.20 + 014157.7 & 35 & 697 & @xmath32 & @xmath33 & 0.06 & ... + & sdss j151155.98 + 040803.0 & 30 & 729 & @xmath34 & @xmath35 & 0.06 & 1 smg ( w/ ) and 1 `` blob '' ( w / o ) + + [ -1.75ex ] faint & sdss j092303.53 + 024739.5 & 38 & 2978 & @xmath36 & @xmath37 & 0.06 & 1 smg ( w/ ) + & sdss j132853.66 - 022441.6 & 36 & 2852 & @xmath38 & @xmath39 & 0.06 & 1 smg ( w/ ) + & sdss j093508.49 + 080114.5 & 35 - 33 & 3230 & @xmath40 & @xmath41 & 0.06 & ... our targets are drawn from a flux - limited sample of 40 luminous , unobscured quasars at @xmath0 , which we have studied in detail in a series of previous publications . here , we only briefly mention the sample selection and the ancillary data available for our targets and refer the reader to our previous papers ( t11 , m12 , n14 ) for additional details . this @xmath0 quasar sample was originally selected from the sixth data release of the sloan digital sky survey ( sdss / dr6 ; * ? ? ? * ; * ? ? ? the sample spans a narrow redshift range of @xmath42 , to enable follow - up near - ir spectroscopy of the broad emission line . such near - ir spectroscopy was indeed performed using the vlt / sinfoni and gemini - north / niri instruments and has provided reliable estimates of the masses ( ) and normalized accretion rates ( ) of the quasars ( t11 ) . these clearly showed that the @xmath0 quasars are powered by fast - growing smbhs with typical masses of @xmath43 and accretion rates of @xmath44 ( median values ) , representing the epoch of fastest growth for the most massive bhs . a follow - up /spire campaign targeted these quasars , probing the peak of the sf - heated dust continuum emission ( m12 , n14 ) . these data provided robust continuum detections for @xmath41/4 of the quasars , with fir luminosities of @xmath45 , suggesting sfrs in the range @xmath46 . a stacking analysis of the remaining @xmath43/4 of the quasars revealed a median sfr of @xmath47 ( see the refined stacking analysis in * ? ? ? * ) . importantly , these `` fir - bright '' and `` fir - faint '' sources are highly uniform in terms of the smbh - related properties that drive the agn emission ( i.e. , , , and hence ) , with a tendency of the fir - bright sources to have somewhat higher and ( by @xmath480.4 dex ) . thus , the data presented in m12 and n14 suggests a wide variety of sf activity among an almost uniform sample of fast - growing smbhs in the early universe , with sfrs that range over an order of magnitude , compared with a significantly narrower range of basic smbh properties . the n14 study also presented the results of a dedicated /irac campaign targeting almost all of the t11 quasars , at observed - frame 3.6 and 4.5 @xmath49 m . these data were used to derive positional priors that allowed us to de - blend the low - resolution /spire data . a comparison of the fir luminosities of the quasar hosts ( implied from the data ) to the agn - related luminosities ( i.e. , the bolometric luminosities from t11 ) , implies that the fir - bright sources are so fir - luminous , that they reach @xmath50 . the fir - faint systems , on the other hand , have @xmath51 , making them `` agn dominated '' . the current study focuses on objects selected from our parent t11 quasar sample , split equally between fir - bright and fir - faint objects . we chose to focus on the lower - redshift sources among our parent sample ( @xmath52 ) to avoid the emission line to be redshifted into the low - atmospheric transmission region near 325 ghz . the redshifts of the selected targets are in the range @xmath53 , with a median of @xmath54 ( as determined from the lines ; see for more details ) . we will nonetheless refer to them here as `` @xmath0 quasars '' , following our previous studies . the targets are all equatorial , so to allow efficient alma observations . in practice , this implies that the nir data for all the quasars studied here were obtained with the vlt / sinfoni . these redshift and declination restrictions leave only 4 of the 10 -detected , fir - bright quasars reported in n14 , of which we finally chose 3 . as for the 3 fir - faint targets , we focused on those with higher quality fits , as assessed in t11.-quality '' and `` -quality '' flags of 1 or 2 in table 2 of t11 . ] we stress that all these selection criteria do not introduce any biases , in terms of the a - priori known smbh- and host - related properties of the systems under study . in particular , the values of the two sub - groups are consistent , within the errors ( except for the relatively high - mass object j1341 ; see ) . the targets were observed in alma band-7 , as part of cycle-2 ( project code 2013.1.01153.s ) , during the period 2014 july 18 to 2015 june 13 . as the capabilities of alma were expanded during this period , the data presented here were collected with a varying number of 12 m antennas , between 29 - 38 . the observations were set up to use the extended c34 - 4 configuration , which provides a resolution of about 03 at 330 ghz ( corresponding to about 2 kpc at @xmath0 ) . we aimed at spectrally resolving the emission line , which is expected to have a width of several hundred ^-1 kms@xmath55 and also to cover a wide spectral range that includes line - free continuum regions . we chose to use the tdm correlator mode , providing four spectral windows , each covering an effective bandwidth of 1875 mhz , corresponding to @xmath56 at the observed frequencies . this spectral range is sampled by 128 channels , providing a spectral resolution of @xmath57 . one such spectral window was centered on the frequency corresponding to the expected peak of the line , given the -based redshifts of our targets ( as determined in t11 ) . the other three bands extended to higher frequencies , with the first being adjacent to the -centered band and the other two separated from this first pair by about 12 ghz . each of these two pairs of bands included a small overlapping spectral region , of roughly 50 mhz . in addition , the rejection of a few of the channels at the edge of the spectral bands , due to divergent flux values ( a common flagging procedure in alma data reduction ) , implies that in some cases a small gap is seen between windows . given this spectral setup of four bands , the alma observations could in principle probe line emission over a spectral region with a width corresponding to roughly @xmath58 . additional details regarding the alma observations , including the full object names of our sources , are given in . for clarity , we use abbreviated object names ( i.e. , `` jhhdd '' ) throughout the rest of the paper . + + the alma band-7 flux densities reach a depth of @xmath59 mjy / beam ( rms ) . this flux limit can be translated to a limit on our ability to detect dusty , sf galaxies , at the redshift range of interest . if we assume the same gray - body fir spectral energy distribution ( sed ) as we do in our analysis of the quasars hosts a dust temperature of @xmath60 and a power - law exponent of @xmath61 , and the scaling between fir luminosity and sfr appropriate for our chosen imf ( see ) , this corresponds to @xmath62 upper limits on the sfr in the range @xmath63 , per beam . given the beam sizes of our alma observations ( ) , these translate to @xmath64 . data reduction was performed using casa package , version 4.5.0 @xcite . the scripts provided by the observatory were used to generate the visibilities . we then applied the ` clean ` algorithm , using a `` natural '' weighting to determine the noise level for each observation by averaging over the three line - free spectral windows . the resulting flux density sensitivities and synthesized beam sizes are presented in . we note that self calibration for the brightest sources did not improve the signal - to - noise ratios and therefore was not implemented . both continuum and emission line images were created by applying the ` clean ` algorithm using a `` briggs '' weighting ( with robustness parameter set to 0.5 ) , to obtain images with the best possible spatial resolution . continuum emission images were constructed using the two highest frequency , line - free spectral windows . emission line images were constructed by subtracting the continuum emission from the other two , lower frequency spectral windows , in the uv space ( using standard casa procedures ) . we verified that the resulting emission line images have no residual continuum signal in them . the sizes of the continuum- and line - emitting regions were determined from the respective images by fitting two - dimensional gaussians . these sizes are given in . velocity and velocity dispersion maps ( i.e. , second- and third - moment maps ) were obtained from the line images using the standard casa procedure . + + the properties of the emission lines were measured from the two lower frequency spectral windows of the continuum - subtracted `` briggs '' weighted cubes . the integrated line fluxes were measured through two different procedures . in the first ( `` spatial '' ) approach , we created zero - moment images ( i.e. , integrated over the spectral axes ) for all sources and fitted the spatial distribution of line emission with 2d gaussian profiles , which are characterized by a peak line flux , semi - major and semi - minor axes , and a position angle . the line fluxes were measured by integrating over these spatial 2d gaussians . in the second ( `` spectral '' ) approach , we extracted 1d spectra of the line from the continuum - subtracted cubes . we used apertures that are based on the aforementioned spatial 2d gaussian profiles , with varying radii ranging @xmath65 , where @xmath66 is the width ( i.e. , equivalent to standard deviation ) of the spatial 2d fitted gaussian profiles . we then fitted the emission line profiles with a simple model of a single gaussian , from which we obtained the integrated line flux . the line fluxes obtained using the two methods are in good agreement , with differences of no more than 0.1 dex for the quasar hosts ( median difference of 0.05 dex ) . we eventually chose to adopt the line fluxes measured through the former , `` spatial '' approach , as it is less sensitive to the low-@xmath67 regions in the outer extended regions of the sources and/or the wings of the line profiles , and since it provides more realistic uncertainties . these line fluxes are reported in . the best - fit emission line profiles obtained through the latter ( `` spectral '' ) approach allow us to obtain the line centers , and therefore -based redshifts , as well as the line widths ( see and ) . we stress that the centers of the line profiles were treated as free parameters and _ not _ constrained to reflect the previously known redshifts of the quasars . we also note that for four of the quasar hosts , the line is observed near the edge of the spectral band , due to the differences between the redshifts based on agn - line- and host - ism - related emission regions ( see ) . the line widths we measure for the quasar hosts are in the range @xmath68 , with a tendency for broader lines among the fir - bright sources ( see discussion in ) . the formal uncertainties on line width measurements are of order of 10% ( i.e. , @xmath69 ) . presents the full - scale continuum emission maps of the sources , extending to about 68 ( @xmath445 kpc ) from the quasars locations , and shows the equivalent line emission maps . all quasar hosts are clearly detected in our new alma data , both in continuum and in line emission , with the least significant detection having s / n@xmath44.5 ( for j0935 ; see ) . the spectra of the quasar hosts are shown in ( together with the companions we discuss below ) . we note that the central frequencies of the lines are shifted , by several hundreds of ^-1 kms@xmath55 , from the redshifts determined using the quasars broad uv emission lines ( i.e. , * ? ? ? we discuss this in more detail in below . + + ' '' '' + + lllcccccccccc bright & j0331 & qso & @xmath70 & 344.47 & @xmath71 & @xmath72 & @xmath73 & @xmath74 & @xmath75 & 3.58 & ... & ... + & j1341 & qso & @xmath76 & 346.84 & @xmath77 & @xmath78 & @xmath79 & @xmath80 & @xmath81 & 4.51 & ... & ... + & j1511 & qso & @xmath82 & 347.99 & @xmath83 & @xmath84 & @xmath85 & @xmath86 & @xmath87 & 3.42 & ... & ... + & & smg & @xmath88 & 347.99 & @xmath89 & @xmath90 & @xmath91 & @xmath92 & @xmath93 & 1.53 & 13.9 & @xmath94 + & & b & @xmath95 & 347.99 & @xmath96 & ... & ... & ... & ... & ... & 25.2 & ... + + [ -1.75ex ] faint & j0923 & qso & @xmath97 & 348.65 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & @xmath102 & 3.21 & ... & ... + & & smg & @xmath103 & 348.65 & @xmath104 & @xmath105 & @xmath106 & @xmath107 & @xmath108 & 2.16 & 36.5 & @xmath109 + & j1328 & qso & @xmath110 & 348.74 & @xmath111 & @xmath112 & @xmath113 & @xmath114 & @xmath115 & 1.63 & ... & ... + & & smg & @xmath116 & 348.74 & @xmath117 & @xmath118 & @xmath119 & @xmath120 & @xmath121 & 2.01 & 44.5 & @xmath122 + & j0935 & qso & @xmath123 & 347.93 & @xmath124 & @xmath125 & @xmath126 & @xmath127 & @xmath128 & 0.80 & ... & ... for one of the fir - bright sources , j1511 , the alma data reveal two faint sub - mm - emitting apparent companions , which are detected at the 7 - 8@xmath66 level in continuum emission . their centers are spatially separated from the quasar host by about 21 and 39 , which corresponds to 14 and 25 kpc ( see ) . these two sources are fainter than the quasar host , in the alma continuum data , by at least a factor of 5 . most importantly , we detect significant line emission from the companion closer to the quasar host , as seen in figures [ fig : cii_maps_lg ] and [ fig : comb_maps_sm ] . the integrated line flux suggests that the detection is significant at the @xmath129 level . the spectrum we extract for this companion ( ) demonstrates that the line emission , if associated with a redshifted transition , is shifted by @xmath475 ^-1 kms@xmath55 from the line of the corresponding quasar host ( see ) . we do not detect any significant line emission from the other , more distant continuum source accompanying j1511 . moreover , we do not detect any other significant emission in the multi - wavelength data available for the @xmath0 quasars , for any of these two companions of j1511 . this may suggest that the most distant source seen in the alma maps of j1511 is a _ projected _ companion source , not physically related with the j1511 system . however , given the low probability of observing such a ( second ) faint sub - mm source within a single alma pointing ( see below ) , it may be indeed tracing a faint , low - mass , and/or low - sfr galaxy that is associated with the quasar host . in such a case , the observed properties suggest that it would be a _ minor _ merger . this latter interpretation is further justified by our findings of clearly _ interacting _ companions , both for j1511 and two of the fir - faint quasar systems , as described below . in particular , we note that the interacting companions show clear emission , while the distant companion of j1511 does not , despite having comparable fir continuum fluxes . we identify two additional emission sources in the maps of two of the fir - faint sources , j0923 and j1328 one around each of these quasars which are clearly detected in the alma continuum maps . these sources are detected at significance levels of @xmath130 , and have continuum flux densities of 1.2 and 0.7 mjy ( for the companions of j0923 and j1328 , respectively ) . the centers of these accompanying sources are located 56 and 68 from the quasars hosts , corresponding to about 36.5 and 44.5 kpc . these two accompanying sources show associated line emission , as seen in the bottom panels of ( and also of ) . similarly to the companion of j1511 described above , the spectra we extract for these two companions ( ) demonstrate that the line emission is shifted by less than @xmath4450 ^-1 kms@xmath55 from the lines of the corresponding quasar hosts . we finally note that , similarly to the aforementioned companion of j1511 , these two sources are not associated with any other ( significant ) emission in the multi - wavelength dataset we have available for the @xmath0 quasars , particularly the data ( at observed - frame 3.6 and 4.5 @xmath49 m ; n14 ) . the ( projected ) spatial and velocity offsets of all companions are given in . presents smaller - scale continuum and line emission maps of all the spectroscopically confirmed systems we identify . the regions where the continuum and line emission is significantly detected are resolved by a few synthesized beams , in all sources . as can be clearly seen , the peaks of the line emission coincides with the peaks of the dust - dominated continuum emission . in the quasars hosts , these peaks of continuum and emission are also consistent with the locations of the quasars , to within 01 , as determined from the sdss optical imaging . before moving on with our analysis of the properties of our quasar hosts and companions and our interpretation of these detections in the context of major galaxy mergers ( ) , we would like to emphasize that the sheer detection of three ( and possibly four ) faint sub - mm companion sources within our alma data are , by itself , highly surprising . the faintest continuum sources in our data ( regardless of detection ) have flux densities on the order of @xmath40.5 mjy . based on recent deep ( alma ) sub - mm surveys ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , we would expect on order of @xmath40.1 such sources per alma pointing ( i.e. , a circular field of view of 18 ) . uv - selected , @xmath131 sf galaxies are yet more rare . the most recent measurements of areal densities in deep fields imply that on the order of 0.01 galaxies with @xmath132 would be observed within a single alma band-7 pointing @xcite . most , but not all such galaxies would have detections ( e.g. , * ? ? ? * ; * ? ? ? * ) number counts of purely -emitting @xmath17 galaxies are highly uncertain , but would probably amount to roughly 0.06 galaxy per each alma pointing ( e.g. , * ? ? ? thus , based on deep ( sub - mm ) surveys that are designed to detect high - redshift sf galaxies our robust detection of continuum - emitting companions in the fields surrounding three of the quasar hosts can not be attributed to chance coincidence . obviously , the robust detection of in three of these companions further strengths this conclusion . we conclude that the host galaxies of all quasars are clearly detected in the new alma data , in both continuum and line emission . we identify three sub - mm galaxies ( smgs hereafter ) accompanying three of the quasar systems - one fir - bright and two fir - faint ( one smg accompanying each of these quasars ) . the companion smgs are located between roughly 14 and 45 kpc ( projected ) from the quasars , with relatively small velocity offsets , @xmath133 . the three companion -emitting smgs detected solely through the new alma data are therefore physically related to , and interacting with the quasar systems . we finally note that no additional line - emitting sources were found in our alma data . + + ' '' '' + + + + ' '' '' + + as clearly seen in , the lines we measure in our sample are often offset towards the lower frequency edge of the observed bands - redshifted with respect to the rest - frame uv emission lines of the quasars . compares the redshifts determined from the line to those determined from the line and also the enhanced sdss - based redshift determinations published in @xcite . the line is believed to be among the best redshift indicators for unobscured agn , with a scatter of about 200 ^ -1 kms@xmath55 compared to the systemic redshift ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . we note that for our @xmath0 quasars the sdss - based redshift determinations rely solely on the ( rest - frame ) uv lines and , both of which may be problematic for the purpose of redshift determinations . the line is known to present significant blueshifts with respect to systemic redshifts ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , while the profile of the line , and particularly the blue wing , is affected by igm absorption ( e.g. , * ? ? ? all quasar hosts present significant shifts of the lines , compared to the broad uv emission lines of the quasars . the fir - bright hosts and one of the fir - faint ones ( j0935 ) show emission redshifted by @xmath134 . interestingly , the only sources presenting _ _ blue__shifts , of @xmath135 and @xmath136 , are the two fir - faint sources , j0923 and j1328 respectively , which have interacting smgs . such significant positive velocity shifts of ism lines , such as , with respect to the quasar s broad emission lines , such as which probe the close vicinity of the smbhs , are not uncommon in high - redshift quasars . for example , the recent study of @xcite shows velocity shifts of @xmath137 in a compilation of seven @xmath138 quasars . @xcite found a shift of @xmath139 for one @xmath140 quasar , but no significant shift for another . negative velocity shifts of several hundred ^-1 kms@xmath55 were also observed among other high - redshift quasars ( e.g. , * ? ? ? * ; * ? ? ? . shows the velocity maps of the quasar hosts and the three smgs accompanying j1511 , j0923 and j1328 . the relatively smooth gradients suggest that the emission originates from a kiloparsec scale , rotating gas structure , with a rotation axis that coincides with the centers of the galaxies and with the quasars themselves ( as shown by the centroid markers ) . the only obvious outlier is the host of j0935 , where the emission is weaker . for the fir - bright quasar hosts , the velocity maps reach maximal velocity values of @xmath141 , while for the fir - faint hosts and their accompanying smgs the corresponding values are significantly lower , @xmath142 this may suggest that the fir - bright quasar hosts are more massive , or gas rich , than the fir - faint ones . + + ' '' '' + + the velocity dispersion maps , presented in , show increased velocity dispersions in the centers of all -emitting systems , reaching @xmath143 . given the beam sizes of our alma data , this trend is probably mostly driven by the effects of beam smearing , as demonstrated in several detailed studies of the kinematics of high - redshift , sub - mm sources ( e.g. , * ? ? ? * ) . a clear signature of this effect is a centrally peaked velocity dispersion , elongated along the minor axis of rotation - which is similar to what is seen in some of our sources ( e.g. , j1341 , j1511 and j0923 ) . thus , the real underlying central velocity dispersions may be significantly lower than what is seen in , resulting in generally more uniform or flatter velocity dispersion profiles , and thus implying rotation - dominated kinematics with @xmath144 . we note that even in such a situation , of @xmath145 on the order of a few , the kinematics may be significantly affected by a turbulent component , as demonstrated in several recent studies of resolved ism kinematics in high - redshift galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) . with these limitations in mind , we cautiously conclude that the _ outer _ parts of the -emitting regions in all sources are dominated by rotation , with @xmath146 . in particular , this is the case in the outer high - velocity regions of the fir - bright systems , where one could have expected to see evidence for dispersion - dominated gas kinematics in the case these systems were driven by major mergers . llccccc bright & j0331 & @xmath147 & @xmath148 & @xmath149 & @xmath150 & @xmath151 + & j1341 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 + & j1511 & @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath161 + + [ -1.75ex ] faint & j0923 & @xmath162 & @xmath163 & @xmath149 & @xmath164 & @xmath165 + & j1328 & @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 + & j0935 & @xmath171 & @xmath172 & @xmath173 & @xmath174 & @xmath175 we next turn to the relative line strength , traced by the ratio between line luminosity and the continuum ( rest - frame ) fir luminosity , /@xmath176 . shows /@xmath176 vs. for the quasar hosts and the interacting smgs in our sample , as well as a large compilation of other galaxies where the line was detected . the compilation , adapted from @xcite , includes inactive star forming galaxies at low and intermediate redshifts ( from * ? ? ? * ; * ? ? ? * ) and smgs and quasar hosts at @xmath16 ( from * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? for the purpose of this comparative analysis , we calculated the fir luminosities assuming a gray - body sed with a dust temperature of @xmath60 and a power - law exponent of @xmath61 , scaled to match the continuum emission of each of the alma - detected sources . these scaled seds are then used to calculate the integrated luminosity between @xmath177 , @xmath178 . as mentioned in @xcite , the measurements of the sources in the compilation were also scaled , to provide consistent estimates of @xmath178 . we present a more detailed analysis of the fir seds of our sources in below . for the quasar hosts , the /@xmath176 ratio is in the range of @xmath179 , spanning a factor of roughly 4.5 . as shows , the fir - bright and fir - faint quasar hosts form a trend of decreasing /@xmath176 with increasing , although the range of /@xmath176 for the two sub - samples overlaps . since all quasar hosts have comparable , this trend of decreasing /@xmath176 is mostly driven by the increase in . the interacting smgs follow the same trend , extending to lower and higher /@xmath176 . we further verified that the trend of decreasing line - to - continuum ratio with increasing fir luminosity is also reflected in the equivalent widths ( ews ; instead of /@xmath176 ; see , e.g. , * ? ? ? the ews of our quasar hosts are in the range @xmath180 @xmath49 m ( @xmath181 ) and follow the same trend with ( continuum ) fir luminosity as that found for /@xmath176 dropping by about 0.65 dex in ew for a 1 dex increase in ( monochromatic ) fir luminosity . the companion smgs extend this trend to @xmath182 @xmath49 m .. several recent studies have demonstrated the wide range of possible /@xmath176 in @xmath17 ( uv - selected ) sf galaxies , covering @xmath183 - similar to the range observed at lower redshifts ( e.g. , * ? ? ? * ; * ? ? ? thus , the so - called `` deficit '' observed in high - redshift smgs and quasar hosts is most probably _ not _ related to simple observational selection effects in the fir or sub - mm regime , but rather to the morphology of the sf activity . specifically , high - redshift quasar hosts exhibit more compact starburst - like sf activity , with /@xmath176 ratios as low as observed in lower - redshift ulirgs . interestingly , our measurements show that the /@xmath176 ratio in the three interacting quasar hosts is significantly lower than that found in their companion smgs . specifically , we find that the /@xmath176 ratio in the quasar hosts of j1511 , j0923 , and j1328 is lower by factors of about 1.7 , 3 and 2.9 , respectively , compared with the companion smgs . as can be clearly seen in , this is consistent with the general trend of decreasing /@xmath176 with increasing , observed for our entire sample of quasar hosts and smgs . moreover , this demonstrates that the -deficit in high - redshift quasar hosts is driven by local properties of the ism and the uv radiation field _ within _ the host galaxies , and not by larger scale effects . indeed , several studies have emphasized that lower /@xmath176 ratios are expected to be found in regions with higher sf densities , similar to starbursts consistent with what is observed ( e.g. , @xcite ; see discussion in @xcite ) . however , the alma data for some of our sources do _ not _ seem to support this explanation . in particular , the hosts of j1328 and j0935 have virtually identical and -emitting region sizes , but ( and therefore /@xmath176 ) in j1328 is twice as high as in j0935 . similarly , the hosts of j0331 and j0923 have very similar , and /@xmath176 in the latter is higher than in the former by merely 30% . however , the -emitting region in j0923 is larger by a factor of roughly four than the one in j0331 . thus , our alma data do not seem to support a simple scenario where /@xmath176 is mainly controlled by the size of the emitting region , however , higher resolution data are needed to critically address this idea . the ( rest - frame ) fir continuum emission observed within the new alma data can be used to estimate the total fir emission and therefore the sfr of the quasar hosts and the accompanying smgs . this obviously requires additional assumptions regarding the shape of the fir sed . we expect that the agn - related contribution to the fir sed is small , on the level of 10% , at most . this assumption is based on several detailed studies of the mid - to - far ir seds of luminous agn across a wide luminosity range ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , although we note that other studies have suggested a higher agn contribution at fir wavelengths , particularly at high agn luminosities ( e.g. , * ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? given the uncertainties related to the sed shape ( see below ) and in order to be consistent with other alma studies of high - redshift quasar hosts , we choose to neglect the ( small ) possible agn contribution to the fir seds of our sources . we have reconstructed the fir seds of our sources following two different approaches . first , we adopted the procedure used in several recent studies of fir emission in high - redshift quasar hosts ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) and have assumed a gray - body sed with dust temperature @xmath60 and @xmath61 . this single temperature dust model is a crude approximation to the more realistic case where dust with a range of temperatures contribute to the observed emission ( for a detailed discussion see , e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . second , we have used the grid of fir seds provided by ( * ? ? ? * ce01 hereafter ) . this grid includes 105 templates spanning a wide range in ir luminosity . as the templates have no free parameters , we have simply identified the template that best matches the alma continuum measurement ( i.e. , monochromatic luminosity ) for each source . this also provides a specific value of total ir ( tir ) luminosity , @xmath184 . in the fir luminosity regime of interest , the uncertainties on the alma continuum measurements are typically consistent with , or smaller than , the differences between adjacent fir templates . the main limitation of this approach is that the template library was constructed to account for low - redshift sf galaxies . however , the small number of available data points does not warrant the use of other sets of templates . in we present the alma continuum measurements and the data from n14 , along with the two previously mentioned types of fir seds . the low spatial resolution of the data means that these flux measurements also include the emission from any accompanying continuum sources for the systems where these are resolved by alma . we therefore also show in a scaled - down version of the data points , assuming the relative flux densities of the different neighboring sources follow those of the alma measurements . as shows , the gray - body seds are generally in good agreement with the previous data . in particular , in the observed - frame 350 @xmath49 m band ( rest - frame wavelength of roughly 60 @xmath49 m ) , the alma - based seds for two of the fir - bright systems ( j1341 and j1511 ) differ from the data by less than 0.05 dex ( for either the gray - body or ce01 seds ) . for the third fir - bright object , j0331 , the luminosities expected from the alma - based seds are significantly lower , by factors of about 2 and 3 , than the fluxes observed with for the gray - body and ce01 seds , respectively . since this source has no detectable companions in the alma data , we suggest that this discrepancy can only be attributed to the uncertainty on the exact shape of the sed , and perhaps to the limited quality of the data . below we investigate how a gray - body sed with a different ( higher ) temperature may account for this discrepancy . for the fir - faint sources , the comparison between the alma - based seds and the data is obviously less straightforward , as the data represent the stacked signal coming from a much larger sample of sources for which the existence of companions that may contribute to the fir fluxes is unclear . nonetheless , the agreement between the new alma data and the stacking measurements is respectable . for two of the sources ( j1328 and j0935 ) , the alma - based gray - body seds are consistent within 0.13 dex of the data ( again at observed - frame 350 @xmath49 m ) . for the third source ( j0923 ) , the gray - body sed over - predicts the luminosity at 350 @xmath49 m by 0.37 dex , while the ce01 sed agrees with the data to within less than 0.1 dex . as a sanity check , if one ignores the scaling factors related to the companions ( in j0923 and j1328 ) , that is - assume that the stacked signal was not significantly affected by companions , then the differences between the alma - based gray - body seds and the data become somewhat smaller . we have also experimented with both colder and warmer gray - body seds , always `` anchored '' to the new alma data . the colder seds , with @xmath185 and @xmath186 , identical to those used in n14 , systematically _ under_-predict the data , by 0.1 - 0.3 dex ( at 350 @xmath49 m ; the discrepancies obviously decrease at observed - frame 500 @xmath49 m ) . among the warmer seds , shown as dotted lines in , we note that an extremely warm gray - body with @xmath187 provides a better agreement between the alma and the data for j0331 . however , this high temperature is beyond what is typically observed , even among the most luminous fir sources ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . this may suggest that , in this source , the radiation emerging from the vicinity of the smbh does contribute to heating of galaxy - scale dust , which is not associated with sf regions in the host ( e.g. , * ? ? ? * ) . we also note that choosing a higher - luminosity template from the ce01 grid would not resolve the discrepancy between the alma and data for this source , as the highest - luminosity ce01 template that closely matches the data would significantly over - predict the alma continuum measurement . we conclude that both the ce01 and fiducial gray - body fir seds ( i.e. , with @xmath60 and @xmath61 ) provide generally good agreement between the alma and data . we therefore chose to use these sets of seds in what follows . we next use the two types of fir seds to calculate the total ir luminosities between @xmath188 , @xmath184 for all sources both quasar hosts and accompanying smgs . , as mentioned in . ] from these , we estimate sfrs following @xmath189 ( following the assumed chabrier imf ) . importantly , we note that the agreement between the integrated tir luminosities obtained with the two types of seds is remarkably good . for the six quasar hosts , the estimates based on the ce01 template seds are lower than those based on gray - body seds by merely 0.1 dex ( median value ) . for the interacting smgs , the difference is only slightly larger , @xmath40.13 dex . the sfrs we obtain for the quasar hosts following this procedure span a wide range of @xmath190 for the gray - body seds or @xmath191 for the ce01 seds ( see ) . the sfrs of the fir - faint sources , @xmath192 ( or @xmath193 using ce01 ) , are in excellent agreement with the value found from stacking analysis performed in n14 , using _ all _ the -undetected sources in the parent @xmath0 sample , of roughly @xmath194 ( see also @xcite for a slightly lower value ) . as noted above , the low spatial resolution stack included the fir emission from both the quasar hosts and the accompanying smgs . total _ sfrs of the different sub - components in the fir - faint systems ( @xmath195 , @xmath196 , and @xmath197 ) are , again , consistent with the stacking result . for the fir - bright sources , the sfrs we derive based on the alma data are high , ranging from @xmath198 for the fiducial gray - body fir seds or @xmath199 for the ce01 ones . we note that the lowest sfrs among this sub - sample are those of j0331 , where an extremely warm gray - body sed is required to match the data . using the @xmath187 gray - body sed we obtain @xmath200 and @xmath201 . these sfrs are in excellent agreement with those derived in n14 , when comparing similar seds . the differences between the ce01-based ir luminosities ( and therefore sfrs ) among the fir - bright quasar hosts are of 0.1 dex , at most . if we instead consider the gray - body seds used in n14 , which assumed @xmath185 and @xmath186 , and employ these sed parameters to our alma data , then the resulting sfrs are , again , in excellent agreement with the n14 ones for two of the systems ( j1341 and j1511 ) . as mentioned above , such a cold sed is inconsistent with the data available for j0331 , and the alma - based for this source is lower than the one obtained in n14 by 0.4 dex . we conclude that the new alma continuum measurements are broadly consistent with the -based ones ( presented in n14 ) and indicate that all quasar hosts and accompanying smgs harbor significant sf activity . the exact values of sfr obviously depend on the assumed shape of the fir sed , but are in good agreement with the ones derived from the data . the only outlier is the fir - bright system j0331 , where the alma - based fir sed and sfr estimates are found to be considerably lower than the -based ones . despite these systematic uncertainties , the new alma data strongly support the picture that the fir - bright sources among the t11 sample of @xmath0 quasars have extreme sfrs , exceeding @xmath12 , while the fir - faint sources have lower , though still intense sfrs , on the order of @xmath202 . most star - forming galaxies are found to populate the so - called `` main sequence of star formation '' on the sfr- plane ( sf - ms hereafter ) . although this relation is not yet well established at @xmath16 , the most recent results from deep surveys suggest a relation of roughly @xmath203 with an intrinsic scatter of @xmath204 dex ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the sfrs we find for the fir - faint quasar hosts , and for the companion smgs , are therefore consistent with those of typical massive , high - redshift sf galaxies , with @xmath205 . several recent studies have highlighted the fact that such sfrs can be sustained without invoking major mergers and instead be driven by the accretion of cold gas onto these galaxies ( e.g. , * ? ? ? * ; * ? ? ? on the other hand , it is challenging to account for the extremely high sfrs found for the fir - bright sources by assuming ms hosts . such an assumption would require stellar masses in excess of @xmath206 . this would imply that the quasar hosts are among the most massive and rarest galaxies ever observed , at any redshift ( e.g. , * ? ? ? * ; * ? ? ? * ) , with number densities on the order of @xmath207 at @xmath208 ( e.g. , * ? ? ? * ; * ? ? ? * ) . in this context , we note that the quasars we study here are among the most luminous in the universe ( by selection ) and therefore also represent a population of rare objects , with number densities on the order of @xmath209 ( e.g. , * ? ? ? * ; * ? ? ? alternatively , it is possible that the fir - bright sources are hosted in galaxies with masses that are comparable to those of the fir - faint systems , but located well above the sf - ms . such systems are typically associated with short periods of intense starburst activity . in particular , sfrs on the order of @xmath210 are often observed in `` classical '' ( i.e. , luminous ) smgs , where they are attributed to late stages of major mergers of massive gas - rich galaxies ( see , e.g. , the recent review by * ? ? ? * and references therein ) . in the next section we use the available data to constrain the ( dynamical ) masses of the quasar hosts , and of their companions , and return to the question of their location in the sfr - mass plane . we discuss the relevance of the major merger interpretation for our sample in . lllccccccccc bright & j0331 & qso & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 & @xmath216 & @xmath217 & @xmath218 & @xmath219 + & j1341 & qso & @xmath220 & @xmath221 & @xmath222 & @xmath223 & @xmath224 & @xmath225 & @xmath226 & @xmath227 & @xmath228 + & j1511 & qso & @xmath229 & @xmath230 & @xmath231 & @xmath232 & @xmath233 & @xmath234 & @xmath235 & @xmath236 & @xmath237 + & & smg & @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath242 & @xmath216 & @xmath243 & @xmath244 & @xmath244 + & & b@xmath245 & @xmath246 & @xmath247 & @xmath248 & @xmath249 & @xmath250 & @xmath250 & @xmath243 & @xmath244 & @xmath244 + + [ -1.75ex ] faint & j0923 & qso & @xmath251 & @xmath252 & @xmath253 & @xmath254 & @xmath255 & @xmath256 & @xmath257 & @xmath258 & @xmath259 + & & smg & @xmath260 & @xmath261 & @xmath262 & @xmath263 & @xmath264 & @xmath265 & @xmath243 & @xmath244 & @xmath244 + & j1328 & qso & @xmath266 & @xmath267 & @xmath268 & @xmath269 & @xmath270 & @xmath216 & @xmath271 & @xmath272 & @xmath273 + & & smg & @xmath274 & @xmath275 & @xmath276 & @xmath277 & @xmath278 & @xmath279 & @xmath243 & @xmath244 & @xmath244 + & j0935 & qso & @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath280 & @xmath281 & @xmath282 & @xmath283 & @xmath284 the measurements may be used to estimate the _ dynamical _ masses ( ) of the quasar host galaxies and the companion smgs . for this purpose , we employ the same prescription as used in several recent studies of ( and co ) emission in high - redshift sources , which assumes the ism is arranged in an inclined , rotating disk ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) : @xmath285 ^ 2\,\sin^2\left(i\right ) \,\,{{\ifmmode m_{\odot } \else $ m_{\odot}$\fi}}\,\ , . \label{eq : mdyn_fwcii}\ ] ] in this prescription , @xmath286 is the size ( deconvolved major axis ) of the -emitting region ( as tabulated in ) . the @xmath287 term reflects the inclination angle between the line of sight and the polar axis of the hosts gas disks , in which the circular velocity is given by @xmath288 . practically , under the assumption of an inclined disk , the inclination angle is often derived from the ( resolved ) morphology of the line - emitting region , following @xmath289 , where @xmath290 and @xmath291 are the semi - minor and semi - major axes of the emitting regions , respectively . such dynamical mass estimates carry significant uncertainties due to the different assumptions required to derive them and given the kind of data available for our systems . in particular , our alma data may not be able to detect the more extended lower surface brightness -emitting regions , thus underestimating @xmath286 , and consequently . in this sense , the estimates derived from our alma data would only trace the very central -emitting regions , on scales of a few kiloparsecs . faint extended emission may also affect the inclination corrections , though this would probably be a subtle effect . more importantly , the inclination corrections for marginally resolved extended sources are somewhat sensitive to non - circular beam shapes , as is the case with some of our data . on the other hand , deeper and higher resolution data may also reveal non - rotating ism components , thus significantly altering the estimates . as noted in , the high central velocity dispersions we observe in our sources are most probably driven by the limited spatial resolution of our alma data and not by such non - rotating components . even if the ism is indeed mostly found in a rotating disk , then determining the underlying ( central ) velocity dispersion would result in lower @xmath292 , and therefore lower . some of these effects are clearly demonstrated whenever increasingly deeper observations were obtained for some @xmath17 sub - mm sources ( e.g. , @xcite for j1148 at @xmath293 ; @xcite for bri 0952 - 0115 at @xmath294 ; and @xcite for br 1202 - 0725 at @xmath18 ) . a more detailed discussion of these , and other uncertainties related to estimates , is also given in @xcite . notwithstanding these uncertainties and caveats , we proceed to estimate the dynamical masses of the sources in our sample , using the prescription given in . for the quasar hosts , the dynamical masses are in the range of @xmath295 ( see ) . the three companion smgs have @xmath296 , @xmath297 , and @xmath298 for the smgs accompanying j1511 , j0923 , and j1328 , respectively . we further derive rough estimates of the inclination angles based on the observed morphology of the emission in all -emitting systems ( given in ) . for the quasar host of j1511 and the accompanying smg , where the -emitting regions are not formally resolved , we use the upper limits on @xmath290 and @xmath299 . the inclination angles we deduce for our sources are in the range @xmath300 . taking these inclination corrections into account , we obtain dynamical masses of @xmath301 for the quasar hosts , while for the companion smgs we have @xmath302 , @xmath303 , and @xmath304 ( for the smgs accompanying j1511 , j0923 , and j1328 , respectively ) . we find no significant difference between the dynamical masses of fir - bright and fir - faint systems . in what follows , we use these inclination - corrected estimates of . we first note that the dynamical masses of the quasar hosts cover a very narrow range , @xmath305 ( i.e. , spanning roughly a factor of 2 ) , and five of the six systems have @xmath306 ( i.e. , spanning less than 0.1 dex ) . interestingly , the latter mass is in excellent agreement with the observed `` knee '' of the _ stellar _ mass function in sf galaxies ( @xmath307 ) , which is known to show very limited evolution up to at least @xmath308 ( e.g. , * ? ? ? * ; * ? ? ? we also note that the dynamical masses of the interacting smgs differ from those of the corresponding quasar hosts by factors of about 0.85 , 0.3 , and 1.8 ( for the j1511 , j0923 , and j1328 systems , respectively ) . given the uncertainties on our estimates mentioned above , these mass ratios are consistent with our interpretation of these interacting systems being _ major _ galaxy mergers ( see below ) . to complement our estimates of dynamical masses , we also derive rough estimates of the _ dust _ and _ gas _ masses in our sources . dust masses are estimated assuming that the fir continuum fluxes measured from our alma data are emitted by optically thin dust , following an sed with @xmath60 and @xmath61 , and further assuming an opacity coefficient of @xmath309 ( following @xcite , for consistency with @xcite ; see also , e.g. , @xcite ) . the dust masses we derive are in the range @xmath310 for the quasar hosts and @xmath311 for the companion smgs . importantly , the dust masses of the quasar hosts comprise @xmath312 of the dynamical masses . this qualitative result is virtually independent of the significant uncertainties involved in the dust mass estimates ( due to the assumptions on the seds and on @xmath313 ) . rough estimates of gas masses can then be inferred by assuming a ( uniform ) gas - to - dust ratio of 100 . these are rather conservative estimates , as several recent studies have shown that the gas - to - dust ratio in high - redshift hosts may be significantly lower ( e.g. , as low as @xmath314 ; * ? ? ? * ; * ? ? ? * and references therein ) . for most of the systems , and particularly the quasar hosts , the gas masses comprise @xmath315 of and reach @xmath316 in only one quasar host ( j1341 ) . adopting the aforementioned lower gas - to - dust ratios would obviously result in yet lower gas - to - dynamical mass ratios . we conclude that the estimates of our sources are dominated , to a large degree , by the stellar components within the galaxies . using our estimates of as proxies for , we again find that all the fir - bright systems are found well above the sf - ms , offset from the relation in by at least 0.5 dex ( j0331 ) and by up to 1.2 dex ( j1341 ) . on the other hand , all the fir - faint quasar hosts , as well as two of the three accompanying smgs ( those of j1511 and j0923 ) , are consistent with the sf - ms , being within about 0.2 dex of the aforementioned relation , which is consistent with the intrinsic scatter associated with it . we now turn to compare the mass growth rates and the masses of the smbhs powering our quasars relative to those of the stellar populations in their host galaxies . for the quasar host galaxies , we assume that the mass grows only due to the formation of new stars at a rate determined by the ce01-based sfrs ( see above ) . for the smbhs , the growth rates are calculated assuming @xmath317 , where is the bolometric luminosity , estimated from the rest - frame uv continuum emission ( see t11 ) , and the radiative efficiency is assumed to be @xmath318 . we find that all systems have @xmath319 ( see ) , with the highest sfr systems j1341 and j1511 having @xmath320 and @xmath321 , respectively . the lower - sfr systems have growth - rate ratios as high as @xmath322 ( median value ) . these growth - rate ratios are consistent with those derived in n14 ( and in * ? ? ? * ) , which is expected given the consistency between the new alma data and the previous measurements . as for the mass comparison , we rely on the dynamical mass estimates derived above and the -based bh masses available from t11 . these estimates used the calibration by @xcite . the more recent calibration by @xcite would have increased by a factor of 1.75 ( @xmath40.24 dex ) , but we chose not to use it for the sake of consistency with our own previous work and with other samples of @xmath323 quasars ( see also * ? ? ? * ; * ? ? ? lists and the bh - to - host mass ratios , which are in the range @xmath324 . shows our estimates of and , along with some similar estimates in local galaxies . for this , we use the subset of elliptical galaxies tabulated in @xcite , for which we take @xmath325 . we note that itself , which is indeed the focus of many studies of various types of local galaxies ( see an extensive discussion in * ? ? ? * ) , is not accessible with our data . the mass and growth - rate ratios we derive can be compared to the typical ratios between the _ masses _ of galaxies and their central smbhs , as observed in the local universe . in the regime of our quasars , these are in the range @xmath326 ( see ; e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? as shows , this is broadly consistent only with the lower end of the @xmath327 range we find in our quasars . moreover , for four of the quasar hosts we find @xmath328 , which is significantly higher than the locally observed value . we stress that the bh - to-_stellar _ mass ratios of our sources would be even higher , thus increasing the discrepancy with the local value , recalling the alternative calibration mentioned above and recalling that @xmath329 ( ) . at the same time , we have shown that itself may be overestimated . similarly high bh - to - host mass ratios were derived for other luminous @xmath17 quasars using similar data and methods ( see , e.g. , @xcite , and references therein , but also @xcite ) . this adds to the evidence for a general trend of increasing @xmath330 with increasing redshift , out to @xmath331 , that is supported by several studies with direct estimates of ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , as well as indirect arguments @xcite . it should be noted , however , that the high agn luminosities of our quasars ( i.e. , above the break in the quasar luminosity function ) may mean that the high @xmath330 values we find do not represent the general ( active ) smbh population at @xmath208 ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? turning back to the ratios , the high - sfr systems in our sample are broadly consistent with what is expected if one assumes that smbhs and their hosts grow `` in tandem '' , obeying the bh - to - stellar mass ratio observed in high - mass systems in the local universe at all times . on the other hand , the high ratios found for the rest of the sample ( four of six quasars ) suggests that in these systems the smbhs are bound to _ _ over__grow the stellar populations . to illustrate the possible evolutionary scenarios for our smbhs and host galaxies , we plot in the expected mass growth in both the bh and stellar components , assuming the observed ( linear ) growth rates are sustained for a short period of 50 myr ( as in * * ) . within this short timespan , most systems will remain broadly consistent with the local range of ratios . if the smbhs would stop growing within a comparably short time - scale , at about @xmath332 , then their host galaxies would have to experience only a slightly longer period of sf activity to reach the corresponding local mass ratios ( @xmath333 ) . however , if the bh growth rates are maintained over longer periods , and/or if the smbhs are instead assumed to be growing exponentially ( i.e. , at constant , instead of constant ) , then the smbhs would significantly overgrow their hosts while approaching the highest bh masses known , @xmath334 , and moving away from the local bh - host relation . in this mass regime , the corresponding local mass ratios are on the order of @xmath335 , which would still require significant sf activity in our objects . our conclusions regarding the relative growth rates are in line with the recent results of @xcite . this study used data for a large sample of some of the most luminous quasars at @xmath336 ( including all of our sources ) and found that the vast majority of systems has @xmath337 , while a small fraction had @xmath338 . assuming that the most luminous quasars at @xmath336 form a continuously evolving population , the @xcite study also suggested that the epoch of fast smbh growth traced by our sources would extend to @xmath331 , perhaps at low duty cycles ( see t11 ) , while the intense sf activity seen in some of the quasar hosts may decrease shortly after @xmath208 . in the context of the @xmath0 quasars we study here , this may indeed mean that our smbhs would reach the high end of the known range , with @xmath334 . the ( decreasing ) sf activity is still required to suffice to reach @xmath335 . we finally highlight the exceptional properties of the j1341 system , which has a high of @xmath339 , an extremely high bh - to - host mass ratio of @xmath340 and a low mass - growth ratio of @xmath341 . all this suggests that the smbh is approaching its final mass , while the host galaxy is forming stars at an intense rate and the system would likely expected to approach the high mass end of the @xmath342 mass ratio ( see ) . this is similar to the over - massive bh cid-947 , recently identified at @xmath343 , and speculated to have experienced an earlier episode of fast eddington - limited growth , to reach the observed @xmath344 and @xmath345 @xcite . and cid-947 at @xmath343 ) are drawn from parent samples of markedly different number densities . ] j1341 may therefore be illustrative of the scenario of early , fast smbh growth to the highest known mass well before @xmath346 , with somewhat longer timescale stellar growth , to eventually reach @xmath335 . given the results of analysis available prior to the new alma observations , the naive expectation for the sample under study was that the fir - bright sources are powered by major mergers between gas - rich galaxies , while the fir - faint sources are evolving secularly or , perhaps , are related to a _ later _ evolutionary phase , where the accreting smbhs may have already affected the sf in their hosts ( see n14 for a detailed discussion ) . moreover , there is strong evidence that the occurrence rate of mergers among agn increases with increasing agn luminosity and redshift @xcite . based on these trends , a high occurrence rate of mergers , in excess of @xmath347 and perhaps as high as @xmath348 , is expected for our @xmath0 , high- ( and high- ) quasars . the new high - resolution alma data allow us to critically revisit these ideas . the small size of our sample naturally limits the scope of our interpretation , however we note that most of the previous studies addressing these questions at comparably high redshifts included yet fewer objects and/or relied on lower - quality data . at face value , our new alma data clearly show that a significant fraction of @xmath0 luminous quasars 50% in our small sample are interacting with companion galaxies of comparable mass , thus supporting the idea that major mergers are a dominant driver of the intense bh and sf activity . moreover , we note that this may constitute a lower limit on the real fraction of interacting quasar hosts , when considering the possibility that additional companions are locate just outside of the alma fields ( i.e. , separated by @xmath349 kpc ) ; that additional close companions are too faint to be detected in our alma data ( i.e. , have @xmath350 ) and/or that some of the isolated quasars are actually in the final stages of a merger with interaction signatures that can only be detected with higher resolution data . however , the properties of the smbhs in our sample , their hosts and companion galaxies under study , highlight several shortcomings of the simplistic merger - driven growth scenario . first , two of the three quasars with robust detections of physically interacting companions ( i.e. , smgs ) are actually among the fir-_faint _ sources , while only one of the fir - bright sources has an interacting companion ( j1511 ) . second , the velocity maps of all the fir - bright systems , and particularly those that lack companions , show evidence for ordered rotation , around an axis that coincides with the centroids of the host galaxies ( and quasars locations ; see and ) . the coincidence of the region of zero velocity with the centroids of the sf activity in the hosts suggests that the redshifted and blueshifted emitting regions are _ not _ tracing ( smaller ) coalescing galaxies . moreover , as noted in , the outer parts of the ism in the fir - bright systems appear to be rotation dominated . this is also seen among the fir - faint systems , although to a lesser extent . we also note that the speculation made in n14 that the fir - faint systems are found in a _ later _ evolutionary stage is disfavored by the new alma data , as ( two of ) the fir - faint systems are seen to be in a rather early stage of a major merger . their lower sfrs will therefore likely increase as the interacting galaxies coalesce . in principle , a possible interpretation for our new alma data may have been that the smbh activity in the fir - faint sources is driven by the `` first passage '' of the interacting quasar hosts with their smg companions . indeed , the high sfrs of the interacting galaxies in the j1511 system may be indicative of what the fir - faint systems would undergo in later stages of the interaction . however , most simulations of major mergers suggest that the smbh would produce a luminous quasar , with high , at the final coalescence phase , and _ not _ in the first passage phase . these same simulations also suggest that the first passage enhances sfr in the interacting hosts , to levels comparable to those found in the later , final coalescence phase ( see , e.g. , @xcite , but also @xcite and @xcite ) . our sample , on the other hand , shows similarly intense smbh growth , with @xmath351 and @xmath352 , for both high- and low - sfr quasar hosts and among systems with and without an interacting companion . any direct comparison with merger simulations is further complicated by the possibility that the companion smgs we detect at larger separations of @xmath353 may have already experienced a much closer passage to the quasar hosts and are observed close to their apocenter . the intense smbh growth may have been triggered during this pericenter passage , when tidal forces were maximal . in this context , we note that any observations of interacting galaxies would be biased toward large separations , due to the longer periods spent at increasingly larger separations . indeed , an inspection of several of the aforementioned simulations suggests that the interacting galaxies are separated by @xmath354 for over 80% of the simulated merger . first and second passages are extremely short , taking up @xmath355 of the time . , may therefore be even higher . ] we therefore conclude that our alma data provide compelling evidence for significant galaxy galaxy interactions ( major mergers ) in some , but not all , quasar hosts . moreover , the links between these interactions and the intense accretion onto the smbhs remains unclear . for the fir - bright sources , we caution that even with the new alma data , we can not completely disprove the possibility that _ all _ these systems are indeed observed in the advanced stages of a major merger . in particular , the evidence for rotation in the ism of some quasar hosts does not by itself disprove a merger scenario , as several studies of low - redshift mergers ( i.e. , ulirgs ) have identified nuclear structures of rotating molecular gas , on scales of a few kiloparsecs ( see , e.g. , @xcite and the discussion in @xcite ) . simulations of ( low - redshift ) major mergers have demonstrated that such gas disks may indeed form several hundreds of myr after the `` peak '' of the merger , once the coalesced nucleus has relaxed ( e.g. , * ? ? ? * ; * ? ? ? on the other hand , the interacting companions we identify among the fir - faint systems can not be related to the sf and smbh activity in a straightforward way , given their large separations . the only way in which _ all _ the quasars in our sample can be explained as triggered by major merger is if the fir - bright systems that lack an interacting companion ( namely j0331 and j1341 ) are observed in the final coalescence stage , while the two interacting fir - faint quasar hosts have already experienced a close first passage that triggered the smbh growth and are observed close to their apocenter . even this scenario fails to account for one of the fir - faint systems ( j0935 ) , which has no companions out to @xmath356 . several sub - mm studies published in recent years have probed the existence of sf galaxies that are accompanying , or indeed interacting with , the hosts of @xmath357 quasars . a prominent example is the interacting system br1202 - 0725 at @xmath18 , which was shown to consist of a pair of interacting smgs , separated by 25 kpc , one of which hosting a luminous quasar , with an additional ( marginal ) detection of a faint agn in the other one @xcite . however , such close interactions are found to be quite rare among high - redshift quasars . several deep optical imaging campaigns in the fields around the known @xmath140 quasars provide no or little robust evidence for physically associated companions ( e.g. , * ? ? ? moreover , the sub - mm ( alma and pdbi ) studies published for most of the known @xmath358 quasars ( by * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) which could have detected companions with sfrs comparable to those we find at separations of up to @xmath4100 kpc report _ no _ such companions . taken at face value , this may suggest a fast increase in the occurrence of mergers among luminous quasar hosts between @xmath359 and @xmath0 that is , within @xmath4350 myr . we note , however , that this discrepancy may be driven by the limited sensitivity and/or limited spatial resolution of the available sub - mm ( alma ) data for some @xmath359 quasars ( e.g. , @xmath44 kpc in @xcite ) . additional deep observations of these @xmath359 sources at sub - kiloparsec resolution may indeed resolve this discrepancy . to conclude the discussion of major mergers , we recall that our smbhs as well as the other @xmath17 quasars mentioned above had to grow continuously and at high rates since very early epochs to account for their high masses ( see t11 , and references therein ) . given the number of interacting companions we find and their separations from the quasars hosts , it seems unlikely that this kind of mergers can be the only driver of such a prolonged period of fast growth . all of the above suggests the epoch of fastest growth of the most massive bhs is driven , at least in part , by mechanisms that are not related to major mergers - such as direct accretion of igm gas ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , minor mergers , and/or galaxy - scale instabilities of the gas or stellar components ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . finally , we note that even if the companion smgs we find are not directly related to the fueling of the fast - growing @xmath0 smbhs , their presence seems to support the idea that rapid early bh growth preferentially takes place in dense large - scale environments ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . to date , such evidence has been highly elusive , with several observational campaigns looking for over - densities of ( rest - frame ) uv - bright galaxies around luminous high - redshift quasars yielding highly ambiguous results ( see , e.g. , @xcite for several examples of such over - dense environments , but also @xcite and some of the systems in @xcite for the contrary ) . our analysis demonstrates that such studies may be significantly biased against dust - obscured high - redshift sf galaxies , thus underestimating the real ( over-)density of galaxies around high - redshift quasars . indeed , the three interacting companions we identify were _ not _ identified in the images of the respective quasars ( n14 ) . a complete census of the cosmic environments of high - redshift , fast - growing smbhs would therefore require a multi - wavelength approach , covering scales of up to a few arc - minutes , and in spectral regimes that are not affected by dust obscuration . this can be done , for example , using compact sub - mm arrays . we have presented alma band-7 observations of luminous quasars at @xmath0 , drawn from a sample of 40 , uv - selected sdss quasars with a wealth of supporting multi - wavelength data . the data probe the rest - frame far - ir continuum emission that arises from cold dust , heated by sf in the host galaxies of the quasars , as well as the emission line that originates from the cold phase of the hosts ism . the alma observations resolve the continuum- and line - emitting regions on scales of @xmath360 kpc . our main findings are as follows . 1 . all quasar hosts are clearly detected and resolved , in both continuum and line emission . the continuum emission suggests intense sf , with the fir - bright sources reaching @xmath361 , consistent with observations of these systems . the quasar hosts exhibit evidence for massive , rotation - dominated gas structures . 2 . three quasar hosts one fir - bright and two fir - faint systems are accompanied by spectroscopically confirmed , interacting companions , with separations in the range @xmath362 kpc and within @xmath363 . the companions themselves are forming stars at rates of a few hundred yr^-1 @xmath5 , slightly lower than the quasar hosts with which they interact . the remaining quasar hosts two fir - bright and one fir - faint lack significant companions . this , combined with the evidence for rotation , may suggest that processes other than major mergers are driving the significant sf activity and fast smbh growth in these systems . the dynamical masses of the quasar hosts , estimated from the lines , are within a factor of @xmath43 of the masses of the interacting companions , supporting our interpretation of these interactions as major mergers . the -based dynamical masses also show that the fir - faint systems are consistent with the `` main sequence '' of star - forming galaxies , while the fir - bright systems are located above it . compared with the bh masses , the -based dynamical host masses are generally lower than what is expected from the locally observed bh - to - host mass ratio . in some of the systems , this discrepancy may grow further , given the high accretion rates of the smbhs . the /@xmath176 ratios in the quasar hosts are consistent with those found in other @xmath17 quasar hosts and smgs and follow the observed trend of declining /@xmath176 with increasing . although our data suggest that the deficit is most probably driven by mechanisms or properties that are intrinsic to the quasar hosts , we do not find evidence for the compactness of the sf regions being the driver of the @xmath364 trend . our analysis clearly demonstrates the wide variety of host galaxy properties , particularly in terms of sfrs and of possible smbh fueling mechanisms , among a relatively uniform population of the fastest - growing smbhs in the early gas - rich universe . it appears that vigorous smbh growth is not necessarily accompanied by extreme sf activity ( i.e. , above what is found in inactive sf galaxies ) and that galaxy galaxy interactions are not a necessary condition for either of the two processes . this broadly supports a scenario where intense smbh and stellar growth in the early universe is driven by secular processes , such as large - scale flows of cold gas , that penetrate into the centers of massive dark matter halos and/or gas or stellar instabilities on smaller scales . our results motivate several paths for follow - up studies to address and test the predictions of the different fueling mechanisms . to robustly determine the role of mergers in the systems that lack companions quasars would require the detection of tidal features ( e.g. , using imaging ) , or mapping the ism kinematics at higher resolution and/or to larger scales ( i.e. , with deeper alma observations ) . obviously , a critical test of the relevance of mergers to the general population of high - redshift quasars necessitates a significantly larger sample , with observations that cover a large field of view while maintaining a high spatial resolution . this can be achieved by extending our analysis to additional @xmath208 and @xmath365 quasars . we have recently guaranteed cycle-4 alma band-7 time to observe 12 additional @xmath0 quasars from the t11 sample , which would allow us to study the host galaxies and close environments of a total of 18 fast - growing , @xmath0 smbhs . we thank the anonymous referee for the very constructive comments that helped us to improve our manuscript . we thank k. schawinski , l. mayer , r. teyssier , p. capelo , m. dotti and d. fiacconi for useful discussions . this paper makes use of the following alma data : ads / jao.alma#2013.1.01153.s . alma is a partnership of eso ( representing its member states ) , nsf ( usa ) and nins ( japan ) , together with nrc ( canada ) , nsc and asiaa ( taiwan ) , and kasi ( republic of korea ) , in cooperation with the republic of chile . the joint alma observatory is operated by eso , aui / nrao , and naoj . this work made use of the matlab package for astronomy and astrophysics @xcite . acknowledges support by the israel science foundation grant 284/13 . c.c . gratefully acknowledges support from the swiss national science foundation professorship grant pp00p2_138979/1 . c.c . also acknowledges funding from the european union s horizon 2020 research and innovation programme under the marie sklodowska - curie grant agreement no 664931 . acknowledges support by the science and technology facilities council ( stfc ) and the erc advanced grant 695671 `` quench '' . mcmullin , j. p. , waters , b. , schiebel , d. , young , w. , & golap , k. 2007 , in asp conf . 376 , astronomical data analysis software and systems xvi , ed . r. a. shaw , f. hill , & d. j. bell ( san francisco , ca : asp ) , 127

we present new alma band-7 data for a sample of six luminous quasars at @xmath0 , powered by fast - growing supermassive black holes ( smbhs ) with rather uniform properties : the typical accretion rates and black hole masses are @xmath1 and @xmath2 . our sample consists of three `` fir - bright '' sources which were individually detected in previous /spire observations , with star formation rates of @xmath3 , and three `` fir - faint '' sources for which stacking analysis implies a typical sfr of @xmath4400 yr^-1 @xmath5 . the dusty interstellar medium in the hosts of all six quasars is clearly detected in the alma data and resolved on scales of @xmath42 kpc , in both continuum ( @xmath6 ) and line emission . the continuum emission is in good agreement with the expectations from the data , confirming the intense sf activity in the quasar hosts . importantly , we detect companion sub - millimeter galaxies ( smgs ) for three sources one fir - bright and two fir - faint , separated by @xmath7 and @xmath8 from the quasar hosts . the -based dynamical mass estimates for the interacting smgs are within a factor of @xmath43 of the quasar hosts masses , while the continuum emission implies @xmath9 . our alma data therefore clearly support the idea that major mergers are important drivers for rapid early smbh growth . however , the fact that not all high - sfr quasar hosts are accompanied by interacting smgs and the gas kinematics as observed by alma suggest that other processes may be fueling these systems . our analysis thus demonstrates the diversity of host galaxy properties and gas accretion mechanisms associated with early and rapid smbh growth .

in recent years it has been assumed that the majority , if not all , large galaxies house at their cores a supermassive black hole ( smbh ) ranging from many hundreds of thousands to billions of times the mass of our sun ( see the recent review by * ? ? ? these objects strongly influence , or are strongly influenced by , the properties of their hosts , as evidenced by certain galaxy - wide properties scaling with the masses of these smbhs , such as in the m-@xmath9 relation @xcite . however , the cause of this link is not well understood . in order to better understand the smbh demographic ( particularly at the low mass end where samples are of very limited size , see e.g. * ? ? ? * ) , and their co - evolution with their hosts , further examples must be studied in dwarf and distant galaxies . however , in the case of a distant / dwarf galaxy lacking an active galactic nucleus ( agn ) , even confirming the existence of an smbh can be difficult . obtaining spatially resolved velocity dispersion measurements across the the galaxy , looking for the gravitational influence of a massive central body , becomes impossible with current instrumentation when the angular size of the galaxy becomes too small . the detection and correct identification of a tidal disruption flare ( tdf ) , however , unequivocally shows the existence of an smbh within the flare s host , irrespective of the host s angular size or apparent magnitude . a tdf is the luminous burst produced by the capture , disruption and subsequent accretion of a star onto a smbh . they are typically characterised by a short - lived ( months to years ) transient with a high temperature ( @xmath10 k ) thermal spectral energy distribution ( sed ) . they occur whenever a star passes within its tidal radius of the central smbh ( @xmath11 ) while remaining outside of the schwarzschild radius ( @xmath12 ) @xcite , since crossing the latter would lead to the star being swallowed whole and thus produce no visible flare . since @xmath13 and @xmath14 , for a given radius ( and mass ) of star , there exists a maximum black hole mass for which the disruption will occur outside @xmath15 ( although in practice the spin of the black hole is also important , e.g. * ? ? ? hence white dwarfs will only be disrupted by intermediate mass black holes ( @xmath16 m@xmath17 ) , while main sequence stars can produce flares with black holes up to @xmath18 m@xmath17 and giants may be disrupted even around the most massive known black holes , though the likely accretion rates and timescales are much longer than for more compact systems @xcite . ultimately , observations of confirmed tidal flares may offer a new approach to measurements of black hole masses @xcite and spins @xcite . a number of tdf candidates have been found in the uv ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , soft x - rays ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the optical ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? . however , most of these were detected at low redshift , often less than @xmath19 due to the necessity of multi - epoch photometry and astrometry , and the relatively shallow surveys from which they were selected . probing these events to much greater distances , thus providing a way to characterise the smbh mass distribution as a function of redshift , would require much more sensitive , high cadence surveys ( such as the lsst , * ? ? ? * ) or possibly chance gravitational lensing events ( e.g. the z=3.3 candidate , * ? ? ? however , a new sub - class of these events , potentially observable out to much larger distances with current observing platforms , has provided us with a new way to observe these transients . the first such event _ swift _ j164449.3 + 573451 ( henceforth _ swift _ j1644 + 57 ) , detected in 2011 march , exhibited extremely unusual high energy behaviour . detected initially as a gamma - ray burst ( grb ) trigger with a long duration ( @xmath71000s , * ? ? ? * ) , the event remained bright and variable for several days , retriggering _ swift_/bat on a further three occasions over the course of 48 hours @xcite making it clear this was not a standard grb , short or long . x - ray monitoring with _ swift_/xrt showed a luminous flaring source that settled into a several - day long plateau before following an approximately power law decay , all with considerable short - term variability superimposed upon it . the source was discovered to lie at a cosmological distance , coincident with the centre ( @xmath20 pc to @xmath21 ) of a faint star - forming host at a spectroscopically confirmed redshift of @xmath22 @xcite , implying the isotropic x - ray luminosity of the event was @xmath23 even at late times . in contrast , the coincident optical / infrared transient peaked at a more modest @xmath24 even after correction for moderate internal extinction @xcite . radio observations of _ swift _ j1644 + 57 with the evla detected a rising unresolved source with an equipartition radius that implied a moderately relativistic expansion with lorentz factor @xmath25 and a formation epoch that coincided with the initial @xmath0-ray detection @xcite . the energetics measured by @xcite , @xmath26 erg at 22 days post burst , also corresponded to the eddington luminosity for accretion onto a @xmath27 black hole . @xcite used a purely observational relation between x - ray luminosity , radio luminosity and the mass of black holes ( ranging from high mass seyferts to stellar mass black holes ) to estimate the mass of the smbh that produced _ swift _ j1644 + 57 . the resulting weak constraint of @xmath28 , was consistent with the black hole mass of @xmath29 estimated by @xcite via the spheroid mass - black hole mass scaling relation of @xcite . these unique broadband properties marked _ swift _ j1644 + 57 as a new class of transient . they suggested the detection of a flare situated in the nuclear region of a dwarf galaxy with energetics and short - term variability consistent with an accretion event onto the central smbh . but the lack of any previous activity in @xmath0-rays during the lifetime of swift @xcite , the spectroscopic classification of the galaxy as star - forming @xcite and the radio formation epoch @xcite all indicated the accretion event was new and not part of any ongoing nuclear activity . thus , the favoured explanation was taken to be that of the tidal disruption of a solar - type star that had also launched a moderately relativistic jet . however , this relativistic tidal disruption flare ( rtdf ) interpretation was not unchallenged , and other mechanisms , perhaps involving the tidal capture of a white dwarf @xcite or massive star core collapse @xcite , were postulated . a second example , _ swift _ j2058.4 + 0516 ( _ swift _ j2058 + 05 , * ? ? ? * ) , was detected in may 2011 . while apparently much fainter than _ swift _ j1644 + 57 , this was largely due to the much greater redshift ( @xmath30 , c.f . @xmath31 for _ swift _ j1644 + 57 ) and this was in fact a more luminous event . the bulk properties ( peak luminosity , total energy , longevity , steep late - time cut - off ) of the event matched well with those of _ swift _ j1644 + 57 @xcite although there were several important differences . the x - ray lightcurve decline was steeper ( a power law with index @xmath32 , while _ swift _ j1644 + 57 was remarkably near the theoretical @xmath33 @xcite , although recent numerical simulations suggest that accounting for stellar structure and closeness of approach , -2.2 is expected in half of all disruptions @xcite ) and the x - ray spectrum was somewhat harder ( photon index @xmath34 , c.f . @xmath35 ) . in addition , the observed radio spectrum of _ swift _ j2058 + 05 was very flat ( @xmath36 ) which constrasts strongly with the optically thick spectrum ( @xmath37 ) of _ swift _ j1644 + 57 @xcite . despite these differences , it was suggested that _ swift _ j2058 + 0516 is the second member of the rtdf class , and in this case the observational differences may offer important diagnostics of the disruption process . another potentially related class of transient is that of the ultra - long grbs ( ulgrbs , @xcite ) . these events exhibit @xmath38ray emission lasting for thousands of seconds ( 1 - 2 orders of magnitude less than the flares described above , but an order of magnitude longer than most grbs ) . while multiple possible paths to their creation have been suggested ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , it is also possible that they are related to tdfs with a relativistic component , although in this case their shorter timescales would imply a white dwarf disruption @xcite . relativistic tdfs are potentially observable out to much greater distances than their non - relativistic thermal tdf cousins due to their beamed emission , analogous to how grbs have been shown to be detectable to extreme redshifts ( e.g. grb090423 , * ? ? ? * ; * ? ? ? @xcite estimated that _ swift _ j1644 + 57 would have been observable out to @xmath39 , while the redshift of _ swift _ j2058 + 05 @xcite ( @xmath40 ) shows certain members may be observable at even larger distances . @xcite suggest that the radio component would be detectable out to @xmath41 , potentially making large scale radio surveys a powerful method for the detection of these events . such rtdfs also evolve on very short timescales compared to agn and so offer a way to study jetted accretion events across their whole lifetimes on human timescales , evidenced by observations of _ swift _ j1644 + 57 showing that the jet has now apparently shut - off @xcite . this is a virtual impossibility in the vastly longer lived agn duty cycles which may last in excess of @xmath42 years ( e.g. * ? ? ? in addition , the relativistic jets emitted by these events have been suggested as a possible source of ultra - high energy cosmic rays ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . given the potential importance of studying these events , it is concerning that it is unclear whether _ swift _ j2058 + 05 would have attracted such detailed follow - up in the absence of _ swift _ j1644 + 57 , considering to its less immediately impressive nature . in addition , the notable temporal coincidence of the two bursts , being only two months apart in the then @xmath43 years that swift had been operating , led to the suggestion at the time that further examples within the _ swift _ archive may have been overlooked ( e.g. * ? ? ? * ) . here we present observations of _ swift _ j1112.2 - 8238 ( henceforth _ swift _ j1112 - 8238 ) which was detected by the _ burst alert telescope in 2011 june and whose nature has to date been uncertain . our spectroscopic observations establish a cosmological redshift for the transient , and we demonstrate optical variability close to the nucleus of a faint galaxy . we analyse the inferred physical properties , comparing them to the properties of previous _ swift _ flare classifications including the recently discovered ultra - long grbs , and to the established relativistic tdf candidates , leading us to suggest that _ swift _ j1112 - 8238 is also a candidate rtdf . all magnitudes presented in this paper are in the ab magnitude system . where necessary , we use a standard @xmath44cdm cosmology with @xmath4570kms@xmath4mpc@xmath4 , @xmath46 and @xmath47 . the outburst of _ swift _ j1112 - 8238 was originally discovered by the _ swift _ telescope @xcite in a four day integration by the burst alert telescope ( bat , * ? ? ? * ) between 2011 june 16 and 19 ( mjd 55728 - 55731 , * ? ? ? we choose to set the trigger time to 2011 june 16 ut 00:01 , although in practice a precise trigger time is poorly defined . the count rate in gamma rays across this period was @xmath48ph s@xmath4@xmath49 with a peak daily average rate of @xmath50 ph s@xmath4@xmath49 recorded on the 16th @xcite . this peak , though high , was not in itself sufficient to trigger the automated transient monitor which has a @xmath51 burst detection threshold @xcite . we utilise the available bat daily average light curves extending back as far as the launch of _ swift _ in 2005 and confirm that there was no pre - trigger or post - burst activity above a @xmath51 threshold level at the flare s position over any 4-day window . we also note that @xcite report no evidence for additional flares from the source . the light curve ( @xmath0-ray , x - ray , optical ) is shown in figure 1 . initial x - ray data was obtained by the _ swift _ x - ray telescope ( xrt , * ? ? ? * ) in a 3000s target of opportunity observation approximately 10 days after the initial trigger ( mjd 55741.7 , * ? ? ? the source was well detected with an observed flux of @xmath52 ergs s@xmath4@xmath49 . an x - ray monitoring programme continued for a further 30 days with all observations obtained in photon counting ( pc ) mode . we obtained reduced xrt products from the _ swift _ archiveobjects ] , created using the techniques outlined in @xcite . the enhanced x - ray position derived from the uvot boresight correction is ra= 11:11:47.32 dec=-82:38:44.2 ( j2000 ) with a 90% error radius of @xmath53 . the combined spectrum of all available pc - mode data is well fit by an absorbed power - law of photon index @xmath54 and n@xmath55 ( int ) = @xmath56 @xmath49 consistent with the galactic value of @xmath57@xmath49 @xcite . although the number of counts is small , there is no evidence for spectral evolution through the observations . the light curve over the same period exhibits a gradual decay but with marked variability ( a factor of @xmath35 in flux ) between individual snapshots ( uninterrupted pointings ) . we obtained further late time observations in april 2014 , with a total xrt exposure time of 6960.3 s ( in pc mode ) . this observation provides an upper limit on the source flux of @xmath58 ergs s@xmath4 @xmath49 ( 99% , determined via the bayesian method of * ? ? ? this is a factor of @xmath59 fainter than the peak luminosity confirming the source s transient nature . -band finding chart for _ swift _ j1112 - 8238 . ( right ) a comparison between the source at @xmath720 days ( top ) and at @xmath71.5 years ( bottom ) post trigger , each panel 15@xmath60 across . the extended host s structure is far clearer in the later epoch , due in part to both the decline of the optical transient and the greatly improved seeing . [ fig : finding],width=317 ] following a uvot non - detection ( b @xmath61 mag , * ? ? ? * ) made at the beginning of the x - ray monitoring programme , we obtained observations in the @xmath62 band with the gemini multiple object spectrograph on gemini south ( gmos - s , * ? ? ? * ) at 2011 july 3 ut 00:58 , 17 days after initial trigger @xcite . we performed later follow - up with gmos - s in the @xmath63 and @xmath62-bands at 1.5 years post - trigger ( starting 2012 december 13 ut 06:50 ) , and with the focal reducer and low dispersion spectrograph 2 ( fors2 ) on the very large telescope ( vlt ) at 2 years post - trigger ( starting 2013 august 31 ut 23:31 ) in @xmath64 and @xmath65 . in addition , as part of the gmos - s spectroscopic follow - up , a number of short exposure acquisition images were taken in @xmath63 ( starting at 2012 december 16 ut 07:30 and 2012 december 23 ut 05:16 ) and in @xmath62 ( starting at 2014 january 3 ut 07:01 , @xmath66 years post trigger ) . all of this imaging was reduced using standard iraf and esorex data reduction techniques . we note that the presence of a nearby bright star ( r = 15.8 mag at an angular distance of @xmath67 , figure [ fig : finding ] ) complicated the analysis of this source . we remove the majority of the flux from this contaminating star by subtracting a model stellar psf , constructed as a median - averaged radial light profile for the star in question . we also considered models for the psf from stars elsewhere in the image , subtracting a rotated copy of the contaminating star , or modelling the star directly via moffat or multiple gaussian fits . all these methods were hampered by the existence of other objects close to the star , or left clear residuals in the data . photometric calibrations for the @xmath62/@xmath64 band was completed through comparison with observations of photometric standards analysed via the esorex fors2 pipeline , the expected systematic offset between the gmos @xmath62 and fors2 @xmath64 filters having been deemed negligible in this low signal to noise regime . the non - standard filter @xmath65 was instead calibrated through comparison with the fors2 standard star , feige 110 , which was observed within a few nights of our observations . finally the @xmath63 band was calibrated with reference to the gemini standard zeropoints . the resultant photometry is detailed in table [ tab : photom ] and plotted in figure [ fig : lightcurve ] . the early time observations showed a point - like source while later observations ( @xmath681yr ) reveal emission with a flux a factor @xmath35 lower than recorded at early times . modelled photometry of the late time emission with a srsic profile using galfit @xcite was consistent with the aperture photometry detailed above , while psf - matched point - source photometry ( completed by scaling a psf built from the image ) yields results a magnitude dimmer , indicating the late time source is extended . the photometry has been corrected for galactic extinction , with e(b - v ) @xmath69 , based on values derived from @xcite and accessed via the nasa / ipac infrared science archive . the individual bandpass corrections were approximated from the corresponding sdss filter corrections and thus have a minor systematic uncertainty not included in table [ tab : photom ] . ._swift _ j1112 - 8238 optical photometry . limits are stated to 3@xmath9 . photometry is presented without host subtraction , although it is likely that the late epochs represent the host ; that is , not significantly contaminated by transient light . note the @xmath62 gmos - s magnitudes were calculated using relative photometry from the vlt @xmath64-band image and so have a minor systematic uncertainty not included here . all observation times are measured from the beginning of the first day of the 4 day _ swift _ trigger observation ( 2011 june 16 ut 00:01 ) . the seeing of each observation is included as it affects the contamination from the nearby bright star [ cols="^,^,^,^,^,^,^ " , ] optical longslit spectroscopy of _ swift _ j1112.2 - 8238 was obtained on gmos - s on 2012 december 16 and 23 using the r400_g5325 grating and independently on fors2 using the 300i+11 grism on 2013 september 5 . the gmos - s spectra had a combined integration time of 2400 seconds ( 4 @xmath70 600 ) with spectral resolution of .17ex7 and a spectral range of 3870 8170 . the fors2 spectrum also had an integration time of 2400 seconds ( 4 @xmath70 600 ) with spectral resolution of .17ex12 and a spectral range of 510011000 . the standard recommended gemini iraf and fors2 esorex data reduction was carried out on the appropriate spectra . in all spectra , a single , weak emission feature was observed at @xmath71@xmath72 ( figure [ fig : spectrum ] ) , with a significance of @xmath73 in the gmos spectrum . no continuum flux , or additional emission lines were seen . the line does not lie at the position of any common zero redshift features . it is offset by @xmath74 km s@xmath4 from the he 7060 line that is sometimes seen in accreting binaries @xcite . however , in these binaries the line is broad , and many other emission features are seen . in addition , the existence of an underlying extended source , interpreted as the host of the transient , greatly reduces the probability of a galactic origin as nebulae are the only galactic source likely to be resolvable , and these typically show multiple emission lines . this indicates that _ swift _ j1112 - 8238 is not a galactic source . the non - detection of other lines proximate in wavelength disfavours the identification of this line as either [ oiii](@xmath754959,5007 ) or h@xmath76 at @xmath77 , since in either case we would expect to observe the other lines . if the line were h@xmath78 at @xmath79 , we may expect to observe either [ nii@xmath80 \lambda 6584 $ ] , or h@xmath76 and [ oiii ] , since all lie within the spectral window covered by our gmos observations . the expected h@xmath76 flux can be calculated directly ( under the assumption the observed line is h@xmath78 , and that the host galaxy extinction is minimal , as implied by the x - ray absorption ) . however , the combination of grating efficiency and galactic reddening mean that we do not expect to observe h@xmath76 in our observations at @xmath81 . the [ oiii ] lines can frequently be substantially brighter than h@xmath76 , and for a galaxy of metallicity @xmath82 , consistent with the inferred absolute magnitude at @xmath79 ( rest frame @xmath83 , * ? ? ? * ) , we would expect [ oiii ] ( @xmath75 5007 ) to be a factor @xmath84 brighter than h@xmath76 . accounting for foreground extinction and grating efficiency as before , we estimate we would expect to observe it at @xmath85 , whereas no line is present at this location . any emission at the location of [ nii ] @xmath86 would be well below the detection limit given this assumed metallicity . we also note that at @xmath79 the absolute magnitude of the galaxy of @xmath87 would be unusually faint . given these combined constraints we disfavour the origin of the line as h@xmath78 . hence we identify the line as oii ( @xmath75 3727 ) at a redshift @xmath88 . in this case , the redward emission lines are beyond the range of our gmos spectroscopy , and lie in bright sky lines in our fors observations , precluding their detection . the low resolution of the spectra means that we are unable to resolve the doublet in this case . this interpretation is supported by the observed galaxy colours . after correction for foreground extinction they are relatively red in @xmath89 , and bluer @xmath90 ( based on the @xmath91 day gemini @xmath62 band photometry ) . although the errors are large , this is consistent with the presence of a balmer break between the @xmath92 and @xmath93bands , as might be expected for @xmath1 . our spectroscopic observations have confirmed the existence of a single emission line that is inconsistent with any zero redshift lines and consistent with our adopted redshift , @xmath88 . in this section we provide a short summary of the inferred rest - frame transient and host properties and then compare them to the properties of possible progenitors . at the time of the first x - ray observations ( 10 days post trigger ) , the isotropic x - ray luminosity of the source was @xmath94ergs@xmath4 . the source showed an approximate power law decay with time of t@xmath95 . however this had considerable short - timescale variability superimposed upon it , with factor of 2 differences in flux on timescales of a few thousand seconds ( @xmath96s after the initial outburst ) . the x - ray spectrum was well fit by a power law spectrum with @xmath97 and there was no evidence of spectral evolution . while multi - component fits to the time series data involving a broken power law or a flare produce statistically better fits , this may simply be due to the intrinsic short - term variability of the source and the sparse sampling of the x - ray lightcurve , precluding the inference of more detailed information about the source . by assuming the late - time optical epochs represent host level flux that is uncontaminated by transient light , we can can subtract this from our earlier observations . this was done through the use of the image subtraction software isis @xcite . we aligned , convolved and subtracted the late time ( @xmath98 years ) image from the early time ( 17 and 21 days ) images in @xmath62 . the subtractions left a clear , point source residual in each image with an inferred position that lay at 0.11@xmath99 and 0.22@xmath100 ( 1@xmath9 ) from the centroid of the host galaxy , determined using a srsic profile fit to the late - time image using the galfit software package , as shown in figure [ fig : centroid ] . the error on the host centroid position is determined under the assumption of a gaussian profile with fwhm equal to the half light radius from the galfit srsic fit . the apparent asymmetry of the host means that this represents a lower limit on the true error in the centroid position . at the inferred redshift , our tightest constraint places the transient @xmath101kpc from the centre of its host , for which the half light radius is @xmath84kpc . the host subtraction also allows us to isolate the optical transient light , determining an absolute magnitude of m@xmath102 , equating to a luminosity of @xmath103ergs@xmath4 . the underlying host has a comparable absolute magnitude , m@xmath104 ( rest frame @xmath105 at @xmath1 ) . based on the luminosity function of galaxies from @xcite this places it near @xmath106 at @xmath107 ( at a redshift of 0.89 , the @xmath62-band equates roughly to rest frame @xmath108-band for which , in the redshift range @xmath109 , the @xmath106 magnitude is -21.7 ) . _ swift _ detected gamma - ray bursts ( grbs ) , are typically detected on timescales much shorter than those for _ j1112 - 8238 . the majority arise from standard rate triggers , although a significant minority are longer - lived and trigger the detector via image triggers , sometimes on timescales of @xmath110 s. however , even the ultra - long grbs @xcite that have durations of @xmath111 s are much shorter than _ swift _ j1112 - 8238 , whose several day long @xmath38ray emission would imply a duration ( if defined as @xmath112 as for grbs ) of closer to @xmath113s . hence on the basis of the @xmath0-ray properties alone , _ swift _ j1112 - 8238 is a much closer analog with _ swift_j1644 + 57 and _ swift _ j2058 + 0516 than with any identified population of grbs . the x - ray properties are also apparently distinct , since the inferred isotropic x - ray luminosity lies an order of magnitude above grbs at a similar epoch ( see e.g. * ? ? ? * ; * ? ? ? * ) , and grb afterglows at such late times seldom show such pronounced variability ( likely due to the lack of engine activity ) . despite the longevity of the gamma - ray emission in ulgrbs , their late time afterglows are generally consistent with , if not slightly fainter than @xcite , those of normal lgrbs , and so the x - ray properties also would suggest a physically distinct system . the optical properties of _ swift _ j1112 - 8238 are rather less conclusive . the optical transient luminosity is comparable with the brightest end of the grb afterglow distribution ( e.g. * ? ? ? * ) , although given the x - ray brightness the inferred x - ray to optical spectral slope is very flat ( @xmath114 ) . if the emission mechanisms were similar to grbs this would identify the counterpart of _ swift _ j1112 - 8238 as a dark burst , and would imply significant extinction @xcite , the correction for which would make the afterglow the brightest seen . alternatively , one may ascribe rather different emission mechanisms to the counterpart to _ swift _ j1112 - 8238 , in which case little extinction may be needed . it is interesting to note in this regard that _ swift _ j2058 + 0516 also has a very flat @xmath115 , despite a strong uv - sed that implied little extinction @xcite . finally , one can also contrast the locations of _ swift _ j1112 - 8238 with those of grbs . long grbs tend to trace the brightest regions of their host @xcite which , given the low spatial resolution of the optical images and small angular size of the galaxy , might show itself as a coincidence of the transient position and the host centroid . indeed , in the study of @xcite approximately 1/6 of the bursts were consistent with the brightest pixels in their host galaxies at _ hst _ resolution . since we currently lack such high resolution images the strength of the association of _ swift _ j1112 - 8238 with its host nucleus is rather weaker than in the cases of _ swift _ j1644 + 57 and _ swift _ j2058 + 0516 . however , in the majority of other regards its properties find a much better match with these events than either with normal long - grbs , of the ultra - long grb population . the apparent coincidence of the transient position and the host centroid makes an association with the central supermassive black hole of the galaxy plausible , and therefore possibly with ongoing agn activity . no catalogued source is consistent with the position of _ swift _ j1112 - 8238 in either the sumss 843ghz survey ( 60% complete down to 6mjy , 100% to 8mjy , * ? ? ? * ; * ? ? ? * ) or the at20 g 20ghz survey ( 91% complete to 100mjy , * ? ? ? this places limits on the pre - flare underlying radio emission of the host to the @xmath116whz@xmath4 level , which is only capable of ruling out the most luminous bl lac type objects @xcite . however , while the x - ray luminosity of the brightest blazar flares can reach the levels observed in _ swift _ j1112 - 8238 , this is generally accompanied by optical emission many magnitudes brighter than presented here , as seen in figure [ fig : x - ray - opt ] . in addition , our late - time x - ray limit places constraints on any underlying activity to a limit of l@xmath117ergs@xmath4 , fainter than the majority of quasars . for these reasons , we disfavour the identification of this flare as a blazar flare . the association of the optical flare with the inferred location of the smbh may indicate the discovery of a new tidal disruption flare . in order to determine if this is plausible , we estimate the mass of the black hole expected to occur within a galaxy of this size . @xcite measure mass to light ratios of galaxies for a given redshift and rest - frame @xmath118-@xmath119 colour . at a redshift of 0.89 , this equates roughly to a @xmath120-@xmath121 band colour in the observer frame . based on an @xmath120-@xmath121 colour of @xmath122 the mass to light ratio is @xmath70.1 , and , coupled with the rest - frame @xmath118-band ( observer frame @xmath120-band ) absolute magnitude of -21.7 , implies a relatively high galaxy mass of @xmath123m@xmath17 . the stellar mass to black hole mass scaling relation of @xcite , produces a lower estimate for the smbh mass of @xmath124m@xmath17 ( although there is considerable scatter in this relation and it is unclear whether the relation is applicable to such low masses ) which is approximately consistent with the result obtained using the method from @xcite of @xmath125 ( by assuming that the stellar mass estimate represents an upper limit on the bulge mass of the host ) . both of these latter estimates are well within the @xmath126m@xmath17 limit for a sun - like star to be disrupted by a smbh and produce a visible tdf , making a tdf origin plausible . from figure [ fig : x - ray - opt ] , we note that the optical absolute magnitude and x - ray luminosity of _ swift _ j1112 - 8238 places it an region of phase space that is devoid of any sources with the exception of the aforementioned relativistic tdf candidates _ swift _ j1644 + 57 and _ swift _ j2058 + 05 . these candidates also match well with this new flare in their late - time x - ray lightcurves as shown in figure [ fig : lightcurve ] , particularly in the case of _ swift _ j1644 + 57 . the overall power law decay observed over the 30 days of _ swift_-xrt follow - up of _ swift _ j1112 - 82 is somewhat shallower than that of the other candidates with an index of @xmath127 . j2058 + 05 had a much steeper decay at a similar epoch with an index of @xmath32 , while _ swift _ j1644 + 57 had a late - time decay remarkably close to the t@xmath128 relation suggested to be a feature of tdf lightcurves @xcite . however , this decay index is somewhat sensitive to the choice of @xmath129 , which in this case is poorly defined , due in part to the unusual trigger method . further , while often @xmath129 is taken to be the time at which the flare becomes observable , the true @xmath129 occurs some time earlier at the point of return of the most bound material which may precede emission by several days . in order to be consistent with a @xmath130 decay , the `` true '' @xmath129 would have to have been @xmath131 before the start of the _ swift _ detection image . this may not be unreasonable , since _ swift _ j1644 + 57 was active at least 4 days prior to its first grb trigger , and had a @xmath132 detection on a single day , 14 days earlier @xcite . constraints on @xmath129 have been attempted in detailed models of previous flares ( e.g. * ? ? ? * ) , however the lack of comprehensive follow - up precludes that possibility in this case . perhaps even more importantly , the short duration over which observations were made also makes it difficult to determine the behaviour of the lightcurve within the context of the longer term emission . indeed , _ swift _ j1644 + 57 s lightcurve was relatively flat at a similar epoch . calculations considering more detailed transport of material through the disc point to a more complex picture , in which the t@xmath128 decline is only present in certain bands and over a rather restricted range of time @xcite , while even more recent calculations suggest that the @xmath133 decay seen in _ swift _ j2058 + 0516 should be present in half of disruptions @xcite . these predictions show that the x - ray flux can plateau over a period of tens of days after the initial disruption meaning the shallow decay of _ swift _ j1112 - 8238 can not place strong constraints on its nature . however it should be noted that these simulations concern the disk emission , whereas , in relativistic tdfs , the x - ray emission is thought to be dominated by the jet . it is unclear if the assumption of a direct correlation between the jet and disc emission is reasonable . spectrally , the low number of counts recorded in _ j1112.2 - 8238 restricts the information that can be extracted . however , the spectrum is well fit with a single , absorbed power - law with a relatively hard spectral index @xmath134 , ( w - stat / dof = 574/586 ) . this is somewhat harder than the late time power - law index in _ swift _ j1644 + 57 ( @xmath25 ) or in _ swift _ j2058 + 0516 ( @xmath135 ) . one area in which previous rtdfs differ is in the apparent correlation between hardness and flux . _ swift _ j1644 + 57 exhibits spectral softening as it fades @xcite , while _ swift _ j2058 + 0516 appears to harden @xcite . for _ swift _ j1112.2 - 8238 it is not possible to discern any variation in spectrum with flux level . the rapid variability observed in the x - ray emission can place constraints on the nature of the emission region . while the variability is not as dramatic as that observed in _ swift _ j1644 + 57 , where factor of 100 changes in flux were observed on timescales of @xmath136 seconds , there is still evidence for factor of 2 variability on timescales of a few thousand seconds . unfortunately the brightness of the source precludes timing at much higher resolution , and so light - travel time arguments would only place weak constraints on the size of the emitting region ( @xmath137 cm , or 100 @xmath15 for a @xmath138 m@xmath17 black hole ) . more compellingly , the gamma ray emission at the time of the first xrt observations is close to the eddington luminosity of a 10@xmath139m@xmath17 black hole , and an extrapolation to early times suggests it was brighter still . the expected black hole mass is a factor of several hundred small than this , it is unlikely that a black hole could accrete at such high super - eddington rates , so while the constraints on beaming are weaker than for _ swift _ j1644 + 57 we still believe this is the most likely explanation for _ swift _ j1112 - 8238 . the lack of more comprehensive optical follow - up precludes the building of an optical sed which would help distinguish between the thermal seds of previous tdfs , ( e.g. asassn-14ae , @xcite ; ps1 - 10jh , @xcite ) , for which the peak absolute magnitudes are loosely consistent , and the differing , non - thermal emission mechanisms suggested in @xcite and @xcite for rtdf candidates . one of these models involves a blazar - analogue combination of inverse compton emission at high frequencies ( x - ray/@xmath0 ) with a second peak at low frequencies ( optical etc . ) from synchotron emission . alternatively the emission in different wavebands may come from spatially separate emission regions . in the lightcurves of _ j1644 + 57 , limits on optical / radio short - term variability set the emission apart from the rapidly varying high energy emission . under the assumption of a spherical emitting region with a blackbody temperature of @xmath140k ( @xmath141k ) , the radius of the region emitting optical light in _ swift _ j1112.2 - 8238 would be about @xmath142 cm ( @xmath143 cm ) . this is approximately consistent with 3 ( 10 ) times the tidal radius of a of sun - like star around a @xmath42m@xmath144 black hole . this result is similar to those obtained from analysis of optical tdfs , perhaps unsurprisingly as the optical luminosity of _ swift _ j1112 - 82 is only a factor of a few higher than seen in some other tdfs ( e.g. * ? ? ? * ; * ? ? ? this may suggest a common mechanism for the optical emission from both relativistic and thermal tdfs . it is also interesting to note _ swift _ j1112 - 8238 shows a sharp decline in its x - ray flux at @xmath145 days post trigger . this may be indicative of dipping as seen in _ swift _ j1644 + 57 @xcite at similar times , or perhaps of a longer term cessation of activity as identified in _ swift _ j1644 + 57 at much later epochs of @xmath98 years @xcite and similarly in _ swift _ j2058 + 05 @xcite . in any case , the final epoch of observations results in a limit which is significantly below the extrapolation of the early emission , requiring either a steepening of the decay or a rapid drop . this suggests broad similarities between the different events , although the sparse sampling of _ swift _ j1112 - 8238 , makes it difficult to rule out alternate interpretations . with the previous rtdf candidates , a variable radio source with a measured lorentz factor of @xmath35 or higher was detected @xcite . in addition the inferred formation epoch from _ swift _ j1644 + 57 implied a recently formed source , consistent with the start of the higher energy emission . this helped lead to the suggestion of a newly formed relativistic jet that accompanied the tdf . the lack of post - flare radio data precludes a similar analysis in the case of _ swift _ j1112 - 8238 . however , a _ swift _ j1644 + 57-like radio lightcurve would be observable even several years after the flare and thus radio constraints may yet be obtainable . , north up , east left ) around _ swift _ j1112 - 8238 , @xmath7 550 days after initial detection , smoothed via a 3 pixel gaussian convolution for clarity . the positions of the optical transient centroid , as measured 17 ( left ) and 21 ( right ) days after trigger with the iraf command imexam , are displayed as red 1-@xmath9 error circles . similarly , the host optical centroid ( white ) as measured via a srsic profile fit with galfit is plotted as a white 1-@xmath9 error circle . this error is a lower limit based on the assumption of a gaussian profile with fwhm equal to twice the half - light radius of the srsic fit . the transient positions are thus coincident with the central position to 1@xmath9 and 2@xmath9 respectively . the consistency of these positions makes an association of the event with the smbh in the galaxy plausible . it should be noted , though , that due to the low surface brightness and possible complex morphology of the host galaxy the host centroid is subject to substantial systematic uncertainty . [ fig : centroid],width=317 ] if _ swift _ j1112 - 8238 is indeed a member of the same class of object as _ swift _ j1644 + 57 and _ swift _ j2058 + 0516 then it brings the total number of such events , as selected by the high energy emission , to three . radio observations of thermal ( non - relativistic ) tdf candidates ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) have been used to attempt to determine the number of `` off - axis '' members and in the case of @xcite , a few candidates may have been discovered . however , it is clear that the detected population is small . nonetheless it is striking that these three outbursts were all discovered by _ swift _ in the space of a 3 month window in 2011 . at first sight it may be argued that the proximity , and consequent brightness , of _ swift _ j1644 + 57 may have motivated the searches that led to the discoveries of the additional candidates . however , the lack of any further examples in the subsequent four years suggests that this is more likely a statistical fluke . it is possible to quantify this via an archival search of _ swift _ grbs and the bat transient monitor @xcite . within the 6.5 years of data reported in @xcite there are two events marked as tdfs ( the previously identified bursts ) , while only a further three are marked as unknown " . two of these ( _ swift _ j1713.4 - 4219 , igr j17361 - 4441 ) lie close to the galactic plane , and are most likely galactic sources . this leaves only the source under discussion , _ swift _ j1112 - 8238 , as a candidate relativistic tdf . it is plausible , though , that some other sources within the catalogue have been misidentified . in particular , _ swift _ j1644 + 57 was initially identified as a galactic fast x - ray transient @xcite . however , the population detected by the bat transient monitor is necessarily small . in total , therefore , it appears that at most a handful of such events have been recorded over _ swift _ s @xmath146 year lifetime . similar to @xcite , we can determine an implied analogous ( i.e. similar _ isotropic _ luminosity ) relativistic tdf rate based on the 3 events observed in @xmath710 years , using the volume bounded by the distance to _ swift _ j2058 + 05 , as the most distant yet observed at @xmath147 giving a comoving volume of 215 gpc@xmath148 , and assuming the local number density of @xmath113@xmath149 smbhs to be @xmath150 mpc@xmath151 @xcite . the resulting rate is found to be @xmath152 per galaxy per year , in stark contrast to the @xmath153 inferred from thermal tdf detections ( e.g. * ? ? ? * ; * ? ? ? even if a significant fraction of the ulgrb population were related to similar phenomena this would be unlikely to constitute the majority of the factor of @xmath154 required . to resolve this discrepancy likely requires a combination of tightly beamed high energy emission , such as that seen in grbs , and that not all tdfs produce relativistic jets . recent late time radio surveys of thermal tdfs by @xcite suggest that @xmath155 of tdfs may have an associated relativistic jet . given this , the required beaming angle for rtdf high energy emission would be of order @xmath156 . at first sight this is not unreasonable , given that , for example , _ swift _ j1644 + 57 produced an isotropic x - ray emission equivalent to the eddington luminosity of a @xmath157m@xmath17 black hole in a galaxy that is only expected to contain an smbh of @xmath158 solar masses @xcite . in this case beaming ( either relativistic and/or geometric ) of a factor @xmath141 , or highly super - eddington accretion would seem to be necessary . however , radio observations of _ swift _ j1644 + 57 point to a rather modest lorentz factor , that would be unlikely to result in such strong collimation ( @xmath25 , @xcite ) , unless the radio and high - energy emission regions are spatially separate , each with their own lorentz factors . it is also possible that the survey of @xcite could be impacted by small number statistics and potential contaminants . one of the two detections made , rxj1420.4 + 5334 has an uncertain host identification due to the large error in the x - ray flare position . the second , ic3599 , may be an agn @xcite , and has recently exhibited repeated flares , either due to repeated partial disruptions of the same star on an @xmath159 year orbit @xcite or due to ongoing agn activity @xcite . because of this , the suggested 10@xmath160 jetted tdf fraction may be overestimated , which would explain the lack of detections in any of the other studies ( e.g. * ? ? ? * ) , thus further contributing to the apparent deficit of detected rtdfs . clearly further observations of larger samples of sources across the electromagnetic spectrum are needed to resolve this question . we have presented multi wavelength observations of _ swift _ j1112 - 8238 , pinpointing it to close to the nucleus of an otherwise quiescent galaxy at @xmath1 . the high x - ray luminosity of the source coupled with its relative optical faintness occupy a region of parameter space which is populated only by the candidate relativistic tdfs . hence we suggest that _ swift _ j1112 - 8238 is the third candidate member of this class of events as detected by their high - energy emission . the discovery of such a small number of events over the lifetime of _ swift _ suggests that they are extremely rare and that , unless the beaming angles are extremely small ( evidence for which may come from the extreme observed luminosities coming from events in hosts with small expected smbhs ) , then their true astrophysical rates are also only a small fraction ( about 1 in @xmath161 ) of the likely non - relativistic tdf rate . the factors which govern the production of a relativistic outflow associated with a given tdf remain unclear , but highlight the needed for rapid dedicated follow - up of the rare examples when they are found . we thank the referee for a careful and thoughtful report that has improved this paper . gcb thanks the midlands physics alliance for a phd studentship . ajl thanks stfc for support under grant i d st / i001719/1 , and the leverhulme trust for support via a philip leverhulme prize . 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 national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia , tecnologia e inovao ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) . based on observations made with eso telescopes at the la silla paranal observatory under programme i d 089.b-0860 . we have made use of the rosat data archive of the max - planck - institut fr extraterrestrische physik ( mpe ) at garching , germany .

we present observations of _ swift _ j1112.2 - 8238 , and identify it as a candidate relativistic tidal disruption flare ( rtdf ) . the outburst was first detected by _ swift_/bat in june 2011 as an unknown , long - lived ( order of days ) @xmath0-ray transient source . we show that its position is consistent with the nucleus of a faint galaxy for which we establish a likely redshift of @xmath1 based on a single emission line that we interpret as the blended [ oii]@xmath2 doublet . at this redshift , the peak x/@xmath0-ray luminosity exceeded @xmath3 ergs s@xmath4 , while a spatially coincident optical transient source had @xmath5 ( m@xmath6 at @xmath1 ) during early observations , .17ex20 days after the _ swift _ trigger . these properties place _ swift _ j1112.2 - 8238 in a very similar region of parameter space to the two previously identified members of this class , _ swift _ j1644 + 57 and _ swift _ j2058 + 0516 . as with those events the high - energy emission shows evidence for variability over the first few days , while late time observations , almost 3 years post - outburst , demonstrate that it has now switched off . _ swift _ j1112.2 - 8238 brings the total number of such events observed by _ swift _ to three , interestingly all detected by _ swift _ over a @xmath73 month period ( @xmath8 of its total lifetime as of march 2015 ) . while this suggests the possibility that further examples may be uncovered by detailed searches of the bat archives , the lack of any prime candidates in the years since 2011 means these events are undoubtedly rare . [ firstpage ] galaxies : nuclei , galaxies : quasars : supermassive black holes , gamma - rays : galaxies

the current flowing out of an ideal cps originates only from split cooper pairs , with one electron being transported over the left and one electron over the right qd . this current is , therefore , subjected to the filtering of spin , valley , and energy of _ both _ qds , and probing the current _ locally _ in one qd contains the _ nonlocal _ information of the filtering effects of both qds . indeed , in this situation , with filters set along the axes @xmath124 ( @xmath125 ) and resonant conditions such that transport is restricted to the selected levels , the density matrix for the outflowing particles takes the form @xmath126 , with @xmath55 the density matrix in the absence of spin - valley filtering . due to the perfect splitting efficiency , the currents through the left and right qd are identical , and we can focus , for instance , on transport through the left qd only . if @xmath127 is the spin and valley independent current operator for transport over the left qd , the property @xmath128=0 $ ] ensures that @xmath129 . in the linear response regime we have furthermore @xmath130 , with @xmath131 the conductance and @xmath132 the voltage applied to both leads with respect to the superconductor . as a function of both qd gate voltages , @xmath131 is resonant at the level crossing @xmath124 . the full amplitude of the transport at this level crossing , denoted by @xmath133 , is obtained by integrating @xmath131 over this resonance . if furthermore the tunneling rates to the qds are independent of the qd gates , the quantities @xmath134 allow us to reconstruct the spin correlators due to the identites @xmath135 and @xmath136 . as a consequence we obtain eq . ( 6 ) in the main text . the relation between conductances and spin correlators , therefore , follows from the same considerations used in the proposed entanglement tests based on noise measurements @xcite . to further test eq . ( 6 ) and its consequences on entanglement detection under realistic conditions , we have implemented the microscopic numerical calculation . as discussed in the main text , the numerical results give an objective demonstration that eq . ( 6 ) and the conclusions for entanglement detection remain robust . in this part of the supplement we illustrate the influence of the coupling of the cnt to the superconductor and the normal leads on the determination of @xmath31 . we provide all parameters used for the tight - binding calculation following ref . . finally we show how the level energies and the spin projections evolve with the magnetic field . figure [ fig : gamma_s ] shows the dependence of @xmath31 on the effective coupling strength @xmath137 between the superconductor and the cnt . the insets show parts of the conductance maps for the @xmath137 values corresponding roughly to the placements of the insets in the plot . for large @xmath137 , the level broadening induced by the superconducting contact mixes the cooper pairs between the qd levels and the conductances are no longer spin projective . this is notable by the similar intensities of all resonances , and corresponds to a strong enhancement of the local distortions discussed in the text . the corresponding values of @xmath31 lie well below @xmath138 . small @xmath137 , on the other hand , lead to a weak cooper pair injection amplitude compared with the hybridization through the superconducting region . as a consequence , the resonance crossings turn into anticrossings . the resulting @xmath31 values sharply increase beyond @xmath39 due to strongly distorted spin correlator reconstructions by the nonlocal hybridization processes . at very small @xmath137 , the anticrossings of different levels overlap , and the spin correlator reconstruction becomes erratic . dependence of @xmath31 on the effective coupling @xmath137 to the superconductor for a ( 20,0 ) cnt with fixed @xmath139 mev , @xmath140 t , @xmath141 , @xmath142 . the insets show a zoom on the conductance maps for the @xmath137 values corresponding to their placement in the figure , with identical logarithmic color scales [ see fig . 3 ( a ) in the main text ] . the center inset represents the valid regime for testing the bell inequality with well resolved resonances of different intensities , and the absence of notable avoided crossings of the resonance peaks . ] a valid measurement of @xmath31 requires @xmath137 corresponding to the central inset in the fig . [ fig : gamma_s ] , represented by well - defined level crossing peaks with unequal intensities . the unequal intensities are a result from the spin filtering of the singlet states , such that spin projection axes that are close to parallel suppress the conductance , while projections that are close to antiparallel allow a maximal transmission . hence unequal , @xmath32 dependent peak intensities are a necessary indicator for spin entanglement , and indeed are the basis for the implementation of the bell test . the dependence on the tunnel coupling to the normal leads , characterized by a tunneling amplitude @xmath143 for @xmath83 , is represented in fig . [ fig : gamma_lr ] . the combination of the @xmath143 with @xmath137 defines the broadening of the qd levels . indeed , in the model of ref . the lateral leads were represented by ideal one - dimensional channels weakly coupled to each end site of the nanotube . in the present calculations the tunneling rates to these leads @xmath144 take values between 10 and 100 mev . the actual broadening introduced to the qd levels becomes then on the order of @xmath145 with @xmath146 the length of qd @xmath147 and @xmath148 the lattice constant . dependence of @xmath31 on the tunneling amplitudes @xmath143 to the normal lead @xmath83 , for @xmath149 for a ( 20,0 ) cnt with fixed @xmath150 mev , @xmath140 t , @xmath141 , @xmath142 . the insets show a zoom on the conductance maps for the @xmath143 values corresponding to their placement in the figure , with identical logarithmic color scales [ see fig . 3 ( a ) in the main text ] . the center inset represents the valid regime for testing the bell inequality with well separated resonances and a high enough pixel resolution such that the integral weight of each peak can be determined with high accuracy . ] in contrast to @xmath137 , the insets in fig . [ fig : gamma_lr ] show that @xmath143 contributes only to a broadening of the levels but leaves the inequality of the peaks unchanged . the @xmath143 values of the insets correspond again roughly to the positions of the insets . at large @xmath143 , the level overlaps lead to strong local overlaps of the projections such that the @xmath151 strongly decrease . since , however , the unequal intensities and so the spin - filtering properties of each qd level are maintained , the value of @xmath152 remains large even for large @xmath143 . yet for larger @xmath143 the influence of the overlaps is well notable by the split off of @xmath114 from the @xmath152 value . for small @xmath143 we notice that most conductances lead to an upturn of @xmath31 . this effect is attributable to the finite resolution of the peaks from the numerics that become only a few pixels wide , and the result is strongly susceptible to the discretization steps of the @xmath153 . the artificial nature of the low @xmath143 behavior is indeed seen by the comparison of @xmath100 and @xmath99 , which show an anomalous opposite behavior in a regime where all resonances are well separated and all couplings to the left and right qds are identical . finally , we notice that since the @xmath143 mainly influence the qd levels locally , an asymmetry @xmath154 has only little impact on the value of @xmath31 as long as all levels can be well resolved . the results shown in the main text represent the optimal values for the chosen cnt and geometry , @xmath150 mev and @xmath155 mev , determined by first identifying a valid @xmath137 leading to well shaped peaks with modulated intensities , and then optimizing the @xmath143 to obtain well resolved resonances . these values , however , are strongly sample and geometry dependent and can be used only as indicative . for the present calculation we have used a cnt of chirality ( 20,0 ) with qd lengths @xmath156 nm and a length of the central superconducting region of 173 nm . yet the same behavior of level separations and @xmath31 values is found for longer system sizes corresponding to experimental situations . a magnetic field of strength @xmath140 t was applied to each qd region with angles @xmath32 on the left qd and angles @xmath157 on the right qd with respect to the cnt axis , for @xmath158 . the soi strengths @xmath15 and the shift @xmath159 have been implemented using the values of refs . , and are given by @xmath160 mev @xmath161 , @xmath162 mev @xmath163 , and @xmath164 mev @xmath165 with @xmath20 the cnt radius in nm , @xmath166 , and @xmath167 the chiral angle , @xmath168 , for a cnt with chiralities @xmath169 . for @xmath170 we have @xmath171 nm , @xmath67 mev , and @xmath172 mev . the induced superconducting gap is @xmath173 mev , and the doping of the central region @xmath174 mev . all further parameters are as described in ref . . for @xmath175 as used for fig . 2 in the main text , we have @xmath176 nm , @xmath177 mev , and @xmath36 mev . levels and spin projections of the left qd as a function of parallel and perpendicular magnetic fields @xmath178 for the ( 18,10 ) cnt described in fig . 2 of the main text . the sketches in the lower left corners indicate the @xmath178 field directions with respect to the left qd . the arrows indicate the spin projections in the @xmath179 plane with the @xmath180 direction pointing upwards and the @xmath181 direction to the right in the plots . at @xmath182 the levels of both valleys are degenerate . at increasing @xmath178 , the levels of one valley increase and the levels of the other valley decrease in energy by the combined effect of orbital and zeeman fields . at parallel field , the spins remain parallel to the cnt axis . at perpendicular field , the level energies are only weakly affected by @xmath178 , yet the spin projections in each valley strongly rotate . the situations at @xmath183 t correspond to the selected angles @xmath184 in the upper right panel of fig . 2 in the main text . ] levels and spin projections as in fig . [ fig : b_1 ] for the right qd at the angle @xmath185 to the left qd . the sketches in the lower left corners indicate the @xmath178 field directions with respect to the right qd . the spins are shown in the global @xmath179 basis corresponding to fig . [ fig : b_1 ] . the situations at @xmath183 t correspond to the selected angles @xmath184 in the lower right panel of fig . 2 in the main text . ] finally , we illustrate the evolution of the qd levels and their spin polarizations as a function of the magnetic field @xmath178 . figure [ fig : b_1 ] displays the 4 spin polarized qd levels of the ( 18,10 ) cnt model used for fig . 2 in the main text , for magnetic fields parallel and perpendicular to the cnt axis of the left qd , respectively . figure [ fig : b_2 ] shows the levels of the right qd for the same fields , which are seen for this qd under the additional angle @xmath186 .

spin - orbit interaction provides a spin filtering effect in carbon nanotube based cooper pair splitters that allows us to determine spin correlators directly from current measurements . the spin filtering axes are tunable by a global external magnetic field . by a bending of the nanotube the filtering axes on both sides of the cooper pair splitter become sufficiently different that a test of entanglement of the injected cooper pairs through a bell - like inequality can be implemented . this implementation does not require noise measurements , supports imperfect splitting efficiency and disorder , and does not demand a full knowledge of the spin - orbit strength . using a microscopic calculation we demonstrate that entanglement detection by violation of the bell - like inequality is within the reach of current experimental setups . the controlled generation and detection of entanglement is a necessary step toward the goal of using quantum states for applications . in a solid state nanostructure this control ideally allows us to manipulate and detect entanglement between selected pairs of electrons . a promising source of entangled electron pairs is the cooper pair splitter ( cps ) . it consists of a superconductor that injects cooper pairs through two quantum dots ( qds ) into two outgoing normal leads , designed such that the cooper pair electrons preferably split and leave the superconductor over different leads but preserve their spin entanglement @xcite . very recently several cps experiments have been performed @xcite and cooper pair splitting efficiencies up to 90% have been reached @xcite . so far , however , a proof that the electrons remain entangled is still lacking . the present experiments do not allow to resolve individual splitting events , and the results of the measurements are time averaged quantities , such as current or noise . these provide information on the average spin correlations of the injected cooper pairs . in this letter we demonstrate that this information can be extracted from the currents alone in a carbon nanotube ( cnt ) based cps , if spin - orbit interaction ( soi ) effects are taken into account @xcite . this allows us to propose a general entanglement test , based on the bell inequality @xcite , which does not require noise measurements @xcite . indeed , the soi in cnts leads to unique spin - energy filtering properties that directly modulate the cooper pair splitting current flowing out of the cps , and ideally suppress any noise . from conductance measurements it is then already possible to reconstruct all spin correlators contained in the bell inequality , thus avoiding the need of ferromagnetic contacts as spin filters , which are challenging to implement . without noise measurements we also avoid the associated problem of electron fluctuations in the detectors @xcite . the built - in energy filtering furthermore leads to an enhanced cooper pair splitting efficiency @xcite . double quantum dot cps based on a bent cnt in an external magnetic field @xmath0 . because of @xmath0 , soi , and the bending angle @xmath1 of the cnt , the spin - valley degeneracy of the qd levels is lifted , and the resulting 4 levels ( boxes ) are spin polarized as indicated by the arrows ( see also fig . [ fig : optimal ] ) . the superconductor sc injects cooper pairs ( hourglass shape ) that split onto the qds and provide a current to the normal leads n that is modulated by the spin projections of the qds ( tunable by the gates @xmath2 ) and can be used to determine the spin correlators for the bell inequality . ] the proposed cps setup is shown in fig . [ fig : setup ] and consists of a regular double - qd cps built from a single - wall cnt , yet made with a ( naturally ) bent cnt such that there is an angle @xmath1 between the qd axes . alternatively , the qds can be built from separate cnts with similar diameters and an angle @xmath1 between them . the soi spin splits the qd levels . in combination with a global magnetic field @xmath0 , the fourfold spin - valley degeneracy of the qd levels is completely lifted . the split levels provide a unique spin filter for electron transport with two spin projection axes per qd , filtering directly the injected cooper pair current . therefore , conductance measurements alone , at fixed @xmath0 , allow a reconstruction of all the spin correlators necessary for the bell inequality . the spin projection axes are different in the two qds due to the bending , and are tunable by @xmath0 . in the following we show that this tunability provides sufficient conditions for obtaining violations of the bell inequality in an ideal cps . we then proceed to a full microscopic calculation and demonstrate that the result remains robust under realistic conditions , as achievable by present experiments . _ soi in cnt quantum dots_. cnts are graphene sheets rolled into a cylinder . they preserve the graphene band structure with two dirac valleys but have enhanced soi contributions due to the curvature . the corresponding model , including the effect of @xmath0 , is described by the sum of the hamiltonians @xcite @xmath3 , \\ h_{cv } & = \hbar v_f \bigl [ \delta k_t^{cv } \sigma_1 + \delta k_z^{cv } \tau_3 \sigma_2 \bigr ] , \\ h_{soi } & = \alpha \sigma_1 s_z + \beta \tau_3 s_z , \label{eq : soi } \\ h_b & = \mu_b g { \mathbf{b}}\cdot { \mathbf{s}}/2 + |e| v_f r b_z \sigma_1 /2 , \label{eq : h_b}\end{aligned}\ ] ] which are matrices in the space spanned by the graphene sublattice indices @xmath4 ( with pauli matrices @xmath5 ) , the valleys @xmath6 ( pauli matrices @xmath7 ) , and the spin projections @xmath8 ( pauli matrices @xmath9 , with @xmath10 oriented along the cnt axis ) . @xmath11 is the fermi velocity , @xmath12 the transverse quantized momentum ( zero for metallic cnts ) , @xmath13 the longitudinal momentum , @xmath14 are momentum corrections induced by the curvature , @xmath15 determine the soi , @xmath16 is the bohr magneton , @xmath17 the land @xmath18-factor , @xmath19 the electron charge , @xmath20 the cnt radius , and @xmath21 the component of @xmath0 along @xmath10 . we have neglected terms leading to the formation of landau levels since at the considered sub - tesla fields they are of no consequence . for a qd , @xmath13 is further quantized by the qd length @xcite . because of its momentum independence , the soi takes the role of an internal valley ( and qd orbital ) dependent zeeman field @xmath22 along @xmath10 , which combines with @xmath0 to the effective field in each valley @xmath23 . these fields lift the spin degeneracy of the qd levels , while the orbital effect of eq . lifts the energy degeneracy between the two valleys for any @xmath24 . the qd levels turn into spin - valley - energy filters . the effective fields define the spin polarization axes @xmath25 , which are nonparallel if @xmath26 , tunable by @xmath0 , and such that the spin - eigenstates in each valley @xmath27 fulfill @xmath28 ( full polarization ) . if @xmath29 , spin measurements can be reconstructed by electron transport over the different qd levels by @xmath30 . values @xmath31 of the bell equation ( left panel ) for an ideal bent cnt - cps as a function of in - plane @xmath0-field rotation angle @xmath32 , for the lowest valence band orbitals in a cnt of chirality ( 18,10 ) , @xmath33 t , @xmath34 , @xmath35 mev , @xmath36 mev , and qd lengths of 200 nm . for this situation , @xmath37 . the horizontal lines mark the threshold @xmath38 and the maximal possible @xmath39 . the right panels show the @xmath32 dependence of the level energies of both qds . the spectra are identical up to the shift by @xmath1 marked by the vertical dashed lines . the arrows indicate the spin polarizations in a global spin basis , as used for the determination of @xmath31 . ] _ bell test in an ideal cnt - cps_. in the double - qd system shown in fig . [ fig : setup ] , the cnt bending changes the orientation of @xmath40 and so of @xmath41 . the spin polarization axes @xmath42 in the left qd become distinct from the axes in the right qd , which we call @xmath43 . we consider an ideal cps , characterized by a perfect cooper pair splitting efficiency with valley - independent pair injection ( see discussion below ) and isolated sharp qd levels . since any injected cooper pair splits onto the different levels in each qd ( the current consists only of split cooper pairs ) , and the tunneling amplitude onto each dot is proportional to the spin projection , the current collected at the normal leads in resonant conditions for a given pair of levels is proportional to @xmath44 and allows us to reconstruct the spin correlators @xmath45 [ see eq . ] . the availability of 2 spin projection axes per qd consequently allows us to test the chsh - bell inequality @xcite @xmath46 any non - entangled state ( including the steady state density matrix considered here ) fulfills this inequality . a violation @xmath47 is sufficient to prove entanglement . for a spin - singlet , a maximal @xmath39 is obtained by orthogonal @xmath48 , @xmath49 , and @xmath50 between @xmath51 and @xmath52 . such optimal axes can not be generally obtained in the cnt - cps , for which @xmath40 and @xmath1 are fixed by the sample fabrication , and only @xmath0 is tunable . yet , as we show in fig . [ fig : optimal ] , this tunability is sufficient to obtain @xmath47 as a function of the angle @xmath32 of a rotating in - plane field @xmath53 ( see fig . [ fig : setup ] ) , for @xmath54 . the shown result is generic and we find similar @xmath47 for most cnt chiralities , diameters , and qd lengths . _ realistic systems_. in a realistic setup , the two qds remain coupled through the superconducting region , their levels are broadened by the contacts , the splitting efficiency is imperfect and electron pairs can tunnel onto the same qd , the tunneling rates depend on the gate voltages , and electrons can interact . any measurement probes the steady state density matrix @xmath55 of the full cps system and not an ideal singlet state . the projections @xmath56 are obtained by narrowing the measurement to an energy window capturing the electron transport through the corresponding level of each qd , typically by differential conductance measurements tuned to the resonances corresponding to the levels . the modified @xmath55 together with the measurement method leads to a distorted reconstruction of the spin correlators , and we need to distinguish between _ local _ and _ nonlocal _ distortion sources . local distortions in one qd are independent of the other qd and modify , e.g. , @xmath57 to @xmath58 . we can write @xmath59 for an intermediate axis @xmath60 , @xmath61 , and a remaining projection @xmath62 . the latter transforms any state into a product state , and local distortions therefore lower the ideal value of @xmath31 by an amount set by the various @xmath63 for the different qd levels . assuming that the level broadening can be kept small so that there is only little overlap between nearby resonances ( assisted also by a charging energy ) , the most important source of local distortions is disorder scattering within each qd . it mixes the wave functions in different valleys @xcite , and the @xmath27 are no longer the eigenstates . while of central importance in metallic cnts , in semiconducting cnts disorder scattering competes with the valley - preserving semiconducting gap of typically @xmath64 100 mev , which has opposite signs in opposite valleys . if the disorder scattering amplitude is smaller it has a negligible influence . therefore , semiconducting cnts are preferable for testing the bell inequality . valley mixing at injection , however , is essential . indeed , if valleys and spins are correlated , for instance , if the singlet splits always into opposite valleys , the transport through other valley combinations does not provide any information on the cooper pairs and the construction of @xmath31 is no longer possible . for a valid spin correlator measurement the injection must mix valleys to produce a detectable signal through all resonances , yet the precise degree of mixing is unimportant . nonlocal distortions of the spin modify the spin projections as an effect of the entire cps system , typically by hybridization between the two qds , and the measured @xmath56 become nonlocal operators . such operators can generate additional entanglement through wave function mixing between the left and right qds . in the cps setup they are a source of error for detecting spin entanglement . yet with the full microscopic calculation discussed next we can see that these nonlocal contributions can be kept under control in realistic conditions . results from the microscopic calculation of a cps , based on a zigzag cnt of chirality ( 20,0 ) with a bending angle @xmath65 , in a field of @xmath66 t ( see the supplemental material @xcite ) . the soi energies @xmath67 mev and @xmath68 mev lead to @xmath69 . ( a ) map of the conductance product @xmath70 ( units of @xmath71 ) as a function of qd gate voltages @xmath2 at @xmath72 . the 4 levels of each qd give rise to the 4 resonances labeled by @xmath73 , @xmath74 . inside the black squares , the cps acts as a spin - valley filter for the projections @xmath75 , and integrating the signal within each black square yields the corresponding observable . ( b ) @xmath2 values marking the positions of the resonances of the levels @xmath73 and @xmath74 of the two qds as a function of @xmath32 . the curves are identical up to the shift by @xmath1 . levels in the same valley @xmath76 see the same field @xmath77 and are identified by having the same curvature as function of @xmath32 . ( c ) @xmath31 as a function of @xmath32 for the conductances @xmath78 given as subscripts of @xmath31 in the figure legend . the @xmath31 values are obtained by analyzing data as shown in panel ( a ) by the method described in the text . the yellow shaded region marks the allowed range of violation of the bell inequality for the spin - singlets in the steady state . ] _ microscopic model_. to quantitatively access a realistic system and to determine the optimal choice of measurements that allows us to gain insight in the effects of local and nonlocal distortions , we have investigated a microscopic tight - binding model of the cnt - cps . our approach follows ref . , which we have complemented to include magnetic fields by terms equivalent to eq . and valley mixing at injection . as a result , we obtain the partial conductances of the cps due to cooper pair splitting ( crossed andreev reflections , @xmath79 ) , elastic cotunneling through the superconducting region ( @xmath80 ) , and the local andreev scattering contributions at each qd ( @xmath81 ) . from these quantities , transport from the superconductor to the normal leads is expressed by the conductances @xmath82 ( @xmath83 ) , and transport between the normal leads by the nonlocal conductance @xmath84 . in fig . [ fig : results ] ( a ) we display a conductance map for a semiconducting cnt as a function of the qd gate voltages @xmath2 that tune the qd levels to resonance . such a result is useful for a bell test if all 4 resonances in each qd are well resolved and their 16 points of intersection , corresponding to the products @xmath85 , form single peaks and not avoided crossings . to access this regime , we have chosen a coupling between the superconductor and the cnt on the order of the superconducting gap ( @xmath86 1 mev ) , and tuned the coupling to the leads such that the resonances are well resolved ( see the supplemental material @xcite ) . similar conditions have been obtained in experiments @xcite , and such a regime can be reached for a wide variety of samples and coupling strengths to the contacts . to analyze the data we integrate the various conductances over regions centered at the crossings as shown by the black squares in fig . [ fig : results ] ( a ) . from the resulting 16 integrals @xmath87 we construct the spin correlators @xmath88 which is a simple consequence from the fact that @xmath89 and @xmath90 is the identity operator ( see the supplemental material @xcite ) . from these @xmath91 we determine @xmath31 by eq . , with the liberty of placing the @xmath92 sign in front of any term in eq . to obtain the maximum @xmath31 . the cooper pair splitting amplitude is directly described by @xmath79 , and the corresponding curve @xmath93 [ fig . [ fig : results ] ( c ) ] captures indeed a similar behavior as the ideal case of fig . [ fig : optimal ] , with @xmath47 in the @xmath32 regions where the levels of different valleys approach each other and the spin projections rotate [ fig . [ fig : results ] ( b ) ] . the measurable conductances @xmath94 , however , contain with @xmath95 contributions that represent strong enough local distortions to suppress @xmath31 below 2 . in the right qd the local distortions are enhanced by level overlaps close to @xmath96 where the @xmath97 and @xmath98 levels become degenerate [ fig . [ fig : results ] ( b ) ] , and indeed @xmath99 decreases in this region . in contrast , the left qd levels remain well separated , and @xmath100 mirrors the upturn of @xmath93 , with @xmath79 overruling the @xmath101 contribution . the same behavior with @xmath102 is found near @xmath103 . on the other hand , @xmath104 corresponds to an experiment of electron injection through a normal lead and contains with @xmath80 a component describing the uncorrelated single - particle transport . since we find that @xmath80 and @xmath79 have a similar amplitude , we expect that @xmath105 . however , @xmath80 contains also the higher order tunneling processes that represent the nonlocal distortions , which may cause @xmath106 to increase again . nonetheless , we find that @xmath107 with a similar shape as @xmath93 , indicating that the nonlocal distortions have a negligible effect . while @xmath79 produces the purest indicator of spin entanglement , it is only indirectly accessible by experiments . on the other hand , the directly measurable @xmath108 are obscured by the local contributions of the @xmath95 . a method of circumventing this problem is to consider products of the @xmath94 , such as @xmath70 . since the projections @xmath109 eliminate all qd degrees of freedom , the product @xmath70 is equivalent to a nonlocal current measurement with a density matrix @xmath110 whose nonlocal contribution is encoded in @xmath111 . by the higher power of @xmath55 and the projections , the relative weight of the local contributions can be reduced , while a spin singlet in @xmath112 remains a spin singlet in @xmath113 . in fig . [ fig : results ] ( c ) we see that the corresponding curve @xmath114 follows almost perfectly @xmath93 , showing that the multiplication @xmath70 is powerful enough to suppress the local distortions in the @xmath94 . therefore , a high splitting efficiency of a cps is not a primary requirement for the proposed bell test . to demonstrate that the large @xmath31 value is indeed an effect of superconductivity , we show with @xmath115 the corresponding curve for @xmath116 obtained for the normal state . the fact that @xmath117 is the strongest indicator that @xmath118 demonstrates indeed the spin entanglement . finally , we have truncated the curves in fig . [ fig : results ] close to @xmath119 and @xmath120 where qd levels strongly overlap [ fig . [ fig : results ] ( b ) ] and spin correlators can no longer be reconstructed . it is indeed important to maintain well separated qd levels . hence the charging energy of the qds , which has been neglected in the microscopic calculation , plays here an important role as it increases the level separation but has much reduced exchange coupling due to the soi induced spin projections of the qd levels . _ conclusions_. we have demonstrated that due to soi effects bent cnt - cps ( or two cnts under an angle ) can be used for entanglement detection in the steady state by a violation of the bell inequality . notable for the bell inequality is that the set of axes @xmath121 along which the spin correlators must be measured can be arbitrary and the precise axis orientations , i.e. , the precise soi strengths , do not need to be known . this is an advantage over entanglement witnesses @xcite or quantum state tomography . although discussed for cnts , the introduced concept of entanglement detection is general and can be implemented in any system allowing tunable spin - energy filtering . for an ideal cnt - cps , a violation of the bell inequality can be achieved for most cnts over a large range of orientations of an external field @xmath0 with strength @xmath54 , which for usual cnts are @xmath122 t. the robustness of this behavior was confirmed by a microscopic calculation that incorporates the local and nonlocal imperfections of a realistic system . from the results we propose the use of the product of conductances @xmath70 as the optimal observable for testing the bell inequality . we have furthermore argued that the spin reconstruction in semiconducting cnts is robust against disorder . to conclude , it should be noted that a bending of the cnt is not an absolute requisite . an equivalent effect can be obtained by applying individual @xmath0 fields on the qds or by providing a constant field offset on one qd by placing a ferromagnet in its vicinity , if sufficient control of the typical field strengths @xmath123 t can be granted . if two separate cnts are connected to the superconductor , they should have similar diameters such that their @xmath40 are comparable . _ we thank a. baumgartner , j. c. budich , a. cottet , n. korolkova , p. recher , and b. trauzettel for helpful discussions and comments . we acknowledge the support by the eu - fp7 project se2nd [ 271554 ] and by the spanish mineco through grant no . fis2011 - 26516 . p.b . also acknowledges the support by the esf under the eurocores programme eurographene . 888 p. recher , e. v. sukhorukov , and d. loss , phys . rev . b * 63 * , 165314 ( 2001 ) . g. b. lesovik , t. martin , and g. blatter , eur . phys . j. b * 24 * , 287 ( 2001 ) ; 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throughout this paper , we assume the truth of the riemann hypothesis ( rh ) , and we let @xmath0 denote the ordinate of the @xmath1-th non - trivial zero of @xmath2 . hejhal @xcite assumed the rh and a weak consequence of montgomery s @xcite pair - correlation conjecture , namely that for some @xmath3 , there is a constant @xmath4 such that @xmath5 holds for all @xmath6 . under these assumptions , he proved the following central limit theorem : for @xmath7 , @xmath8 so under these assumptions , @xmath9 , suitably normalized , converges in distribution over fixed ranges to a standard normal variable . to obtain more precise information about the tails of the distribution , we consider the moments @xmath10 where @xmath11 is the zero counting function . notice that @xmath12 is defined for all @xmath13 provided the zeros of @xmath2 are simple , as is widely believed . gonek @xcite @xcite carried out an extensive study of @xmath12 . he proved , under the assumption of the rh , that @xmath14 as @xmath15 . it was suggested by gonek @xcite , and independently by hejhal @xcite , that @xmath12 is on the order of @xmath16 . ng @xcite proved , under the rh , that @xmath17 is order of @xmath18 , which is in agreement with that suggestion . hughes , keating , and oconnell @xcite , applied the random matrix philosophy ( e.g. see @xcite ) , which predicts that certain behaviors of @xmath19-functions are mimicked statistically by characteristic polynomials of large matrices from the classical compact groups . this led them to predict that for re(@xmath13 ) @xmath20 , @xmath21 where @xmath22 is the barnes g - function , and @xmath23 is an `` arithmetic factor . '' the conjecture ( [ eq : hkc ] ) is consistent with previous theorems and conjectures . recently , conrey and snaith @xcite , assuming the ratios conjecture , gave lower order terms in asymptotic expansions for @xmath24 and @xmath17 . they conjectured the existence of certain polynomials @xmath25 , for @xmath26 and @xmath27 , such that @xmath28 the conjecture for the case @xmath26 was subsequently proved by milinovich @xcite , assuming the rh . it is expected that such polynomials exist for other integer values of @xmath29 as well . the purpose of this article is to study numerically various statistics of the derivative of the zeta function at its zeros . in particular , we consider the distribution of @xmath9 , moments of @xmath30 , and correlations among moments . the goal is to obtain more detailed information about the derivative at zeros , and to enable comparison with various conjectured and known asymptotics . our computations rely on large sets of zeros at large heights that are described in detail in @xcite . we find that the empirical distribution of @xmath9 , normalized to have mean zero and standard deviation one , agrees generally well with the limiting normal distribution proved by hejhal , as shown in figure [ distder23 ] . but the empirical mean and standard deviation pre - normalization are noticeably different from predicted ones . also , as shown in figure [ tailder23 ] , the frequency of very small normalized values of @xmath9 is higher than predicted by a standard normal distribution , while the frequency of very large normalized values is lower than predicted . since these differences appear to decrease steadily with height , however , they are probably not significant . to examine the tails of the distribution of @xmath31 , we present data for the moments of @xmath30 over short ranges : @xmath32 for large @xmath13 , the empirical values of @xmath33 deviate substantially from the values suggested by the leading term prediction ( [ eq : hkc ] ) . this is not surprising . because for @xmath13 large relative to @xmath34 , the contribution of lower order terms is likely to dominate , and so the leading term asymptotic on its own may not suffice . furthermore , the said deviations decrease steadily with height and they occur in a generally uniform way for roughly @xmath35 , so they are consistent with the effect of `` lower order terms '' still being felt even at such relatively large heights . in the specific cases of the second and fourth moments of @xmath30 , the conjectures of conrey and snaith @xcite supply lower order terms , and the agreement with the data is much better , as shown in table [ csmomentder23 ] . for sufficiently many values of @xmath34 , then solving the resulting system of equations . however , this did not yield good approximations of the coefficients ( even for small @xmath13 ) , which is not surprising , since the scale is logarithmic and the conrey and snaith expansion is only asymptotic . ] as @xmath13 increases , the observed variability in the moments of @xmath30 is more extreme , but it is still significantly less than we previously encountered in the moments of @xmath36 ( see @xcite ) . to illustrate , our computations of the twelfth moment of @xmath30 over 15 separate sets of @xmath37 zeros each ( near the @xmath38-rd zero ) show that the ratio of highest to lowest moment among the 15 twelfth moments thus obtained was 2.36 . in contrast , that ratio for the twelfth moment of @xmath36 was 16.34 , which is significantly larger ( see @xcite ) . in general , the variability in statistical data for @xmath30 is considerably less than the variability in statistical data for @xmath36 . it is not immediately clear why this should be so , considering , for instance , that the central limit theorem for @xmath9 is only conditional , while that for @xmath39 is not , and both theorems scale by the same asymptotic variance . in the case of negative moments , our data is in agreement with gonek s conjecture ( @xcite ) @xmath40 as @xmath15 . but starting at @xmath41 , and as @xmath13 decreases , the empirical behavior of negative moments becomes rapidly more erratic . for example , using the same 15 zero sets near the @xmath38-rd zero mentioned previously , the ratio of highest to lowest negative moment among them gets very large as @xmath13 decreases ; we obtain : 1.03 , 8.45 , 178.49 , and 17240.99 , for @xmath42 and @xmath43 , respectively ( this can be deduced easily from table [ nmomentder23 ] ) . notice that the point @xmath41 is special because it is where the leading term prediction ( [ eq : hkc ] ) first breaks down due to a pole of order 1 in the ratio of barnes g - functions . extreme values of negative moments are caused by very few zeros . when @xmath41 , for instance , the largest observed moment among our 15 sets is @xmath44 . about 87% of this value is contributed by 4 zeros where @xmath30 is small and equal to 0.002439 , 0.002453 , 0.004388 , and 0.004365 . by computing them in two ways , using the odlyzko - schnhage algorithm , and using the straightforward riemann - siegel formula ; the results from the two methods agreed to within @xmath45 such small values of @xmath46 typically occur at pairs of consecutive zeros that are close to each other . for example , the values 0.002439 and 0.002453 occur at the following two consecutive zero ordinates : @xmath47 the above pair of zeros is separated by 0.00032 , which is about 1/400 times the average spacing of zeros at that height ( which is @xmath48 ) . to investigate possible correlations among values of @xmath49 , we studied numerically the ( shifted moment ) function : @xmath50 we plotted @xmath51 , for several choices of @xmath13 , @xmath34 , and @xmath52 , and as @xmath53 varies . the resulting plots indicate there are long - range correlations among the values of the derivative at zeros . unexpectedly , the tail of @xmath54 ( figure [ corrder23 ] ; right plot ) strongly resembles the tail for the shifted fourth moment of @xmath36 ( figure 4 in @xcite ) . to better understand these correlations , we considered the `` spectrum '' of @xmath9 ; see ( [ eq : specf ] ) for a definition . a plot of the spectrum reveals sharp spikes , shown in figure [ fftder23 ] . these spikes can be explained heuristically by applying techniques already used by fujii @xcite and gonek @xcite to estimate sums involving @xmath55 . conjecture ( [ eq : derclt1 ] ) suggests the mean and standard deviation of @xmath56 for zeros from near @xmath57 ( i.e. near the @xmath38-rd zero ) are about 2.0 and 1.4 , respectively . this is far from the empirical mean and standard deviations listed in table [ sum0 ] , which are 3.4907 and 1.0977 . change very little across different zero sets near the same height . for example , using a different set of @xmath58 zeros near the @xmath38-rd zero , the empirical mean is 3.4907 and the empirical standard deviation is 1.0978 , which are very close the numbers listed in table [ sum0 ] . we note that the empirical mean and standard deviation are closer to the values suggested by the central limit theorem for characteristic polynomials of unitary matrices ( see @xcite ) , which are 3.47 and 1.12 . ] since these quantities grow very slowly ( like @xmath59 ) , these differences are probably not significant . .summary statistics for @xmath9 using sets of @xmath60 zeros from different heights the column `` zero '' lists the zero number near which the set is located . sd stands for standard deviation . [ sum0 ] [ cols="^,^,^,^,^",options="header " , ] starting with the investigations of @xcite , several long - range correlations have been found experimentally in zeta function statistics . such correlations are not present in random matrices , but do appear in some dynamical systems that for certain ranges are modeled by random matrices . so far all the zeta function correlations of this nature have been explained ( at least numerically and heuristically ) by relating them to known properties of the zeta function , such as explicit formulas that relate primes to zeros . a natural question is whether such correlations arise among values of @xmath55 . in order to detect correlations among values of @xmath30 , consider @xmath61 we computed this shifted moment function for various choices of @xmath53 , @xmath34 , and @xmath52 . ( we also considered similar sums with exponents other than 4 , but for simplicity do not discuss them here . ) figure [ corrder23 ] presents some of our results near the @xmath62-th and @xmath38-rd zeros , and with @xmath52 spanning about @xmath60 zeros in both cases . the figure shows that correlations do exist and persist over long ranges . also , the shape of @xmath54 near the @xmath62-th zero is similar to that near the @xmath38-rd zero , except the former has higher peaks , and covers the range @xmath63 , as opposed to @xmath64 , which suggests oscillations scale as @xmath65 . we remark the plot of @xmath54 in figure [ corrder23 ] ( right plot ) is similar to a plot in @xcite of the shifted fourth moment of the zeta function on the critical line : @xmath66 which we reproduce here in figure [ smg2 ] for the convenience of the reader . using @xmath60 zeros near the @xmath62-th ( left plot ) and the @xmath38-rd zero ( right plot),title="fig : " ] using @xmath60 zeros near the @xmath62-th ( left plot ) and the @xmath38-rd zero ( right plot),title="fig : " ] , with @xmath67 , near the @xmath38-rd zero , drawn for @xmath68 a multiple of @xmath69 . the dashed line is a sine kernel . ] to explain observed correlations , we numerically calculated the function : @xmath70 which is related to long - range periodicities in @xmath55 . assuming the rh , fujii @xcite supplied the following asymptotic formula in the case @xmath71 : @xmath72 where @xmath73 ( the euler constant ) and @xmath74 . empirical values of @xmath75 agree well with formula ( [ eq : fuj ] ) . for example , with @xmath52 spanning @xmath76 zeros , we obtain @xmath77 near the @xmath78-zero , and we obtain @xmath79 near the @xmath38-rd zero . but as @xmath80 increases , @xmath81 experiences sharp spikes for certain @xmath80 , as shown in figure [ fftder23 ] , which depicts the segment @xmath82 ( in the remaining portion @xmath83 , the spikes get progressively denser ) . the sharp spikes in figure [ fftder23 ] show the existence of long - range periodicities among values of @xmath55 . these spikes , as well as the correlations described above , are not unexpected . they can be demonstrated to follow from the properties of the zeta function , by estimating proper contour integrals . such methods were used for continuous averages by ingham @xcite and even others before him , and for discrete averages over zeros by gonek @xcite and fujii @xcite . the main step involves integration of @xmath84 , and estimates of such integrals . applying such methods to @xmath85 suggests that the function @xmath86 experiences large spikes at approximately @xmath87 . for by a heuristic argument involving the ( very ) regular spacing of zeros one expects that @xmath88 in the definition of @xmath89 can be replaced by @xmath1 without too much error ( see @xcite for a similar argument in the context of long - range correlations in zero spacings ) . therefore , @xmath81 should behave similarly to @xmath89 . are almost unchanged if instead of plotting @xmath81 we plot @xmath89 . ] in particular , we expect the @xmath90-th spike in figure [ fftder23 ] to occur at approximately @xmath91 , and that agrees well with the evidence of the graphs . , defined in ( [ eq : specf ] ) , using @xmath76 zeros near the @xmath62-th zero ( upper left ) , @xmath78-rd zero ( upper right ) , and @xmath38-rd zero ( lower left ) . the lower right plot is another plot near the @xmath38-rd zero , except it uses a different set of @xmath92 zeros.,title="fig : " ] , defined in ( [ eq : specf ] ) , using @xmath76 zeros near the @xmath62-th zero ( upper left ) , @xmath78-rd zero ( upper right ) , and @xmath38-rd zero ( lower left ) . the lower right plot is another plot near the @xmath38-rd zero , except it uses a different set of @xmath92 zeros.,title="fig : " ] , defined in ( [ eq : specf ] ) , using @xmath76 zeros near the @xmath62-th zero ( upper left ) , @xmath78-rd zero ( upper right ) , and @xmath38-rd zero ( lower left ) . the lower right plot is another plot near the @xmath38-rd zero , except it uses a different set of @xmath92 zeros.,title="fig : " ] , defined in ( [ eq : specf ] ) , using @xmath76 zeros near the @xmath62-th zero ( upper left ) , @xmath78-rd zero ( upper right ) , and @xmath38-rd zero ( lower left ) . the lower right plot is another plot near the @xmath38-rd zero , except it uses a different set of @xmath92 zeros.,title="fig : " ] as usual , define the rotated zeta function on the critical line by @xmath93 the rotation factor @xmath94 is chosen so that @xmath95 is real . in our numerical experiments , @xmath96 . since @xmath97 , it suffices to compute @xmath98 . to do so , we used the numerical differentiation formula ( taylor expansion ) @xmath99 where the remainder term in ( [ eq : numdiff ] ) satisfies @xmath100 we chose @xmath101 , and approximated the derivative by @xmath102 to evaluate @xmath95 at individual points , we used a version of the odlyzko - schnhage algorithm @xcite implemented by the second author @xcite . if the point - wise evaluations of @xmath103 and @xmath104 via this implementation are accurate to within @xmath105 each , then the approximation ( [ eq : numdiff1 ] ) is accurate to within @xmath106 . numerical tests suggested @xmath107 is normally distributed with mean zero and standard deviation @xmath108 . therefore , @xmath107 is typically around @xmath108 . also , varying the choice of @xmath109 in ( [ eq : numdiff1 ] ) suggested the approximation is accurate to about 4 decimal digits with @xmath110 and @xmath111 . in principle , our computations of @xmath55 can be made completely rigorous by carrying them out in sufficient precision . if one plans on calculating @xmath55 with very high precision , however , it will likely be better to first derive a riemann - siegel type formula for @xmath112 itself , with explicit estimates for the remainder . such a formula will be useful on its own as it can be be used to check other conjectures about @xmath113 . numerical data from high zeros of the zeta function generally agrees well with the asymptotic results that have been proved , as well as with several conjectures . there are some systematic differences between observed and expected distributions , but the discrepancies decline with growing heights . the results of this paper provide additional evidence for the speed of convergence of the zeta function to its asymptotic limits . they also demonstrate the importance of outliers , and thus the need to collect extensive data in order to obtain valid statistical results . the long - range correlations that have been found among values of the derivative of the zeta function at zeros can be explained by known analytic techniques .

the derivative of the riemann zeta function was computed numerically on several large sets of zeros at large heights . comparisons to known and conjectured asymptotics are presented . [ multiblock footnote omitted ] _ dedicated to professor akio fujii on his retirement . _

in the context of the predictive maintenance of the french railway switches ( or points ) which enable trains to be guided from one track to another at a railway junction , we have been brought to parameterize switch operations signals representing the electrical power consumed during a point operation ( see figure [ signal_intro ] ) . the final objective is to exploit these parameters for the identification of incipient faults . the method we propose to characterize signals is based on a regression model incorporating a discrete hidden process allowing abrupt or smooth switchings between various regression models . this approach has a connection with the switching regression model introduced by quandt and ramsey @xcite and is very linked to the mixture of experts ( me ) model introduced by jordan and jacobs @xcite by the using of a time - dependant logistic transition function . the me model , as discussed in @xcite , uses a conditional mixture modeling where the model parameters are estimated by the expectation maximization ( em ) algorithm @xcite@xcite . other alternative approaches are based on hidden markov models in a context of regression @xcite . a dedicated em algorithm including a multi - class iterative reweighted least - squares ( irls ) algorithm @xcite is proposed to estimate the model parameters . section 2 introduces the proposed model and section 3 describes the parameters estimation via the em algorithm . the fourth section is devoted to the experimental study using simulated data and real data . we represent a signal by the random sequence @xmath0 of @xmath1 real observations , where @xmath2 is observed at time @xmath3 . this sample is assumed to be generated by the following regression model with a discrete hidden logistic process @xmath4 , where @xmath5 : @xmath6 in this model , @xmath7 is the @xmath8-dimensional coefficients vector of a @xmath9 degree polynomial , @xmath10 is the time dependant @xmath8-dimensional covariate vector associated to the parameter @xmath7 and the @xmath11 are independent random variables distributed according to a gaussian distribution with zero mean and variance @xmath12 . this section defines the probability distribution of the process @xmath4 that allows the switching from one regression model to another . the proposed hidden logistic process supposes that the variables @xmath13 , given the vector @xmath14 , are generated independently according to the multinomial distribution @xmath15 , where @xmath16 is the logistic transformation of a linear function of the time - dependant covariate @xmath17 , @xmath18 is the @xmath19-dimensional coefficients vector associated to the covariate @xmath20 and @xmath21 . thus , given the vector @xmath14 , the distribution of @xmath22 can be written as : @xmath23 where @xmath24 if @xmath25 i.e when @xmath2 is generated by the @xmath26 regression model , and @xmath27 otherwise . the pertinence of the logistic transformation in terms of flexibility of transition can be illustrated through simple examples with @xmath28 components . as it can be shown in figure [ logistic_function_k=2_p=012 ] ( left ) , the dimension @xmath29 of @xmath30 controls the number of changes in the temporal variation of @xmath31 . more particularly , if the goal is to segment the signals into convex homogenous parts , the dimension @xmath29 of @xmath30 must be set to @xmath32 . the quality of transitions and the change time point are controlled by the components values of the vector @xmath30 ( see figures [ logistic_function_k=2_p=012 ] ( middle ) and ( right ) ) . [ cols="^,^,^ " , ] from the model given by equation ( [ eq.regression model ] ) , it can be proved that the random variable @xmath33 is distributed according to the normal mixture density @xmath34 where @xmath35 is the parameter vector to be estimated . the parameter @xmath36 is estimated by the maximum likelihood method . as in the classic regression models we assume that , given @xmath14 , the @xmath37 are independent . this also involves the independence of @xmath2 @xmath38 . the log - likelihood of @xmath36 is then written as : @xmath39 since the direct maximization of this likelihood is not straightforward , we use the expectation maximization ( em ) algorithm @xcite@xcite to perform the maximization . the proposed em algorithm starts from an initial parameter @xmath40 and alternates the two following steps until convergence : [ [ e - step - expectation ] ] * e step ( expectation ) : * + + + + + + + + + + + + + + + + + + + + + + + this step consists of computing the expectation of the complete log - likelihood @xmath41 , given the observations and the current value @xmath42 of the parameter @xmath36 ( @xmath43 being the current iteration ) : @xmath44\nonumber\\ & = & \sum_{i=1}^{n}\sum_{k=1}^k t^{(m)}_{ik}\log ( \pi_{ik}({\mathbf{w}})\mathcal{n}(x_{i};{\boldsymbol{\beta}}^t_k{\boldsymbol{r}}_{i},\sigma^2_k ) ) \enspace,\end{aligned}\ ] ] where @xmath45 is the posterior probability that @xmath2 originates from the @xmath26 regression model . as shown in the expression of @xmath46 , this step simply requires the computation of @xmath47 . [ [ m - step - maximization ] ] * m step ( maximization ) : * + + + + + + + + + + + + + + + + + + + + + + + + in this step , the value of the parameter @xmath36 is updated by computing the parameter @xmath48 maximizing the expectation @xmath46 with respect to @xmath36 . the maximization of @xmath46 can be performed by separately maximizing @xmath49 the maximization of @xmath50 with respect to @xmath51 is a multinomial logistic regression problem weighted by the @xmath47 . we use a multi - class iterative reweighted least squares ( irls ) algorithm @xcite@xcite@xcite to solve it . maximizing @xmath52 with respect to @xmath53 consists of analytically solving a weighted least - squares problem . in addition to providing a signal parametrization , the proposed approach can be used to denoise and segment signals . the denoised signal can be approximated by the expectation @xmath54 where @xmath55 is the parameters vector obtained at the convergence of the algorithm . on the other hand , a signal segmentation can also be deduced by computing the estimated label @xmath56 of @xmath2 : @xmath57 . this section is devoted to the evaluation of the proposed algorithm using simulated and real data sets . two evaluation criteria are used in the simulations : the misclassification rate between the simulated partition and the estimated partition and the euclidian distance between the denoised simulated signal and the estimated denoised signal normalized by the sample size @xmath1 . the proposed approach is compared to the piecewise regression approach @xcite . each signal is generated according to the regression model with a hidden logistic process defined by eq ( [ eq.regression model ] ) . the number of states of the hidden variable is fixed to @xmath58 and the order of regression is set to @xmath59 . the order of the logistic regression is fixed to @xmath60 what guarantees a segmentation into convex intervals . we consider that all signals are observed over @xmath61 seconds . for each size @xmath1 we generate 20 samples . the values of assessment criteria are averaged over the 20 samples . figure [ fig.error rates ] ( left ) shows the misclassification rate obtained by the two approaches in relation to the sample size @xmath1 . it can be observed that the proposed approach is more stable for a few number of observations . figure [ fig.error rates ] ( right ) shows the results obtained by the two approaches in terms of signal denoising . it can be observed that the proposed approach provides a more accurate denoising of the signal compared to the piecewise regression approach . for the proposed model , the optimal values of @xmath62 has also been estimated by computing the bayesian information criterion ( bic ) @xcite for @xmath63 varying from @xmath64 to @xmath65 and @xmath9 varying from @xmath27 to @xmath66 . the simulated model , corresponding to @xmath58 and @xmath59 , has been chosen with the maximum percentage of @xmath67 . this section presents the results obtained by the proposed model for signals of switch points operations . one situation corresponding to a signal with a critical defect is presented . the number of the regressive components is chosen in accordance with the number of the electromechanical phases of a switch points operation ( @xmath68 ) . the value of @xmath29 has been set to @xmath32 , what guarantees a segmentation into convex intervals , and the degree of the polynomial regression has been set to @xmath69 which is adapted to the different regimes in the signals . figure [ resultat_signal_aig_2 ] ( left ) shows the original signal and the denoised signal ( given by equation ( [ eq . signal expectation ] ) ) . figure [ resultat_signal_aig_2 ] ( middle ) shows the variation of the proportions @xmath70 over time . it can be observed that these probabilities are very closed to @xmath32 when the @xmath26 regressive model seems to be the most faithful to the original signal . the five regressive components involved in the signal are shown in figure [ resultat_signal_aig_2 ] ( right ) . in this paper a new approach for signals parametrization , in the context of the railway switch mechanism monitoring , has been proposed . this approach is based on a regression model incorporating a discrete hidden logistic process . the logistic probability function , used for the hidden variables , allows for smooth or abrupt switchings between polynomial regressive components over time . in addition to signals parametrization , an accurate denoising and segmentation of signals can be derived from the proposed model . b. krishnapuram , l. carin , m.a.t . figueiredo and a.j . hartemink , sparse multinomial logistic regression : fast algorithms and generalization bounds , _ ieee transactions on pattern analysis and machine intelligence , _ 27(6 ) : 957 - 968 , june 2005 . s. r. waterhouse , _ classification and regression using mixtures of experts , _ phd thesis , department of engineering , cambridge university , 1997 . v. e. mcgee and w. t. carleton , piecewise regression , _ journal of the american statistical association _ , 65 , 1109 - 1124 , 1970 .

a new approach for signal parametrization , which consists of a specific regression model incorporating a discrete hidden logistic process , is proposed . the model parameters are estimated by the maximum likelihood method performed by a dedicated expectation maximization ( em ) algorithm . the parameters of the hidden logistic process , in the inner loop of the em algorithm , are estimated using a multi - class iterative reweighted least - squares ( irls ) algorithm . an experimental study using simulated and real data reveals good performances of the proposed approach .

note : in the published version this document accompanies the above paper as supplementary material in the form of an epaps archive - i.e it is the same material as can be found at reference [ 15 ] above . the references in this section are self contained and can be found at the end of this document . in this section we compute the emitted photon wavepacket , taking into account the fact that the electron spin is precessing during the emission . this effect and the finite @xmath95 time of the electron , are the dominant source of imperfect cluster state production in the machine gun . any notation not defined here is defined in the paper . the hamiltonian we consider is @xmath96where @xmath97 . here @xmath98 is a projector on the dot excited state ( trion ) manifold and @xmath99 is the free em field hamiltonian . the zeeman interaction for the electron is @xmath100 , where @xmath101 is the bohr magneton , and @xmath102 is the effective gyromagnetic ratio of the electron in the qd . for the low magnetic fields we are dealing with , we can assume @xmath103 for the heavy hole [ 1 ] . we abbreviate @xmath104 . we take @xmath105 . the interaction hamiltonian of the qd with the photon field , @xmath106 is given in the rotating wave and dipole approximation by @xmath107where @xmath108 are the coupling constants which depend on the details of the specific quantum dot in question , and the ( heavy ) trion states are denoted by @xmath109 , @xmath110 respectively . we want to calculate matrix elements of the form : @xmath111 here @xmath112 and @xmath113 are left and right circularly polarised single photon states created by @xmath114 and @xmath115is the resolvent of the hamiltonian . denote by @xmath116 the projector on ground state manifold . then the matrix block inversion formula gives @xmath117with @xmath118 the operator @xmath119 can only couple the state @xmath120 ( or @xmath121 to itself ( recalling that the hole is assumed to not precess ) . therefore@xmath122where @xmath123 , and @xmath124 . when performing the contour integral ( @xmath125 ) , we can replace @xmath126 with the lamb shift @xmath127 and the inverse decay rate @xmath128 . in the following we shall take @xmath129 for simplicity , and absorb @xmath127 into the definition of @xmath130 this gives @xmath131 next , we consider @xmath132 where @xmath133 we now have the final result:@xmath134 writing the matrix elements explicitly : @xmath135we now consider the contour integral . the two poles contributing in the limit @xmath136 are @xmath137 and @xmath138 we denote @xmath139 and a standard complex integration gives@xmath140\]]where @xmath142 above and in the following we omit the phase factors @xmath143 . using the notation @xmath144this simplifies to @xmath145similarly @xmath146which yields @xmath147 \\ & = & \sin ( g_{e}bt)g(k)+i\cos ( g_{e}bt)f(k)\end{aligned}\]]an analogous calculation shows that the amplitudes for starting with the down trion are@xmath148and @xmath149 starting from the ground state manifold , the excitation and subsequent photon emission and spin precesion can be broken into `` good '' and `` bad '' parts , by introducing the operators : @xmath150 the operator @xmath151 corresponds to a correct application of a cnot gate . this happens with an amplitude @xmath51 , which depends on the photon s energy @xmath52 : @xmath152 the operator @xmath153 corresponds to an errored gate which is applied with amplitude @xmath59 , where @xmath154 we get @xmath155 which we break @xmath156 into the good and bad states : @xmath157 as follows : @xmath158 and @xmath159 note that @xmath160 where @xmath161 and @xmath162 the relation to the operators @xmath163 and @xmath164 is @xmath165 and @xmath166 where @xmath167 denotes the free propagation . similarily , if we start from the state @xmath168 the resulting state would be @xmath169lets break @xmath156 into good and bad states : @xmath170 as follows:@xmath171and@xmath172the relation to the operator @xmath163 and @xmath164 is similar to the one described above . note that again @xmath173 if we start from the state @xmath174 the resulting state would be @xmath175 where @xmath176 denotes spin rotation around the @xmath177 axis . the bad part of the wavefunction is therefore equivalent to a pauli @xmath27 error on the ideal cluster state - and as discussed in the paper this error on the spin can be turned into errors on the either the two photons emitted the error occurs on the spin , or as @xmath28 errors on the photon which has already been emitted and the first emitted photon . this latter possibility shows that this quantum scapegoat effect can act backwards in time ! note that while the error is coherent , the coherence between @xmath178 and @xmath179 can be removed by applying stabilizers randomly to the photons . in fact the good and bad wavepackets have non - trivial overlap , and in the next section we consider how our knowledge of this can allow us to apply a simple unitary correction which increases the amplitude of ideal cluster state produced . considering the bad amplitude @xmath59 we can clearly write it as @xmath180 with @xmath181 note that first term is orthogonal to @xmath182then @xmath183\ ] ] ( note that @xmath184 is purely imaginary . ) we can now make the correction @xmath185 where @xmath186 figure 2 of the paper is plotted assuming this simple form of unitary correction has been carried out . in the paper we only briefly mentioned the spectral dephasing which will occur while the system is excited . our intuition contrasted with that of others , namely we felt that this process would not affect the entanglement of the state - in particular with respect to the polarization degrees of freedom we are interested in - but would only lead to the emitted photon wavepackets being in a mixture of different frequencies . this in turn would only affect the ( small fraction ) of photons which have to go through fusion gates , and such photons can be filtered before entering the gates in a way which will only lead to a change in the success probability of the ( non - deterministic ) gate . that is , such filtering need not even lead to a loss error ( as explained below ) . as such the only effect will be that we need to use more photons - but the overhead is some constant factor . nevertheless , to be sure of how the device behaves and of the potential magnitude of this effect , we turn now to the necessary calculations . we begin with a description of the results and a heuristic discussion . there are two excited state ( exciton ) energy levels to consider , which in the paper we denote @xmath187 , @xmath188 . these levels are degenerate . we consider both pure dephasing and cross dephasing . in _ pure dephasing _ of these levels , their couplings to the environment are identical - in practise this means that both states could evolve the same ( random ) phase . note that this will not affect the relative phase between the two states ( such a relative phase would ultimately become a relative phase between the two terms in the entangled photonic cluster state ) . therefore , it will have no bearing on the quality of the entanglement in the polarization degrees of freedom . it will , however , cause the emitted photons to have a broader range of frequencies ( broadened linewidth ) . in [ 2 ] such pure dephasing was measured , and from their results one can infer that the new linewidth would be approximately double the natural linewidth . however , we stress that this broadening is not of particular importance to the operation of our protocol , for two reasons . firstly the majority of the emitted photons will never go through any optical element where ( for instance ) they may be required to interfere with other photons . rather they will simply be measured directly , and this polarization measurement will not care about their frequency . for the photons that do need to undergo type - ii fusion gates ( to cross link with other qubit lines ) the incoherence over the emitted frequencies can be removed ( if necessary ) by suitable spectral filtering . this filtering can be done in an active manner with a suitable prism wherein any photon which does not pass through the filter is simply diverted into another path and measured in an appropriate basis - in terms of the cluster state this simply removes the qubit . as such the only effect of such imperfections is to change the probability of success of the type - ii fusion gate . however efficient quantum computation is obviously possible regardless how small this probability is , as long as it is finite . let us now show some calculations to bolster the above discussion . we treat the pure exciton dephasing as a markovian process and describe it using a lindblad operator @xmath189 where we use the same notations as in the paper . the density matrix of the system evolves according to a standard master equation . we compare the probabilities described by this density matrix to the probabilities @xmath190 in the above @xmath191 is the zeeman splitting , and we have shifted @xmath192 . the off - diagonal matrix elements corresponding to pauli errors have the same form as eq . but with @xmath193 functions replacing the @xmath194 functions . although the function @xmath195 is wider then @xmath196 , the total probability is equal to the total probability without dephasing @xmath197for any value @xmath198 . in figure 1 we plot the probabilities @xmath199 versus @xmath196 . the parameters chosen for the plot are @xmath200 ( as in the inset of fig . 2 of our manuscript ) and @xmath201 [ 1 ] , which of course enters only into @xmath199 . likewise , figure 2 plots @xmath202 vs. @xmath203 . in figure 3 , we compare @xmath199 and @xmath204 showing that spectral filtering is still possible also in the presence of pure exciton dephasing . a potentially more serious source of error would be _ cross dephasing_. experimentally such dephasing has not been observed , despite an attempt to measure it in [ 2 ] , where they could only lower bound the dephasing to be at least 20 times smaller than the pure dephasing . even if this cross dephasing were exactly equal to the lower bound obtained in [ 2 ] , the effect on the emitted photons would ultimately lead to @xmath28-errors on the qubits , which also localize . this potential increase in the pauli error rate is negligible compared to the other sources of pauli error we considered in the paper . to calculate the error probablilty on the photonic qubits we have assumed markovian dynamics for the interaction of the spin with its environment . previous studies [ 3 - 7 ] suggest a non - markovian decay in time of the coherence in the reduced density matrix of a spin coupled to a nuclear spin bath . we note that for the short times we are interested in , the markovian assumption actually overestimates the single qubit error rate . the resulting non markovian correlations of the errors on the different qubits will be studied in a forthcoming publication . however , we note that these correlations can be dealt with , see for example the discussion in [ 4 ] .

we present a method to convert certain single photon sources into devices capable of emitting large strings of photonic cluster state in a controlled and pulsed `` on demand '' manner . such sources would greatly reduce the resources required to achieve linear optical quantum computation . standard spin errors , such as dephasing , are shown to affect only 1 or 2 of the emitted photons at a time . this allows for the use of standard fault tolerance techniques , and shows that the photonic machine gun can be fired for arbitrarily long times . using realistic parameters for current quantum dot sources , we conclude high entangled - photon emission rates are achievable , with pauli - error rates per photon of less than @xmath0 . for quantum dot sources the method has the added advantage of alleviating the problematic issues of obtaining identical photons from independent , non - identical quantum dots , and of exciton dephasing . the primary challenge facing optical quantum computation is that of building suitable photon sources . the majority of effort has been directed at single photon sources . four single photons can be used in an interferometer to produce a maximally entangled bell pair of photons @xcite , and given a source of bell pairs it is in principle possible to fuse them @xcite into larger so - called _ cluster states _ @xcite . these somewhat magical quantum states can be used for performing quantum computation via the simple procedure of making individual ( single - qubit ) measurements on the photons involved . recently a promising new approach has been to produce bell pairs directly @xcite via a radiative cascade in quantum dots . however , even an ideal such source would only reduce the overall resources required for a full optical quantum computation by a small factor . we will show that with current technology it is possible to manipulate certain single photon sources , in particular quantum dots , so as to generate a continuous stream of photons entangled in long strings of ( various varieties of ) 1-dimensional cluster states . using these strings , cluster states capable of running arbitrary quantum algorithms can be very efficiently generated by fusion . we analyze all error mechanisms and show that the error rates can be very low - close to fault tolerant thresholds for quantum computing - even if the source is operated for timescales much longer than the typical decoherence times . we begin with a highly idealized description of the proposal . consider a source with a degenerate spin @xmath1 ground state manifold . the basis @xmath2 , @xmath3 denotes the spin projection along the @xmath4 axis . furthermore , imagine that optical transitions at frequency @xmath5 are possible _ only _ to a doubly degenerate excited state manifold . the excited states @xmath6 , @xmath7 have @xmath8 , thus only the ( single photon ) transitions @xmath9 and @xmath10 are allowed . such transitions are well known to occur , for example , in quantum dots ( qds ) which emit single photons via charged - exciton decay @xcite . we only consider the emitted photons propagating along the @xmath4 axis . therefore , if the initial state of the source is @xmath2 ( @xmath3 ) , an excitation to the state @xmath6 ( @xmath7 ) followed by radiative decay , results in the emission of a single right ( left)-circularly polarized photon @xmath11 ( @xmath12 ) and leaves the source in the state @xmath2 ( @xmath3 ) . now , consider the initial state @xmath13 , and a coherent excitation pulse with a linear polarization along the @xmath14 direction . ( the exciting pulse itself need not necessarily propagate along the @xmath4 direction , which is useful for separation of the coherent and emitted light ) . such a pulse couples equally to both transitions . therefore , the processes described above happen in superposition , and the emitted photon will be entangled with the electron : the joint state of both systems would be the bell pair @xmath15 . repeating such a procedure would produce ghz - type entangled states , which are not useful for quantum computing , and for which disentangling the photons from the electron spin is difficult . moreover , the ghz state is highly vulnerable to decoherence . by contrast , the cluster states suffer none of these problems . to see how to create cluster states , we now imagine that before the second excitation of the system , when the state of the spin and the first photon is @xmath16 , the spin undergoes a @xmath17-rotation about the y - axis . under this operation , described by @xmath18 the state evolves to @xmath19 . a second pulse excitation , accompanied by a second photon emission , will now result in the two photons and the electron spin being in the state @xmath20 . in terms of abstract ( logical ) qubit encodings we will take @xmath21 . it can be readily verified that rotating the spin with another @xmath17 rotation , now leaves the spin and two photons in the state : @xmath22 which is exactly the 3 qubit linear cluster state . repeating the process of excitation followed by @xmath17 rotation , will produce a third photon such that the electron and three photons are in a 4-qubit linear cluster state . the procedure can , in principle , be repeated indefinitely , producing a continuous chain of photons in an entangled linear cluster state . note that one advantage of producing a cluster state is that the electron can be readily disentangled from the string of entangled photons , for example by making a computational ( @xmath23 ) basis measurement on the most recently created photon . in fact , since in general the initial state of the spin will be mixed , such a detection of a photon in state @xmath24 ) polarization can also be used to project the spin to the @xmath2 @xmath25 state , and initializes the cluster state ( either outcome is ok ) . it can be readily verified that the whole idealized procedure just described is equivalent to the qubit quantum circuit depicted in fig . [ circuit ] . ( requiring the careful tracking of certain phases ) , and the physical process of creating a photon with left / right circular polarization conditioned on the state of the electron spin becomes the controlled not gate which leaves the qubit ( photon ) in state @xmath26 ( i.e. @xmath11 ) if the electron spin is in state @xmath26 ( i.e. @xmath2 ) , but otherwise flips it . crucially , as depicted , a pauli @xmath27 error on the spin localizes ; i.e it is equivalent to @xmath27 and @xmath28 errors on the next two photons produced.,width=340 ] a general analysis of how cluster states are generated by evolution of atoms in cavities undergoing general pumping and decay can be found in @xcite , and interesting cavity qed proposals can be found in @xcite . we will primarily focus on a specific implementation of our proposal , namely photon emission from a quantum dot , via the process of creation and subsequent decay of a charged exciton ( trion ) . in practise the expressions we derive , such as the structure of the emitted photon wavepackets , can be easily applied to any systems which obey similar selection rules , and the imperfections we discuss are , for the most part , generic . the importance of the selection rules arises as follows . in semiconductor quantum dots , the @xmath29 states are naturally split off from the @xmath30 ones primarily due to confinement . they correspond to trions containing two electrons in the singlet state and a _ light _ hole or _ heavy _ hole respectively . we can consider only the _ heavy _ trions and neglect the mixing between them . in other systems , while the transitions to @xmath31 may be energetically split off by an external field , or may simply have different couplings , generically they will still lead to imperfections equivalent to non - orthogonality of the emitted photons . moreover processes in other systems tend to be slower , and temporally longer pulses may well also be required because of nearby energy levels . although these problems can be remedied somewhat by applying proper filtration protocols to the output cluster state ( at the expense of larger loss rates ) we focus on quantum dots for which the suppression is essentially perfect , the processes are fast , and the energy levels well separated . although other options exist , we will consider from now on the situation where the @xmath17 rotations on the spin are performed by placing the quantum dot in a constant magnetic field of strength @xmath32 which is directed along the y - direction ( i.e. in the plane of the dot ) . the spin precession at frequency @xmath33 in the z - x plane therefore implements the desired rotation every @xmath34 . suitably timed strobing of the dot by the excitation pulse , followed by the rapid exciton decay , will therefore enable the machine - gun - like generation of 1d cluster state described above . the potential imperfections to be considered are as follows : ( i ) the non - zero lifetime of the trion @xmath35 means that the magnetic field causes precession of the electrons during the emission process . this leads to errors induced on the quantum circuit of fig [ circuit ] ; however we shall find that they can be understood as implementing an error model on the final output cluster state which takes the form of pauli errors occurring with some independent probability on pairs of ( photonic ) qubits . ( ii ) interaction of the electron spin with its environment results in a non - unitary evolution of the spin . this evolution consists of two parts : decoherence ( in which we include both dephasing and spin flips ) and spin relaxation . decoherence is characterized by a @xmath36 time . fortunately we will see that both these processes also lead only to errors occurring independently on two ( photonic ) qubits at a time . efficient cluster state quantum computation can proceed even if every qubit has a finite ( though small ) probability of undergoing some random error @xcite . this implies that the protocol s running time is not limited by @xmath36 , while the errors are amenable to standard quantum error correction techniques for cluster states . spin relaxation is characterized by a @xmath37 time , and is a process which projects the spin to the ground state . in semiconductor quantum dots @xmath37 times are extremely long @xmath38 @xcite . therefore , we shall not discuss the effects of this process further here . we point out , however , that it can be shown this process also leads to errors of a localized form , and so in principle is no obstacle to the continuous operation of the device even for times much longer than @xmath37 . ( iii ) the last source of error is related to the issue of ensuring the photons are emitted into well - controlled spatial modes . in practise this technological issue of mode matching ( say by placing the dot in a microcavity ) results in some amount of photon loss error in the final state . significant progress on this issue is being made for a variety of quantum dots @xcite , although we emphasize that for our proposal strong coupling to the cavity is _ not _ required . fortunately photonic cluster state computation can proceed even in the presence of very high ( up to 50% ) loss @xcite , and we will not consider this source of error further . we now turn to detailed calculations of the error rate inflicted by imperfections ( i ) and ( ii ) discussed above . we first calculate the effect of a finite ratio of the trion decay time @xmath39 time to the spin precession time . we denote by @xmath40 the state of the system ( the quantum dot and photons ) at time @xmath41 after the @xmath42 excitation pulse , @xmath43 . by @xmath44 we mean the state of the system _ just before _ the @xmath45 excitation pulse ( we assume the excitation is instantaneous ) . following the excitation , the trion state decays , emitting a photon and leaving an electron in the qd , the spin of which then precesses in the magnetic field . these lead to an evolution of the quantum state described by the following map ( see @xcite for details ) : ( t_n+)=u^()(g+f)^(t_n^-)(g+ f)u ( ) [ rhon ] the unitary operator @xmath46 describes the precession of the electron spin and the free propagation of the photons . the generalized creation operators @xmath47 , @xmath48 , [ photon error ] describe the excitation and decay process , adding a photon to the state . the trion states decay exponentially with @xmath41 , therefore we have omitted them from eq . ( [ rhon ] ) ( which describes the state of the system at times greater than the trion decay time , _ i.e. _ @xmath49 ) . note that the photons created in each cycle are well separated from the ones created in the previous cycles ( formally , this is taken into account by the free propagation of the photons ) . equation ( [ rhon ] ) describes a circuit isomorphic to the one in fig . [ circuit ] . the operator @xmath50 corresponds to a correct application of a cnot gate . this happens with an amplitude @xmath51 , which depends on the photon s energy @xmath52 : @xmath53 here @xmath54 and @xmath55 . the complex energy of the trion states is denoted by @xmath56 , where @xmath57 is their lifetime . the operator @xmath58 corresponds to a cnot gate followed by a @xmath27 error on the spin qubit . this errored gate is applied with amplitude @xmath59 , where @xmath60 let us for the moment treat the processes described by @xmath50 and @xmath58 as incoherent with each other . then the resulting state is described by the circuit of fig . [ circuit ] with a probability @xmath61 that each cnot gate is followed by @xmath27 error on the spin qubit . as noted in fig . 1 , a state with a @xmath27 error on the spin after generation of the @xmath42 photon ( _ i.e _ , after the @xmath42 cnot ) , is equivalent to a state @xmath27 and a @xmath28 error on the @xmath62 and @xmath63 photons , with no error on the spin . note that the error probability increases with magnetic field strength , because the spin can precess more during the lifetime of the trion @xcite . therefore it is advantageous to consider relatively low magnetic fields , for which @xmath64 . taking the coherence between @xmath50 and @xmath58 into account , it can be seen that a unitary correction @xmath65 with @xmath66 yields a further improvement of the error rate . we also point out that as @xmath51 is more localized around @xmath5 then @xmath59 ( inset of fig . [ errordiagram ] ) , selection of photons with energy @xmath67 would yield a lower error rate at the expense of ( heralded ) loss . and @xmath68 . a stronger field causes faster precession which increases a chance of the error during decay , but reduces the standard dephasing error due to a finite @xmath36 . the inset is a plot of the good mode function @xmath69 ( blue ) and error mode function @xmath70 ( red ) at @xmath71 , from which it can be deduced that spectral filtering can reduce the error rates further.,width=340 ] [ errordiagram ] the calculation above ignores the possibility of the exciton dephasing @xcite during the decay process . pure dephasing , in which both excited levels evolve the same ( random ) phase , will have no affect on the entanglement in polarization with which we are concerned . cross dephasing ( experimentally seen to be very small @xcite ) will lead to @xmath28-errors on the qubits , which also localize ( see @xcite for a detailed discussion ) . we now turn to the issue of decoherence of the spin as a result of its interaction with the nuclei in the quantum dot . assuming markovian dynamics ( discussed further in @xcite ) , it is well known @xcite that the resulting dephasing and spin flip dynamics are equivalent to the action of random pauli operations @xmath72 with some probabilities @xmath73 . the probabilities @xmath74 are suppressed due to the presence of the magnetic field , while @xmath75 is characterized by @xmath36 , the dephasing time . this can readily be shown to give @xmath76 as the probability of a given spin error in 1 cycle . we already noted that a @xmath27 error on the spin becomes a pauli error on the next 2 photons . similarly a @xmath28 error at the end of the @xmath42 cycle is equivalent to a @xmath28 error on the @xmath62 photon , as can be seen from fig . [ circuit ] and the fact that the operator @xmath77 and @xmath78 have similar actions on the states @xmath79 and @xmath80 . as @xmath81 , an @xmath82 error again affects only the next two photons to be generated . in fig . [ errordiagram ] we plot the total probability of error on any given qubit , @xmath83 , as a function of the two dimensionless parameters @xmath84 and @xmath85 . we include the aforementioned easily implemented unitary correction . although free induction decay @xmath36 may be relatively short in low magnetic fields , using a spin echo pulse at half way along the cycle time can extend @xmath36 considerably , and remove the dephasing caused by a wide distribution of nuclear ( overhauser ) magnetic fields ( often termed inhomogeneous broadening and characterized by a @xmath86 time ) . to estimate an achievable error rate , we consider a decay time of @xmath87 , and a dephasing time of @xmath88 with the addition of the spin echo pulses ( a lower bound of @xmath89 have been measured in high magnetic fields @xcite ) . this gives @xmath90 . from fig . [ errordiagram ] , one can deduce that a probability of error less then @xmath0 can be achieved by applying a magnetic field of @xmath91 ( we take @xmath92 ) . we note that even without the spin echo pulses , error rates of about @xmath93 are achievable , which enables the production of considerable longer and higher quality optical cluster states than those produced by current methods . so far we have considered pulse excitations that are timed to coincide with ( integer multiples of ) @xmath17 rotations of the spin . in fact it can be advantageous to sometimes wait for a full @xmath94 rotation to occur . this has the effect of emitting subsequent photons which are redundantly encoded @xcite . fusing together such qubits gives a highly efficient method for producing higher - dimensional cluster states which are universal for quantum computing . photons which undergo fusion can be spectrally filtered ( via a suitable prism ) , such that if they fail to pass the filter they can still be measured and removed from the cluster state . this filtering does not lead to an increase in loss error rates , but simply decreases the overall success probability of the fusion gates . current experiments produce photonic cluster states via spontaneous parametric downconversion @xcite , and would seem to be limited to producing 6 to 8 photon cluster states . our proposal in principle can produce strings of thousands of photons ; however initial experiments will be limited by collection efficiency . with the parameters above a simple analysis shows that we would need a collection+photodetection efficiency of about 18% for a demonstration of on - demand 12-photon cluster states , where the full 12 qubits are expected to be detected about once every 10 seconds . finally , our proposal is suggestive of an efficient mechanism for entangling matter qubits @xcite , and we feel this is a topic worthy of further investigation . _ acknowledgments . _ tr acknowledges the support of the epsrc , the qip - irc , and the us army research office . we acknowledge numerous useful conversations with s. economou and d. gershoni . 99 q. zhang , _ et . al _ , phys . rev . a * 77 * , 062316 ( 2008 ) d. e. browne and t. rudolph phys . rev . lett . * 95 * , 010501 ( 2005 ) . r. raussendorf and h. j. briegel , * 86 * , 5188 ( 2001 ) . n. akopian _ et . al _ , phys . rev . lett . * 96 * , 130501 ( 2006 ) , m. stevenson _ et . al . _ , nature * 439 * , 179 ( 2006 ) , a. gilchrist _ et . al . _ , nature * 445 * , e4-e5 . e. meirom _ et . al_. , phys . rev . a 77 , 062310 ( 2008 ) , j. e. avron _ et . al_. , phys . rev . lett . * 100 * , 120501 ( 2008 ) . m. bayer _ et . al . _ phys . rev . b * 65 * 195315 ( 2002 ) . c. schoen et al , phys . rev . lett . 95 , 110503 ( 2005 ) . j. metz _ et . al . _ , phys . rev . a * 76 * , 052307 ( 2007 ) ; c. schoen _ et . al . _ , phys . rev . a * 75 * , 032311 ( 2007 ) ; a. beige _ et . al . _ , j. mod . opt . 54 , 397 ( 2007 ) . r. raussendorf _ et . al . _ , annals of physics * 321 * , 2242 ( 2006 ) ; c. m. dawson _ et . al . _ , phys . rev . lett . * 96 * , 020501 ( 2006 ) ; m. a. nielsen and c. m. dawson , eprint : quant - ph/0601066 ; panos aliferis and debbie w. leung , phys . rev . a * 73 * , 032308 ( 2006 ) m. varnava _ et . al _ , phys . rev . lett . * 97 * , 120501 ( 2006 ) ; ibid , phys . rev . lett.,*100 * , 060502 ( 2008 ) . m. kroutvar _ et . al_. , nature * 432 * , 81 ( 2004 ) . k. hennessy _ et . al . _ , nature * 445 * , 896 ( 2006 ) ; i. fushman _ et . al . _ , phys . stat . sol . 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the goal of this paper is to shed some light on the connection between min - sum algorithm ( msa ) decoding and the formulation of decoding as a linear program . in particular , we address the problem of bounding the performance of linear programming ( lp ) decoding with respect to word error rate . the bounds reflect similar analytic bounds for msa decoding of low - density parity - check ( ldpc ) codes due to wiberg @xcite and establish the existence of an snr threshold for lp decoding . while highly efficient and structured computer - based evaluation techniques , such as density evolution ( see e.g. @xcite ) , provide excellent bounds on the performance of iterative decoding techniques , to the best of our knowledge , the best analytic performance bound in the case of msa decoding is still the bound given by wiberg in his thesis based on the weight distribution of a tree - like neighborhood of a vertex in a graph . a similar bound was also derived by lentmaier et al . we derive the equivalent bound for lp decoding of regular ldpc codes . in this paper we are interested in binary ldpc codes where a binary ldpc code @xmath0 of length @xmath1 is defined as the null - space of a sparse binary parity - check matrix @xmath2 , i.e. @xmath3 . in particular , we focus on the case of regular codes : an ldpc code @xmath0 is called @xmath4-regular if each column of @xmath2 has hamming weight @xmath5 and each row has hamming weight @xmath6 . the rate of a @xmath4-regular code is lower bounded by @xmath7 . to an @xmath8 parity - check matrix @xmath2 we can naturally associate a bipartite graph , the so - called tanner graph @xmath9 . this graph contains two classes of nodes : variable nodes @xmath10 and check nodes @xmath11 . both variable nodes and check nodes are identified with subsets of the integers . variable nodes are denoted as @xmath12 and check nodes are denoted as @xmath13 . whenever we want to express that an integer belongs to the set of variable nodes we write @xmath14 ; similarly , when an integer belongs to the set of check nodes we write @xmath15 . the tanner graph @xmath9 contains an edge @xmath16 between node @xmath14 and @xmath17 if and only if the entry @xmath18 is non - zero . the set of neighbors of a node @xmath19 is denoted as @xmath20 ; similarly , the set of neighbors of a node @xmath17 is denoted as @xmath21 . in the following , @xmath22 will be the set of edges in the tanner graph @xmath9 . the convex hull of a set @xmath23 is denoted by @xmath24 . if @xmath25 is a subset of @xmath26 then @xmath24 denotes the convex hull of the set @xmath25 after @xmath25 has been canonically embedded in @xmath27 . the inner product between vectors @xmath28 and @xmath29 is denoted as @xmath30 . finally , we define the set of all binary vectors of length @xmath6 and even weight as @xmath31 . in the rest of this paper we assume that the all - zeros word was transmitted an assumption without any essential loss of generality because we only consider binary linear codes that are used for data transmission over a binary - input output - symmetric channel . given a received vector @xmath32 we define the vector @xmath33 of log - likelihood ratios by @xmath34 maximum likelihood ( ml ) decoding may be cast as a linear program once we have translated the problem into @xmath35 . to this end we embed the code into @xmath35 by straightforward identification of @xmath36 with @xmath37 . in other words , a code @xmath0 is identified with a subset of @xmath38 . this description is usually not practical since the polytope @xmath39 is typically very hard to describe by hyperplanes ( or as a convex combination of points ) . given a parity - check matrix @xmath2 , the linear program is relaxed to @xcite here , @xmath40 is the so - called fundamental polytope @xcite which is defined as @xmath41 where @xmath42 where @xmath43 is the @xmath44-th row of @xmath2 . since @xmath45 is always a feasible point , i.e. @xmath46 holds , zero is an upper bound on the value of the lp in lp decoding . in fact , we can turn this statement around by saying that whenever the value of the linear program equals zero then the all - zeros codeword will be among the solutions to the lp . thus , motivated by the assumption that the all - zeros codeword was transmitted , we focus our attention on showing that , under suitable conditions , the value of the lp is zero which implies that the all - zeros codeword will be found as a solution . for simplicity we only consider channels where the channel output is a continuous random variable . in this case a zero value of the lp implies that the zero word is the unique solution with probability one . the main idea now is to show that the value of the dual linear program is zero . this technique , dubbed `` dual witness '' by feldman et al . in @xcite will then imply the correct decoding . first , however , we need to establish the dual linear program . to this end , for each @xmath47 , we associate the variable @xmath48 with the edge between variable node @xmath49 and check node @xmath44 in the tanner graph @xmath9 . in other words , we have a variable @xmath48 if and only if the entry @xmath18 is non - zero . for each @xmath17 we define the vector @xmath50 that collects all the variables @xmath51 . also , for each @xmath17 , we associate the variable @xmath52 with the check node @xmath44 . we have is an inequality ( @xmath53 ) . however , there always exists a maximizing assignment of dual variables that satisfies this conditions with equality . ] the dual program has a number of nice interpretations . any @xmath52 is bounded from above by zero and can only equal zero if the vector @xmath50 has minimal correlation with the all - zeros codeword . would have to be replaced by the corresponding code . ] thus the dual program will only get a zero value if we find an assignment to @xmath48 such that the local all - zeros words are among the `` best '' words for all @xmath44 . we are constraint in setting the @xmath48-values by the second equality constraint . while msa decoding is not the focus of interest in this paper , it turns out that the msa lies at the core of the proof technique that we will use . the msa is an algorithm that is being run until a predetermined criterion is reached . with each edge in the graph we associate two messages : one message is going towards the check - node and one is directed towards the variable node . let the two messages be denoted by @xmath54 and @xmath55 , respectively , where , as in the case of the single variable @xmath48 in the section above , variables are only defined if the entry @xmath18 is non - zero . the update rules of msa are then rather than the quantity @xmath55 we will consider its negative value . moreover , we keep track of the messages that were sent by message numbers in the superscript . thus we modify the msa update equations as clearly , the sign change leaves the algorithmic update steps essentially unchanged . ( note that e.g. when all @xmath56 are non - negative then all @xmath57 will be non - positive . ) still , we may e.g. write @xmath58 which more closely reflects the structure of the dual program above . we will need the notion of a computation tree ( ct ) @xcite . we can distinguish two types of cts , rooted either at a variable node or at a check node . our cts will be rooted at check nodes which is more natural when dealing with the dual program . a ct of depth @xmath59 consists of all nodes in the universal cover of the tanner graph that are reachable in @xmath60 steps . in particular , we will most of the time assume that the leaves in the ct are variable nodes . assume we have run the msa for @xmath59 iterations , corresponding to a ct of depth @xmath59 . for the moment let us also assume that the underlying graph has girth larger than @xmath61 . based on the iterations of the msa and fixed ct root node @xmath62 we can assign values to the dual variables in the following way . was assign values to @xmath48 according to the distance of the edge @xmath16 to the root node of the ct . so , if @xmath16 is at distance @xmath63 from the root node @xmath64 then @xmath48 is assigned the value @xmath65 and if @xmath16 is at distance @xmath66 from the root node @xmath64 then @xmath48 is assigned the value @xmath67 . to the root @xmath64 is larger than @xmath68 then @xmath69 . ] let us denote this assignment to variables @xmath48 as @xmath70 indicates that the assignment is based on the ct rooted at node @xmath64 . ] . note that the assignment @xmath71 does not satisfy the constraints of the dual linear program , i.e. itself it is not dual feasible . in particular , any edge of distance more than @xmath68 from the root is assigned the value @xmath72 and hence at any variable node @xmath49 at distance more than @xmath68 from the root we do not satisfy the constraint @xmath73 unless @xmath74 happens to be @xmath72 . however , we have the following lemma . for each @xmath62 let an assignment @xmath71 be given based on @xmath59 iterations of the msa . the sum @xmath75 is a multiple of a dual feasible point . more precisely , for the number @xmath76^{(\ell-1)}$ ] the vector @xmath77 is a dual feasible point . each variable node @xmath19 is part of @xmath78^{(\ell-1)}$ ] cts for different root nodes @xmath64 and so one can verify that we must have @xmath79^{(\ell-1)}\lambda_i$ ] . using the abbreviation @xmath80^{\ell-1}$ ] we see that @xmath77 is a dual feasible point . the above lemma gives a structured way to derive dual feasible points for lp decoding from the messages passed during the operation of the msa . however , these points are not very good since the overall assignment @xmath81 is again dominated by the leaves of the ct with all the pertaining problems . the problem becomes obvious when we write out the assignment @xmath81 as a function of the msa messages directly . if we perform @xmath59 steps of iterative decoding , for any edge @xmath47 we can write @xmath82 written in form of a telescoping sum we get @xmath83 while the above sums show that the dual feasible point can be easily computed alongside the msa recursions it also shows the problem that messages @xmath84 and @xmath85 are weighted exponentially more for small values of @xmath86 . we will have to attenuate the influence of the leaves in the cts in order to make interesting statements to this end , let @xmath87 be a vector with positive entries of length @xmath59 and let a generalized assignment @xmath88 to dual variables be derived from @xmath71 by multiplying the message on each edge at distance @xmath63 or @xmath66 by @xmath89 . is said to be at distance one from @xmath44 ; @xmath90 is set to one . ] in other words , values assigned to edges at distance three or four from the root node are multiplied with @xmath91 , values at distance five and six are multiplied with @xmath92 etc . again we can form the multiple of a dual feasible point as is shown in the next lemma . each variable node @xmath19 is part of @xmath78^{(\ell-1)}$ ] cts for different root nodes @xmath64 . because all edges incident to a variable node are attenuated in the same way , one can verify that we must have @xmath94^{(\ell-1)}\lambda_i$ ] . using the abbreviation @xmath95^{\ell-1}$ ] we see that @xmath77 is a dual feasible point . optimizing the vector @xmath87 gives us some freedom and we want to choose the vector @xmath87 appropriately . first we have to learn more about the dual feasible point that we construct in this way . while we kept the feasibility of an assignment @xmath96 by identically scaling the values @xmath48 that are adjacent to a variable node in a ct , we scale values @xmath48 that are adjacent to check nodes differently . given a vector @xmath87 , the dual feasible point may be easily computed together with the messages of the msa . to this end define a vector @xmath97 with components @xmath98 . writing again the dual variable @xmath99 as functions of @xmath84 and @xmath85 we get @xmath100 written in form of a telescoping sum we obtain @xmath101 let @xmath104 and fix some @xmath17 . assume the msa yields messages where @xmath84 is positive for all @xmath105 for some @xmath86 . the inner product @xmath106 is non - negative for all @xmath107 , in particular it is positive for all @xmath108 . are given by @xmath21 . ] recall that @xmath85 is negative for all @xmath47 ( this is in line with the modification of the msa ) . one can easily verify the following fact about the vector containing @xmath109 for all @xmath105 : there is only one negative entry and the absolute value of this entry matches the absolute value of the smallest positive entry . the statement follows . with the choice of @xmath110 , which results in @xmath111 , we get the following expression for the dual feasible point @xmath112 or @xmath113 we are still in a situation where @xmath114 is weighted by a factor that grows exponentially fast in @xmath59 . however , we note that , once the msa has converged , @xmath84 also grows exponentially fast in @xmath86 and this offsets , to some extend , the exponential weighing of @xmath114 . in order to exploit this fact more systematically we initialize the msa s check to variable messages @xmath115 , @xmath47 , with @xmath116 , where @xmath117 is a large enough positive number . with this initialization we can guarantee ( for @xmath118 ) for all @xmath47 that the value of @xmath84 is strictly positive .. ] thus we can apply lemma [ lemma : inner : product:1 ] . it remains to offset the choice @xmath115 with @xmath119 . to this end we consider a ct of depth @xmath59 rooted at check node @xmath64 . consider the event @xmath120 that the all - zeros word on this ct is more likely than any word that corresponds to a local nonzero word assigned to the root node .. is defined on the ct without the change in initialization ] let @xmath118 and assume the event @xmath120 is true . moreover , assume that we initialize the msa with check to variable messages @xmath121 , @xmath47 , for a large enough number @xmath117 . the inner product @xmath122 is non - negative for all @xmath107 , in particular it is non - negative for all @xmath108 . we exploit the fact that summaries sent by the msa can be identified with cost differences of log - likelihood ratios . consider a message @xmath123 on edge @xmath124 . this message may be written as @xmath125 for some @xmath126 . since the msa propagates cost summaries along edges , we can interpret @xmath126 as the summary of the cost due to the @xmath74 inside the subtree that emerges along the edge @xmath124 . similarly , @xmath127 is the cost contributed by the leaf nodes of this sub - tree . here we use the fact that the minimal codeword which accounts for a one - assignment in edge @xmath124 contains exactly @xmath128 leaf nodes with a one - assignment . but then the vector @xmath129 equals the vector @xmath130 . the event @xmath120 is true only if the inner product @xmath131 is positive for all @xmath132 . hence event @xmath120 implies the claim of the lemma . let @xmath133 be the averaged assignment to the dual variables obtained from the msa messages with @xmath115 set to @xmath116 . lemmas [ lemma : inner : product:1 ] and [ lemma : assignment:3 ] imply that the sum , @xmath134 has a non - negative value for any @xmath107 , and , in particular , the value equals zero for @xmath135 . it follows that @xmath136 in the dual lp can be chosen as zero . for each check node @xmath44 for which event @xmath137 is true we can be sure that the correlation of any codeword in @xmath31 with @xmath138 is non - negative . if we can be sure that the event @xmath137 is true for all check nodes we would , thus , have exhibited a dual witness for the optimality of the all - zeros codeword . we have to estimate the probability of the event @xmath137 and set it in relation to the number of checks in the graph @xmath9 . in order to estimate the latter we employ a result by gallager @xcite that guarantees the existence of @xmath4-regular graphs in which we can conduct @xmath59 steps of msa decoding without closing any cycles provided that @xmath59 satisfies @xmath139 where the term @xmath140 in this expression is independent of @xmath141 . finally , we can estimate the probability of the event @xmath137 from the known weight distribution of the code on the ct provided the underlying graph has girth at least @xmath61 . the minimal codewords have weight @xmath142 and there are a total of @xmath143 minimal - tree codewords . based on a union bound we thus get an expression @xmath144 which means that @xmath145 decreases doubly exponentially in @xmath59 if the bhattacharyya parameter @xmath146 satisfies @xmath147 . let a sequence of @xmath4-regular ldpc codes be given that satisfies equation . under lp decoding this sequence achieves an arbitrarily small probability of error on any memoryless channel for which the bhattacharyya parameter @xmath146 satisfies @xmath148 . for such a channel the word error probability @xmath149 decreases as @xmath150 for some positive parameters @xmath151 and @xmath152 . most of the proof is contained in the arguments leading up to the theorem . in order to see the explicit form of the word error rate we employ a union bound for the @xmath153 check nodes combining and . we find that the word error rate is bounded by @xmath154 where @xmath140 does not depend on @xmath141 . the statement of the theorem is obtained by simplifying this expression . we conclude this paper with an intriguing observation concerning the awgn channel . in @xcite it is proved that no @xmath4-regular ldpc code can achieve an error probability behavior better than @xmath155 for constants @xmath156 and @xmath157 that are independent on @xmath141 . the result of the theorem thus shows that there exist sequences of ldpc codes whose error probability behavior under lp decoding is boxed in as a function of @xmath141 between : @xmath158 m. lentmaier , d. v. truhachev , d. j. costello , jr . , and k. zigangirov , `` on the block error probability of iteratively decoded ldpc codes , '' in _ 5th itg conference on source and channel coding _ , ( erlangen , germany ) , jan . 14 - 16 2004 . j. feldman , _ decoding error - correcting codes via linear programming_. phd thesis , massachusetts institute of technology , cambridge , ma , 2003 . available online under ` http://www.columbia.edu/ ` ` ~jf2189/pubs.html ` . r. koetter and p. o. vontobel , `` graph covers and iterative decoding of finite - length codes , '' in _ proc . 3rd intern . symp . on turbo codes and related topics _ , ( brest , france ) , pp . 7582 , sept . 15 2003 . o. vontobel and r. koetter , `` graph - cover decoding and finite - length analysis of message - passing iterative decoding of ldpc codes , '' _ submitted to ieee trans . inform . theory , available online under ` http:// ` ` www.arxiv.org/abs/cs.it/0512078`_ , dec . j. feldman , t. malkin , c. stein , r. a. servedio , and m. j. wainwright , `` lp decoding corrects a constant fraction of errors , '' in _ proc . ieee intern . symp . on inform . theory _ , ( chicago , il , usa ) , p. 68 , june 27july 2 2004 .

in his thesis , wiberg showed the existence of thresholds for families of regular low - density parity - check codes under min - sum algorithm decoding . he also derived analytic bounds on these thresholds . in this paper , we formulate similar results for linear programming decoding of regular low - density parity - check codes .

the spin - statistics theorem , according to which identical particles with integer spin are bosons whereas those with half - integer spin are fermions , breaks down if the particles are confined to one or two dimensions . realization of this fact had its origin over forty years ago when it was shown @xcite that the many - body problem of hard - sphere bosons in one dimension can be mapped exactly onto that of an ideal fermi gas , so that many properties of such bose systems are fermi - like . it is now known that this `` fermi - bose duality '' is a very general property of identical particles in 1d , not restricted to the hard - sphere model , and relating strongly interacting bosons to weakly - interacting fermions and vice versa . in recent years this esoteric subject has become highly relevant through experiments on ultracold atomic vapors in atom waveguides @xcite . an understanding of their properties is important for atom interferometry @xcite and integrated atom optics @xcite , which are potentially important for development of ultrasensitive detectors of accelerations and gravitational anomalies . when an ultracold atomic vapor is placed into an atom waveguide with sufficiently tight transverse confinement , its two - body scattering properties are strongly modified and short - range correlations are greatly enhanced . this occurs in a regime of low temperatures and densities where both the chemical potential @xmath0 and the thermal energy @xmath1 are less than the transverse oscillator level spacing @xmath2 , so transverse oscillator modes are frozen and the dynamics is described by a one - dimensional ( 1d ) hamiltonian with zero - range interactions @xcite . this 1d regime has already been reached experimentally @xcite , as has a regime with @xmath3 but @xmath4 @xcite . we assume herein that both @xmath3 and @xmath5 as in @xcite . nevertheless , _ virtually _ excited transverse modes renormalize the effective 1d coupling constant via a confinement - induced resonance , as first shown for bosons @xcite and recently for spin - polarized fermionic vapors @xcite . at the very low densities of ultracold atomic vapors , the 3d interatomic interactions are usually adequately described by the s - wave scattering length and a corresponding 3d zero - range pseudopotential , the input for the original derivation @xcite of an effective 1d interaction between waveguide - confined spinless bosonic atoms . a more detailed and comprehensive theory has been developed recently @xcite . in the simplest case of spinless bosons the resultant effective 1d interaction is of the form @xmath6 where @xmath7 is an explicit and nontrivial function @xcite of the 3d s - wave scattering length , @xmath8 , and @xmath9 and @xmath10 are 1d coordinates of the interacting atoms measured along the longitudinal axis of the waveguide . this derivation @xcite will be described in sec . [ subsec : scattering ] , and in sec . [ subsec : pseudopotentials ] the derivation of effective 1d pseudopotentials will be presented , for application not only to the spinless bose gas but also to the case of spinor fermi and bose gases , for which the definition of pseudopotentials is much more delicate due to wave function discontinuities induced in the zero - range limit by 1d odd - wave interactions derived from 3d p - wave scattering . in the spatially uniform case ( no longitudinal trapping potential ) the zero - temperature properties of an @xmath11-atom spinless bose gas are determined by a dimensionless coupling constant @xmath12 @xcite where @xmath13 is the atomic mass and @xmath14 is the 1d number density , @xmath15 being the length of the periodic box . the exact @xmath11-boson ground state was found in the spatially uniform case by the bethe ansatz method in a famous paper of lieb and liniger ( ll ) @xcite , and spawned later development of a powerful and more general approach @xcite . an exact solution in the presence of longitudinal trapping is not known , but is well approximated by a local equilibrium approach @xcite . since the density is in the denominator , at sufficiently _ low _ densities one enters the regime of strong interactions and strong short - range correlations where effective - field approaches fail , exactly the opposite of the situation in 3d where the density is in the numerator of the dimensionless coupling constant . for @xmath16 the scattering reduces to specular reflection ( negligible transmission coefficient ) and the hamiltonian reduces to that of impenetrable point bosons , whose exact @xmath11-particle ground state was found in 1960 by an exact mapping to the _ ideal _ bose gas @xcite , leading to `` fermionization '' of many properties of the bose gas and related to breakdown of the symmetrization postulate and the spin - statistics theorem @xcite . this fermi - bose mapping method and its application to determination of exact n - atom ground states will be described in sec . [ subsec : fba ] . in sec . [ subsec : fbb ] a very powerful generalization of this mapping to arbitrary coupling strength @xcite will be described and its application to determination of the ground state of the magnetically trapped , spin - aligned fermi gas will be reviewed , and in sec . [ subsec : fbc ] application of two further generalizations of the mapping to determination of the ground state of the optically trapped spinor fermi gas will be described . we begin from the hamiltonian for two atoms under transverse harmonic confinement and subject to an arbitrary interaction potential @xmath17 where @xmath18 and @xmath19 are the atomic masses , @xmath20 is the transverse trap frequency , and @xmath21 and @xmath22 are the laplacian and radial coordinate of the @xmath23 atom , respectively . this hamiltonian is separable in relative and center - of - mass coordinates @xmath24 , @xmath25 , @xmath26 being the total mass , yielding @xmath27 , where @xmath28 and @xmath29 where @xmath30 is the reduced mass , and @xmath31 and @xmath32 are the relative and center - of - mass radial coordinates , respectively . the center - of - mass hamiltonian is that of a simple harmonic oscillator whose solution is known , hence we focus only on the relative motion of the two particles . this reduces the problem to a single particle of mass @xmath0 , subject to transverse harmonic confinement , which is scattered by an external potential @xmath33 . the central equation which must be solved is therefore schrdinger s equation for the state of relative motion of the two atoms @xmath34|\psi(e)\rangle = 0\ ] ] where @xmath35 and @xmath36 is the interatomic potential . determining the eigenstates of this hamiltonian , in particular within the s - wave scattering approximation for @xmath33 , is the central goal of this section . in this subsection we briefly review the t - matrix formulation of scattering theory , which provides a convenient framework for approaching the present problem . let us first introduce the retarded green s function for a system with the hamiltonian @xmath37 and energy @xmath38 @xmath39 we then define the t - matrix at energy @xmath38 of the scatter @xmath36 in the presence of the background hamiltonian @xmath37 in the usual manner as @xmath40^{-1 } \hat{v}\nonumber\\ & = & \sum_{n=0}^\infty \left[\hat{v}\hat{g}_{\hat{h}}(e)\right]^n\hat{v},\end{aligned}\ ] ] the summation form being valid provided that there are no difficulties with convergence . two relations on which we will rely heavily are the lippman - schwinger relation @xmath41 which relates the full green s function of the system @xmath42 to the unperturbed green s function @xmath43 and the t - matrix , and the lupu - sax formula @xcite @xmath44 \right]^{-1}\hat{t}_{\hat{h}',\hat{v}}(e ) , \nonumber\end{aligned}\ ] ] which relates the t - matrix of the scatter @xmath36 in the background hamiltonian @xmath37 to the t - matrix for the same scatter but in a different background hamiltonian @xmath45 . in the continuous part of the spectrum of the total hamiltonian @xmath42 its eigenstates can be expressed as a sum of an incident and a scattered wave according to @xmath46 where @xmath47 , the `` incident '' state vector , satisfies @xmath48 we can then express the schrdinger equation for the total system as @xmath49 \left(|\psi_0(e)\rangle+|\psi_s(e)\rangle\right)=0.\ ] ] this equation is readily solved for the scattered wave in terms of the unperturbed green s function and the t - matrix , yielding @xmath50 which will serve as the basis for our treatment of the present scattering problem . at first glance it may seem that finding the t - matrix is no easier than a direct solving of the schrdinger equation ( [ hpsie ] ) . we will demonstrate , however , that the t - matrix formulation allows for a self - consistent description of the low - energy part of the spectrum that uses the _ free - space _ low - energy scattering properties of the interaction potential as the _ only _ input . in addition the low - energy ( s - wave ) limit is isolated to a single well - defined approximation without requiring the ad - hoc introduction of regularization via a pseudo - potential . in this section we first outline this self - consistent low - energy treatment . we then solve for the t - matrix using the standard huang - fermi pseudo - potential , showing that the pseudo - potential reproduces the exact result in this situation . let the unperturbed hamiltonian @xmath37 be a hamiltonian for a single nonrelativistic particle in presence of a trapping potential @xmath51 : @xmath52\langle{\bf r}|\psi\rangle \quad , \end{aligned}\ ] ] assume also that the particle is ` perturbed ' by a scatterer given by @xmath53 localized around @xmath54 . in what follows we will derive a _ low - energy approximation _ for the t - matrix of the scatterer @xmath55 in presence of @xmath37 . it is important to note that , by definition , the t - matrix acts only on eigenstates of the unperturbed hamiltonian , which we can safely assume to be regular everywhere . ( this is of course a constraint on the properties of the unperturbed hamiltonian . ) in this case the zero - range s - wave scattering limit does not require any regularization of the t - matrix . by making use of the lupu - sax formula ( [ lupusax ] ) , we first derive the correct form of the t - matrix in the low - energy s - wave regime without the introduction of a regularized pseudo - potential . in the following section , however , we will see that the results we obtain are in agreement with the standard huang - fermi pseudopotential approach to s - wave scattering . we begin our derivation by first specifying a `` reference '' background hamiltonian @xmath45 as @xmath56\langle{\bf r}|\psi\rangle.\end{aligned}\ ] ] this hamiltonian is that of a free particle , but with an explicit energy dependence included so that the eigenstates have zero wavelength at all energies . we note that this reference hamiltonian agrees with the free - space hamiltonian in the zero - energy limit . while this hamiltonian may seem strange , it is a valid reference hamiltonian which turns out to be useful because the resulting t - matrix is energy independent for any scattering potential . the green s function for this hamiltonian is given by @xmath57 as can be verified by direct substitution into @xmath58\ , \hat{g}_{\hat{h}'}(e)=\hat{i}$ ] . in turn the @xmath59-matrix of the interaction potential @xmath55 in presence of @xmath45 is independent of energy and can therefore be expressed as @xmath60 where the kernel @xmath61 is defined as normalized to unity , @xmath62 the normalization coefficient @xmath63 is then related , through the zero - energy scattering amplitude , to the three - dimensional scattering length @xmath64 according to @xmath65 imagine that the kernel @xmath66 is well localized within some radius @xmath67 . in perturbative expansions at low energies this kernel only participates in convolutions with slow ( as compare to @xmath67 ) functions , in which case it can be approximated by a @xmath68-function , @xmath69 this straightforward approximation is the key to the s - wave scattering approximation . this effectively replaces the exact reference t - matrix by its long - wavelength limit , so that the reference t - matrix assumes the form @xmath70 which is equivalent to @xmath71 where @xmath72 is the position eigenstate corresponding to the location of the scatterer . in expression ( [ tdelta ] ) @xmath73 and @xmath74 refer to the wavevectors of any matrices which multiply the t - matrix from the left and right , respectively . if we now substitute the above expression for the reference t - matrix into the lupu - sax formula ( [ lupusax ] ) for the t - matrix under the background hamiltonian @xmath37 we arrive at @xmath75^ng|0\rangle\langle 0| \nonumber \\ & & \quad = \left[1-g\langle 0|\hat{g}_{\hat{h}}(e)|0\rangle+g\langle 0|\hat{g}_{\hat{h}'}(e)|0\rangle\right]^{-1 } g|0\rangle\langle 0| . \nonumber\end{aligned}\ ] ] making use of eq . ( [ g ] ) , we introduce the function @xmath76 , defined as @xmath77,\end{aligned}\ ] ] from which we obtain the following simple expression for the t - matrix of the scatterer @xmath36 in presence of the trap : @xmath78 from comparing the equations the free - space and bound green s functions obey one can show that the singularity in bound green s function is the same as that in the free - space green s function . hence , @xmath79 is the value of the regular part of the bound green s function at the origin . for the case of transverse harmonic confinement function @xmath79 has been explicitly computed in @xcite . it reads @xmath80 where @xmath81 is the generalized riemann zeta function described in the mathematical literature @xcite : @xmath82 \nonumber \\ & & \mbox{re}(s)>0 , \quad -2\pi < \arg(n+\alpha ) \le 0.\end{aligned}\ ] ] note that no established convention for choosing the branch of the irrational power functions exist : the choice above is just the most suitable for the needs of this paper . by a direct substitution to the equation for the green s function of the relative motion of two particles in a waveguide @xmath83 it is easy to show that the green s function can be decomposed to a sum over the transverse modes in the following way : @xmath84 where @xmath85 , @xmath86 are the zero - angular - momentum eigenstates of the transverse oscillator , and @xmath87 is the green s function of a free one - dimensional particle : @xmath88 assume now that our energy belongs to the single mode window : @xmath89 in this case all the @xmath90 terms in the expansion ( [ green_decomposition ] ) exponentially decay at large @xmath91 . accordingly the solution of the scattering problem ( [ psispis0 ] ) , ( [ tehv ] ) now resembles a solution of a _ one - dimensional _ scattering problem : @xmath92 where @xmath93 is the strength of the three - dimensional t - matrix , i.e. , @xmath94 and @xmath95 . comparing the expression ( [ tmp10 ] ) and the general form of a scattering solution ( [ psispis0 ] ) , ( [ tehv ] ) it is natural to interpret the expression in the braces in the l.h.s . of ( [ tmp10 ] ) as a solution of a _ one - dimensional _ scattering problem , subject to a scattering potential whose ( one - dimensional ) t - matrix reads @xmath96 with @xmath97 for low energies @xmath98 the strength of the one - dimensional t - matrix is approximately @xmath99 -\frac{i}{k } } \label{tau1dapprox } \ ] ] the relative error of this approximation scales as @xmath100 . one can now attempt to introduce an effective one - dimensional scatterer whose t - matrix is close or equal to the one given above ( [ t1d ] ) , ( [ tau1dapprox ] ) . a straightforward calculation shows that the t - matrix of a one - dimensional delta - potential @xmath101 has a form @xmath102 where the one - dimensional scattering length @xmath103 is related to the potential strength by @xmath104 comparison of the t - matrices ( [ tau1dapprox ] ) and ( [ tau1ddelta ] ) leads to a conclusion that the waveguide scattering t - matrix ( [ tau1dapprox ] ) can be exactly reproduced by a delta - potential ( [ delta - potential ] ) of a scattering length @xmath105 , \end{aligned}\ ] ] where @xmath106 is the riemann zeta function and @xmath107 . the above expression reproduces the result of @xcite where the effective one - dimensional potential has been obtained via a straightforward solution of the scattering problem . notice that at @xmath108 the coupling constant diverges , signifying the so - called `` confinement induced resonance '' ( cir ) . the significance of this resonance has been confirmed in _ ab initio _ two - body numerical calculations with finite - range realistic interatomic potentials @xcite and in many - body monte - carlo simulations @xcite . further extensions of the zero - range model for the effective one - dimensional scatterer can be envisioned . for example , if one chooses to to reproduce the scattering properties ( more precisely the denominator of the one - dimensional t - matrix ( [ tau1dexact ] ) ) with a relative error of @xmath109 , the one - dimensional delta - potential can be replaced by a rectangular potential of a finite width @xmath110 and hight / depth @xmath111 . in the case of repulsive interaction the model potential is a rectangular barrier , and for attractive interactions it is a rectangular well . in the limit of @xmath112 their half - widths @xmath113 are given by @xmath114 and in both cases the strength @xmath111 of the potential is @xmath115 attempts to carry through a program analogous to the above for the case of p - wave scattering between polarized fermions meet numerous ( hopefully technical ) obstacles : most of the limiting procedures become mutually nonuniformly convergent and no clear way to identify a correct order is visible . in any case the closest candidate for the p - wave analog of the free - space three - dimensional t - matrix is the pseudopotential introduced in @xcite @xmath116 that can be shown to reproduce correctly the low - energy behavior of the p - wave scattering amplitude . here @xmath117 is the p - wave scattering volume , that defines the low - energy behavior of the p - wave scattering phase via @xmath118 @xcite . an elegant way around these difficulties has been fond recently by granger and blume @xcite , who used a k - matrix technique that does not explicitly involve any zero - range objects . the analysis of polarized fermi gases presented below is heavily based on the granger and blume findings . this is the simplest case . the 1d scattering length @xmath103 is defined in terms of the ratio of derivative @xmath119 and value @xmath120 of the relative wave function just outside the range @xmath121 of the interaction : @xmath122 which is equivalent in the zero - range limit @xmath123 to the familiar ll _ contact condition _ @xmath124 for the delta function interaction @xmath125 provided that the scattering length and coupling constant are related by @xmath126 where @xmath0 is the effective mass @xmath127 . it follows from the expression for @xmath128 derived in @xcite and eq.([bose - a1d ] ) herein that @xmath129^{-1}\ ] ] implying the existence of a confinement - induced resonance cir @xcite of the coupling constant as @xmath130 is tuned via a 3d feshbach resonance @xcite past the resonance point @xmath131 . hence the whole range of 1d coupling constants from @xmath132 to @xmath133 is experimentally achievable by tuning @xmath130 over a narrow range in the neighborhood of the 1d resonance . it was shown recently @xcite that this is a 1d feshbach resonance between ground and excited transverse vibrational manifolds . it was shown in @xcite and in sec . [ e1d ] herein that at low longitudinal energies @xmath134 the 1d scattering amplitude generated by the interaction @xmath125 reproduces the exact 3d scattering amplitude in the waveguide to within a relative error @xmath100 . consider next a magnetically trapped , spin - aligned atomic vapor of spin-@xmath135 fermionic atoms in a tight waveguide . the @xmath11-fermion spin wave function is magnetically frozen in the configuration @xmath136 , so the space - spin wave function must be _ spatially _ antisymmetric , s - wave scattering is forbidden , and the leading interaction effects at low energies are determined by the 3d p - wave scattering amplitude . such p - wave interactions are usually negligible at the low densities of ultracold atomic vapors , but they can be greatly enhanced by p - wave feshbach resonances @xcite . granger and blume derived an effective one - dimensional k - matrix for the corresponding two - fermion problem @xcite in a tight waveguide . in the low - energy @xcite domain the k - matrix can be reproduced , with a relative error @xmath137 , by the contact condition @xcite @xmath138 where @xmath139 is the p - wave `` scattering volume '' @xcite , @xmath140 is the p - wave scattering length , and @xmath141 is the hurwitz zeta function evaluated at @xmath142 @xcite . the expression ( [ fermi - renorm ] ) has a resonance at a _ negative _ critical value @xmath143 . note that @xmath144 as @xmath145 approaches this critical value , implying that the exterior wave function ( i.e. , outside the interaction region @xmath146 ) satisfies the _ free - particle _ schrdinger equation . this is the opposite of the bosonic case , where it follows from eq . ( [ bose - a1d ] ) that @xmath147 at resonance . in accordance with ( [ fermi - contact ] ) , the low - energy fermionic wavefunctions , eq . ( 20 ) of @xcite , are discontinuous at contact , but left and right limits of their derivatives coincide . @xmath145 is tunable via a 3d feshbach resonance @xcite , allowing experimental realization of all values of @xmath148 from @xmath132 to @xmath133 . it will be shown that in this fermionic case the effective 1d coupling constant is @xmath149 , which can be compared and contrasted with the previously defined bosonic 1d coupling constant @xmath126 . the dimensionless fermionic coupling constant is @xmath150 . note that the density @xmath151 is in the numerator , whereas it is in the denominator of the bosonic analog @xmath152 . for @xmath153 one has a `` fermionic tg gas '' @xcite , a fermionic analog of the impenetrable bose gas , called the `` tonks - girardeau '' ( tg ) gas in recent literature @xcite . as previously noted , @xmath144 in the fermionic tg limit , implying an interaction - free exterior wave function . it will be shown in sec . [ subsec : fbb ] that in this limit the fermi gas maps to an _ ideal bose _ gas , providing a physical explanation of the interaction - free nature of the exterior wave function . this can be compared and contrasted with the bosonic tg gas , which maps to an _ ideal fermi _ gas . although a discontinuity in the derivative is a well - known consequence of the zero - range delta function pseudopotential and plays a crucial role in the solution of the lieb - liniger model @xcite , discontinuities of @xmath154 itself have received little attention , although they have been discussed previously by cheon and shigehara @xcite and are implicit in the recent work of granger and blume @xcite . for a fermionic wave function @xmath155 the discontinuity @xmath156 is a trivial consequence of antisymmetry together with the fact that a nonzero odd - wave scattering length can not be obtained in the limit @xmath157 unless @xmath158 these discontinuities are rounded off when @xmath159 , since the interior wave function interpolates smoothly between the values at @xmath160 and @xmath161 . this is illustrated in fig . [ fig : one ] for the special case of the fermionic tg gas , the limit @xmath153 @xcite . the potential is chosen to be a square _ well _ because we will find later that stability of the ground state against collapse requires that the corresponding effective zero - range interaction ( @xmath123 ) be _ negative _ definite , which can be shown to be the case when @xmath162 and hence both @xmath163 and @xmath164 . the energy is taken as zero so the exterior solution is @xmath165 ; an interior solution fitting smoothly onto this is @xmath166 with @xmath167 , the critical value where the last bound state passes into the continuum , a zero - energy resonance . a fermionic contact condition with a finite scattering length can be obtained in the limit @xmath157 if @xmath168 scales with the width @xmath169 as @xmath170 $ ] . untrapped fermionic tg gas ground state ( dashed line ) compared with zero - energy scattering solution for a square well with range @xmath121 and depth @xmath171 corresponding to the boundary between no bound state and one bound state , a zero energy resonance ( solid line ) , as function of relative coordinate z. units are such that @xmath172 . ] following the bosonic case , where the @xmath68-interaction can be introduced naturally to cancel the @xmath68-functions resulting from double differentiation of functions with discontinuous derivatives , in the case of fermions whose wave function is discontinuous it is tempting to introduce @xmath173 interactions . however , @xmath173-functions and second derivatives are ill - defined if used in a convolution with discontinuous functions . this difficulty is resolved by realizing that ( as will be shown ) , the @xmath173 is associated with the _ interior _ wave function ( @xmath174 ) whereas the contact condition ( [ fermi - contact ] ) refers to the @xmath123 limit of the wave function at @xmath175 , just _ outside _ the range of the interaction potential . take the hamiltonian to be @xmath176 where @xmath177 is a pseudopotential operator to be determined . @xmath178 is nonsingular for @xmath179 , but at the origin there are singular contributions . the first derivative is @xmath180\delta(z)$ ] . the second derivative then has an additional singular contribution from differentiation of the delta function : @xmath181\delta^{'}(z)\quad .\ ] ] in the second term on the right , @xmath182 is associated with the interior wave function , as we will eventually verify explicitly from the @xmath183 limit of a finite - range interaction , whereas its prefactor arises from the contact condition just outside the range of the interaction . define a linear regularizing operator @xmath184 by @xmath185\quad .\ ] ] then the desired pseudopotential is @xmath186 . it satisfies a convenient projection property @xmath187 where @xmath188 is any even wave function . although @xmath155 is always purely odd ( antisymmetric ) for the spin - aligned fermi gas , this projection property is useful for the spinor ( spin - free ) fermi gas to be discussed in sec . [ subsubsec : spin - free ] , where the spatial dependence has , in general , both even and odd contributions . this is the reason for the general form @xmath189 $ ] of the factor occurring in eq . ( [ fermi - regularize ] ) , even though it could be simplified to @xmath190 or @xmath191 when acting on @xmath155 , for which @xmath192 as a consequence of antisymmetry . it is easy to show that terms in @xmath182 cancel from @xmath193 as a consequence of the contact condition ( [ fermi - contact ] ) provided that @xmath149 . the physical significance is clarified by starting from the same square well already discussed in connection with fig . [ fig : one ] , i.e. , @xmath194 when @xmath174 and zero when @xmath195 . the antisymmetric solution @xmath196 of the zero - energy scattering equation @xmath197\psi_{f}(z)=0 $ ] inside the well is @xmath166 with @xmath198 the scattering length @xmath199 is defined by @xmath200 which is satisfied in the limit @xmath123 if @xmath171 scales with @xmath121 as @xmath170 $ ] . in that limit the boundary conditions reduce to eq . ( [ fermi - contact ] ) . inside the well the kinetic and potential energy terms are @xmath201 and @xmath202 . for @xmath203 , @xmath204 is proportional to a representation of @xmath205 as @xmath157 , since @xmath206 . then its derivative @xmath207 is a representation of @xmath182 . noting that @xmath208 as @xmath157 we have @xmath209 which agrees with the kinetic energy term @xmath210\delta^{'}(z)$ ] since @xmath211 and @xmath212 are to be interpreted as @xmath213 and @xmath214 as @xmath123 . next consider the potential energy term inside the well as @xmath123 : @xmath215 . comparing this with @xmath216 , using the expression for @xmath217 , noting that @xmath218 are to be interpreted as @xmath219 for @xmath157 , one finds that the two expressions for the potential energy term agree in that limit . it is clear from this derivation that the @xmath182 in the effective 1d hamiltonian can be interpreted as the `` ghost of the vanished interior wave function '' , which plays a crucial role in the odd - wave interaction even in the limit @xmath157 . in this section we assume that the spinor fermi gas is optically trapped , so the spins are unconstrained . this case is more complicated than the spin - aligned fermi gas even in the absence of explicit spin - spin interactions or external spin - dependent potentials , because the requirement of antisymmetry under combined space - spin exchanges @xmath220 induces implicit space - spin coupling leading to nontrivial spin dependence of the wave functions . here each spin z - component argument @xmath221 takes on the values @xmath222 or @xmath223 , or equivalently @xmath224 . consider 3d two - body scattering in the waveguide . there are both s - wave scattering states , which are space symmetric and spin antisymmetric with spin eigenfunctions of singlet form @xmath225 , and p - wave scattering states , which are space antisymmetric and spin symmetric with spin eigenfunctions of triplet form @xmath226 or @xmath227 or @xmath228 . assume that the hamiltonian does not depend on spin . then the spin dependence of fermionic wave functions @xmath155 need not be indicated explicitly and they can be written as the sum of spatially even and odd parts @xmath188 and @xmath229 . the odd part decomposes further into three components going with the three triplet spin eigenfunctions , but one need not complicate the notation at this point since the hamiltonian acts in the same way on each of them . the effective 1d interactions are determined by 1d scattering lengths @xmath230 for spatially even waves @xmath231 related to 3d s - wave scattering and spatially odd waves @xmath232 related to 3d p - wave scattering . the contact condition for 1d even - wave scattering is the same as the previously - given one ( [ bose - contact ] ) with @xmath233 and @xmath103 replaced by @xmath188 and @xmath230 , and the one for 1d odd - wave scattering is @xmath234 , the @xmath159 version of eq . ( [ fermi - contact ] ) , with @xmath155 and @xmath148 replaced by @xmath229 and @xmath235 . in the zero - range limit @xmath123 these can be combined into @xcite @xmath236\nonumber \\ \psi(0+)-\psi(0-)= -a_{1d}^{o}[\psi'(0 + ) + \psi'(0-)]\end{aligned}\ ] ] where @xmath237 . in this section @xmath238 is a fermionic function @xmath155 , but we use a more general notation because the same equations apply to a spinor bose gas . @xmath230 and @xmath239 are related to the 3d s - wave scattering length @xmath130 and the 3d p - wave scattering volume @xmath145 by eqs . ( [ bose - a1d ] ) and ( [ fermi - renorm ] ) , with @xmath103 replaced by @xmath230 and @xmath199 replaced by @xmath239 . take the hamiltonian to be @xmath240 here @xmath241 differs from @xmath177 of sec . [ subsubsec : spin - aligned ] only by the obvious substitutions , i.e. , @xmath242 with @xmath184 defined by eq . ( [ fermi - regularize ] ) with @xmath155 replaced by @xmath229 . the even - wave pseudopotential is more complicated than the simple delta - function interaction of sec . [ subsubsec : spinless - bosons ] , because the delta function is ambiguous at the point @xmath243 , where @xmath154 is discontinuous due to the discontinuity in its odd - wave component @xmath229 . in order to determine the correct form , start from the first derivative @xmath244\delta(z)$ ] as before . now the second derivative has _ two _ singular contributions in addition to the nonsingular term @xmath245 , one because in general @xmath246 and the other from the derivative of the delta function : @xmath247\delta(z)\nonumber\\ & + & [ \psi(0+)-\psi(0-)]\delta^{'}(z)\ .\end{aligned}\ ] ] with proper choice of @xmath248 the odd - wave pseudopotential @xmath241 cancels the @xmath182 term from the kinetic energy , and we define the even - wave pseudopotential @xmath249 so as to cancel the @xmath205 term : @xmath250 where the linear operator @xmath251 is defined by @xmath252\delta(z)\quad .\ ] ] these pseudopotentials satisfy convenient projection properties @xmath253 on the even and odd parts of @xmath154 , and their matrix elements are @xmath254 $ ] and @xmath255^{*}[\psi^{'}(0+)+\psi^{'}(0-)]$ ] . they connect only even to even and odd to odd wave functions if we stipulate that @xmath256 [ the average of @xmath257 and @xmath258 if @xmath259 is odd and @xmath260 [ the average of @xmath261 and @xmath262 if @xmath259 is even . in fact , the wave function and its derivative at @xmath243 refer to the _ internal _ wave function as modified by the potential , whereas @xmath263 and @xmath264 refer to the wave function _ just outside _ the range of the potential , and the above values at @xmath243 follow from the way the internal wave function interpolates between the contact conditions on the _ exterior _ wave function . using ( [ general - contact ] ) one finds that terms in @xmath205 and @xmath182 cancel from @xmath265 if the even and odd - wave coupling constants are related to the scattering lengths by @xmath266 and @xmath267 . in this section we will review the theory of fermi - bose mappings relating the exact @xmath11-particle energy eigenstates of systems of fermions and bosons in 1d with effective zero - range interactions , and application of these mappings to determination of the @xmath11-atom ground states . this will be done first for the original mapping for impenetrable bosons ( @xmath268 ) @xcite , then for a very powerful generalization to arbitrary values of @xmath269 due to cheon and shigehara@xcite , and finishing with a further generalization to the case of spinor fermi and bose gases , important for applications to optically trapped fermionic atoms whose spins are unconstrained . it was already pointed out in the famous paper of lieb and liniger on the 1d bose gas with delta - function interactions @xcite and in secs . [ section : intro ] and [ subsubsec : spin - aligned ] that the 1d gas of impenetrable point bosons is the limit @xmath16 of the ll gas , the `` tg limit '' . tonks gave the first treatment of the statistical mechanics of a 1d hard - sphere gas @xcite , which was restricted to the classical high - temperature regime and provided no information about the extreme quantum limit characteristic of ultracold atomic vapors . the formula for the exact quantum - mechanical ground - state energy of the 1d hard - sphere bose gas appeared in a paper of bijl where it is quoted without derivation @xcite , and a derivation was published by nagamiya @xcite . then in 1960 one of us @xcite and stachowiak @xcite independently rederived the ground - state energy . the fermi - bose mapping method was first introduced in @xcite , although nagamiya had previously noted @xcite that in the `` fundamental sector '' @xmath270 the ground state wave function of a spatially uniform , 1d hard - core bose gas can be written as an ideal fermi gas determinant , continuation into other permutation sectors being effected by imposing overall bose symmetry under all permutations @xmath271 in spite of the fermionic _ _ anti__symmetry under permutations of _ orbitals _ ( _ not _ coordinates ) in the fundamental sector . the mapping theorem is much more general , also holding in the presence of external potentials and/or finite two - particle or many - particle interactions in addition to the hard core interaction @xcite . it also applies to the 1d time - dependent many - body schrdinger equation and has been used to treat some time - dependent interference properties of the 1d hard core bose gas @xcite . we now briefly review the mapping theorem . the @xmath11-boson hamiltonian is assumed to have the structure @xmath272 where the real , symmetric function @xmath55 contains all external potentials ( e.g. , a longitudinal trap potential ) as well as any finite interaction potentials _ not including _ the hard - sphere repulsion , which is instead treated as a constraint on allowed wave functions @xmath273 : @xmath274 let @xmath275 be a fermionic solution of @xmath276 which is antisymmetric under all pair exchanges @xmath277 , hence all permutations . one can consider @xmath155 to be either the wave function of a fictitious system of `` spinless fermions '' , or else that of a system of real , spin - aligned fermions . define a unit antisymmetric function " @xcite @xmath278 where @xmath279 is the algebraic sign of the coordinate difference @xmath280 , i.e. , it is + 1(-1 ) if @xmath281(@xmath282 ) . for given antisymmetric @xmath155 , define a bosonic wave function @xmath233 by @xmath283 which defines the fermi - bose mapping . @xmath233 satisfies the hard core constraint ( [ eq2 ] ) if @xmath155 does , is totally symmetric ( bosonic ) under permutations , obeys the same boundary conditions as @xmath155 , and @xmath284 follows from @xmath285 @xcite . in the case of periodic boundary conditions ( no trap potential , spatially uniform system ) one must add the proviso that the boundary conditions are only preserved under the mapping if @xmath11 is odd , but the case of even @xmath11 is accomodated by imposing periodic boundary conditions on @xmath155 but _ anti_periodic boundary conditions on @xmath233 . the mapping theorem leads to explicit expressions for all many - body energy eigenstates and eigenvalues under the assumption that the only two - particle interaction is a zero - range hard core repulsion , represented by the @xmath286 limit of the hard - core constraint , the `` tg gas '' . such solutions were obtained in sec . 3 of @xcite for periodic boundary conditions and no external potential . at the low densities of ultracold atomic vapors it is usually sufficient to consider this case , although it has been shown by astrakharchik _ _ @xcite that for a longitudinally trapped ll gas with attractive interaction @xmath287 there is a regime within which the equation of state is well approximated by taking @xmath288 . the exact ground state is also known in this case @xcite . here we limit ourselves to the usual strongly repulsive tg limit @xmath16 . since wave functions of `` spinless '' or spin - aligned fermions are antisymmetric under coordinate exchanges , their wave functions vanish automatically whenever any @xmath289 , the constraint has no effect , and the corresponding fermionic ground state is the ground state of the _ ideal _ gas of fermions , a slater determinant of the lowest @xmath11 single - particle plane - wave orbitals . the exact many body ground state was found @xcite to have energy @xmath290 where @xmath14 is the linear number density , and the wave function was found to be a pair product of bijl - jastrow form @xmath291| , \ ] ] where @xmath15 is the perimeter of the annular trap . in spite of the very long range of the individual pair correlation factors @xmath292|$ ] , the pair distribution function @xmath293 , the joint probability density that if one particle is found at @xmath294 a second will be found at @xmath295 , was found to be of short range @xmath296 ^ 2 $ ] . clearly , @xmath297 which reflects the hard core nature of the two - particle interaction . by examination of the excited states the system was found to support propagation of sound with speed @xmath298 @xcite , and it was shown that this agrees with the thermodynamic formula in terms of the compressibility of the ground state . `` fermionization '' holds only for those properties expressible in terms of the configurational probability density @xmath299 . the momentum distribution depends on the single - particle correlation function @xmath300 ( reduced single - particle density matrix ) , which is very different from that of the ideal fermi gas and very difficult to evaluate . its eigenfunctions are plane waves @xmath301 because of translational invariance of the system , and the corresponding eigenvalues define the momentum distribution function @xmath302 , the discrete fourier transform of @xmath303 , the allowed values of @xmath73 being @xmath304 with @xmath305 . in a classic _ tour de force _ lenard found @xcite @xmath306 to be of order @xmath307 , large but much less than the @xmath308 value required for bose - einstein condensation . more generally , @xmath303 was found @xcite to be of order @xmath309 at small @xmath73 . the corresponding momentum distribution is sharply peaked at low @xmath73 and falls like @xmath310 at large @xmath73 @xcite , very different from the filled fermi sea of the ideal fermi gas , for which @xmath302 is unity for @xmath311 and zero for @xmath312 . the exact ground state of the tg gas is also known in the presence of a longitudinal trap potential @xmath313 @xcite . it follows from the mapping theorem that the exact n - boson ground state @xmath314 is @xmath315 where @xmath316 is the ground state of @xmath11 spinless fermions with the same hamiltonian and impenetrability constraint . the fermionic ground state is a slater determinant of the lowest @xmath11 single - particle eigenfunctions @xmath317 of the harmonic oscillator ( ho ) , where @xmath318 with @xmath319 the hermite polynomials and @xmath320 , @xmath321 being the longitudinal oscillator length . by factoring the gaussians out of the determinant and carrying out elementary row and column operations , one can cancel all terms in each @xmath322 except the one of highest degree @xcite , yielding a simple but exact analytical expression of bijl - jastrow pair product form for the @xmath11-boson ground state : @xmath323 \prod_{1\le j < k\le n}|z_{k}-z_{j}|\ ] ] with @xmath324 . it is interesting to note the strong similarity between this exact 1d @xmath11-boson wave function and the famous laughlin variational wave function of the 2d ground state for the quantized fractional hall effect @xcite , as well as the closely - related wave functions for bosons with weak repulsive delta - function interactions in a harmonic trap in 2d found by smith and wilkin @xcite . both the single particle density and pair distribution function depend only on the absolute square of the many - body wave function , and since @xmath325 they reduce to standard ideal fermi gas expressions . the single particle density , normalized to @xmath11 , is @xmath326 and the pair distribution function , normalized to @xmath327 , is @xmath328 although the hermite polynomials have disappeared from the expression ( [ sho - ground ] ) for the many - body wave function , they reappear upon integrating @xmath329 over @xmath330 coordinates to get the single particle density @xmath331 and over @xmath332 to get the pair distribution function @xmath333 , and these expressions in terms of the ho orbitals @xmath317 are the most convenient ones for evaluation . some qualitative features of the pair distribution function are apparent : in the first place it vanishes at contact @xmath334 , as it must because of impenetrability of the particles . furthermore , the correlation term @xmath335 is a truncated closure sum and approaches @xmath336 as @xmath337 , as is to be expected since the healing length in a spatially uniform 1d hard core bose gas varies inversely with particle number @xcite . as a result the width of the null around the diagonal @xmath338 decreases with increasing @xmath11 , and vanishes in the limit . for @xmath339 much larger than the healing length , @xmath61 reduces to the uncorrelated density product @xmath340 , so the spatial extent of the pair distribution function is that of the density and varies as @xmath341 @xcite . detailed gray - scale plots of @xmath333 in the @xmath342 plane for the cases @xmath343 , @xmath344 , and @xmath345 are shown in fig . 1 of @xcite . the reduced single - particle density matrix with normalization @xmath346 is @xmath347 for @xmath348 it can not be expressed in terms of @xmath349 , and is therefore very different from that of the ideal fermi gas . the multi - dimensional integral can not be evaluated analytically , but in @xcite it was evaluated numerically by monte carlo integration for not too large values of @xmath11 , and grayscale plots are shown in fig . 2 of @xcite . more accurate numerical results were found in @xcite , and highly accurate results for large values of @xmath11 were found in @xcite . in a macroscopic system , the presence or absence of bec is determined by the behavior of @xmath350 as @xmath351 . off - diagonal long - range order is present if the largest eigenvalue of @xmath352 is macroscopic ( proportional to @xmath11 ) , in which case the system exhibits bec and the corresponding eigenfunction , the condensate orbital , plays the role of an order parameter @xcite . although this criterion is not strictly applicable to mesoscopic systems , if the largest eigenvalue of @xmath352 is much larger than one then it is reasonable to expect that the system will exhibit some bec - like coherence effects . thus we examine here the spectrum of eigenvalues @xmath353 and associated eigenfunctions @xmath354 ( `` natural orbitals '' ) of @xmath352 . the relevant eigensystem equation is @xmath355 @xmath353 represents the occupation of the orbital @xmath356 , and one has @xmath357 . accurate values of the @xmath353 have been determined in @xcite . in particular , the largest eigenvalue @xmath358 was shown to be of order @xmath307 for large @xmath11 , as in the spatially uniform case . next we examine the momentum distribution , which can be shown @xcite to be a double fourier transform of @xmath352 : @xmath359 the spectral representation of the density matrix then leads to @xmath360 where the @xmath361 are fourier transforms of the natural orbitals : @xmath362 . the key features are that the momentum spectrum maintains the sharp peaked structure reminiscent of the spatially uniform case @xcite and that the peak becomes sharper with increasing atom number @xmath11 . by way of contrast , for a 1d fermi gas the corresponding momentum spectrum is a filled fermi sea and can be expressed as @xmath363 . in a recent paper @xcite we devised a scheme to measure the momentum spectrum based on raman outcoupling and showned that the angular cross section accurately mirrors the momentum distribution . figure [ fig : two ] shows the angular cross section versus angle for both @xmath345 impenetrable bosons ( dashed line ) and the corresponding system of non - interacting fermions ( solid line ) ; see @xcite for details . for @xmath345 . the dashed line is for the 1d gas of impenetrable bosons and the solid line is for the corresponding system of non - interacting fermions ] consider first the case @xmath343 . the spins are frozen in the configuration @xmath226 by the magnetic field , so the spatial relative wave function @xmath364 is antisymmetric ( odd in @xmath8 ) . the hamiltonian @xmath365 and corresponding odd - wave pseudopotential were derived in sec . [ subsubsec : spin - aligned ] . defining a mapped bosonic ( even in @xmath91 ) wave function by @xmath366 and mapped scattering length @xmath367 where @xmath279 is @xmath368 if @xmath281 and @xmath369 if @xmath282 , one finds that @xmath233 satisfies the usual bosonic contact condition @xmath370 , the zero - range limit @xmath123 of eq . ( [ eq1 ] ) . since the kinetic energy contributions from @xmath179 also agree , one has a mapping from the fermionic to bosonic problem which preserves energy eigenvalues and dynamics . the relationship between coupling constants @xmath217 in @xmath365 and @xmath371 in @xmath372 is @xmath373 , and by ( [ fermi - renorm ] ) this relationship agrees with the low - energy limit of eq . ( 25 ) of @xcite . in the limit @xmath374 arising when @xmath375 , this is the @xmath343 case of the original mapping @xcite from hard sphere bosons to an ideal fermi gas , but now generalized to arbitrary coupling constants and used in the inverse direction . this generalizes to arbitrary @xmath11 : antisymmetric fermionic solutions @xmath275 are mapped to symmetric bosonic solutions @xmath273 via eqs . ( [ eq3 ] ) and ( [ eq4 ] ) . the fermi contact conditions are @xmath376 and imply the bose contact conditions @xmath377 , and these are the usual ll contact conditions @xcite . the fermionic hamiltonian is @xmath378 where @xmath379 $ ] . although well - defined in the exact schrdinger equation and in first - order perturbation theory , this fermionic pseudopotential becomes ambiguous in higher - order perturbation theory . however , after mapping to the bosonic hilbert space one has the usual lieb - liniger interaction @xmath380 which is well - behaved in all perturbation orders and in second quantization . this generalization of the fermi - bose mapping theorem , due to cheon and shigehara @xcite , extends the useful domain of the mapping of eqs . ( [ eq3 ] ) and ( [ eq4 ] ) to the whole range of coupling constants @xmath7 and @xmath381 . the first application to the spin - aligned fermi gas is due to blume and granger , who were led to the mapping by consideration of the zero - range limit of a k - matrix formulation @xcite . they treated only the case @xmath343 but did not restrict themselves to the low - energy limit considered here . the exact ground state @xcite of @xmath382 is known for all positive @xmath7 if no external potential or nonzero range interactions are present , and the mapping then generates the exact @xmath11-body ground state of @xmath383 . the dimensionless bosonic and fermionic coupling constants @xmath152 and @xmath150 introduced in sec . [ subsubsec : spin - aligned ] satisfy @xmath384 . the energy per particle @xmath385 is related to a dimensionless function @xmath386 available online @xcite via @xmath387 where @xmath269 is related to @xmath388 herein by @xmath389 . this is plotted as a function of @xmath388 in fig . [ fig : three ] . for the spin - aligned fermi gas , versus dimensionless fermionic coupling constant @xmath388 . ] for @xmath390 as occurs at a p - wave feshbach resonance , one has the `` fermionic tg gas '' discussed in sec . [ subsubsec : spin - aligned ] and @xcite . for bosons the tg regime , which maps to the _ ideal fermi _ gas , is reached when @xmath371 is large enough and/or the density @xmath151 _ low _ enough that @xmath16 . a similar simplification occurs in the fermionic case , where a fermionic tg regime is reached when @xmath217 is large enough and/or @xmath151 _ high _ enough that @xmath153 . the corresponding fermionic tg gas then maps to the _ ideal bose _ gas since @xmath384 . in this section we assume that the hamiltonian is spin - independent and that the fermionic vapor is trapped in a tight _ optical _ atom waveguide . the spins are then free to assume whatever configuration minimizes the ground - state energy . first assume @xmath343 . the two - body relative wave functions are @xmath391 where the spatially even part @xmath392 contains an implicit spin - odd singlet spin factor , and the spatially odd part @xmath393 contains implicit spin - even triplet spin factors . states of combined space - spin bosonic symmetry can be defined by the same mapping @xmath366 used in the previous section , which now maps the spatially even fermionic function @xmath394 to a spatially odd bosonic function @xmath395 and the spatially odd fermionic function @xmath396 to a spatially even bosonic function @xmath397 while leaving the spin dependence unchanged . then the even - wave contact conditions for @xmath398 follow from the odd - wave contact conditions for @xmath399 and the odd - wave contact conditions for @xmath400 follow from the even - wave contact conditions for @xmath401 . as before , one has a mapping from the fermionic to bosonic problem which preserves energy eigenvalues and dynamics . the bosonic hamiltonian is of the same form as the fermionic one ( [ h_1d ] ) but with mapped coupling constants @xmath402 and @xmath403 . this generalizes to arbitrary @xmath11 : fermionic solutions @xmath404 are mapped to bosonic solutions @xmath405 by the usual mapping ( [ eq3 ] ) , ( [ eq4 ] ) , where the spin z - component variables @xmath406 take on the values @xmath222 and @xmath223 . the n - fermion and n - boson hamiltonians are both of the form @xmath407 $ ] . on fermionic states @xmath155 , @xmath408 and @xmath248 are @xmath409 and @xmath410 , whereas on the mapped bosonic states @xmath411 they are @xmath402 and @xmath403 . this is discussed in more detail in @xcite . assume that both @xmath412 and @xmath413 . if @xmath410 is zero then it follows from a theorem of lieb and mattis @xcite that the fermionic ground state has total spin @xmath414 ( assuming @xmath11 even ) , as shown in the spatially uniform case by yang @xcite and with longitudinal trapping by astrakharchik _ if @xmath410 is not negligible then the ground state may not have @xmath414 . in fact , if @xmath409 is zero then one can apply a theorem of eisenberg and lieb @xcite to the mapped spinor boson hamiltonian , with the conclusion that one of the degenerate bose ground states is totally spin - polarized , has @xmath415 , and is the product of a symmetric spatial wave function @xmath314 and a symmetric spin wave function . the ground state is then the same as the one discussed in the previous section , except that now there is an @xmath416-fold directional degeneracy since @xmath417 can range from @xmath418 to @xmath419 . any @xmath414 state has a higher energy ; in fact , for @xmath420 the mapped bose gas is partially space - antisymmetric , raising its energy by the exclusion principle . so far we have considered only the extreme cases where either the even - wave or odd - wave coupling constant vanishes . assume now that they may take on any non - negative values . consider first the case @xmath343 of a longitudinally trapped spinor fermi gas , with relative spatial wave function @xmath364 and hamiltonian differing from eq . ( [ h_1d ] ) by addition of a harmonic trap potential @xmath421 . the even and odd - wave coupling constants are @xmath422 and @xmath423 as before . @xmath155 may be taken to be either spatially even with associated singlet spin function @xmath225 which has @xmath414 , or else spatially odd with associated spin function which is one of the @xmath424 triplets @xmath226 , @xmath227 , or @xmath228 . the singlet case has spatially even wave functions identical with those of trapped @xmath343 bosons . the odd - wave pseudopotential then projects to zero so the @xmath414 eigenstates are independent of @xmath425 , and the even - wave pseudopotential reduces to @xmath426 . the exact eigenstates are known @xcite , being of the form @xmath427 where @xmath428 is a weber ( parabolic cylinder ) function @xcite and @xmath429 . the absolute value in the argument leads to a cusp at @xmath243 and the ll cusp condition of eq . ( [ bose - contact ] ) and @xcite ( with @xmath430 replaced by @xmath409 ) leads to a transcendental equation for the allowed values of @xmath431 : @xmath432 in terms of the dimensionless parameter @xmath433 . the energy eigenvalues are @xmath434 , the ground state is that solution for which @xmath431 vanishes as @xmath435 , and its energy @xmath436 is a monotonically increasing function of @xmath408 . next consider the @xmath424 ( triplet ) solutions , for which @xmath155 is spatially odd . then the even - wave pseudopotential projects to zero , and on carrying out the fermi - bose mapping @xmath366 the odd - wave pseudopotential is changed to @xmath437 with @xmath402 . thus the @xmath424 ground state energy is the same function of @xmath438 that the @xmath414 ground state energy is of @xmath409 , and is therefore a monotonically _ decreasing _ function of @xmath410 . it follows that the @xmath414 and @xmath424 ground - state energies are equal on the hyperbola @xmath439 in the ( @xmath440 ) plane , which forms a phase boundary between the region where the absolute ground state has @xmath414 and that where it has @xmath424 . for @xmath441 ( below the phase boundary ) the ground state has @xmath414 and its energy is independent of @xmath410 , and for @xmath442 ( above the phase boundary ) it has @xmath424 and its energy is independent of @xmath409 . since it depends only on a symmetry argument and is independent of @xmath443 , it is reasonable to conjecture that the above phase boundary is valid for all @xmath11 and for both spatially uniform and longitudinally trapped spinor fermi gases . in other words we make the following _ conjecture_. the ground state of a spinor one - dimensional fermi gas undergoes a quantum phase transition at @xmath444 , being paramagnetic ( @xmath414 ) for @xmath445 and ferromagetic , with @xmath446 for @xmath447 ; here @xmath448 and @xmath449 are the dimensionless even and odd - wave coupling constants @xmath450 and @xmath451 . as a first step in motivating this conjecture , it is convenient to change from the representation in terms of space - spin antisymmetric wave functions @xmath404 to a representation in which states with @xmath452 are represented by functions @xmath453 , where we assume @xmath11 even and @xmath454 . @xmath455 is antisymmetric under permutations of the z - coordinates @xmath456 of up - spin atoms and also under permutations of those @xmath457 of down - spin atoms , but it has no particular symmetry under exchanges @xmath458 of up - spin with down - spin atoms . formally treating up and down - spin particles as different species can be shown @xcite to be physically equivalent to the usual representation @xmath404 . the hamiltonian is similar to that of eq . ( 3 ) of @xcite , but also includes odd - wave interactions : @xmath459\end{aligned}\ ] ] where the even and odd - wave pseudopotential operators are those of eq.([h_1d ] ) expressed in terms of relative coordinates @xmath460 , etc . even - wave interactions @xmath461 and @xmath462 between like - spin particles are absent because @xmath455 is antisymmetric in both @xmath456 and in @xmath457 , so the corresponding even - wave interactions project to zero . however , both even and odd - wave interactions between up and down - spin particles are present because the states @xmath455 are neither symmetric nor antisymmetric under exchanges @xmath463 between up and down - spin atoms . the like - spin odd - wave interactions @xmath464 and @xmath465 can be transformed into much simpler even - wave interactions by a generalization of the previous fermi - bose mapping . define a mapped wave function @xmath466 of a _ two - component bose gas _ by @xmath467 where @xmath468 is the previously - defined `` unit antisymmetric function '' of eq . ( [ eq3 ] ) , equal to @xmath469 everywhere , antisymmetric in its arguments , and changing sign here only at `` same - species '' collisions @xmath470 and @xmath471 . then @xmath466 satisfies a mapped schrdinger equation @xmath472 with the same eigenvalue @xmath38 and with @xmath473\quad .\end{aligned}\ ] ] now there are only interactions of ll type @xcite between like - spin atoms , and these are straightforward to treat ; they have a mapped coupling constant @xmath402 . on the other hand , the mapping does not affect the contact condition for collisions @xmath474 of unlike - species atoms , so these interactions are the same as in eq . ( [ 2-species ] ) and retain the original fermionic coupling constants @xmath409 and @xmath410 . consider now the gross - pitaevskii ( gp ) approximation , the variational energy with a trial function @xmath475 [ \prod_{j=1}^{n - k}u_{\downarrow}(y_{j})]$ ] . in the absence of longitudinal trapping one has @xmath476 where @xmath15 is the length of the periodic box . when acting on these trivial wave functions the odd - wave interactions @xmath477 in ( [ 2-component - bose ] ) project to zero and the even - wave interactions @xmath478 reduce to simple lieb - liniger interactions @xmath479 , so the variational energy per particle in the thermodynamic limit ( @xmath480 ) is @xmath481\ ] ] where @xmath482 with @xmath452 . thus @xmath483 increases with @xmath417 if @xmath484 and decreases with @xmath417 if @xmath485 . it follows that the ground state has @xmath486 if @xmath484 or equivalently @xmath441 , and @xmath487 ( the maximal value ) if @xmath442 . this is the same result obtained previously for the exact @xmath343 ground state in a trap . the ground state has @xmath486 if @xmath488 and @xmath487 if @xmath489 , so there is a hyperbolic phase boundary @xmath490 in the ( @xmath491 ) plane . in the @xmath492 ( @xmath493 ) phase the ground state energy depends only on @xmath438 , hence on @xmath449 but not on @xmath448 , a result which we will show holds also for the _ exact _ @xmath11-atom ground state . the gp approximation is valid when both @xmath494 and @xmath495 are small enough , more precisely when both @xmath496 and @xmath497 . since the product @xmath498 can be made to assume any desired value without violating these inequalities , it is reasonable to suppose that the phase boundary @xmath490 holds for the _ exact _ ground state . we now show that this is true . to see this , note first that since @xmath466 is symmetric in @xmath456 and in @xmath457 , it follows that when acting on @xmath466 , @xmath499 is equivalent to @xmath500 defined in connection with eq . ( [ even - interaction ] ) with @xmath501 , and similarly @xmath502 is equivalent to @xmath503 . furthermore , one may formally add interactions @xmath464 and @xmath465 since these vanish on @xmath466 because of its symmetry in @xmath456 and @xmath457 . thus in ( [ 2-component - bose ] ) we may replace @xmath499 by @xmath504 $ ] and @xmath502 by @xmath505 $ ] without changing its action on @xmath466 . after this has been done , we note that when @xmath506 , the resultant hamiltonian reduces to that of a _ one_-component bose gas with both even and odd - wave interactions , with ground state energy @xmath507 which is independent of @xmath73 , hence independent of @xmath417 . hence , on the line @xmath490 ground states of all values of @xmath417 from @xmath508 to @xmath419 are degenerate , and it is easy to show that this remains true if spin - independent longitudinal trap potentials are added to @xmath509 . to complete the proof we need to convert this into a statement about the total spin @xmath510 of the ground state , not merely its value of @xmath417 . to do this , first map back to the two - component fermi gas states @xmath453 . they are not in general eigenstates of @xmath510 , but they can be converted into such eigenstates @xmath155 , with the same eigenvalues of both energy and @xmath417 , by antisymmetrizing with respect to all combined space - spin exchanges @xmath511 after renaming the variables as follows : @xmath512 and @xmath513 . in this standard representation the hamiltonian , total spin , and its z - component are mutually commuting , so nondegerate energy eigenstates are also eigenstates of @xmath514 and @xmath515 , and degenerate ones can be chosen to be such simultaneous eigenstates . let @xmath516 be the lowest such state for given values of @xmath517 and @xmath518 . we have shown that if @xmath488 then such a ground state has @xmath486 , if @xmath489 it has @xmath519 , and if @xmath490 it can have any value of @xmath417 from @xmath508 to @xmath520 . if @xmath489 this state also has @xmath492 since @xmath519 implies @xmath492 . if @xmath488 then the ground state also has @xmath414 because if it had @xmath521 than there would exist states with @xmath522 of the same energy , contradicting the monotonic dependence of energy on @xmath523 demonstrated in ( [ gp - energy ] ) , which we assume to hold also for the _ exact _ ground state . it follows that the hyperbola @xmath490 is the boundary between a ground state @xmath414 phase ( @xmath488 ) and a @xmath492 phase ( @xmath489 ) . it follows from the above that the dependence of the ground - state energy on total spin @xmath510 is more complicated than envisioned in either @xcite or @xcite : it is indeed true that if @xmath518 is _ exactly _ zero , then the ground state has @xmath414 as assumed in @xcite , but if @xmath524 then by approaching the confinement - induced resonance ( cir ) of @xmath448 implied by eq . ( [ cir - bose ] ) , one can in principle always make @xmath489 even if @xmath525 , thereby inducing a phase transition from the paramagnetic @xmath414 phase to the ferromagnetic @xmath415 phase . in the opposite case @xmath497 encountered near the cir of @xmath449 implied by eq . ( [ fermi - renorm ] ) a phase transition in the opposite direction ( @xmath415 to @xmath414 ) would in principle occur if @xmath526 , but in practice there is no way of making @xmath448 that small . in the region @xmath489 where @xmath415 the exact ground state is totally spatially antisymmetric and hence spin - aligned , so its energy is independent of @xmath448 and given by the results in @xcite and sec . [ subsubsec : spin - aligned ] herein . in the region @xmath488 where @xmath414 the ground state is thus far only known for the case @xmath527 , where it was determined by yang @xcite in the spatially uniform case and by astrakharchik _ @xcite in the longitudinally trapped case . no analytical or numerical results are yet known for the ground - state energy in the region @xmath488 if @xmath528 , but it should be investigated by numerical calculations . these will be more complicated than the previous ones @xcite due to the presence of both even and odd - wave interactions between up and down - spin atoms . however , if the ground state is real and nodeless @xcite they should be feasible , perhaps by using an interaction potential consisting of a narrow and deep well to represent the odd - wave interaction ( see sec . [ subsubsec : spin - aligned ] ) together with a somewhat broader `` soft rod '' potential to represent the even - wave interaction . our proof of the exact ground - state phase boundary @xmath490 is a `` physicist s proof '' and we make no claim of mathematical rigor . in the first place , as pointed out in sec . [ subsubsec : p - wave ] , a rigorous derivation of the zero - range , 1d limit of the odd - wave interaction between fermions in tight waveguides does not exist at present , although we believe that eq . ( [ fermi - renorm ] ) is a correct zero - range , low - energy consequence of the k - matrix treatment of granger and blume @xcite . furthermore , for @xmath420 our proof of the phase boundary is not completely rigorous since we had to assume that the ground state energy is a single - valued function of @xmath417 both for @xmath488 and for @xmath489 . we feel that this is justified since we proved it to be true in the gp approximation and showed that the criteria for validity of the gp approximation can be satisfied on the phase boundary @xmath490 . however , a truly rigorous proof of this phase boundary does not yet exist , and we hope that one will be forthcoming . we are very grateful to doerte blume for helpful comments and for communications regarding her closely - related works with brian granger @xcite and with astrakharchik _ @xcite , and to ewan wright for references @xcite . we thank michael moore and thomas bergeman for their invaluable assistance in preparation of the les houches lecture notes @xcite on which sec . 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derivation of effective zero - range one - dimensional ( 1d ) interactions between atoms in tight waveguides is reviewed , as is the fermi - bose mapping method for determination of exact and strongly - correlated many - body ground states of ultracold bosonic and fermionic atomic vapors in such waveguides , including spin degrees of freedom . odd - wave 1d interactions derived from 3d p - wave scattering are included as well as the usual even - wave interactions derived from 3d s - wave scattering , with emphasis on the role of 3d feshbach resonances for selectively enhancing s - wave or p - wave scattering so as to reach 1d confinement - induced resonances of the even and odd - wave interactions . a duality between 1d fermions and bosons with zero - range interactions suggested by cheon and shigehara is shown to hold for the effective 1d dynamics of a spinor fermi gas with both even and odd - wave interactions and that of a spinor bose gas with even and odd - wave interactions , with even(odd)-wave bose coupling constants inversely related to odd(even)-wave fermi coupling constants . some recent applications of fermi - bose mapping to determination of many - body ground states of bose gases and of both magnetically trapped , spin - aligned and optically trapped , spin - free fermi gases are described , and a new generalized fermi - bose mapping is used to determine the phase diagram of ground - state total spin of the spinor fermi gas as a function of its even and odd - wave coupling constants .

the short dynamical time scales and coherence of quasi - periodic oscillations ( qpos ) observed in accreting xrb s strongly suggest that qpos can * * * * provide valuable information on the accretion dynamics in the innermost parts of these systems . in this respect the discovery of low 20 - 50 hz qpos in luminous neutron star ( ns ) binaries by van der klis et al . ( 1985 ) , kilohertz qpos in ns s by strohmayer et al . ( 1996 ) and hectohertz qpo s in bh s by morgan , remillard & greiner ( 1997 ) opened a new era in the study of the dynamics near compact objects . following these discoveries psaltis , belloni & van der klis ( 1999 ) , hereafter pbk , demonstrated that these ns and bh low and high frequencies follow a remarkably tight correlation . these features are the horizontal branch oscillations ( hbo ) along with low frequency noise lorentzian , @xmath7 and the lower khz qpo @xmath5 for ns and bh respectively . belloni , psaltis & van der klis ( 2002 ) , hereafter bpk , have updated pbk s correlation adding data from nowak ( 2000 ) , boirin et al . ( 2000 ) , homan et al . ( 2001 ) , di salvo et al . ( 2001 ) and nowak et al . pbk suggest that the low and high frequencies correlate in a way that seems to depend only weakly on the properties of the sources , such the mass , magnetic field , or possibly the presence of a hard surface in compact object . mauche ( 2002 ) has reported low and high frequency qpo s in the dwarf nova ss cygni and vw hyi ( see wouldt & warner 2002 for for the vw hyi observations ) and has called attention to the fact that these qpo s extend the correlation of pbk downward in frequency by more than two orders of magnitude . this frequency correlation observed over such a broad range of sources indicates that a common phenomenon is responsible for qpos in compact objects and raises the question whether one can find a observational relations which would distinguish bh candidates from other compact sources . the comparative study of spectral and qpo features in bh candidate sources has yielded important information which can help answer this question . as far as spectral properties are concerned , there is increasing observational evidence that the large complex of spectral states originally devised by belloni ( 2000 ) to classify the highly variable x - ray spectra of bh galactic sources like grs 1915 + 105 , can be reduced to a few simple configurations [ see e.g. mcclintock and remillard ( 2003 ) and belloni ( 2003 ) ] : ( 1 ) a low hard power law dominated state where the photon spectral index is near 1.5 ; ( 2 ) a steep power law state in which the index is about 2.7 ; ( 3 ) and a thermally dominated state . the observed variability is explained by transitions between these three canonical configurations . low frequency ( 1 - 10 hz ) qpo s are generally observed to present in states ( 1 ) and ( 2 ) , during transitions between states ( 1 ) and ( 2 ) but never in ( 3 ) . both high and low frequency qpo s have been thought to be associated with the accretion disk , though their specific origin has not widely understood ( although see titarchuk & wood 2002 , hereafter tw02 , for explanation of their origin ) . attempts to find consistent correlations with observable disk parameters have lead to mixed results . in particular , there are so many exceptions to the observed correlations of frequency to disk spectral values sometimes observed that no general statement about the relation of the qpo with any spectral parameter solely associated with the disk , such as thermal flux , temperature , inner radius , etc . can be made . for example , exceptions to one of the most widely quoted correlations , i.e. between disk flux and frequency , have been presented for the well studied micro - quasar grs 1915 + 105 by fiorito , markwardt , & swank ( 2003 ) and for xte 1550 - 564 by remillard et . al . ( 2002a , b ) . on the other hand , a number of recent studies have revealed consistent and robust correlations between the photon index of the power - law component of the x - ray spectrum and low qpo frequency . such strong correlations are observed the well - studied bh binary , grs 1915 + 105 in almost all situations in which the qpo is observable and in several other bh s over a wide range of observations and states . these studies include : \a . a comprehensive study of correlations between photon index and qpo frequency in bh sources by vignarca et al . ( 2003 ) ; these results show strong correlations between index and qpo frequency as well as saturation of the index between at the values @xmath9 and @xmath10 for several bh sources and for the same source in different states . recent studies by kalemci ( 2002 ) for a number of galactic bh x - ray transients observed during outburst decay which confirm the persistence of the correlation between frequency and index during state transitions . our own studies of grs 1915 + 105 of @xmath11 states ( fiorito , markwardt , & swank 2003 ) , in which strong correlation of qpo frequency with power - law index have been observed ; such states have been traditionally noted as low brightness , power law dominated states ( see e.g. belloni et al . 1999 ) where the disk is at least tenuous if not undetectable altogether . studies tracing the qpo frequency and index over time between state transitions shows that the frequency tracks the index during state evolution ( homan , et . al . 2001 ) and sobczak et al . ( 1999 , 2000 ) from a value of about @xmath9 to about @xmath10 , as the qpo frequency varies from about 1 to 10 hz . the persistence of the correlation of index with qpo frequency and its tracking with respect to time suggests that the underlying physical process or condition which gives rise to the low frequency qpo are tied to the corona ; and , furthermore , that this process varies in a well defined manner as the source progresses from one state to another . moreover , the fact that the same correlations are seen in so many galactic x - ray binary bh sources , which vary widely in both luminosity ( presumably with mass accretion rate ) and state , suggests that the physical conditions controlling the index and the low frequency qpos are characteristics of these sources , and by virtue of the low - high - frequency correlations of pbk - bpk , may be a universal property of _ all _ accreting compact systems . the above data have motivated us to develop a detailed model of the physics of the corona surrounding a bh s which directly predicts the behavior of the spectral index with fundamental properties of the corona . the model we have developed incorporates fundamental principles of fluid mechanics , radiative transfer theory and oscillatory processes . it identifies the origin of the qpo as a fundamental property of a compact coronal region near the bh and shows how the photon index of this corona changes as a function of mass accretion rate . it has been already shown in a number of papers ( see below ) that the absence of the firm surface in the bhs leads to the development of the very strong converging flow when the mass accretion rate is higher than eddington rate @xmath12 . the main observational features of the converging flow should be seen in high / soft phase , while the thermal comptonization spectrum of the soft ( presumably disk ) photons should be seen in the hard phase . in series of papers ( chakrabarti & titarchuk 1995 , hereafter ct95 ; titarchuk , mastichiadis & kylafis 1996 , 1997 ; ebisawa , titarchuk & chakrabarti 1996 ; titarchuk & zannias 1998 ; shrader & titarchuk 1998 , 1999 ; borozdin et al . 1999 ; laurent & titarchuk 1999 , 2001 , hereafter lt99 and lt01 respectively ; titarchuk & shrader 2002 ; turolla , zane & titarchuk 2002 ) the authors argue that the drain properties of black horizon are necessarily related to the bulk inflow in bh sources and that the spectral and timing features of this bulk inflow are really detected in the x - ray observations of bhs . the signatures of the inflow are : \i . an extended steep power law where the photon index saturates to the value @xmath10 as the mass accretion rate increases - the precise value of the index is a function of the temperature of the flow but its relatively high value is a result of inefficient photon upscattering in the converging flow due to photon trapping . lt99 demonstrated that the spectral photon index varies in narrow range from 3 to 2.7 with mass accretion rate ( in eddington units ) increases from 2 to 7 . moreover , lt99 demonstrated using monte carlo simulations that in the low hard state when the plasma temperature is of order 50 kev the bulk inflow spectrum is practically identical to thermal comptonization spectrum ( there is no any noticeable effect of the bulk comptonization in the spectrum ) . in fact , the effect of the bulk comptonization compared to thermal one is getting stronger when the plasma temperature drops below 10 kev . the small variation of the photon index around 1.7 is a characteristic signature of the low hard state that was pointed out in earlier work by ct95 and later it was confirmed by lt99 . \ii . the qpo high frequency ( 100 - 300 hz ) , which is inversely proportional to bh mass . it is worth noting that the qpo frequency scales as @xmath13 is a generic feature of any qpo model in which qpo frequency is scaled with the schwarzchild radius . furthermore , titarchuk , osherovich & kuznetsov ( 1999 ) , hereafter tok , presented observational evidence that the variability of the hard x - ray radiation in ns s and bh s occurs _ in a bounded configuration_. ford & van der klis ( 1998 ) found that the break frequency in the power density spectrum ( pds ) is correlated with qpo frequency for some particular ns sources ( 4u 1728 - 32 ) . wijnands & van der klis ( 1999 ) later found a similar correlation in bh sources . titarchuk & osherovich ( 1999 ) , hereafter to99 , and tok explained this correlation in nss and bhs respectively . they argue the observed pds is the power spectrum of an exponential shot , @xmath14 which is the response of the diffusion propagation to any perturbation in a bounded medium . the inverse of the characteristic diffusion time @xmath15 and the normal mode qpo frequency @xmath16 has a well defined relation to the size of the bounded configuration , because @xmath17 and @xmath18 where @xmath19 is the specific pertrubation ( magneto - sonic ) velocity and @xmath20 is a charactestic size of the configuration ( see to99 ) . titarchuk , lapidus & muslimov ( 1998 ) , hereafter tlm98 proposed that this bounded configuration ( cavity ) surrounding compact objects is the transition layer ( tl ) that is formed as a result of dynamical adjustments of a keplerian disk to the innermost sub - keplerian boundary conditions . they argued that this type of adjustment is a generic feature of the keplerian flow in the presence of the sub - keplerian boundary conditions near the central object and that it does not necessarily require the presence or absence of a hard surface . tlm98 concluded that an isothermal sub - keplerian transition layer between the ns surface and its last keplerian orbit forms as a result of this adjustment . the tl model is general and is applicable to both ns and black hole systems . the primary problem in both ns and bh systems is understanding how the flow changes from pure keplerian to the sub - keplerian as the radius decreases to small values . tlm98 suggested that the discontinuities and abrupt transitions in their solution result from derivatives of quantities such as angular velocities ( weak shocks ) . they were first to put forth the possibility of the transition layer formation to explain most observed qpos in bright low mass x - ray binaries ( lmxbs ) . in figure 1 we illustrate the main idea of the transition layer concept for ns s ans bh s . in this _ paper _ , we explain the general correlation between photon index and low frequency in the frameworks of the transition layer model and we give more arguments for the nature of the spectral phases ( states ) and phase transition observed in bhs . the main features of the tl model are given in 2.1 the formulation of the problem of low frequency oscillations and the relationship with ma oscillations in the transition layer are also described in 2.1 . the coronal model and the spectral index - optical depth relation is given in 2.2 . the derivation of the low frequency - index correlation and the details of modeling of the spectral phase transition are present in 2.3 . we analyze specific qpo and spectral data in terms of the tl model in 3 . we discuss the signatures and the methods of the identification of bh and ns sources using timing and spectral characteristics in 4 . our summary and conclusions also follow in 4 . tlm98 define the transition layer as a region confined between the the inner sub - keplerian disk boundary and the first keplerian orbit ( for the tl geometry , see fig . 1 and fig . 1 in tlm98 and tok ) . the main idea at the basis of our investigation is that a keplerian accretion disc is forced to attain sub - keplerian rotation close to the central object ( a neutron star or a black hole ) . the transition between the keplerian and the sub - keplerian flow takes place in a relatively narrow , shock - like region [ the size of the region of order radius of the central object ( ns , bh ) ] where dissipation occurs . as a consequence , the gas temperature increases and the disc puffs up , forming a hot corona . then the corona intercepts the soft disc photons , up - scatter them via thermal and dynamical comptonization finally giving rise to the hard power - law tail . the power - law index value strongly depends on the coronal optical depth and temperature ( see ct95 , tmk97 , lt99 ) . when the mass accretion rate in the disk increases and consequently the disk soft photon flux increases , the corona is drastically cooled down to a temperature of order 5 - 10 kev ( in 2.2 we present a detailed analysis of this effect ) . for such a low plasma temperature the bulk motion of the converging flow is more efficient in upscattering disk photons than thermal comptonization . furthermore , the indices saturates to the asymptotic values around 2.7 as the mass accretion rate of the converging flow increases . the exact asymptotic value of the index is determined by the plasma temperature only ( lt99 ) . tlm98 evaluate the size of the transition layer @xmath20 as a function of a nondimensional paramerer @xmath21 ( often called the reynolds number ) which is the inverse of the @xmath22 parameter for the accretion flow ( shakura & sunyaev 1973 , hereafter ss73 ) , @xmath23 where @xmath24 is the accretion rate in the disk , @xmath25 is a half - thickness of a disk , @xmath26 is a characteristic radial velocity at a given radius @xmath27 in the disk , @xmath28 is electron ( proton ) number density , @xmath29 is the proton mass and @xmath30 is the diffusion coefficient . @xmath30 can be defined as @xmath31 using the turbulent velocity and the related turbulent scale , respectively or as @xmath32 for the magnetic case where @xmath33 is the conductivity ( e.g. see details of the @xmath34definition in lang 1998 ) . it is worth noting that the viscosity @xmath35 is a function of @xmath36 . because @xmath21 is related to @xmath37 and the tl thomson optical depth @xmath38 , which is a product of @xmath39 ( @xmath40 , @xmath41 are the tl inner and the outer radius respectively ) , electron number density @xmath28 and thomson cross - section @xmath42 ( namely , @xmath43 ) are both related to @xmath37 one can , in principle , determine @xmath38 as a function of @xmath44parameter . because of the uncertainties of the disk equation of the state , of the viscosity @xmath35 in the disk there is still uncertainty in the precise determination of this @xmath45relation . we address to this issue in 2.3 . tlm98 determine the @xmath46dependence using the equation of motion in the disk where the radial motion is controlled by friction and the angular momentum exchange between adjacent layers , resulting in the loss of initial angular momentum by the accreting matter ( e.g. ss73 ) . we define an adjustment point where a keplerian accretion disc is forced to attain sub - keplerian rotation close to the central object ( ns or bh ) . in order to determine the adjustment radius one should solve this equation of motion subject to three boundary conditions one at the innermost disk radius @xmath40 and two at the adjustment radius @xmath41 namely , at @xmath40 the disk rotational frequency matches the rotational frequency of the central object ( spin ) , at @xmath41 there is no break in the rotational frequency and the left - hand and right - hand side derivatives of r are equal . thus , for a given value of @xmath44parameter the rotational frequency profile @xmath47 and the outer radius of the transition layer are uniquely determined by the three boundary conditions and the equation of motion ( see for example , eqs . 8 - 9 in t099 ) . the adjustment of the keplerian disk to the sub - keplerian inner boundary creates conditions favorable for the formation of a hot plasma outflow at the outer boundary of the transition layer , because the keplerian motion ( if it is followed by sub - keplerian motion ) must pass through the super - keplerian centrifugal barrier region ( tlm98 ) . the equation ( see tlm98 , eq . 11 , or to99 , eq . 9 ) determines @xmath41 as a function of @xmath21 . for given values of @xmath48 one can find values of the keplerian frequencies @xmath49 ( see eq . 1 in to99 ) . for example for the observed range of the high qpo frequencies in bh sources , from 1 to 100 hz ( the low qpo frequencies from 0.1 to 10 hz ) ( see fig . 2 ) , and for a bh mass of ten solar masses one can find from tlm98 eq . ( 11 ) that @xmath21 varies from 2 to 20 ( a variation which is related to the mass accretion rate @xmath50 see eq . [ eq : rey ] ) . we remind a reader that in the framework of the tlm these high qpo frequencies are interpreted as keplerian frequencies @xmath51 at @xmath41 . it is worth noting that in the literature @xmath52 is treated as a constant and a free parameter independent of the mass accretion rate ( or the spectral state ) , whereas we find that @xmath53 varies from 0.5 to 0.05 with @xmath37 , i.e. @xmath53 is a very strong function of @xmath37 . soria ( 1999 ) inferred the effective value of the viscosity @xmath54parameter as a function of the magnetic field . he showed that regardless of the true viscosity in the disk the @xmath54 parameter is always higher than @xmath55 if the ratio of the magnetic pressure @xmath56 to the gas pressure @xmath57 is about 10% and more . below ( see also tw02 for details ) we show that using the low frequency - high frequency correlation the inferred ratio of @xmath58 that is consistent with the values of @xmath59 and the soria s predictions of the high values of @xmath53 as @xmath60 the tl model identifies the low frequency qpo as that associated with the viscous magnetoacoustic ( ma ) oscillation of the bounded tl ; this mode is common to both bh s and ns s . the correlation of the ma frequency with the gravitational ( keplerian ) frequency @xmath61 were derived by to99 , titarchuk , bradshaw & wood ( 2001 ) for ns and then it was generalized by tw02 , for bh and wd . the ma frequency is derived as the eigenfrequency of the boundary - value problem resulting from a mhd treatment of the interaction of the disk with the magnetic field . the problem is solved for two limiting boundary conditions which encompass realistic possibilities . the solution yields a velocity identified as a mixture of the sound speed and the alfvn velocity . the tbw treatment does not specify how the eigenfrequency is excited or damped . however , it makes clear that the qpo is a readily stimulated resonant frequency [ see titarchuk , cui & wood ( 2002 ) and titarchuk ( 2002 ) for details of excitation of the eigenfrequencies ] . a linear relation was derived between @xmath8 and @xmath6 ( see fig . 2 and tw02 ) @xmath62 where @xmath63^{1/2}(h / r_{out})$ ] is a proportionality coefficient which is universal to the extent that @xmath64 and @xmath65 remain about the same from one source to the next . here @xmath66 is the ratio of the gas pressure to the magnetic pressure and the coefficient @xmath67 for stiff and free boundary conditions respectively and @xmath25 is the half - width of the keplerian part of the disk . identification of @xmath68 , with @xmath69 respectively , leads tw02 to determination of @xmath70 and @xmath71 using the observed value of @xmath72 . on the other hand @xmath73 at the adjustment radius @xmath41 as of function of @xmath44parameter can be readily obtained using eq . ( 9 ) in to99 [ see above for details of @xmath74 determination ] . thus , as a result of the identification of @xmath7 with @xmath8 , we obtain the relation of @xmath75 as a function of @xmath21 , namely @xmath76 in figure 3 we present the inferred @xmath7 as a function of @xmath21 . we derive this relation using equations ( 9 ) in to99 with the assumptions that @xmath77 ( see eq . 7 in to99 for definition @xmath78 ) and @xmath79 . it is evident that the inferred @xmath68 are inversely proportional to @xmath80 because @xmath81 and @xmath82 where @xmath20 is the tl characteristic size which is proportional to @xmath80 . we calculate the outer radius of the tl , @xmath83 as a function of @xmath7 using relations [ eq.[eq : ma ] ( or eq . [ high - low ] ) here and eqs . 1 , 9 in to99 ] , keeping in mind that @xmath84 . because the radius of interest , which in our case is the outer tl boundary radius @xmath41 , is related to @xmath6 one can relate @xmath41 to @xmath7 using formula ( [ high - low ] ) . we present this function in figure 4 . the spectral index@xmath85frequency correlation can be derived if we find a spectral index@xmath85tl optical depth correlation . because @xmath21 is is a function of the mass accretion rate @xmath37 and the optical depth @xmath86 is a function of @xmath37 , we are lead to a correlation of @xmath7 and @xmath86 and ultimately to the index@xmath85frequency correlation . it is worth noting that @xmath21 and @xmath38 dependences on @xmath36 implies that a correlation between these quantities exist _ but does not mean a linear correlation between @xmath21 and @xmath38 because @xmath44parameter also depends on the accretion flow viscosity @xmath35 _ which is also a function of @xmath36 . the tl model first proposed by tlm98 naturally provides a compact corona for comptonization upscattering of the illuminating accretion disk photons which is just the region of adjustment of keplerian accretion flow ( disk ) to the sub - keplerian innner boundary condition . the essential sub - keplerian rotation of ns in lmxbs is a well established observational fact due to the direct measurements of ns spins ( which are in the range of 200 - 400 hz , see the review by van der klis 2000 for details ) . thus it is quite reasonable that the presence of a tl for nss exists . in the case of bhs no direct spin measurements are yet available and hence the presence of the tl may at first glance seem less clear . on the other hand , the bulk motion comptonization model which requires the existence of the innermost region of non keplerian flow onto a slowly rotating or static schwarschild bh , is very consistent with observations and in fact its signature is the saturation of the photon index which is observed ( see lt01 for details ) and thus one can conclude that fast rotating kerr bh is ruled out by the observations . in fact , there is no space between the the inner edge of the disk and the horizon in the fast rotating kerr bh to upscatter the disk soft photons by the bulk motion to the energies of order 500 kev that are detected in the high energy observations ( grove et al . thus we can reasonably conclude that bh may also undergo sub - keplerian rotation and the existence of a tl for slowly rotating bhs . it is likely that this adjustment to sub - keplerian flow is not smooth and that near the adjustment radius the strong or weak shocks can be formed . the adjustment radius ( shock ) region can be treated as a potential wall at which the accreting matter releases its gravitational energy in a geometrically thin target . as a consequence of this the shock plasma temperature is much higher than that in the surrounding regions of the keplerian disk . the shock region gets puffed up oscillating with frequencies close to @xmath6 ( titarchuk 2003 ) . additional oscillations can appear ( at least in ns case ) because of the rotational configuration above the disk which is a magnetosphere in ns case ( see details in tok ) . the shock formation leads to the formation of the hot inner bounded region . the hard photons in the shocked region illuminate the surrounding material evaporating some part of the disk . it is obvious that the evaporated fraction depends on the mass accretion rate in the disk ( see ct95 ) . some small fraction of the hard photons ( at maximum 6 % ) can be reflected by the cold parts of the disk ( ct95 and see also basko , sunyaev & titarchuk 1974 ) . let us assume the column density of this region @xmath87 is a few grams or that the thomson optical thickness @xmath86 is a few ( see ss73 for the thickness estimate of the innermost part of the disk ) . the rate of energy release at this region @xmath88 is a few percent of the eddington luminosity since the adjustment radius is located within 4 - 100 schwarzschild radii ( see fig . the heating of gas due to the gravitational energy release should be balanced by the photon emission . for high plasma temperature , comptonization and free - free emission is the main cooling channel , and the heating of electrons is presumably due to the coulomb collisions with protons ( see fig . 1 for the picture of disk corona ) . under such physical conditions the energy balance can be written as [ see also zeldovich & shakura ( 1969 ) , hereafter zs69 and tlm98 ] @xmath89 where @xmath90 is the optical depth within the shock , @xmath91 is a distribution function for the radiative energy distribution , @xmath92 is the relativistic correction factor , @xmath93 is the electron temperature in k , @xmath94 cm s@xmath95 k@xmath95 and @xmath96 erg k@xmath97 cm s@xmath95 are a dimensional constants . in this formula we neglect the gas heating due to recoil effect . the distribution @xmath91 is obtained from the solution of the diffusion equation ( zs69 ) , @xmath98 subject to two boundary conditions . here we assume that the region of the gravitational energy release ( corona ) is a spherical shell surrounding the central object and the total flux in the corona is a sum @xmath99 of the gravitational energy release @xmath88 and the illumination flux from outside of the corona ( disk ) @xmath100 . the inner and outer boundary conditions for a bh corona are that there are no scattered radiation from outside of the corona . then in the eddington approximation ( see sobolev 1975 ) the boundary conditions can be written as follows : @xmath101 where @xmath102 and @xmath103 are at the inner and outer boundaries respectively . for a ns corona one should assume that inside of the inner corona boundary the energy density @xmath104constant , namely that the radiation flux emitted towards the central object at some point of the inner boundary returns back ( reflected by the ns surface ) , namely this condition is equivalent to the reflection condition : @xmath105 below we present a general formula of the coronal temperature which combines the bh and ns cases . the solution of equations ( [ eq : diff]-[eq : bdcon ] ) provides us with the distribution function for the energy density in the bh corona @xmath106\}.\ ] ] we neglect the dependence of @xmath107 on @xmath90 to derive this formula . furthermore , we take a representative value of @xmath108 for the maximum value in the corona which is very close to mean value of @xmath109 i.e. @xmath110.\ ] ] in order to approximate the coronal temperature . in figure 5 we present the results of calculations of the temperatures @xmath111 ( kev ) and the energy spectral indices of the comptonization spectrum @xmath53 ( photon index @xmath112 ) as a function of the optical depth of the shell @xmath86 . the different curves of @xmath113 are related to the different ratios of @xmath114 . the calculations have been made for values of @xmath115 g @xmath116 and @xmath117 erg @xmath118 s@xmath95 which are characteristic values of the density and luminosity for the standard disk model ( ss73 ) . in fact , the calculation results weakly depend on @xmath119 and @xmath88 if @xmath120 is of order a few or less . in this case the equation for the temperature is simplified @xmath121 which has the solution @xmath122^{-1}. \label{form : temp}\ ] ] it is easy to show that the case for which @xmath123 is identical to the ns case ( the inner reflective boundary ) when the disk illumination is neglected . one can see from eq . ( [ form : temp ] ) that @xmath93 is a function of @xmath124 ( only ) if @xmath125 . it is also worth noting that @xmath126 is insensitive to the total luminosity @xmath107 if @xmath125 ( eq . [ eq : temp ] ) . the spectral index @xmath53 as a function of the product of @xmath127 are insensitive to @xmath107 too for @xmath128 because the the comptonization parameter @xmath129 and @xmath130 ( see rybicki & lightman 1979 and sunyaev & titarchuk 1980 ) . this inferred property of the comptonization spectra reproduces the observed independence of @xmath53 on the bolometric luminosity ( see e.g. tanaka 1995 ) when @xmath53 varies within the range @xmath131 ( @xmath132 ) . this defines the so called the low / hard state in bhs . below we show that the observed plateau of the index - frequency correlation at low values of @xmath133 hz is the result of this behaviour because the lower frequency values are related to the low mass accretion rates @xmath134 and @xmath86 is of order one when @xmath135 . we calculate the exact values of the spectral indices @xmath53 for a given @xmath93 and @xmath86 using the relativistic formulas developed by titarchuk & lybarskij ( 1995 ) [ see formulas ( 17 ) , ( for plane geometry , note their @xmath86 is our @xmath136 ) and ( 24 ) there ] . in figure 5 we present the spectral indices as a function of @xmath86 for bh and ns cases when the disk illumination is negligible small with respect to the coronal energy release ( a condition which can be a model for the low / hard state in these systems ) . the plasma temperature values of 20 - 150 kev for @xmath137 and @xmath138 ( for @xmath139 ) are typical values of these quantities for low / hard state in bhs . when @xmath100 is comparable with @xmath88 the cooling becomes more efficient due to comptonization and free - free processes and therefore @xmath93 unavoidably decreases [ see @xmath140values for @xmath141 in fig . this illumination effect and mass accretion rate increase may explain the hard - soft transition when the compton temperature drops substantially with increase of the soft ( disk ) photon flux ( for illustration , see figure 1 ) . in figure 5 we show that temperature drops from 50 - 60 kev for @xmath142 to 7 - 10 kev for @xmath143 . the temperature may drop also due to an increase in the optical depth ( presumably because @xmath144 increases ) even when @xmath142 . lt99 studied the comptonization of the soft radiation in the converging inflow ( ci ) into a black hole using monte carlo simulations . the full relativistic treatment has been implemented to reproduce the spectra . lt99 show that spectrum of the soft state of bhs can be described as the sum of a thermal ( disk ) component and the convolution of some fraction of this component with the ci upscattering spread ( green s ) function . the latter boosted photon component is seen as an extended power law at energies much higher than the characteristic energy of the soft photons . lt99 demonstrate the stability of the power law index ( the photon index , @xmath145 ) over a wide range of the plasma temperature 0 - 10 kev , and mass accretion rates ( higher than 2 in eddington units ) due to upscattering and photon trapping in the converging inflow . the spectrum is practically the same as that produced by standard thermal comptonization when the ci plasma temperature is of order 50 kev ( the typical ones for the bh hard state ) and the photon index @xmath146 is around 1.7 lt99 also demonstrate that the change of the spectral shapes from the soft state to the hard state is clearly related to the temperature and optical depth of the bulk inflow . we combine our results of calculations of index and compton cloud temperatures for thermal comptonization case ( see 2.2 and fig.5 ) and lt99 s results of the spectral calculations ( see table 2 in lt99 for the spectral index values ) to evaluate the power law index as a function of the optical depth of the compton cloud @xmath86 . we use the values of the index and of the temperature for @xmath147 ( see fig . 5 ) when @xmath86 varies from 1 to 3 to decribe the index behavior and the plasma temperature variation in the hard state . figure 6 presents the photon index @xmath146 as a function of @xmath86 . this function @xmath148 can be fitted analytically as follows : @xmath149@xmath150 where @xmath151 and @xmath152 , @xmath153 and @xmath154 . in order to derive the index dependence @xmath146 on @xmath7 one should relate the parameter @xmath21 to @xmath86 and then combine the two derived relationships @xmath146 vs @xmath86 and @xmath7 vs @xmath21 ( see figures 3 - 6 ) . the optical depth @xmath86 and @xmath44parameter are functions of @xmath37 . to fit the data we assume that @xmath155 . we have three free parameters , black mass @xmath156 , @xmath157 and @xmath158 to fit the model to the data if we assume that the converging inflow ( ci ) is essentially cold ( the ci temperature @xmath159 kev ) . this relationship is all that is needed to fit the observed @xmath160 correlations . we analyze data points for photon index vs low frequency correlation obtained from trudolyubov et al . ( 1999 ) and vignarca et al . ( 2003 ) for grs 1915 + 105 and sobczak et al . ( 1999 , 2000 ) for xte j1550 - 564 . also we make use data for grs 1915 + 105 by fiorito , markwardt & swank ( 2003 ) hereafter fms , which is a detailed study of qpo and spectral features of the steady low hard states of grs 1915 + 105 for representative 1996 and 1997 observations . fms computed the power spectra of combined binned and event mode data for pca channels 0 - 255 and sampled at 0.25 - 64 hz for 4-second intervals . the spectra were averaged over the entire observation duration . the pds spectra were fit with a combination of lorentzian lines and exhibited a break frequency , a central qpo frequency and in some case the presence of the first harmonic . background subtracted energy of each obsid were obtained and time average over the entire observation duration using standard 2 pca and hexte data where available ( both clusters ) . the xspec model used to fit the energy spectra consisted of a multi - temperature thermal disk component ( diskbb ) , a simple power law and a small iron line component . for the 1996 data ( obsid10408 ) , it was found that the disk component could be neglected without affecting the fit . acceptable @xmath161 were obtained in all cases . the first part of the fms study was to first to look for correlations of different spectral components , presumably thermal disk components , with qpo frequency to try to determine the reason for the non - unique total flux correlations previously observed . in the case of obsid 20402 ( 1997 ) we did observe the expected positive correlation of qpo frequency with disk flux in agreement with previous analyses of muno et al . ( 1999 ) and markwardt et al . ( 1999 ) , which were done in other states . however , we observed an anti - correlation of frequency with disk flux for obsid 10408 ( 1996 ) ; and no correlation of frequency with disk flux for obsid 10258 ( 1996 ) . furthermore , no correlation of qpo frequency with disk temperature is seen for any of the obsids we studied . these results are at variance with the often - quoted result that disk parameters are closely associated with qpo frequency . remillard , et . al . ( 2002b ) show a comparison of low qpo frequency and the apparent disk flux for the source xte j1550 - 564 during its 1998 - 1999 outburst . the qpos are reported with differences in phase lags . again , it is clear that a frequency - disk flux correlation is observed for this source is only observed at lower ( @xmath162 hz ) qpo frequencies and fluxes than usually observed for grs 1915 + 105 . however , robust correlation between index and the central frequency of low frequency qpos are also seen in a wide variety of bh sources radiating in different states and during transition between states . the qpo frequencies in such studies are observed in the pds obtained by integrating over time intervals ranging from the complete time of the observation to segmented time bins where the luminosity remains constant , [ see e.g. markwardt , et . al ( 1999 ) ] . strong frequency - index correlations are observed by kalemci ( 2002 ) in outburst decay ( presumably state transitions ) of a number of sources and by vignarca , et . al . ( 2003 ) , in a comprehensive study which focuses its attention on the behavior of the power law index and qpo frequency in several bh candidate sources . we will use the data of vignarca , our studies and that of other as well in our analysis ( see 3.2 below ) . observation of low term tracking of the qpo frequency with the variations of the photon index has been observed in at least two bh sources . tracking is seen in the data of rossi ( 2003 ) and homan ( 2001 ) in observations of xte j1560 - 500 taken over a 30-day period . in this period the source apparently transits from the low hard state ( @xmath2 ) to the intermediate state and the soft state where the index apparently remain close to the constant at a value ( @xmath163 ) . in this transition the qpo frequency varies from about 1 to 10 hz . similar tracking is observed in xte j1550 - 564 by sobzcak et al . ( 1999 ) , ( 2000 ) . however , the index and qpo frequency does not track with the total x - ray flux ( vignarca et al . 2003 , figure 7 ) . _ grs1915 + 105 _ in figure 7 we present the results of our fits to the data of vignarca , et . al ( 2003 ) for the plateau observations of the microquasar grs 1915 + 105 using the tl model . the best fit parameters are @xmath164 , @xmath165 and @xmath166 . we note that the value @xmath164 is consistent with the bh mass evaluation @xmath167 obtained by greiner et al . ( 2001 ) using the ir spectra and by shrader and titarchuk ( 2003 ) ( see also borozdin et al . 1999 ) using the x - ray spectra . the observable index saturations at low and high values of @xmath168 are nicely reproduced by the model . the low and high frequency plateau regions of the data and fit are signatures of the two spectral phases which are explained by two different regimes of comptonization ( upscattering ) : ( 1 ) bulk flow comptonization in the soft state ( saturation at @xmath169 ) and ( 2 ) thermal comptonization in the hard state ( the photon index tends to level at @xmath170 ) . a comparison of the index - frequency correlation of plateau observations with vignarca s @xmath64 and @xmath171 data [ belloni ( 1999 ) classification scheme ] and the model fits are plotted in figure 8 . the only difference in the data is different saturation values at high @xmath168 that can be readily explained by a change of plasma temperature in the converging inflow @xmath172 . for upper value of @xmath173 the temperature @xmath174 kev as for @xmath175 the temperature @xmath176 kev . the best - fit parameters for the latter data are the same as for the plateau observations except that in the model function @xmath148 ( see eq . [ eq : gam - tau ] ) the coefficents @xmath177 , @xmath178 and @xmath179 are replaced by @xmath180 , @xmath181 and @xmath182 for @xmath183 . in figure 9 we present a plot of power - law index versus qpo centroid frequency for the observations of class @xmath53 and @xmath171 of grs 1915 + 105 from vignarca et al . ( 2003 ) ( black points = obs . blue points correspond to the values for observations by fiorito et al ( 2003 ) for obsid 20402 and obsid 10258 . magenta points correspond to positions where the obsid 20402 data points are shifted to one half the measured qpo low frequency values . this suggests that the latter data points are related to the second harmonics of the @xmath168 frequencies . it is well known that data for ns qpos is a mixture of the first and the second harmonics of horizontal branch oscillation ( hbo ) frequencies ( see e.g. van der klis 2000 ) . red points correspond to positions where should be points with half of frequencies for vignarca s obs . . a theoretical curve ( blue solid line ) obtained using @xmath148 ( see eq . [ eq : gam - tau ] ) ( for coefficents @xmath180 , @xmath181 and @xmath182 for @xmath183 ) for @xmath164 and @xmath184 fits the fms obsid 10258 data points . it is worth noting that relation between @xmath21 and @xmath38 is slightly different from that we obtained for the best - fits presented in fig namely , the index of @xmath21 , increases from 1.25 to 1.5 . it means that for the same value of @xmath21 the optical depth @xmath38 is slightly higher than that in state presented in fig . this change can be explained by a slightly higher accummulation of plasma in the transition layer . for example , if we fix the disk viscosity @xmath35 , then the @xmath44parameter is determined by @xmath36 only . but the mass accretion rate @xmath36 depends on the product of number density @xmath28 and the flow radial velocity @xmath185 , i.e @xmath186 . thus , for a given value of @xmath187 the optical depth @xmath188 varies as @xmath185 varies . smaller values of @xmath185 corresponds to the higher values of @xmath28 and consequently to higher values of @xmath38 , namely higher plasma accummulation in the transition layer can be related to smaller velocities @xmath185 . it is interesting to note that the suggested half - frequency points ( magenta and red ones ) are located in a narrow corridor between these inferred curves ( presented in fig . 8 and fig . 9 respectively ) . _ xte j1550 - 564 _ figure 10 shows a plot of power law index versus qpo frequency for xte j1550 - 564 . all data points presented in figures 10 , 11 see also ( see also fig . 6 , 8 in vignarca et al . ) were obtained by sobczak et al . ( 1999 ) , ( 2000 ) ; remillard et al . ( 2002a , b ) . the data again show a flattening out or saturation of the index at a value @xmath4 . it is clearly seen from fig . 8 of vignarca et al . that most of the points presented there can be fit by a smooth line except for a few points on the right that we temporarily exclude from our consideration . the shift transformation @xmath189 of the grs 1915 + 105 curve , @xmath190 ( see fig . 7 ) into @xmath191 produces a fit for the xte 1550 - 564 data . this means that if two sources are in the same accretion regime [ i.e. when @xmath192 is the same ] their relative bh masses can measured by the simple shift transformation @xmath193 . in our case we use @xmath194 for grs 1915 + 105 and @xmath195 for xte 1550 - 564 . these values are very close to the values that have been obtained by shrader & titarchuk ( 2003 ) for these sources using x - ray spectroscopic methods ( shrader & titarchuk 1999 ) . thus the tl model provides an independent way to estimate the mass of one or more bh s in the same accretion state . a comparison of all data points of correlating photon index vs qpo low frequency for xte 1550 - 564 ( black points ) with the inferred correlation is shown in figure 11 . the saturation index value in the theoretical curve is related to specific value of the plasma temperature of the converging inflow @xmath172 , which changes from 5 kev to 20 kev from the top curve to the bottom respectively . slightly different cooling regimes of the site of the converging bulk inflow can explain the different saturation levels of the index observed in xte 1550 - 564 . the temperature of the converging inflow is determined by the disk illumination geometry and by the intrinsic heating of the flow by pairs produced very close to the bh event horizon ( see laurent & titarchuk 2004 ) . all these effects may be manifested by the observed index - frequency correlation at high values of @xmath7 . _ 4u 1630 - 47 _ figure 12 shows a plot of index versus qpo frequency for 4u 1630 - 47 [ trudolyubov et al . ( 1999 ) ; tomsick & kaaret ( 2000 ) , kalemci ( 2002 ) ] . the data again show a similar behavior to that of xte j1550 - 564 with an index saturation at @xmath196 and @xmath197 . one can see two phases with the index level at @xmath198 and @xmath199 and a transition between them that is much sharper than that observed in the other sources dicussed above . the best - fit curve @xmath200 ( solid line ) is obtained using @xmath201 for @xmath202 and @xmath203 for @xmath204 and @xmath205 function ( see formula 14 ) for the best - fit bh mass @xmath206 which is close to the estimate made by borozdin et al . ( 1999 ) using the x - ray spectroscopic method . the solid and dash curves are for saturation index value @xmath197 ( related to @xmath207 kev ) and @xmath196 ( related to @xmath208 kev ) respectively . coefficients @xmath209 , @xmath210 and @xmath211 for @xmath212 and @xmath213 for solid and dash curves respectively . our explanation for the observed anomalously sharp transition of the spectral index is as follows . because at the sharp spectral transition qpo frequencies do not change we can speculate that the tl size also does not change at the sharp transition . but if the tl size is almost the same then the @xmath44parameter is almost the same because the tl size depends on @xmath44 parameter only ( see section 2.1 ) . but the spectral index is a function of @xmath38 and the tl plasma temperature . if the index rises drastically this means that upscattering is also drastically suppressed during the transition . then we have only one possibility i.e. that we are observing a sharp switching from the thermal comptonization to the bulk motion comptonization regimes . at the end of this transition we see a signature of the bulk inflow in terms of the index saturation . the bulk motion comptonization is dominant when the plasma temperature drops significantly to the value of order 10 kev . this can happen when a strong cooling emerges either because the optical depth @xmath38 increases due the accumulation matter in the transition layer or because an additional source of cooling such as a soft photon supply from the disk appears ( see investigation of these cases in section 2.2 ) . the latter case can be realized if the corona intercepts more disk soft photons due to a puffing up of the disk or a build up of the disk beneath the corona . we have developed a model of accretion onto a black hole which greatly simplifies and reclassifies the plethora of `` states '' observational assigned to categorize the x - ray observations of variable bhc into two generic phases ( states ) : \i ) _ soft state _ where we see the effect of the bh as a drain . bulk inflow upscattering of disk photons dominates the behavior of the bh spectrum . the power law spectrum is steep in this situation . the observed high energy photons are emitted from a compact region , where soft energy photons of the disk are upscattered by bulk matter inflow forming the steep power law with photon index around 2.8 , and low qpo frequencies ( above 10 hz ) and high qpo frequencies ( of the order of 100 hz ) are observed . in terms of the relativistic particle acceleration in the region of the corona and its effect on upscattering the soft disk photons , the soft state spectrum is a result of the first order fermi acceleration with respect to @xmath214 , i.e. a relative photon energy change @xmath215 . we would like to emphasize that bulk inflow is present in bh when the high mass accretion is high _ but not in ns _ , where the presence of the firm surface leads to the high radiation pressure which eventually stops the accretion . the bulk inflow and all its spectral features are absent in nss , in particular , the saturation of the index @xmath4 with respect to qpo frequency , which is directly related to the optical depth and mass accretion rate , observed in the soft spectral state of only bhs and is therefore a particular signature of a bh . \ii ) _ hard state , _ which is comparatively starved for accretion . the hard phase ( state ) is related to an extended thermal compton scattering cloud ( cavity ) characterized by a photon index @xmath216 and the presence of low qpo frequencies ( below 1 hz ) . the low - hard spectrum is a result of the fermi acceleration of the second order with respect to @xmath214 , i.e. @xmath217 . the effect of the first order on @xmath214 is smeared out by the quasi - symmetry of the particular dynamic , predominantly thermal motion of the compton cloud plasma . the transition layer componization model : i. specifies the precise behavior of the power law index for each state ; ii . identifies the origin and nature of low frequency and high frequency qpos ; iii . predicts and explains the observed relationship between them and their dependence mass accretion rate and spectral photon index for each of the phases ; iv . predicts and explains the observed robust correlation of low frequency qpos with spectral index observed in bhcs ; v. predicts and explains the observed correlation between low and high frequency qpos in bhs , neutron stars and white dwarfs . we interpret the correlation between low frequency @xmath7 and power - law photon index @xmath146 investigated by sobczak et al . ( 1999 ) , ( 2000 ) ; remillard ( 2000b ) ; homan et al . ( 2001 ) ; kalemci ( 2002 ) , vignarca et al . ( 2003 ) and fiorito , markwardt & swank ( 2003 ) for a variety of bh sources and states . the observed correlation strongly constrains theoretical models and provides clues to understanding the nature of the qpo phenomena particularly in bhs and in compact objects in general . \i . a sub - keplerian rotation of central object ( ns and bh ) which is observationally well established fact in nss of lmxb but it still needs verification for bhs . however the existence of an adjustment region i.e. the tl is quite reasonable assumption for all but very rapidly rotating bhs . unfortunately up to now there is no direct measurement of a bh spin . our fit to the data uses the model predicted dependence of the index on the tl optical depth @xmath38 and the model predicted dependence of the qpo frequencies on the @xmath44parameter . it is clear from the model that @xmath38 and @xmath21 are correlated but within tlm model one can not determine the exact @xmath45dependence because of the uncertainty of the disk equation of the state and the viscosity in the disk . therefore we have determined the @xmath218 relation by fitting the observed data , i.e. the correlation between the qpo low - frequency and the photon spectral index . as a result of our analysis of the observed index - qpo correlation we conclude that the function @xmath218 is almost the same for grs 1915 + 105 and xte j1550 - 564 but it is drastically different for 4u 1630 - 47 . for this source the inferred @xmath38 shows a very strong dependence of @xmath21 on the qpo frequencies produced during a sharp spectral state transition . we speculate that this peculiar behavior of the index vs qpo frequency and consequently the inferred @xmath38 vs @xmath21 is a result of the plasma accumulation in the transition layer . it is possible that a similar index - frequency correlation can be detected in other sources . the model fits to the data gives us the relatively low values of @xmath219 ( corresponding to viscosity parameter @xmath52 from to 0.05 to 0.5 ) . such low values of @xmath21 can be understood in the framework of mhd treatment of the disk viscosity . in fact , soria ( 1999 ) inferred @xmath53 of order 0.1 ( regardless of the true disk viscosity ) and higher if in the disk a magnetic pressure is 10% of the gas pressure or higher . \iv . within tlm one can estimate the absolute normalization of the magnetoacoustic ( qpo low ) frequencies ( see e.g. tbw01 ) but a more precise normalization can be only obtained using the observed low - high frequency correlation ( see fig . 2 ) . \v . in some very soft ( or extended power law ) regimes the positive correlation of frequency and index appears to be unbounded , i.e. there is no indication of saturation of the index value ( fms ) . explanation for this effect is out of scope of the present tl model . we can only suggest here that the extended power law regime occurs when strong outflows obscure the bulk inflow and the relatively cold outflow from winds downscatters the emerging high energy photons and soften the observed spectrum ( we will provide the details of this picture elsewhere ) . vignarca s figure 6 plots the index@xmath85frequency for two sources xte j1550 - 564 and gro j1655 - 40 together . the behavior of the later sources is quite distinct from all others analyzed - it shows a reverse or _ negative _ correlation between photon index and frequency . this is the only case known to exhibit this type of behavior . we propose that at a high accretion rate regime the pair production heating adds to and can dominate the heating in the cavity causing a transition back to thermal comptonization phase . in this phase a reverse correlation between low qpo frequency can result and is our explanation of the single case where this phenomena is observed . further theoretical investigation of this case will be addressed in future studies . \(1 ) we find that the observed low frequency - the index correlation is a natural consequence of an adjustment of the keplerian disk flow to the innermost sub - keplerian boundary conditions near the central object . this ultimately leads to the formation of the sub - keplerian transition layer ( tl ) between the adjustment radius and the innermost boundary ( the horizon for bh ) . \(2 ) in the framework of the tl model @xmath5 is related to the gravitational frequency at the outer ( adjustment ) radius @xmath220 and @xmath7 is related to the magnetoacoustic oscillation frequency @xmath221 . using a relation between @xmath8 and the mass accretion rate and the photon index @xmath146 and the mass accretion rate we infer a correlation @xmath222between @xmath8 and the spectral index @xmath146 . \(3 ) identification of @xmath7 with @xmath8 , allow us to make a comparison of the theoretically predicted correlation with the observed correlation . for this identification we use the one temperature plasma assumption . we apply the plasma temperature obtained from the comptonization spectra ( electron temperatures ) for calculations of magnetoacoustic frequencies which strongly depend on the proton temperature . the one temperature assumption is quite consistent with the data . \(5 ) we found that a hard phase ( state ) related to an extended compton cloud ( cavity ) characterized by the photon index around 1.7 and the low qpo frequencies below 1 hz . this is the regime where thermal comptonization dominates the upscattering of soft disk photons and the spectral shape ( index ) is almost independent of mass accretion rate . \(6 ) we find that the soft phase ( state ) is related to the very compact region where soft energy photons of the disk are up scattered forming the steep power law spectrum with photon index saturating around 2.8 . this is the regime where bulk motion comptonization dominates and the effect of an increase in the mass accretion is offset by the effect of photon trapping in the converging flow into the bh . \(7 ) in the context of distinquishing between ns and bh sources we would like to note the following for low hard states . the source can only be a ns if a soft blackbody - like component with the color temperature of order of 1 kev is observed in the spectrum ( torrejon et al . 2004 ) . on the contrary , such a high temperature is necessarily related to a high disk luminosity in the case of a bh and this case is never observed for bh s in the low hard state \(8 ) we offer a new method of bh mass estimation using the index - frequency correlations ; namely , if the theoretical curve of the index - frequency dependence @xmath223 related to the bh mass parameter @xmath224 fits the data for a given source then the simple slide of the frequency axis @xmath225 with respect to @xmath7 may allow us to obtain the mass @xmath226 by fit of @xmath227 to the observed correlation for another source . we acknowledge tomaso belloni for kindly supplying us data published in vignarca et al . we also acknowledge and thank the referee for his / her suggestions to clarify our results and explain the limitations of our model . , neutron star ( open circles ) and , black hole candidate ( filled circles ) sources . neutron star and black hole data are from belloni , psaltis & van der klis ( 2002 ) . the dashed line represents the best - fit of the observed correlation ( see tbw01 and tw02 where this correlation is predicted and explained using the tl model ) . this plot also appears in mauche ( 2002).,width=480,height=576 ] and @xmath171 ( black points ) and @xmath53 and @xmath171 ( red points = obs . 15,16 ) of grs 1915 + 105 from vignarca et al . ( 2003 ) along with a fit using the tl model with @xmath164 and @xmath228 . values for the plauteau observations ( see previous figure ) are plotted for comparison ( blue points ) . , width=576,height=576 ] and @xmath171 of grs 1915 + 105 from vignarca et al . ( 2003 ) ( black points = obs . blue points correspond to the values for observations by fiorito et al ( 2003 ) ( see text ) . magenta points correspond to positions where should be points with half of those frequencies . red points correspond to positions where should be points with half of frequencies for obs . 18 ( red points ) . a curve ( blue solid line ) is for @xmath164 and @xmath184 . , width=576,height=576 ] . black points and line for grs 1915 + 105 and red points and line for xte j1550 - 564 . the xte j1550 - 564 curve is produced by sliding the grs 1915 + 105 curve along the frequency axis with factor @xmath229 ( see text for details ) . , width=576,height=576 ] with the inferred correlation . the saturation index value in the theoretical curve is related to specific value of the plasma temperature of the converging inflow @xmath172 , which changes from 12 kev to 20 kev from the top to the bottom respectively.,width=576,height=576 ]

recent studies have shown that strong correlations are observed between the low frequencies ( 1 - 10 hz ) of quasiperiodic oscillations ( qpos ) and the spectral power law index of several black hole ( bh ) candidate sources , in low hard state , steep power - law ( soft ) state and in transition between these states . the observations indicate that the x - ray spectrum of such state ( phases ) show the presence of a power - law component and are sometimes related to simultaneous radio emission indicated the probable presence of a jet . strong qpos ( @xmath0 rms ) are present in the power density spectrum in the spectral range where the power - law component is dominant ( i.e. 60 - 90% ) . this evidence contradicts the dominant long standing interpretation of qpos as a signature of the thermal accretion disk . we present the data from the literature and our own data to illustrate the dominance of power - law index - qpo frequency correlations . we provide a model , that identifies and explains the origin of the qpos and how they are imprinted on the properties of power - law flux component . we argue the existence of a bounded compact coronal region which is a natural consequence of the adjustment of keplerian disk flow to the innermost sub - keplerian boundary conditions near the central object and that ultimately leads to the formation of a transition layer ( tl ) between the adjustment radius and the innermost boundary . the model predicts two phases or states dictated by the photon upscattering produced in the tl : ( 1 ) hard state , in which the tl is optically thin and very hot ( @xmath1 50 kev ) producing photon upscattering via thermal componization ; the photon spectrum index @xmath2 for this state is dictated by gravitational energy release and compton cooling in an optically thin shock near the adjustment radius ; ( 2 ) a soft state which is optically thick and relatively cold ( @xmath3 5 kev ) ; the index for this state , @xmath4 is determined by soft - photon upscattering and photon trapping in converging flow into bh . in the tl model for corona the qpo frequency @xmath5 is related to the gravitational ( close to keplerian ) frequency @xmath6 at the outer ( adjustment ) radius and @xmath7 is related to the tl s normal mode ( magnetoacoustic ) oscillation frequency @xmath8 . the observed correlations between index and low and high qpo frequencies are readily explained in terms of this model . we also suggest a new method for evaluation of the bh mass using the index - frequency correlation .

interest in x - ray emission in starburst galaxies ( sbgs ) stems from their diverse radiative activity which is powered by an abundant population of massive , young stars . the enhanced star formation rate in these gas and dust rich galaxies leads to intense far - infrared ( fir ) radiation from warm interstellar dust heated by the massive stars . the large number of x - ray binaries and supernova ( sn ) remnants , as well as hot winds driven by sn shocks , imply that x - ray emission from a sbg is more intense than from a normal spiral ( bookbinder 1980 ) . moreover , since sn remnants are prime sites for shock - acceleration of particles to high energies , compton scattering of relativistic electrons by the local radiation fields will enhance x - ray emission in a sbg . this expectation is heightened by the fact that in a sbg the energy density of the fir radiation field can be much higher than that of the cosmic microwave background radiation ( schaaf 1989 , rephaeli 1991 ) . the sb phase may occur in various types of galaxies ( triggered , perhaps , by galactic mergers ) . as in other active galaxies , a sbg may also have an active , compact nucleus , and one of the primary issues in the study of sbgs is the possible connection between low - luminosity active galactic nuclei ( agn ) and sb activities . since both phenomena may occur in at least some active galaxies , it is important to determine some of their observational manifestations in order to better characterize sbgs . this obviously is a prerequisite in the assessment of the significance of the sb phase in galactic evolution , and its cosmological ramifications , such as the enrichment of the intergalactic medium by metal - rich gas ejected during the sb phase , and the contributions of sbgs to the x - ray background ( rephaeli 1995 ) . high energy phenomena are triggered by the massive star formation activity , and because of the obscuration of optical emission and considerable re - processing of ir emission , x - rays allow a more direct and penetrating view of the inner regions of sbgs . detailed x - ray spatial and spectral information is very valuable for determining the basic properties of sbgs . however , considerable knowledge on the emission mechanisms and environments may be gained from good quality spectral data alone , particularly so if these include temporal information . m82 and ngc 253 are the closest sbgs . these ` archetypical ' sbgs have been observed with all the major x - ray satellites ( e.g. , fabbiano 1988a , tsuru 1990 , ohashi 1990 , boller 1992 , matsumoto & tsuru 1999 , cappi 1999 , pietsch 2001 ) . their x - ray emission is quite substantial , in the range @xmath6 erg / s ; reported best - fit spectra include both thermal ( cappi 1999 ) and power - law forms ( e.g. weaver 2000 ) . both galaxies have already been observed also with _ ( strickland 2000 , matsumoto 2001 , kaaret 2001 ) . at low x - ray energies , emission from these galaxies extends well beyond their optical disks , so the sb activity is not manifested only in the central nuclear sb region ( @xmath7 kpc ) . however , even in these nearby galaxies the characteristics of the emission have not yet been determined in much detail . it is important to determine the relative contributions of the sb - powered versus nuclear - powered emission , and the respective significance of thermal and nonthermal processes . to better establish the x - ray properties of m82 and ngc 253 , we have initiated observations with the proportional counter array ( pca ) and the high energy x - ray timing experiment ( hexte ) aboard the rossi x - ray timing explorer ( rxte ) satellite . our main motivation has been to use the good temporal and wide - energy capabilities of the rxte in order to assess the possible role of an agn - like variable emission , and the level of contribution of compton scattering to the integrated emission in sbgs . these rxte observations and the results of the spectral and temporal analysis are reported here . the rxte was inserted into low earth orbit on 1995 dec 31 . the pca was described by jahoda ( 1996 ) and the hexte by rothschild ( 1998 ) . the pca has energy resolution of 16% at 6 kev and a useful energy range of 2.5 kev to about 25 kev , limited at the high end by uncertainties of determining the internal background . the hexte has resolution of 15% at 60 kev and an energy range from 15 to 250 kev . for the present observations the errors of background determination with hexte were negligible . for both instruments calibration of energy response has been extensive and is accurate to better than 2% . the instrumental time resolution of microseconds was not needed , but the very flexible rxte scheduling of observations was exploited to permit sampling on all time scales from one orbit ( 90 minutes ) to ten months . both the pca and hexte are non - imaging with field - of - view ( fov ) one degree . over the period february - november 1997 , thirty one - orbit observations each were made of m82 and ngc 253 . to sample possible variability as well as temporal changes through the entire campaign , a somewhat intricate plan ( table 1 ) was devised consisting of four observing sequences . eight of the orbits were spaced over one day , seven orbits over one week , another seven over eight weeks , and the remaining eight over 42 weeks . the spacing of observations in each sequence was not uniform but moderately irregular : each sequence was divided into 15 equal segments , and observations were placed in either seven or eight of these segments in patterns based on the cyclic difference set of size 15 ( baumert 1971 ) . such patterns allow uniform sensitivity to variability on a somewhat broader range of time scales or frequencies than with evenly spaced samples . with the two longest sequences the corresponding m82 and ngc 253 observations occurred within a day of each other . three pca detectors were employed for all observations . analysis of m82 was complicated by variability and by the presence of another known source with appreciable x - ray emission , m81 , in the fov . the separation of m81 by @xmath8 from m82 resulted in a transmission of about 38% for m81 in the rxte fields . although the rxte field rotated considerably during the 10-month campaign , the pca fov , nominally a hexagonal pyramid , is sufficiently conical that the area presented to m81 varied by not more than 10% . we have also analyzed contemporaneous archival asca and rosat data to provide limits to the m81 contribution . .observation sequences [ cols="<,<,<,<",options="header " , ] the best - fit thermal forms for the late extra emission have temperatures that are modestly but significantly lower than for the earlier emission . if the entire data set samples a single source and process , then this process must generate somewhat softer spectra at higher emission levels . we note that whether the extra emission is assumed to have a thermal or a power law form , the addition of an absorbing column results in a significant improvement in @xmath9 , dramatically so for the power law form . the appearance of an extra column with increased flux levels may not be unphysical even if the high and low levels result from a single process , but it is simpler to interpret the need for absorption as an indication of an obscured source . thus , based on this spectral analysis it would appear that the origin of the extra late emission is in a source , most likely a single source , which did not emit the earlier lower flux . a similar high - low analysis has been performed by matsumoto & tsuru ( 1999 ) and by ptak & griffiths ( 1999 ) , both using essentially the same asca data collected from 1993 to 1996 . matsumoto & tsuru give the more useful summary of spectral fitting of the high - low flux : a thermal form with best - fit @xmath10 kev , fits slightly better than a power law with best - fit number index 1.8 . both forms require strong absorption with column near @xmath11 @xmath12 . these rxte results , with much better high energy sensitivity , strongly favor a thermal form over power law , but the best fit values for column and kt , @xmath13 and 4.7 kev , respectively , each differ from the corresponding asca value by a factor of two . differences of this scale probably can not be dismissed as intercalibration errors : asca and rxte saw something rather different . if the same single source was seen , it varies on the scale of roughly a year in both column and general spectral shape . much of the absorbing column must lie near to the source . given the varied nature of x - ray emitting environments in sbgs , significant thermal emission is clearly expected from the relatively high abundance of various stellar sources such as x - ray binaries , hot gas in snrs , and sn - driven winds in the is space and inner halo . the main issues that can be addressed based directly on our spectral analysis have been briefly discussed in the previous section . here we elaborate further on the implications of these results . a detailed comparison between the results of the rxte and bepposax observation is complicated , given that the pca has a larger fov than that of the mecs experiment . also , no temporal comparison is possible because the observations were made over non - overlapping periods , and due to the fact that bepposax observed m82 over only two days . it is nonetheless of considerable interest that bepposax measurements do show some flux variability in the central region of m82 . this is a clear indication that at least part of the variability seen in the pca data is intrinsic to m82 , and is not fully attributable to the variable ( seyfert 1 ) galaxy m81 , which is in the fov of the pca but not in that of the mecs . some , seemingly periodic , variability of the m81 emission on a timescale of up to @xmath1 days , has been seen recently by bepposax ( pellegrini 2000 ) . if this is characteristic of m81 , then it is clear that the stronger variability detected by the rxte over a longer timescale has a different origin . we also note from this general comparative point of view that there is no indication of a second , low - temperature component in the rxte data , whose lower threshold energy at @xmath14 kev makes rxte insensitive to it . an important result of our analysis is the large secular variation in the flux of m82 during the last four observation segments in july 1997 . such a large temporal change of galactic - scale emission is commonly attributed to the characteristics of the accretion process onto a massive black hole . these include also power law noise spectra which have been observed for several agn sources ( edelson & nandra 1999 ) with spectral indices of 1.5 1.7 , only mildly inconsistent with the present case of 1.0 @xmath15 0.4 . the variable emission would then be expected to have an agn - like power law energy spectrum with very significant photoelectric absorption . indeed , our spectral analysis yields evidence for the presence of such a component in the variable emission from m82 . the deduced values of @xmath16 are rather high , @xmath17 @xmath12 ( about an order of magnitude higher than column densities in seyfert i galaxies such as m81 ) , perhaps indicative of a seyfert ii nucleus ( risaliti 1999 ) . however , the best - fit power law index , @xmath18 , is much higher than the more typical value of @xmath19 . the thermal fit of the residual emission with the high absorption yields a very low iron abundance , perhaps too low to be acceptable . alternatively , the flux increase in july 1997 may possibly be due to variable emission from an x - ray binary system . if in m82 , the high level of emission ( @xmath20 erg - s@xmath21 in the 2 - 10 kev band ) would make this unusually luminous when compared with the more common luminosity range ( @xmath22 erg - s@xmath21 ) of binaries . although rare , a few such x - ray luminous binaries are known ( in the spiral galaxies ngc3628 and ngc4631 - fabbiano 1988b ) . the spectral shapes of these sources are usually fit with power - law indices @xmath23 , or @xmath24 values of a few kev . arguing against a binary origin for the extra emission is the very low upper limit of 15 ev for the equivalent width of an fe k@xmath25 line . the 320 ev equivalent width of hercules x-1 fe k@xmath25 line ( gruber 2001 ) reasonably typical of galactic accreting binaries , is a factor @xmath26 higher . > from asca monitoring in 1993 and 1999 it was concluded ( matsumoto & tsuru 1999 , ptak & griffiths 1999 ) that the m82 flux varied by a factor @xmath27 over this period . they argue that the varying emission originates from within a central 10 region , and that if due to accretion onto a massive black hole , the implied mass is @xmath28 . a compact source in the nucleus with the deduced level of luminosity , @xmath29 erg / s in the 2 - 10 kev band ( based on a distance of 3.6 mpc ) , would imply that m82 is a low luminosity agn . however , we consider the identification of the source of flux variability with an agn in m82 to be insecure . continued monitoring of the central emission in m82 is needed in order to better establish its temporal , spectral , and spatial properties . more recently , high - resolution observations of the central 1 region of m82 with the _ chandra _ satellite revealed 4 sources whose flux exhibits significant temporal variability ( matsumoto 2001 , kaaret 2001 ) . in particular , the 0.510 kev flux of one of these sources ( cxom82 j095550.2 + 694047 ) varied by a factor of @xmath30 ( during the period october 1999 january 2000 ) . matsumoto ( 2001 ) suggest that this source is the origin of the variability detected by asca . in its high flux state , the emission from this source is comparable to the high - low flux we deduce from the rxte measurements . irrespective of the origin of the variable spectral component in m82 , it is important to note that about half of the measured emission is at most weakly varying . even though the superposed emission from many variable sources is non - varying , given the thermal ( unabsorbed ) character of the non - varying emission , and previous evidence for its spatial extent , it is reasonable to conclude that this emission is powered by starburst activity . cappi ( 1999 ) have suggested that the main high - temperature spectral components , which account for most of the observed 2 - 10 kev emission in both ngc 253 and m82 , are largely due to emission from hot galactic winds . while we do expect the intense star formation activity to drive hot galactic winds , it is clear that thermal emission in sbgs is a superposition of emissions from a population of x - ray binaries , sn remnants , and galactic winds . the determination of the exact origin of the main high - temperature spectral component will be possible only when more sensitive spectral and imaging measurements are made of ngc 253 and m82 . both bepposax and rxte find no evidence for variability in the emission from ngc 253 , so this galaxy is not dominated by an agn . the best - fit temperature derived from the bepposax measurements ( cappi 1999 ) , @xmath31 kev ( 90% confidence ) , is roughly consistent with our best - fit value of @xmath32 ( 68% confidence ) , when the uncertainty in the background subtraction is included . even though the value of the iron abundance we determine here is lower than that deduced from the bepposax measurements , both are substantially uncertain and ( therefore ) in rough agreement . power law and thermal models were previously found to provide acceptable fits to the emission observed by asca from both m82 & ngc 253 ( ptak 1997 . ) in fact , moran & lehnert ( 1997 ) , who analyzed 1993 asca observations , concluded that the main component of the emission in m82 is nonthermal , and based on the similar values of the x - ray and radio power law indices , they suggested that the emission is from compton scattering of relativistic electrons by the local radiation fields . indeed , the low iron abundances we deduce here are puzzling given the intense star formation activity , and the associated processing of metal enriched interstellar gas in these galaxies . higher abundances are obtained not only when the dominant spectral component of the continuum emission is a power law , but in any case where nonthermal emission contributes appreciably to the overall emission . for example , if the 2 - 10 kev flux in the power law component is taken to be 25% of the thermal component , then the iron abundance in m82 increases from its best - fit value @xmath33 ( table 4 ) to 0.23 . it has previously been shown by goldshmidt & rephaeli ( 1995 ) that compton scattering of radio producing relativistic electrons by the far infrared radiation field can account for the substantial high energy ( @xmath34 kev ) emission detected by osse ( aboard cgro ) from ngc 253 ( bhattacharya 1994 ) . the similarities in radio properties of m82 and ngc 253 and their common starburst nature are sufficiently strong indications that compton scattering may play an appreciable role also in m82 . however , neither the bepposax pds nor the hexte observations were sufficiently sensitive to clearly detect this component at high energies where it can dominate the emission . we acknowledge useful comments by the referee on an earlier version of the paper . this work was supported by nasa grant nas5 - 4623 at ucsd . archival data analysis reported in this paper was aided by tools provided by nasa / heasarc .

the two nearby starburst galaxies m82 and ngc 253 were observed for @xmath0 ksec over a 10-month period in 1997 . an increase of the m82 flux by a factor @xmath1 was measured during the period july - november , when compared with the flux measured earlier in 1997 . the flux measured in the field centered on m82 includes @xmath2 of the emission from the seyfert 1 galaxy m81 . the best - fitting model for the earlier emission from m82 is thermal with @xmath3 kev . in the high flux state , the emission additionally includes either an absorbed second thermal component or absorbed power - law component , with the former providing a much better fit . a likely origin for the temporal variability is a single source in m82 . the flux of ngc 253 , which did not vary significantly during the period of observations , can be well fit by either a thermal spectrum with @xmath4 kev , or by a power law with photon index of @xmath5 . we have also attempted fitting the measurements to more realistic composite models with thermal and power - law components , such as would be expected from a dominant contribution from binary systems , or compton scattering of ( far ) ir radiation by radio emitting electrons . however , the addition of any amount of a power - law component , even with cutoff at 20 kev , only increases chi - square . the 90% confidence upper limit for power law emission with ( photon ) index 1.5 is only 2.4% of the 2 10 kev flux of m82 ; the corresponding limit for ngc 253 , with index 2.0 , is 48% . a&a

the conjecture that _ mean field phase diagrams are well approximated by systems with long range interactions _ can not be taken literally as it obviously fails in one dimensional systems ( if the second moment of the interaction is finite ) , moreover the mean field critical exponents are [ believed to be ] different from those computed for finite range interactions . with proper caveat however the conjecture is generally regarded as correct and indeed there are mathematical proofs mainly referring to specific models and focused on the occurrence of phase transitions . the choice of the approximating hamiltonian is not at all arbitrary and the results so far have been obtained for reflection positive interactions , @xcite , and for kac potentials , @xcite . the former choice is clearly motivated by a powerful and well developed theory , the latter class seems more general , in particular includes systems of particles in the continuum as the one considered in the present paper . we will in fact study here a continuum version of the classical potts model . its mean field free energy is @xmath6\ ] ] @xmath7 , @xmath8 represents the density of particles with spin @xmath9 , @xmath10 , @xmath2 ; @xmath11 the inverse temperature ; @xmath12 the chemical potential . despite the simplicity of the model its thermodynamics , which is defined by minimizing @xmath13 over @xmath14 , has a rather interesting structure . in @xcite and @xcite it is proved that the resulting phase diagram is characterized by a critical curve @xmath15 , as in figure 1 . @xmath16 has @xmath5 minimizers @xmath17 , @xmath18 . there are positive numbers @xmath19 , @xmath20 so that @xmath21 furthermore @xmath22 so that the total density of the state @xmath23 is smaller than the total density in any of the ordered critical points @xmath24 , @xmath25 , which is in fact equal to @xmath26 . when @xmath27 , only the ordered states survive and there are @xmath28 minimizers , when @xmath29 , only the disordered state survives and there is a unique minimizer . therefore when crossing vertically the critical curve the total density jumps , a phenomenon which can be related to magnetostriction as argued in @xcite . the kac proposal applied to leads to hamiltonians of the form @xmath30 where @xmath31 , @xmath32 , @xmath33 , @xmath34 , is a finite configuration of particles with spin ; @xmath12 the chemical potential and @xmath35 , @xmath36 a symmetric probability kernel , say with range 1 . an analysis a la lebowitz and penrose , @xcite ( see also gates and penrose , @xcite ) proves that the mesoscopic ( @xmath37 ) behavior of the system with hamiltonian @xmath38 is described by @xmath39 as a functional defined on functions @xmath40 with compact support . let @xmath41 a torus in @xmath42 , call @xmath43 the functional on @xmath44 , then obviously @xmath45 ( just restrict the inf on the l.h.s . to constant functions ) . thus a preliminary condition for the particle model to have mean field behavior is to require that holds with equality , which ( we suspect ) requires extra conditions on @xmath46 . in @xcite the kac proposal has been modified in such a way that the above condition is automatically satisfied . call @xmath47 the mean field energy , in our case @xmath48 ( i.e. the first two terms on the r.h.s . of , the third one is the contribution of the entropy to the free energy ) and set @xmath49 @xmath50 with @xmath51 a smooth , symmetric , translational invariant probability kernel with range 1 , @xmath52 if @xmath53 . namely the `` modified kac proposal '' we are adopting is to suppose that _ the particle hamiltonian has an energy density at point @xmath54 given by the mean field free energy computed on the empirical density @xmath55_. analogous prescription can be applied whenever the mean field order parameter is a density ( or as in this case a collection of densities ) . the free energy functional associated to is , supposing @xmath41 a torus in @xmath42 , @xmath56 which can be rewritten as @xmath57 by convexity the second integral is non negative and 0 on the constants ; the first one is minimized by taking @xmath58 constantly equal to the minimizer of @xmath59 . thus holds in this case with equality . notice that the hamiltonian @xmath38 of has the form because it can be written as @xmath60 thus the lmp prescription in this case is just a positivity assumption on the kernel @xmath46 ( more precisely @xmath61 ) . in the sequel we will restrict to the choice . the main result in this paper is [ thme1.1 ] for any @xmath1 , @xmath2 , @xmath62 there is @xmath63 and for any @xmath64 there are @xmath65 and @xmath5 dlr measures at @xmath66 , denoted by @xmath67 , with the following properties . @xmath68 each @xmath69 is a translational invariant , extremal dlr measure ( with trivial @xmath70-algebra at infinity ) ; @xmath68 any translational invariant dlr measure is a convex combination of @xmath71 ; @xmath68 calling @xmath72 the average density of particles with spin @xmath73 in @xmath69 and @xmath74 the mean field values , @xmath75 ; @xmath68 any measure @xmath69 , @xmath25 , is invariant under any exchange of spin labels which does not involve @xmath76 while @xmath77 is invariant under any exchange of spin labels . the proof of theorem [ thme1.1 ] uses specific features of the model besides the property that is true with equality . which properties are of general nature and which ones are instead truly specific of the model is difficult to say . to a great extent the proof follows from the analysis ( a la pirogov - sinai ) of the lmp model in chapter 11 and 12 of @xcite , but there are several points where we need to overcome important difficulties not present in the lmp model . among them the main one is about the exponential decay of correlations in the restricted ensemble , theorem 3.1 of the companion paper @xcite . how to go from such a result to the proof of theorem [ thme1.1 ] is the content of the present paper . theorem [ thme1.1 ] does not claim anything away from @xmath66 , this allows to simplify the traditional pirogov - sinai approach . the conjecture is that when @xmath12 varies in @xmath78 , @xmath79 suitably small , then we go from uniqueness @xmath80 to @xmath28 extremal states , @xmath81 , ( always referring to translational invariant dlr states ) . the potts model does not exactly fall in the class considered in @xcite but presumably the analysis in @xcite can be extended to prove the above conjecture . it is also plausible that the estimates are uniform in a small neighborhood of @xmath11 , in such a case we would have local closeness of the mean field and the finite @xmath82 phase diagrams , thus partially confirming the validity of the conjecture in the beginning of the introduction . .5 cm in part i we define the model and establish the main notation , section [ sec:2 ] , and then prove theorem [ thme1.1 ] , section [ sec : z31 ] , supposing that the peierls estimates on contours are valid . in part ii we prove the peierls estimates , this being the more technical part of the paper . we start with the basic definitions . they are quite standard and consistent with those of the companion paper @xcite . .5 cm we give the following definitions . @xmath68 we denote by @xmath83 , @xmath84 , the partition @xmath85 of @xmath0 into the cubes @xmath86 ( @xmath87 and @xmath88 the cartesian components of @xmath54 and @xmath89 ) . we call @xmath90 the cube which contains @xmath91 . .5 cm @xmath68 a set is @xmath83-measurable if it is union of cubes in @xmath83 . a function @xmath92 is @xmath83-measurable if its inverse images are @xmath83-measurable sets , or , equivalently , if it is constant on the cubes of @xmath83 . .5 cm @xmath68 calling two sets connected if their closures have non empty intersection , given a @xmath93-measurable region @xmath41 we call @xmath94 $ ] the union of all cubes of @xmath83 in @xmath95 which are connected to @xmath41 . analogously we call @xmath96 $ ] the union of all cubes of @xmath83 in @xmath41 which are connected to @xmath97 . we start with the definition of the phase space . @xmath68 it is convenient to represent the phase space @xmath98 of the potts model as a spin system on the lattice , the spins taking values in a non compact space . with @xmath99 the cubes of the partition @xmath100 , we then define @xmath101 and @xmath102 . thus an element @xmath103 is a sequence @xmath104 , if @xmath105 we will then say that in @xmath106 there are @xmath107 particles at positions @xmath108 with spins @xmath109 , @xmath110 . as particles are undistinguishable , physical observables are functions symmetric under exchange of particles labels and the actual physical phase space is @xmath111 which is obtained by taking the quotient under permutation of indices . to simplify notation in the sequel we will just write @xmath98 being clear from the context if we are referring to @xmath112 . since labels are unimportant we can write a configuration @xmath103 as a sequence @xmath113 , @xmath114 , @xmath115 , indeed @xmath116 , namely the set of all @xmath117 , identifies the component of @xmath118 in @xmath119 . given @xmath120 we write @xmath121 and we call @xmath122 . we finally denote by @xmath123 the configuration which collects all the particles in @xmath118 and @xmath124 , evidently referring here to indistinguishable particle configurations . .5 cm @xmath68 we consider @xmath125 equipped with its natural topology and @xmath98 with the product topology calling @xmath126 the corresponding borel @xmath70-algebra . while the product topology in @xmath98 is not physically correct ( the path of a particle moving continuously from a cube @xmath106 to another one is not continuous in the product topology ) yet the borel structure is not changed and since we are interested in measure theoretically properties the above definition becomes acceptable . .5 cm @xmath68 we denote by @xmath127 the measure on @xmath128 which restricted to @xmath129 is equal to @xmath130 , such that if @xmath131 is a bounded measurable function on @xmath128 @xmath132 if @xmath41 is a bounded @xmath100 measurable region we define _ the free measure _ @xmath133 on @xmath134 observing that for any measurable set @xmath135 @xmath136 we have already defined the energy @xmath38 ( of a finite configuration ) , see . the energy in a bounded set @xmath41 with boundary condition @xmath137 is defined as usual as @xmath138dr\ ] ] the expression on the r.h.s . depends only on the particles of @xmath137 at distance @xmath139 . in the sequel we will sometimes replace @xmath140 by @xmath70-finite measures by setting @xmath141 @xmath142 any non negative @xmath70-finite measure on @xmath143 . by identifying @xmath144 as a sum of dirac deltas we may regard the convolution @xmath145 as a particular case of . in particular we will often consider @xmath146dr\ ] ] where @xmath147 were @xmath148 , @xmath149 is one of the minimizers of the mean field free energy @xmath150 . the gibbs measure in @xmath41 ( @xmath41 a bounded , measurable set in @xmath0 ) with boundary conditions @xmath151 is @xmath152 where the partition function @xmath153 is the normalization factor in . we will also consider more general boundary conditions with @xmath151 replaced by @xmath70-finite measure , the formula is again with the energy defined using . .5 cm as usual in statistical mechanics local equilibrium and deviations from equilibrium are defined in terms of `` averages '' and of `` coarse grained '' variables . we briefly recall the main notion adapted to the present context . given a configuration @xmath154 we denote by @xmath155 the number of particles in the configuration @xmath118 which are in the cube @xmath156 and have spin @xmath9 , namely @xmath157 we also define the density of particles in @xmath158 , @xmath159 the phase indicators are introduced using two scales @xmath160 and @xmath161 and an accuracy parameter @xmath162 . all these numbers depend on @xmath82 and there is much flexibility about their choice , for the sake of definitiveness we fix them as follows : ( choice of parameters ) . [ contorni3 ] _ we choose @xmath160 and @xmath161 as functions of @xmath163 : @xmath164 supposing for simplicity that @xmath3 and @xmath165 are both in @xmath166 . we also choose @xmath167 _ we require that @xmath168 and @xmath169 @xmath170 thus for @xmath82 small , @xmath160 is much larger than 1 and much smaller than @xmath171 equal to the range of the interaction ; it defines a scale large enough to make statistics reliable . indeed , the scale @xmath160 is used together with the accuracy parameter @xmath162 to determine if a configuration ( or a density ) is close to a mean field equilibrium value in a cube @xmath172 . this will be done via the _ phase indicator _ that we denote by @xmath173 . local equilibrium is instead present when the above closeness extends to regions in the scale @xmath161 thus regions with a diameter much larger than the interaction range . to quantify the local equilibrium we use the _ phase indicator on the scale @xmath161 _ that we denote by @xmath174 . .5 cm for any @xmath175 we then define in analogy to @xmath176 and @xmath177 and @xmath178 , $ } \\0 & { \rm otherwise}. \end{cases}\ ] ] recalling , the previous definitions extend to particle configurations @xmath118 by setting @xmath179 we will often drop the suffix @xmath180 by writing @xmath173 instead of @xmath181 , analogously for @xmath174 . given @xmath182 and @xmath183 , we define `` the @xmath76-restricted ensemble '' as @xmath184 if @xmath185 we simply write @xmath186 . by an abuse of notation we also denote by @xmath187 the space of densities @xmath188 such that @xmath189 in @xmath41 . .5 cm first observe that @xmath190 in fact the two regions are separated by the set @xmath191 which is what we call spatial support of a contour . given a configuration @xmath118 such that @xmath192 is bounded , we call contour a pair @xmath193 where @xmath194 , the spatial support of @xmath195 is @xmath196 and @xmath197 , its specification . abstract contours @xmath195 are the pairs which arise from some configuration as above . we decompose the complement of sp@xmath198 as @xmath199 where @xmath200 is the unbounded , maximal connected component of @xmath201 . we denote by @xmath202 we omit the proof of the following proposition ( which is a straightforward consequence of the definition of phase indicators and contours ) . .5 cm [ thm2.6 ] suppose @xmath118 has a contour @xmath195 , then there is @xmath203 such that @xmath204 for all @xmath205 $ ] , @xmath206 as in . moreover if @xmath41 is any maximal connected component of @xmath207 then there is @xmath208 such that @xmath209 for all @xmath210 $ ] . .5 cm proposition [ thm2.6 ] implies that given any @xmath195 , @xmath211 , @xmath212 $ ] is determined by @xmath213 and assumes the same value for all @xmath214 is a contour for @xmath215 . we will then say that @xmath195 has color @xmath76 if @xmath204 for all @xmath205 $ ] and denote by @xmath216 the union of the maximal connected components @xmath217 of int@xmath198 where @xmath209 for all @xmath218 $ ] . given a color @xmath76 and a bounded , simply connected @xmath219-measurable region @xmath41 , we denote by @xmath220 the collection of all sequences @xmath221 of contours of color @xmath76 with spatial support in @xmath222 $ ] and such that the spatial supports are mutually disconnected . from a technical point the main results in this paper are theorem [ thmz3.1 ] and theorem [ thmz3.2 ] below . their statements involve the notion of @xmath76-boundary conditions , diluted gibbs measure and diluted partition functions . .5 cm @xmath68 let @xmath223 and @xmath41 a bounded @xmath224-measurable region . a configuration @xmath151 is a @xmath76-boundary condition relative to @xmath41 if there is a configuration @xmath225 ( see ) which is equal to @xmath151 in the region @xmath226 . .5 cm @xmath68 let @xmath41 , @xmath151 and @xmath227 as above , then the @xmath76-diluted gibbs measure in @xmath41 with b.c.@xmath151 is @xmath228$}\}}\,\ , \nu(dq_{\lambda})\ ] ] where @xmath229$}\ } } e^{-\beta h_{{\lambda},{\lambda } } ( q_{\lambda}| \bar q_{{\lambda}^c } ) } \ , \nu(dq_{\lambda})\ ] ] is the diluted partition function . [ thmz3.1 ] for any @xmath230 there are @xmath231 , @xmath232 and for any @xmath233 there is @xmath65 such that for any bounded , simply connected , @xmath234 measurable region @xmath41 , any @xmath76- boundary conditions @xmath151 and any @xmath235 , @xmath236 the proof of theorem [ thmz3.1 ] is a corollary of theorem [ thmz3.2 ] below , which involves the fundamental notion of contour weights : .5 cm @xmath68 given a @xmath76-colored contour @xmath195 and a @xmath76-boundary condition @xmath227 relative to @xmath206 , we define the true weight @xmath237 as @xmath238 where @xmath239 ; @xmath240 decomposes into @xmath241 maximal connected components @xmath242 ; @xmath243 denotes the value of @xmath174 on @xmath244 $ ] . .5 cm the above are called true weights to distinguish them from fictitious weights introduced in the proof of theorem [ thmz3.2 ] . .5 cm [ thmz3.2 ] in the same context of theorem [ thmz3.1 ] , for all @xmath82 small enough and recalling definition , @xmath245 .5 cm as already pointed out theorem [ thmz3.2 ] is the main technical result in this paper , its proof follows the pirogov - sinai strategy and it is reported in part ii . we will next show that theorem [ thmz3.1 ] follows from theorem [ thmz3.2 ] . .5 cm * proof of theorem [ thmz3.1 ] * ( using theorem [ thmz3.2 ] ) . by definition the @xmath76-diluted gibbs measures in @xmath41 have support on configurations where @xmath246 on @xmath247 $ ] . therefore if @xmath248 there must be a contour @xmath195 such that @xmath249 . thus @xmath250 is bounded by @xmath251 where @xmath252 ranges over all possible values of @xmath194 such that @xmath253 ; @xmath254 is the number of @xmath224 cubes in @xmath252 . @xmath255 is the number of possible values of @xmath256 , @xmath257 the number of @xmath258 cubes in @xmath252 . the above is bounded by @xmath259 the sum vanishes as @xmath260 , see for instance the proof of theorem 9.2.8.1 in @xcite , such that for @xmath82 small enough the above is bounded by @xmath261 . .5 cm in the following sections we will see that the proof of theorem [ thme1.1 ] follows from theorem [ thmz3.2 ] and theorem 3.1 of @xcite using the same general arguments as in @xcite for the analogous proof in the lmp model . in part ii we will prove theorem [ thmz3.2 ] . a probability @xmath262 on @xmath263 is dlr at @xmath264 if for any bounded , measurable cylindrical function @xmath131 and any bounded measurable set @xmath265 , @xmath266 we fix @xmath11 and set by default @xmath267 and @xmath268 , see theorem [ thmz3.1 ] . @xmath11 and @xmath12 in the sequel will be often omitted from the notation . we will start by proving : .5 cm [ thmz4.0.1 ] the set of dlr measures at @xmath269 is a non empty , convex , weakly compact set . .5 cm the proof is made simpler by the assumption that the interaction is non negative . we follow closely section 12.1 of @xcite where the analogous statement is proved for the lmp model and where the reader may find more details . the basic estimate is below . let @xmath270 , @xmath271 the gibbs measure on @xmath272 at @xmath269 with boundary conditions @xmath151 , @xmath273 then , using the non negativity of the interaction , @xmath274 and therefore there are @xmath275 and @xmath276 ( decreasing with @xmath107 ) such that for any @xmath277 , @xmath278 by supposing ( without loss of generality ) @xmath275 large enough , there exist configurations @xmath225 , @xmath18 , such that @xmath279 call @xmath280 the set of all probabilities on @xmath263 and @xmath281 define also for any bounded @xmath100-measurable set @xmath265 , @xmath282 if @xmath41 is @xmath224-measurable and @xmath227 as in , then by @xmath283 which is therefore non empty . a stronger statement actually holds : .5 cm [ lemmaz4.0.1 ] @xmath284 is a non empty , convex , weakly compact set and if @xmath285 then @xmath286 . .5 cm * proof . * for any @xmath287 the set @xmath288 is compact and if @xmath289 , @xmath290 then , by the prohorov theorem , the weak closure of @xmath291 is weakly compact . since @xmath292 is closed , the inequalities @xmath293 are preserved under weak limits such that @xmath284 is weakly closed , hence weakly compact . convexity and the inclusion @xmath286 are obvious and the lemma is proved . .5 cm [ coroz4.0.1 ] let @xmath294 be an increasing sequence of @xmath295-measurable sets invading @xmath0 , then + @xmath296 is a non empty , convex , weakly compact set independent of the sequence @xmath294 . .5 cm [ lemmaz4.0.2 ] any measure in @xmath297 is dlr and any dlr measure is in @xmath297 . .5 cm * proof . * let @xmath298 be a bounded , measurable ( but not necessarily @xmath100-measurable ) set , and @xmath299 a bounded @xmath100-measurable set . then if @xmath300 , @xmath301 and since @xmath302 , it then follows that @xmath303 , hence that @xmath262 is dlr . viceversa if @xmath262 is dlr then by and the dlr property , @xmath304 . by @xmath305 , hence @xmath306 and by the arbitrariness of @xmath41 in @xmath297 . .5 cm corollary [ coroz4.0.1 ] and lemma [ lemmaz4.0.2 ] prove theorem [ thmz4.0.1 ] . moreover .5 cm [ thmz4.0.2 ] let @xmath294 be an increasing sequence of @xmath307-measurable regions invading @xmath0 and @xmath227 , @xmath18 , configurations satisfying . then @xmath308 converges weakly to a measure @xmath309 and ( with @xmath231 as in theorem [ thmz3.1 ] ) @xmath310 .5 cm * proof . * call @xmath311 $ ] , @xmath312 $ ] . then by and @xmath313 . since @xmath314 is increasing , by lemma [ lemmaz4.0.1 ] for @xmath315 , @xmath316 which is weakly compact . then @xmath308 converges weakly by subsequences to an element @xmath69 of @xmath317 . thus @xmath318 and by corollary [ coroz4.0.1 ] @xmath319 . follows because it is satisfied by @xmath308 , @xmath308 converges weakly to @xmath69 by subsequences and @xmath320 is closed . the title means that @xmath76-boundary conditions , @xmath183 , select a unique measure in the thermodynamic limit . the precise results are stated in theorem [ thmz5.0.1 ] and its corollary theorem [ thmz5.0.3 ] . .5 cm [ thmz5.0.1 ] there are @xmath321 and @xmath322 positive such that for all @xmath82 small enough , for all @xmath203 , for all bounded , @xmath224-measurable , simply connected regions @xmath323 , @xmath324 , for all @xmath76-boundary conditions @xmath325 , for all @xmath224-measurable sets @xmath298 in @xmath326 and for all bounded , measurable cylindrical functions @xmath131 in @xmath298 , @xmath327 .5 cm the proof will be obtained after rewriting the expectations @xmath328 in a way which allows to exploit the couplings introduced in @xcite . .5 cm _ notation . _ we fix @xmath298 and @xmath131 as in theorem [ thmz5.0.1 ] . let @xmath329 be a bounded , @xmath307-measurable set , @xmath330 and ( recall ) @xmath331 denote by @xmath332 the subset of @xmath220 of collections @xmath333 made exclusively of external contours , namely such that all @xmath334 are mutually disconnected . let @xmath335 and call ( dependence on @xmath131 , @xmath41 and @xmath298 is not made explicit ) : @xmath336\}\\ & & \hskip-2 cm f(q,{\underline}{\gamma})= \frac{n^{(k)}({\underline}{\gamma};q;f)}{n^{(k ) } ( { \underline}{\gamma};q;1 ) } , \;\;\ ; q : q_{{\rm ext}({\underline}{\gamma})}\in \mathcal x^{(k)}_{{\rm ext}({\underline}{\gamma } ) } \label{z5.0.5 } \end{aligned}\ ] ] where , calling @xmath337\}$ ] and @xmath338 , @xmath339 [ thm14n.2.1 - 1 ] with the above notation @xmath340 @xmath341 where @xmath342 is the subset of external contours in @xmath343 ( obtained by deleting from @xmath343 all @xmath344 with @xmath345 for some other @xmath346 ) ; and , recalling the definition , @xmath347 .5 cm the proof is completely analogous to that of theorem 12.5.1.1 in @xcite and omitted . * proof of theorem [ thmz5.0.1 ] . * by theorem [ thmz3.2 ] the contour weights satisfy the assumptions in theorem 3.1 of @xcite which can then be applied . it then follows that there is a coupling @xmath348 of @xmath349 and @xmath350 with the following property . @xmath351 where @xmath352 is the set of all @xmath353 for which there exists a @xmath224-measurable region @xmath354 such that : if @xmath355 then @xmath356=\emptyset$ ] ; the contours of @xmath357 and @xmath358 with spatial support in @xmath354 are identical as well as the restrictions to @xmath354 of @xmath124 and @xmath359 ; finally @xmath360 $ ] . by , @xmath361 and by the definition of @xmath362 , @xmath363 on @xmath352 , hence theorem [ thmz5.0.1 ] . as an immediate corollary of theorem [ thmz5.0.1 ] we have : .5 cm [ thmz5.0.3 ] in the same context of theorem [ thmz5.0.1 ] , latexmath:[\[\label{z5.0.11 } \big| g^{(k)}_{{\lambda}_1,q_1}(f)- \mu^{(k)}(f)\big| \le c \|f\|_{\infty } where @xmath69 is the dlr measure defined in theorem [ thmz4.0.2 ] . in this section we will prove that the dlr measures @xmath69 have all trivial @xmath70-algebra at infinity ( also called the tail field ) and they are therefore extremal dlr measures . the property follows from the peierls bounds , theorem [ thmz3.2 ] , and the exponential decay of correlations , theorem [ thmz5.0.3 ] . the particular structure of the model is at this point rather unimportant and indeed we will be able to avoid many proofs by referring to their analogues @xcite . .5 cm [ definz6.0.1 ] let @xmath365 be an arbitrary but fixed increasing sequence of @xmath366-measurable cubes of sides @xmath367 which invades the whole space and @xmath368 the collection of sequences @xmath369 of the form @xmath370 , where @xmath371 , is the translation by @xmath372 . .5 cm when proving in the next subsections that the measures @xmath69 are translational invariant , we will need translates of the sequence @xmath365 , hence the definition of @xmath368 . observe that sequences in @xmath368 are not necessarily @xmath373-measurable . .5 cm [ definz6.0.2 ] the @xmath76-tail field , @xmath203 , is defined as @xmath374 .5 cm [ thmz6.0.1 ] for all @xmath82 small enough @xmath375 , @xmath203 . .5 cm by taking countably many intersection we can reduce the proof of theorem [ thmz6.0.1 ] to the proof that for any sequence @xmath376 and any @xmath234-measurable cube @xmath298 @xmath377 this would be direct consequence of theorem [ thmz5.0.3 ] if we had @xmath378 instead of @xmath379 in and the whole point will be to reduce to such a case . .5 cm @xmath68 we call @xmath380 the union of all @xmath366 cubes contained in @xmath294 ( recall @xmath294 may not be @xmath234-measurable ) and define the random set @xmath381 as follows . @xmath382 is the union of @xmath383 with all the maximal connected components @xmath384 of the set @xmath385 such that @xmath386\cap { \lambda}_{n;0}^c\ne \emptyset$ ] . we call @xmath387 the complement of @xmath381 and observe that by construction , @xmath388 for all @xmath389 $ ] . .5 cm @xmath68 given @xmath54 call @xmath390 the maximal connected component of @xmath391 which contains @xmath54 ( @xmath390 may be empty ) . calling @xmath392 the diameter of the set @xmath384 , we define @xmath393 $ } \big\}\ ] ] notice that @xmath394 [ thmz6.0.2 ] let @xmath395 and @xmath396 then @xmath397 and for all @xmath398 and all measurable , bounded functions @xmath131 cylindrical in @xmath298 , @xmath399 with @xmath322 and @xmath321 as in . .5 cm the proof of theorem [ thmz6.0.2 ] is completely analogous to the proof of theorem 12.2.2.5 in @xcite and it is omitted , we just outline its main steps . to prove we write @xmath400 and follows from and the inequality @xmath401 . to prove the latter we observe that @xmath402 is a cylindrical set and therefore @xmath403 . we can then use the peierls bounds in and after some standard combinatorial arguments prove the desired inequality and hence . let @xmath404 then @xmath405 hence @xmath406 such that follows from , the bound being uniform in all @xmath131 cylindrical in @xmath298 . _ proof of theorem [ thmz6.0.1 ] . _ we use a borel - cantelli argument . let @xmath407 as above , then @xmath408 by @xmath409 and theorem [ thmz6.0.1 ] follows from and . in this section we will prove that any translational invariant dlr measure can be written as a convex combination of the measures @xmath69 , this is not yet the decomposition into ergodic dlr measures because we do not know that the @xmath69 are translational invariant ( a statement proved in the next section ) . however it follows directly from theorem [ thmz5.0.3 ] that any @xmath69 is translational invariant under @xmath410 . indeed by theorem [ thmz5.0.3 ] for any @xmath411 , @xmath412 weakly , then @xmath413 and the latter is equal to @xmath414 which by theorem [ thmz5.0.3 ] is equal to @xmath69 . we have : .5 cm [ thmz7.0.1 ] for all @xmath82 small enough the following holds . let @xmath415 be any dlr measure invariant under @xmath416 , then there is a unique sequence @xmath417 of numbers in @xmath418 $ ] such that @xmath419 .5 cm * proof . * the proof is an adaptation of the classical argument by gallavotti and miracle - sole for the analogous property in the ising model at low temperatures . its extension to ising models with kac potentials has been carried out in @xcite and adapted in @xcite to the lmp model . all these proofs are basically the same as the original one and we think it useless to repeat it once more here . the argument shows ( see for instance section 12.3 in @xcite ) that there are numbers @xmath420 $ ] , @xmath18 with the following property ; for any bounded cylindrical function @xmath131 there is a function @xmath421 vanishing as @xmath422 and satisfying , for any @xmath366-measurable cube @xmath41 of side @xmath423 : @xmath424 by compactness there is a sequence @xmath425 ( independent of @xmath131 ) such that @xmath426 , for all @xmath76 . then @xmath427 and is proved since @xmath415 is determined by expectations of bounded cylindrical functions @xmath131 . in this section we will complete the proof of theorem [ thme1.1 ] by proving that the measures @xmath69 are translational invariant , since their tail field is trivial they are then ergodic and the decomposition becomes the decomposition into ergodic dlr measures . let @xmath368 be as in definition [ definz6.0.1 ] and @xmath428 , @xmath429 , define @xmath430(f),\ ; \text{for any $ \{{\lambda}_n\}\in \mathcal s$}{\nonumber}\\&&\text { and any bounded , measurable cylindrical function $ f$ } \big\ } \label{z8.0.1 } \end{aligned}\ ] ] such that @xmath431 the tail set of definition [ definz6.0.2 ] . it then follows ( see the proof of the analogous lemma 12.4.1.1 in @xcite for details of the proof ) that : .5 cm [ lemmaz8.0.1 ] for any @xmath432 , @xmath429 , @xmath433\big ( \mathcal q_{k,{\rm tail};\;i}\big)=1\ ] ] moreover @xmath434 if and only if @xmath435 . [ lemmaz8.0.2 ] for all @xmath82 small enough the following holds : for any @xmath432 , @xmath429 , @xmath436 and @xmath437 , @xmath438 , are mutually singular and @xmath439 . .5 cm * proof . * by lemma [ lemmaz8.0.1 ] it suffices to show that @xmath440 which is proved by the same argument used to prove lemma 12.4.1.3 of @xcite , we just report the main steps . suppose ( without loss of generality ) that @xmath441 , i.e. @xmath69 an ordered state . since @xmath437 is invariant by translations of @xmath442 , for any @xmath432 , @xmath429 , we may also restrict to @xmath443 with @xmath444 . let @xmath41 be a @xmath366-measurable cube , @xmath445 the number of particles in @xmath446 with spin @xmath447 , then @xmath448 because @xmath436 is invariant under translations in @xmath442 . @xmath449 is then bounded from above by @xmath450^{1/2 } \end{aligned}\ ] ] @xmath451 having used cauchy - schwartz . since the energy is non negative , @xmath452 thus in conclusion @xmath453 for any @xmath454 , @xmath455 , @xmath456 $ ] . let @xmath457 and @xmath458 the number of @xmath366-cubes in @xmath41 and @xmath459 , then for any @xmath460 @xmath461(\frac { |q_{\lambda}(k)|}{|{\lambda}|})\ge \mu^{(k)}(\frac { |q_{{\lambda}_0}(k)|}{|{\lambda}|})=\frac{n_{{\lambda}_0}}{n_{\lambda}}\mu^{(k)}(\frac { |q_{c_0^{(\ell_{+})}}(k)|}{\ell_{+}^d } ) \end{aligned}\ ] ] we then write @xmath462 and get @xmath463(\frac { |q_{\lambda}(k)|}{|{\lambda}|})\ge \frac{n_{{\lambda}_0}}{n_{\lambda}}(\rho^{(k)}_{k}-\zeta)(1-e^{-\beta ( c^*/4 ) \zeta^{2 } \ell_{-}^d } ) \\ & & \hskip.2cm\ge \rho^{(k)}_{k}-\big(\zeta + \rho^{(k)}_{k}\{e^{-\beta ( c^*/4 ) \zeta^{2 } \ell_{-}^d } + \frac{n_{\lambda}-n_{{\lambda}_0}}{n_{\lambda}}\}\big ) \end{aligned}\ ] ] which is strictly larger than the r.h.s . of for @xmath41 large and @xmath82 small . . we will prove translational invariance for special values of the mesh , @xmath465 , the general case follows by a density argument completely analogous to the one used for the lmp model , see subsection 12.4.2 of @xcite , which is therefore omitted . .5 cm [ thmz8.0.1 ] for all @xmath466 and all @xmath82 small enough the measures @xmath69 are invariant under translations by @xmath443 , for any @xmath467 . .5 cm * proof . * fix @xmath465 . since @xmath437 is invariant under translations in @xmath468 , the measure @xmath469 is invariant under the group of translations @xmath470 . then by @xmath471 by lemma [ lemmaz8.0.2 ] @xmath472 for any @xmath438 , such that @xmath473 . on the other hand @xmath474 , therefore in @xmath475 for all @xmath438 and hence @xmath476 . if there is @xmath477 such that @xmath478 , then again by lemma [ lemmaz8.0.2 ] , @xmath479 and @xmath480(\mathcal q_{k,{\rm tail};i})=1 $ ] which contradicts ( as we have proved that @xmath476 ) . [ sec : z9 ] in this part we prove theorem [ thmz3.2 ] , thus we show that there is a constant @xmath231 such that the peierls bounds are satisfied with constant @xmath481 where we say that the peierls bound holds with constant @xmath322 if @xmath482 for all @xmath76 , for all bounded contours of color @xmath76 and for all @xmath76-boundary conditions @xmath227 . as explained in subsections 11.4 and 11.5 of @xcite , following the approach of zahradnik,@xcite we introduce the cutoff contours weights . given any @xmath76 and any @xmath76-colored contour @xmath195 we are going to define the weight @xmath483 for any configuration @xmath118 which is a @xmath76-boundary condition for @xmath206 in below , the definition will imply that @xmath483 depends only on the restriction of @xmath118 to @xmath484 . for @xmath485 we call @xmath486 and for any bounded , simply connected @xmath307-measurable region @xmath41 and any @xmath76-boundary condition @xmath487 we introduce the @xmath76-cutoff partition function in a region @xmath41 with b.c . @xmath137 as @xmath488 with same notation as in we then define @xmath489 @xmath490 all the above quantities depend on the weights @xmath491 which we define ( implicitly ) by introducing first a constant @xmath492 and then setting @xmath493 is not a closed formula because the r.h.s.still depends on the weights , however the contours on the r.h.s.are `` smaller '' and , by means of an inductive procedure , it is possible to prove there is a unique choice of @xmath494 such that holds for all @xmath76 , all @xmath195 and all @xmath495 , see theorem 10.5.1.2 in @xcite . the important point of these definitions is that if the estimate below holds , then the cut - off weights are equal to the true ones . this is the content of the next theorem whose proof is omitted being completely analogous to theorem 10.5.2.1 in @xcite . [ thm3.2 ] suppose that for any @xmath76 , any contour @xmath195 of color @xmath76 and any @xmath76-boundary conditions @xmath495 for @xmath206 , @xmath496 then @xmath497 we will prove that if @xmath492 is small enough then holds for all @xmath82 correspondingly small . the main ingredient in the proof of is the exponential decay in restricted ensembles proved in @xcite . the proof of is based on an extension of the classical pirogov - sinai strategy , we refer to chapter 10 of @xcite for general comments and proceed with the main steps of the proof . most of it follows from chapter 11 of @xcite and theorem 3.1 of @xcite . precise quotations will be given in complementary sections where we will also add proofs to fill in parts not covered by the above references . the first step is the following theorem . .5 cm [ energy estimate ] [ thm:2.9 ] there is @xmath498 such that the following holds . given any @xmath499 there is @xmath322 such that for all @xmath12 : @xmath500 , for all @xmath76 , for all @xmath76-contour @xmath195 , for all @xmath76 boundary conditions @xmath495 and for any @xmath492 , the following estimate holds for all @xmath82 small enough : @xmath501\ell_-^d n_{\gamma } } \prod_{j=1}^p \frac{e^{\beta \mathbf{i}_{k_j}({\rm int}_{j}({\gamma } ) ) } z^{(k_j)}_{{\rm int}_{j}({\gamma})),{\lambda}}(\chi^{(k_j)}_{{\rm sp}({\gamma } ) } ) } { e^{\beta \mathbf{i}_{k}({\rm int}_{j}({\gamma } ) ) ) } z^{(k)}_{{\rm int}_{j}({\gamma } ) ) , { \lambda}}(\chi^{(k)}_{{\rm sp}({\gamma})})}\ ] ] where for any bounded @xmath224-measurable set @xmath502 , @xmath503-\int_{{\omega}}e_{\lambda}^{\rm mf } ( j_{\gamma}\star\chi^{(k)}_{\omega^c})\ ] ] @xmath504 and @xmath24 a minimizer of @xmath150 , see ) . in classical pirogov - sinai models with nearest neighbor interactions the analogue of theorem [ thm:2.9 ] follows directly from the extra energy due to presence of the contour , here contours have a non trivial spatial structure which leads , after a coarse graining a la lebowitz - penrose , to a delicate variational problem . theorem [ thm:2.9 ] will be proved in section [ sec:4 ] . we exploit the arbitrariness of @xmath505 in theorem [ thm:2.9 ] and fix @xmath506 such that the first factor on the r.h.s . of is consistent with but we need a good control of the ratio of partition functions in . as typical in the pirogov - sinai theory a necessary requirement comes from demanding that the pressures in the restricted ensembles are equal to each other , a request which will fix the choice of the chemical potential . [ equality of pressures ] [ plabeta ] for any chemical potential @xmath507 $ ] and any van hove sequence @xmath508 of @xmath307-measurable regions the following limits exist @xmath509 and they are independent of the sequence @xmath294 and of the @xmath76 boundary condition @xmath140 . moreover there is @xmath510 and , for all @xmath4 small enough , there is @xmath65 , @xmath511 such that @xmath512 theorem [ plabeta ] is proved in appendix [ sec : a2.10 ] . the existence of the thermodynamic limits , , is not completely standard because there is the additional term given by the weights of the contours . however the peierls bounds ( automatically satisfied by the cut - off weights ) imply that contours are rare and small and can then be controlled . the equality is more subtle , it is proved by showing that ( i ) @xmath513 and @xmath514 depend continuously on @xmath12 ; ( ii ) by a lebowitz - penrose argument they are close as @xmath260 to the mean field values ; ( iii ) the difference of the mean field pressures changes sign as @xmath12 crosses @xmath515 . by theorem [ plabeta ] at @xmath268 the volume dependence in the ratio disappears and to conclude the estimate we need to prove that the next surface term `` is small '' . this is the hardest part of the whole analysis where theorem 3.1 of @xcite enters crucially . .5 cm [ surface corrections to the pressure ] [ thm:2.14 ] with @xmath492 as in there are @xmath516 and @xmath322 such @xmath322 such that for any @xmath517 , all bounded @xmath307-measurable @xmath518 and all @xmath519 @xmath520 with @xmath65 and @xmath521 as in theorem [ plabeta ] . theorem [ thm:2.14 ] is proved in section [ sec:5 ] . .5 cm by theorem [ thm:2.14 ] with @xmath522 , @xmath523| \label{2.29}\ ] ] @xmath524| \label{2.29a}\ ] ] hence by @xmath525 the last holding for all @xmath82 small enough . by we have then proved and yields @xmath526 the proof of the energy estimate , is divided into two steps . the first step ( theorem [ thm2.2 - 11 ] below ) is the proof that it is possible to reduce to `` perfect boundary conditions '' , namely to reduce the analysis to partition functions with `` perfect boundary conditions '' , i.e. with boundary conditions @xmath24 , one of the minimizers of the mean field free energy functional . this implies that we can factorize with a negligible error the estimate in int@xmath198 from the one in sp@xmath198 . the second step in the proof of involves a bound on the constrained partition function in @xmath194 which yields the gain factor @xmath527 . .5 cm without loss of generality we fix @xmath76 and restrict to contours @xmath528sp@xmath529 with color @xmath76 and define regions in @xmath206 as follows . the construction is the same as that used in chapter 11 , subsection 11.2.1 of @xcite to which we refer . we denote by @xmath530 the maximal connected components @xmath531 int@xmath198 such that @xmath209 for all @xmath532 $ ] . we call @xmath533,\;\ ; \delta_2:=\delta_{\rm out}^\ell[c({\gamma})^c\cup \delta_1],\;\ ; \delta_3:=\delta_{\rm out}^\ell[c({\gamma})^c\cup \delta_1\cup \delta_2],\qquad \ell:=\ell_+/8\ ] ] these are successive corridors that we meet when we move from @xmath534 into @xmath194 . in all of them @xmath535 and the region where @xmath536 is far away , by @xmath537 . when approaching sp@xmath198 from int@xmath538 we see : @xmath539\cup \delta_{\rm in}^\ell[{\rm int}_{h}({\gamma})],\;\ ; \delta^{(h)}_5:=\delta_{\rm out}^\ell[\delta^{(h)}_4],\qquad h \text { may also be } = k\ ] ] by the definition of contours the above @xmath540 corridors are in a region where @xmath541 , and the distance from where @xmath542 , is @xmath543 . we then call @xmath544 where @xmath545 @xmath546 and finally , letting @xmath547 , @xmath548 , we define @xmath549 observe that the points in @xmath41 interact only with those in @xmath550 . after a lebowitz - penrose coarse graining in @xmath551 we will reduce to a variational problem with the free energy functional @xmath552 defined in below . we will prove existence and uniqueness of minimizers and their stability properties concluding that with `` negligible error '' we can eliminate " the corridor @xmath553 in @xmath554 sp@xmath555 $ ] which separates int@xmath198 from the rest of sp@xmath198 , their interaction with @xmath553 being replaced by an interaction with perfect boundary conditions . .5 cm given any set @xmath556 , any b.c . @xmath557 and any measurable set @xmath558 , we call @xmath559 [ reduction to perfect boundary conditions ] [ thm2.2 - 11 ] there are @xmath560 such that for all @xmath12 : @xmath561 , for all @xmath82 small enough for any @xmath223 , any contour @xmath193 of color @xmath76 and any boundary condition @xmath562 the following holds . recalling , we call @xmath563 then , with @xmath552 defined in below , and @xmath41 as in @xmath564 .5 cm we first rewrite @xmath565 as follows . [ lemma13n.2.1 - 1 ] there is a non negative , bounded function @xmath566 whose explicit expression is given in below , which vanishes unless the phase indicator @xmath567 verifies @xmath568 this function @xmath569 is such that @xmath570 .5 cm * proof . * the argument is the same as lemma 11.2.2.3 of @xcite , for the reader convenience we report it . we drop the dependence on @xmath12 from the notation . we call @xmath571 and we define @xmath572 recall that @xmath573 depends only on the restriction of @xmath574 to @xmath575 where @xmath575 is the set of all @xmath54 at distance @xmath576 from @xmath577 $ ] . in fact all contours @xmath195 which contribute to @xmath578 have spatial support in @xmath579 . for this reason we can change arbitrarily @xmath574 in the complement of @xmath580 leaving unchanged @xmath578 . thus @xmath581 because @xmath582 contains all @xmath583)\le { \gamma}^{-1}$ ] . analogously , calling @xmath584 we define @xmath585 since @xmath586 by setting @xmath587 and recalling , becomes an identity . we postpone the proof of theorem [ thm2.2 - 11 ] since it uses a coarse graining in @xmath551 that reduce to a minimization problem for the free energy functional that we define in the next subsection . .5 cm the lp ( lebowitz penrose ) free energy functional @xmath588 , defined in below , is a @xmath589-discretization of . we start by recalling properties of the mean field free energy proved in @xcite . recalling , we call @xmath590 and we observe that the minimizers @xmath24 , @xmath591 are critical points of @xmath592 , namely they satisfy @xmath593 we will denote the common minimum value of @xmath594 by @xmath595 furthermore @xmath596 is strictly convex , namely there is a constant @xmath597 such that @xmath598 let @xmath41 be a @xmath599- measurable bounded region of @xmath0 . given two non negative functions , @xmath188 and @xmath600 defined in @xmath601 and @xmath599- measurable , we call @xmath602 the function equal to @xmath188 in @xmath41 and equal to 0 in @xmath97 . analogous definition for @xmath603 . we define @xmath604 where , setting @xmath605 , @xmath606 and the matrix @xmath607 is given by @xmath608 @xmath609 @xmath610 finally @xmath611 in is @xmath612\ ] ] the relation of this functional with the model is given in the theorem [ propa.2 ] below . let @xmath41 be a @xmath613-measurable bounded region of @xmath0 . recalling the definition and , we shorthand @xmath614 , and @xmath615 @xmath616 , @xmath617 . ( space of densities ) . [ dens ] we call @xmath618 the space of non negative , @xmath83-measurable functions defined in @xmath619 . thus the elements @xmath620 are actually functions of finitely many variables , i.e. @xmath621 . given any @xmath622 we denote by @xmath623 and we call @xmath624 analogously , given any @xmath625 , we call @xmath626 . given a configuration @xmath140 and recalling the definition of the constrained partition function given in , we have : [ propa.2 ] there is @xmath627 such that the following holds . for any @xmath628@xmath629 and for any @xmath630 @xmath631 the proof of theorem [ propa.2 ] is the same as the proof of theorem 11.1.3.3 of @xcite and it is omitted . in theorem below we state what we need in order to prove , its proof is postponed to section [ sec : i.2 ] . [ thm:4.3 ] let @xmath195 be a contour of color @xmath76 and @xmath632 . let @xmath551 be the set defined in . given any @xmath633 such that @xmath634 in @xmath635 $ ] , for all @xmath540 , let @xmath636 . there are positive constants @xmath322 and @xmath321 and for all @xmath540 there are positive functions @xmath637 , such that @xmath638 and furthermore for all @xmath540 and all @xmath639 , @xmath640 theorem [ thm:4.3 ] is the main model - dependent estimate needed for the proof of theorem [ thm2.2 - 11 ] the others arguments are the same as in subsection 11.2.2 of @xcite . we thus only sketch them . going back to , we first observe that if @xmath641 then @xmath642 ( see ) with @xmath643 . then by theorem [ propa.2 ] @xmath644 where we have set @xmath645 , @xmath646 and @xmath647 if @xmath438 . since the dependence on @xmath12 in @xmath648 is given by the term @xmath649 , for all @xmath12 : @xmath561 , we have @xmath650 thus from , and theorem [ thm:4.3 ] we get @xmath651 .5 cm using the formula @xmath652 setting @xmath653 and using , we have @xmath654 by using that @xmath561 we replace @xmath515 by @xmath12 in the last two terms with an error bounded by @xmath655 and then use theorem [ propa.2 ] `` backwards '' to reconstruct partition functions . we then have @xmath656 since an analogous bound holds for @xmath657 , we then get from and from thus proving theorem [ thm2.2 - 11 ] . the main step to conclude the proof of the energy estimate , is to bound the last term in , namely @xmath658 with @xmath41 as in . also here we use coarse graining , but since the number of particles is not bounded we can not apply directly theorem [ propa.2 ] . the same difficulty is present in the lmp model and the arguments used there can be straightforwardly adapted to the present contest . the outcome is theorem [ thm:4.5 ] below which is the same as proposition 11.3.0.1 of @xcite to which we refer for proofs . we need some definitions . let @xmath659 be a positive number such that @xmath660,\qquad \rho_{\max}>\max_h\max_s\rho^{(h)}_s+\zeta,\qquad \sum_{n\ge \rho_{max}\ell_-^d}\frac { ( s\ell_-^de^{\beta{\lambda}})^n}{n!}\le e^{-4s\rho_{max}\ell_-^d}\ ] ] @xmath661 for any @xmath662 , recalling we call @xmath663 we also call @xmath664 the collection of all pairs of cubes @xmath665 both in @xmath666 and such that @xmath667 , @xmath668 , @xmath669 , @xmath670 , @xmath668 , @xmath671 . we require that @xmath664 must be maximal , namely any pair @xmath672 that verify the same property must have at least one among @xmath673 and @xmath674 appearing in @xmath664 . we denote by @xmath675 the number of cubes ( cubes not pairs of cubes ! ) appearing in @xmath664 . .5 cm [ thm:4.5 ] let @xmath676 be as in and . there is @xmath677 such that for all @xmath82 small enough and for all @xmath12 such that @xmath678 . @xmath679 where , recalling , @xmath680 the main result in this section is theorem [ thm3.1 ] below which gives a bound of the @xmath681 on the right hand side of . this estimate is the same as the one given in theorem 11.3.2.1 of @xcite to which we refer for proofs . .5 cm in the sequel @xmath682 denotes a bounded , connected @xmath683-measurable region such that its complement is the union of @xmath684 ( maximally connected ) components @xmath685, ... ,@xmath686 at mutual distance @xmath687 ( such that they do not interact ) : @xmath688 . the boundary conditions are chosen by fixing arbitrarily @xmath689 , @xmath690 , and setting @xmath691 on @xmath692 , analogously to we will denote for any @xmath616 , @xmath693 the reason why the region @xmath502 is @xmath694-measurable is that we are going to use theorem [ thm3.1 ] with @xmath695 , @xmath41 as in . recalling we set for any given @xmath696 , @xmath697}\ ] ] so that recalling , @xmath698 given a kernel @xmath699 , @xmath700 , we denote by @xmath701 as a consequence @xmath702 with this notation , we the following holds . [ lemma4.16 ] let @xmath703 and @xmath704 as above . let @xmath705 be any @xmath706-measurable function . letting @xmath707 we get @xmath708 where ( recall ) , @xmath709 finally @xmath710 .5 cm * proof . * recalling and we rewrite as follows : @xmath711 by adding and subtracting @xmath712 , we then have @xmath713 we add and subtract @xmath714 to the second term in and we add and subtract @xmath715 to the last term . we also use that by , for any @xmath696 , @xmath716 . we thus get . @xmath717 , where @xmath718 is @xmath719 where @xmath720 , see . by convexity the second sum in the definition of @xmath718 is non negative . also the first term is bounded from below by replacing replacing @xmath721 by @xmath691 , such that @xmath722 all the entropy terms cancel with each other , so we get . .5 cm given a function @xmath188 defined in @xmath723 and @xmath724-measurable , let @xmath725 and @xmath726 be the numbers defined in subsection [ sec : i ] . [ thm3.1 ] let @xmath727 be such that @xmath728 with @xmath676 as in theorem [ thm:4.5 ] . for any @xmath729 , @xmath704 and @xmath705 as in lemma [ lemma4.16 ] , if @xmath730 , then @xmath731 \zeta^2\ell_-^d\ ] ] where @xmath569 is defined in and @xmath732 , \text{\bf 1}_{\omega_i } \big ) - \big(e_{{\lambda}_\beta}^{\rm mf}(\hat j_{\gamma } * \chi^{(k_i)}_{\omega_i}),\text{\bf 1}_{\omega}\big)\ ] ] .5 cm the proof is the same as the one of theorem 11.3.2.1 of @xcite and thus is omitted . in this subsection we conclude the proof of theorem [ thm:2.9 ] , the arguments we use are taken from subsection 11.3.3 of @xcite . by and there is a constant @xmath322 such that @xmath733 for all @xmath82 small enough ( recall that @xmath734 ) , the min is achieved at @xmath735 such that ( @xmath736 ) , @xmath737 which inserted in yields with a new constant @xmath322 : @xmath738 since @xmath739 , we have @xmath740\ ] ] therefore the exponent in the last term of becomes @xmath741 going backwards we write @xmath742 thus after some cancelations @xmath743 let @xmath744 $ ] , @xmath745 $ ] be the element of @xmath746 closest to @xmath24 and let @xmath747\big\}$ ] . then for all @xmath12 such that @xmath748 , @xmath749 let @xmath151 be any configuration in @xmath750 such that @xmath751 $ ] , then @xmath752 analogous bounds hold for the other partition functions such that @xmath753 is bounded by @xmath754 from and , follows with @xmath727 as in theorem [ thm3.1 ] . in this section we prove theorem [ thm:4.3 ] , by analyzing a variational problem for the free energy functional . we need a similar result in section [ sec:5 ] in the proof of theorem [ thm:2.14 ] for an interpolated functional . we thus state the result in a way that includes both cases . we study the variational problem @xmath755\ ] ] where , recalling , @xmath756\ ] ] where @xmath588 is defined in and @xmath757 we prove that away from @xmath97 the minimizer is exponentially close to @xmath24 . the constraint @xmath758 , namely @xmath759 , is essential as it localizes the problem in a neighborhood of the ( stable ) minimum where it is possible to prove that the critical points , i.e. the solutions of the euler - lagrange equations , converge exponentially to @xmath24 . thus in this section we prove the following result . [ thmi.3.1 ] let @xmath41 be a @xmath224-measurable set and let @xmath760 . then , for any @xmath761 $ ] , there is a unique minimizer @xmath762 of the variational problem . furthermore there are @xmath322 and @xmath763 both positive such that @xmath764 .5 cm observe that theorem [ thm:4.3 ] is a corollary of the above theorem for @xmath765 . in section 5 of @xcite we have studied a similar variational problem but there the constraint was on the single variable @xmath766 because the functional considered there had as main term the lebowitz - penrose free energy on the scale @xmath160 . here we have a simpler functional but we have to face the new problem of controlling the fluctuations of @xmath602 from its average on cubes of site @xmath160 . in this section we will use both the lattices @xmath767 and @xmath768 . we thus define the following . .5 cm @xmath68 @xmath769 denotes the euclidean space of vectors @xmath770 with the usual scalar product @xmath771 . for @xmath772 we simply write @xmath773 . @xmath68 for any @xmath774 we denote by @xmath775 the point such that @xmath776 . for @xmath777 we let @xmath778 and we observe that @xmath779 @xmath68 we fix a @xmath224-measurable region @xmath41 and @xmath761 $ ] and omitting the dependence on @xmath41 and @xmath780 , we rewrite the functional as follows . notice that there are two equal entropy terms : one in @xmath588 multiplied by @xmath780 and the other , explicitly written on the r.h.s . of , multiplied by @xmath781 . the same holds for the terms multiplied by @xmath515 . calling @xmath782 the vector with components @xmath783 as in , we have @xmath784 where @xmath785 , @xmath786 @xmath787 as in and @xmath788 @xmath68 we call @xmath789 the kernel obtained by averaging over the cubes of @xmath258 while @xmath790 is over those of @xmath706 , thus @xmath791 we also define @xmath792 observe that there is @xmath498 such that @xmath793 @xmath68 we introduce a @xmath794-relaxed constraint @xmath795)_+\}^4+\{({\rm av}(\rho_{{\lambda}};x , s)- [ \rho^{(k)}_{s}-\zeta])_-\}^4 \big ) \end{aligned}\ ] ] where @xmath796 , @xmath797 and @xmath798 , @xmath799 @xmath800 is such that @xmath801 and @xmath802 if @xmath803 . @xmath68 we denote by @xmath804 and @xmath805 the minimizers of @xmath806 in @xmath807 and of @xmath131 in @xmath808 . @xmath68 for any @xmath706-measurable function @xmath809 , we denote by @xmath810 we say that a @xmath706-measurable function @xmath602 is a critical point of @xmath806 , respectively @xmath131 , if @xmath811 , respectively @xmath812 . in this subsection we prove some a - priori bounds on the fluctuations of the minimizers @xmath804 and @xmath805 from their averages . we start from the latter proving the following result . [ prop6.1 ] let @xmath41 be a @xmath224-measurable set and let @xmath813 . there is @xmath231 such that for all @xmath82 small enough the following holds . * ( i)*. let @xmath814 any @xmath258 cube . let @xmath815 and denote by @xmath816 . if @xmath817 is a minimizer of @xmath818 in @xmath819 then @xmath820 for all @xmath821 . * ( ii)*. if @xmath822 minimizes @xmath823 then @xmath824 for all @xmath825 . we postpone the proof of proposition [ prop6.1 ] giving first some preliminary lemmas . for any @xmath826^s$ ] and any @xmath827 $ ] for @xmath828 and @xmath829 as in ( i ) of proposition [ prop6.1 ] , we let @xmath830 where @xmath831 we regard @xmath832 as a function of @xmath833 . @xmath834 has the interpretation of a chemical potential for the species @xmath9 , @xmath835 is an auxiliary parameter , we will eventually set @xmath836 , in which case @xmath837 . we let @xmath838 and define the map @xmath839 on @xmath840 by setting for @xmath841 @xmath842 [ lemma6.3 ] there is @xmath516 such that for all @xmath517 and all @xmath843^s$ ] , @xmath832 has a unique minimizer @xmath844 . furthermore for all @xmath262 and @xmath835 , @xmath845 .5 cm * proof . * since @xmath846 are compact for any @xmath847 and @xmath832 is smooth , then @xmath832 has a minimum . any minimizer ( which is strictly positive because of the entropy term in @xmath832 ) is also a critical point , namely a fixed point of the map @xmath839 , such that recalling , @xmath848 since @xmath849 , @xmath850 , @xmath851 and @xmath852 , then holds for any minimizer . thus any minimizer belongs to the set @xmath853\}$ ] and @xmath854 is invariant under @xmath839 . moreover , for any @xmath841 , @xmath855 such that @xmath856 for @xmath82 small enough , @xmath857 such that @xmath839 is a contraction and the fixed point is unique . [ lemmaappk.1.3 ] there is @xmath677 such that for all @xmath82 small enough and with @xmath844 as in lemma [ lemma6.3 ] , @xmath858 .5 cm * proof . * recalling the definition we use the estimate to replace @xmath859 with @xmath860 . thus , recalling the definition of @xmath696 in , since @xmath861 there are @xmath862 and @xmath863 positive such that ( below we shorthand @xmath864 for @xmath865 ) @xmath866 - c_2({\gamma}\ell_-)= t \,r^{(k)}_{s}-(s-1)\zeta - c_2({\gamma}\ell_-)\ ] ] and by , @xmath867+e^{\beta ( { \lambda}_\beta+1)}c_3({\gamma}^d\ell_-^d)+ c_2({\gamma}\ell_-)\ ] ] recalling that @xmath868\}}$ ] , we then obtain from and for @xmath82 small enough . in the next lemma we prove that the minimizer @xmath844 is smooth . [ lemmaappk.1.4 ] @xmath869 is a smooth function of @xmath843^s$ ] and @xmath827 $ ] ( derivatives of all orders exist ) and there is a constant @xmath322 such that @xmath870 @xmath871 .5 cm * proof . * the minimizer @xmath872 is implicitly defined by the critical point equation @xmath873 , see . thus we call @xmath874 and we observe that @xmath875 where @xmath876 by this is a positive definite matrix and by the definition of @xmath877 @xmath878 by the implicit function theorem we then conclude that the function @xmath879 , such that @xmath880 is differentiable and its derivative verifies @xmath881 thus from and we get . same argument applies for the derivative @xmath882 . we next define @xmath883 since @xmath884 , the number of cubes in @xmath885 contained in @xmath828 are @xmath886 , thus @xmath887 is the average density of the species @xmath9 when the chemical potential is @xmath262 . our purpose is to prove that for any @xmath888 , s=1, .. ,s$ ] , there is @xmath889^s$ ] such that @xmath890 [ lemmaappk.1.4.1 ] for any @xmath891 $ ] , @xmath10 , the equation has a solution @xmath892^s$ ] and there is a constant @xmath322 such that for all @xmath9 , @xmath893 . * * we fix a vector @xmath891 $ ] , @xmath10 , and we first observe that the equation @xmath894 has obviously a solution @xmath895 , which , recalling , is obtained by solving @xmath896 and by the same arguments used in the proof of lemma [ lemmaappk.1.3 ] , @xmath897 . we then for look for a function @xmath898 , @xmath899 $ ] such that @xmath900 by differentiating the above equation we get the following cauchy problem for @xmath898 : @xmath901 by the @xmath902 matrix @xmath903 has diagonal elements @xmath904 while the non diagonal elements are bounded by @xmath905 thus ( for @xmath82 small ) the matrix @xmath903 is positive definite and invertible and depends smoothly on @xmath262 . as a consequence the cauchy problem has a unique solution and since by @xmath906 the solution @xmath898 satisfies the bound @xmath907 for all @xmath9 . we now have all the ingredients for the proof of the proposition [ prop6.1 ] stated at the beginning of this subsection . * proof of proposition [ prop6.1]*. _ proof of ( i)_. let @xmath908 be a minimizer of @xmath909 and call @xmath910 since @xmath911 , ( see ) , @xmath912 is as in lemma [ lemmaappk.1.4.1 ] and therefore , for @xmath82 small enough , there is @xmath913 such that @xmath914 then writing @xmath915 for @xmath916 and using , @xmath917 and using @xmath918 hence @xmath919 and since by lemma [ lemma6.3 ] the minimizer is unique we get that @xmath920 . on the other hand by , there is @xmath231 such that for all @xmath82 small enough , @xmath921 for all @xmath841 . _ proof of ( ii ) . _ let @xmath822 be a minimizer of @xmath922 and let @xmath923 be a @xmath924 cube . we write @xmath925 and we prove that @xmath926 minimizes @xmath927 . in fact @xmath928 and if @xmath929 were not a minimizer than for any minimizer @xmath930 , calling @xmath931 , we would have @xmath932 and this would be a contradiction . thus @xmath929 is a minimizer and by ( i ) @xmath933 , @xmath841 . the proof of ( ii ) then follows from the arbitrariness of @xmath814 . we will next consider @xmath79 and start by proving the analogue of lemma 5.2 of @xcite : [ lemmaappk.1.5 ] there is a constant @xmath677 such that for all @xmath79 small enough any minimizer @xmath934 of @xmath806 is also a critical point and for all @xmath935 , and all @xmath936 @xmath937 .5 cm * proof . * we denote by @xmath938)_+\}^4 + \{({\rm av}(\rho_{\lambda};x , s)- [ \rho^{(k)}_{s}-\zeta])_-\}^4\ ] ] then for all @xmath939 , @xmath940 and since @xmath696 vanishes on @xmath941 : @xmath942 and , calling @xmath943 , @xmath944 we have @xmath945)_+\}^4 + \{({\rm av}(\hat\rho_{{\lambda},{\epsilon}};x , s)- [ \rho^{(k)}_{s}-\zeta])_-\}^4 \le 4{\epsilon}(\phi'-\phi'')\ ] ] let @xmath946 given @xmath947 @xmath948 we denote by @xmath949 the point such that @xmath776 and analogously @xmath950 is such that @xmath951 . then for @xmath860 as in , there is a constant @xmath322 such that @xmath952 observe that @xmath953(z_x)\ ] ] thus by jensen inequality , @xmath954(z_x)\ ] ] then , from we get that for all @xmath955 , @xmath956 by using we then get @xmath957 an upper bound for @xmath958 is @xmath959 . thus there is @xmath322 such that @xmath960 from and , follows . from and by choosing @xmath794 so small that @xmath961 , we conclude that the minimizer is in the interior of @xmath808 and thus is a critical point . [ propappk.1.3 ] there is @xmath231 such that for any @xmath4 small enough , for all @xmath79 small enough ( depending on @xmath82 ) , if @xmath962 minimizes @xmath806 in @xmath808 then @xmath963 .5 cm * proof . * by lemma [ lemmaappk.1.5 ] given any @xmath82 if @xmath794 is small enough , then @xmath804 is a critical point , namely it satisfy the following equation . @xmath964)_+\}^3 + \{({\rm av}(\hat\rho_{{\lambda},{\epsilon}};x , s)- [ \rho^{(k)}_{s}-\zeta])_-\}^3 \label{6.40 } \end{aligned}\ ] ] where @xmath965 let @xmath814 a @xmath924 cube and let @xmath966 . then @xmath967 and from we get @xmath968 we use and we get @xmath969 thus for all @xmath970 @xmath971 where we have used and the fact that for all @xmath794 small enough , @xmath972 . [ lemmaappk.1.5bis ] @xmath804 converges by subsequences and any limit point @xmath805 is a minimizer of @xmath131 in @xmath941 . .5 cm * proof . * the proof is exactly the same as that of lemma 5.3 in @xcite . compactness implies convergence by subsequences and by any limiting point is in @xmath973 . since for any @xmath974 @xmath975 , by taking the limit @xmath976 along a subsequence converging to some @xmath977 , we get @xmath978 , thus @xmath977 is a minimizer . in this subsection we prove convexity of @xmath131 and @xmath806 and from this we will deduce the uniqueness of their minimizers . .5 cm [ thmappk.2.1 ] given any @xmath979 and @xmath980 ( @xmath981 as in ) , for all @xmath82 small enough the following holds . let @xmath766 and @xmath982 for all @xmath983 , @xmath984 , then the matrix @xmath985 , @xmath986 , is strictly positive : @xmath987 same statements holds for @xmath988 , @xmath79 . .5 cm * proof . * the proof is analogous to that of theorem 5.5 in @xcite for completeness we sketch it . we will prove the theorem only in the case @xmath989 . denoting by @xmath990 below the diagonal matrix with entries @xmath991 @xmath992 extend @xmath993 and @xmath551 as equal to 0 outside @xmath41 and set @xmath994 then , by and using that @xmath787 is symmetric , @xmath995 u(x , s)^2\big\ } { \nonumber}\\ & & \hskip2 cm - \sum_s\sum_{x\in { \gamma}^{-1/2}\mathbb z^d}[\frac{1}{\beta\rho^{(k)}_s } -\kappa^ * ] u(x , s)^2 + \frac{1}{\beta } ( u,\rho_{\lambda}^{-1 } u ) \end{aligned}\ ] ] by the curly bracket is non negative as well as @xmath996 $ ] . since , by cauchy schwartz inequality , for each @xmath9 @xmath997 then @xmath998 u(x , s)^2 \le ( u,[\frac{1}{\beta\rho^{(k)}}-\kappa^*]u)\ ] ] thus @xmath999 u)\ ] ] and is proved . a minimizer @xmath1000 of @xmath131 is not necessary a solution of @xmath1001 . however a property analogous to the one states in lemma 5.4 of @xcite holds . [ lemmaappk.2.1 ] any minimizer @xmath977 of @xmath1002 , @xmath1003 , is `` a critical point '' in the sense : @xmath68 if for some @xmath1004 , @xmath935 , @xmath936 , @xmath1005 ( strictly ! ) , then @xmath1006 @xmath68 if instead @xmath1007 , then for all @xmath1008 with positive average , i.e. @xmath1009 @xmath1010 while for all @xmath1008 with null average , i.e. @xmath1011 @xmath1012 the following holds . .5 cm [ thmappk.2.2 ] given any @xmath979 and @xmath980 ( @xmath981 as in ) for all @xmath82 and @xmath794 small enough , as well as for @xmath989 the following holds . let @xmath1013 be either a minimizer of @xmath1014 , @xmath1015 or , if @xmath989 , a `` critical point '' ( in the sense of lemma [ lemmaappk.2.1 ] ) satisfying the inequality @xmath1016 for all @xmath1004 , @xmath1017 @xmath1018 . then for any @xmath602 such that @xmath1019 for all @xmath1004 as above , @xmath1020 * proof . * the proofs is the same as the one of theorem 5.6 in @xcite . given any @xmath602 as in the statement of the theorem , we interpolate by setting @xmath1021 , @xmath1022 $ ] , then with @xmath1023 @xmath1024,\rho_{\lambda}- \hat\rho_{{\lambda},{\epsilon}}\big ) \,d\theta ' \,d\theta + \big(d_{\lambda}f_{\epsilon}(0),\rho_{\lambda}- \hat\rho_{{\lambda},{\epsilon}}\big ) \end{aligned}\ ] ] since @xmath1013 is a minimizer , then @xmath1025 . this is immediate in the case @xmath79 , while it follows from lemma [ lemmaappk.2.1 ] applied to @xmath1026 in the case @xmath989 . hence follows from theorem [ thmappk.2.1 ] . .5 cm as shown in @xcite , an immediate consequence of theorem [ thmappk.2.2 ] is : .5 cm [ coroappk.2.1 ] for any @xmath82 and @xmath79 small enough the minimizer of @xmath806 is unique , same holds at @xmath989 for @xmath131 . for @xmath79 ( and small enough ) there is a unique critical point in the space @xmath1027 ; such a critical point minimizes @xmath806 . analogously , when @xmath989 there is a unique critical point in the sense of lemma [ lemmaappk.2.1 ] . such a critical point minimizes @xmath131 . the minimizer of @xmath806 , @xmath79 , converges as @xmath976 to the minimizer of @xmath131 . .5 cm [ propappk.3.1 ] for all @xmath79 small enough , the minimizer of @xmath1028 is @xmath1029 which is also the minimizer of @xmath1030 . .5 cm * proof . * case @xmath989 . if @xmath1031 , then for all @xmath1032 and @xmath984 @xmath1033\big\}\ ] ] where @xmath1034 on the r.h.s . is equal to @xmath1035 if @xmath1036 and to @xmath1037 if @xmath1038 . the function @xmath1039 solves the above equation and by corollary [ coroappk.2.1 ] it is unique and the only minimizer of @xmath1040 . case @xmath79 and small . being a critical point of @xmath806 as well , @xmath1029 is by corollary [ coroappk.2.1 ] also a minimizer of @xmath806 . [ thmappk.3.1 ] there are @xmath1041 and @xmath677 such that for all @xmath82 small enough the minimizer @xmath1042 of @xmath1043 satisfies @xmath1044 .5 cm * proof . * by proposition [ propappk.3.1 ] , @xmath1029 is the minimizer of @xmath1045 , ( both for @xmath79 and for @xmath989 ) thus the difference @xmath1046 @xmath1047 , can be seen as the difference of two minimizers relative to two different boundary conditions and we can then proceed as in the proof of theorem 5.9 of @xcite . however the proof is different : the complication comes from the fact that the constraint is here on the averages and not on the elementary variables @xmath1035 as it was in @xcite . thus the strategy of the proof of is to reduce to the case already treated in @xcite . define for @xmath1048 $ ] , @xmath1049 and call @xmath1050 , @xmath1051 , the minimizer of @xmath1052 ( both for @xmath79 and for @xmath989 ) . the same argument used in the proof of theorem 5.9 of @xcite shows that @xmath1053 is differentiable in @xmath1054 such that @xmath1055 we now estimate @xmath1056 uniformly in @xmath794 and @xmath1054 and this will give . shorthand @xmath1057 we have that @xmath1058 where @xmath1059 so we need to bound @xmath1060 , @xmath1061 . explicitly , ( recall the notation in subsection [ subsec:6.1 ] ) @xmath1062 and @xmath1063)_+\}^2 + \{({\rm av}(\hat\rho_{{\lambda},{\epsilon}}(\cdot;\theta);x , s ) - [ \rho^{(k)}_{s}-\zeta])_-\}^2 \big)\ ] ] finally @xmath1064\ ] ] recalling the definition of @xmath860 in we define for all @xmath1065 and the corresponding @xmath1066 , @xmath1067 calling @xmath1068 . we extend these three matrices to all @xmath1069 by setting their entries equal 0 outside @xmath41 . we denote by @xmath1070 , the norm of the matrix @xmath1071 in @xmath1072 ( @xmath1073 ) , and we observe that @xmath1074 by theorem [ thmappk.2.1 ] , @xmath551 is a positive matrix such that the inverse @xmath1075 is well defined . since the same proof applies also to the matrix @xmath384 , we have that @xmath384 as well is a positive matrix with a well defined inverse @xmath1076 and @xmath1077 . moreover by and we get , for @xmath82 small enough , @xmath1078 thus , from we have that @xmath1079 . since @xmath1080 , for @xmath82 small enough , we have that the series below is convergent and @xmath1081 for a proof of this statement see theorem a.1 in appendix a of @xcite . we are going to prove that there are @xmath321 and @xmath322 positive such that for @xmath1051 , @xmath1082 we first show why the estimate concludes the proof of the theorem . from the definition and using , we have that for all @xmath1051 , and for all @xmath616 , @xmath1083 [ e^{-{\omega}{\gamma}|x - y|}v(y , s ' ) ] \\&&\hskip1 cm \le c^ * \max_{s ' , y\in { \lambda}\cap { \gamma}^{-1/2}\mathbb{z}^d } [ e^{-{\omega}{\gamma}|x - y|}v(y , s ' ) ] \label{6.59 } \end{aligned}\ ] ] recalling @xmath1084\le c\frac{{\gamma}^{-d}}{{\gamma}^{-d/2 } } \max_{y\in { \lambda}\cap { \gamma}^{-1/2}\mathbb{z}^d } [ e^{-{\omega}{\gamma}|x - y|}\mathbf{1}_{\text{dis}(y,{\lambda}^c)\le { \gamma}^{-1}}]\ ] ] thus follows from and . . from we have @xmath1085 we will prove that @xmath1086 then , by @xmath1087 such that we are reduced to the proof of . we define @xmath1088 , then @xmath1089 . thus @xmath1090 we say that two pairs @xmath1004 , @xmath1091 are equivalent , @xmath1092 we call @xmath1093 , and for any @xmath1004 , @xmath1094 we define @xmath1095 with these definitions , @xmath1096 and then , according to we need to solve @xmath1097 . recalling and observing that for all @xmath1098 and all @xmath9 @xmath1099 , we call @xmath1100 and we observe that @xmath1101 assume the same value in all points equivalent to @xmath1004 and to @xmath1091 . thus @xmath384 can be considered also as an operator from @xmath1102 onto itself in the following way . let @xmath1103 , @xmath1104 , and let @xmath1105 its extension to @xmath1106 given by @xmath1107 for all @xmath1108 . then for any @xmath1109 and any @xmath9 , @xmath1110 where @xmath1111 observe that @xmath1112 where @xmath1113 is a vector in @xmath1114 . thus we find @xmath1115 by solving separately @xmath1116 and @xmath1117 with @xmath1118 , @xmath1119 and @xmath1120 . by direct inspection we have @xmath1121 it is easy to see that @xmath1122 we prove below that there is a constant @xmath1123 so that @xmath1124 concludes the proof of in fact @xmath1125 * proof of * the matrix @xmath1126 acting on functions constant on the scale @xmath160 is the same operator considered in theorem 5.9 of @xcite where a statement stronger than has been proven . thus the proof of is contained in the proof of theorem 5.9 of @xcite , but , for the reader convenience we sketch it . to have the same notation as in @xcite , we use the label @xmath372 for a pair @xmath1004 , @xmath1127 , writing @xmath1128 , @xmath1129 if @xmath1130 and shorthand @xmath1131 for @xmath1132 and @xmath1133 for @xmath1134 . we call @xmath1135 the matrix @xmath1136 so that @xmath1137 and @xmath1138 is a diagonal matrix in @xmath1102 . we need to distinguish values @xmath372 where @xmath1139 is large , and so we define @xmath1140 let @xmath1141 be the orthogonal projection on @xmath1142 and @xmath1143 , thus @xmath1141 selects the sites where @xmath1138 is large and @xmath1144 those where it is small . let @xmath1145 we look for @xmath1146 such that @xmath1147 . in @xcite ( see eqs ( 5.34)(5.38 ) in @xcite ) it is proven that @xmath828 is invertible on the range of @xmath1144 and that @xmath1148 @xmath1149 the matrix @xmath828 satisfies the hypothesis of theorem a.1 and theorem a.2 of @xcite so that there are @xmath727 , @xmath862 and @xmath1150 such that @xmath1151 furthermore if @xmath800 is large enough and @xmath1152 , there is @xmath1153 such that @xmath1154 by observing that for any @xmath1155 , @xmath1156 , from , inequality follows . in this section we fix the chemical potential @xmath268 such that holds and we prove theorem [ thm:2.14 ] . we often omit to write explicitly the dependence on @xmath65 . we write an interpolation formula for the partition functions @xmath1157 , @xmath1158 defined in . here @xmath41 is a @xmath1159- measurable set and the boundary condition @xmath1160 . the reference hamiltonian @xmath1161 , @xmath1162 , is the one defined in and it has been chosen in such a way that for all @xmath89 @xmath1163 and all @xmath616 , @xmath1164 that can be proved by using the mean field equation . recalling , we call @xmath1165 for any @xmath1166 we let ( @xmath1167 below ) @xmath1168 and ( @xmath1169 below ) @xmath1170 we thus have a formula for our partition function @xmath1171 , namely @xmath1172 where @xmath1173 is the expectation with respect to the following probability measure @xmath1174 with support on the set @xmath1175 @xmath1176 we call @xmath1177 in this subsection we prove the following theorem . [ thm52.1 ] there is @xmath1178 such that @xmath1179\big)\big| \le c_b{\gamma}^{1/4}\,|{\lambda}\setminus{\lambda}_{00}|\ ] ] where @xmath1180 furthermore letting @xmath1181 , @xmath1182 \big)\big|\le c_b{\gamma}^{1/4}\,|{\lambda}\setminus{\lambda}_{00}|\ ] ] we first observe that ( recall ) @xmath1183 where , recalling , @xmath1184 we write @xmath1185(x)= e^{\rm mf}\big(\hat j_{\gamma}\star [ \rho-\rho^{(k)}]\big)(x)+\sum_{s , s';s\ne s ' } \rho^{(k)}_s \hat j_{\gamma}\star [ \rho-\rho^{(k)}](x , s ' ) \label{5.14b}\ ] ] we call @xmath1186\cap { \lambda}\big)\ ] ] we define the set ( recall the definition of @xmath188 ) @xmath1187\big|\le \bar{c}{\gamma}^{1/4 } \label{5.15b}\ ] ] where @xmath1188 is some large constant . we also call @xmath1189 from , and the fact that the interaction range is @xmath171 , we have that @xmath1190\big)\big|\le c[{\gamma}^{1/2}+\bar{c}{\gamma}^{1/4}+ \mu_{{\lambda},t , k}(\bar{\mathcal{a}}_\delta ) ] \,\big|\delta\big|\ ] ] we are left with the estimate of the probability on the r.h.s of . we first write @xmath1191 and we estimate separately the numerator and the denominator in starting from the numerator . the following estimate holds for the weights of the contours ( see lemma [ lemma2 ] in appendix [ app : bounds ] ) @xmath1192^{| \delta\cup \delta_{\rm { out}}^{(2\ell_+)}(\delta)|/\ell^d_+}\ ] ] we partition @xmath41 in @xmath1193 where @xmath1194\cap{\lambda}_{00 } \label{5.20b}\ ] ] we also define the set @xmath1195 such that @xmath1196 we write @xmath1197 and we notice that from we get @xmath1198^{[| \delta\cup \delta_{\rm { out}}^{(2\ell_+)}(\delta)|]/\ell^d_+ } \\&&\hskip2.5cm\,\,\times\hat z_{\delta\cup b , t}(\bar{\mathcal a}_\delta|q_{{\lambda}_{00}\setminus b } \cup \chi^{(k)}_{{\lambda}^c } ) \label{5.22b } \end{aligned}\ ] ] where @xmath1199 is as in but with the energy given by @xmath1200 . we call @xmath1201 the density on the scale @xmath589 corresponding to the configuration @xmath1202 and we call @xmath1203 the density corresponding to the configuration @xmath1204 . a result analogous to theorem [ propa.2 ] holds also for the partition function with the interpolated hamiltonian . thus , recalling definition [ dens ] , we have that @xmath1205 were analogously to , for any region @xmath502 @xmath1206 \label{flat}\ ] ] since we want to use theorem [ thmi.3.1 ] where @xmath1207 , we first change the chemical potential by using that there is @xmath322 such that @xmath1208 thus holds with @xmath1209 in place of @xmath1210 and with a new constant @xmath322 . we have @xmath1211 recalling that @xmath1212 , we can use theorem [ thmi.3.1 ] concluding that there is a function @xmath1213 ( that depends on @xmath1214 ) such that @xmath1215 in @xmath553 and @xmath1216 calling @xmath1217 the subset of @xmath1218 that is connected to @xmath298 , we observe that the functional on @xmath1219 depends on the boundary conditions only in @xmath553 and @xmath97 where they are equal to the pure phase @xmath24 . thus @xmath1220 to get rid of the dependence on @xmath1214 of @xmath1221 , we minimize the second term on the right hand side of using again theorem [ thmi.3.1 ] . we thus get the existence of @xmath1222 such that , going back to , @xmath1223 we are left with the estimate of the @xmath681 on the r.h.s . of . recalling , we estimate separately the first term , namely @xmath1224 and the second namely @xmath1225 $ ] . for the former we use lemma [ lemma4.16 ] with @xmath1226 . then , calling @xmath1227 , from , , , and the fact that @xmath1228 , we get that @xmath1229 where @xmath1230 , the mean field free energy , is quadratic around @xmath24 . since @xmath1231 , we can take taylor expansion up to the second order , getting @xmath1232 ^ 2 \label{5.29b } \end{aligned}\ ] ] if @xmath1233 , observing that the integral over @xmath1234 in can be restricted to a sum over @xmath1235 we get @xmath1236 ^ 2\big\}^{1/2 } \label{5.30b } \end{aligned}\ ] ] we next observe that @xmath1237 using again that @xmath1238 and taking @xmath1239 , from , , and we get that @xmath1240 to estimate the other term in @xmath1241 we observe that the function of @xmath1242 , @xmath1243 is strictly convex and by and the only minimum is in @xmath24 . thus , calling @xmath1244 there is @xmath231 such that @xmath1245\ge \phi^*_k|\delta\cup b_1|+c^*\sum_s { \gamma}^{-d/2}\sum_{\delta\cup b_1\cap\mathfrak l_{\gamma}}[\rho-\rho^{(k)}]^2 \label{5.32b } \end{aligned}\ ] ] for @xmath1246 , observing that the integral over @xmath1247 in can be restricted to a sum over @xmath1248 , and using that @xmath1249 in @xmath1250 and that @xmath1251 , ( see ) we get @xmath1252 ^ 2\big\}^{1/2 } \\ & & { \nonumber}\le \sum_s \big\{{\gamma}^{-d/2}\sum_{x\in\delta \cap \mathfrak l_{\gamma}}{\gamma}^{-d/2}\sum_{y\in \delta \cup b_1}\hat j_{\gamma}(x , y)\big[\rho(y , s)-\rho^{(k)}\big]^2\big\}^{1/2 } \\ & & \le \sum_s \big\{{\gamma}^{-d/2}\sum_{y\in \delta \cup b_1 } \big[\rho(y , s)-\rho^{(k)}\big]^2{\gamma}^{-d/2}\sum_{x\in \mathfrak l_{\gamma}}\hat j_{\gamma}(x , y)\big\}^{1/2 } \label{5.33b } \end{aligned}\ ] ] using again that @xmath1253 and provided @xmath1254 , from and , we get that @xmath1255\ge \phi^*_k|\delta\cup b_1|+ 2\bar{c}{\gamma}^{1/2}|\delta|,\qquad \forall \rho\in \bar{\mathcal a}^*_\delta\ ] ] from and , we thus get for all @xmath1256 @xmath1257 + 4\bar{c } |\delta|{\gamma}^{1/2 } \\ & & \hskip3 cm \geq f_{\delta\cup b_1,t } ( \chi^{(k)}_{\delta\cup b_1}|\chi^{(k)}_{(\delta\cup b_1)^c})+ 4\bar{c } |\delta|{\gamma}^{1/2 } , \label{5.41b } \end{aligned}\ ] ] .5 cm by estimating @xmath1258 , and calling @xmath1259 from and we get that for all @xmath1260 , and @xmath1261 , @xmath1262 .5 cm going back to - and observing that we can choose @xmath1263 so large that @xmath1264\big)\le - 2\bar{c}{\gamma}^{1/2}|\delta| , \end{aligned}\ ] ] we get @xmath1265 observe that the function @xmath1266 is in the set @xmath1267 , thus from theorem [ propa.2 ] we get that @xmath1268 we next observe that the denominator in can be bounded using the first inequality in getting ( recall that @xmath1203 is the density corresponding to the configuration @xmath1269 ) @xmath1270 thus from , , using again that @xmath1271 , we get @xmath1272 using that there is @xmath322 such that @xmath1273 , , and , conclude the proof of . the proof of is similar : with @xmath298 as in and @xmath1274 as in , we get that @xmath1275 \big)\big| \le c[{\gamma}^{1/4}+ \mu_{{\lambda},t , k}(\mathcal{a}_\delta^c ) ] \,|\delta| \\&&\hskip4 cm \le c\,\,|\delta|\,[{\gamma}^{1/4}+ e^{-\bar{c}{\gamma}^{1/2}|\delta| } ] , \end{aligned}\ ] ] the following result will be used to control the first term on the r.h.s . of . [ thm : press ] for any van hove sequence @xmath1276 of @xmath1277- measurable region the following limit exists @xmath1278 furthermore for any @xmath1279 there is @xmath322 and for any @xmath82 small enough @xmath1280 for any @xmath1277- measurable region @xmath41 .5 cm * proof . * for any @xmath224 measurable region @xmath41 we write @xmath1281 where @xmath1282 denotes expectation w.r.t . @xmath1283 and @xmath1284 . let @xmath1285 . since @xmath1286 and since for @xmath1287 in @xmath343 , @xmath1288\cap \delta_{\rm out}^{2{\gamma}^{-1}}[{\rm sp}({\gamma}')]=\emptyset}$ ] @xmath1289 we have @xmath1290 because @xmath1291 is the expectation of @xmath1292 which satisfies the same bound independently of @xmath118 . by as a direct consequence of the cluster expansion , see for instance theorem 11.4.3.1 of @xcite , for all @xmath82 small enough @xmath1293 where the `` hamiltonian '' @xmath1294 can be written as @xmath1295 , @xmath298 ranging over the connected @xmath224-measurable sets ; the potentials @xmath1296 are translational invariant ( in @xmath468 ) and satisfy the bound : for any @xmath1297 and for any @xmath1298 @xmath1299 for all @xmath82 small enough ( @xmath1300 being the number of @xmath224 cubes in @xmath298 ) . we are now ready to conclude the proof of theorem [ thm : press ] that we will prove with @xmath1301 because @xmath1302 . we have @xmath1303 } \sum_{\delta\supset c^{(\ell_+)}}|u^{(k)}_{{\gamma},\delta}|}$ ] such that follows from . the next results are the main tools for dealing with the bulk part of the expectation on the integral on the r.h.s . of . the following theorem is a corollary of theorem 3.1 of @xcite whose statement is given in the proof below . [ thm : infgibbs ] there are @xmath516 , @xmath1041 and @xmath322 such that for all @xmath517 and all @xmath1304 $ ] there is a probability measure @xmath1305 on @xmath1306 which is invariant under translations in @xmath1307 and such that the following holds . for any bounded @xmath1308- measurable region @xmath1309 and for any @xmath1310 , ( @xmath1311 is defined in ) and any cylinder function @xmath131 with basis in @xmath298 , @xmath1312 where @xmath1313 is the smallest @xmath1314-measurable set that contains @xmath298 and where @xmath1315 , respectively @xmath1316 denote the expectation w.r.t @xmath1174 , respectively @xmath1305 . .5 cm * proof . * in theorem 3.1 of @xcite it has been proved that for any bounded , @xmath224-measurable regions @xmath41 and @xmath1317 and any boundary conditions @xmath1318 and @xmath1319 the following holds . let @xmath1320 and @xmath1321 be the probabilities on @xmath1175 and respectively on @xmath1322 defined as in but with the boundary conditions @xmath1318 and @xmath1319 instead of @xmath1323 . then there is a coupling @xmath1324 of @xmath1320 and @xmath1321 such that if @xmath298 is any @xmath224-measurable subset of @xmath41 : @xmath1325 where @xmath1326 agrees with @xmath1327 in @xmath298 if all @xmath346 such that the closure of sp@xmath198 intersects @xmath298 are also in @xmath1328 and viceversa and moreover @xmath1329\}\ ] ] inequality implies that for all cylinder functions @xmath131 with basis @xmath1330 , for @xmath1331 as in the statement of the theorem , @xmath1332 then for any sequence @xmath1276 of @xmath1277- measurable region , and for any @xmath131 as above , the sequence @xmath1333 is a cauchy sequence and the limit @xmath1334 defines a probability on @xmath1306 which is invariant under translations in @xmath1307 . from the uniformity on the boundary conditions defining @xmath1335 it follows that for any @xmath1336 ( @xmath294 an element of the sequence defining @xmath1305 ) @xmath1337 and this implies . we will use the following consequence of theorem [ thm : infgibbs ] . [ thm : la00 ] for any @xmath1308- measurable region @xmath619 , for any @xmath1338 @xmath1339 where @xmath1340 is defined in . .5 cm * proof . * we denote by @xmath1341 the smallest @xmath1277- measurable set that contains the set @xmath1342 . we have that @xmath1343 where @xmath1344 is a bound for the number of particles in a cube of @xmath1345 and @xmath1346 is a bound for the number of cubes in @xmath1345 that intersect the set @xmath1341 . thus from and we get @xmath1347 since @xmath1348 , follows from . as a consequence of theorem [ thm : press ] and corollary [ thm : la00 ] we have the following result . [ thm5.5 ] let @xmath521 and @xmath1349 be the pressures defined in theorem [ plabeta ] and theorem [ thm : press ] respectively . then @xmath1350 where , @xmath1351 - \sum_{s}(\varphi_{k}(s)+{\lambda}_{\beta,{\gamma } } ) \frac{| q(s)\cap c_{0}^{(\ell_+ ) } |}{\ell_+^d}.\ ] ] with @xmath1352 we will get by dividing by @xmath1353 and letting @xmath1354 . we first write ( below we set @xmath1355 ) , @xmath1356 we write @xmath1357)\setminus{\lambda}_{00 } } e^{\rm mf}[j_{{\gamma}}\star q_{\lambda}\cup \chi_{{\lambda}^c}^{(k ) } ] - e^{\rm mf}[j_{{\gamma}}\star \chi_{{\lambda}^c}^{(k ) } ] \\ & & + \sum_{x\in\mathfrak{l}_{\ell_+}\cap{\lambda}_{00 } } \int_{c_{x}^{(\ell_+ ) } } e^{\rm mf}[j_{{\gamma}}\star q_{\lambda } ] \label{5.54b } \end{aligned}\ ] ] taking expectation w.r.t @xmath1174 we get @xmath1358)\setminus{\lambda}_{00 } } \mathbb{e}_{{\lambda},t , k}\big(e^{\rm mf}[j_{{\gamma}}\star q_{\lambda}\cup \chi_{{\lambda}^c}^{(k ) } ] - e^{\rm mf}[j_{{\gamma}}\star \chi_{{\lambda}^c}^{(k)}]\big ) \\&&+ \ell_+^d \sum_{x\in\mathfrak{l}_{\ell_+}\cap{\lambda}_{00 } } \mathbb{e}_{{\lambda},t , k}\big(\mintone{c_{x}^{\ell_+ } } e^{\rm mf}[j_{\gamma}\star q ] - \sum_{s}(\varphi_{k}(s)+{\lambda}_{\beta,{\gamma } } ) \frac{| q(s)\cap c_{x}^{\ell_+ } |}{\ell_+^d}\big ) \label{5.55b } \end{aligned}\ ] ] as in the proof of we have that @xmath1359 thus @xmath1360)\setminus{\lambda}_{00 } } \mathbb{e}_{{\lambda},t , k}\big(e^{\rm mf}[j_{{\gamma}}\star q_{\lambda}\cup \chi_{{\lambda}^c}^{(k ) } ] - e^{\rm mf}[j_{{\gamma}}\star \chi_{{\lambda}^c}^{(k)}]\big)=0\ ] ] from theorem [ thm : infgibbs ] , using the invariance under @xmath161-translation of the limiting measure @xmath1305 and noticing that @xmath1361 , we get @xmath1362 from theorem [ plabeta ] , and , follows . .1 cm [ cor5.7 ] we call @xmath1363 -e^{\rm mf}\big[j_{\gamma}\star\chi^{(k)}_{{\lambda}^c}\big]\right ) -\int_{\lambda}e^{\rm mf}\big[j_{\gamma}\star\chi^{(k)}_{{\lambda}^c}\big]\ ] ] then the following holds . @xmath1364\ ] ] where @xmath1365 is defined in and @xmath1366 in , @xmath1367\big)\\&&\hskip1.6cm- \mathbb{e}_{\infty , t , k}\big(\big [ e^{\rm mf}\big(j_{\gamma}\star q_{\lambda}\cup \chi^{(k)}_{{\lambda}^c } \big ) -e^{\rm mf}(\rho^{(k)})\big]\big ) \\&&\hskip-1 cm \mathcal{r}_3=\mathbb{e}_{{\lambda},t , k } \big(\psi(j_{\gamma}\star q_{\lambda};\chi^{(k)}_{{\lambda}^c})\big)-\mathbf{i}_k({\lambda}^c ) \end{aligned}\ ] ] finally , recalling , @xmath1368{\gamma}^{-d/2 } \sum_{x\in { \lambda}_{00}\cap\mathfrak{l}_{\gamma}}[\mathbb{e}_{{\lambda},t , k}\big(\hat j_{\gamma}\star \rho_{\lambda}\big)-\mathbb{e}_{\infty , t , k}\big(\hat j_{\gamma}\star \rho_{\lambda}\big ) ] \\&&\hskip-1 cm \mathcal{r}_5=\sum_s[\varphi_{k}(s)+{\lambda}_{\beta,{\gamma}}]{\gamma}^{-d/2 } \sum_{x\in { \lambda}\setminus{\lambda}_{00}\cap\mathfrak{l}_{\gamma}}[\mathbb{e}_{{\lambda},t , k}\big(\hat j_{\gamma}\star ( \rho_{\lambda}-\rho^{(k)})\big)-\mathbb{e}_{\infty , t , k}\big(\hat j_{\gamma}\star ( \rho_{\lambda}-\rho^{(k)})\big ) ] \end{aligned}\ ] ] * proof . * coming back to and using and we get @xmath1369dt \\\ ] ] recalling and definition , we get @xmath1370 \big ) + \psi(j_{\gamma}\star q_{\lambda};\chi^{(k)}_{{\lambda}^c})\ ] ] thus adding and subtracting @xmath1371 we get that the term corresponding to the energy @xmath1340 in the integral on the r.h.s . of is given by @xmath1372\big)- \mathbb{e}_{\infty , t , k}(e^{\rm mf}\big(j_{\gamma}\star q\big ) + \mathcal{r}_3+\mathbf{i}_k({\lambda}^c ) \\&&=\int_{{\lambda}_{00}}\big[\mathbb{e}_{{\lambda},t , k}(e^{\rm mf}\big(j_{\gamma}\star q_{\lambda}\big)- \mathbb{e}_{\infty , t , k}(e^{\rm mf}\big(j_{\gamma}\star q_{\lambda}\big)\big ] + \mathcal{r}_2 + \mathcal{r}_3+\mathbf{i}_k({\lambda}^c ) \end{aligned}\ ] ] we have used `` back '' @xmath161-translation invariance of @xmath1305 and to get the term @xmath1373 we have added and subtract @xmath1374 . for the remaining terms in the expectation on the right hand side of , we consider @xmath1162 as a function defined on the whole lattice @xmath1375 by setting @xmath1376 outside @xmath41 , so as to we write @xmath1377{\gamma}^{-d/2 } \sum_{x\in \mathfrak{l}_{\gamma}}\hat j_{\gamma}\star \rho_{\lambda}\ ] ] where we used that @xmath1251 . we then split the sum for @xmath1378 in a sum over @xmath1311 plus the sum over @xmath1379 . in this last sum we add and subtract @xmath24 thus getting @xmath1380 and @xmath1381 . from it follows that in order to conclude the proof of theorem [ thm:2.14 ] , namely of , we need to show that @xmath1382 that are in @xmath1383 $ ] is bounded by @xmath1384}{\ell^d_+}\frac{\ell_+^{d-1 } } { ( { \gamma}^{-1})^{d-1}}\frac{({\gamma}^{-1})^d}{\ell_-^d}\le c\,\delta_{\rm{in}}^{\ell_+}[{\lambda}]\,{\gamma}^{(1-\alpha_-)d + \alpha_+}\ ] ] from we then get @xmath1385 while yields @xmath1386 in order to estimate @xmath1387 we observe that it is equal to the expectation of @xmath1388 } \{e^{\rm { mf}}(j_{\gamma}*[q_{\lambda}\cup \chi^{(k)}_{{\lambda}^c}])- e^{\rm { mf}}(\rho^{(k)})\ } \\ & \quad + \int_{\delta_{\rm out}^{\ell_+}[{\lambda } ] \cup \delta_{\rm in}^{\ell_+}[{\lambda } ] } \{e^{\rm { mf.}}(\hat j_{\gamma}*\chi^{(k)}_{{\lambda}^c})- e^{\rm { mf } } ( j_{\gamma}*\chi^{(k)}_{{\lambda}^c } ) \}\end{aligned}\ ] ] by , the expectation of the first term is bounded by @xmath1389 , while shows that the expectation of the second term is bounded by @xmath1390 so @xmath1391 . in this appendix we will prove lower and upper bounds on the weights @xmath1392 defined in , see lemma [ lemma1 ] and lemma [ lemma2 ] below . these results are quite general thus their proof is equal to the one for the lmp model given in @xcite . we also give bounds on the energy in lemma [ lemma3 ] below . the subsets of @xmath601 that we consider here are all bounded @xmath1393 measurable regions . we will often drop the dependence on @xmath12 and @xmath1394 when no ambiguity may arise , thus calling @xmath1395 . we extend the definition by setting for @xmath1396 and @xmath1397 , @xmath1398 observe that , since sp@xmath1399 if and only if sp@xmath1400\ne \emptyset$ ] , then @xmath1401 , hence indeed extends the definition . .5 cm [ lemma1 ] [ lower bounds ] for any @xmath1402 @xmath1403 @xmath1404 @xmath1405 .5 cm * proof . * see lemma 11.1.1.1 in @xcite . .5 cm [ lemma2 ] [ upper bounds ] @xmath1406 , @xmath1407 , is a non decreasing function of @xmath1408 , namely @xmath1409 and for any @xmath1410 , @xmath1411 moreover there is a constant @xmath1298 such that @xmath1412 for any @xmath285 @xmath1413 @xmath1414|/\ell_+^d}\ ] ] finally , @xmath1415^ n\big)^{|{\lambda}|/\ell_+^d}\ ] ] .5 cm * proof . * see lemma 11.1.1.2 of @xcite .5 cm we now give bounds on the energy . .5 cm [ lemma3 ] let @xmath659 be as in . there is @xmath1153 such that for any @xmath446 such that @xmath1416 , for all @xmath235 , and @xmath984 and for any particle configuration or density function @xmath140 such that @xmath1417 , @xmath1418 ( in particular if @xmath1419 and @xmath1420 for some @xmath76 ) , @xmath1421 if also @xmath1422 is such that @xmath1423 , @xmath1418 , then @xmath1424 finally for all @xmath446 , @xmath140 and all @xmath12 , @xmath1425 * proof . * first notice that for any @xmath1426 @xmath1427 fix @xmath1428 , since there are at most @xmath1429 cubes in @xmath683 at distance @xmath576 from @xmath54 , we have @xmath1430 thus recalling we have that @xmath1431 } \{e_{\lambda}^{\rm mf } ( j_{\gamma}*(q_{\lambda}\cup\bar q_{{\lambda}^c}))-e_{\lambda}^{\rm mf } ( j_{\gamma}*\bar q_{{\lambda}^c})\}\big| \\&&\hskip2 cm \le c 3^d\|j\|_\infty \rho_{\max}\,|{\lambda}| \end{aligned}\ ] ] thus proving with a constant @xmath322 independent of @xmath12 if @xmath1432 . analogously we have @xmath1433 \cup \delta_{\rm out}^{{\gamma}^{-1}}[{\lambda } ] } \big| \{e^{\rm mf } ( j_{\gamma}*(q_{\lambda}\cup\bar q_{{\lambda}^c}))-e^{\rm mf } ( j_{\gamma}*\bar q_{{\lambda}^c})\ } \\&&\hskip1 cm -\{e^{\rm mf } ( j_{\gamma}*(q_{\lambda}\cup\bar q'_{{\lambda}^c}))-e^{\rm mf } ( j_{\gamma}*\bar q'_{{\lambda}^c})\ } \big| \le c ' |\partial { \lambda}|{\gamma}^{-1 } \end{aligned}\ ] ] finally notice that for @xmath1434 and @xmath1435 , we can write @xmath1436 thus proving . . in this appendix we prove theorem [ plabeta ] . the proof is the same as the analogous one for the lmp model in subsection 11.7 of @xcite . the latter in fact is based on bounds on the energy and on the weights of the contours which are the same as those proved in appendix [ app : bounds ] . existence of pressure when the phase space is non compact it is not an easy problem in general , the simplifying feature in the lmp and the potts hamiltonian being the bound @xmath1438 uniform on the boundary conditions , see , which in general can not be expected to hold . in this way the problem is essentially reduced to the case of compact spins . with the bounds proved in appendix [ app : bounds ] the contours weights are also easily controlled , the argument is standard in statistical mechanics . let @xmath1439 be the configuration obtained from @xmath118 by interchanging spin @xmath1440 and spin @xmath76 , leaving all the other spins and all the positions unchanged . by the symmetry of the hamiltonian @xmath1441 and the jacobian @xmath1442 . moreover @xmath1443 @xmath1444 one - to - one and onto and @xmath1445 . then @xmath1446 , hence the thesis as we have already proved independence on the boundary conditions . .5 cm by the bounds in appendix [ app : bounds ] we reduce to the same setup as in lmp and the proof becomes the same as in subsection 11.7.3 of @xcite . notice that the dependence on @xmath12 is explicit in the hamiltonian but also implicit in the contours weights . the dependence on the former is differentiable while the dependence of the cutoff weights on @xmath12 is only proved to be continuous . the whole argument is quite standard . by and @xmath1448 by theorem [ propa.2 ] @xmath1449 postponing the proof that @xmath1450 we get @xmath1451 choose @xmath41 as a cube of side @xmath1452 , then @xmath1453 and follows letting @xmath422 . it thus remains to prove that for any @xmath1454 and @xmath82 small enough , @xmath1450 . the proof is taken from proposition 11.1.4.1 in @xcite . call @xmath188 the function equal to @xmath602 on @xmath41 and to @xmath1455 on @xmath1456 . since @xmath1457 , @xmath1458 , @xmath1459 we can write the integral of the sum as the sum of the integrals and in the integral with @xmath1460 we can replace @xmath1461 by @xmath1462 . then @xmath1463 becomes @xmath1464 since @xmath1465 , for all @xmath82 small enough the first curly bracket is minimized by setting @xmath1466 ; the second curly bracket by convexity is non negative and vanishes when @xmath1466 ; the third one is independent of @xmath602 , hence @xmath1450 . the proof of follows because : @xmath1467 is continuous and there is @xmath677 such that for all @xmath82 small enough @xmath1468 holds because : @xmath1469 . by and the smoothness of @xmath1470 , there is @xmath1471 such that for any @xmath1150 and all @xmath82 correspondingly small , @xmath1472\ ] ] hence @xmath1473 . * acknowledgments * one of us ( yv ) acknowledges very kind hospitality at the math . depts . of roma torvergata and laquila , partially supported by prin @xmath1474 and @xmath1475 , grefi - mefi ( gdre 224 cnrs - indam ) , cpt ( umr 6207 ) and universit de la mditerrane . i.m . acknowledges partial support of the gnfm young researchers project statistical mechanics of multicomponent systems " .

this is the second of two papers on a continuum version of the potts model , where particles are points in @xmath0 , @xmath1 , with a spin which may take @xmath2 possible values . particles with different spins repel each other via a kac pair potential of range @xmath3 , @xmath4 . in this paper we prove phase transition , namely we prove that if the scaling parameter of the kac potential is suitably small , given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist @xmath5 mutually distinct dlr measures .

recently , the lhcb collaboration has reported on the observation of two hidden - charm pentaquark states in @xmath5 decays , the @xmath6 with a mass of @xmath7 mev and a width of @xmath8 mev and the @xmath0 with a mass of @xmath9 mev and a width of @xmath10 mev @xcite . the preferred @xmath11 assignments are of opposite parity . the best fit yields spin - parity @xmath11 values of @xmath12 , but other possibilities , either @xmath13 or @xmath14 , are also acceptable . the decay branching ratios @xmath15 and @xmath16 have been measured as well @xcite . in the past decade , many mesonic exotic states have been observed experimentally and many of them clearly contain more than the minimum quark content dictated by the naive quark model , such as the charged @xmath17 @xcite and @xmath18 @xcite states . the @xmath19 states are the first exotic states observed in the heavy - flavor baryonic sector . the observation of the @xmath19 states has aroused a lot of interest in the theoretical community . they have been studied in various frameworks , such as the molecular picture @xcite , the diquark picture @xcite , the qcd sum rules @xcite , and the soliton model @xcite . on the other hand , questions have been raised regarding whether the observed enhancement could be due to kinematical effects or singularities @xcite . in a recent publication , it was suggested that the existence of exotic @xmath20 states can imitate broad bumps in the @xmath21 invariant mass distributions and thus affects the interpretation of the @xmath6 @xcite . further discussions on the nature of the two @xmath19 states can be found in refs . @xcite . one should note that even before the lhcb announcement , the existence of hidden - charm pentaquark states have been explored both in the molecular picture @xcite and in the quark models @xcite ) . the discovery potential of such states has been explored in @xmath22 @xcite and @xmath23 @xcite induced reactions . it is clear that at this moment , both the experimental and theoretical studies are not yet conclusive . experimentally , the ambiguities in the spin - parity assignments should be clarified . theoretically , different interpretations are often not consistent with each other , not only in terms of the dominant fock components of these states but also in terms of the predicted or obtained spin - parities . the only way to clarify the situation , given the rather large statistics already achieved by lhcb , is to study complementary reactions or decay modes where the @xmath19 states or their counterparts ( predicted by various models ) can be observed . in refs . @xcite , photoproduction of the @xmath19 states off a proton target have been studied . in ref . @xcite , assuming the @xmath19 states are genuine pentaquark states belonging to either an octet or a decuplet representation , the decays of @xmath24 , @xmath25 , and @xmath26 into a pentaquark state and a pseudoscalar meson have been studied . the decays of @xmath27-baryons into a pentaquark state and a pseudoscalar meson have also been examined in the diquark model @xcite . experimental studies of either the photoproductions or the decay modes of @xmath27-baryons into all available final states will definitely help us better understand the nature of the pentaquark states . one should note that most theoretical approaches predicted the existence of the @xmath19 counterparts . in particular , in the unitary approach of ref . @xcite that in addition to an isospin 1/2 and strangeness zero state , two more states are predicted in the isospin zero and strangeness @xmath28 sector . if the @xmath0 state corresponds to the non strange state(s ) as shown in ref . @xcite , there is good reason to believe that its strange counterparts exist as well . the hidden - charm pentaquark states with strangeness can decay into @xmath4 . therefore , in the present work , we propose to study the decay of the @xmath29 state into @xmath30 . the mechanism of this reaction resembles that of the @xmath31 suggested in ref . @xcite , which can naturally explain the lhcb data using the input from the unitary model of refs . the experimental observation or invalidation of the existence of such a state can help clarify the nature of the @xmath19 states and improve our understanding of low - energy strong interactions . in this section , we describe the weak decay process of @xmath32 . following the formalism first proposed in ref . @xcite and also used to study the @xmath5 decay @xcite , we can separate the decay of the @xmath26 into two steps , weak decay and hadronization , and final state interactions . at the quark level , the decay of @xmath33 is depicted in fig . [ fig : mechanism ] . the quark content of @xmath26 is @xmath34 , and the @xmath35 and @xmath36 quarks are in a state of spin zero . to have the color degree antisymmetric , the flavor part of the wave function should be antisymmetric with respect to @xmath35 and @xmath36 . as a result , the @xmath26 wave function can be written as @xmath37 in the second equality , we have adopted a simplified notation for the wave functions . in fig . [ fig : mechanism ] , the @xmath27 quark first decays into a @xmath38 quark by emitting a @xmath39 meson , then the @xmath39 translates into a pair of @xmath40 and @xmath36 quarks , which is cabibbo favored . the pair of @xmath41 hadronizes into the @xmath42 , while the @xmath36 quark picks up an anti - quark from the vacuum to form a @xmath43 meson or an @xmath44 meson , the spectator pair @xmath45 then hadronizes into a baryon with the remaining quark from the vacuum . in the following , we have to find out how the quark combinations @xmath46 hadronizes into a pair of ground state meson and baryon . the hadronization into a meson baryon pair can be achieved by replacing the @xmath47 matrix in su(3 ) flavor space with its counterpart @xmath48 using hadronic degrees of freedom , namely @xmath49 where we have used standard @xmath50 mixing @xcite and have neglected the @xmath51 because of its heavy mass . with such a replacement , we obtain @xmath52+\bar{k}^0 \left [ \frac{1}{\sqrt{2}}d(ds - sd)\right]\nonumber\\ & & -\frac{\eta}{\sqrt{3}}\left [ \frac{1}{\sqrt{2}}s(ds - sd)\right].\end{aligned}\ ] ] the combinations of three quarks can be written in terms of the ground - state baryon wave functions with a bit of algebra @xcite @xmath53 where we have introduced the isospin 1/2 combination @xmath54 , @xmath55 , and @xmath56 . the process described above corresponds to the tree - level feynman diagram of fig . [ fig : decaymodel](a ) . . , scaledwidth=45.0% ] other hadronization processes , for instance , that of the @xmath35 or @xmath36 quark of the @xmath26 hardronizing into a meson is penalized compared to the one described above , because of the involvement of large momentum transfer @xcite . decay : ( a ) direct @xmath57 vertex at tree level , ( b ) final state interaction of @xmath58 , and ( c ) final state interaction of @xmath59.,scaledwidth=45.0% ] in addition to the tree level diagram [ fig . [ fig : decaymodel](a ) ] , we need to take into account the final state interactions of these meson - baryon pairs , which are known to be very strong and are depicted in fig . [ fig : decaymodel](b ) and ( c ) . the amplitude @xmath60 for the transition can be written as , @xmath61 \ , \nonumber \\ & = & v_p\left ( h_{\bar k\lambda } + t_{\bar k\lambda } + t_{j/\psi\lambda}\right ) , \label{eqn : fullamplitude}\end{aligned}\ ] ] where @xmath62 expresses the hadronization strength , and @xmath63 ( the channel indices @xmath64 ) denotes the one - meson - one - baryon loop function , chosen in accordance with the unitary model for the scattering matrix @xmath65 that will be described in the following subsection . @xmath66 and @xmath67 are the invariant masses of the final states @xmath68 and @xmath4 , respectively , and @xmath69 stands for the weights of the transition , which are given by eq . ( [ eq : hadronization ] ) , @xmath70 final state interactions between a ground state octet baryon and a pseudoscalar meson in the strangeness @xmath71 and isospin @xmath72 channel has been studied in detail in the unitary model of ref . @xcite . in this work , a pole is found on the complex plane and identified as the experimentally observed @xmath73 . a subsequent work along the same lines showed that in addition to the @xmath73 state also the @xmath74 was generated @xcite . in the present work , we choose this approach to describe the interactions among the coupled channels @xmath75 , @xmath76 , and @xmath77 . in the unitary approach of ref . @xcite , the transition amplitudes are written as @xmath78^{-1}v , \label{eq : tvg}\ ] ] where the matrix @xmath79 is obtained from the lowest order meson baryon chiral lagrangian , @xmath80 where the magnitudes @xmath81 and @xmath82 are the energy and mass of the baryon in channel @xmath83 , and the coefficients @xmath84 are shown in table [ tab : cij ] . the optimal choice for the decay constant @xmath85 is used , and @xmath86 mev @xcite . .coefficients @xmath84 of the meson baryon amplitudes for isospin @xmath3 ( @xmath87 ) @xcite . [ cols="<,>,>,>,>",options="header " , ] [ tab : cij ] the ( diagonal ) matrix @xmath88 in eq . ( [ eqn : fullamplitude ] ) and eq . ( [ eq : tvg ] ) accounts for the loop integral of a meson and a baryon propagator and depends on the regularization scale , @xmath89 , and a subtraction constant for each channel , @xmath90 , that comes from a subtracted dispersion relation . the analytical expression of @xmath88 is given as follows @xcite , @xmath91 \right\ } \ , \label{eq : gpropdr}\end{aligned}\ ] ] where @xmath92 and @xmath93 are the masses of baryon and meson in the @xmath94-th channel , and the regularization scale @xmath95 mev @xcite and the subtraction constant @xmath96 , @xmath97 , @xmath98 and @xmath99 ) in table 2 of ref . @xcite gives similar results as our choice . ] are used for @xmath100 , @xmath75 , @xmath76 , and @xmath77 channels . for the @xmath4 channel , we take @xmath101 mev and @xmath102 @xcite . , @xmath103 , @xmath104 , and @xmath105 as a function of the invariant mass of @xmath100 , @xmath75 , @xmath76 , @xmath77 system : ( a ) @xmath106 for the channels of @xmath100 , @xmath75 , @xmath76 , and @xmath77 , and ( b ) the same as ( a ) but with @xmath107 and @xmath97 . the insets show the magnified @xmath108 and @xmath109 amplitudes . , title="fig:",scaledwidth=45.0% ] , @xmath103 , @xmath104 , and @xmath105 as a function of the invariant mass of @xmath100 , @xmath75 , @xmath76 , @xmath77 system : ( a ) @xmath106 for the channels of @xmath100 , @xmath75 , @xmath76 , and @xmath77 , and ( b ) the same as ( a ) but with @xmath107 and @xmath97 . the insets show the magnified @xmath108 and @xmath109 amplitudes . , title="fig:",scaledwidth=45.0% ] assuming the existence of the strangeness counterpart of the @xmath0 as shown in refs . @xcite and following the approach proposed in ref . @xcite to describe the lhcb data , we can parameterize the transition matrix element for @xmath4 in eq . ( [ eqn : fullamplitude ] ) as @xmath110 in refs . @xcite , two states are found in the strangeness @xmath28 and isospin 0 channel with the following pole positions @xmath111 and @xmath112 . these numbers are obtained without any fine tuning . if now we assume that one of these states corresponds to the strange counterpart of the @xmath0 and use the experimental measurement of its mass as a reference , we can imagine that the counterpart of the @xmath0 , @xmath113 , should appear at @xmath114 , where @xmath115 can be estimated using either the @xmath116 or @xmath117 mass difference , which are 175 and 257 mev , respectively . as an rough estimate , one can take @xmath118 mev and obtain @xmath119 mev . as for @xmath120 , it should be of order 10 mev @xcite , and therefore we take @xmath121 mev . for the coupling @xmath122 , we use a value of @xmath123 , as given in ref . @xcite . finally , the invariant mass distribution of the process @xmath124 reads @xmath125 where @xmath67 and @xmath66 are the invariant masses of @xmath4 and @xmath75 . for a given value of @xmath67 , the range of @xmath126 is defined as , @xmath127 where @xmath128 and @xmath129 are the energies of @xmath130 and @xmath43 in the @xmath4 rest frame . similar formulas are obtained for the range of @xmath131 when we fix @xmath66 . invariant mass distributions for @xmath132 .,scaledwidth=45.0% ] in this section , we present our results for the process @xmath133 . first , we show the absolute value of the transition amplitudes @xmath134 for the @xmath100 , @xmath75 , @xmath76 , and @xmath77 in @xmath3 and @xmath2 in fig . [ fig : modulet](a ) . we also show the corresponding results with @xmath107 and @xmath97 in fig . [ fig : modulet](b ) . the results for both choices look very similar . the @xmath73 can be clearly seen in the @xmath100 invariant mass distributions , and @xmath75 distributions in fig . 3(b ) , but not so prominently in fig . 3(a ) . as a result , we anticipate that an experimental study of the @xmath135 can yield valuable information on the poorly known @xmath73 as well . next , we predict the @xmath4 invariant mass distribution for the @xmath132 decay in fig . [ fig : dwidth ] , up to an arbitrary normalization ( @xmath136 ) . the red solid line shows the result without the @xmath4 interaction [ only the term @xmath137 of eq . ( [ eqn : fullamplitude ] ) ] , and the blue dashed - dotted line stands for the result of our full model . we observe a prominent structure around 4650 mev on top of the background when the @xmath4 interaction is taken into account . a variation of @xmath138 shifts the peak position accordingly , but a clear signal can still be observed . furthermore , as long as the width is smaller than 100 mev , experimental observation should not be too difficult . we stress however that the strength of the signal will depend strongly on the coupling of the hidden charm state to the @xmath59 , i.e. , @xmath122 . a much smaller value of the coupling will diminish the signal as naively expected . therefore , indeed an experimental study of the decay mode we propose can help to verify or disprove the unitary approach for this particular case . in fig . [ fig : dwidth2 ] , we show the invariant mass distribution of the @xmath26 decay as a function of the @xmath75 invariant mass . it is seen that the @xmath4 final state interaction does not affect much the predicted shape . however , the two cusps reflecting the @xmath76 and @xmath139 thresholds can be easily recognized . the strong enhancement around @xmath140 gev can be identified with the @xmath74 as in ref . @xcite . depending on the particular parameter set for the @xmath75 interaction , the @xmath73 is also visible . it is to be noted that the decay mechanism of the present process is the same as that of the @xmath5 @xcite . in both decays , the involved ckm matrix element is the product of @xmath141 . therefore , as a crude estimate , we would like to guess that the decay rate of the @xmath135 is of the same order of magnitude as that of @xmath142 , neglecting the difference in phase space and final state interactions . this is somehow consistent with the study of ref . @xcite , where it is found that @xmath143 , where @xmath144 denotes a pentaquark state having the same light quark composition as that of the proton or @xmath130 , while the first number is for spin 3/2 and that in the parenthesis for spin 5/2 . therefore , both studies show that the decay mode proposed in the present work should and can be studied at lhcb . we have proposed to study the @xmath145 decay to measure a hidden - charm pentaquark state with strangeness , predicted to exist in the unitary approach . this model has predicted the existence of two non - strange hidden - charm pentaquark states in the energy region where the @xmath0 has been seen . the decay mechanism we employed has been previously adopted to successfully describe the lhcb @xmath5 invariant mass distributions . our study showed that the strange hidden - charm pentaquark state can be clearly seen on top of the background . given the fact that both the unitary model and the reaction mechanism have been tested in describing the lhcb @xmath146 decay , we strongly encourage our experimental colleagues to study the @xmath1 decay proposed here , which can be very helpful to test the existence of the pentaquark states and their nature . one of us , e. o. , wishes to acknowledge support from the chinese academy of science ( cas ) in the program of visiting professorship for senior international scientists ( grant no . 2013t2j0012 ) . l.s.g thanks the institute for nuclear theory at university of washington for its hospitality and the department of energy for partial support during the completion of this work . this work is partly supported by the national natural science foundation of china under grant nos . 11475227 , 1375024 , 11522539 , 11505158 , 11475015 , and 11165005 , the open project program of state key laboratory of theoretical physics , institute of theoretical physics , chinese academy of sciences , china ( no.y5kf151cj1 ) , the spanish ministerio de economia y competitividad and european feder funds under the contract number fis2011 - 28853-c02 - 01 and fis2011 - 28853-c02 - 02 , and the generalitat valenciana in the program prometeo ii-2014/068 . we acknowledge the support of the european community - research infrastructure integrating activity study of strongly interacting matter ( acronym hadronphysics3 , grant agreement n. 283286 ) under the seventh framework programme of eu . 99 r. aaij _ et al . _ [ lhcb collaboration ] , phys . lett . * 115 * , 072001 ( 2015 ) [ arxiv:1507.03414 [ hep - ex ] ] . r. aaij _ et al . _ [ lhcb collaboration ] , arxiv:1509.00292 [ hep - ex ] . s. k. choi _ et al . _ [ belle collaboration ] , phys . * 100 * , 142001 ( 2008 ) [ arxiv:0708.1790 [ hep - ex ] ] . k. chilikin _ et al . _ [ belle collaboration ] , phys . d * 88 * , no . 7 , 074026 ( 2013 ) [ arxiv:1306.4894 [ hep - ex ] ] . r. aaij _ et al . _ [ lhcb collaboration ] , phys . * 112 * , no . 22 , 222002 ( 2014 ) [ arxiv:1404.1903 [ hep - ex ] ] . m. ablikim _ et al . _ [ besiii collaboration ] , phys . lett . * 110 * , 252001 ( 2013 ) [ arxiv:1303.5949 [ hep - ex ] ] . z. q. liu _ et al . _ [ belle collaboration ] , phys . lett . * 110 * , 252002 ( 2013 ) [ arxiv:1304.0121 [ hep - ex ] ] . t. xiao , s. dobbs , a. tomaradze and k. k. seth , phys . b * 727 * , 366 ( 2013 ) [ arxiv:1304.3036 [ hep - ex ] ] . 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assuming that the recently observed hidden - charm pentaquark state , @xmath0 , is of molecular nature as predicted in the unitary approach , we propose to study the decay of @xmath1 to search for the strangeness counterpart of the @xmath0 . there are three ingredients in the decay mechanism : the weak decay mechanism , the hadronization mechanism , and the finite state interactions in the meson - baryon system of strangeness @xmath2 and isospin @xmath3 and of the @xmath4 . all these have been tested extensively . as a result , we provide a genuine prediction of the differential cross section where a strangeness hidden - charm pentaquark state , the counterpart of the @xmath0 , can be clearly seen . the decay rate is estimated to be of similar magnitude as the @xmath5 observed by the lhcb collaboration .

neutron beta decay provides one of the most sensitive means for exploring details and limits of our understanding of the weak interaction . thanks to its highly precise theoretical description @xcite , neutron decay is sensitive to contributions from processes not included in the standard model ( sm ) of particles and interactions ( for comprehensive reviews see refs . @xcite ) . neglecting recoil , radiative and loop corrections , the differential decay rate for unpolarized neutrons is given by parameters @xmath0 and @xmath1 : @xmath5 , where @xmath6 , @xmath7 , @xmath8 and @xmath9 are the electron momentum , energy , and e@xmath10 opening angle , respectively @xcite . the e@xmath11 correlation parameter @xmath0 , and the asymmetry parameters with respect to the neutron spin : @xmath12 ( beta ) , @xmath13 ( neutrino ) , and @xmath14 ( proton ; @xmath15 in leading order ) possess complementary dependencies on the ratio of fermi constants @xmath16 , as well as on operators that depart from the @xmath17 form of the sm charged current ( cc ) weak interaction . additionally , @xmath1 , the fierz interference term , offers an independent test of scalar and tensor admixtures arising in broad classes of l - r mixing susy extensions . thus precise measurements of neutron decay parameters offer the distinct advantage of overconstrained independent checks of the sm predictions , as well as the potential for indicating or ruling out certain types of extensions to the sm @xmath17 form @xcite . hence , a set of appropriately precise measurements of the neutron decay parameters @xmath0 , @xmath1 , @xmath12 , and @xmath13 will have considerably greater physics implications than the erstwhile predominant experimental focus on @xmath12 , i.e. , @xmath18 . at a minimum , such a data set combined with new measurements of the neutron lifetime , @xmath19 , will enable a definitive resolution of the persistent discrepancies in @xmath18 and cabibbo kobayashi - maskawa ( ckm ) matrix element @xmath20 @xcite . the nab collaboration @xcite has undertaken to carry out precise measurements of @xmath0 , the e@xmath11 correlation parameter , and @xmath1 , the so far unmeasured fierz interference term , in neutron decay . goal accuracies are @xmath21 and @xmath22 . a novel @xmath23 field - expansion spectrometer based on ideas outlined in ref . @xcite will be used in the fundamental neutron physics beamline ( fnpb ) at the spallation neutron source ( sns ) at oak ridge , tennessee . the nab experiment constitutes the first phase of a program of measurements that will continue with second - generation measurements of spin correlations in neutron decay . the next experiment , named ` abba ' , will measure parameters @xmath12 and @xmath13 in addition to @xmath0 and @xmath1 . in addition , the proton asymmetry @xmath14 will be measured with the same apparatus . together , nab and abba form a complete program of measurements of the main neutron decay parameters in a single apparatus with shared systematics and consistency checks . the experiments are complementary : nab is highly optimized for the measurement of @xmath0 and @xmath1 , while abba focuses on @xmath12 and @xmath13 with a lower - precision consistency check of the @xmath0 and @xmath1 parameters . nab joins two existing experiments , aspect @xcite and acorn @xcite , which also study @xmath0 . the correlation parameter of interest , @xmath0 , measures the dependence of the neutron beta decay rate on the cosine of the e@xmath10 relative angle . the nab method of determination of @xmath0 relies on the linear dependence of @xmath24 on @xmath25 , the square of the proton momentum for a given electron momentum ( or energy ) . conservation of momentum gives the relation @xmath26 to to where , to a very good approximation , @xmath27 depends only on @xmath28 ( or @xmath29 ) . thus , eq . [ eq : psq_cos_enu ] reduces to a linear relation between @xmath30 and @xmath25 for a fixed @xmath29 . the mapping of @xmath30 and @xmath25 is shown graphically in fig . [ fig : proton_ps ] . in this plot , the phase space alone distributes proton events evenly in @xmath25 between the lower and upper bounds for any fixed value of @xmath31 . given the linear relationship between @xmath25 and @xmath30 , the slope of the @xmath25 probability distribution is determined by the correlation parameter @xmath0 ; in fact it is given by @xmath32 , where @xmath33 ( see fig . [ fig : resp_slope ] ) . this observation leads to the main principle of measurement of @xmath0 which involves measurement of the proton momenta via the proton time of flight ( tof ) , @xmath34 , in a suitably constructed magnetic spectrometer . ideally , the magnetic field longitudinalizes the proton momentum and @xmath35 ; @xmath34 is measured as the difference between the arrival times of the electron and the proton at the detector(s ) . in the present discussion we neglect the electron tof . parameter @xmath0 is determined from the slopes of the @xmath36 distributions for different values of @xmath37 . if @xmath0 were null , all distributions would have a slope of zero . having multiple independent measurements of @xmath0 for different electron energies provides a powerful check of systematics , as discussed below . the fierz interference term @xmath1 is determined from the shape of the measured electron energy spectrum . for fixed @xmath8 , a perfect spectrometer would record a trapezoidal distribution of @xmath36 with sharp edges . the precise location of these edges is determined by well - defined kinematic cutoffs that only depend on @xmath8 . however , a realistic time - of - flight spectrometer will produce imperfect measurements of the proton momenta due to the spectrometer response function , discussed in sect . [ sec : uncerts ] . the measured locations and shapes of edges in @xmath36 distributions will allow us to examine the spectrometer response function and verify that the fields have been measured correctly . the main requirements on the spectrometer are : 1 . the spectrometer and its magnetic ( @xmath38 ) and electric ( @xmath39 ) fields are designed to be azimuthally symmetric about the central axis , @xmath40 . 2 . neutrons must decay in a region of large @xmath38 . the resulting protons and electrons spiral around a magnetic field line . an electric field is required to accelerate the proton from the ev - range energies to a detectable energy range prior to reaching the detector . this field imposes , however , an energy threshold on e@xmath41 detection . 4 . the proton momentum must rapidly become parallel to the magnetic field direction to ensure that the proton time of flight @xmath42 . this requirement dictates a sharp field curvature ( @xmath43 ) at the origin , followed by a sharp falloff of @xmath44 . the basic concept of the spectrometer consists of collinear solenoids with their longitudinal axis oriented normal to the neutron beam , which passes through the solenoid center . the solenoidal magnetic field starts out high at the position of the neutron beam , typically 4 t , dropping off quickly to parallelize the momenta as protons enter the long `` drift '' region . in the detection region at either end of the solenoid the field is increased to 1/4 of its central peak value . cylindrical electrodes ( consisting of three sections ) maintain the neutron decay region at a potential of + 30kv with respect to the ends of the solenoid where detectors are placed at ground potential . the magnetic field strength is sufficiently high to constrain both electrons and protons from neutron decay to spiral along the magnetic field lines with the component of the spiral motion transverse to the field limited by cyclotron radii of the order of a few millimeters . ) and magnetic field ( @xmath13 ) profiles on axis for 1/2 of the nab spectrometer length.,title="fig : " ] ) and magnetic field ( @xmath13 ) profiles on axis for 1/2 of the nab spectrometer length.,title="fig : " ] hence , two segmented si detectors , one at each end of the solenoid , view both electrons and protons in an effective @xmath23 geometry . the time of flight between the electron and proton is accurately measured in a long , @xmath45 meter , drift distance . the electron energy is accurately measured in the si detectors . the proton momentum and electron energy determine the electron neutrino opening angle . we note that by sorting the data on proton time of flight and electron energy , @xmath0 can be determined with a statistical uncertainty that is only 4% greater than the theoretical minimum @xcite . , shown for different proton momenta , the magnetic field from fig . [ fig : spect_1 ] and a centered neutron beam with a width of 2 cm . the calculation assumes full adiabaticity of the proton motion . ] a not - to - scale schematic view of the field expansion spectrometer is shown in fig . [ fig : spect_1 ] . electrons and protons spiral around magnetic field lines and are guided to two segmented si detectors , each having a @xmath46100@xmath47 active area , and depicted schematically in fig . [ fig : det_sketch ] . in the center of the spectrometer the axial field strength is 4 t , in the drift region 0.1 t , and near the si detectors 1 t ( see fig . [ fig : spect_1 ] ) . in a realistic spectrometer , however , the perfect one - to - one correspondence of proton momentum and time of flight is lost , due to imperfect momentum longitudinalization and other systematic effects , such as the lateral size of the neutron beam in the decay region . in other words , the detector response function instead of being a delta function in @xmath36 for each value of @xmath25 , becomes a broadened function , such as the ones calculated for three proton momenta and depicted in fig . [ fig : nabspectrometerresponse ] . the key challenge of the nab approach to measuring @xmath0 is to minimize the width of the detector response function while keeping the relevant systematics under control . the resulting tof distributions no longer have sharply cut off edges as in fig . [ fig : resp_slope ] . a sample set of results of geant4 @xcite monte carlo calculations for three electron energies is shown in fig . [ fig : realspec ] . , for electron kinetic energies @xmath48 , 500 and 700kev , generated in a realistic geant4 monte - carlo simulation using the @xmath38 field from fig . [ fig : spect_1 ] and a centered neutron beam with a width of 2 cm . ] strictly speaking , determining @xmath1 requires detecting only the electron and reliably measuring its kinetic energy . nevertheless , there are a number of challenges associated with this measurement , commented on in the following section . the statistical sensitivity of our measurement method is primarily determined by the spectrometer acceptance and imposed energy and tof restrictions . the statistical uncertainties for our measurements of the @xmath0 and @xmath1 parameters in neutron decay are listed in tab . [ tab : stat_unc ] , reflecting the dependence on @xmath49 , the electron kinetic energy detection threshold , and @xmath50 , the maximum proton tof accepted . additionally , the electron energy calibration @xmath51 and the precise length @xmath52 of the low - field drift region represent important sources of systematic uncertainty . thus , parallel analyses will be performed keeping @xmath51 and @xmath52 free , in order to study and remove their systematic effects . table [ tab : stat_unc ] shows that the reduction in statistical sensitivities under these conditions is modest . .top : statistical uncertainties @xmath53 for the e-@xmath10 correlation parameter @xmath0 . a perfect spectrometer would obtain @xmath54 . bottom : statistical uncertainties @xmath55 for the fierz interference term @xmath1 . [ cols="<,^,^,^,^ " , ] + @xmath56 with @xmath51 and @xmath52 variable . @xmath57 with @xmath51 and @xmath0 variable . [ tab : stat_unc ] the calculated fnpb neutron decay rate under sns full - power conditions of @xmath4619.5/(@xmath58s ) , and with the nab fiducial decay volume of 20@xmath58 , yields @xmath46400 detected decays / sec @xcite . in a typical 10-day run of @xmath59s of net beam time we would achieve @xmath60 and @xmath61 . since we plan to collect several samples of @xmath62 events in several 6-week runs , the overall nab accuracy will not be statistics - limited . controlling the measurement systematics presents by far the greatest challenge in the nab experiment . the most basic task is to specify the spectrometer fields with precision sufficient for an accurate determination of the spectrometer response function @xmath63 . we have adopted two methods of addressing this problem . in the first approach ( method a ) , we determine the shape of the spectrometer response function from theory , leaving several parameters free , to be determined by fits to the measured spectra . the second approach ( method b ) relies on obtaining the detection function with its uncertainties _ a priori _ from a full description of the neutron beam and electromagnetic field geometry . subsequently , the experimental data are fitted with only the physics observables as free parameters . below we summarize some of the main challenges along with strategies for their control at the required level . a much more detailed discussion of both methods and the experimental challenges is given in the nab experiment proposal @xcite . * _ precise specification of the neutron beam profile : _ a mere 100@xmath64 m shift of the beam center induces @xmath65% . however , this effect cancels when averaging over the two detectors on opposite sides of the solenoid ; measuring a nonzero up - down proton counting asymmetry pins it down sufficiently . * _ magnetic field map : _ the field expansion ratio defined as @xmath66 must be controlled at the level of @xmath67 in order to keep @xmath3 under @xmath2 . this will be mapped out using a calibrated hall probe . field curvature must be determined with an accuracy of @xmath68 in dedicated measurements . average mapping accuracy @xmath69 must be kept below @xmath70 . * _ flight path length : _ an uncertainty of order @xmath71 m results in @xmath3 at our limit . hence , @xmath52 will be kept as a fitting parameter . additionally , we will perform a consistency check by making differential measurement using segmented electrodes . * _ homogeneity of the electric field : _ electric potential will have satisfy stringent limits on inhomogeneities as discussed in the nab proposal @xcite . * _ rest gas : _ requires vacuum of @xmath72pa or better . * _ adiabaticity _ of the magnetic field configuration is not an absolute requirement . detailed monte carlo analysis has shown excellent efficiency of proton momentum longitudinalization for certain relatively non - adiabatic fields . however , an adiabatic design makes the evaluation of systematic errors simpler and more reliable . * _ doppler effect : _ adverse effects of the doppler effect will apparently be controlled sufficiently by the spectrometer design , but a thorough analysis will be made in conjunction with the final design . * _ detector alignment : _ the spectrometer imaging properties provide for a self - consistent calibration in the data . * _ electron energy calibration _ is required at the @xmath73 level . to achieve it we ll use radioactive sources , evaluate directional count rate asymmetries , and also leave it as a fitting parameter with acceptably small loss of statistical sensitivity ( see tab . [ tab : stat_unc ] ) . * _ trigger hermiticity _ is affected by the particle impact angle on the detector , backscattering , and tof cutoff ( planned in order to reduce accidental backgrounds ) . several consistency checks will be evaluated from the data to quantify and characterize the various aspects of trigger hermiticity . * _ tof measurement uncertainties : _ the requirement is @xmath74ps . while it is not necessary to reach this timing accuracy for each event , it has to be achieved for the event sample average , a realistic goal given the planned event statistics . * _ edge effects _ introduce important systematics . thanks to the imaging properties of the spectrometers , these can be controlled and corrected for to a sufficient degree with appropriate cuts on the data . + sources of uncertainties in the measurement of @xmath1 are fewer than for @xmath0 since accurate proton momentum measurement ( via its tof ) is not required . the dominant sources are electron energy calibration ( discussed above ) and electron backgrounds . * _ neutron beam related backgrounds _ are notoriously hard to calculate and model _ a priori _ , and will ultimately have to be measured and characterized _ in situ_. reasonable estimates place the beam - related background rates below the signal rate . while we have plans for shielding and lining surfaces with neutron absorbing @xmath75lif material , the coincident technique of detecting e p pairs helps to reduce substantially the effect of beam - related accidental backgrounds . * _ particle trapping : _ electrons can be trapped in the decay volume , expansion , and tof regions . these regions form an electrode - less penning trap . the potential well trap does not cause a problem for electrons above our energy threshold . the longitudinalization of the electron momentum due to the magnetic field allows all of them to escape and to reach the detector . low energy electrons from neutron decay , from field ionization or from rest gas interactions are a concern since trapped particles ionize the rest gas , and the ions form a time - dependent background . several strategies are under consideration to remove the trapped particles ; they will be refined under real running conditions . the nab collaboration plans simultaneous high - statistics measurements of neutron decay parameters @xmath0 , the e@xmath10 correlation coefficient , and @xmath1 , the fierz interference term , with @xmath21 and @xmath76 . development of the abba / nab si detectors is ongoing and remains a technological challenge . each of the target properties of the detector have been realized separately ; the remaining task is to realize them simultaneously in one piece of silicon . p. r. huffman , et al . , `` beamline performance simulations for the fnpb '' , private communication , 2005 . initial measurements performed in the fnpb up to the time of this writing are in agreement with the simulation .

the nab collaboration will perform a precise measurement of @xmath0 , the electron - neutrino correlation parameter , and @xmath1 , the fierz interference term in neutron beta decay , in the fundamental neutron physics beamline at the sns , using a novel electric / magnetic field spectrometer and detector design . the experiment is aiming at the @xmath2 accuracy level in @xmath3 , and will provide an independent measurement of @xmath4 , the ratio of axial - vector to vector coupling constants of the nucleon . nab also plans to perform the first ever measurement of @xmath1 in neutron decay , which will provide an independent limit on the tensor weak coupling . , , , , , , , , , , , , , , , , , , , , , , , , , neutron beta decay , correlations , precision measurement 13.30.-ce , 14.20.dh , 23.40.-s , 24.80.-y

the apollo - type near - earth asteroid ( 4179 ) toutatis was originally discovered on 10 february 1934 and remained a lost asteroid until it was once again detected by c. pollas and colleagues on 4 january 1989 in caussols , france . from a dynamical viewpoint , the asteroid moves on an approximately 4:1 resonant orbit at a large eccentricity with the earth and has passed through a close encounter with the earth every four years since 1992 @xcite . dating back to the decadal years of near - earth flybys of the asteroid , the radar observations obtained from arecibo and goldstone reveal that toutatis appears to be an irregularly shaped asteroid with two distinct lobes @xcite . various types of ground - based observations indicate that toutatis is a tumbling , non - principal axis ( npa)-rotating small body @xcite . these effects , as observed in earth - approaching flybys , have also been reported based on optical observations and extensive radar measurements @xcite . the first near - earth flyby for toutatis occurred in december 1992 at a distance of 0.242 au , when the asteroid again came into view . optical observations were gathered from at least 25 sites around the world through an international campaign . the observed rotational light curves of toutatis appeared to be highly unusual , with a large amplitude and a non - periodic long rotation period . subsequently , @xcite reported two major periods of complex rotation of approximately 7.3 and 3.1 days , as estimated from analysis of the data . in addition , they reported that toutatis was the first asteroid to show strong photometric evidence of complex rotation . however , the authors did not clarify this complex rotation phenomenon . furthermore , radar observations were performed by goldstone in california and by arecibo observatory during toutatis approach in 1992 . the delay - doppler images achieved a spatial resolution of 19 meters in range and 0.15 millimeters per second in radial velocity . @xcite suggested a rotational period between 4 and 5 days based on these radar data . according to the investigations of @xcite , the damping timescale for the slow non - principal axis rotation of toutatis exceeds the age of the solar system . thus , @xcite noted that the spin state of toutatis may be primordial . however , recent investigation suggests that yorp effects may slow the spin states of asteroids . thus , toutatis spin state remains a mystery . based on these high - resolution delay - doppler radar observations , @xcite used a least - squares estimation to calculate toutatis three - dimensional shape , spin states , and moment - of - inertia ratios . they showed that the dimensions along the three principal axes are 1.92 , 2.40 and 4.60 kilometers and that toutatis rotates in a long - axis mode . the two major periods were found to be 5.41 days for the rotation about the long principal axis and 7.35 days for the long - axis precession about the angular momentum vector . the results derived from the radar data were inconsistent with the solutions presented by @xcite . moreover , @xcite adopted the published optical light curves @xcite and a radar - derived shape and spin - state model @xcite to estimate the hapke parameters of toutatis . the hapke photometric model was applied , and a @xmath6 minimization proposed by @xcite was performed . the synthetic light curves that were generated based on their model provided a good fit to the optical data , with an rms residual of 0.12 mag . they showed that the combination of the optical data and radar observations led to an estimation of the spin - state parameters for toutatis that was superior to the radar - derived outcomes . the two parameters describing the moment - of - inertia ratios were determined to be 3.22 and 3.02 , respectively . based on the triaxial ellipsoid shape and spin state given by @xcite , @xcite presented the results of modeling the light curve variations of this unusual rotating asteroid by numerically integrating euler s equation in combination with the explicit expression for an asteroid s brightness as a function of euler angles . they achieved good agreement between the observed and calculated light curves . they emphasized that the light curves of toutatis were dominated by the precession effect and by the superposition of precession and rotation , which resulted in an unapparent relationship between the rotation period alone and the light curves . this understanding yielded an appropriate explanation for the inconsistency between the rotational period of toutatis determined from optical data @xcite and that determined from radar observations @xcite . during the 1996 near - earth approach , toutatis was observed by the goldstone 8510-mhz radar system . based on the physical model derived from the observations of the 1992 approach , @xcite analyzed the radar measurements and refined the estimates of the spin state of toutatis . the combination of optical and radar data was proven to better predict the orientational sequence displayed in the images captured in 1996 . after refinement , the two periods of toutatis were updated and estimated to be @xmath7 days for the rotation about the long principal axis and @xmath8 days for the uniform precession of the long principal axis about the angular momentum vector . these two parameters yielded moment - of - inertia ratios of @xmath9 and @xmath10 . thus , the orientation at the 2004 approach could be predicted in both inertial and geocentric coordinate systems . @xcite determined that mutual gravitational interactions between an asteroid and a planet or another asteroid can play a significant role in shaping the asteroid s spin state . they analyzed the interactions of a sphere with an arbitrary mass and with toutatis based on the radar - derived shape model . the results thus obtained could partially explain the phenomenon of toutatis current unusual rotational state . it was demonstrated that the tumbling spin state of toutatis might have been caused by near - earth flybys over its lifetime . this hypothesis enabled the estimation of the mass distribution and moment - of - inertia for toutatis @xcite , thereby allowing the likely internal structure to be inferred . using radar observations of five flybys from 1992 to 2008 , @xcite modeled the rotational dynamics and estimated toutatis spin - state parameters using the least - squares method . they calculated the euler angles , angular velocities , and moment - of - inertia ratios as well as the center - of - mass ( com)-center - of - figure ( cof ) offset . by directly relating the com - cof offset and the moment - of - inertia ratios to the spherical harmonic coefficients of the first- and second - degree gravity potential , they could determine the driving force of the external torque due to an external spherical body and evaluate the spin state . the terrestrial and solar tidal torques were considered in their dynamical models , and all aforementioned parameters were included in the variable state vector to be estimated in the study . furthermore , the spin states and uncertainties were propagated to the 2012 flyby epoch . on 13 december 2012 , the first space - borne close observation of toutatis was achieved by the second chinese lunar probe , change-2 , at a distance of @xmath11 meters from toutatis surface @xcite . optical images of the asteroid were acquired by one of the onboard engineering cameras during the outbound flyby . through analysis of over 400 images , @xcite estimated toutatis osculating orbit , its dimensions along the major axes , and its orientations . the highest resolution of the images was better than 3 meters . new discoveries were made , including the presence of a giant depression at the large end , a sharply perpendicular silhouette near the neck region , and direct evidence of boulders and regoliths . the geological features suggest that toutatis may have a rubble - pile structure . the physical length and width were determined to be @xmath12 , respectively , and the direction of the @xmath13 axis was calculated to be ( 234.1@xmath14 , 60.7@xmath14 ) . they showed that the bifurcated configuration may indicate that toutatis is of contact binary origin and that it is composed of two major lobes ( head and body ) . in this work , we perform an extensive investigation of the optical images of toutatis captured by change-2 , and we determine the orientation of the asteroid at the flyby epoch . in combination with radar observations ( @xcite and references therein ) , we estimate the rotational parameters of toutatis . moreover , the solar and terrestrial tidal torques are considered in the establishment of the rotational dynamics model . the torque due to the misalignment of the center of mass and the origin of the body - fixed frame is evaluated to be insignificant at first order @xcite . furthermore , we incorporate the external gravitational tidal effects from the moon and jupiter in our dynamical model . compared with the previous prediction @xcite , our results for toutatis orientation , derived for change-2 s flyby epoch from both radar data and optical images , demonstrate good consistency with the observational results of the spacecraft @xcite . our simulations reproduce the trajectory of the long axis in space , with a precession amplitude of approximately @xmath15 . this high amplitude of toutatis precession is supportive of its tumbling attitude as observed from earth . the characteristics of the angular momentum variations is investigated in detail , and the variation induced by the near - earth flyby in 2004 is estimated to be @xmath16 . the orientation of its angular momentum in space is found to be described by @xmath4 and @xmath5 , and therefore , this orientation has remained nearly constant over the past two decades . the rotational periods are estimated from the simulations to be 5.38 and 7.40 days for the rotation and precession , respectively . these values are in good agreement with the work of @xcite . this work is structured as follows : section 2 presents the observational data , which comprise ground - based measurements and optical images acquired by change-2 . in this section , we also analyze the optical data to derive the orientation of toutatis at the flyby epoch . in section 3 , we model the rotational dynamics of toutatis based on euler s equation . the least - squares and multiple shooting methods are employed to fit the variable state vector and the corresponding results . the simulation results are presented in section 4 . finally , we conclude by discussing the innovations of our investigation compared with previous works . as described above , toutatis progrades on an approximately 4:1 resonant eccentric trajectory with the earth . orbital determination and rotational parameters for toutatis have been documented since the asteroid began to be continually observed in 1992 . since that time , ground - based observations have been performed for its every near - miss of earth . as is well known , on 13 dec 2012 , change-2 completed the first successful close flyby of toutatis and acquired numerous images of this asteroid @xcite . using the released data from the minor planet center and hundreds of optical observations from the ground - based observational campaign that lasted from july to december of 2012 , the orbital determination of toutatis was precisely achieved within uncertainties on the order of several kilometers , and the orbital parameters at the flyby epoch were calculated to be @xmath17=2.5336 au , @xmath18=0.6301 , @xmath19=0.4466@xmath14 , @xmath20=124.3991@xmath14 , @xmath21=278.6910@xmath14 and @xmath22=6.7634@xmath14 . hence , the initial orbit can be integrated to calculate the relative positions of toutatis with respect to the sun , earth , moon and other major planets in the solar system , which are required for computing the external torques from the solar tides , the terrestrial tides and the gravitational tides from other bodies . the positions of the major planets and the moon are calculated based on the de405 ephemerides released by jpl . the gravitation of the sun , the major planets and 67 asteroids in the main belt as well as post - newtonian effects are considered in the dynamical model to achieve the orbital integration of toutatis . in addition , chebyshev polynomial fitting is numerically implemented to obtain the position of the asteroid at any given epoch . radar measurements of toutatis acquired by goldstone and arecibo from 1992 to 2008 are used to solve for the asteroid s rotational parameters . @xcite presented 33 sets of radar observations . together with the orientation obtained by change-2 at the flyby epoch ( see section 2.2 ) , we have 33 sets of ground - based observation outcomes , including euler angles , angular velocities and one space - borne orientation parameter . the observational data are summarized in table 1 . the observational errors of the radar data are estimated to be between @xmath23 and @xmath24 @xcite for the euler angles and between 2@xmath25 day@xmath26 and 10@xmath25 day@xmath26 for the components of angular velocity ; these errors are taken into account in our fitting . [ cols="^,^,^,^,^,^,^,^",options="header " , ] the residuals of the euler angles ( 34 sets ) and angular velocity ( 33 sets ) were normalized with respect to the maximum radar observational errors ( @xmath24 for the euler angles and @xmath27/day for the angular velocities ) and are shown in figure 6 . because the previous prediction of the orientation at the change-2 flyby epoch based on the radar - derived results differs from that observed by change-2 , the use of the optical data might have degraded the convergence of the simulation algorithm . therefore , the magnitude of the residuals is larger than that observed in the results from the radar data ( takahashi et al , 2013 ) . the residual errors in the simulations were normalized with respect to the observational uncertainties . because of the inconsistency of the observational data , the magnitudes of the residuals are slightly higher than those of the previous results . however , all deviations lie within the @xmath28 region . the largest bias is found in the roll angle , which exhibits a remarkable difference between the prediction obtained from the radar measurements and the authentic spin state of toutatis that is directly indicated by change-2 s observations at the flyby epoch . according to the numerical results derived from radar observations collected before 2008 @xcite , we considered a render effect for toutatis and generated a predicted imaging outcome prior to change-2 s flyby , as shown in figure 7a @xcite . in addition , based on the images acquired by change-2 , we corrected the attitude of toutatis by rotating the radar - derived shape model ( see section 2.2.1 ) to search for a good match with the change-2 images acquired at the flyby epoch ( fig . 7b ) , which provide the only space - borne optical data regarding toutatis orientation . furthermore , the present simulations yielded another solution for toutatis attitude during the near - earth flyby in 2012 . figure 7c shows the outcomes derived from our rotational model using space- and ground - based observations . in comparison with the results obtained from the optical images ( fig . 7b ) , the radar - derived results ( fig . 7a ) exhibit a dramatic deviation in the roll angle ; hence , these results yield a different profile of the asteroid . the simulation results derived from our dynamical model ( fig . 7c ) differ from those of figure 7b with a pitch angle bias of within @xmath29 . thus , we may safely conclude that our outcomes represent a good improvement in the understanding of toutatis spin state . the orientation of the long axis in the inertial frame likely reflects the precession of toutatis . based on the dynamical model of rotation , we calculated the variation in the direction of the long axis . figure 8 shows the trajectories of the long axis with respect to the j2000 ecliptic coordinate system in a unit sphere over the past two decades . the motions of the long axis are projected onto the x - y , x - z and y - z planes ( see figs . 8a , 8b and 8c , respectively ) . the figure reveals that the long - axis motion of the asteroid has remained ellipsoidal in the x - y and y - z planes , whereas it has rectilinearly precessed in the x - z plane . all curves lie outside the ecliptic plane , implying that the small lobe of toutatis is always located above the large end from a viewpoint close to the ecliptic . moreover , the orientation of the center axis of precession of toutatis can be approximately determined from figure 8 , and the derived spherical coordinates can be estimated to be @xmath30 in the ecliptic coordinate system . as a result of toutatis clockwise rotation and precession , the center axis of precession points nearly along the opposite direction to the angular momentum ( see section 4.3 ) . the amplitude of the precession is approximately @xmath15 , which may shed light on the significantly different attitudes of the asteroid that have been observed from earth . considering the radar - derived shape model and the components of the inertial matrix inferred by @xcite , we will now explore the angular momentum of toutatis induced by various external gravitational torques . figure 9 shows the variations in the external gravitational torques acting on the spin state of toutatis from 1992 to 2012 . the solar torque is on the order of @xmath31 , indicating that its value is 2 - 3 orders higher at the perigee than at the apogee . its periodic variation is clearly associated with toutatis orbital period . the variation tendencies of the gravitational torques arising from the earth and moon are similar , as shown by the red and blue curves , respectively . there is a @xmath32 difference in the orders of these torques because of the magnitudes of the masses of the bodies from which they arise . the periods of both torques are consistent with that of the black curve because of toutatis resonance orbit with the earth . at present , toutatis is also in a 3:1 mean motion resonance orbit with jupiter @xcite ; thus , one entire period of the torque induced by jupiter is displayed by the green curve @xcite . based on the rotational dynamical equation and the integrated orbit , the overall influence of these external torques on the variation in the magnitude of toutatis rotational angular momentum from 1992 to 2012 was normalized with respect to the initial magnitude @xmath33 @xcite , as shown in figure 10a . the terrestrial tidal torque ( see the red curve in fig . 10b ) causes a considerable change in angular momentum when the asteroid approaches earth at the perihelion or during the earth flyby that occurs every four years . the most significant change , with a variation in angular momentum magnitude on the order of 0.03% , occurred in 2004 as a result of toutatis passing the earth within 4.02 lunar distances . similarly , the tendency of the effect of the lunar torque is consistent with that of the terrestrial torque , as shown in figure 10c . the solar tides always have a predominant influence on the rotational variation . however , the terrestrial tides also play an important role in the variation in angular momentum during toutatis regular nearby visits to earth . figure 10d shows the influence on the angular momentum exerted by jupiter . the order of magnitude of this effect slightly changed after the 2004 near - earth flyby , and the amplitude continually remains lower than that of the terrestrial torque . as our simulation results indicate , the angular momentum orientation of toutatis is determined to be described by ( @xmath4 and @xmath5 ) and has remained nearly unchanged in space over the past two decades . figure 11 shows the variations in toutatis angular momentum orientation from 1992 to 2012 in the j2000 ecliptic frame . the amplitude of this change is shown to be less than one degree in both longitude and latitude . jumps in the angular momentum orientation occur at the perihelion of each orbit . a small change in behavior is evident in figure 11b as a result of the 2004 near - earth flyby , consistent with figure 10b . the variation with solar distance that is apparent in figure 11c and 11d indicates that the solar and terrestrial torques predominantly affect the rotational motion of the asteroid . the misalignment of the curves in figure 11d is also a result of the 2004 near - earth flyby . figure 11e and 11f show the angular momentum orientations of 33 sets of radar observations . compared with the numerical results , the observation data fall within a reasonable error range , with the exception of a few points at large bias . the motions of the short or middle axis reflect the status of the rotation about the long axis , whereas the long axis motion represents precession . to calculate the two periods associated with the spin states of toutatis , we determined the latitudinal variations of the asteroid s long and middle axes in the j2000 ecliptic frame , as shown in figure 12 . we applied fourier transform to analyze the periods of the two oscillation parameters and found that they are 5.38 days for the rotation about the principal axis and 7.40 days for the precession of the principal axis . these results are in good agreement with the previous results @xcite . let @xmath34 and @xmath35 indicate the latitudes of the asteroid s long and middle axes , respectively , in the j2000 ecliptic coordinate system . figures 12 and 13 show the latitudinal variations of these axes during the 1992 and 1996 flybys , respectively . additionally , the numerical results ( represented by dotted lines in figs . 12 and 13 ) that were calculated from our dynamical model are found to be in good agreement with the radar observations within the error bars ( marked by stars ) @xcite , as listed in table 4 . hence , we may conclude that the orientation parameters of toutatis obtained from our investigation are very reliable . this evidence provides further confirmation that our proposed rotational model can be used to correctly evaluate the spin status of toutatis or other asteroids . in this work , we apply the observations collected during change-2 s outbound flyby to model the rotational dynamics and determine the spin state of toutatis . based on flyby images , we utilize the radar - derived shape model to calculate toutatis orientation at the flyby epoch . in addition , we estimate the 3 - 1 - 3 euler angles to be @xmath1 , @xmath2 , and @xmath3 , respectively . consequently , our results have greatly improved the estimation of the orientational parameters of toutatis with respect to the previous predictions . in combination with ground - based observations , we investigated the evolution of the spin parameters using numerical simulations . in addition to the solar and terrestrial torques , the tidal effects arising from the moon and jupiter are extensively considered in our dynamical model . the magnitude and influence of these gravitational torques were analyzed in this work . the solar tide appears to always be the dominant torque acting on the angular momentum of toutatis . furthermore , the contribution to the external gravitation torque due to the com - cof offset appears to be negligible in the first - order approximation . we also found that the closest near - earth flyby , at 4.02 lunar distances , resulted in a 0.03% change in the magnitude of the angular momentum of toutatis . the dynamical influence exerted by saturn on the angular momentum was also assessed in further simulations and found to be approximately @xmath36 lower than that of jupiter . hence , we can safely conclude that saturn plays a less important role in the variation of toutatis angular momentum . the attitude at the change-2 flyby epoch that was derived from the numerical simulations yielded a better approximation to the optical results than that previously obtained from radar data alone . the largest deviation in the euler angles is observed in the pitch angle , with a bias of less than @xmath29 . the uncertainties corresponding to observational uncertainties and data processing error were considered in the simulations . the inconsistency in the different types of observational data may have led to higher residuals compared with previous results . simulations based solely on radar observations were also performed , and the corresponding rms magnitude was much lower . however , the results obtained using a higher - accuracy dynamical model and a combination of the various types of observations yielded a good result that is highly consistent with the optical images acquired during the change-2 flyby in 2012 . the precession of toutatis was investigated by considering the motion of its long axis . the behavior of the orbit in the inertial frame was found to be circular , with a center axis pointing along ( @xmath37 ) . the precession amplitude was estimated to be up to @xmath38 , which may be responsible for the significantly different attitude of the asteroid as observed by ground - based facilities . moreover , by exploring the motions of the long axis and middle axis , we determined the rotation period of toutatis using fourier analysis . the two major periods were found to be 5.38 days for the principal axis rotation and 7.40 days for the precession , in agreement with the results reported by @xcite . toutatis angular momentum orientation was determined to be described by @xmath4 and @xmath5 , indicating that it has remained nearly unchanged for the last two decades . because of the increasing magnitudes of the solar and terrestrial torques , tiny jumps in the angular momentum orientation occur at perihelion in each orbital period . however , the dynamical effects caused by the near - earth flyby in 2004 slightly changed the latitude of toutatis angular momentum orientation . hence , our simulation results are in good agreement with previous radar observations . in a word , based on the combination of change-2 s observations and radar data , our investigation offers an improved understanding of the rotational dynamics of toutatis . the authors greatly acknowledge m. w. busch and y. takahashi for their helpful discussions and suggestions . this work is financially supported by national natural science foundation of china ( grants no . 11303103 , 11273068 , 11473073 ) , the strategic priority research program - the emergence of cosmological structures of the chinese academy of sciences ( grant no . xdb09000000 ) , the innovative and interdisciplinary program by cas ( grant no . kjzd - ew - z001 ) , the natural science foundation of jiangsu province ( grant no . bk20141509 ) , and the foundation of minor planets of purple mountain observatory .

in this work , we investigate the rotational dynamics of the ginger - shaped near - earth asteroid 4179 toutatis , which was closely observed by change-2 at a distance of @xmath0 meters from the asteroid s surface during the outbound flyby @xcite on 13 december 2012 . a sequence of high - resolution images was acquired during the flyby mission . in combination with ground - based radar observations collected over the last two decades , we analyze these flyby images and determine the orientation of the asteroid at the flyby epoch . the 3 - 1 - 3 euler angles of the conversion matrix from the j2000 ecliptic coordinate system to the body - fixed frame are evaluated to be @xmath1 , @xmath2 and @xmath3 , respectively . the least - squares method is utilized to determine the rotational parameters and spin state of toutatis . the characteristics of the spin - state parameters and angular momentum variations are extensively studied using numerical simulations , which confirm those reported by @xcite . the large amplitude of toutatis precession is assumed to be responsible for its tumbling attitude as observed from earth . toutatis angular momentum orientation is determined to be described by @xmath4 and @xmath5 , implying that it has remained nearly unchanged for two decades . furthermore , using fourier analysis to explore the change in the orientation of toutatis axes , we reveal that the two rotational periods are 5.38 and 7.40 days , respectively , consistent with the results of the former investigation . hence , our investigation provides a clear understanding of the state of the rotational dynamics of toutatis . [ firstpage ] minor planets , asteroids : individual ( toutatis ) - planets and satellites : dynamical evolution and stability - planets and satellites : interiors

gravitational lensing is an invaluable tool in astrophysics . it has been exploited in the past to study properties of both the background lensed galaxies as well as the foreground lenses ( see reviews by * ? ? ? * ; * ? ? ? the variety of studies enabled by lensing has motivated searches for lensing events in various multi - wavelength astronomical datasets , from optical ( e.g. , slacs : @xcite ) , sub - mm ( e.g. , _ herschel _ : @xcite ) to radio ( class : @xcite ) . given that most galaxy - scale lenses involve @xmath11 image separations , most successful searches for gravitational lenses have involved the use of high resolution imaging . in this respect , _ hst _ surveys have been successful in uncovering lensed galaxies even in relatively small areas on the sky @xcite . the candels is now obtaining wfc3 and acs imaging data of several well - known extragalactic fields at unprecedented depth and resolution @xcite . the early data include f125w , f160w ( wfc3 ) , and f814w ( acs ) imaging over an area of 249 arcmin@xmath12 in goods - s and uds fields . these initial imaging data are adequate for a first study of lensed galaxies . in comparison to the work we present here @xcite identified 10 candidate lensed galaxies using acs imaging over 0.22 deg@xmath12 of the ecdfs . based on their success rate at identifying candidate lensing events , we expect about 3 to 4 lensed galaxies in current candels data and about 15 when completed with imaging over 0.35 deg@xmath12 . due to the ground - based and acs data in candels fields , we do expect some fraction of the lensed galaxies to be identified already . the paper is organized as follows : in section 2 we present a brief summary of the candidate lens selection . section 3 describes surface brightness models , while in section 4 we outline the lens models and discuss results on the three lens systems . for lensing models we assume the best - fit concordance cosmology consistent with wmap-7 year data @xcite . lcccccccc goods - s01 & 53.019907@xmath13 & -27.770704@xmath13 & & @xmath14 & & @xmath15@xmath16 & @xmath17@xmath16 & @xmath18@xmath16 + @xmath19 & & & & & & & & + ( dark matter ) & & & & & @xmath20@xmath21 & & & @xmath22@xmath21 + ( lensed source ) & & & @xmath23 & & & @xmath24@xmath25 & @xmath26@xmath25 & @xmath27@xmath25 + goods - s02 & 53.026810@xmath13 & -27.791320@xmath13 & & @xmath28 & & @xmath29@xmath16 & @xmath30@xmath16 & @xmath31@xmath16 + @xmath32 & & & & & & & & + ( dark matter ) & & & & & @xmath33@xmath21 & & & @xmath34@xmath21 + ( lensed source ) & & & @xmath35 & & & @xmath36@xmath25 & @xmath37@xmath25 & @xmath38@xmath25 for the lens candidate search we make use of the candels v0.5 drizzled mosaics at the pixel scale of 0.03 and 0.06@xmath39/pixel for wfc3 and acs bands , respectively , and involving two epochs of uds and five epochs of goods - s ( wide ) @xcite . the uds mosaic has 5@xmath40 point source depths of 27.3 , 26.3 and 25.9 ( ab ) magnitudes in f814w ( @xmath41-band ) , f125w ( @xmath2-band ) , and f160w ( @xmath7-band ) , respectively . the corresponding depths for goods - s mosaics are 27.7 , 26.6 , and 26.5 ( ab ) . as an initial search for lensed galaxies in the uds and goods - s fields we created individual false - color postage stamps of @xmath42 potential lensing galaxies with @xmath43 mag . our selection of lensed candidates is simply based on morphological information only . for this , we inspected individual image cutouts searching for both extended structure and color differences at the outskirts of galaxies that could be evidence for lensing . this process allowed us to make an initial selection of 20 candidate gravitationally lensed galaxies . through an internal poll within the candels team we ranked the list of 20 and selected three high priority targets for further studies reported in this paper . the three lensing candidates are goods - s01 and goods - s02 ( table 1 ) and uds-01 ( table 2 ) . goods - s02 was identified by @xcite as a candidate lensing system and is included as their second highest priority lensing system from acs images ; the highest priority lensing system from @xcite is not in our candidate list as candels tiles do not overlap with it . goods - s01 was previously identified as a partial - ring galaxy in gems and goods @xcite . uds-01 was previously identified as a lensing galaxy by @xcite as part of a radio source followup . the lensed source was later identified in the cfht legacy survey and was included in a snapshot _ hst_/acs lensing program for strong lenses in the cfhtls @xcite . the candels wfc3 imaging complements existing acs data in @xmath44- ( f606w ) and @xmath41-band . furthermore this field has both cfhtls megacam imaging in the optical and ukidss wfcam ultra - deep survey ( uds ) imaging in the near ir . cfhtls images reach 5@xmath40 depths of 25.9 , 26.5 , 25.8 , 26.0 and 24.6 magnitudes ( ab ) , respectively , in @xmath45 , @xmath46 , @xmath47 , @xmath48 , and @xmath49 respectively with typical seeing in the uds-01 region of about 0.8 arcsec ( fwhm ) . we also make use of @xmath50 , @xmath51 and @xmath52-band images from ukidss uds data release 8 ( almaini et al . in prep . ) , which reach @xmath53 , @xmath54 and @xmath55 magnitudes ( ab , @xmath56 with @xmath57 aperture ) respectively . the seeing fwhm in the region of uds-01 range from @xmath58 for the @xmath52-band data to @xmath59 for @xmath51-band . uds-01 involves a foreground lens galaxy which is a member of a group / cluster potential at @xmath60 and with a velocity dispersion of 770 km / sec @xcite . the system involves two background lensed sources with one at a measured spectroscopic redshift of 1.847 and another with a previously estimated photometric redshift of 2.90@xmath61 in @xcite using cfhtls and acs @xmath44- and @xmath41-band data . we revise this estimate to be @xmath62 here with 12-band photometry involving cfht ( @xmath45 , @xmath46 , @xmath47 , @xmath48 , @xmath49 ) , ukidss ( @xmath50 , @xmath51 , @xmath52 ) , acs ( @xmath44 , @xmath41 ) , and wfc3 ( @xmath2 , @xmath7 ) . lcl + ra & 34.404790@xmath13 & acs @xmath41-band + dec & @xmath635.224868@xmath13 & acs @xmath41-band + redshift & 0.6459@xmath640.0003 & tu et al . 2009 + @xmath65 & @xmath66 km s@xmath6 & lens model + @xmath67 & @xmath68 & lens model + @xmath69 & @xmath70 & lens model + @xmath71 & @xmath72 & lens model + @xmath73 & @xmath74@xmath39 & galfit + & @xmath75 kpc & + @xmath76 & @xmath77 & galfit + @xmath78 ( srsic ) & 8.0@xmath79 & galfit + + ra & 34.389725@xmath13 & geach et al . 2007 + dec & @xmath635.220814@xmath13 & geach et al . 2007 + redshift & @xmath80 & geach et al . 2007 + @xmath81 & @xmath82 km s@xmath6 & geach et al . 2007 + + redshift & @xmath83 & tu et al . 2009 + acs / f606w & @xmath84 & photometry + acs / f814w & @xmath85 & + wfc3/f125w & @xmath86 & + wfc3/f160w & @xmath87 & + ra@xmath88 & 34.402409@xmath13 & lens model + dec@xmath88 & @xmath635.224189@xmath13 & lens model + @xmath89 ( a ) & @xmath90 & lens model + @xmath78(a ) & @xmath91 & galfit / lens reconstuction + @xmath73(a ) & @xmath92 & galfit / lens reconstuction + & @xmath93 kpc & ( in f125w ) + @xmath94 & @xmath95 & galfit / lens reconstuction + @xmath89(b ) & @xmath96 & lens model + @xmath78(b ) & @xmath97 & galfit / lens reconstuction + @xmath73(b ) & @xmath98 & galfit / lens reconstuction + & @xmath99 kpc & ( in f125w ) + @xmath100 & @xmath101 & galfit / lens reconstuction + m@xmath102 & @xmath103 m@xmath4@xmath104 & sed / lens magnification + sfr & @xmath5 m@xmath4yr@xmath6@xmath104 & sed / lens magnification + e(b - v ) & @xmath105 & sed / lens magnification + + redshift & @xmath106 & photo - z + acs / f606w & @xmath107 & photometry + acs / f814w & @xmath108 & + wfc3/f125w & @xmath109 & + wfc3/f160w & @xmath110 & + ra@xmath88 & 34.402245@xmath13 & lens model + dec@xmath88 & @xmath635.223925@xmath13 & lens model + @xmath89(a ) & @xmath111 & lens model + @xmath78(a ) & @xmath112 & galfit / lens reconstuction + @xmath73(a ) & @xmath113 & galfit / lens reconstuction + & @xmath114 kpc & ( in f125w ) + @xmath94 & @xmath115 & galfit / lens reconstuction + @xmath89(b ) & @xmath116 & lens model + @xmath78(b ) & @xmath117 & galfit / lens reconstuction + @xmath73(b ) & @xmath118 & galfit / lens reconstuction + & @xmath119 kpc & ( in f125w ) + @xmath100 & @xmath95 & galfit / lens reconstuction + m@xmath102 & @xmath120 m@xmath4@xmath104 & sed / lens magnification + sfr & @xmath9 m@xmath4@xmath121@xmath104 & sed / lens magnification + e(b - v ) & @xmath122 & sed / lens magnification the two - dimensional surface brightness profile of each of the three foreground lensing galaxies were modeled using the galfit 3.0 routine @xcite in order to perform the foreground lens - background source separation shown on figs . 1 and 2 . a 12@xmath12312@xmath39 region was cut out from each of the uds and goods - s mosaics and was used as the input image for galaxy profile modeling . for galfit modeling the foreground lensing galaxies were assumed to have a _ srsic _ profile of the form @xmath124\ , , \ ] ] where @xmath73 is the scale radius and @xmath78 is the index . since the only object of interest for galfit modeling was the lensing galaxy for each image , all other sources including the lensed sources were masked for this purpose . to improve the accuracy and decrease the level of degeneracy of the morphological parameters such as the integrated magnitude , _ srsic _ index , and effective radius , psfs for wfc3 data generated internally by candels @xcite were used as an input for convolution in the modeling . _ hst_/wfc3 noise rms maps were also utilized instead of galfit s internal sigma image generating algorithm to improve the uncertainties reported in the output parameters . the residual after accounting for the central lensing galaxy profile is taken to be the lensed flux of the background source . these residual images are used directly in lens modeling . we use the publicly available lenstool software @xcite in combination with the idl routine amoeba_sa to optimize the lens model and the shape of the lensed galaxy . for the candidate lenses goods - s01 and goods - s02 , we assume that each of the foreground lens galaxies is embedded in a dark matter halo that can be described as an elliptical singular isothermal sphere ( sis ) . the halos are fixed to the centroids of the galaxies as determined from galfit . for the lensed galaxies , we assume exponential disk profiles with @xmath125 . there are a total of nine free parameters the position , the intrinsic magnitude , the scale length , the ellipticity , and the position angle ( pa ) of the source ( @xmath126 , @xmath127 , @xmath128 , @xmath129 , @xmath130 , and @xmath131 ) , and the ellipticity , the pa , and the velocity dispersion of the dark matter halo ( @xmath69 , @xmath132 , and @xmath81 ) . uds-01 is a more complicated case . the central lensing galaxy at @xmath133 belongs to an x - ray detected galaxy group at @xmath134 . hence , besides the elliptical sis halo centered on the lensing galaxy , we also include a circular sis halo at the center of the galaxy group 56 nw of uds-01 with its velocity dispersion fixed to that of the member galaxies ( 774 km s@xmath6 at @xmath134 * ? ? ? in addition , there are two _ pairs _ of background galaxies that are strongly lensed by the joint gravitational potential of the central galaxy and the group : one at @xmath135 corresponding to the tangential arc , the other at a photometric redshift of @xmath136 corresponding to the double system further out ( see fig . [ fig : uds ] ) . because the tangential system provides better constraints to the dark matter distribution , we use the tangential system to find the parameters of the sis halo associated with the central lensing galaxy , then we use this solution to constrain the positions and shapes of the double system . again we assume exponential profiles for the background galaxies . our fitting procedure is as follows . for an initial set of parameters describing the source and the deflector , we use lenstool to generate a deflected image of the source . we then convolve it with the wfc3 psfs at each band and subtract it from the observed lens - subtracted image to compute the @xmath137 value . this process is iterated with amoeba_sa to find the parameters that minimize the @xmath137 value . amoeba_sa is based on the idl multidimensional minimization routine amoeba with simulated annealing added ( e. rosolowsky , private communication ) . we adopt an initial temperature " of 100 and decrease it by 40% in each subsequent calls of amoeba_sa . 1 shows the best - fit lensing models for the einstein ring candidate goods - s01 and the cusp lens goods - s02 and table 1 summarizes lens model properties . for lens modeling , we assume that the background galaxies in both systems are at @xmath138 . we attempted to extract the photometric redshift of the background galaxies by sed fitting to candels and archival residual fluxes . both systems , however , are compact with each having an einstein radius of @xmath139 0.5@xmath39 and an accurate separation of the lens profile from the background source flux was challenging . the reduced-@xmath137 value for the best - fit lens model is 1.1 and 1.5 for goods - s01 and -s02 , respectively . the dark matter profile of goods - s02 lens has an ellipticity of @xmath34 , while the the einstein radius is @xmath28@xmath39 . these are consistent with values of 0.48 and 0.4reported in @xcite , respectively . the lens galaxies have effective radii of @xmath140 kpc ( goods - s01 ) and @xmath141 kpc ( goods - s02 ) . using the best - fit velocity dispersion and axis ratios , we estimate enclosed masses within critical ( einstein ) radii of @xmath142 m@xmath143 and @xmath144 m@xmath143 for goods - s01 and s02 , respectively . the corresponding @xmath41-band luminosities for apertures matched to the critical radii of the two galaxies are @xmath145 l@xmath143 and @xmath146 l@xmath143 . we find total mass - to - light ratios of @xmath147 and @xmath148 out to critical radii of goods - s01 and s02 , respectively . note that there is an additional systematic error of @xmath149 to 2.0 in the @xmath150 ratio associated with the unknown background source redshifts for the two lens systems . 2 shows the best - fit lens model and the lensed galaxy image reconstruction directly in the source plane . table 2 summarizes lens modeling , profile fitting , and sed modeling results . to measure source properties we make use of wfc3/f125w image . the arc and counterimage , taking the form of a cusp lens , involve an extend source with at least two peaks of emission . the source responsible for double image is also made up of two separate galaxies in the source plane . in table 2 we separate the properties of each of these galaxy pairs with labels @xmath151 and @xmath152 , while the two components are identified in fig . 2(b ) and ( c ) panels . in fig . 3 we show the seds of the sources responsible for the lensed arc and the north - west double source , with the luminosity scale corrected for the magnification in making these sed plots , instead of using galfit to remove the lens galaxy , we take advantage of the asymmetry of the lensed system and create residual images by subtracting a mirrored image from the original image . this removes the need for accurate galaxy profiles for the lens . irregular - sized apertures are made to enclose most of the flux from the arc and the nw - double in the cfht and ukidss images as highlighted in fig . the wfc3/f160w photometry of the arc is abnormally high . this is probably due to the redshifted [ oiii]4959 and 5007 lines , as both fall within the wfc f160w band for a source at @xmath0 . this hypothesis is supported by the fact that the bandpass in wfc3 begins at 1.4 @xmath89 m , while ukirt wfcam @xmath51-band filter used in ukidss has effectively a zero throughput at @xmath153 @xmath89 m and misses the [ oiii]emission lines . stellar population synthesis models are built for exponentially declining star formation histories with sfr(t ) @xmath154 exp(@xmath155 ) with various @xmath156 and ages @xcite , with the restriction that template ages are less than the age of the universe at the corresponding redshift . for each template we fit for stellar mass and intrinsic extinction to match the observed seds , assuming the extinction law of @xcite and parameterized by @xmath157 . the templates that gives the minimum @xmath137 values are chosen as the best fit . since the sed templates do not include emission lines , we did not include the f160w data point for @xmath0 source in the modeling . the quoted best - fit stellar masses and sfrs are corrected for magnification using amplification factors calculated for the photometry apertures using the best - fit lensing model ( table 2 ) . we measure the difference of sed predicted flux scaled from f125w filter to f160w and the observed f160w flux as a measurement of the [ oiii]intensity . 2(d ) shows that this difference is not localized to a specific region on the arc suggesting that the [ oiii]emission is spatially distributed broadly across the two components responsible for the arc . assuming [ oiii]4959 to 5007 ratio is 0.33 , we estimate @xmath158ergs s@xmath6 @xmath159 @xmath6 and a rest - frame equivalent width for the [ oiii]emission of @xmath139 700 . the suggestion that the arc has high [ oiii]emission is consistent with the discovery of star - bursting dwarf galaxies in candels with strong [ oiii]emission lines falling in the wfc3 @xmath2-band @xcite . our source is at a slightly higher redshift with [ oiii]in the wfc3 @xmath7-band . the @xcite sources have rest - frame equivalent widths as high as 1000 and [ oiii]fluxes a factor of 3 to 4 higher than that of the arc after correcting for magnification . the stellar mass and the sfr we find the arc source with sed fits are consistent with the suggestion that the two sources are dwarf galaxies with a star - burst phase with @xmath160 of around 9 to 10 myr . based on galfit modeling of the reconstructed source profile , the sources responsible for the arc are compact with effective radii around 0.05 to 0.5 kpc , while the two sources responsible for the double images have effective radii of 0.2 to 0.6 kpc . in fig . 3 right panel we summarizes these sources in the stellar mass - size and compare to other samples of galaxies . as summarized there the arc sources are not only dwarf galaxies , or smaller than typical `` dwarf '' galaxies with a few times 10@xmath161 m@xmath143 stellar mass , the galaxies are bursting with stars and have high [ oiii]emission . the lensing system is radio bright @xcite and thus a high resolution map of the radio emission will be useful to have to narrow down the source of [ oiii]emission , especially if either or both of the two compact , star - bursting galaxies magnified to the arc host an agn . to summarize this _ letter _ , we have presented results from a search for lensed galaxies in early candels imaging data involving two candidates in goods - south and a bona fide lensing system in the uds field . we reconstruct the lensed sources in the uds system and through galfit and sed model fits to the total aperture fluxes establish their properties including effective radius , stellar mass , star - formation rate , among others . the pair of galaxies lensed to the arc and a counterimage show bright [ oiii]line emission in the wfc @xmath7-band . the sources are compact and are undergoing a star - bursting phase . they are some of the smallest galaxies with lowest stellar masses at @xmath162 and falls below in size and stellar mass when compared to even dwarf galaxies at these redshifts . financial support for this work was provided by nasa through grant hst - go-12060 from the space telescope science institute , which is operated by associated universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . partly based on observations obtained with cfht / megacam the ukidss project is defined in @xcite .

we present results from a search for gravitationally lensed galaxies present in the _ hubble space telescope ( hst)_/wide field camera-3 ( wfc3 ) images of the cosmic assembly near - ir deep extragalactic legacy survey ( candels ) . we present one bona fide lens system in uds and two compact lens candidates in the goods - s field . the lensing system in uds involves two background galaxies , one at @xmath0 lensed to an arc and a counterimage , and the second at a photometric redshift of @xmath1 lensed to a double image . we reconstruct the lensed sources in the source plane and find in each of the two cases the sources can be separated to a pair of galaxies . the sources responsible for the arc are compact with effective radii of 0.3 to 0.4 kpc in wfc3 @xmath2-band and a total stellar mass and a star - formation rate of @xmath3 m@xmath4 and @xmath5 m@xmath4 yr@xmath6 , respectively . the abnormally high @xmath7-band flux of this source is likely due to [ oiii ] emission lines with a rest - frame equivalent width about 700 for [ oiii ] 5007 . the sources responsible for the double image have corresponding values of about 0.4 to 0.5 kpc , @xmath8 m@xmath4 and @xmath9 m@xmath4 yr@xmath6 . once completed candels is expected to contain about 15 lensing systems and will allow statistical studies on both lensing mass profiles and @xmath10 lensed galaxies . [ firstpage ]

graphene is a solid surface in three - dimensional ( 3d ) space.@xcite the area per atom , @xmath3 , is a thermodynamic property difficult to be measured . in fact the accessible observable is its projection , @xmath4 , onto the mean plane of the membrane , with @xmath5 . the equality is achieved in a strictly plane layer . the existence of two different areas , @xmath3 and @xmath4 , suggests a duplicity of physical properties . for example , the negative thermal expansion coefficient of graphene refers only to @xmath4 , while the thermal expansion of @xmath3 is positive.@xcite an internal tension conjugated to the actual membrane area @xmath3 should be distinguished from a mechanical frame tension , @xmath6 , conjugated to the projected area , @xmath4 . it is the tension @xmath6 , the lateral force per unit length at the boundary of @xmath4 , the magnitude that defines the thermodynamic ensemble in computer simulations.@xcite @xmath6 is measurable in fluid membranes by micropipette aspiration experiments.@xcite in addition , graphene elastic moduli , as the bulk or young modulus , may have different values if they are defined from fluctuations of either @xmath3 or @xmath4 . to avoid misunderstandings one should specify unambiguously the kind of variable to which one is referring . differences between @xmath3 and @xmath4 originate from the existence of ripples or wrinkles , that are a manifestation of the perpendicular acoustic ( za ) vibrational modes of the layer . the _ harmonic _ long - wave limit ( @xmath7 ) of the za phonon dispersion is @xmath8 here @xmath9 is the atomic mass density and @xmath10 the bending rigidity of the layer . @xmath1 is the fluctuation tension,@xcite that depends on the applied mechanical tension as @xmath11 @xcite the anharmonicity of the out - of - plane fluctuations causes a renormalization of the harmonic parameters . room temperature simulations of free standing graphene at _ zero mechanical tension _ @xmath12 reveal a _ finite fluctuation tension _ of @xmath13 n / m.@xcite this result agrees with analytical treatments of anharmonic effects by perturbation theory,@xcite with a study of the coupling between vibrational and electronic degrees of freedom by density functional calculations , @xcite and with the analysis of symmetry constraints in the phonon dispersion curves of graphene.@xcite all these studies are compatible with an anharmonic relation between fluctuation and mechanical tensions as @xmath14 however , the long - wave limit predicted by a membrane model with anomalous exponents deviates from this relation.@xcite in this paper , the anharmonicity of a free standing graphene layer is studied by molecular dynamics ( md ) simulations in the @xmath15 ensemble ( @xmath16 being the number of atoms in the simulation cell and @xmath17 the temperature ) . the fluctuation tension , @xmath1 , and the bending rigidity , @xmath10 , of the layer are studied at @xmath18 k as a function of both tensile @xmath19 and compressive @xmath20 stresses . the analytic long - wave limit , @xmath0 , of the za phonons allows us the formulation of a finite - size correction to the simulations . the amplitude of transverse fluctuations , @xmath21 , the projected area , @xmath22 and the bulk moduli , @xmath23 and @xmath24 , associated to the fluctuation of the areas @xmath3 and @xmath4 , are studied in the thermodynamic limit ( @xmath25 as a function of @xmath6 . the bulk moduli ( @xmath23 and @xmath24 ) are observables with different behavior . while @xmath23 remains finite for all studied tensions , @xmath26 for a critical compressive tension , @xmath27 . this is the maximum tension that a planar layer can sustain , before making a transition to a non - planar wrinkled structure . our findings provide light into the variability of experimental data on the young modulus of graphene based either on high - resolution electron energy loss spectroscopy ( hreels ) @xcite , on interferometric profilometery,@xcite or on indentation experiments with an atomic force microscope ( afm).@xcite the simulations are performed in the classical limit with a realistic interatomic potential lcbopii.@xcite the original parameterization was modified to increase the @xmath28 limit of the bending rigidity from @xmath29 ev to a more realistic value , @xmath30 ev.@xcite a supercell @xmath31 of a 2d rectangular cell @xmath32 including 4 carbon atoms was employed with 2d periodic boundary conditions.@xcite the supercell is chosen so that @xmath33 . runs consisted of @xmath34 md steps ( mds ) for equilibration , followed by @xmath35 mds for the calculation of equilibrium properties . the time step amounts to 1 fs . full cell fluctuations were allowed in the @xmath15 ensemble . atomic forces were derived analytically by the derivatives of the potential energy @xmath36 . the stress tensor estimator was similar to that used in previous works@xcite @xmath37 where @xmath38 is the atomic mass , @xmath39 is a velocity coordinate , and @xmath40 is a component of the 2d strain tensor . the brackets @xmath41 indicates an ensemble average . the derivative of @xmath36 with respect the strain tensor was performed analytically . the mechanical tension is given by the trace of the tensor @xmath42 the analyzed trajectories are subsets of @xmath43 configurations stored at equidistant times during the simulation run . the fourier analysis of transverse fluctuations was applied to simulations with @xmath44 atoms to obtain @xmath1 and @xmath10 as a function of @xmath6 . some simulations with @xmath45 were performed to check the convergence of the @xmath1 and @xmath10 calculation . the finite - size effect in transverse fluctuations and projected area was studied with additional simulations up to @xmath46 atoms . the discrete fourier transform of the heights of the atoms is @xmath47 the position of the @xmath48th atom is @xmath49 , where @xmath50 is a 2d vector in the @xmath51 plane and the height of the atom is @xmath52 . without loss of generality , the average height of the layer is set as @xmath53 the wavevectors , @xmath54 , with wavelengths commensurate with the simulation supercell , are @xmath55 with @xmath56 and @xmath57 . @xmath58 is an integer scaling factor to be defined below that unless otherwise specified is identical to one . assuming energy equipartition the mean - square amplitude @xmath59 of the za modes is related to the phonon dispersion as @xmath60 where @xmath61 is the boltzmann constant . our analysis of the long - wave limit of @xmath62 is reminiscent of the simplest atomic model with an acoustic flexural mode , namely a 1d chain of atoms with interactions up to second - nearest neighbors . the dispersion relation for this model is@xcite @xmath63\;.\label{eq : rho_w2}\ ] ] @xmath64 is the module of the vector @xmath65 . the parameters ( @xmath66 , @xmath67 , and @xmath68 ) are obtained by a least - squares fit of the simulation results for @xmath69 with the expression obtained by inserting eq . ( [ eq : rho_w2 ] ) into the r.h.s . of eq . ( [ eq : a2 ] ) followed by multiplication by @xmath70 . the fit is done for all wavevectors with @xmath71 nm@xmath72 . the first two coefficients in the taylor expansion of @xmath73 as a function of @xmath74 provide @xmath1 and @xmath10 as@xcite @xmath75 @xmath76 the results of our md simulations are divided into three subsections dealing with the long - wave limit of the za vibrations , the finite - size correction of observables depending on the za modes , and the elastic moduli of graphene . the dependence of @xmath1 and @xmath10 with the mechanical tension @xmath6 is displayed in fig . [ fig : sigma_kappa ] . @xmath6 varies between 0.3 n / m , a value close to the maximum compressive stress ( @xmath770.5 n / m ) sustained by a planar layer with @xmath44 , and a tensile stress of @xmath78 n / m . the fluctuation tension obeys an anharmonic relation , @xmath79 , with @xmath80 n / m . the value of @xmath1 in the vicinity of @xmath81 ( see fig . [ fig : sigma_kappa]b , solid line ) shows a clear shift from the harmonic expectation ( @xmath82 , dotted line ) . @xmath10 decreases monotonically as the mechanical tension becomes more tensile ( see fig . [ fig : sigma_kappa]c ) . the rate of decrease is smaller for tensions @xmath83 n / m . finite - size effects are significant in graphene simulations.@xcite the amplitude of the out - of - plane fluctuations , @xmath84 is a function of @xmath1 and @xmath10 , as these variables define the long - wave limit of the za modes . let us study the finite - size error of the average @xmath85 obtained in a @xmath86 simulation . the @xmath54-grid in eq . ( [ eq : k_n ] ) for the size @xmath87 is made up of elementary rectangles @xmath88 . let @xmath89 be the rectangle having the @xmath90 point at one vertex . the values of @xmath91 for the vertices of @xmath89 are ( 0,0 ) , ( 0,1 ) , ( 1,0 ) , and ( 1,1 ) , with ( 0,0 ) as the @xmath90-point . let us now consider successively larger cells defined with @xmath92 , where @xmath93@xmath94 is an integer _ scaling factor_. geometry in @xmath95-space dictates that the larger the cell size , the denser the @xmath95-grid . the number of @xmath95-points in the elementary area @xmath89 increases as @xmath96 , i.e. , it grows as @xmath97 for @xmath16 increasing as @xmath98 . the finite - size correction for @xmath85 is based on a discrete sum in reciprocal space . the sum is over the @xmath96 @xmath95-points in @xmath89 @xmath99 the prime indicates that the @xmath90-point ( @xmath100 ) is excluded from the sum . the multiplicative factor is the number of elementary areas , @xmath89 , in the brillouin zone . it is equal to the multiplicity of a general position in @xmath74-space , i.e. , 4 ( 6 ) for a 2d rectangular ( hexagonal ) unit cell . the amplitudes @xmath101 are calculated by eq . ( @xmath102 ) with the _ analytic _ long - wave approximation @xmath103 . the weight factors @xmath104 are unity except for those @xmath54 points at the vertices ( @xmath105 and sides ( @xmath106 of @xmath89 . the finite - size correction to the average @xmath85 is then @xmath107 to check the reliability of this analytical model , we have compared results for @xmath108 derived from @xmath109 using eq . ( [ eq : h2_correction ] ) , with those obtained directly from simulations with @xmath16 atoms . results for @xmath110 at 300 k and @xmath81 with @xmath16 varying between 24 and 33600 atoms are displayed in fig . [ fig : h2_ap]a as open circles . the finite - size correction for @xmath109 is shown as a broken line . the agreement with the simulation data is very good . note that the average @xmath111 nm@xmath112 for @xmath113 increases by two orders of magnitude for @xmath46 . the dispersion law @xmath0 correctly predicts the finite - size effect in @xmath21 . the values of @xmath1 and @xmath10 at @xmath81 are 0.094 n / m and 1.7 ev ( see fig . [ fig : sigma_kappa ] ) . the finite - size correction obtained with @xmath114 is nearly indistinguishable from that with @xmath109 , an indication of the consistency of our approach . the dispersion law , @xmath115 , implies that @xmath110 increases with the size of the sample as @xmath116.@xcite a similar scheme applies to the size correction of @xmath117 . differential elements of the real and projected areas are related by the surface metric as@xcite @xmath118 where @xmath119 ( @xmath120 ) denotes the partial derivative of the height @xmath121 with respect to @xmath122 ( @xmath123 ) , and the r.h.s . is a first - order approximation when deviations from planarity are small . by integration of the r.h.s and after fourier transform one derives@xcite @xmath124 the sum @xmath125 in reciprocal space provides the finite - size correction to the projected area @xmath117 as @xmath126 simulation results for the projected area @xmath127 are presented in fig . [ fig : h2_ap]b as open circles . size effects are significant . @xmath127 decreases with increasing @xmath16 , and converges to a finite area per atom for @xmath128 . this behavior is in good agreement to previous monte carlo ( mc ) simulations with the lcbopii model.@xcite the finite - size correction for @xmath109 using eq . ( [ eq : ap_n ] ) is shown as a broken line . a remarkable agreement to the simulation data the correction with @xmath114 is nearly indistinguishable from that with @xmath129 simulation results of the real area @xmath130 with @xmath113 are shown in fig . [ fig : h2_ap]b . @xmath3 is calculated by triangulation of the surface , with c atoms and hexagon centers as vertices of the triangles . hexagon centers are located at the average position of their six vertices . the area @xmath130 , in contrast to @xmath127 , displays a small finite - size error , not visible at the scale of fig . [ fig : h2_ap]b . for @xmath113 , the relative finite - size error in @xmath130 amounts to @xmath131 % , while that of @xmath127 is two orders of magnitude larger , @xmath132 % . a larger size , @xmath133 , is needed to reduce the finite - size error of @xmath127 to the small error achieved for @xmath130 with @xmath113 . note that our finite - size correction considers only the acoustic za mode . the obtained results imply that the rest of vibrational modes ( namely the in - plane and optical out - of - plane ( zo ) modes of the layer ) display a comparatively small size effect . the difference between @xmath3 and @xmath4 for a _ continuous _ membrane in the @xmath128 limit can be calculated by integration and fourier transform of the r.h.s . of eq . ( [ eq : da]).@xcite with the za dispersion law , @xmath0 , one gets @xmath134\:,\end{aligned}\ ] ] with @xmath135 . a quadratic term @xmath136 in @xmath73 is a sufficient condition for the convergence of the integral . we focus now on the elastic moduli of graphene . first , the finite - size correction @xmath137 is derived with @xmath109 at mechanical tensions @xmath6 in the range @xmath138 to @xmath139 n / m , using the values of @xmath1 and @xmath10 shown in fig . [ fig : sigma_kappa ] . for each tension @xmath6 and size @xmath16 , @xmath137 is then obtained at two close tensions @xmath1400.016 n / m , in order to calculate numerically the derivative @xmath141 . the bulk modulus @xmath24 for size @xmath16 is then obtained as @xmath142 the values of @xmath24 for the sizes @xmath143 , with @xmath144 ( @xmath145 ) and @xmath146 ( @xmath147 ) are plotted in fig . [ fig : b]a as dotted and full lines , respectively . for comparison , open circles display @xmath24 from @xmath15 simulation with @xmath44 atoms , as derived from the fluctuation formula@xcite @xmath148 it is remarkable the agreement between the values of @xmath24 from the simulations with @xmath44 atoms and from the finite - size extrapolation with @xmath109 . this agreement is more demanding than that of @xmath110 and @xmath127 in fig . [ fig : h2_ap ] , because of the wide range of studied mechanical tensions . the bulk modulus , @xmath23 , calculated from the fluctuation of the real area , @xmath130 , in simulations with @xmath113 is shown as open squares in fig . [ fig : b ] . the finite - size effect in @xmath23 is negligible at the scale of fig . [ fig : b]a , in line with the negligible finite - size effect in the real area @xmath130 ( see fib . [ fig : h2_ap]b ) . @xmath23 and @xmath24 behave quite differently . the anharmonicity causes a finite derivative of @xmath23 with respect to @xmath6 , @xmath149 @xmath24 is close to @xmath23 at the largest studied negative tensions , when out - of - plane fluctuations are small , but @xmath24 becomes much smaller than @xmath23 as @xmath6 increases . at a critical compressive tension , @xmath150 the bulk modulus @xmath24 vanishes . @xmath27 represents the stability limit for a planar layer before the stable configuration becomes wrinkled . @xmath27 displays a strong size effect that is shown in fig . [ fig : b]b . the critical mechanical tension , @xmath151 decreases with the number of atoms as @xmath152 . in the thermodynamic limit we get @xmath153 , i.e. , a planar layer is able to sustain a compressive tension of about @xmath154 n / m before becoming wrinkled . the structural plots in fig . [ fig : b]b shows that wrinkles are generated along a preferential direction . the young modulus , @xmath155 , of a 2d layer is related to the bulk modulus by @xmath156 , where @xmath157 is the poisson ratio . we have calculated @xmath158 for the employed lcbopii model in the classical @xmath28 limit . using this value , the young moduli , @xmath155 and @xmath159 , of graphene have been plotted in fig . [ fig : y ] as a function of @xmath6 . @xmath159 was derived in the thermodynamic limit ( @xmath128 ) by applying the finite - size correction to simulations with @xmath109 . @xmath155 has a small size effect and it was derived by a least - squares fit of simulations with @xmath113 atoms . the young modulus @xmath155 , related to the real area @xmath3 , shows a monotonic dependence with @xmath6 . for @xmath81 we find @xmath160 n / m . on the other side , @xmath159 displays a maximum ( @xmath161 n / m ) at @xmath162 n / m , decreases rapidly for @xmath163 n / m and vanishes at the critical tension @xmath164 n / m . experimental hreels results of the young modulus of both planar and corrugated graphene supported on a variety of metal substrates are displayed in fig . [ fig : y ] as open circles.@xcite hreels provides in - plane phonon dispersion curves . the elastic constants derived from the sound velocities of the acoustic in - plane branches are a property related to the real area of the layer that should correspond to the observable @xmath155 . in fact , we find good agreement between the hreels results and our simulation results for @xmath155 . results from afm indentation experiments , shown as triangles in fig . [ fig : y ] , also agree with our simulation results for @xmath155.@xcite the lack of correlation between the elastic modulus and the mechanical tension , reported in the experiments of ref . , is in line with the weak dependence of the simulation results of @xmath155 on the value of @xmath6 . however , elastic constants from interferometric profilometery are derived by fitting the experimental data to an _ average surface_.@xcite these elastic constants , plotted as squares in fig . [ fig : y ] , should correspond to the observable @xmath159 . the two interferometric profilometery results are displayed at mechanical tensions of 0.04 and 0.09 n / m . the tension of the graphene layer depends on the sample processing by factors that can not be controlled experimentally . thus the previous tensions have been chosen to fit the experimental data to our @xmath159 curve . we have analyzed the long - wave limit of the acoustic transverse fluctuations of graphene at 300 k. a finite - size correction for the out - out - plane amplitude , @xmath21 , the projected area , @xmath4 , and the bulk modulus , @xmath24 , has been based on the dispersion relation , @xmath0 . the size correction has small computational cost , displays excellent agreement to simulations with larger cells , and strongly supports the validity of the acoustic dispersion law in graphene . the fluctuation tension , @xmath1 , depends on the external mechanical tension , @xmath6 , by an anharmonic relation , @xmath79 . at 300 k we find @xmath13 n / m . the finite value of @xmath165 has a large influence in the amplitude of the out - of - plane fluctuations and in the mechanical stability of the crystalline membrane against wrinkling . the young modulus , @xmath159 , related to the projected area varies between 0 and @xmath166 n / m depending upon the mechanical tension sustained by the layer . however , the young modulus , @xmath155 , related to the real area , amounts to @xmath167 n / m in the absence of external mechanical tension , and decreases to @xmath166 n / m for large tensile stresses of -4 n / m . the existence of two different observables , @xmath155 and @xmath159 , provides a reliable explanation for the experimental values of the young modulus of graphene as measured by hreels , afm , and interferometric profilometery . this work was supported by direccin general de investigacin , mineco ( spain ) through grants no . fis2012 - 31713 , and fis2015 - 64222- c2 - 1-p . we thank the support of j. h. los in the implementation of the lcbopii potential .

room temperature simulations of graphene have been performed as a function of the mechanical tension of the layer . finite - size effects are accurately reproduced by an acoustic dispersion law for the out - of - plane vibrations that , in the long - wave limit , behaves as @xmath0 . the fluctuation tension @xmath1 is finite ( @xmath2 n / m ) even when the external mechanical tension vanishes . transverse vibrations imply a duplicity in the definition of the elastic constants of the layer , as observables related to the real area of the surface may differ from those related to the in - plane projected area . this duplicity explains the variability of experimental data on the young modulus of graphene based on electron spectroscopy , interferometric profilometery , and indentation experiments .

the study of weak @xmath1 decays is turning into an unexpected very valuable source of information on hadron dynamics @xcite . the @xmath1 and @xmath5 decays into @xmath6 and a pair of pions brought surprises showing that in the @xmath7 decay the two pions produced a big signal of the @xmath8 resonance and no trace of the @xmath9 @xcite , while in the @xmath10 decay the two pions contributed strongly in the @xmath9 region and made only a small contribution in the @xmath8 region @xcite . a study of these processes taking into account explicitly the final state interaction of the pions , together with its coupled channels , gave a good interpretation of these features @xcite , providing support for the picture of the chiral unitary approach , where these two resonances emerge as a consequence of the interaction of pseudoscalar mesons in coupled channels @xcite . similarly , the study of @xmath1 and @xmath5 decays into @xmath6 and a vector meson @xcite gave support to the picture in which the vector mesons are basically made of @xmath11 @xcite . for the @xmath12 decays , only upper bounds for their branching ratios of the order of @xmath13 are available so far @xcite . theoretical studies of these decays are available and they use the naive factorization , or qcd factorization @xcite , perturbative qcd with the @xmath14 factorization @xcite , or other kinds of factorization approximations @xcite . in the present paper we address the problem in a different way , establishing a link to the @xmath15 decays , with v a vector meson , which were studied in @xcite . the link to the @xmath12 is then established by converting the vector meson v into a photon , using for it the vector meson dominance ( vmd ) hypothesis @xcite , which is most practically implemented using the local hidden gauge approach @xcite . the intricate form factors stemming from the weak decay and qcd matrix elements are taken into account by using the experimental value of the @xmath16 decay width . this new way of addressing the problem provides rates which should be rather accurate , and they come at a moment where the rates obtained from the other theoretical approaches differ sometimes by two orders of magnitude . also significant is the fact that , while the rates obtained are below the present upper bounds , the branching ratio for @xmath17 is only one order of magnitude smaller than the present experimental bound . this should serve as a motivation to push the experimental limits to get absolute values for this rate that could shed some light on the theoretical methods to address the problem . more problematic is the @xmath18 decay , where we predict a rate about two orders of magnitude smaller than the experimental bound , but should this rate be measured it would also help us understand better the relevant physics behind these processes . the paper is organized as follows . in the next section we present the formalism . iii considers further theoretical uncertainties and compares the final results with those of other theoretical approaches . we finish with some conclusions in sec . the idea that we follow is to link the @xmath19 and @xmath20 reactions to @xmath21 , @xmath22 , and @xmath23 by converting the neutral vector mesons into a photon . for this purpose we use the sakurai vmd hypothesis @xcite , which is most practically implemented using the local hidden gauge approach @xcite . the @xmath24 production is addressed following the work of @xcite , which we describe briefly below . the @xmath25 and @xmath26 decay mechanisms at the quark level are given in fig . [ bdecay ] @xcite . the first thing to note is that in diagram ( a ) for the @xmath25 decay one has the quark transition @xmath27 , which requires the cabibbo - kobayashi - maskawa ( ckm ) matrix element , @xmath28 , and hence is cabibbo suppressed . on the other hand , in the decay of @xmath29 , diagram ( b ) , one has the @xmath30 transition that goes with the ckm matrix element @xmath31 , and hence the process is cabibbo favored . in both decays we create a @xmath32 pair that will make the @xmath6 meson and an extra pair of quarks , @xmath33 in the @xmath25 decay and @xmath34 in the @xmath26 decay . in @xcite this extra pair of quarks is hadronized , including a further @xmath35 pair with the quantum numbers of the vacuum , in order to have two mesons in the final state , in addition to the @xmath6 . but here we are interested in the production of @xmath36 , @xmath37 , @xmath38 in addition to the @xmath6 . then it is most opportune to mention that the studies conducted to determine the nature of mesons in terms of quarks conclude that the low - lying scalar mesons come from the interaction of pseudoscalars , but the low - lying vector mesons are basically @xmath35 states . this has been thoroughly tested by using large @xmath39 arguments in @xcite or applying an extension of the compositeness weinberg sum rule @xcite to states not so close to threshold @xcite and in particular in @xmath40 partial waves @xcite . indeed , in @xcite , using experimental data and the generalized sum rule , one concludes that the amount of @xmath41 in the @xmath36 wave function is very small , of the order of 10% . the same conclusion is obtained for the @xmath42 component in the @xmath43 in @xcite . this means that the @xmath35 component is the basic one in the wave function of the vector mesons and we shall adhere to this picture . in view of this , from fig . [ bdecay ] , we can already describe the production of the @xmath36 , @xmath37 , and @xmath38 mesons in addition to the @xmath6 . for this we recall that the wave functions of these mesons in terms of quarks are given by @xmath44 next , as done in @xcite , we refrain from doing an elaborate evaluation of the matrix elements involved in the weak decay and factorize them in terms of a factor @xmath45 , in view of which , the amplitudes for @xmath46 production are given by @xmath47 where we have explicitly spelled out the ckm matrix elements that distinguish one process from the other . in @xcite it was shown that using eqs . ( [ bjv1])([bjv3 ] ) and similar ones for @xmath48 and @xmath49 , the rates obtained for these decays , relative to one of them to eliminate the factor @xmath45 , were in very good agreement with experiment @xcite . the same conclusions were reached in the study of the @xmath50 and @xmath51 in @xcite and in the study of the semileptonic @xmath52 , @xmath53 , and @xmath54 into @xmath36 , @xmath37 , @xmath43 , and @xmath38 done in @xcite . in view of this , we proceed to convert the vector mesons @xmath55 , @xmath37 , @xmath38 into a photon in order to have @xmath56 in the final state . for this we need the lagrangian for this conversion , that is obtained from the local hidden gauge general lagrangian @xcite as @xcite @xmath57 where @xmath58 is the electron charge , @xmath59 , @xmath60 is the universal coupling in the local hidden gauge lagrangian , @xmath61 , with @xmath62 the vector meson mass , which we take as @xmath63 mev , and @xmath64 93 mev the pion decay constant . in eq . ( [ lvmd ] ) , @xmath65 , @xmath66 are the photon and vector meson fields and @xmath67 is the charge matrix of the @xmath68 , @xmath69 , and @xmath70 quarks . the diagrams for the @xmath71 production are now shown in fig . [ vmd ] . the vector meson field in eq . ( [ lvmd ] ) is an su(3 ) matrix and the symbol @xmath72 stands for the trace of @xmath73 . the field @xmath66 is given by @xmath74 and then the @xmath75 lagrangian can be written more simply as @xmath76 where now @xmath77 stands for the @xmath55 , @xmath37 , @xmath38 fields and @xmath78 and ( b ) @xmath26 decays into @xmath32 , making @xmath6 , and a pair of light quarks.,title="fig:",scaledwidth=40.0% ] and ( b ) @xmath26 decays into @xmath32 , making @xmath6 , and a pair of light quarks.,title="fig:",scaledwidth=40.0% ] and ( b ) @xmath20 ( b ) using vector meson dominance.,title="fig:",scaledwidth=40.0% ] and ( b ) @xmath20 ( b ) using vector meson dominance.,title="fig:",scaledwidth=40.0% ] the matrix elements for the diagrams of fig . [ vmd ] are given by @xmath79 a bit of algebra , summing over the @xmath80 polarization , yields @xmath81 where @xmath82 stands for the vector ( equal to the photon ) momentum , and for the moment we do not put the structure of the @xmath83 in terms of the vector polarization . we simply show that the polarization of @xmath80 gets converted into the one of the photons with some factors . omitting the polarization of the photon in eq . ( [ tbgamma ] ) , as we have done in eqs . ( [ bjv1])([bjv3 ] ) , we can write @xmath84 @xmath85 the decay widths for @xmath86 and @xmath0 are given by @xmath87 and similar ones for @xmath25 decays , where @xmath88 , @xmath89 are the absolute value of the @xmath38 , @xmath71 momentum in the @xmath26 rest frame . since we do not explicitly evaluate @xmath45 , we perform ratios of widths with respect to @xmath90 , and we take @xmath91 from experiment @xcite , i.e. , @xmath92 hence , the ratios we are interested in are @xmath93 @xmath94 taking into account the ckm matrix elements , @xmath95 , @xmath96 , we obtain @xmath97 @xmath98 where the errors are the same relative errors of eq . ( [ exbra ] ) . for practical purposes , we can take @xmath99 ( which differ only by a few percent @xcite ) and , thus , eqs . ( [ branchb0])([branchbs0 ] ) give branching ratios . it is interesting to compare the results of eqs . ( [ branchb0])([branchbs0 ] ) with the present experimental bounds for these branching ratios . the lhcb s recent work @xcite quotes at 90% confidence level @xmath100 @xmath101 as we can see , the results that we obtain from eqs . ( [ branchb0])([branchbs0 ] ) are consistent with the experimental bounds of eqs . ( [ lhcb1])([lhcb2 ] ) . it is interesting to note that the rate we obtain for @xmath102 is much smaller than the present bound , but the rate that we get for @xmath103 is only one order of magnitude smaller than the present bound . : ( a ) short range processes ; ( b)-(d ) long range processes ( @xmath104 indicates possible intermediate states ) . ] transition responsible for the short range part in @xmath105 . ] made explicit for @xmath20 . ] at this point , and before we go into a discussion of the spin structure of the amplitudes in the next section , we would like to situate the work into a more general context . the mechanism that we use for the decay classifies into what is denoted as long range processes in @xcite . in these works , the processes @xmath106 , @xmath107 have been studied and the mechanisms are separated into a short range part and a long range part . the long range part is evaluated using the concept of vector meson dominance , much as it has been done here . schematically , the mechanisms are depicted in fig . [ fig : new ] for the @xmath105 decay . in @xcite it was found that the short range diagram ( a ) dominated the amplitude . the explicit mechanism responsible for the large contribution of diagram ( a ) is depicted in fig . [ fig : short ] @xcite . the penguin diagrams of fig . [ fig : short ] are dominated by the two - quark intermediate states @xcite and they lead to a branching fraction @xmath108 @xcite . the rate is about a factor 30 larger than the boundary for @xmath19 quoted in eq . ( [ lhcb1 ] ) , indicating that the equivalent short range mechanism might be absent in the @xmath19 reaction . this would not be an exception since in @xcite it was found that the short range terms are much smaller than those of the long range in the radiative decay of charm mesons . in order to shed some light on this issue , we plot in fig . [ fig : meca ] the four mechanisms of fig . [ fig : new ] for @xmath109 considering the explicit form of fig . [ fig : short ] for the short range mechanism . we can see that diagram ( a ) , which includes the @xmath110 transition that was found to have a large value in @xcite for the @xmath106 transition , does not lead to @xmath6 in the final state . it would instead contribute to @xmath111 and actually we can see that the branching fraction for this mode is indeed large , @xmath112 . on the other hand , diagrams ( b ) , ( c ) , and ( d ) , described as long range in @xcite , all can lead to @xmath6 in the final state through the combination of @xmath32 . the diagram that we have calculated corresponds to diagram ( b ) . with this perspective we can justify the suppression of the mechanisms of ( c ) and ( d ) with respect to ( b ) . indeed , diagram ( b ) has the weak process in just one quark of the original @xmath113 , while ( c ) and ( d ) involve two quarks . these processes , including two body matrix elements , are usually penalized with respect to those including only one body ( see discussions in sec . 4 of @xcite ) . on the other hand , in diagram ( c ) , one has an intermediate @xmath114 state which is off shell by the energy carried out by the photon ( about 1.7 gev ) , and in diagram ( d ) the @xmath32 intermediate state is off shell by about the same amount . the double penalization should make these two mechanisms small compared to diagram ( b ) , which would be the dominant term for this reaction . so far , we have not paid attention to the structure of the @xmath115 vertex . in fact , we can have two possible structures , one that conserves parity ( pc ) and another one that violates parity ( pv ) , both of which are allowed in the weak decay . the structures are @xmath116 where @xmath117 , @xmath118 are the momenta of the @xmath6 and @xmath80 , respectively . for the case of photon production , @xmath119 and @xmath118 will then stand for the photon . the other structure is given by @xmath120 and again @xmath119 , @xmath118 would be the polarization and momenta of the photon for the case of photon production . note that in the case of photon production the two structures are gauge invariant . these two structures are explicitly used in the theoretical works @xcite . for the case of @xmath75 in the final state , the two structures guarantee gauge invariance , but for the case of @xmath121 such restriction is not necessary in principle . this issue was widely discussed in @xcite since by starting with a more general amplitude for @xmath121 and implementing the vector meson dominance ( vmd ) @xmath75 conversion lagrangian of eq . ( [ vmdvgamma ] ) , the resulting amplitude might not be gauge invariant . some prescription is given in @xcite , eliminating the longitudinal - longitudinal @xmath121 helicities in the @xmath121 process and then applying the vmd conversion . while this can be a reasonable approach , we would like to recall that a systematic study of the @xmath121 and @xmath75 processes using the local hidden gauge approach @xcite to deal with vector mesons and their interaction produces gauge invariant amplitudes after the @xmath75 conversion via eq . ( [ vmdvgamma ] ) . this comes after subtle cancellations due to a contact term and vector exchange interactions . this has been shown explicitly in the radiative decay of axial vector mesons in @xcite and in the @xmath122 decay of the @xmath123 and @xmath124 resonances @xcite . in view of this , and to guarantee the forms of eqs . ( [ vpc ] ) and ( [ vpv ] ) for the case of @xmath125 decay , we assume the same structure for the @xmath121 decay , which leads to eqs . ( [ vpc])([vpv ] ) upon the vmd transition of eq . ( [ vmdvgamma ] ) . in @xcite a final state interaction of the @xmath126 in the @xmath127 process is taken into account . we do not do that explicitly since this would be accounted for by our @xmath128 amplitude of eqs . ( [ vpc])([vpv ] ) . explicit evaluations of this interaction are done in @xcite following the @xmath121 interaction of the local hidden gauge approach in @xcite . nonetheless , in our case with @xmath6-light vector meson interaction , this interaction is weak and proceeds through coupled channels , since the tree level @xmath46 interaction is zero because we can not exchange a @xmath35 pair from the @xmath32 pair to the light vectors . in the evaluation of the rates of @xmath86 in @xcite , the explicit structure of the vertices was irrelevant , as far as one takes the @xmath80 vector masses to be equal , which is a good approximation . however , here , the structures can give some different weights depending on whether one has a vector meson or a photon in the final state . hence we evaluate the weights of these structures for the particular case that we have . we find after summing over polarization of the vector mesons or the photon ( we get the same structure in both cases ) @xmath129 @xmath130 where @xmath131 , @xmath132 or 0 ( for @xmath80 or @xmath71 production ) , and @xmath133 the fact that eqs . ( [ vpc])([vpv ] ) are gauge invariant guarantees that only the transverse polarizations of the photon contribute . this can be easily shown by explicitly taking the sum over the transverse photon polarizations @xmath134 instead of the covariant one @xmath135 valid for gauge invariant amplitudes . in both cases , one reproduces the results of eqs . ( [ sum1])([sum2 ] ) . so far , in the results we have shown in eqs . ( [ branchb0])([branchbs0 ] ) , the structures eqs . ( [ sum1])([sum2 ] ) are not taken into account . in order to evaluate the ratios of eqs . ( [ rate1])([rate2 ] ) taking into account these vector structures , we would have to multiply these ratios by the following ratio , @xmath136 which is shown in table i. .values of the @xmath137 correction factor of eq . ( [ rfactor ] ) . [ cols="<,<,<",options="header " , ] one should note that the approaches seem totally different , but they are not so . the elaborate calculations done in @xcite would go in our approach in the evaluation of @xmath86 which we do not do explicitly . instead we use the experimental value of @xmath4 . then , with the help of ref . @xcite , where one relates theoretically the different @xmath86 decays , and the vmd hypothesis , we can evaluate finally the rates of eqs . ( [ final1])([final2 ] ) . one should note that the form factors used in @xcite also rely on some other observables for their determination . in this sense , it is not so much the physics , but the strategy to get the rates , which changes from our approach to the other ones . the fact that we obtained very good rates for the @xmath138 decays in @xcite and the reliability of the vmd in the range of energies studied here should made our predictions rather solid . indeed , explicit application of the local hidden gauge approaches and vector meson dominance gives good rates for @xmath139 and @xmath140 @xcite , the two - photon and one - photon - one - vector decay widths of @xmath123 , @xmath124 , @xmath141 , @xmath142 and @xmath143 @xcite , and others @xcite . it is interesting to note that the branching ratio that we get for @xmath144 is just one order of magnitude smaller than the experimental bound . with increasing statistics in present facilities , this should serve as an incentive for extra experimental efforts in this reaction to determine an absolute rate . mechanism . ] through @xmath145 intermediate production . in diagram ( a ) , the four momenta of the particles are given in the parentheses . ] the mechanism of fig . 1(b ) for @xmath146 is color suppressed compared with the mechanism for @xmath147 depicted in fig . [ fig : loop1 ] , which is color favored . in view of this , one may wonder why the loop correction @xmath148 could not compete with the tree level process studied so far . to answer this question we evaluate the contribution of the loop of fig . [ fig : loop2 ] . the evaluation requires the use of the lagrangians @xmath149v^\mu\rangle,\ ] ] where @xmath150 with @xmath151 mev and @xmath152 mev . the matrices @xmath38 and @xmath66 are extended to su(4 ) to accommodate the @xmath153 and @xmath6 mesons and are given in @xcite . as a consequence we find @xmath154 the coupling of the photon to the pseudoscalar mesons is equally given by @xmath155 with @xmath156 . since there is much phase space for the @xmath157 decay , the loop function is dominated by the positive energy part of the @xmath52 , @xmath158 propagators emerging from the @xmath26 and the loop function is also dominated by its imaginary part @xcite . then we can write in the rest frame of the @xmath159 @xmath160 with @xmath161 the coupling of @xmath147 , @xmath162 , @xmath163 , and @xmath164 a form factor to account for the off - shellness of the @xmath6 and @xmath71 couplings with the @xmath165 @xmath153 meson off shell . we take an empirical coupling of the type @xmath166 and @xmath167 gev or less . by performing the @xmath168 integration analytically we obtain @xmath169 where we keep explicitly the @xmath71 polarization vector spatial and we also neglect the three - momentum of the @xmath6 versus its mass . the integral gives a result of the type @xmath170 ( @xmath171 ) but the second term vanishes with transverse photons . the second diagram of fig . [ fig : loop2 ] gives the same contribution as the first one and , considering explicitly that we have only transverse photons , we have @xmath172 and the sum over polarizations , taking eq . ( [ eq : photonpol ] ) into account , gives @xmath173 with @xmath174 by taking the imaginary part of @xmath175 we find @xmath176 with @xmath177 , and @xmath178 the @xmath153 on - shell momentum for @xmath147 . the two denominators in eq . ( [ eq : imi ] ) do not lead to poles in @xmath179 . the coupling @xmath161 of @xmath26 to @xmath180 is taken from experiment @xcite and we have @xmath181 from which @xmath182 and @xmath183 altogether we find now @xmath184 by taking @xmath167 gev , we find @xmath185 which is about a factor of 20 smaller than what we obtained from vector meson dominance in eq . ( [ final2 ] ) . but taking @xmath186 gev the branching ratio is @xmath187 , still one order of magnitude smaller than what was found before . certainly , one can think of similar loops with @xmath153 , @xmath188 intermediate states , but the exercise done indicates that these loops should be reasonably smaller than what has been calculated before . there is more to it : we can look at the diagrams of fig . [ fig : loop2 ] and replace a @xmath71 by a @xmath38 ( or @xmath6 ) meson . in the vector meson dominance picture the @xmath71 production amplitude is obtained from an amplitude producing @xmath38 and @xmath6 followed by conversion of @xmath38 and @xmath6 into a photon . if we ignore the @xmath6 contribution , the @xmath38 contribution alone is already accounted for in our formalism , since we take the @xmath23 process from experiment and convert the @xmath38 into a @xmath71 . the empirical process also accounts for this loop contribution . hence , what one is missing is only the fraction of the loop diagram that has the @xmath71 formed from @xmath6 . their contributions have strength @xmath189 , @xmath190 for @xmath38 , @xmath6 , summing to the @xmath58 charge , and hence what is missing is still smaller than the loop function that we have calculated . there is another empirical proof that these loop corrections are small . indeed , as we have commented , replacing the @xmath71 in fig . [ fig : loop2 ] by a @xmath38 gives a contribution to @xmath38 production coming from loops . the same diagram does not contribute to @xmath36 and @xmath37 production , since neither of them couples to @xmath153 . this means that the loop contributions are very selective to the vector mesons produced . if the loop corrections to @xmath46 ( @xmath191 ) were important , then one would not obtain good results for these processes omitting the loops . yet , the works done in @xcite on @xmath1 and @xmath192 decays , taking only the tree level diagram of fig . 1 and considering the vector mesons as coming solely from the final @xmath35 pair , indicate that this picture is rather accurate for the ratios of branching ratios , in agreement with experiment within errors . radiative @xmath1 decays are potentially sensitive to both the standard model physics and beyond the standard model physics . recent studies based on a novel nonperturbative mechanism , which includes a primary quark level transition , and the following hadronization and final state interactions of the produced hadrons , are capable of explaining very successfully a large variety of experimental data with a minimum amount of input . in the present work , we have extended such an approach and utilized the vector meson dominance hypothesis to predict the branching ratios of the radiative @xmath1 decays . the resulting parameter - free predictions not only are consistent with the present experimental upper bounds but also show a characteristic pattern that can be verified experimentally . our results show that although the @xmath193 decay rate is too small to be detected in the near future , the @xmath20 is much closer to the capacity of the lhcb detector . these results should serve to encourage our experimental colleagues to continue their efforts to obtain an absolute rate for this decay process . it should be stressed that unlike earlier theoretical studies based on qcd factorization or perturbative qcd , the approach developed in the present work relies on experimental information mostly and , as a result , should be free of uncertainties inherent in earlier studies . l.s.g thanks the nuclear theory group of valencia university for its hospitality during his visit . this work is partly supported by the national natural science foundation of china under grants no . 11375024 and no . 11522539 , the spanish ministerio de economia y competitividad and european feder funds under contract no . fis2011 - 28853-c02 - 01 and no . fis2011 - 28853-c02 - 02 , and the generalitat valenciana in the program prometeo ii-2014/068 . s. stone , arxiv:1509.04051 [ hep - ex ] . e. oset _ et al . _ , int . j. mod . e * 25 * , 1630001 ( 2016 ) [ arxiv:1601.03972 [ hep - ph ] ] . r. aaij _ et al . _ [ lhcb collaboration ] , phys . b * 698 * , 115 ( 2011 ) [ arxiv:1102.0206 [ hep - ex ] ] . r. aaij _ et al . _ [ lhcb collaboration ] , phys . d * 86 * , 052006 ( 2012 ) [ arxiv:1204.5643 [ hep - ex ] ] . t. aaltonen _ et al . _ [ cdf collaboration ] , phys . d * 84 * , 052012 ( 2011 ) [ arxiv:1106.3682 [ hep - ex ] ] . v. m. abazov _ et al . _ [ d0 collaboration ] , phys . rev . d * 85 * , 011103 ( 2012 ) [ arxiv:1110.4272 [ hep - ex ] ] . r. aaij _ et al . _ [ lhcb collaboration ] , phys . rev . d * 90 * , 012003 ( 2014 ) [ arxiv:1404.5673 [ hep - ex ] ] . w. h. liang and e. oset , phys . b * 737 * , 70 ( 2014 ) [ arxiv:1406.7228 [ hep - ph ] ] . j. a. oller and e. oset , nucl . phys . a * 620 * , 438 ( 1997 ) [ erratum - ibid . a * 652 * , 407 ( 1999 ) ] [ hep - ph/9702314 ] . j. a. oller , e. oset and j. r. pelaez , phys . d * 59 * , 074001 ( 1999 ) [ erratum - ibid . d * 60 * , 099906 ( 1999 ) ] [ erratum - ibid . d * 75 * , 099903 ( 2007 ) ] [ hep - ph/9804209 ] . n. kaiser , eur . j. a * 3 * , 307 ( 1998 ) . m. p. locher , v. e. markushin and h. q. zheng , eur . j. c * 4 * , 317 ( 1998 ) [ hep - ph/9705230 ] . j. nieves and e. ruiz arriola , nucl . a * 679 * , 57 ( 2000 ) [ hep - ph/9907469 ] . e. golowich and s. pakvasa , phys . b * 205 * , 393 ( 1988 ) . h. y. cheng , phys . d * 51 * , 6228 ( 1995 ) [ hep - ph/9411330 ] . e. golowich and s. pakvasa , phys . d * 51 * , 1215 ( 1995 ) [ hep - ph/9408370 ] . j. f. donoghue , e. golowich and a. a. petrov , phys . d * 55 * , 2657 ( 1997 ) [ hep - ph/9609530 ] . g. burdman , e. golowich , j. l. hewett and s. pakvasa , phys . d * 52 * , 6383 ( 1995 ) [ hep - ph/9502329 ] . s. bertolini , f. borzumati and a. masiero , phys . * 59 * , 180 ( 1987 ) . b. grinstein , r. springer , m. b. wise phys . b * 202 * , 138 ( 1988 ) r. molina , d. nicmorus and e. oset , phys . d * 78 * , 114018 ( 2008 ) [ arxiv:0809.2233 [ hep - ph ] ] . l. s. geng and e. oset , phys . d * 79 * , 074009 ( 2009 ) [ arxiv:0812.1199 [ hep - ph ] ] . t. branz , l. s. geng and e. oset , phys . d * 81 * , 054037 ( 2010 ) .

radiative @xmath0 decays provide an interesting case to test our understanding of ( non)perturbative qcd and eventually to probe physics beyond the standard model . recently , the lhcb collaboration has reported an upper bound , updating the results of the babar collaboration . previous theoretical predictions based on qcd factorization or perturbative qcd have shown large variations due to different treatment of nonfactorizable contributions and meson - photon transitions . in this paper , we report on a novel approach to estimate the decay rates , which is based on a recently proposed model for @xmath1 decays and the vector meson dominance hypothesis , widely tested in the relevant energy regions . the predicted branching ratios are @xmath2=\left(3.50\pm0.34^{+1.12}_{-0.63}\right)\times10^{-8}$ ] and @xmath3=\left(7.20\pm0.68^{+2.31}_{-1.30}\right)\times10^{-7}$ ] . the first uncertainty is systematic and the second is statistical , originating from the experimental @xmath4 branching ratio .

n - body numerical simulations have established a link between the existence of boxy - shaped galaxy bulges and the presence of a bar instability in the disk ( combes and sanders , 1981 ; combes et al 1990 , hereafter called * c90 * ; raha et al 1991 ; pfenniger and friedli , 1991 ; friedli and martinet , 1993 ; friedli and benz , 1993 ) . the bar can thicken , driven by vertical resonances , giving the bulge a boxy - shaped or peanut appearance , provided that the galaxy is seen at a high inclination ( i@xmath570@xmath6 ) , and depending on the orientation of the major axis of the bar with respect to the line of nodes ( measured by the azimuth @xmath7 ) : an edge - on bar ( @xmath7=90@xmath6 ) adopts a boxy shape and the peanut feature appears for a large range of intermediate azimuths . the thickening is supported by star orbits that leave the galaxy plane and it occurs only in a _ privileged _ region along the bar where there is coincidence between the horizontal resonance ilr ( @xmath8 + = @xmath9-@xmath10/2 ) and its vertical counterpart ( @xmath8=@xmath9-@xmath11/2 ) . other mechanisms , not related to the presence of a bar potential in the disk , can explain the formation of peanut bulges . several authors support the merger / accretion interpretation ( binney and petrou 1985 , shaw 1993 , shaw et al 1993 ) . the occurrence of dissipationless collapse of an isolated system ( lima - neto and combes 1995 ) , possibly in the presence of a bound companion ( may et al ( 1985 ) ) have also accounted for the existence of boxy shapes in galaxy bulges . in view of the large variety of scenarios it is necessary to envisage a detailed comparison between the different models and the observations for specific objects . there are very few examples where the connection between bars and box - peanut bulges have been investigated from an observational point of view . bettoni and galletta 1994 found photometric evidence of a bar in the peanut spiral ngc4442 ; kuijken and merrifield 1995 studied a sample of edge - on spirals using h@xmath4 and optical lines as tracers of gas kinematics . however extinction effects can be severe in highly inclined disks where the column densities of gas and the associated dust can reach very high values . the use of _ macroscopically _ optically thin tracers such as co can help us to get a more clear picture of the gas kinematics , especially in the central regions of these spirals . in particular , in this work we investigate the link between bars and box - peanut bulges using co as a fair tracer of the molecular gas kinematics in the nuclear disk of a candidate galaxy : ngc4013 . the sbc edge - on spiral ngc4013 has been identified as an extreme box - peanut bulge object , as seen in the optical pictures of van der kruit and searle ( 1982 ) and jarvis ( 1986 ) . rand ( 1996 ) ( hereafter referred as * r96 * ) have taken deep h@xmath4 images of ngc4013 searching for extraplanar diffuse ionized gas ( dig ) , discovering an impressive set of filaments coming out the plane in the nuclear region . he suggests this is the signature of a nuclear outflow , after a starburst episode related with the distorted morphology of the nucleus . ngc4013 was mapped in the j=21 and 10 lines of @xmath0co with the iram 30 m telescope ( hpbw 13@xmath3 and 21@xmath3 , respectively ) by gmez de castro and garca - burillo , 1997 ( hereafter referred as * gcgb97 * ) . the co disk extends to r=100@xmath3 and it consists of a ring - like source ( of radius r=30 - 40@xmath3 ) and an unresolved fast - rotating nuclear disk . the high - velocity co component has no hi counterpart ( see bottema , 1987 , 1995 and 1996 ) and it is best explained by * gcgb97 * invoking the presence of a non - axisymmetric potential . still the spatial resolution of the observations was insufficient to undertake a detailed analysis of gas kinematics in the nucleus of the galaxy . furthermore the single - dish data suggested also the existence of _ out - of - the - plane _ gas that might be connected either with the vertical structure of the box - peanut bulge stable orbits , or the occurrence of an outflow and local ejections of material , as the dig measurements of * r96 * seem to indicate . high - resolution ( @xmath23@xmath3 ) interferometer co data of ngc4013 are presented in this work , intending to study the distribution and kinematics of molecular gas in the nucleus of this galaxy and hopefully clarify the nature of the link between bars and box / peanut bulges . the study of dynamics of the gas in a peanut potential will deserve detailed numerical simulations involving both the stars and the gas , to be fully presented in a forthcoming publication ( paper ii ) . in this paper we limited ourselves to analyse whether there is a spontaneous and stable vertical response of the gas to a peanut potential , using a first run of numerical simulations . the @xmath0co(10 ) observations were made with the iram interferometer of plateau de bure ( see description in guilloteau et al 1992 ) , between december 1995 and march 1996 . during less than one third of the observing period we observed simultaneously the 21 line of @xmath0co . we used the compact ( cd ) 4-antenna configurations , consisting of 18 baselines ranging from 24 to 180 m. the declination of the source ( @xmath1244@xmath13 ) allowed good uv coverage and made possible to synthesize an ellipsoidal beam of fwhm=3.4@xmath143.3@xmath3 in the 10 line , adopting natural weighting and no taper on the visibilities ( fwhm=1.2@xmath3 + @xmath151@xmath3 in the 21 line ) . the @xmath0co(10 ) antenna half - power primary beam was 43@xmath16 ( 22@xmath16 in the @xmath0co(21 ) ) . the primary beam field was centered at @xmath4(j2000)=11@xmath1758@xmath1831.7@xmath19 , @xmath20(j2000)=43@xmath1356@xmath21 + 48.1@xmath3 . assuming a distance of 11.6 mpc ( bottema 1995 ) , the 43@xmath16 primary beam corresponds to 2.4 kpc ( 1@xmath3=56pc ) . a total bandwidth of 430 mhz ( 1120 and 560 kms@xmath1 for the 10 and 21 lines respectively ) , centered at v=839kms@xmath1 ( lsr ) and largely covering the range of velocities observed in ngc4013 ( @xmath22250kms@xmath1 , according to * gcgb97 * ) , was observed with a resolution of 2.5 and 1.25 mhz ( 6.5 kms@xmath1 and 3.3 kms@xmath1 at 115 and 230 ghz , respectively ) . the central 140 mhz of this band was also observed at 115 ghz with a resolution of 0.63 mhz ( 1.6 kms@xmath1 ) . data calibration was made in the standard way using the clic software package ( lucas 1992 ) . the correlator was calibrated every 20 min with a noise diode , and the rf passband once at the beginning of each observing run on 3c273 . the relative phase of the antennas was checked every 20 min on the nearby quasars 1308 + 256 and 1156 + 295 . the rms atmospheric phase fluctuations were typically between 10@xmath13 and 25@xmath13 at 115ghz . the data were then cleaned to yield channel maps . the rms noise in a 2.5 mhz - wide channel is 2.3mjy ( @xmath23=0.018k ) per 3.35@xmath24 beam at 115ghz ( the 1-@xmath25 noise is of 20mjy per beam in the 21 line or equivalently 0.35k ) . in the following , we will denote by @xmath26 and @xmath27 the kinematical major and minor axis of ngc4013 ( @xmath26 increasing to the ne and @xmath27 to the nw ) . all coordinates will be referred to the dynamical center ( x@xmath28,z@xmath28 ) which has been determined as follows : we first fit the position angle of the galaxy ( pa ) together with the z@xmath28 position of the disk , applying a standard least - squares method to the integrated intensity map of fig2a . the inferred value for the orientation of the galaxy plane is @xmath295@xmath13 , in agreement with the value derived from optical imaging . finally both x@xmath28 and the systemic velocity @xmath30 are derived imposing the maximum symmetry on the the inner 10@xmath3 region of major axis p v plot taken at z = z@xmath28 . we calculate ( x@xmath28,z@xmath28)=(0,0)= @xmath31.36 , @xmath32.9 . this position coincides within 1@xmath3 with the radio - continuum sources detected at 6 cm and 21 cm ( note that bottema ( 1995 ) reports a wrong position for the radio - continuum source ) as well as with the optical center determined by palumbo et al 1988 . similarly , the velocities ( @xmath33 ) will be relative to @xmath30=840@xmath22 + 10 kms@xmath1 ( lsr ) . fig.1 shows the @xmath0co(10 ) velocity - channel maps observed with the interferometer ( oriented along x and z axes ) . emission appears from v=143 to 170 kms@xmath1 , concentrated in a rotating edge - on disk whose vertical structure is marginally resolved with our 3.4@xmath3 beam . however we notice out of the plane gas excursions mostly in the bottom - left quadrant ( within the range @xmath34x=(5@xmath3,25@xmath3 ) and @xmath34z=(3@xmath3,8@xmath3 ) ) , visible at certain velocities ( within the range @xmath34v=(20kms@xmath1,105kms@xmath1 ) ) . at the derived dynamical center offset ( 0,0 ) , co emission displays a large spread of velocities : @xmath22130kms@xmath1 . as shown in fig.2a , which represents the @xmath0co(10 ) velocity - integrated temperature map , the observed high - velocity gas at the center is linked with the presence of a conspicuous co nuclear disk in the inner 100pc(2@xmath3 ) . the nuclear disk shows an east - west asymmetry : the maximum of emission is located at x@xmath22@xmath3 , i.e. , eastwards with respect to the dynamical center . the remarkable asymmetry of the inner 100pc of ngc4013 is more clearly shown by the higher resolution 21 data : fig.2b shows the velocity - integrated map in the 21 line which allows to resolve the nuclear disk radially . the latter displays an asymmetrical ring - like structure of radius r@xmath21.7@xmath3 ( 95pc ) . the overall distribution of molecular gas in the disk shows a similar e - w asymmetry with respect to ( 0,0 ) : co is stronger and it is more extended in the eastern side , up to x=35@xmath3 , than in the western side of the disk , up to x=-25@xmath3 ( the single - dish data of * gcgb97 * already showed this asymmetry holds for radii 40@xmath35r@xmath36100@xmath3 ) . the same asymmetry is echoed by other star formation tracers ( h@xmath4 and non - thermal radiocontinuum ) . a similar asymmetry in the co distribution is present in the inner nucleus of our galaxy ( bally et al 1987 , 1988 ) and other spirals ( see the case of ngc891 : garca - burillo and gulin , 1995 ) . if we take a co to h@xmath37 conversion factor of x = n(h@xmath37)/i@xmath38=2.3@xmath15 + 10@xmath39@xmath40k@xmath1km@xmath1s(strong et al 1988 ) , the h@xmath41 mass derived from the @xmath0co(10 ) interferometer map is m(h@xmath37)=3@xmath1510@xmath42m@xmath43 . for this we assumed the distance to be d=11.6 mpc and integrated the co flux within a rectangular area of dimensions 70@xmath4418@xmath3 , centered on the position ( 0,0 ) . including the mass of helium , the total molecular gas mass in the bure field is m@xmath45=m(h@xmath37+he)=4@xmath1510@xmath42m@xmath43 . the nuclear disk mass content is derived to be close to m@xmath460.6@xmath1510@xmath42m@xmath43 ( this is an upper limit as we integrated i@xmath38 within r=2@xmath3 for all velocities , including the contribution of gas seen in projection close to ( 0,0 ) but located far from the nucleus in the plane of the galaxy ) . taking into account that the shortest spacing measured by the interferometer is @xmath220 m , we expect to filter scales @xmath36l@xmath4720 - 25@xmath3 at 115ghz ( equivalently , @xmath36l@xmath4710 - 15@xmath3 at 230ghz ) . we have derived the fraction of the single - dish @xmath0co(10 ) and ( 21 ) fluxes included in the bure maps . the zero - spacing flux filtered out by the interferometer is kept very low in the 10 line : we detect nearly @xmath2100@xmath48 of the 30m - flux within the interferometer primary beam . on the contrary , the filtering is severe at 230ghz : we estimate that @xmath21/2 of the 30 m flux is missing in the 21 primary beam . assuming the vertical distribution to be gaussian - like we have derived the thickness of the co disk ( @xmath49 ) deconvolving the measured fwhm on the @xmath0co(10 ) velocity - integrated intensity map of fig.2a by our 3.4@xmath143.3@xmath3 synthesized beam . the inferred @xmath49 shows no systematic radial trend along the major axis and it varies between 1.5@xmath3 and 2.5@xmath3 , i.e. translated into spatial scales @xmath49=80 - 130pc . these values are comparable with the thickness of the thin molecular gas disk in the galaxy : 60100pc ( bronfman et al 1988 ) . the existence of co emission at high @xmath27 is clearly visible in the bottom left quadrant of fig.2a ( see also fig.3a ) . note that the lowest contour corresponds to 5@xmath50 , where a value of @xmath51=0.2jybeam@xmath1kms@xmath1 , has been obtained through the expression @xmath51=@xmath52v , taking @xmath25=0.002jybeam@xmath1 and @xmath34v=100 + kms@xmath1 . the mass of molecular gas at high @xmath27 is m(h@xmath37)@xmath21.5@xmath1510@xmath53 + m@xmath43 , ( @xmath2510@xmath48 of the total emission ) . the latter is a conservative lower limit as the co - to - h@xmath37 conversion factor might be significantly higher . the existence of vertical extensions of molecular gas in the nucleus were already suggested by * gcgb97 * using coarse resolution single - dish data . the connection between the dig filaments discovered by * r96 * and the above reported co extraplanar emission is best illustrated at certain velocities . figs.3b - c - d represent an overlay of a k - band image of ngc@xmath544013 , showing the peanut bulge distortion of the nucleus in the region ( @xmath55 , @xmath49)= ( @xmath2215@xmath3 , @xmath2210@xmath3 ) with the co - velocity channel maps at v=26,52 and 104kms@xmath1 ( figs.3b , 3c and 3d , respectively ) . a sketch of the dig filaments of * r96 * is included for comparison . * r96 * interpreted the impressive set of 4 dig filaments above and below the nucleus of ngc@xmath544013 as the signature of a massive nuclear starburst . supernova explosions might inject a huge amount of energy to the gas finally lifted to high z. at close sight , filament a is associated with co emission breaking out of the plane ( v=26kms@xmath1 ) , although extraplanar gas appears to be more spread above the plane than the dig feature ( see v=52kms@xmath1 and fig 3a ) . filament b is also associated with extraplanar co ( v=104kms@xmath1 ) . we have estimated the gravitational potential energy of the extraplanar molecular gas associated with filament a ( the most impressive ) , using the inferred positions of co emission above the mid - plane . taking the two isothermal stellar components model of jacobi and kegel ( 1994 ) , fitted on the luminosity distribution out of the plane in ngc4013 , the resultant gravitational energy per unit mass @xmath7 is @xmath56 @xmath57 where @xmath58 and @xmath59 stand for the stellar mid - plane densities and velocity dispersions of the i=1,2 components and h is the fitted scale height ( h=1.72kpc and @xmath60=58.3kms@xmath1 for ngc4013 , according to jacobi and kegel ( 1994 ) ) . the average height of the co extraplanar emission observed within the interval v=13,78kms@xmath1 is z@xmath2270pc , which gives a total potential energy of 50@xmath1510@xmath61ergs for @xmath62=4@xmath1510@xmath63m@xmath43 . we emphasize that the above equation allows to calculate the total potential energy of the material at height z , and therefore should be taken as a lower limit to the total input energy that must be given to the molecular gas in the plane to reach the observed values : first we have not considered the kinetic energy of these gaseous structures at present and we made no assumptions on the efficiency of the mechanism which is responsible for injecting the energy to the gas in the plane . whatever the nature of this energy source is , at least 50 type ii supernovae are required ( assuming that a type ii supernova releases @xmath210@xmath61 ergs to the ism ) . this lower limit might transform into an order of magnitude higher number of required sn , once the radiative and dissipative energy losses of the process are taken into account , henceforth suggesting that the nucleus of ngc 4013 might be experiencing a starburst episode . on the other hand , the restoring force of the galaxy plane is diminished near the position of the maximum distortion of the potential and this limit can be slightly lowered . the fir indices in ngc4013 do not indicate a massive starburst at the global scale of the disk : neither the normalised infrared fluxes , log(fir / a@xmath64=-13.8 ( where a@xmath65 is the galaxy major axis in arcminutes ) and log(fir / l@xmath66)=0.3 , nor the iras colors ( s@xmath67/s@xmath68=1.5 ; s@xmath69/s@xmath67=30 ) are indicative of typical starbursts ( see huang et al , 1996 for discussion of a large galaxy sample ) . however there is evidence that supports a starburst event in the nucleus : first , a huge fraction ( @xmath230@xmath48 ) of the total radio - continuum flux at 21 cm is emitted by the nuclear disk ; this fraction is the highest among the sample of galaxies published by hummel et al 1991 . most noticeably , there is an association between the h - shape of the dig filaments and extraplanar co emission with the box - peanut appearance of the bulge . a similar set of filaments has been discovered by rand in a galaxy also classified as a box - peanut : ngc3079 . connection between the nuclear starburst , the distorted stellar potential and the co kinematics is discussed below ( sections 5 and 6 ) . hi observations ( bottema 1995 ) showed the existence of a highly warped disk in ngc4013 . co emission at high z present in our observations could come from regions of the outer warped disk seen in projection near the nucleus . however a comparison of velocity channel maps of atomic and molecular gas leads us to reject this explanation : whereas co emission at high z appears at negative velocities in the bottom left quadrant ( figs . 2 - 3 ) , the hi warp shows up in the top left quadrant ( ne in fig . 2 of bottema 1995 ) within the same velocity range . fig.4a shows the @xmath0co(10 ) position - velocity diagram taken along the major axis . the high - inclination of the galaxy allows us to get the whole range of radial velocities in the plane . molecular gas emission is spread in a romboid - like region where we distinguish three velocity components : * [ i]@xmath54the most outstanding component displays a large spread of velocities ( v=@xmath22130kms@xmath1 ) towards the dynamical center ( referred as the straight ridge * c * ) and it has no hi counterpart . the high - velocity gas corresponds to the central co nuclear disk identified in figs.2a - b and it extends from x=3@xmath3 to x=3@xmath3 . it is fully resolved in the 21 line where it extends from x=2@xmath3 to 2@xmath3 ; note that @xmath70 reaches a maximum value of @xmath21000kms@xmath1kpc@xmath1 in the inner 100 pc(2@xmath3 ) . * [ ii]@xmath54part of the emission is concentrated in a straight ridge ( * r * ) which slowly drifts in velocity when we move along the major axis : it extends from ( x=30@xmath3,v=80kms@xmath1 ) to ( x=30@xmath3,v=100kms@xmath1 ) . hi emission is detected in this region of the p - v plot according to bottema 1987 , 1995 and 1996 . co emission fills unevenly the p v space between * c * and * r*. * [ iii]@xmath54we detect gas at non - circular velocities ( or velocities _ forbidden _ by circular rotation ) in the inner @xmath225@xmath3 . this region ( * f * ) extends symmetrically on both sides of the nucleus : at x@xmath23.5@xmath3 , we measure co emission up to v@xmath260kms@xmath1 , whereas circular rotation would impose v@xmath50 for x@xmath360 . the same applies for the offset x@xmath23.5@xmath3 where v@xmath260kms@xmath1 . we have derived the co - based rotation curve ( v@xmath71 ) from the terminal velocities method , applied to the p v major axis diagram . we have used the 21 data in the inner x=@xmath222@xmath3 , the 10 data from 2@xmath35x@xmath3625@xmath3 , and finally added the 30 m data of * gcgb97 * for the outer disk ( 25@xmath35x@xmath36100@xmath3 ) . in the derivation , we have taken into account the effect of channel smearing ( @xmath34v=6.5kms@xmath1 ) and assumed a typical cloud - cloud velocity dispersion of @xmath210kms@xmath1 . finally we have assumed that @xmath33 is translatable into v@xmath71 , as circular - motions should be the main contributor to @xmath33 ( sofue 1996 ; garca - burillo 1997 ) . note that the hi - based rotation curve ( v@xmath72 ) can not account for the co high - velocity gas component * c*. the scarcity of atomic hydrogen in the nucleus , together with the low resolution of hi observations can explain the differences between v@xmath72 and v@xmath71 . we therefore conclude that the real rotation curve is certainly much steeper in the inner 20@xmath3 than inferred using hi data : v@xmath71=110kms@xmath1 at r=110pc , this implies a dynamical mass of m@xmath73=r@xmath74= 2.8@xmath1510@xmath42m@xmath43 , assuming a spheroidal component in the nuclear region ( a factor 0.6 lower in the case of a flat disk distribution ) . therefore , we estimate m@xmath45/m@xmath73 to be of 22@xmath48 inside the nuclear disk . the latter ratio goes down monotonically to reach 10@xmath48 at r=500pc . although the dynamics is still dominated by the stars , the contribution of molecular gas to the total mass content of the nuclear disk is significantly high and the effects of gas self - gravitation might not be negligible . as stated above , the * f * component indicates the existence of emission at non - circular velocities . the high - resolution of the present observations precludes any explanation of * f * in terms of beam dilution : at x=@xmath223.5@xmath3 we are off by more than one synthesized beam from the center . the presence of the * f * component is more readily explained by invoking a deviation from axisymmetry in the inner ( r@xmath2200pc ) mass distribution of ngc4013 . more precisely , we have observational evidence that the gas flow in this galaxy might be driven by a barred potential : * the major axis p - v diagram shown in fig.4a displays the _ figure - of - eight _ shape typical of gas flowing along a bar ( binney et al 1991 ; garca - burillo and gulin 1995 ; kuijken and merrifield 1995 ) . components * c * and * r * , producing a double - peaked line - of - sight velocity plot , would correspond to molecular gas populating two different families of orbits : x@xmath37 inner orbits for * c * , and higher energy x@xmath75 orbits for * r*. the * f * component is best explained as the projection of the outer x@xmath75 orbits close to the cusped orbit . x@xmath37 orbits develop between two inner linblad resonances ilrs ( the outer ( oilr ) and the inner ( iilr ) ) . x@xmath75 orbits exist between the oilr and the corotation of the bar ( cor ) . this particular morphology of the p - v plot can not be reproduced if we impose axisymmetry in the potential ( kuijken and merrifield 1995 ) . * moreover we expect to see the imprint of a bar in the radial distribution of molecular gas as a reflect of secular evolution . gravitational torques induce a radial redistribution of gas which accumulates in rings between the ilrs , the 4:1 or uhr resonance and the outer linblad resonance ( olr ) . as a result , the corotation region is progressively emptied of gas . the radial distribution of gas in ngc4013 shows a compact central source of radius r@xmath22@xmath3 ( * c * ) , a region relatively emptied of gas between * c * and * r * ( r@xmath210 - 30@xmath3 ) and a secondary maximum towards r@xmath250@xmath3 ( according to the single - dish data of * gcgb97 * ) . this distribution , showing the existence of several rings , is well accounted by the bar hypothesis . * fig.6 shows the @xmath0co(21 ) isovelocity contours in the nuclear disk region . the z - distribution of molecular gas is spatially resolved at 1@xmath3 resolution : we detect the presence of a slight velocity gradient along the minor axis of the galaxy and a westwards tilt of @xmath260@xmath6 in the isovelocities of the nuclear disk , suggesting that we are seeing in projection a non - axisymmetric gas distribution . although the arguments enumerated above can not be taken as a proof of the existence of a barred potential , it all points out to this scenario as the simplest explanation . to explore on more firm grounds the consequences of this hypothesis , in the following section we try to locate the main resonances of the bar in the disk . the standard first - order approach consists of deriving the principal frequencies of the disk from the fitted rotation curve ( v@xmath71 ) : @xmath9 , @xmath9-@xmath10/2 , @xmath9+@xmath10/2 and @xmath9+@xmath10/4 ( see fig.5 ) . assuming the epicyclic approximation , we can figure out the value of the bar pattern speed ( @xmath76 ) and consequently the position of the main resonances , based on observational and theoretical arguments . this procedure is not intended to provide an accurate fit of @xmath76 , but to test if the basic results of numerical simulations are supported by the present observations . as shown in fig.5 , the @xmath9-@xmath10/2 curve presents a strong maximum ( @xmath2180kms@xmath1kpc@xmath1 ) at r@xmath23.5@xmath3 and it goes monotonically down to 1/10 of its peak value ( @xmath220kms@xmath1kpc@xmath1 ) at r@xmath220@xmath3 ; farther out , it stays quite flat until the edge of the optical disk . self - consistent numerical simulations of bars , including only the stellar component ( * c90 * ) , predict that the formation of the peanut occurs when the bar breaks up the plane near a _ marginal _ ilr ( i.e. , the bar pattern speed remains always close to the maximum of @xmath9-@xmath10/2 ) . if this is to apply in ngc4013 , we would require the bar to have an unlikely high pattern - speed ( @xmath77180 + kms@xmath1kpc@xmath1 ) which also implies a _ marginal _ ilr at r@xmath24@xmath3 . the latter is not compatible with the observations : the maximum distortion of the peanut bulge in ngc@xmath78 is at r@xmath212@xmath3 . moreover , as we ignore a priori the orientation of the bar major axis along the line of sight , r@xmath212@xmath3 should be taken as a lower limit for the ilr . most of the available numerical simulations have treated the appearance of the peanut distortion and its evolution including only the stars . although the influence of a dissipative component on the fate of the peanut remains to be studied thoroughly , a plausible evolutionary sequence can be advanced here . the peanut instability first sets in at a _ marginal _ ilr of the bar , but this should be taken as a starting point . the subsequent inwards flux of gas towards the original ilr , driven by the bar s gravitational torque , can change the @xmath9-@xmath10/2 curve in the inner region . the curve can become progressively steeper and finally two inner linblad resonances ( outer ( oilr ) and inner ( iilr ) ) are bound to appear . the stellar bar itself is expected to slightly slow down in the process . the peanut distortion is the relic of an old _ marginal _ ilr ; however a _ strong _ ilr region will develop in the course of time . any reasonable value of @xmath76 in ngc4013 implies we have two ilrs at present : an inspection of fig.5 leads to that conclusion unavoidably . therefore , the existence of a peanut instability and a _ strong _ ilr region in the nucleus may not be in contradiction , but should be taken as a result of secular evolution . numerical simulations to be developed in paper ii will allow us to test this scenario . fitting the bar pattern speed is beyond the scope of the present paper , however a lower limit on @xmath76 can be tentatively set . the radial distribution of molecular gas ( with a relative depression or hole between r=10@xmath3 and r=30@xmath3 ) suggest that corotation can not be at a radius larger than r@xmath250@xmath3 , implying that @xmath7760 - 70kms@xmath1kpc@xmath1 . the latter implies the following loci of principal resonances : r@xmath792@xmath3 , r@xmath8012@xmath3 , r@xmath8150@xmath3 , r@xmath8270@xmath3 and r@xmath83100@xmath3 . corotation of the bar is kept well inside the optical disk . this is expected for bars in spirals of early or intermediate hubble types such as ngc4013 , classified as sbc ( see combes and elmegreen 1993 ) . secondly , if @xmath7760 - 70kms@xmath1kpc@xmath1 , the oilr appears inside the old marginal ilr ( at 12@xmath3 ) , as proposed lines above . finally the major axis distribution of co showing a maximum at r@xmath250@xmath3 is well accounted for if a ring forms near the uhr resonance ( r@xmath8270@xmath3 ) . in a barred and peanut - shaped potential , the gas in principle tends to follow the periodic orbits , and is dragged through vertical resonances out of the plane , like the stars . however , the gas is dissipative , on a short time - scale ; cloud - cloud collisions occur with a characteristic collision time of 10@xmath84 yr . while the peanut and perpendicular elevation time scale is a long process , of time - scale almost 10@xmath85 yr ( * c90 * ) . therefore , the gas component looses its disordered kinetic energy , and in particular in the z - direction , rather quickly . the gas settles down in a very thin disk , irrespective of the peanut - shape potential . in order to study if there can be a stable and spontaneous vertical response of gas in a peanut potential , we have performed self - consistent numerical simulations involving gas and stars . in this first run , the gas mass fraction is kept very low on purpose ( @xmath22% ) . the gas clouds are considered as sticky particles and we neglected the effects of self - gravitation . no effect of star formation on the gas dynamics is considered either . the code used is fft - based particle - mesh , with the gas treated as sticky particles ( combes & gerin 1985 ) . the useful grid is 128@xmath15128@xmath1564 , and the suppression of the fourier images is done through the algorithm of james ( 1977 ) . the run included 150k stellar particles and 40k gas particles . a bar forms spontaneously in the stellar component , since the bulge to disk mass ratio is 1/3 , and there is no other spheroidal component that could stabilize the disk . after 2@xmath1510@xmath85 yr , the bar has developped a characteristic peanut shape ( see contours of fig.7 ) . the main result is that , at any time , the gas disk remains very thin ( @xmath2 200pc at half - power ) , and is never perturbed by the peanut / shaped potential or only in a very transient manner ( see fig . 8) . the vertical thickening of the bar and the subsequent formation of the peanut can be caused by instabilities associated with resonances between the bar motion and the vertical oscillations of the stars(*c90 * ; see also binney(1981 ) ) ; this effect has also been attributed to the buckling or fire - house instability by raha et al 1991 . the bar instability first acts to increase the eccentricity of stellar orbits and align their principal axes ; this causes the buckling instability , precisely about the vertical resonance region , which increases vertical velocity dispersion and thickness . once the bar is thickened , the nature of the stellar orbits in the peanut can be described with orbits trapped around the 2:2:1 periodic orbit family : these correspond to the vertical lindblad resonance , where in the frame of the bar , the particles just perform two z - oscillations in one turn ( @xmath86 , the bar pattern speed ) . this happens to occur in the region of the in - plane inner lindblad resonance ( where @xmath87 ) , and therefore the resonant orbits have the 3-d shape of a banana ( projected into an ellipse in the plane ( * c90 * , pfenniger & friedli 1991 ) . a detailed orbital study has emphasized the bifurcations of the main periodic orbits @xmath88 onto the banana and anti - banana orbit families ( pfenniger & friedli 1991 ) . these periodic orbits , in a strong bar , very often possess loops , and the gas is not likely to follow them because of dissipation . this may explain the small propensity of the gas to be elevated vertically , at least for directly rotating orbits . for retrograde orbits , on the contrary , the orbit family @xmath89 bifurcates due to vertical instability into the anomalous orbits ( corresponding this time to a 1/1 resonance , or one vertical oscillation for one turn , pfenniger & friedli 1991 ) . this family does not possess loops , and the gas can be maintained more easily on these ( friedli & benz 1993 ) . these retrograde orbits could be populated in particular during an accretion event , where gas clouds arrive with a given angular momentum , un - correlated with that of the galaxy . the case of ngc128 is a remarkable example . it is a peanut - shaped galaxy seen edge - on , where an inclined gaseous disk is observed to counter - rotate with respect to the stars ( emsellem & arsenault 1997 ) . although this galaxy has apparently no sign of perturbed morphology , it has a companion nearby ( ngc127 ) which could have provided the retrograde gas . these stable retrograde orbits are characteristic of a tumbling triaxial potential . already van albada et al ( 1982 ) had remarked that in the rotating frame of a triaxial object , the coriolis forces on retrograde particles produce a torque , which is opposite to and compensates the torque of the restoring forces towards the plane . the latter would have made the retrograde orbits to precess , and since they differentially precess with radius , the clouds collisions and dissipation would have forced the gas to settle in the main plane . the coriolis forces , therefore , stabilise the gas into the inclined retrograde orbits . these orbits are not particularly related to the peanut shape , but both phenomena indicate the presence of bars . in ngc4013 , the presence of a prodigious warp ( bottema et al 1987 ) suggests that a large amount of gas has recently been accreted with un - related angular momentum . part of it could have gone towards the center . if spontaneously , directly - rotating gas will not stay at large height above the plane due to the peanut shape of the potential , it is possible that accreted gas with a different angular momentum , not aligned with the principal axis , could be trapped in banana orbits , or more likely in retrograde anomalous orbits . more simulations are needed , taking into account the accretion origin of the gas , and also its larger mass fraction and self - gravity ( see paper ii ) . an alternative mechanism to explain the presence of gas at high altitude above the plane , is the galactic fountain effect , due to massive star formation . this could explain the morphology of the h@xmath4 filaments , that appear much higher above the plane than the neutral gas , traced by the co emission in the center . the fact that the filaments seem to coincide with the peanut region might not be a coincidence : the gas is easier to expel when the restoring force of the plane is less , i.e. in the thicker stellar plane . extraplanar neutral+ionized gas and dust have been found in another edge - on spiral : ngc891 . the existence of a thick molecular disk , containing m(h@xmath37)@xmath210@xmath42m@xmath90 up to z@xmath21kpc , was first established by garca - burillo et al ( 1992 ) using single dish co observations . * r96 * also reports the detection of an extended dig halo with some prominent h@xmath4 filaments in this galaxy . recently howk and savage ( 1997 ) have discovered the dusty counterpart of the thick co disk : the hst and wiyn optical bvr images of ngc891 show hundreds of dust filaments lying far from the mid - plane . the derived neutral gas mass content of these absorbing structures closely agrees with the co - based estimates . although some of the extraplanar dust features are interpreted as supernova - driven galactic chimneys , other are less clearly linked with highly energetic phenomena in the disk . in either cases , the existence of an extended neutral+ionized gas with a set of dust filaments is related to the high star formation efficiency in the disk ( garca - burillo et al 1992 ; * r96 * ; howk and savage 1997 ) . we have observed with the iram interferometer the emission of the 10 and 21 lines of @xmath0co in the nucleus of ngc4013 , an edge - on sbc spiral possessing a box - shaped bulge , with spatial resolutions of @xmath23.3@xmath3 and @xmath21.2@xmath3 , respectively . our maps show the presence of a distinct fast - rotating ( v@xmath91 + @xmath2130kms@xmath1 ) nuclear disk of radius r@xmath22@xmath3(100pc ) and gas mass m@xmath460.6@xmath1510@xmath42m@xmath43 . the high - velocity component ( absent in the hi map ) is accompanied by gas emission at non - circular velocities within @xmath223@xmath3 from the dynamical center , indicating that the gas flow is driven by a non - axisymmetrical potential . an analysis of the gas kinematics , derived from the @xmath0co(10 ) major axis position - velocity plot and the @xmath0co(21 ) isovelocities map , supports a bar model for ngc4013 . the observed ring - like distribution of molecular gas ( at r@xmath22@xmath3(100pc ) and r@xmath250@xmath3(2.7kpc ) ; the outer ring inferred from the 30 m map ) is interpreted as the imprint of a bar . although a link between the bar and the box - shaped bulge in ngc4013 is suggested , there are apparent discrepancies between the results of numerical simulations and the model proposed here . disagreement concerns the basic parameters of the bar generating the peanut . in self - consistent simulations of the star component the peanut forms near the locus of a _ marginal _ ilr of the bar . instead , the derived @xmath9-@xmath10/2 curve in ngc4013 leaves little doubt of the existence of a _ strong _ ilr response , irrespective of the chosen pattern speed of the bar ( @xmath76 ) . the nuclear disk would be located in the vicinity of the iilr , with no gaseous ring counterpart in the oilr . the present observations suggest that the inclusion of a dissipative component in simulations might probably change the evolution of the stellar peanut : although the latter appears near a marginal ilr , the inflow of gas driven by the bar , makes two ilrs appear and accelerates the secular evolution of the gaseous disk in the inner @xmath21kpc region of the galaxy . numerical simulations , involving both the stars and the gas in a peanut potential , will analyse the response of the gas in the disk of the galaxy fully testing the bar hypothesis ( see paper ii ) . we analysed the vertical distribution of molecular gas , showing that , although the bulk of co emission comes from a thin disk ( with deconvolved thickness fwhm@xmath280 - 130pc ) , there is a non - negligible amount of molecular gas ( m(h@xmath37)=1.5@xmath1510@xmath84m@xmath43 ) at large z distances from the plane ( z@xmath2200 - 300pc ) . the close relationship between the dig filaments seen in h@xmath4 coming out of the plane and the presence of molecular gas emission , suggests that both share a common origin : gas ejected by a massive nuclear starburst . a preliminary run of simulations has restricted to study the vertical response of the gas to a peanut potential , that spontaneously forms in a disk of stars . gas clouds are treated as test particles and we neglect here the effects of star formation and self - gravity in the gas dynamics . due to its dissipative nature , the gas forms a thin disk very quickly . contrary to the stars , the gas can not be maintained at high altitude above the galaxy plane along stable orbits . cloud - cloud collisions make impossible the population of banana and antibabana self - intersecting orbits . moreover , it remains to be studied if gas clouds can populate the vertical bifurcation of the retrograde x@xmath92 family in a peanut , after an accretion episode ( see paper ii ) . however this mechanism is unlikely to hold for ngc4013 , where the bulk of the gas in the disk is directly rotating . among the different explanations for the gas at high z inclined resonant orbits connected to the peanut , gas accretion in the course of an interaction and , finally , the galactic fountain model the latter seems the best to account for the h@xmath4 and co filaments . although the peanut distortion formed in the stars comes from a bar in the disk ( the presence of the latter being suggested by the observed co kinematics ) , gas is being ejected in the nucleus after a bar driven starburst . the filaments come from the inner 200pc(4@xmath3 ) and reach a height of several kpc , coinciding with the maximum peanut distortion where the strength of the restoring forces of the plane is diminished . this work has been partially supported by the spanish cicyt under grant number pb96 - 0104 . the authors heartly thank richard rand for giving us the h@xmath4 image of ngc4013 used in this paper . we also would like to thank martin shaw for making available his k - band image . we finally thank james binney , the referee , for his encouraging comments and criticisms .

the nucleus of the box - shaped galaxy ngc4013 has been observed with the iram interferometer in the j=10 and j=21 lines of @xmath0co . our maps show the existence of a fast rotating ( 130 kms@xmath1 ) molecular gas disk of radius r@xmath22@xmath3 ( 110pc ) . several arguments support the existence of a bar potential in ngc4013 . the _ figure - of - eight _ pattern of the major axis p - v plot , the ring - like distribution of gas , and the existence of gas emission at non - circular velocities are best accounted by a bar . we have also detected gas at high z distances from the plane ( z@xmath2200 - 300pc=4@xmath3 - 5@xmath3 ) . the latter component is related to a system of 4 h@xmath4 filaments of diffuse ionized gas that come out from the nucleus . the galactic fountain model seems the best to account for the h@xmath4 and co filaments . although the peanut distortion can be spontaneously formed by a stellar bar in the disk , gas at high z might have been ejected after a nuclear starburst . the h@xmath4 filaments start in the plane of the disk at r@xmath2200pc(4@xmath3 ) , and reach several kpc height at r@xmath2600pc(10@xmath3 ) , coinciding with the maximum peanut distortion where the strength of the restoring forces of the plane have a minimum . we have critically examined other alternatives judged less probable : the existence of a co warp ( connected to the hi warping disk ) , the accretion of gas along stable inclined orbits and , finally , a vertical gas response near the resonances of the peanut ( the latter is tested by numerical simulations ) . although a link between the bar and the box shaped bulge in ngc4013 is suggested we find noticeable differences between the results of previous numerical simulations and the present observations . the discrepancy concerns the parameters of the bar generating the peanut . we see in ngc4013 the existence of a _ strong _ ilr region . the inclusion of a dissipative component , which remains to be thoroughly studied , may change the evolution of the stellar peanut : although in simulations the peanut appears initially near a _ marginal _ ilr , the inflow of gas driven by the bar , can make two ilrs appear .

spin glass models are simplified , although still quite complex , models retaining the main features of physical systems which show glassy behavior in some region of their phase diagram . they may be considered as theoretical laboratories where the combined effects of disorder and frustration can be investigated . their phase diagram and critical behavior can be used to interpret the experimental results for complex materials . ising - like spin glasses , such as the @xmath0 ising model,@xcite model disordered uniaxial magnetic materials characterized by random ferromagnetic and antiferromagnetic short - ranged interactions , such as fe@xmath8mn@xmath9tio@xmath10 and eu@xmath8ba@xmath9mno@xmath10 ; see , e.g. , refs . . the random nature of the short - ranged interactions is mimicked by nearest - neighbor random bonds . three - dimensional ( 3d ) ising spin glasses have been widely investigated . at low temperatures they present ferromagnetic and glassy phases , depending on the amount of frustration . the critical behaviors along the finite - temperature paramagnetic - ferromagnetic and paramagnetic - glassy ( pg ) transition lines have been accurately studied.@xcite on the other hand , the low - temperature behavior , in particular the nature of the glassy phase and of the boundary between the ferromagnetic and glassy phases , is still debated . in this paper we focus on the low - temperature transition line which separates the ferromagnetic phase , characterized by a nonzero magnetization , and the spin - glass ( glassy ) phase in which the magnetization vanishes while the overlap expectation value remains nonzero . we consider the 3d @xmath0 ising model , defined by the hamiltonian@xcite @xmath11 where @xmath12 , the sum is over the nearest - neighbor sites of a cubic lattice , and the exchange interactions @xmath13 are uncorrelated quenched random variables with probability distribution @xmath14 the usual bimodal ising spin glass model , for which @xmath15=0 $ ] ( brackets indicate the average over the disorder distribution ) , corresponds to @xmath16 . for @xmath17 we have @xmath15=2p-1\neq 0 $ ] , and ferromagnetic ( or antiferromagnetic ) configurations are energetically favored . ising model . the phase diagram is symmetric under @xmath18 ( but for small values of @xmath2 the system is antiferromagnetic ) . ] the phase diagram of the cubic - lattice @xmath0 ising model is sketched in fig . [ phadiad3 ] . we only consider @xmath19 because of the symmetry @xmath18 . while the high - temperature phase is always paramagnetic ( p ) , at low temperatures there is a ferromagnetic ( f ) phase for small frustration , i.e. , small values of @xmath20 , and a glassy ( g ) phase with vanishing magnetization for sufficiently large frustration . in fig . [ phadiad3 ] we do not report any low - temperature mixed phase with simultaneous glassy and ferromagnetic behavior as found in mean - field models @xcite , for which , at present , there is no evidence.@xcite the different phases are separated by transition lines belonging to different universality classes . they meet at a magnetic - glassy multicritical point m located along the so - called nishimori line@xcite @xmath21 $ ] , where the magnetic and the overlap two - point correlation functions are equal . scaling arguments@xcite show that the transition lines must be all parallel to the @xmath22 axis at the multicritical point m. the paramagnetic - ferromagnetic ( pf ) transition line starts at the ising transition of the pure system at @xmath23 , at @xcite @xmath24 , with a correlation - length exponent @xmath25 ( @xmath26 from ref . and @xmath27 from ref . ) . along the pf line the magnetic critical behavior is universal,@xcite and belongs to the randomly - dilute ising universality class,@xcite characterized by the correlation - length critical exponent @xmath28 . it extends up to the multicritical point m , located at@xcite @xmath29 , @xmath30 , whose multicritical behavior is characterized by two even relevant renormalization - group ( rg ) perturbations with rg dimensions @xmath31 and @xmath32 . the paramagnetic - glassy ( pg ) transition line runs from m to the finite - temperature transition at @xmath16 , at@xcite @xmath33 . the glassy critical behavior is universal along the pg line;@xcite the overlap correlation - length exponent is quite large,@xcite @xmath34 . finally , ( at least ) another transition line is expected to separate the ferromagnetic and glassy phases . this is the ferromagnetic - glassy ( fg ) transition line that marks the onset of ferromagnetism and which runs from m down to the point d at @xmath35 . the nature and the general features of this transition line in ising spin glasses are not known . beside a few numerical works at @xmath35,@xcite this issue has never been investigated at finite temperature . an interesting issue concerning the fg transition line is whether it is reentrant , which would imply the existence of a range of values of @xmath2 for which the glassy phase is separated from the paramagnetic phase by an intermediate ferromagnetic phase . as proved in refs . , ferromagnetism can only exist in the region @xmath36 , which implies that @xmath37 . we also mention that , using _ entropic _ arguments applied to frustration , the fg phase boundary was argued to run parallel to the @xmath22 axis,@xcite i.e. , @xmath38 for any @xmath39 , with the critical behavior controlled by a @xmath35 _ percolation _ fixed point.@xcite the fg transition was numerically investigated at @xmath35 in ref . , obtaining the estimate @xmath40 for the critical disorder , which is slightly larger than @xmath41 . thus , it suggests a slightly reentrant fg transition line , although its apparent precision is not sufficient to exclude @xmath38 . in this paper we study the nature of the fg transition . in particular , we investigate whether the magnetic variables show a continuous and universal critical behavior from m to d , and whether hyperscaling is violated as it occurs in some systems whose critical behavior is controlled by a zero - temperature fixed point , like the 3d random - field ising model.@xcite note that we focus on the low - temperature ferromagnetic transition line , which marks the onset of ferromagnetism moving from the glassy phase with zero magnetization . there is also the possibility that a second low - temperature transition line exists for larger values of @xmath2 . in this case there would be a mixed low - temperature phase , in which ferromagnetism and glassy order coexist . this occurs in mean - field models @xcite such as the infinite - range sherrington - kirkpatrick model.@xcite however , numerical @xmath35 ground - state calculations in the 3d @xmath0 ising model on a cubic lattice@xcite and in related models@xcite do not seem to show evidence of a mixed phase and are consistent with a unique transition . in this paper we present a monte carlo ( mc ) study of the critical behavior along the fg transition line . we perform simulations of finite systems defined on cubic lattices of size @xmath42 . a finite - size scaling ( fss ) analysis of numerical data at @xmath43 and @xmath44 as a function of @xmath2 shows that magnetic correlations undergo a continuous transition along the fg line . the critical behavior is universal , i.e. , independent of @xmath22 along the line . for the magnetic critical exponents we obtain @xmath4 and @xmath5 . moreover , hyperscaling is verified . the fg transition line turns out to be slightly reentrant . indeed , we find @xmath45 at @xmath43 and @xmath46 at @xmath44 , which are definitely larger than the disorder parameter @xmath47 at the multicritical point . therefore , for a small interval of the disorder parameter , around @xmath3 , the phase diagram presents three different phases : a low - temperature glassy phase , an intermediate ferromagnetic phase , and a high - temperature paramagnetic phase . note that the critical behavior of the magnetic correlations along the fg transition line shows a new universality class of ferromagnetic transitions in ising - like disordered systems , which differs from the randomly - dilute ising universality class describing the critical behavior along the pf transition line , and from the random - field ising universality class characterized by hyperscaling violation . the general features of the phase diagram presented in fig . [ phadiad3 ] should also characterize the temperature - disorder phase diagram of other 3d ising spin glass models with tunable disorder parameters . for example , one may consider models with gaussian bond distributions , such as @xmath48 , \label{gauss}\ ] ] where the parameters @xmath49 and @xmath50 control the amount of disorder ( the pure ferromagnetic model corresponds to @xmath51 and @xmath52 ) . this distribution is also characterized by the presence of a nishimori line @xmath53 , where the magnetic and the overlap two - point correlation functions are equal . we also mention that an analogous temperature - disorder phase diagram , with three transition lines meeting at a multicritical point like fig . [ phadiad3 ] , is also found in 3d xy gauge glass models.@xcite a similar phase diagram is also expected for other continuous spin glasses , like xy and heisenberg spin glasses with bond distributions ( [ pmjdi ] ) or ( [ gauss ] ) . the paper is organized as follows . in sec . [ mcsim ] we describe the mc simulations , and provide the definitions of the quantities we consider . [ numres ] presents the fss analysis of the mc data , reporting the main results of the paper . finally , in sec . [ conclusions ] we draw our conclusions . in the appendix we report some details of the fss analyses . in order to study the fg transition line , which connects points m and d in fig . [ phadiad3 ] , we perform mc simulations of the @xmath0 ising model on cubic lattices of size @xmath54 with periodic boundary conditions . we use the metropolis algorithm , the random - exchange method , and multispin coding . implementation details can be found in ref . . in the random - exchange simulations we consider @xmath55 systems at the same value of @xmath2 and at different temperatures in the range @xmath56 , with @xmath57 and @xmath58 . the value @xmath59 is chosen so that the thermalization at @xmath59 is sufficiently fast typically we take @xmath60while the intermediate values @xmath61 are chosen such that the acceptance probability for the temperature exchange is at least @xmath62 . we require one of the @xmath61 to be along the nishimori line.@xcite the results for this temperature value can be compared with the known exact results and thus provide a check of the mc code and the thermalization . finally , one of the temperatures always corresponds to @xmath44 . the parameter @xmath55 increases with @xmath54 and varies from @xmath63 for @xmath64 to @xmath65 for @xmath66 . thermalization is checked by verifying that disorder averages are stable when increasing the number of mc steps for each disorder realization . we average over a large number @xmath67 of disorder samples : @xmath68 samples for @xmath69 , @xmath70 for @xmath71 , @xmath72 for @xmath73 , @xmath74 for @xmath75 , and @xmath76 for @xmath66 . the simulations are quite costly , because of the very slow dynamics for low temperatures . this makes the computational effort increase with a large power of the lattice size . in our range of values of @xmath54 , the number of iterations which must be discarded for thermalization apparently increases as @xmath77 for our largest lattices ( with an increasing trend with increasing @xmath54 ) . hence , taking into account the volume factor , the cpu time for each disorder realization apparently increases as @xmath78 ( but we should warn that its large-@xmath54 asymptotic behavior may be even worse ) . in total , simulations took approximately 40 years of cpu time on a single core of a recent standard commercial processor . we consider the magnetization and the magnetic correlation function defined as @xmath79},\label{mgdef}\\ & & g(x ) \equiv [ \langle \sigma_0 \sigma_x \rangle ] , \nonumber\end{aligned}\ ] ] where the angular and the square brackets indicate the thermal and the quenched average over disorder , respectively . we define the magnetic susceptibility and the second - moment correlation length , respectively as @xmath80 where @xmath81 , @xmath82 , and @xmath83 is the fourier transform of @xmath84 . moreover , we consider the cumulants @xmath85\over [ \mu_2]^{2 } } , \label{rdef}\\ & & u_{22 } \equiv { [ \mu_2 ^ 2 ] -[\mu_2]^2 \over [ \mu_2]^2},\nonumber \end{aligned}\ ] ] where @xmath86 at the critical point @xmath87 , @xmath88 , and @xmath89 ( in the following we call them phenomenological couplings and denote them by @xmath90 ) are expected to approach universal values in the large-@xmath54 limit ( within cubic @xmath91 systems with periodic boundary conditions ) . in the ferromagnetic phase we have @xmath92 , @xmath93 , and @xmath94 , while in the glassy phase we expect @xmath95 . we also define analogous quantities using the overlap variables @xmath96 , where @xmath97 and @xmath98 are two independent replicas corresponding to the same couplings @xmath13 . in particular , we consider @xmath99 and @xmath100 defined by replacing the magnetic variables with the overlap variables in eqs . ( [ xidefffxy ] ) and ( [ rdef ] ) . in this section we present a finite - size scaling ( fss ) analysis of the mc data close to the fg transition line . we consider two values of the temperature , @xmath101 and @xmath102 , below the temperature @xmath103 of the multicritical point m , and perform a fss analysis as a function of @xmath2 . at @xmath43 . the vertical lines show the location of the multicritical point m : @xmath104 . ] to begin with , we analyze the data at @xmath43 . in fig . [ rxidata ] we show the mc estimates of @xmath105 as a function of @xmath20 . analogous plots are obtained for @xmath88 and @xmath89 . the data for different lattice sizes clearly show crossing points , providing evidence for a continuous transition . they cluster at values of @xmath2 which are definitely larger than @xmath106 , ruling out a vertical transition line from m to the @xmath35 axis . ( bottom ) and @xmath89 ( top ) vs @xmath105 at @xmath43 . , title="fig : " ] ( bottom ) and @xmath89 ( top ) vs @xmath105 at @xmath43 . , title="fig : " ] in the critical limit , the phenomenological couplings @xmath90 scale as @xmath107 , \label{r - fss}\ ] ] where we have neglected analytic and nonanalytic scaling corrections . equivalently , one can test fss by considering two different couplings @xmath108 and @xmath109 . in the fss limit @xmath110 , where the function @xmath111 is universal , i.e. , identical in any model that belongs to a given universality class . clear evidence of fss is provided in fig . [ rxiut0p5 ] , where the phenomenological couplings @xmath88 and @xmath89 are reported versus @xmath87 . the data appear to rapidly approach a nontrivial limit with increasing the lattice size . scaling corrections are only visible in the case of @xmath89 , but they decrease with increasing @xmath54 . in order to determine the critical parameter @xmath112 and the exponent @xmath113 , we fit @xmath88 , @xmath89 , and @xmath87 to eq . ( [ r - fss ] ) . details are reported in app . we obtain @xmath114 where @xmath115 is the value of the phenomenological coupling @xmath90 at the critical point . scaling corrections turn out to be small . an analogous fss analysis can be performed at @xmath102 , with the purpose of checking universality , i.e. , of determining whether all transitions along the fg line belong to the same universality class . for this purpose , we use the fact that , given any pair of rg invariant quantities @xmath108 and @xmath109 , the fss function @xmath110 is universal . in fig . [ rxiu ] we plot @xmath88 and @xmath89 versus @xmath105 for both @xmath43 and @xmath44 . the plot of @xmath88 provides good evidence of universality : all data fall onto a single curve with remarkable precision . the results for @xmath89 show instead significant scatter , but they are also consistent with universality if one takes into account scaling corrections : indeed , as @xmath54 increases the data for @xmath102 approach the @xmath43 results . for a more quantitative check , we must explicitly take into account scaling corrections at @xmath44 , since they are significantly larger than those observed at @xmath43 . for instance , fits of the phenomenological couplings at @xmath44 to eq . ( [ r - fss ] ) show a somewhat large @xmath116/dof ( dof is the number of degrees of freedom of the fit ) . moreover , the estimates show systematic trends as the lattices with smaller values of @xmath54 are discarded in the fit , see app . [ appa ] for details . to include scaling corrections , we fit the data to @xmath107 + l^{-\omega } g_r[(p - p_c ) l^{1/\nu}].\ ] ] the smallest @xmath116/dof is obtained for @xmath117 . correspondingly @xmath118 , in substantial agreement with the estimate ( [ fgexp ] ) . also the estimates of @xmath119 , @xmath120 , and @xmath121 , see app . [ appa ] , are in agreement with the estimates ( [ rexp ] ) at @xmath43 . therefore , all results strongly support the universality of the critical behavior along the fg line . it is difficult to estimate reliably the exponent @xmath122 from the data . it we assume universality and fit the results at @xmath44 fixing @xmath123 , we obtain @xmath124 . note that the fits of the data at @xmath43 give much larger values for @xmath122 , i.e. , @xmath125 , see app . this is probably due to the fact that corrections with @xmath126 have very small amplitudes at @xmath43 , so that we are simply measuring an effective exponent that mimicks the behavior of several correction terms . the fss fits also provide estimates of @xmath112 at @xmath44 . we obtain @xmath127 note that @xmath128 , conferming the reentrant nature of the fg transition line . ( bottom ) and @xmath89 ( top ) vs @xmath105 at @xmath44 and at @xmath43 ( only data with @xmath129 ) . , title="fig : " ] ( bottom ) and @xmath89 ( top ) vs @xmath105 at @xmath44 and at @xmath43 ( only data with @xmath129 ) . , title="fig : " ] versus @xmath130 for @xmath44 and @xmath43 . ] as discussed at length in ref . , in the critical limit the magnetic susceptibility scales as @xmath131 , \label{chi - sc1}\ ] ] where @xmath132 is related to the magnetic scaling field and is an analytic function of @xmath2 ( and also of the temperature ) . fits of @xmath133 at @xmath44 and @xmath43 are good ( @xmath116/dof of order 1 ) if we include all data such that @xmath134 , provided that @xmath132 is taken into account ( see app . [ appb ] for details ) . we end up with the final estimate @xmath135 since @xmath130 is a function of @xmath136 in the fss limit , see eq . ( [ r - fss ] ) , we can rewrite eq . ( [ chi - sc1 ] ) as @xmath137 the function @xmath138 is universal apart from a multiplicative constant , which takes into account the freedom in the normalization of the function @xmath132 . in fig . [ chisc ] we show the quantity @xmath139 for @xmath44 and @xmath43 . for each temperature the function @xmath132 is determined by fitting the susceptibility data to eq . ( [ chi - sc2 ] ) , fixing @xmath140 . moreover , the scaling fields are normalized so that @xmath141 for @xmath142 . if we discard the data with @xmath64 and 8 at @xmath43 , all points fall on top of each other , confirming universality . since the fg transition line extends up to @xmath35 , hence the critical behavior may be controlled by a zero - temperature fixed point , hyperscaling might be violated , as it happens in the 3d random - field ising model.@xcite in order to check whether hyperscaling holds along the fg line , we consider the magnetization , which is expected to behave as @xmath143 at the critical point , and the magnetic susceptibility , which scales as @xmath144 . if hyperscaling holds , @xmath145 and @xmath146 are related by @xmath147 ( in the present case @xmath148 ) , which guarantees that @xmath149 scales as @xmath150 . in order to verify whether eq . ( [ hypersc ] ) holds , we consider @xmath151 and assume that it behaves as @xmath152 . \label{chiom2}\ ] ] if hyperscaling holds , @xmath153 vanishes . a fss analysis of the data at @xmath43 and @xmath44 gives the rather stringent bound ( details in app . [ appc ] ) @xmath154 which allows us to conclude , quite confidently , that hyperscaling holds . if this the case , using estimates ( [ etaest ] ) and ( [ fgexp ] ) of @xmath146 and @xmath113 , we obtain @xmath155 as a further check , we consider the sample distribution @xmath156 of the thermal averages of the magnetization @xmath157 at the critical point @xmath158 , @xmath43 , which is expected to behave asymptotically as @xmath159 in fig . [ hyst ] we plot @xmath160 using @xmath161 . the data clearly show the expected scaling behavior . in conclusion , the numerical results do not show evidence of hyperscaling violations in the critical behavior of magnetic correlations . and @xmath158 . we set @xmath161 . ] our data for @xmath162 can also be used to provide further evidence of universality . indeed , if we use the fact that @xmath130 is a function of @xmath136 , see eq . ( [ r - fss ] ) , we can rewrite eq . ( [ chiom2 ] ) for @xmath163 as @xmath164 where @xmath165 should be the same at @xmath43 and at @xmath102 if all transitions along the fg transition line belong to the same universality class . the plot of the data , see fig . [ hyperscfig ] , clearly confirms universality : all points fall onto a single curve . versus @xmath130 for @xmath44 and @xmath43 . ] ( bottom ) and @xmath100 ( top ) , defined in terms of the overlap variables , at @xmath43 and @xmath44 . , title="fig : " ] ( bottom ) and @xmath100 ( top ) , defined in terms of the overlap variables , at @xmath43 and @xmath44 . , title="fig : " ] in our numerical study we also consider quantities involving the overlap variables , such as @xmath166 and @xmath100 , defined at the end of sec . [ mcsim ] . in fig . [ rxio ] we show mc data up to @xmath73 ( since their computation turned out to be significantly more demanding , we restricted the measurements for the lattices @xmath167 to the magnetic correlations ) . unlike the magnetic quantities , the overlap data do not show crossings in the interval of @xmath2 we have investigated . apparently @xmath100 decreases continuously , while @xmath166 increases as @xmath168 . this may reflect the fact that the fg transition line separates two _ ordered _ phases with respect to the overlap variables . note that the differences between data at the same @xmath2 and @xmath22 and at different values of @xmath54 decrease as @xmath20 increases . hence , if there is a line in the @xmath169 plane where the overlap variables show crossings , it must be such that @xmath170 , i.e. , it must lie in the region @xmath171 , where no ferromagnetism is possible . we investigate the critical behavior along the ferromagnetic - glassy transition line of the @xmath22-@xmath2 phase diagram of the cubic - lattice @xmath172 ( edwards - anderson ) ising model , cf . ( [ lh ] ) , which marks the low - temperature boundary between the ferromagnetic phase and the glassy phase where the magnetization vanishes , i.e. , the transition line that runs from m down to the point d at @xmath35 in fig . [ phadiad3 ] . we present a numerical study based on mc simulations of systems of size up to @xmath66 , obtaining mc estimates of several quantities at @xmath43 and @xmath44 ( which are well below the temperature @xmath103 of the multicritical point m ) as a function of the disorder parameter @xmath2 . the results of the fss analyses are consistent with the two continuous magnetic transitions belonging to the same universality class . the corresponding critical exponents are @xmath4 and @xmath5 . since the critical line extends up to @xmath35 , the critical behavior may be controlled by a zero - temperature fixed point . correspondingly , it is possible to have hyperscaling violations , as it occurs in the 3d random - field ising model . our mc results show that the hyperscaling relation @xmath6 is satisfied , so that @xmath173 and @xmath174 . the fss results provide a robust evidence of a universal magnetic critical behavior along the fg transition line . a reasonable hypothesis is that also the zero - temperature transition belongs to the same universality class . this is supported by the available numerical data at @xmath35 . the numerical study of ref . for the @xmath0 ising model at @xmath35 , using lattice sizes up to @xmath175 , provided evidence of a magnetic transition at @xmath40 , with critical exponents @xmath176 and @xmath177 . numerical analyses@xcite for other ising spin - glass models at @xmath35 give consistent values of the critical exponents , @xmath178 and @xmath179 using data up to @xmath73 . these estimates are substantially consistent with our results along the fg transition line , supporting a universal critical behavior along the fg transition from the multicritical point m down to the @xmath35 axis . we also investigate the behavior of overlap correlations . they do not appear to be critical and show an apparently smooth behavior across the fg transition . our numerical results do not show evidence of other transitions close to the transition line where ferromagnetism disappears . thus , they do not hint at the existence of a mixed ferromagnetic - glassy phase , as found in mean - field models,@xcite in agreement with earlier @xmath35 numerical studies.@xcite the fg transition line is slightly reentrant . indeed , we find that @xmath45 at @xmath43 and @xmath46 at @xmath44 , which are definitely larger than @xmath47 , although they are quite close . this implies that there exists a small interval of the disorder parameter , around @xmath3 , showing three different phases when varying @xmath22 : with increasing the temperature , the system goes from the low - temperature glassy phase with zero magnetization , to an intermediate ferromagnetic phase , and finally to the high - temperature paramagnetic phase . correspondingly , it first undergoes a glassy - ferromagnetic transition with @xmath4 and then a ferromagnetic - paramagnetic transition with @xmath180 . we mention that a slightly reentrant low - temperature transition line , where ferromagnetism disappears , also occurs in the phase diagram of the 2d @xmath0 ising model.@xcite the main features of the fg transition line are not expected to depend on the particular discrete bond distribution of the @xmath0 ising model , cf . ( [ pmjdi ] ) . they should also apply to more general distributions with tunable disorder parameters , such as the gaussian distribution reported in eq . ( [ gauss ] ) , and also to experimental spin glass systems with tunable disorder . we conclude showing fig . [ tp ] which reports all available numerical results for the phase boundaries of the cubic - lattice @xmath172 ising model ( [ lh ] ) in the @xmath22-@xmath2 plane , taken from ref . for the pf transition line , from ref . for the multicritical point along the nishimori ( n ) line @xmath181 $ ] , from ref . for the data along the pg line , from this paper along the fg line , and from ref . for the @xmath35 transition point . the dashed lines are interpolations of the data along the transition lines which satisfy the expected scaling behavior at the multicritical point where they meet , controlled by the crossover exponent @xmath182 , see refs . for details . @xcite ising model ( [ lh ] ) in the @xmath22-@xmath2 plane . the dashed lines are interpolations of the data @xcite . ] the mc simulations were performed at the infn pisa grid data center , using also the cluster csn4 . lcccc + & & & & + @xmath116/dof & 5594/289 & 567/229 & 203/169 & 79/109 + @xmath113 & 0.971(4 ) & 0.964(5 ) & 0.954(8 ) & 1.00(2 ) + @xmath112 & 0.77230(1 ) & 0.77275(2 ) & 0.77284(3 ) & 0.77281(5 ) + @xmath119 & 0.7453(2 ) & 0.7564(3 ) & 0.7592(6 ) & 0.759(2 ) + @xmath120 & 1.3450(2 ) & 1.3364(4 ) & 1.3343(6 ) & 1.334(2 ) + @xmath121 & 0.3046(2 ) & 0.3045(3 ) & 0.3057(6 ) & 0.310(2 ) + + & & & & + @xmath116/dof & 10593/289 & 1842/229 & 365/169 & 98/109 + @xmath113 & 1.054(5 ) & 0.995(5 ) & 0.963(8 ) & 0.982(20 ) + @xmath112 & 0.76819(2 ) & 0.76920(2 ) & 0.76975(3 ) & 0.76994(5 ) + @xmath119 & 0.6826(2 ) & 0.7019(3 ) & 0.7147(6 ) & 0.7220(17 ) + @xmath120 & 1.3973(3 ) & 1.3779(4 ) & 1.3662(6 ) & 1.3613(19 ) + @xmath121 & 0.3094(3 ) & 0.3020(4 ) & 0.2977(5 ) & 0.3013(17 ) + in order to determine the exponent @xmath113 and the critical parameter @xmath112 , we analyze the phenomenological couplings @xmath183 , @xmath89 , and @xmath87 . in the critical limit each quantity @xmath90 behaves as @xmath184 + u_\omega(p ) l^{-\omega } g_r[u_p(p ) l^{1/\nu } ] , \label{scaling - r}\ ] ] where the nonlinear scaling fields @xmath185 and @xmath186 are analytic functions of @xmath2 . we have @xmath187 while , in general , we expect @xmath188 . for both temperatures our data belong to a small interval of values of @xmath2 , so that we expect the approximations @xmath189 and @xmath190 to work well . to check it , we also performed fits assuming @xmath191 . we did not find any significant difference . we first analyze the results at @xmath101 . we perform combined fits of the three quantities to eq . ( [ scaling - r ] ) without scaling corrections ( we set @xmath192 ) . if the scaling functions @xmath193 are approximated by fourth - order polynomials , we obtain the results reported in table [ table - r ] . we report estimates for different @xmath194 : in each fit we only include data satisfying @xmath195 . corrections are quite small and indeed the results corresponding to @xmath196 and @xmath197 mostly agree within errors . we also perform fits that take into account scaling corrections . we fix @xmath122 , approximate @xmath198 by a second - order polynomial , and repeat the fit for several values of @xmath122 between 1 and 5 . if we perform a combined fit of @xmath88 and @xmath105 ( we include all results with @xmath199 ) , the smallest @xmath116/dof ( dof is the number of degrees of freedom of the fit ) is obtained for @xmath200 and one would estimate @xmath201 and @xmath202 . if instead we use @xmath88 , @xmath105 , and also @xmath89 we obtain @xmath203 , @xmath204 , and @xmath205 . these results indicate that scaling corrections are quite small , and quite probably can not be parametrized be a single correction term . our best estimates of @xmath122 are simply effective exponents that parametrize the contributions of several different correction terms , which are all relevant for our small lattice sizes . if we compare all results , we end up with the estimates @xmath206 and @xmath4 , reported in eq . ( [ fgexp ] ) . for the phenomenological couplings at criticality , @xmath207 , we obtain the estimates reported in eq . ( [ rexp ] ) , i.e. , @xmath208 , @xmath209 and @xmath210 . the final estimates and their errors take into account the results of the fits with and without scaling corrections . the same analyses can be performed at @xmath102 . combined fits to eq . ( [ scaling - r ] ) without scaling corrections give the results reported in table [ table - r ] . it is quite clear that scaling corrections at @xmath102 are larger then those at @xmath101 . the goodness of the fit is worse and the fit results show systematic trends . it is however reassuring that they apparently converge towards the estimates ( [ fgexp ] ) and ( [ rexp ] ) , in agreement with universality . it is interesting to check whether scaling corrections can explain the differences which occur among the results for @xmath44 reported in table [ table - r ] and the results obtained at @xmath101 . since the results for @xmath121 at @xmath102 are nonmonotonic as a function of @xmath194 , at least two correction terms must be included to explain the observed trend of the data . therefore , the fit of the @xmath89 data with a single scaling correction makes no sense . in any case , the estimate obtained for @xmath211 differs from the one reported in eq . ( [ rexp ] ) by one combined error bar , and therefore is in agreement with universality . we then perform combined fits of @xmath88 and @xmath130 to eq . ( [ scaling - r ] ) , approximating @xmath198 by a second - order polynomial and fixing @xmath122 to several values between 0.5 and 1.5 . the smallest @xmath116/dof is obtained for @xmath212 . correspondingly , we obtain @xmath213 , @xmath214 , and @xmath215 . the estimates of the phenomenological couplings at criticality are now in very good agreement with the estimates at @xmath101 . as for @xmath113 we obtain @xmath118 , which is sligthly smaller than , but still consistent with the estimate at @xmath101 . if we fix @xmath216 as obtained at @xmath43 , we find @xmath124 , @xmath217 , @xmath218 , @xmath219 . these fits provide an estimate of @xmath112 at @xmath44 . we quote the estimate @xmath220 already reported in eq . ( [ fgexpt1 ] ) , which satisfies the inequality @xmath221 , which one would obtain from the results reported in table [ table - r ] . it is unclear how reliable our estimates of @xmath122 are . in any case , they suggest a value close to 1 . .estimates of the exponent @xmath146 obtained by fits to eq . ( [ fit - chi ] ) , where @xmath222 is approximated by a fourth - order polynomial and @xmath223 by a second - order polynomial . in each fit we only include the data which satisfy @xmath195 . we fix @xmath123 and the value of @xmath112 : @xmath206 at @xmath101 and @xmath220 at @xmath102 . [ cols="<,^,^,^,^ " , ] in order to study hyperscaling we consider the ratio @xmath224 if hyperscaling holds , it should behave as @xmath225 \approx f_h[(p - p_c ) l^{1/\nu } ] , \label{scaling - h}\ ] ] where we have neglected scaling corrections . in order to allow for a possible hyperscaling violation we introduce a new exponent @xmath153 and assume that @xmath226.\ ] ] to determine @xmath153 we perform fits to @xmath227,\ ] ] where @xmath228 is approximated by a second - order polynomial . fit results are reported in table [ table - zeta ] . here we fix @xmath113 and @xmath112 to the values determined above . the quality of the fits is very good and scaling corrections are apparently small for both values of the temperature . the exponent @xmath153 is clearly compatible with zero , proving that hyperscaling is satisfied . more precisely , we obtain the bound @xmath229 , already reported in eq . ( [ zetaest ] ) . m. hasenbusch , f. parisen toldin , a. pelissetto , and e. vicari , j. stat . mech . : theory exp . ( 2007 ) p02016 ; p. calabrese , v. martn - mayor , a. pelissetto , and e. vicari , phys . e * 68 * , 036136 ( 2003 ) . an interpolation of the pf data with the correct scaling behavior at the multicritical point is provided by @xmath230 , with @xmath231 and @xmath232 . in fig . [ tp ] the fg line with @xmath233 is approximated by the line of equation @xcite @xmath234 , where @xmath235 ( which is fixed using our numerical estimate of @xmath112 at @xmath44 ) , @xmath236 , and @xmath106 is the position of the multicritical point . for @xmath237 we report the straight lines connecting the data at @xmath238 and @xmath239 . analogously , we proceed for the pg line , reporting the curve @xmath240 for @xmath241 and a straight line for @xmath242 .

we investigate the ferromagnetic - glassy transitions which separate the low - temperature ferromagnetic and spin - glass phases in the temperature - disorder phase diagram of three - dimensional ising spin - glass models . for this purpose , we consider the cubic - lattice @xmath0 ( edwards - anderson ) ising model with bond distribution @xmath1 , and present a numerical monte carlo study of the critical behavior along the line that marks the onset of ferromagnetism . the finite - size scaling analysis of the monte carlo data shows that the ferromagnetic - glassy transition line is slightly reentrant . as a consequence , for an interval of the disorder parameter @xmath2 , around @xmath3 , the system presents a low - temperature glassy phase , an intermediate ferromagnetic phase , and a high - temperature paramagnetic phase . along the ferromagnetic - glassy transition line magnetic correlations show a universal critical behavior with critical exponents @xmath4 and @xmath5 . the hyperscaling relation @xmath6 is satisfied at the transitions , so that @xmath7 . this magnetic critical behavior represents a new universality class for ferromagnetic transitions in ising - like disordered systems . overlap correlations are apparently not critical and show a smooth behavior across the transition .

nowadays , the diffused innovation policies require frequent survival estimates based on necessarily small samples . that may happen when the reliability of technological products continuously improved must be monitored ; or when the efficacy of always - new chemotherapy must be promptly checked . in helping statisticians to choose a suitable survival model , careful consideration of the generative mechanisms of the involved random variable ( rv ) plays an important ( often neglected ) role . such consideration can supplement or even prevail over usual model selection procedures , when the observations are extremely few and , consequently , the information about the effective shape of the `` parent '' distribution ( i.e. the population distribution ) is very scarce . in this context , the paper provides the mathematical models of three typical generative mechanisms of the inverse weibull ( iw ) rv . so , the paper helps exploiting the iw model to give correct answers for some specific survival problems , found in biometry and reliability , for which it appears the natural interpretative stochastic model . doubtless , the iw rv is not widely known and so scarcely identified . the iw model is referred to by many different names like `` frechet - type '' ( johnson et al . 1995 ) , `` complementary weibull '' ( drapella 1993 ) , `` reciprocal weibull '' ( lu and meeker 1993 ; mudholkar and kollia 1994 ) , and `` inverse weibull '' ( erto 1982 ; erto 1989 ; johnson et al . 1994 ; murthy et al . . an early study of the iw model is reported in the unprocurable paper ( erto 1989 ) . however , it seems to be no comprehensive reference in the literature that studies the iw as survival model . this paper tries to do that specifically exploring its peculiar probabilistic and statistical characteristics . the peculiar heavy right tail of probability density as well as the upside - down bathtub ( ubt ) shaped hazard function of the iw model has been really found in several applications ( nelson 1990 ; rausand and reinertsen 1996 ; gupta et al . 1997 ; gupta et al . 1999 ; jiang et al . 2003 ) . also the inverse gamma , inverse gaussian , log - normal , log - logistic , and the birnbaum - saunders models show similarly shaped hazard rates ( glen 2011 ; klein and moeschberger 2003 ; lai and xie 2006 ) . however , a model incorrectly fitted to iw data may lead to very wrong critical prognoses , even despite its good fitting to the empirical distribution . in fact , especially when few observations are available , the empirical distribution contains scarce information about the shape of the far - right tail , which is the main and unusual feature of the iw distribution . so , the knowledge of primary generative mechanisms leading to the iw rv can help one not to miss its proper application in some real life peculiar circumstances , analytically shown in the following . obviously , the inverse of the iw data follows a weibull distribution . so the parameter estimates of the iw distribution can be easily obtained by applying to its reciprocal data the same standard procedures implemented in packages for the weibull model ( see murthy et al . the probability density function ( pdf ) of the iw rv @xmath0 with scale parameter @xmath1 and shape parameter @xmath2 is : @xmath3 it is skewed and unimodal for @xmath4 . the @xmath5th moment of the iw rv is @xmath6 and it exists if @xmath7 then the mean @xmath8 and the variance @xmath9 follows . the most distinctive applicative feature of the iw model is its heavy right tail . that is highlighted by the _ property n. 1 _ : `` the pdf of the iw model is infinitesimal of lower order than the negative exponential as @xmath10 goes to infinity . '' in fact , the ratio of the iw pdf ( [ eq1 ] ) ( setting @xmath11 , for simplicity ) to the negative exponential function goes to infinity as @xmath10 goes to infinity . the cumulative distribution function ( cdf ) @xmath12 , the survival function ( sf ) @xmath13 and the hazard rate ( hr ) @xmath14 are easily derived from ( [ eq1 ] ) : @xmath15 @xmath16 the hr is infinitesimal as @xmath10 goes to infinity . it is unimodal and belongs to the ubt class ( see glaser 1980 ) with only one change point : _ property n. 2 _ : `` the hr of the iw model has a unique global maximum between the mode @xmath17 and the value @xmath18 . '' the condition of maximum for the iw hr does not lead to a closed - form solution . however , taking the derivative of the logarithm of the iw hr ( and appropriately arranging the terms ) the necessary condition for the maximum of the hr implies that : @xmath19 the auxiliary functions @xmath20 and @xmath21 corresponding to the first and second members of this equation , have a unique intersection point . in the first quadrant these two functions are both increasing up to their maximum point , whose abscissa is for both functions equal to @xmath22 and then they are both decreasing and infinitesimal to the same order as @xmath10 goes to infinity . moreover , it is possible to verify that @xmath20 is null as @xmath10 goes to 0 , while @xmath23 is null for the iw mode @xmath24 . because of the following inequalities : @xmath25 we derive that the intersection point of the two auxiliary functions , that is the maximum point of the hr , falls between the mode @xmath26 and @xmath18 . the mean residual life ( @xmath27 , also called the life expectancy of the @xmath28 fraction of items lived longer than @xmath29 is : @xmath30 being @xmath31 the lower incomplete gamma function . the following _ property n. 3 _ stands : `` the @xmath27 function of the iw model is bathtub - shaped . '' this property can be deduced from the general results given in gupta and akman ( 1995 ) and is in agreement with the properties of the hr . so , the iw model belongs to the class of distribution for which the reciprocity of the shape of the hr and @xmath27 functions holds . specifically , the @xmath27 decreases from the initial value @xmath32 ( as @xmath10 goes to 0 ) to its minimum at the change point @xmath33 and then increases infinitely as @xmath10 goes to infinity . being @xmath34 ( e.g. , see lai and xie , 2006 , chap . 4 ) , the change point @xmath33 must solve the equation @xmath35 necessarily . in practice , this peculiar @xmath27 shape can be found , for example , in some biometry problems when the longer the patient s survival time from his tumor ablation the better his prognosis . if @xmath36 are i.i.d . random variables , the limit distribution for their maximum is the iw distribution ( [ eq2 ] ) ( johnson _ et al . _ 1995 ) . therefore , for instance , when a disease or failure is related to the maximum value of a critical non - negative variable , this generative mechanism can be considered . this generative mechanism differs from the following three new ones , since for these the time variable does play an explicit role in their modeling . let @xmath37 be a system deterioration index that , as such , is a strictly increasing function of the run time @xmath10 . at every intercept with the vertical line passing through @xmath10 , suppose that the uncertainty about @xmath37 can be reasonably fitted by a weibull pdf , with shape parameter constant and scale parameter @xmath38 , function of @xmath10 , modeled by a generic power law : @xmath39 if a threshold ( maximum , positive ) value allowed for @xmath37 exists , the system has the iw sf . in fact , consider a weibull random variable @xmath37 with pdf : @xmath40 , \\ y\ge 0,\quad v,\;u>0 \\ \end{array}\ ] ] where @xmath41 , the shape parameter , is constant , and @xmath42 , the scale parameter , is the drift function ( [ eq8 ] ) . if @xmath43 is the threshold ( maximum , positive ) value for @xmath37 , then : @xmath44 } .\ ] ] substituting @xmath45 back into the previous relationship , we obtain : @xmath46.\ ] ] on putting @xmath47 and @xmath48 the iw sf follows . this mechanism is found in many technological corrosion phenomena that give rise to failures only when they reach a threshold deepness @xmath49 the mechanism is found also in many biologic degenerative phenomena ( i.e. , gradual deterioration of organs and cells ) where the loss of function appears when the deterioration deep @xmath37 reaches a fixed threshold value . besides , this mechanism is found when tumors spread potential metastases with a dissemination probability proportional to their size @xmath37 . hence , a tumor size greater than a given threshold value @xmath43 causes a rate of occurrence of metastases which is really first increasing and then decreasing ( see le cam and neyman 1982 , p. 253 ) like the iw one ( [ eq3 ] ) . if the stress @xmath50 ( in the broad sense ) is a rv with distribution that can be reasonably fitted by a weibull model and the strength @xmath51 that opposes @xmath52 is a decreasing function of time @xmath10 that can be modeled by a generic power law : @xmath53 the resulting sf is the iw one . in fact , if the stress @xmath50 is a weibull random variable : @xmath54 and the strength @xmath51 that opposes @xmath52 follows the decreasing function of time ( [ eq12 ] ) : @xmath55 . \\ \end{array}\ ] ] substituting @xmath56 back into the previous relationship , we obtain : @xmath57\ ] ] then , renaming @xmath58 and @xmath48 the iw sf follows . this mechanism is common for many mechanical components ( see , for example , bury 1975 , p. 593 ; shigley 1977 , p. 184 ) as well as it is found in patients with a decreasing vital strength following the ( [ eq12 ] ) ( e.g. , because they are subjected to intensive and prolonged chemotherapy ) and subjected to a relapse having a random virulence or gravity @xmath59 in these cases , an hr first quickly increasing and then slowly decreasing , is sometimes surprisingly observed ( see carter et al . 1983 , p. 79 ) . suppose that a disease ( or failure ) is latent and the physiological defensive attempts averse to it occur randomly according to a poisson model . if the probability of one successful defensive attempt depends on the incubation time @xmath10 ( but not on the number of previously occurred defensive actions ) according to a generic power law decreasing function : @xmath60 the iw cdf follows . in fact , suppose that the random variable @xmath61 describing the physiological defensive attempts against a latent disease ( or failure ) , occurs according to a poisson law : @xmath62 let @xmath63 be the probability of one successful defensive attempt , which depends on the incubation time @xmath10 ( but not on the number of previously occurred defensive actions ) according to the function ( [ eq16 ] ) . consequently , the probability of manifest disease ( or failure ) is : @xmath64 then , on putting @xmath65 and @xmath66 the iw cdf follows . this mechanism is found in biometry when the immune system works randomly against antigens , and its effectiveness decreases as the disease expands ( see le cam and neyman 1982 , p. 15 ) . in reliability , this mechanism is found when a technological system is randomly ( i.e. , without any definite plan ) maintained : the smaller the time from the beginning of the failure process ( up to the maintenance action ) the greater the maintenance efficacy . consider the following 50 pseudo random ( ordered ) data generated from a close - to - standard " parent cdf ( [ eq2 ] ) with @xmath67 and @xmath68 ( we can not put @xmath69 since , in general , the @xmath5th moment of the iw pdf exists if @xmath70 ) : 0.2776 , 0.2931 , 0.3384 , 0.4321 , 0.4739 , 0.4771 , 0.5331 , 0.5424 , 0.5482 , 0.5571 , 0.6139 , 0.6451 , 0.6523 , 0.6587 , 0.7166 , 0.7838 , 0.8466 , 0.8892 , 0.9278 , 0.9651 , 1.008 , 1.051 , 1.123 , 1.203 , 1.213 , 1.366 , 1.529 , 1.795 , 1.947 , 2.093 , 2.143 , 2.189 , 2.246 , 2.453 , 2.526 , 2.858 , 2.924 , 3.381 , 3.383 , 3.587 , 4.964 , 5.101 , 5.139 , 6.753 , 10.11 , 11.37 , 12.68 , 16.88 , 17.25 , 19.07 . the anderson - darling statistic ( anderson and darling 1954 ) @xmath71 , with a @xmath72-value equal to 0.94333 , shows the high conformity of this sample to the parent cdf . incidentally , in this paper , we chose this specific goodness - of - fit test since it emphasizes the tails of the presumed parent distribution . however , in the above case , also tests that give less weight to the tails lead to similar results . suppose that we want to identify a generic cdf model being very well fitted to both the data and the parent cdf , but we do nt have any strong information about the latter . we decide to adopt a `` less informative model '' which is coherent with our poor information . we chose a polynomial cumulative hr ( hr ) model of order 3 , since it is the minimum able to fit a non - monotone model too . in our ( simulated ) condition , we can define an excellent `` a priori '' model by fitting the polynomial to 50 points ( vertically equally spaced ) of the known parent cdf . the resulting model is : @xmath73 which has a coefficient of determination @xmath74 . moreover , being the anderson - darling statistic @xmath75 , with a @xmath72-value equal to 0.2856 , this `` a priori '' model appears very well fitted to data too . incidentally , the maximum likelihood ( ml ) estimates of its three parameters give the following polynomial hr model very close to the former ( [ eq19 ] ) : @xmath76 which has a coefficient of determination @xmath77 . suppose now that the analysis of the generative mechanism suggests us to fit the iw model to the 50 data . the ml estimates of its parameters are @xmath78 and @xmath79 . the coefficient of determination of the hr function estimated from this iw model is @xmath80 . the anderson - darling statistic is @xmath81 with a @xmath72-value equal to 0.9530 . although the previous analysis has shown that the two cdf models fit the data very well , some important characteristics could be different . to highlight that , we compare some critical estimates obtained from the `` a priori and less informative '' model ( [ eq19 ] ) with those obtained using the last `` fitted and informative '' iw model . from these two models we obtain the @xmath27 estimates reported in [ tab1 ] , where the true values are those of the parent population . .@xmath27 estimates for the polinomial and iw fitted models [ cols="<,<,<,<",options="header " , ] [ tab2 ] this example is representative of the critical real - world situations in which only tiny data sets are available . the dataset consists of 15 times to breakdown ( in minutes ) of an insulating fluid between electrodes at a constant voltage @xmath82 ( 36 kv ) , provided in nelson ( 1982 , p. 105 ) : 0.35 , 0.59 , 0.96 , 0.99 , 1.69 , 1.97 , 2.07 , 2.58 , 2.71 , 2.90 , 3.67 , 3.99 , 5.35 , 13.77 , 25.50 . unfortunately , due to small size of the sample , we can not rely on the sample point @xmath83 on the graph of [ fig1 ] to start the selection of a reasonable model . however , analyzing the experiment ( aiming to derive the lifetime distribution of the insulating fluid ) we come to the conclusion that it shows an example of the `` deterioration '' mechanism close to the one described in section 3.1 . in fact , the mean of the insulating resistance @xmath84 of the fluid decreases according to a positive ( and less than one ) power function of time . this model belongs to the arrhenius class of cumulative damage relationships , widely found in life tests with constant stress ( see , e.g. , nelson 1990 ) . consequently , the mean of the resistive leakage current @xmath85 ( i.e. , the system deterioration index @xmath86 increases with a positive ( and greater than one ) power of time to the dielectric failure , which occurs when a threshold value @xmath43 ( fixed by the operating and environmental conditions supposed constant ) is exceeded . moreover , the nature of the failure mechanism is stationary and does not induce any change in the shape of the @xmath87 pdf . then , a pdf model with mean increasing as a power function of time and with constant shape is well rendered by the weibull model ( [ eq9 ] ) . in fact , being constant the shape parameter @xmath41 , its mean @xmath88 is effectively a positive ( and greater than one ) power function of the time . hence we decide to assume the iw model as our weighted hypothesis . however , we consider also the log - logistic model because , as shown in [ fig1 ] , it plays the role of a frontier separating the iw model and many other alternative models . the ml estimates of the iw parameters are @xmath89 and @xmath90 ; the anderson - darling statistic is @xmath91 with a @xmath72-value equal to 0.596 ; the _ mll _ is @xmath92 . the ml estimates of the log - logistic parameters are @xmath93 and @xmath94 and the anderson - darling statistic is @xmath95 with a @xmath72-value equal to 0.870 ; the maximized log - likelihood is @xmath96 . the comparison of the two alternative models by means of the anderson - darling statistic and the _ mlls _ ( both at their ml value ) would support the log - logistic model . however , we think that the differences are not enough large ( e.g. only 0.3 unit separates the two _ mlls _ ) to contradict the previous choice based on a careful and detailed technological analysis . the paper proves that the iw distribution is another of the relatively few ubt survival distributions . so , when dealing with ubt distributions , it is helpful to have an alternative model that has , moreover , a distinctive heavy right tail . this paper demonstrates how the iw distribution is the natural candidate , among all the survival models , to face three unreported classes of real and well defined degenerative phenomena . so the practitioners are helped to choose this model by profiting from the knowledge of the involved phenomena , such as a disease or failure , rather than exclusively on the usual analysis of goodness - of - fit . some illustrative examples show that the polynomial cumulative hazard model and the log - logistic one can both fit the cdf of iw data very well . the polynomial model is used as antithetic benchmark because : a ) differently from the iw model , it is capable of giving a wide range of hr shapes ; b ) it is used in situations where strong assumptions about the parent distribution are unavailable . the log - logistic model has been considered because : a ) it is the closest model which shares the upside - down bathtub ( ubt ) shaped hazard function ; b ) it plays the role of a frontier separating the iw model from many other alternative models . however , all the illustrative examples show that the above models even though very well fitted to iw data may be very misleading because they entail highly incorrect assessments concerning , for instance , the mean residual life . the paper proves that when any knowledge about generative mechanism is unavailable selecting between the iw and the log - logistic models that one which minimizes the anderson - darling statistic or , even better , maximizes the likelihood is a very effective procedure . finally , we show that for the iw and log - logistic models both selection criteria are independent of hypothetical distribution parameters , and the corresponding probabilities of correct selection are respectively greater than 0.85 and 0.93 when the size of the available sample is greater than 50 . instead , when the size of the available sample is less than 30 ( i.e. , in a very frequent situation in the technological and biological fields ) selecting the correct model purely on the basis of the empirical distribution remains a highly risky procedure , since the probabilities of wrong selection are respectively greater than 0.23 and 0.12 . + * references *

the peculiar properties of the inverse weibull ( iw ) distribution are shown . it is proven that the iw distribution is one of the few models having upside - down bathtub ( ubt ) shaped hazard function . three real and typical degenerative mechanisms , which lead exactly to the iw random variable , are formulated . so a new approach to proper application of this relatively unknown survival model is supported . however , we consider also the case in which any knowledge about generative mechanism is unavailable . in this hypothesis , we study a procedure based on the anderson - darling statistic and log - likelihood function to discriminate between the iw model and others alternative ubt distributions . the invariant properties of the proposed discriminating criteria have been proven . based on monte carlo simulations , the probability of the correct selection has been computed . a real applicative example closes the paper . = 4 mean residual life , model selection , ubt shaped hazard rate

recent discoveries of extra - solar giant planets stars have raised questions about their formation ( e.g. * ? ? ? * ; * ? ? ? indeed , their characteristics have been a surprise : they orbit much closer to the stars than the planets in our own solar system and their masses range from that of saturn up to 10 times the mass of jupiter ( @xmath20 m@xmath21 ) . these planets are expected to contain a solid core surrounded by a shell of metallic hydrogen and helium and an outer low pressure atmosphere where hydrogen is in the form of h@xmath16 @xcite . to build such gaseous giant planets , a large reservoir of h@xmath16 gas is needed at the time of their formation , most likely in the form of a circumstellar disk ( e.g. * ? ? ? * ; * ? ? ? most studies of circumstellar material associated with young stars and debris - disk objects rely on continuum observations of the infrared to millimeter emission produced by heated dust ( e.g. * ? ? ? * ; * ? ? ? dust particles represent only a trace component of disks , however , which have 99% of their mass initially in the form of h@xmath16 gas . line imaging of trace molecules such as co with millimeter interferometers reveals the presence of gas in circumstellar disks with sizes of @xmath22100 - 400 au , but the inferred masses are up to two orders of magnitude lower than those deduced from the dust continuum assuming a standard gas / dust ratio and co / h@xmath16 conversion factor as in molecular clouds ( e.g. * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? the millimeter observations have nevertheless provided compelling evidence for gas in keplerian rotation around the central star ( e.g. * ? ? ? * ; * ? ? ? we report here the result of the first spectral survey of the pure - rotational h@xmath23 emission lines from circumstellar disks , the only molecule hich can directly constrain the reservoir of warm molecular gas . a related question is the temperature structure of the circumstellar disks . the radial temperature structure is usually constrained by modeling of the spectral energy distribution assuming either a thin , flat disk geometry ( e.g. * ? ? ? * ) or a flaring disk ( e.g. * ? ? ? * ; * ? ? ? the dust in these models is heated by radiation from the central star and by the release of energy through accretion . recent calculations by different groups show substantial differences , however ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . specifically , flared disks may have surface layers with temperatures in excess of 100 k out to @xmath24100 au @xcite . the fitting of spectral energy distributions is known to give ambiguous answers and many disk parameters are still debated because of the non - uniqueness of the fits ( e.g. * ? ? ? * ; * ? ? ? h@xmath13 emission line data can provide direct measurements of the temperature of the warm gas . according to standard models ( e.g. * ? ? ? * ; * ? ? ? * ) , giant planet formation by core accretion of gas occurs in the first few millions years . thus , the timescale for the disappearance of the gas compared with that of the dust is of interest . based on continuum data , @xcite , @xcite , @xcite and @xcite suggested that dust disks around t tauri stars disappear at an age of a few million years . @xcite searched for evolutionary trends in the outer disk dust mass around herbig ae stars . they found no evidence for changes between 10@xmath25 and 10@xmath18 years , but an abrupt transition seems to occur at 10@xmath18 years from massive dust disks to tenuous debris disks . @xcite conducted a survey of co emission from a - type stars with ages between 10@xmath1710@xmath18 years and concluded that the gaseous disks disappear within 10@xmath18 years . determination of the gaseous mass from co data is hampered , however , by several difficulties compared to that from h@xmath26 . provided that h@xmath16 traces the bulk of molecular gas , it can constrain the time scale for gas dissipation from the disk directly . observations of the pure - rotational lines such as the h@xmath16 @xmath9=20 s(0 ) 28.218 @xmath27 m and @xmath9=31 s(1 ) 17.035 @xmath27 m lines are difficult from the ground because of the low terrestrial atmospheric transmission in the mid - infrared . the short wavelength spectrometer ( sws ) on board the _ infrared space observatory _ ( iso ) has allowed the first opportunity to observe a sample of t tauri and herbig ae stars , as well as a few young debris - disk objects . the small mass of h@xmath16 implies that the two lowest rotational lines have upper states which lie at rather high energies , 510 k and 1015 k above ground , respectively . the @xmath9=20 and @xmath9=31 transitions are thus excellent tracers of the ` warm ' ( @xmath8 80200 k ) component of disks . the mid - infrared h@xmath28 data provide complementary information to ultraviolet h@xmath28 emission @xcite or absorption @xcite data toward circumstellar disks , which either probe only a small fraction of the h@xmath28 or depend on the line of sight through the disk and foreground material . h@xmath28 has also been detected at near infrared wavelengths @xcite , but since these lines are excited by ultraviolet radiation , x - rays or shocks , they also can not be used as a tracer of mass . spectroscopic observations of h@xmath16 have several advantages over other indirect methods . first , since it is the most abundant gaseous species , no conversion factor is needed . also , contrary to co , which has a condensation temperature of @xmath2920 k @xcite , it does not freeze effectively onto grain surfaces unless the temperatures fall below @xmath292 k @xcite lower than the minimum temperature that a disk reaches . its photophysics and high abundance allow h@xmath16 to self - shield efficiently against photodissociation by far - ultraviolet photons , such as those produced by a - type stars @xcite . moreover , because the molecule is homonuclear , its rotational transitions are electric quadrupole in nature , and thus possess small einstein a - coefficients . on the one hand , this presents an observational problem since high spectral resolution is required to see the weak line on top of the usually strong mid - infrared continuum . on the other hand , the benefit is that the lines remain optically thin to very high column densities , making the radiative transfer simple . another disadvantage is that the lines are only sensitive to warm gas and can not probe the bulk of the ( usually ) cold circumstellar material probed by co @xmath30 and @xmath31 interferometric observations . also , the high continuum optical depths at 28 @xmath32 m prevent observations into the inner warm mid - plane of the disk . as a complement , the same stellar sample has therefore been observed in the @xmath6co and @xmath7co @xmath33 lines with the _ james clerk maxwell telescope _ and the @xmath6co @xmath34 line with the _ caltech submillimeter observatory_. these transitions probe lower temperatures than h@xmath0 , about 2080 k in the regime where the dust is optically thin . the combination of h@xmath16 and co observations is sensitive to the full temperature range encountered in disks . along with millimeter continuum observations taken from the literature , such data can provide a global picture of the structure and evolution of both the gas and dust components of circumstellar disks . the paper is organized as followed . we first justify the choice of the objects in our sample ( 2 ) . in 3 , a description of the observations is provided with emphasis on the special data reduction method used for the h@xmath16 lines . in 4 and 5 , the data are presented and physical parameters such as mass and temperature are derived from our observations of h@xmath16 and co lines , as well as from 1.3 millimeter continuum fluxes taken from the literature . the accuracy of each method is assessed . in 6 , the different results are compared and possible trends with effective temperature of the star or age are investigated and the possible origin of the warm gas is mentioned briefly . finally , a discussion of the gas content in debris - disk objects is given . the results for one object , the double binary gg tau , have been presented by @xcite . earlier accounts of this work may be found in @xcite and @xcite , whereas the debris - disk sources are discussed in @xcite . @xcite present searches for h@xmath16 emission in a complementary set of weak - line t tauri objects . our study focuses on two classes of pre - main - sequence stars with transitional ages spanning 10@xmath1710@xmath18 years . t tauri stars in the sample have spectral types of me and ke , corresponding to stellar masses in the range from 0.25 to 2 m@xmath21 and are probably younger analogs to the sun . the higher - mass herbig ae stars ( 23 m@xmath21 ) share the spectral type of debris - disk sources and may be considered as younger counterparts to the debris - disk objects . in addition , three young debris - disk objects , namely 49 ceti , hd 135344 and @xmath35 pictoris are included in our sample . the choice of objects is based on several criteria in order to maximize the chance to detect the faint h@xmath16 lines on top of the mid - infrared continuum and to avoid confusion with emission from remnant molecular cloud material . first , the observed stars exhibit the strongest 1.3 millimeter fluxes in the survey of t tauri stars by @xcite and herbig ae stars by @xcite , i.e. , they possess the highest dust disk masses among the t tauri and herbig ae stars in the taurus - auriga cloud . second , they have all been imaged with millimeter interferometers in co and dust continuum and show evidence for keplerian disks . the only exceptions are ux ori and ww vul , where no co is detected . third , the sample is biased toward sources with a weak mid - infrared continuum at 1030 @xmath32 m to improve the line - to - continuum contrast . this also prevents instrumental fringing problems . a faint mid - infrared excess suggests that a ` dust hole ' exists in the disk close to the star , which may be caused by settling and coagulation of dust particles in the mid - plane @xcite , to clearing of the inner part of the disk by small stellar companion(s ) or proto - planet(s ) ( e.g. * ? ? ? * ) or to shadowing of part of the disk @xcite . finally , most of these stars are located in parts of the taurus cloud where the co emission is very faint or absent . our original sample also included objects in ophiuchus @xcite , but these have been discarded from this sample because of confusion by cloud material . hd 135344 , 49 ceti and @xmath19 pictoris have been identified as debris - disk objects based on their far - infrared excess above the expected photospheric flux level ( e.g. backman & paresce 1993 ) . keck 20 @xmath27 m images reveal the presence of dust disks around the first two sources ( koerner 2000 , silverstone et al . , private communication ) whereas @xmath19 pictoris has been imaged at many wavelengths ( e.g. * ? ? ? hd 135344 , however , shows strong single - peaked h@xmath36 emission @xcite , suggesting that it also has herbig ae - type characteristics . the three debris - disk sources are objects located far from any molecular cloud . this work does not constitute a statistical study since the sample is limited in number and biased toward the highest disk masses and low mid - infrared continuum . in table [ sources ] the stellar properties of objects of our sample are tabulated , including coordinates , effective temperature , luminosity , and distance , together with references to relevant literature . the h@xmath16 @xmath37 s(0 ) line at 28.218 @xmath27 m and the @xmath38 s(1 ) line at 17.035 @xmath27 m were observed with the iso - sws grating mode aot02 @xcite . typical integration times were 6001000 s per line , in which the 12 detectors were scanned several times over the 28.0528.40 and 16.9617.11 @xmath27 m ranges around the lines . the h@xmath16 @xmath39 s(3 ) 9.66 @xmath27 m and @xmath40 s(5 ) 6.91 @xmath27 m lines were measured in parallel with the s(0 ) and s(1 ) lines , respectively , at virtually no extra time . the spectral resolving power @xmath41 for point sources is @xmath222000 at 28 @xmath27 m and @xmath222400 at 17 @xmath27 m . the sws aperture is @xmath42 at s(0 ) , @xmath43 at s(1 ) , and @xmath44 at s(3 ) and s(5 ) . for a few sources , observations of the s(1 ) line at a @xmath45 off position have been obtained as well . the s(2 ) j=42 12 @xmath27 m line was also searched for toward 49 ceti and hd 135344 . the continuum provides narrow band photometry . since the observing procedure does not perform spatial chopping , no zodiacal or background emission is subtracted . the zodiacal background component has a continuous spectrum corresponding to a dust temperature of about 260 k @xcite with an estimated flux density in the sws aperture of about 0.3 jy , so that it can contaminate the continuum emission in some of our faintest objects . continuum fluxes above 3 jy are considered as coming essentially from the sources ( star+disk ) alone . the expected peak flux levels of the h@xmath16 lines are close to the sensitivity limit of the instrument . in order to extract the h@xmath16 lines , special software designed to handle weak signals on a weak continuum was used for the data reduction in combination with the standard interactive analysis package . the details and justification of the methods used in the software are described elsewhere @xcite and summarized below ( see also the iso - sws manual at http://www.iso.vilspa.esa.es$/$users$/$expl_lib$/$sws_top.html ) . the raw data consist of 12 non - destructive measurements per elementary integration ( reset ) corresponding to the 12 single - pixel detectors , hence 24 observed points for a 2 second reset . a single scan lasts 200 seconds and typically 35 scans per line have been obtained , corresponding to 720012000 data points . since the readout system acts as a capacitor , the signal has the form of an exponential decay , and this curvature is first corrected using the ac time constant obtained during the pre - flight calibration phase . then a correction of the instantaneous response function , or ` pulse - shape ' , is applied with the level of the correction determined from the data themselves because the shape varies in time , a procedure called ` self - calibration ' . finally , a cross - talk correction is performed . this chain of calibration results in removing the curvature and improving the straightness of the observed slope which is in fact the measure of the flux . it also increases appreciably the photometric accuracy and allows a better subsequent determination of the noise . other factors , such as dark current drifts , influence the sensitivity limit of the instrument as well and have to be corrected . the majority of noisy data points are actually caused by impacts of cosmic rays , called glitches , either on the detectors or on the readout electronics . the level of cosmic ray hits fluctuates markedly , depending on the position of the satellite and the activity of the sun . the rate of glitches may vary from scan to scan . at the level we are interested in , up to 50% of the data points can be rendered unusable by cosmic rays or other instrumental artifacts . cosmic rays not only affect the sensitivity of the detectors instantaneously , but also for some longer recovery time , a phenomenon called the post - glitch effect . most of the time , the glitches are secondary electron - hole pairs created by the interaction of the energetic particles with the detector elements ; while the lifetime of these pairs is short , other consequences of the impact can last longer . the decay of this effect is observed to have an exponential form . the observing procedure used by the sws allows investigators to track events emerging simultaneously in more than one detector . these so - called ` correlated - noise events ' appear as a spurious feature in emission or sometimes in absorption with a gaussian profile whose width is close to the resolution of the instrument . the gaussian - like profile comes from the fact that the glitch affects several detectors simultaneously , which results in a shift in wavelength in the final spectrum . in order to detect and circumvent the glitches , four types of statistical filters have been defined . the first two are standard filters also employed in the sws pipeline software ; the last two are additions by us . the software is written in _ idl _ ( interactive data language ) . each of these filters generates an array of non - valid points detected by the adopted statistical method characterized by a unique parameter . thus , careful choices of filter parameters are crucial in determining the quality of the resulting spectrum . the arrays are then cross - correlated . most of the time , the glitches are detected by more than one filter and those points are immediately discarded . the first filter consists in removing points which have a flux outside a specified range defined by the user . this procedure may seem artificial , but is justified by the fact that both line and continuum fluxes are faint . in our data , this method removes points 5 sigma above the continuum standard deviation calculated using all points . the second filter searches for data points with a standard deviation of the slope - fitting higher than the standard value adopted in the sws pipeline . this filter is efficient when used after the self - calibration procedure described above . the third filter has been set up specifically in this work to detect correlated noise . this filter detects the glitches which are discrete stochastic events in the time domain . the data from the 12 detectors observed at a single time are summed , and the mean and standard deviations are computed . if the standard deviation is higher than a specified parameter @xmath46 , the data points are considered as glitches and are discarded in all 12 detectors . the value of @xmath46 , which is a multiple of the standard deviation @xmath47 , @xmath48 , is difficult to determine _ a priori _ and can vary from scan to scan . indeed , the computed @xmath47 is affected by the number of cosmic ray hits a high rate of glitches results in a high standard deviation so that @xmath46 has to be small . we have therefore used an automated procedure to find the optimum values of @xmath49 , in which each spectrum is examined with a range of values of @xmath49 from @xmath50=16 times the standard deviation for each individual scan @xmath51 with a step of 0.5 . thus , for a typical case of 3 scans , 10@xmath52 versions of the reduced spectrum are generated . the fourth step removes additional points one or two resets after a glitch is detected by the previous technique . the data reduction procedure results in a `` dot cloud '' of observed fluxes as functions of wavelength . as a final step , convolution with a gaussian whose fwhm is set by the theoretical resolution of iso - sws at the relevant wavelength is done . we have chosen to use a flux - conserving interpolation which can modify the resolution but does not change the total integrated flux . since the lines are not spectrally resolved , the line profile is not relevant . small velocity shifts of the line of order @xmath53 km s@xmath54 compared with the rest wavelengths are frequent . many parameters can cause such a shift , including the low signal - to - noise of the data or pointing offsets . the latter problem not only affects the peak position but also the flux since the beam profile is highly dependent on the position in the entrance slit of the spectrometer . because the h@xmath16 emission can arise from a region 12@xmath55 offset compared to the position of the star , additional shifts of the order of a few tens of km s@xmath56 are possible the 1000 spectra are then sorted by number of remaining data points . generally , the noise level due to glitches tends to decrease significantly as the number of points decreases until a minimum is reached when the statistical noise takes over because of the small number of data points left . with this non - standard data reduction procedure , it is difficult to devise an objective detection criterion . therefore , we adopt the following definition of the level of confidence in our detections , depending on the final @xmath57 of the spectrum as well as the fraction of reduced spectra in which the line is clearly seen . a line is considered to be detected when the @xmath57 is 3 or higher and if its profile lies within a gaussian mimicking the line profile of an extended source filling the entire beam . observations which are only slightly affected by cosmic ray hits show detections in a large number of the reduced spectra ( @xmath58 75% of the 1000 spectra ) . the level of confidence of the detection is considered high " in those cases . the level becomes medium " when the detection is present in about 5075% of the spectra . in cases of non - detection , the line is seen in less than 50% of the reductions . ultimately , we can not rule out possible instrumental artifacts which are not detected by our filters . of all the possible reductions , the spectrum with the lowest continuum fluctuation ( fringing ) and noise and the highest @xmath57 of the line is kept as our best reduced spectrum and plotted in this paper . the criterion of high peak flux and @xmath57 comes from the fact that the filters described above eliminate not only noisy data points but also some valid points to a certain level . to keep this level as low as possible , a compromise between quality ( i.e. , @xmath57 ) and flux level is adopted . the non - gaussian nature of the noise makes the overall error difficult to estimate , and we assume a fiducial 30% photometric uncertainty in the rest of the paper . this error is propagated into all the resulting temperatures and masses . the actual uncertainty may be larger due to the low @xmath59 of the data , but can not be quantified in a consistent way for different sources . note that the above procedure only throws away data points and therefore can not create artificial lines . this is confirmed by the absence of lines at blank sky , or off - source , positions reduced with the same procedure . the above method was adopted for all sources with a weak continuum level ( @xmath603 jy ) . for sources with a strong mid - infrared continuum ( ab aur , hd 163296 , ry tau , cq tau , mwc 863 ) , the fringing effect on the continuum becomes the limiting factor for detection . errors in the dark current subtraction are a possible cause of this fringing . for these sources , the fringes have been minimized by varying the dark current level . as a complement to the iso - sws data , we have observed the same sample of t tauri and herbig ae stars in various moderate- to high-@xmath9 co transitions between 1998 and 2000 with submillimeter single - dish telescopes . previous studies have observed the lowest @xmath9=10 and/or @xmath61 transitions , either with interferometers @xcite or with single dishes ( e.g. * ? ? ? * ) , but no homogeneous data set using the same line , isotope and telescope exists for our sources . we focus here on the higher-@xmath9 32 and 65 transitions to probe gas with @xmath62=2080 k. observations of the @xmath6co and @xmath7co @xmath33 lines were carried out at the _ james clerk maxwell telescope _ ( jcmt ) using the dual polarization receiver b3 as the frontend and the digital autocorrelator spectrometer ( das ) as the backend . data were acquired with a beam switch of 180@xmath55 and , in cases of extended emission , also a position switch up to @xmath63 . to check for extended emission , several positions offset by @xmath64 were observed as well . since the fwhm beam size of the jcmt at 345 ghz is 14@xmath55 and the extent of the disks at the distance of taurus is at most 5@xmath55 , the observations suffer from large beam dilution , as do the h@xmath16 data . the receiver was tuned single sideband , with typical system temperatures above the atmosphere ranging from 400600 k. the spectral resolution was typically 0.13 km s@xmath56 , sufficient to resolve the line profiles , but the data are hanning - smoothed once to improve the signal - to - noise . the final spectral resolution is 0.26 km s@xmath65 . integration times were typically 1020 minutes for @xmath6co and up to 2 hrs for @xmath66co reaching a typical rms noise of @xmath29 15 mk . the antenna temperatures have been converted to main beam temperatures using a main beam efficiency at 330 ghz of @xmath67=0.62 obtained from observations of planets by the jcmt staff ( see http://www.jach.hawaii.edu/jacpublic/jcmt/rx/b3/cal.html ) . the data reduction was performed using the specx and class software . the @xmath6co @xmath34 data were obtained with the _ caltech submillimeter observatory _ ( cso ) using the sensitive 650 ghz receiver of @xcite in double - sideband mode . two acousto - optical spectrometers with resolutions of 0.05 and 0.5 km s@xmath56 were used as the backends . typical system temperatures under excellent weather conditions were @xmath68 k. the cso beam size at 650 ghz is @xmath69 , comparable to the jcmt beam at 330 ghz , and the main beam efficiency is @xmath67=0.40 . the final continuum subtracted h@xmath16 spectra are presented in figures [ h2s0 ] and [ h2s1 ] for the @xmath9=20 s(0 ) and 31 s(1 ) lines respectively . the typical rms noise level is 0.20.3 jy . the dash - dotted lines in figure [ h2s0 ] and [ h2s1 ] indicate the wavelength range in which the h@xmath16 line is expected , taking into account the possible velocity shifts discussed in 3 . as explained in 3 , our line profiles may differ from the nominal instrumental profile in width because of the adopted interpolation scheme . the h@xmath16 line positions and basic molecular data are listed in table [ h2data ] , whereas the h@xmath16 s(0 ) and s(1 ) integrated fluxes are reported in table [ h2iso ] . the s(3 ) and s(5 ) lines are not detected in any of the objects with an upper limit of @xmath704@xmath7110@xmath72 erg s@xmath56 @xmath73 ( 3@xmath74 ) . the level of confidence of a detection is indicated in the right - hand column of table [ h2iso ] . similarly , the s(2 ) line is not detected toward 49 ceti or hd 135344 with an upper limit of @xmath299@xmath7510@xmath72 erg s@xmath54 @xmath73 . both the s(2 ) and s(3 ) lines are located in a wavelength region where silicate features in emission or absorption are strong . lines are detected in several disks around t tauri and herbig ae stars , with no apparent trend with age or spectral type ( see 6.2 ) . there are also likely detections of lines toward the debris - disk objects , especially from hd 135344 and @xmath76 pictoris . the s(1 ) line shows a wider spread in observed fluxes and is more readily detected for several reasons . first , the einstein-@xmath77 coefficient for the @xmath9=@xmath78 line is a factor of 16.5 larger than that of the @xmath79 line . also , the spectral resolution is somewhat higher at 17 @xmath32 m than at 28 @xmath32 m and the continuum lower , so that the line - to - continuum ratio is larger . finally , the sensitivity of the 17 @xmath32 m detectors is better . all of these factors explain why the s(1 ) line is more easily seen than the s(0 ) line , in spite of the fact that the @xmath9=3 level has a factor of 40 lower population than the @xmath9=2 level in gas with an estimated temperature of around 100 k. except for the case of @xmath35 pictoris , the iso - sws beam is much larger than the typical sizes of the circumstellar disks of @xmath605@xmath81 . thus , care has to be taken that the h@xmath16 emission is not affected by any remnant cloud or envelope material in the beam . observations of the s(1 ) line have been obtained at several off source positions @xmath45 south . toward 49 ceti and hd 135344 , which are far away from any molecular cloud , no emission is detected off source at the level of 8@xmath8210@xmath72 erg s@xmath56 @xmath73 rms , consistent with the expectation that diffuse atomic gas does not emit in h@xmath16 lines . a weak s(1 ) line of @xmath29 10@xmath83 erg s@xmath56 @xmath73 is seen @xmath45 south of lkca 15 . this flux probably comes from a background cloud at a different velocity than that of the source ( see below ) . strong h@xmath16 lines have been detected with the sws toward embedded herbig ae and t tauri stars where ultraviolet photons and shocks interact with the surrounding material , but in these cases the observed excitation temperatures of 500700 k are much higher than those found for our objects @xcite . searches for h@xmath16 lines toward diffuse molecular clouds with @xmath84 mag have been performed by @xcite , but no lines are detected at the level of 8@xmath8210@xmath72 erg s@xmath56 @xmath73 rms for clouds with densities less than @xmath85 @xmath86 and incident radiation fields less than 30 times the standard interstellar radiation field . the strengths of the s(0 ) and s(1 ) lines from diffuse clouds can also be estimated from ultraviolet observations of h@xmath26 obtained with the _ copernicus _ satellite and the _ far - ultraviolet space explorer _ ( fuse ) consider as an example the recent fuse results for the translucent cloud toward hd 73882 ( @xmath87=2.4 mag ) by @xcite . the observed column densities in @xmath9=2 , 3 and 5 translate into fluxes of @xmath88 , @xmath89 and @xmath90 erg s@xmath56 @xmath73 for the s(0 ) , s(1 ) and s(3 ) lines , respectively , assuming the gas fills the iso - sws beam . the s(0 ) and s(1 ) fluxes are comparable to our observed values , but the s(3 ) flux is significantly higher than our upper limits . indeed , both the _ copernicus _ and fuse data give typical excitation temperatures for the @xmath9=27 levels of @xmath29300 k , significantly larger than the values of @xmath29100200 k found here . moreover , such thick clouds as those toward hd 73882 or @xmath91 oph emit significant co emission ( e.g. * ? ? ? * ; * ? ? ? * ) , which is generally not observed at the off source positions in our sample . to check for the presence of molecular gas at off - source positions , mini - maps in @xmath6co 3 - 2 have been obtained in steps of 30@xmath55 for most of our sources ( see also 5 ) . for gg tau , lkca 15 , mwc 480 and gm aur , no emission is found off source down to 30 mk rms at the velocity of the sources . in other cases such as go tau , dr tau , hd 163296 , weak off - source emission is seen in @xmath6co 32 with velocities shifted compared to the source velocity , but this off - source emission is not seen in @xmath7co 32 . the off - source co emission is at least a factor of 30 lower than found for translucent clouds such as hd 73882 . thus , the bulk of the molecular gas for these sources is clearly located in the disks , but some lower - density cloud material may be present . based on the above arguments combined with the absence of s(3 ) emission , this diffuse gas is expected to make only a small contribution to the s(0 ) and s(1 ) lines . finally , two of our objects , ab aur and ry tau , show single dish co data which are clearly dominated by more extended remnant envelope material . in these cases , a significant fraction of the h@xmath16 emission may arise from extended gas although the temperature in the envelope ( 1020 k ) may be too low to produce substantial rotational excitation . in summary , for most of our sources , the h@xmath16 emission is unlikely to be contaminated by extended emission from diffuse molecular gas , but this can not be ruled out for cases such as ab aur . in fact , h@xmath28 ultraviolet absorption toward ab aur has been detected by fuse @xcite and arises in an extended low - density envelope around the star or from general foreground material . the integrated flux @xmath92 of a rotational emission line @xmath93 of h@xmath16 , assuming that the line is optically thin and not affected by dust extinction , and that the gas is at a single temperature @xmath94 , is given by @xmath95 where @xmath96 is the wavelength of the transition , @xmath97 is the spontaneous transition probability , @xmath98(h@xmath16 ) the total column density of h@xmath16 and @xmath99 the population of level @xmath100 . @xmath101 corresponds to the source size , which is not known since the h@xmath16 data are spatially unresolved . for gas densities larger than 10@xmath52 @xmath86 , the lines are thermalized and the population @xmath99 follows the boltzmann law @xmath102 with @xmath103 being the energy of the upper level , @xmath104 the partition function of h@xmath16 at @xmath94 and @xmath105 the nuclear statistical weight factor , which is 1 for para - h@xmath16 ( even @xmath9 ) and 3 for ortho - h@xmath16 ( odd @xmath9 ) . the lines are optically thin up to column densities of @xmath106 @xmath73 owing to the low values of the einstein @xmath107coefficients . when both the s(0 ) and s(1 ) lines are detected , the excitation temperature can be obtained from the relation @xmath108 in lte , @xmath94 is equal to the kinetic temperature @xmath109 . @xmath110 and @xmath111 are the integrated s(0 ) @xmath37 and s(1 ) @xmath38 fluxes , respectively . since no data are available to constrain the ortho - h@xmath16 to para - h@xmath16 ratio , we assume that the ortho / para ratio is in lte at the temperature @xmath112 . at @xmath109=100 k , the ortho / para ratio is 1.6 . no correction for differential extinction between the s(0 ) and s(1 ) lines is applied . the inferred temperatures range from 100 to 200 k ( see table [ h2iso ] ) . the uncertainty of 30% in the fluxes propagates into a @xmath2210% error in the temperature . if the emission were affected by @xmath2230 magnitudes of extinction , the derived temperatures would be increased by typically @xmath2220 k , illustrating that this does not have a large effect . the upper limits on the s(3 ) line translate into upper limits on the gas temperature of typically @xmath60 250 k if no correction for differential extinction is made . similarly , the upper limits on the s(2 ) line imply temperatures @xmath60200 k for hd 135344 and 49 ceti . the detection of either the s(0 ) or s(1 ) lines combined with the upper limits on s(3 ) imply a probable temperature range of 100200 k for the gas . because the lines are optically thin , the measured fluxes can be translated directly into a beam - averaged column density of warm gas using eq . ( 1 ) with @xmath101 equal to the solid angle of the iso - sws beam at the observed wavelength . the gas mass can be computed from @xmath113 where in addition to the above assumptions , @xmath114 is the distance in pc which is provided by the hipparcos satellite or taken from the literature ( see table [ sources ] ) . the derived masses are presented in table [ h2iso ] and depend strongly on the population @xmath99 and thus on the temperature @xmath109 . this is illustrated in figure [ masstemp ] , which shows the inferred mass as a function of temperature for a s(1 ) line flux of 10@xmath115 erg s@xmath56 @xmath73 at the distance of taurus ( 140 pc ) : for temperatures between 100 and 200 k , the mass changes by approximately one order of magnitude . assuming an error on the flux of @xmath2230% and including an error on the distance of 10% , the error on the mass reaches @xmath2255% in cases where both the s(0 ) and s(1 ) lines have been measured . when only one line is detected , a range of masses is obtained by assuming that the excitation temperature lies between 100 and 200 k and consequently shows a large spread . if the line emission were affected by 30 magnitudes of extinction , the derived masses are changed by less than 20% : the increase in mass due to the extinction correction is compensated by its decrease owing to the higher inferred temperature ( see 4.3 ) . the continuum around the lines can be used to perform narrow band photometry and the resulting absolute fluxes are given in table [ mircont ] . moreover , data in the 3.4 @xmath27 m region have been obtained in parallel and are included . the values at 28 @xmath27 m are consistent , within the errors ( estimated to be @xmath24 30% ) , with the iras point source catalog fluxes extrapolated from observations at 25 @xmath27 m . as mentioned in 3 , the sources with faint mid - infrared continuum fluxes ( @xmath60 1 jy ) can be contaminated by zodiacal emission and include a contribution from the star . typically , for a star located in the taurus cloud with an effective temperature @xmath116=8700 k and luminosity @xmath117=1.5 , the stellar monochromatic fluxes are 0.26 , 0.04 , 0.16 , 3 10@xmath118 and 7 10@xmath119 jy at 3.4 , 6.9 , 9.6 , 17 and 28 @xmath27 m respectively . thus , the stellar contributions at 17 and 28 @xmath27 m are negligible compared to the zodiacal emission . a complete understanding of the mid - infrared line and continuum emission requires a detailed radiative transfer code and a specific disk model implying many assumptions . we adopt here a simplified picture based on the @xcite model of irradiated passive disks . the disk is divided into 3 components : ( i ) a hot part giving rise to the near - infrared emission ; ( ii ) a warm part ( @xmath12080300 k ) responsible for the mid - infrared emission , and perhaps also the h@xmath16 emission ; and ( iii ) a cold part ( @xmath12180 k ) giving the submillimeter continuum and the co emission . component ( ii ) corresponds to the warm surface layer in the chiang & goldreich models . component ( i ) is not present in those models , but may be due to very hot thermal emission in an inner boundary layer , or to non - thermal emission by very small grains or pahs @xcite , from fe - containing grains @xcite or to due a very hot inner layer @xcite . our main reason for this partition is to compare separately the ` warm ' and ` cold ' gas and dust components . to obtain a rough estimate of the temperature of the warm dust , the 17/28 @xmath122 m flux ratios have been fitted with an optically thin dust model , as may be appropriate for the surface layers of disks . the grain emissivities of @xcite have been used . figure [ figcont ] shows the resulting warm dust temperature for different values of the 17/28 @xmath32 m ratio . the observed values are included in figure [ figcont ] and the resulting fits are summarized in table [ mircont ] . the observational errors on the temperature are @xmath2410% . interestingly , the beam average warm gas temperatures derived from h@xmath16 are higher by 2050 k ( see figure [ td_g ] ) . there are several possible explanations of this difference , e.g. the location of the emitting gas and dust may be different or a gas heating mechanism other than gas - grain collisions has to be invoked . observations of @xmath6co 32 lines have been performed with the jcmt toward most of the sources observed with iso , plus a few other t tauri and herbig ae stars . in all but a few cases , the @xmath6co 32 and @xmath7co 32 lines are detected with good signal - to - noise ( @xmath125 5 @xmath126 ) , and the spectra are presented in figures [ 12co ] and [ 13co ] . the lines show the typical double - peaked profile consistent with emission from a disk in keplerian rotation seen at a certain angle ( e.g. * ? ? ? * ; * ? ? ? the full width at half maximum of the line profile is typically 2.53 km s@xmath56 and the separation between the two peaks is of order of 1.22 km s@xmath127 . the integrated fluxes are computed by fitting two gaussians , which are tabulated in table [ co3_2 ] . the uncertainty in the integrated fluxes is dominated by the calibration error of @xmath2930 % . the mean integrated area of the @xmath6co 32 line for the t tauri stars is @xmath290.5 k km s@xmath56 higher than that for herbig ae stars . the clear presence of the double peak suggests that the microturbulence in these disks is no more than 0.2 - 0.3 km s@xmath56 , comparable to the thermal width of @xmath290.22 km s@xmath56 at 30 k. co 32 is not detected toward cq tau . this non - detection is compatible with the co 21 flux detected by @xcite using the owens valley millimeter array . a clear @xmath6co 32 disk line profile is also detected toward the debris - disk object hd 135344 , previously studied in @xmath6co 21 by @xcite . no 32 searches have been performed toward 49 ceti , but zuckerman et al . ( 1995 ) report a detection of the co 21 line . a deep search for @xmath6co 21 has been performed toward @xmath128 pictoris by @xcite with a limit of 11 mk rms in the 23@xmath55 sest beam . co is seen by ultraviolet absorption lines , however , and @xcite infer a column density of ( 6.3 @xmath1290.3 ) 10@xmath130 @xmath73 of co gas at a temperature of 2050 k. figure [ coratio ] shows the @xmath7co 32 versus @xmath6co 32 integrated line fluxes normalized to a distance of 100 pc . the different regimes of optical depth are indicated . the data fall in the region where @xmath7co 32 is thin whereas @xmath6co 32 is optically thick . no difference is found between the t tauri and the herbig ae stars . assuming a [ @xmath6c]/[@xmath7c ] ratio of 60 and the same excitation temperature for @xmath6co and @xmath7co , the beam - averaged optical depths @xmath131 of the @xmath7co 32 line can be calculated , and are given in the last column of table [ co3_2 ] . if the excitation temperature of @xmath6co is higher than that of @xmath7co , as suggested by models of @xcite , @xmath131 could be increased by a factor of two . nevertheless , a low optical depth @xmath132 of @xmath7co 32 is confirmed by the non - detection of c@xmath133o 32 emission @xcite . thus , the @xmath7co 32 line could constitute a tracer of the total gas mass in the outer part of disks provided the excitation temperature can be determined and the @xmath7co / h@xmath16 conversion factor is known . as mentioned in 4.4 , @xmath6co 32 observations have also been obtained at positions offset from the sources , in particular for a 30@xmath55 offset ( two jcmt beams ) . in all cases , the double - peaked line profile disappears completely at the off position , confirming that it arises from the circumstellar disk . in some sources , however , a narrow profile at a velocity slightly offset from that of the disk remains . this emission is due either to remnant envelope material or the general molecular cloud from which the star formed . its strength is uncertain up to a factor of two since it was not possible to find a good off - position in all cases . for the specific case of lkca 15 , no emission was found at 30@xmath55 offset , but a weak @xmath6co 32 line with @xmath134=0.22 k appeared at @xmath135 km s@xmath56 at the 1@xmath136 south position , where the h@xmath16 s(1 ) off - source spectrum was taken . this co emission is more than 10 km s@xmath56 offset from the velocity of the star and is most likely the result of a chance coincidence with a background cloud . @xmath6co 65 emission is detected toward several sources using the cso ( see figure [ co_6_5 ] ) . weak , but clear double - peaked profiles are seen from disks such as those around lkca 15 and mwc 480 ( see also * ? ? ? the line is particularly strong toward ab aur , likely because of the extended envelope . indeed , a small @xmath6co 32 and 65 map around the source shows strong lines even at one beam offset . the integrated fluxes are reported in table [ co6_5 ] . the 65 line probes preferentially gas at higher temperatures around 100 k , but its high optical depth decreases the excitation conditions to lower temperatures . the ratio of the 65/32 line intensity is a measure of this temperature @xcite . a full analysis requires a 2-d radiative transfer calculation for disk models with different radial and vertical temperature profiles . however , a rough estimate can be obtained from a simple 1-d escape probability formalism assuming an iso - thermal and iso - density slab in which the abundances are chosen such that the @xmath6co 65 and 32 lines are optically thick . this slab would be representative of the intermediate and surface layers of disks , from which most of the emission is thought to arise . this simple analysis shows that the sources have a range of temperatures . the upper limit on the co 65 line for ry tau indicates a cool emitting region of about 10 k. the sources lkca 15 , ab aur , gg tau and gm aur all have relatively low temperatures between 20 and 50 k , but these ranges can be extended considerably if typical calibration errors of 20% are taken into account , especially for gg tau and ab aur . the sources go tau , mwc 480 , v892 tau and dr tau have lower limits to the temperatures of 30 k and in general suggest high temperatures up to a few hundred kelvin . such high temperatures indicate that the upper layers of the disks are heated efficiently by the stellar light and are most probably flared so that they capture the radiation far from the star . note however that the derived temperatures are extremely sensitive to the errors in the line ratios . dr tau is surrounded by extended cloud emission and the observed lines may well be emitted in different regions . in summary , the combined detection of co 65 and h@xmath28 in several sources suggests that these sources may posses a warm upper layer , consistent with a flared disk geometry . two sources ( ry tau and lkca 15 ) could have lower temperatures on average which could either mean that the disk is flatter , or that dust - settling is taking place , reducing the heating of the gas in the upper layers of the disks ( e.g. * ? ? ? higher @xmath59 co 65 data and more accurate calibration are needed to use the 65/32 ratio as an effective temperature probe ( see @xcite for a detailed discussion ) . disk masses can be derived from the observed @xmath7co 32 data assuming that most of the flux arises from the outer part of the disk at a constant temperature . the simplification of an isothermal outer disk is supported by detailed modeling along the lines of @xcite . these models have a power - law decrease of the temperature with radius to explain the behavior of the spectral energy distribution , but the gradients in the outer disk are quite small . the reason could be that the ambient interstellar radiation field incident on the outer disk regulates the temperature structure with radius . because of the large beam dilution , our observations are not sensitive to the warm inner gas , but only probe the outer cold gas . a common outer gas temperature of 30 k is therefore assumed for all our objects . this is slightly higher than the temperature fixed by the local interstellar field , which is around 1015 k in quiet molecular cloud environments such as found in taurus and ophiuchus . it is consistent with the observed @xmath7co 32/10 line ratios @xcite . in the optically thin limit , the gas mass derived from @xmath7co 32 is given by : @xmath138/[^{13}\rm{c}]}{60}\right ) \left ( \frac{\rm{h}_{2}/^{12}\rm{co}}{10^{4}}\right ) \frac{t_{\rm{ex}}+0.89}{e^{-16.02/t_{\rm{ex}}}}\frac{\tau } { 1-e^{-\tau } } \left ( \frac{d}{100\ , \ , \rm{pc}}\right ) ^{2}\int t_{\rm{mb}}\ , \ , dv\ , \ , \rm{m}_{\odot } \ ] ] the derivation of this formula is similar to that for co 10 by @xcite . the mass varies by a factor of 2 for excitation temperatures between 20 and 100 k , so that the exact value of the excitation temperature is not crucial . two main parameters must be assumed : the [ @xmath139c]/[@xmath12c ] elemental isotope ratio and the h@xmath13/@xmath139co conversion factor . this factor is certainly not constant from source to source and we adopt here a reference value of 10@xmath140 , typical for dense molecular gas in which co is not depleted . as will be shown below , this factor is likely to be much larger in disks due to the combined chemical effects of freeze - out and photodissociation . for sources for which no @xmath12co data are available ( cq tau , aa tau , 49 ceti , hd 135344 ) , the @xmath139co data have been used to determine the cold gas masses . the cold gas masses can also be determined from the millimeter continuum emission emitted by the cold dust , assuming a gas / dust ratio . all sources in our sample have been previously observed in the millimeter continuum with single - dish telescopes , usually at 1.3 or 1.1 millimeter ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? for some sources , millimeter interferometer data exist at the same wavelengths , giving similar flux levels @xcite , indicating that most of the single - dish emission indeed comes from the disk rather than any remnant envelope . to compute the cold dust mass , @xmath141 k is adopted , similar to that found for co. the value for the mass absorption coefficient @xmath142 ( gas + dust ) is taken to be @xmath143 @xmath144 g@xmath56 from @xcite and assumes @xmath145 . the disk mass ( gas + dust ) is then given by : @xmath146 where @xmath147 is the observed flux at 1.3 mm in jy . the observational data and resulting masses are summarized in table [ mmcont ] . the errors in the observed fluxes are taken to be @xmath2430% . in the previous section , we applied three methods to estimate the masses of disks around pre - main - sequence and debris - disk stars , summarized in table [ summary ] . the derived masses differ considerably , well beyond the error bars . we now discuss the strengths and weaknesses of each of these methods . in the upper panel of figure [ co_h2_dust ] , the masses obtained from the @xmath7co 32 spectra are compared to those computed from the 1.3 millimeter continuum emission assuming a mean disk temperature of 30 k for the t tauri and herbig ae stars . the dust around hd 135344 and @xmath35 pictoris has been taken to be warmer at 95 and 85 k respectively ( see * ? ? ? * ; * ? ? ? * ) . the results for sources for which only @xmath6co data are available are included , as well as those for tw hya studied by @xcite . as found in previous studies based on lower-@xmath9 transitions ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the masses derived from co are in general factors of 10200 lower than those found from the millimeter continuum . no distinction can be made between t tauri and herbig ae stars . the debris - disk objects as well as tw hya seem to suffer very strong co depletion , more than a factor of 10@xmath52 , in agreement with previous studies ( e.g. * ? ? ? * ; * ? ? ? many explanations have been put forward , including depletion of co onto grains and dispersal of the disk gas . as argued in 5.1 , optical depths effects are unlikely to be the main cause . @xcite show that the underabundance of co is plausibly caused by a combination of freeze - out in the coldest regions of the disk near the mid - plane , as well as photodissociation of co in the upper layers of the disk by stellar and interstellar ultraviolet radiation @xcite . substantial depletions due to freeze - out have been found in dense , cold molecular cloud cores and in young stellar objects environments ( e.g. * ? ? ? * ; * ? ? ? the millimeter continuum method is not exempt from difficulties either . in particular , it suffers from the poor knowledge of the dust opacity constants @xmath148 . theoretically , @xmath149 should be well determined for particles that are much smaller than the wavelength of observation , but its value depends strongly on the assumed particle composition ( silicates , amorphous carbon , ice mantle ) and also on particle size , shape and fluffyness . within the range of possible values , however , @xmath148 remains small enough to ensure that the emission is optically thin beyond a few au , so that the determination of the total mass is quite straightforward . other assumptions in this method include a constant @xmath148 for the whole sample ( i.e. , no evolution of the opacity constant ) and the gas / dust ratio of 100:1 in the disks . in the lower panel of figure [ co_h2_dust ] , the warm gas masses derived from the h@xmath13 lines are plotted as functions of the total gas masses obtained from the 1.3 millimeter continuum . for the pre - main sequence stars , the warm gas masses are a fraction ( 110% ) of the total gas masses , assuming a gas / dust ratio of 100 . some sources such as lkca 15 , however , show a much larger fraction , of order 30% . @xcite modeled the spectral energy distribution of lkca 15 and concluded that this source shows the strongest vertical dust settling . in the case of go tau , contamination by surrounding emission is possible . the warm gas masses have also been plotted versus the cold gas masses derived from @xmath12co , but no correlation is found , as expected . in order to search for evolutionary trends in our results , the ages of the stars need to be known . this is usually done by comparing the positions of the stars on a hertzsprung - russell diagram with theoretical evolutionary tracks . these tracks have many implicit assumptions , however , and give different results depending on the choice of the equations of state , the model used for convection , the opacities , etc . ( see * ? ? ? * for a review ) . on the observational side , there are also uncertainties in the distance estimates of the sources , the extinction and to what extent the intrinsic luminosity is affected by disk accretion . the precise spectral type of few sources like mwc 480 remains controversial : a4 or a3ep+sh according to @xcite and @xcite respectively . moreover , all t tauri stars exhibit photometric variability , preventing a precise determination of their characteristics . some stars such as ry tau and gg tau are binary systems and thus their stellar characteristics must be corrected . although these factors result in significant absolute uncertainties , the relative ages may be less affected . to obtain a consistent set of relative ages , we have re - estimated the ages of the stars in our sample using the recent pre - main - sequence evolutionary models of @xcite , which take the accretion history into account . the results are shown in figure [ hr_siess ] . we take any binary systems to be single stars , so that their ages should be considered rough estimates . the newly evaluated ages are consistent with previous determinations and are listed in table [ ages ] . if the tracks of @xcite are used , a similar age ordering is obtained . the discrepancies are largest for brown dwarfs and stars younger than 10@xmath150 years . since our stars have higher mass ( @xmath1250.5 m@xmath151 ) and ages greater than one million years , the differences between the models are not significant . it is not the purpose of this paper to discuss the validity of the different tracks . the errors in our derived ages are of order 12 millions years , increasing for the older objects . the ages of the intermediate mass stars are less well determined because their effective temperature and luminosity do not vary significantly over a large range of ages . in particular , the age of @xmath76 pictoris is controversial . recently , @xcite argue that @xmath76 pictoris is only ( 20@xmath12910 ) @xmath152 years old with the error bar reflecting the uncertainties in the isochrones used to derive the age . the young age of @xmath76 pictoris is consistent with the view that it is part of a cluster of recent nearby star formation ( e.g. * ? ? ? * ) . whatever its actual age , @xmath76 pictoris is the oldest member in our sample . figure [ mass_age ] shows the total disk masses deduced from the three methods plotted against the ages of the stars . no strong evolutionary trend appears but the behavior seems to be similar for the three methods . figure [ gas_to_dust ] presents the total warm + cold gas masses derived from the h@xmath13 + co data relative to the total dust mass derived from the 1.3 millimeter continuum versus age . as discussed in 6.4 , only the debris - disk objects show a gas / dust ratio close to 100:1 , but this may be coincidental ; for the younger objects , a significant amount of cold h@xmath13 is likely present , but is not traced by co. care has to be taken in the interpretation of these data , however . as mentioned before , the choice of objects in our sample is biased toward the higher disk masses and some of the detections are marginal . in fact , so - called weak - line t tauri stars are surrounded by disks with lower masses @xcite . this is consistent with the non - detection of h@xmath13 in these objects by @xcite . our data are not sensitive to masses as small as @xmath153 m@xmath154 for objects at a distance of 140 pc . similarly , @xmath76 pictoris may be unusual since it is one of the dustiest members of the debris - disk family . finally , it is difficult to compare different absolute masses since the mass of the disk at a given time of its evolution likely depends on the initial mass available . the derived amount of warm gas is significant and raises the question of the source of heating . @xcite discussed several possibilities , including photon heating by stellar and interstellar radiation and shock - heating caused by the interaction between a stellar wind and the surface of disks . here we investigate whether the observed trends provide further clues to the dominant mechanisms . quantitative discussions and detailed modeling are left for future work . since the disks in our sample have negligible accretion onto the star ( typically @xmath155 10@xmath156 m@xmath157 yr@xmath65 ) , the irradiation of the central object should control , at least partially , the temperature profile of the disks . to study this scenario , we plot in figure [ corr4 ] the excitation temperatures derived from the h@xmath13 s(0 ) and s(1 ) lines as functions of the effective temperature of the star . obviously , no significant correlation is found in this figure . we can , however , distinguish three groups : t tauri , herbig ae and debris - disk stars . the t tauri stars have gas at @xmath24100 k , whereas their higher mass counterparts are surrounded by gas at 150 k or more . the higher @xmath158 observed in disks around herbig ae stars suggests that the harder stellar ultraviolet radiation can be transformed more efficiently into heat for these objects . however , if the number of photons with wavelengths @xmath1551100 capable of ionizing atomic carbon is also increased , this results in a larger c@xmath159 abundance , increasing the cooling as well . detailed modeling of the surface heating as a function of radiation field is needed . note that classical models of photon - dominated - regions ( pdrs ) do not show any increase of the h@xmath13 excitation temperature versus strength of the incident radiation field for the normal interstellar field typical of a b0 star , even though the surface temperatures increase ( e.g. * ? ? ? however , the variation of @xmath158 with effective temperature of the star has not yet been modeled . a link seems to exist between the excitation temperature and the continuum flux at 28 @xmath122 m normalized at 100 au ( figure [ corr4]@xmath160 ) . above a certain threshold , the excitation temperature increases as the continuum flux becomes higher . as a consequence , the warm gas mass drops with continuum flux or @xmath161 because of the steep dependence of the mass on temperature ( eq . ( 3 ) and figure [ masstemp ] ) ( see figure [ corr4]@xmath162 and [ corr4]@xmath163 ) . moreover , the fraction of warm gas to the total gas mass around typical a stars like hd 163296 or ab aur is small compared to that around t tauri stars . the role of ultraviolet radiation in heating the surfaces of flared disks is taken into account in recent models by @xcite , @xcite and @xcite . as shown by @xcite , these models fall short of explaining the observed masses of warm h@xmath16 gas by factors of at least a few . it is not yet clear whether this discrepancy is significant , since the same models also fail for normal molecular clouds unless the grain formation rate of h@xmath16 is significantly enhanced ( e.g. * ? ? ? * ; * ? ? ? the presence of a thin envelope can enhance the scattered stellar radiation and thus also the warming @xcite . at the edges of pdrs , the main heating agent is the photoelectric effect on grains , with small grains and pahs being particularly effective @xcite . @xcite investigated the influence of the effective temperature of the central illuminating star on the gas heating efficiency by very small grains ( grain radius between 4 and 180 ) and pahs . they showed that the efficiency drops only by a factor of 4 from a star at 10000 k to one at 4000 k. adding the fact that most t tauri stars exhibit ultraviolet excess and a strong lyman alpha emission line , the effective heating by low mass stars compared to intermediate mass stars should be similar . the detections of pahs around ab aur @xcite and hd 135344 @xcite suggest that these large molecules can play a role in the heating of the disks , but quantitative models have not yet been performed . the gas can attain higher temperatures than the dust in these layers , consistent with our observations , and its emission can emerge from the surfaces even if the mid - plane is optically thick in the mid - infrared continuum . the efficiency of photoelectric heating decreases significantly , however , if the size of the grains is increased , so the dust size distribution also plays an important factor in this analysis @xcite . alternatively , the line emission can escape through ` holes ' or ` gaps ' in the disk created by low - mass companion(s ) , e.g. planets or brown dwarfs @xcite . such gaps could also result in a larger surface area intercepted by the radiation . in any case , the detection of ultraviolet emission from fluorescent h@xmath13 toward other pre - main - sequence stars indicates that ultraviolet radiation plays some role in these systems @xcite . note that this fluorescent h@xmath28 seen in the ultraviolet must arise from much hotter gas , of order 2000 k , probably located in an inner boundary layer close to the star . possible heating of h@xmath13 by shocks created by the interaction of a stellar wind with the surface of disks was discussed by @xcite . a significant constraint is however provided by the non - detection of the h@xmath13 s(3 ) lines in our sources , since shocks tend to populate the high-@xmath164 h@xmath13 levels as well . shock models by @xcite give much higher h@xmath13 excitation temperatures than observed , making them less plausible . the most interesting cases are formed by the debris disk objects hd 135344 , @xmath76 pictoris and 49 ceti . the disks around these objects are considered gas poor based on co observations assuming a h@xmath13/co conversion factor of 10@xmath140 @xcite . for hd 135344 and @xmath165 pictoris , two lines are possibly detected ( in more than 50% of the spectra obtained by our data reduction procedure ) , giving a measure of the temperature and the mass with 10% and 55% uncertainty , respectively , if only the standard 30% calibration errors are considered . for 49 ceti , only the s(0 ) line is seen , leading to an h@xmath13 mass of ( 3.5@xmath1661.9)@xmath8210@xmath167 m@xmath154 if @xmath168 k is assumed . the derived mass of gas around @xmath76 pictoris is @xmath169 m@xmath154 or @xmath170 0.17@xmath1710.09 m@xmath172 , and it is @xmath24 6.4 @xmath173 m@xmath154 around hd 135344 . the amount of gas in the @xmath76 pictoris disk is significantly smaller than that for other disks . the disks are not resolved within the iso - sws beam for hd 135344 and 49 ceti and barely resolved for @xmath76 pictoris . therefore , the location of the emitting gas is unknown . the detection of h@xmath28 gas in the @xmath76 pictoris disk may seem surprising since this disk has a very low co / dust mass ratio ( e.g. * ? ? ? * ; * ? ? ? the presence of some neutral gas was , however , invoked by @xcite in order to slow down ions like or leaving the disk since these ions suffer strong radiation pressure from the star . they considered only as a major species of the stable ring . the detection of lines together with the measured @xmath174 ratio implies densities of 10@xmath17510@xmath150 @xmath86 @xcite . this is consistent with the density at 40 au in the disk model of @xcite which has m@xmath176 @xmath24 10@xmath177 m@xmath21 , similar to that found here . in their model of debris disks , @xcite consider the balance of the formation and destruction of co and h@xmath28 . their main conclusion is that the co molecule can exist only in the dense part of disks protected from photodissociation whereas h@xmath13 is widely spread . moreover , co freezes out onto dust in the coldest parts near the midplane , making it a poor tracer of the gas . if the h@xmath16 detections are valid , nearly all of the gas is at high temperatures ( @xmath178 k ) in the debris disks . since the disk of @xmath35 pictoris has an optical depth less than unity at optical wavelengths , the ultraviolet photons can warm the gas in the entire disk through the photoelectric effect and other processes @xcite . these gas temperatures are only slightly higher than those derived for the dust components : the spectral energy distributions of @xmath76 pictoris and hd 135344 are well fitted by a single dust temperature of 8090 k @xcite , indicating that these disks are globally warmer than those around t tauri or herbig ae stars . the estimated total dust mass around @xmath76 pictoris ranges from 0.3@xmath17910@xmath180 m@xmath154 @xcite to 10@xmath180 m@xmath154 @xcite . the gas - to - dust mass ratio lies therefore between 45 and 380 , and is much higher than the value of 0.02 derived from co ultraviolet observations . note that if these co molecules are the evaporation products of infalling comets onto @xmath76 pictoris @xcite , it is not possible to derive the primordial h@xmath28 content of the disk from co. a pertinent consequence of the presence of gas in debris disks is that it affects the dynamics of the dust in those disks ( e.g. * ? ? ? * ) . for sufficiently large gas masses , dust generation by collisions of planetesimals will not be possible . for @xmath76 pictoris , however , the gas mass of @xmath24 0.1 m@xmath172 is small enough that it does not prevent a collisional cascade . @xcite modeled the evolution of dust grains in disks with gas masses up to a few tens of earth masses , comparable to that found for @xmath76 pictoris . they show that although grains migrate radially due to radiative pressure and gravity , equilibrium orbits exist for a specific range of grain sizes . most interestingly , their models can reproduce ring - like disk morphologies with an inner disk clear of small grains . the similarity of our derived gas / dust ratio for @xmath76 pictoris with the interstellar value of @xmath24100:1 is therefore likely coincidental : some of the dust and gas may have accumulated into ( gaseous ) planets and planetesimals , been expelled from the disk due to radiation pressure , or fallen onto the star by poynting - robertson drag . we have conducted the first survey of h@xmath13 rotational line emission from disks around a sample of t tauri and herbig ae stars and from young stars with debris disk using the iso - sws . the observed spectra reveal the presence of an unexpectedly large amount ( 0.110 @xmath179 10@xmath118 m@xmath154 ) of molecular gas at @xmath24100 k. no correlations between the warm gas masses with disk masses derived from @xmath7co and 1.3 millimeter emission were found . whereas the bulk of the gas around t tauri and herbig ae stars is cold , the warm gas may constitute the major gaseous component of debris - disk objects like hd 135344 and @xmath76 pictoris . there is no apparent difference between the low and the intermediate mass pre - main - sequence stars . the possible heating mechanisms responsible for the warm gas are discussed . no process can adequately explain the large amount of warm gas , but the ubiquitous presence of warm h@xmath16 , the higher gas than dust temperatures , and the detection of pahs in few of the objects suggest that a common mechanism like photoelectric heating by ultraviolet radiation could be the main heating agent . further modeling is needed . complementary observations of @xmath139co 32 , 65 and @xmath12co 32 have been performed . the line profiles are resolved and exhibit double - peaked features consistent with gas emitted from a disk in keplerian rotation around a central object . ratios of integrated fluxes of the two isotopomers @xmath139co and @xmath12co show that the @xmath12co 32 line is not highly optically thick and potentially a tracer of the cold component of disks . the presence of warm gas is supported by the detection of @xmath139co 65 toward a few sources where h@xmath28 has also been found . the gaseous masses inferred from the @xmath12co intensities are much smaller than those found from the dust continuum emission . co is likely strongly affected by photodissociation via the stellar and interstellar ultraviolet radiation in the surface layers and freeze - out onto grain surfaces in the midplane . the h@xmath13 , co and millimeter continuum data together with rough age estimates of our stars allow evolutionary trends to be investigated . no strong evolution in the masses derived from co , h@xmath13 or dust is found . there is a large diversity among the stars studied in the ( 110)@xmath181 years range . the limited number of objects , the limited quality of the iso data and uncertainties in the derived masses prevent definitive conclusions on the gas survival time scale . the analysis of the h@xmath13 data presented here suffers greatly from limited spatial and spectral resolution as well as sensitivity . ground - based spectrometers soon to be operational on 810 m class telescopes will be able to study the s(1 ) line at vastly higher spectral and spatial resolution , but will not have access to the ground state para - h@xmath13 transition at 28 @xmath122 m . moreover , the surface brightness sensitivities of these warm large telescopes is only marginally improved compared with small cryogenic space observatories such as iso . more complete studies with future air- and space - borne mid - infrared spectrometers on sirtf , sofia , and eventually ngst will greatly improve on our ability to examine the h@xmath13 emission lines from young stars , and properly address the many interesting questions associated with the structure of circumstellar disks and the formation of giant gaseous planets raised in this paper . this work was supported by the netherlands organization for scientific research ( nwo ) grant 614.41.003 and a spinoza grant , and by grants to gab from nasa ( nag58822 and nag59434 ) . a.n . is supported in part by asi ars-98 - 116 grant . discussions with eugene chiang , peter goldreich , michiel hogerheijde , frank shu and doug johnstone are appreciated . the authors thank the staff of the cso and jcmt , in particular fred baas and remo tilanus , for their support and the dutch iso data analysis center ( didac ) at sron - groningen , especially edwin valentijn and fred lahuis , for their help during the data reduction of the iso - sws spectra . habart , e. , boulanger , f. , verstraete , l. , falgarone , e. , pineau des forts , g. , & abergel , a. 2000 , in iso beyond the peaks : the 2nd iso workshop on analytical spectroscopy , held 2 - 4 february 2000 , at vilspa . , 103 llllllll + + aa tau & k7 & 04 34 55.5 & @xmath18224 28 54 & 3.60 & @xmath1830.15 & 140 & 1 + dm tau & m0.5 & 04 33 48.7 & @xmath18218 10 12 & 3.56 & @xmath1830.5 & 140 & 1 + dr tau & k7 & 04 47 06.3 & @xmath18216 58 41 & 3.64 & @xmath1820.025 & 140 & 2 + gg tau & k7 & 04 32 30.3 & @xmath18217 31 41.0 & 3.58 & @xmath1830.22 @xmath1660.23 & 140 & 3 + go tau & m0 & 04 43 03.1 & @xmath18225 20 19 & 3.58 & @xmath1830.43 & 140 & 1 + ry tau & k1 & 04 21 57.41 & @xmath18228 26 35.6 & 3.76 & @xmath1820.81 & 133 & 2 , 4 + gm aur & k7 & 04 55 10.2 & @xmath18230 21 58 & 3.59 & @xmath1830.12 & 140 & 2 + lkca 15 & k7 & 04 39 17.8 & @xmath18222 21 03 & 3.64 & @xmath1830.27 & 140 & 2 + ux ori & a3iiie & 05 04 29.9 & @xmath18303 47 14.3 & 3.94 & @xmath1821.51 @xmath184 & 430 & 7 + hd 163296 & a3ve & 17 56 21.26 & @xmath18321 57 19.5 & 3.94 & @xmath1821.41 @xmath185 & 122 @xmath186 & 8 + cq tau & f5ive & 05 35 58.47 & @xmath18224 44 54.1 & 3.84 & @xmath1830.21 @xmath187 & 100 @xmath188 & 8 + mwc 480 & a3ep+sh & 04 58 46.27 & @xmath18229 50 37.0 & 3.94 & @xmath1821.51 @xmath184 & 131 @xmath189 & 8 + mwc 863 & a1ve & 16 40 17.92 & @xmath18323 53 45.2 & 3.97 & @xmath1821.47 @xmath190 & 150 @xmath191 & 8 + hd 36112 & a5ive & 05 30 27.53 & @xmath18225 19 57.1 & 3.91 & @xmath1821.35 @xmath192 & 200 @xmath193 & 7 + ab aur & a0ve+sh & 04 55 45.79 & @xmath18230 33 05.5 & 4.00 & @xmath1821.68 @xmath194 & 144 @xmath195 & 8 + ww vul & a0 & 19 25 58.75 & @xmath18221 12 31.3 & 3.97 & @xmath1820.73 & 550 & 9 + v892 tau & a0 & 04 18 40.61 & @xmath18228 19 16.7 & 3.90 & @xmath1821.75 & 140 & 10 + 49 ceti & a1v & 01 34 37.78 & @xmath18315 40 34.9 & 3.97 & @xmath1821.37 & 61 & 11 + hd 135344 & f8v & 15 15 48.44 & @xmath18337 09 16.0 & 3.79 & @xmath1820.60 & 80 & 12 + @xmath76 pictoris & a5v & 05 47 17.09 & @xmath18351 03 59.5 & 3.91 & @xmath1820.94 & 19.28 @xmath196 & 13 + lccccc h@xmath13 s(0 ) 2@xmath1970 & 28.218 & 509.88 & 2.94 10@xmath198 & 54 + h@xmath13 s(1 ) 3@xmath1971 & 17.035 & 1015.12 & 4.76 10@xmath199 & 1.1 10@xmath175 + h@xmath13 s(2 ) 4@xmath1972 & 12.278 & 1814.43 & 2.76 10@xmath200 & 2.0 10@xmath140 + h@xmath13 s(3 ) 5@xmath1973 & 9.662 & 2503.82 & 9.84 10@xmath200 & 1.9 10@xmath201 + llllll aa tau & @xmath1551.5 & 8.1@xmath1660.25 & 100200 & 20.60.2 & medium + dr tau & @xmath1551.5 & @xmath1550.8 & & & + gg tau & 2.5@xmath1660.8 & 2.8@xmath1660.8 & 110@xmath16611 & 3.6@xmath1661.8 & high + go tau & 5.6@xmath1661.7 & 7.1@xmath1662.1 & 113@xmath16611 & 6.4@xmath1663.2 & medium + ry tau & @xmath1551.5 & @xmath1550.8 & & & + gm aur & @xmath1551.5 & @xmath1550.8 & & & + lkca 15 & 5.7@xmath1662.2 & 5.3@xmath1661.6 & 105@xmath16610 & 8.6@xmath1664.3 & medium + + + ux ori & 6.8@xmath1662.0 & @xmath1550.8 & 100200 & 1179 & high + hd 163296 & 1.9@xmath1660.6 & 22@xmath1666 & 220@xmath16622 & 0.4@xmath1660.2 & high + cq tau & 5.9@xmath1661.8 & 40@xmath16612 & 180@xmath16618 & 2.0@xmath1661.0 & high + mwc 480 & @xmath1551.5 & 10@xmath1663 & 100200 & 78.80.7 & + mwc 863 & 6.9@xmath1662.1 & 24@xmath1667 & 146@xmath16614 & 1.5@xmath1660.8 & high + hd 36112 & @xmath1551.5 & 3.6@xmath1661.1 & 100200 & 18.70.2 & medium + ab aur & 4.1@xmath1661.2 & 30@xmath1669 & 185@xmath16618 & 1.3@xmath1660.7 & high + ww vul & @xmath1551.5 & @xmath1550.8 & & & + + + 49 ceti & 6.6@xmath1662.0 & @xmath1550.8 & 100200 & 2.30.3 & medium + hd 135344 & 9.0@xmath1662.7 & 5.5@xmath1661.7 & 97@xmath16610 & 6.4@xmath1663.2 & medium + @xmath76 pictoris & 7.0@xmath1662.1 & 7.7@xmath1662.3 & 109@xmath16611 & 0.17@xmath1660.08 & medium + lrrrrrlc aa tau & @xmath1550.1 & @xmath1550.1 & 0.1 & 1.1 & 1.2 & 93@xmath1669 & + dr tau & 2.0 & 1.7 & 2.4 & 4.3 & 5.4 & 90@xmath1669 & + gg tau & 0.5 & 0.4 & 0.8 & 1.1 & 2.1 & 81@xmath1668 & + go tau & 0.2 & 0.2 & 0.3 & @xmath1550.1 & 1.4 & 88@xmath1669 & + ry tau & 5.2 & 5.1 & 15.9 & 17.0 & 16.9 & 92@xmath1669 & + gm aur & @xmath1550.1 & @xmath1550.1 & 0.5 & 0.1 & 1.2 & 56@xmath1666 & + lkca 15 & 0.3 & 0.4 & 0.5 & 0.4 & 0.2 & 120@xmath16612 & + + + ux ori & 1.0 & 0.7 & 3.5 & 1.4 & 4.6 & 71@xmath1667 & + hd 163296 & 8.5 & 7.0 & 18.0 & 16.9 & 15.4 & 78@xmath1668 & + cq tau & 2.6 & 2.1 & 7.1 & 13.1 & 21.6 & 155@xmath16615 & + mwc 480 & 3.8 & 3.7 & 8.7 & 4.7 & 7.2 & 85@xmath1669 & + mwc 863 & 6.1 & 5.5 & 22.3 & 16.7 & 16.2 & 98@xmath16610 & + hd 36112 & 3.6 & 2.5 & 6.3 & 4.8 & 6.4 & 81@xmath1668 & + ab aur & 13.2 & 9.6 & 29.9 & 24.4 & 45.4 & 96@xmath16610 & + ww vul & 0.9 & 0.7 & 2.3 & 1.8 & 2.1 & 84@xmath1668 & + + + 49 ceti & 1.8 & 0.6 & 0.3 & 0.8 & 0.2 & 161@xmath16616 & + hd 135344 & 3.2 & 2.1 & 1.3 & 2.8 & 8.0 & 74@xmath1667 & + @xmath76 pictoris & 12.4 & 3.4 & 2.7 & 3.0 & 6.6 & 96 @xmath16610 & + llccclccll [ tab_codata ] dm tau & 1.02@xmath1660.30 & 6.2 & 1.0 & & 0.25@xmath1660.07 & 6.4 & 0.6 & 0.56 & 0.18@xmath1660.05 + & & & & & 0.19@xmath1660.06 & 5.6 & 0.6 & & + dr tau & 3.45@xmath1661.03 & 6.8 & 0.7 & & 0.21@xmath1660.06 & 6.9 & 0.3 & 0.06 & 0.07@xmath1660.02 + & 1.93@xmath1660.58 & 9.1 & 0.5 & & & & & & + & 6.84@xmath1662.05 & 10.3 & 1.0 & & & & & & + & 1.38@xmath1660.41 & 10.0 & 0.9 & & & & & & + gg tau & 1.28@xmath1660.38 & 5.7 & 1.0 & & 0.21@xmath1660.06 & 5.5 & 1.2 & 0.19 & 0.16@xmath1660.05 + & 1.41@xmath1660.42 & 7.0 & 1.0 & & 0.27@xmath1660.08 & 7.2 & 1.2 & & + go tau & 0.77@xmath1660.23 & 5.2 & 1.0 & & 0.11@xmath1660.03 & 4.3 & 0.8 & 0.18 & 0.09@xmath1660.03 + & 1.31@xmath1660.39 & 7.1 & 1.0 & & 0.09@xmath1660.03 & 7.0 & 0.3 & & + & 0.18@xmath1660.05 & 6.2 & 0.4 & & & & & & + & 0.09@xmath1660.03 & 5.5 & 0.7 & & & & & & + ry tau & 3.94@xmath1661.18 & 6.3 & 0.3 & & 0.21@xmath1660.06 & 6.4 & 0.3 & & 0.06@xmath1660.02 + & 2.48@xmath1660.74 & 6.9 & 0.3 & & & & & + gm aur & 0.62@xmath1660.18 & 4.8 & 1.0 & & 0.24@xmath1660.07 & 4.6 & 1.6 & 0.35 & 0.16@xmath1660.05 + & 0.89@xmath1660.27 & 6.4 & 0.9 & & 0.21@xmath1660.06 & 6.9 & 1.5 & & + lkca 15 & 0.58@xmath1660.17 & 5.4 & 1.3 & & 0.16@xmath1660.05 & 5.2 & 1.4 & 0.38 & 0.14@xmath1660.04 + & 0.61@xmath1660.18 & 7.0 & 1.3 & & 0.22@xmath1660.06 & 7.1 & 1.4 & & + + + hd 163296 & 1.44@xmath1660.43 & 4.7 & 1.5 & & 0.43@xmath1660.13 & 4.5 & 1.5 & 0.62 & 0.56@xmath1660.16 + & 1.63@xmath1660.49 & 6.9 & 1.5 & & 0.51@xmath1660.15 & 7.3 & 1.5 & & + & 0.75@xmath1660.22 & 5.5 & 7.0 & & & & & + & 0.83@xmath1660.25 & 5.5 & 7.0 & & & & & & + cq tau & @xmath155 0.06 & & & & @xmath155 0.06 & & & & + mwc 480 & 1.25@xmath1660.37 & 4.2 & 1.1 & & 0.27@xmath1660.08 & 4.0 & 1.2 & 0.27 & 0.17@xmath1660.05 + & 1.12@xmath1660.33 & 6.0 & 1.1 & & 0.30@xmath1660.09 & 6.2 & 1.2 & & + hd 36112 & 1.03@xmath1660.31 & 4.9 & 4.7 & & 0.31@xmath1660.09 & 5.9 & 1.8 & 0.36 & 0.23@xmath1660.07 + ab aur & 26.1@xmath1667.8 & 5.8 & 1.5 & & 5.00@xmath1661.50 & 5.8 & 1.6 & 0.21 & 1.72@xmath1660.51 + v892 tau & 2.18@xmath1660.65 & 7.0 & 1.1 & & & & & & + & 2.27@xmath1660.68 & 8.2 & 1.2 & & & & & & + + + hd 135344 & 0.39@xmath1660.12 & 6.4 & 1.0 & & & & & & 2.1@xmath1660.6@xmath17910@xmath167 + & 0.41@xmath1660.12 & 7.7 & 1.0 & & & & & & + llll dl tau & @xmath1551.8 & & + dm tau & @xmath1551.8 & & + dr tau & 11.6@xmath1661.5 & 10.0 & 1.6 + gg tau & 1.9@xmath1660.4 & 4.9 & 3.5 + go tau & 4.7@xmath1661.3 & 5.2 & 2.4 + ry tau & @xmath1552.0 & & + gm aur & 2.8@xmath1660.7 & 5.1 & 1.6 + lkca 15 & 1.9@xmath1660.8 & 6.8 & 3.3 + cq tau & @xmath1552.8 & & + mwc 480 & 2.3@xmath1660.8 & 4.9 & 2.5 + ab aur & 51.7@xmath1662.2 & 5.9 & 2.1 + v892 tau & 11.7@xmath1660.7 & 7.4 & 1.2 + llll aa tau & 1.7@xmath1660.8 & 88@xmath16626 & 1 + dm tau & 2.1@xmath1660.9 & 109@xmath16633 & 1 + dr tau & 3.1@xmath1661.4 & 159@xmath16648 & 1 + gg tau & 11.6@xmath1665.2 & 593@xmath166178 & 2 + go tau & 1.6@xmath1660.7 & 83@xmath16625 & 1 + ry tau & 4.0@xmath1661.8 & 229@xmath16669 & 1 + gm aur & 4.9@xmath1662.2 & 253@xmath16676 & 1 + lkca 15 & 3.3@xmath1661.5 & 167@xmath16650 & 3 + tw hya & 1.5@xmath1660.7 & 784@xmath166235 & 4 + + + ux ori & 4.2@xmath1661.9 & 23@xmath1667 & 6 + hd 163296 & 6.5@xmath1662.9 & 441@xmath166132 & 7 + cq tau & 2.2@xmath1661.0 & 221@xmath16666 & 7 + mwc 480 & 2.2@xmath1661.0 & 131@xmath16639 & 7 + mwc 863 & 1.0@xmath1660.5 & 45@xmath16613 & 7 + hd 36112 & 2.9@xmath1661.3 & 72@xmath16621 & 7 + ab aur & 2.1@xmath1660.9 & 100@xmath16630 & 8 + ww vul & 3.2@xmath1661.4 & 10.5@xmath1663.1 & 9 + v892 tau & 5.6@xmath1662.5 & 289@xmath16687 & 8 + + + 49 ceti & 0.04@xmath1660.018 & 12.7@xmath1663.8 & 10 + hd 135344 & 0.28@xmath1660.126 & 142@xmath16642 & 11 + @xmath76 pictoris & 0.003@xmath1660.00135 & 24@xmath1667 & 12 + llll aa tau & 2.4 & 1.2 & 1 + dm tau & 2.5 & & + dr tau & 3.8 & 2.5 & 1 + gg tau a & 1.7 & 0.82 & 1 + go tau & 3.2 & & + ry tau & 7.8 & 6.5 & 1 + gm aur & 1.8 & 1.3 & + lkca 15 & 11.7 & 8.3 & 1 + tw hya & 9.3 & 15 & 2 + + + ux ori & 4.6 & 2 & 3 + hd 163296 & 6.0 & 5 & 4 + cq tau & 8.9 & 10 & 4 + mwc 480 & 4.6 & 6 & 4 + mwc 863 & 6.0 & 5 & 4 + hd 36112 & 6.0 & 6 & 4 + ab aur & 4.6 & @xmath20235 & 4 + ww vul & & & + v892 tau & & & + + + 49 ceti & 7.8 & & + hd 135344 & 16.7 & & + @xmath76 pictoris & 20 & @xmath20220100 & 5 + llll aa tau & 17@xmath1668 & & 0.220 + dm tau & 21@xmath1669 & 0.18@xmath1660.05 & + dr tau & 31@xmath16614 & 0.07@xmath1660.02 & + gg tau a & 116@xmath16652 & 0.16@xmath1660.05 & 3.6@xmath1661.8 + go tau & 16@xmath1667 & 0.09@xmath1660.03 & 6.4@xmath1663.2 + ry tau & 40@xmath16618 & 0.06@xmath1660.02 & + gm aur & 49@xmath16622 & 0.16@xmath1660.05 & + lkca 15 & 33@xmath16615 & 0.14@xmath1660.04 & 8.6@xmath1664.3 + + + ux ori & 42@xmath16619 & & 9117 + hd 163296 & 65@xmath16629 & 0.56@xmath1660.16 & 0.4@xmath1660.2 + cq tau & 22@xmath16610 & & 2.0@xmath1661.0 + mwc 480 & 22@xmath16610 & 0.17@xmath1660.05 & 0.778.8 + mwc 863 & 10@xmath1665 & & 1.5@xmath1660.8 + hd 36112 & 29@xmath16613 & 0.23@xmath1660.07 & 0.218.7 + ab aur & 21@xmath1669 & 1.72@xmath1660.51 & 1.3@xmath1660.7 + + + 49 ceti & 0.4@xmath1660.2 & 10@xmath167 & 0.32.3 + hd 135344 & 2.8@xmath1661.3 & 2.1@xmath1660.6@xmath17910@xmath167 & 6.4@xmath1663.2 + @xmath76 pictoris & 0.03@xmath1660.015 & & 0.17@xmath1660.08 +

we present iso short - wavelength - spectrometer observations of h@xmath0 pure - rotational line emission from the disks around low and intermediate mass pre - main - sequence stars as well as from young stars thought to be surrounded by debris disks . the pre - main - sequence sources have been selected to be isolated from molecular clouds and to have circumstellar disks revealed by millimeter interferometry . we detect ` warm ' ( @xmath1 k ) h@xmath0 gas around many sources , including tentatively the debris - disk objects . the mass of this warm gas ranges from @xmath2 m@xmath3 up to @xmath410@xmath5 m@xmath3 , and can constitute a non - negligible fraction of the total disk mass . complementary single - dish @xmath6co 32 , @xmath7co 32 and @xmath6co 65 observations have been obtained as well . these transitions probe cooler gas at @xmath8 2080 k. most objects show a double - peaked co emission profile characteristic of a disk in keplerian rotation , consistent with interferometer data on the lower-@xmath9 lines . the ratios of the @xmath10co 32/@xmath11co 32 integrated fluxes indicate that @xmath10co 32 is optically thick but that @xmath12co 32 is optically thin or at most moderately thick . the @xmath11co 32 lines have been used to estimate the cold gas mass . if a h@xmath13/co conversion factor of 1@xmath1410@xmath15 is adopted , the derived cold gas masses are factors of 10200 lower than those deduced from 1.3 millimeter dust emission assuming a gas / dust ratio of 100 , in accordance with previous studies . these findings confirm that co is not a good tracer of the total gas content in disks since it can be photodissociated in the outer layers and frozen onto grains in the cold dense part of disks , but that it is a robust tracer of the disk velocity field . in contrast , h@xmath16 can shield itself from photodissociation even in low - mass ` optically thin ' debris disks and can therefore survive longer . the warm gas is typically 110 % of the total mass deduced from millimeter continuum emission , but can increase up to 100% or more for the debris - disk objects . thus , residual molecular gas may persist into the debris - disk phase . no significant evolution in the h@xmath16 , co or dust masses is found for stars with ages in the range of 10@xmath1710@xmath18 years , although a decrease is found for the older debris - disk star @xmath19 pictoris . the large amount of warm gas derived from h@xmath16 raises the question of the heating mechanism(s ) . radiation from the central star as well as the general interstellar radiation field heat an extended surface layer of the disk , but existing models fail to explain the amount of warm gas quantitatively . the existence of a gap in the disk can increase the area of material influenced by radiation . prospects for future observations with ground- and space - borne observations are discussed .

the scattering of two pions at low energies is the simplest of all dynamical strong - interaction processes from the theoretical standpoint . as pions are the pseudo - goldstone bosons associated with the spontaneous breaking of chiral symmetry , their low - momentum interactions are constrained by the approximate chiral symmetries of quantum chromodynamics ( qcd ) , and the scattering lengths for @xmath10 in the s - wave are uniquely predicted at leading order in chiral perturbation theory ( @xmath11-pt ) . higher orders in the chiral expansion give contributions to these scattering lengths that are suppressed by powers of @xmath12 , where @xmath13 is the mass of the pion , and @xmath14 is the scale of chiral symmetry breaking . however , part of these higher - order contributions are from local counterterms with coefficients that are not constrained by chiral symmetry alone . lattice qcd is the only known technique with which one can rigorously calculate these strong - interaction quantities . a lattice calculation of the @xmath1 scattering length as a function of the light - quark masses , @xmath15 , will allow the determination of the relevant counterterms that appear in the chiral lagrangian . of course , this is only true for values of @xmath13 within the chiral regime where @xmath16 . even though it is likely that in the future lattice calculations will be performed at the physical values of the light - quark masses , and while a chiral extrapolation of observables will not be necessary , the values of the counterterms will be essential as the chiral lagrangian will still be used to compute more - complex strong - interaction processes that are too costly to compute directly on the lattice . a second important motivation for studying @xmath1 scattering with lattice qcd is its impact upon weak - interaction processes ( and physics beyond the standard model ) , such as @xmath17 , and the determination of fundamental constants of nature associated with electroweak physics , such as cp violation in the standard model . lattice qcd calculations are performed on a euclidean lattice , and the maiani - testa theorem demonstrates that s - matrix elements can not be determined from @xmath18-point green s functions computed on the lattice at infinite volume , except at kinematic thresholds @xcite . however , lscher has shown that by computing the energy levels of two - particle states in the finite - volume lattice , the @xmath19 scattering amplitude can be recovered @xcite . the energy levels of the two interacting particles are found to deviate from those of two non - interacting particles by an amount that depends on the scattering amplitude and varies inversely with the lattice spatial volume . a number of lattice qcd calculations of @xmath1 scattering in the @xmath0 channel have been performed previously using lscher s method . for obvious reasons , calculations were initially performed in quenched qcd @xcite but in a recent tour - de - force study , the cp - pacs collaboration exploited the finite - volume strategy to study @xmath0 , s - wave @xmath1 scattering in fully - dynamical lattice qcd with two flavors of improved wilson fermions @xcite , with pion masses in the range @xmath20 . a lattice qcd calculation of @xmath1 scattering should fulfill several conditions : ( i ) quark masses small enough to lie within the chiral regime so that the extrapolation to the physical point is reliable , ( ii ) a lattice volume large enough to avoid finite - volume effects from pions `` going around the world '' but small enough for the energy shifts to be measurable and ( iii ) a lattice spacing small enough so that discretization effects are under control . based on these requirements we have performed a fully - dynamical lattice qcd calculation with two degenerate light quarks and a strange quark ( 2 + 1 ) with pion masses in the chiral regime , @xmath21 ( 319 configurations ) , @xmath22 ( 649 configurations ) and @xmath4 ( 453 configurations ) . the configurations we have used are the publicly - available milc lattices with dynamical staggered fermions of spatial dimension @xmath23 , and we have used domain - wall @xcite propagators generated by lhpc at the thomas jefferson national laboratory ( jlab ) . this paper is organized as follows . in section [ sec : finvol ] we discuss lscher s finite - volume method for extracting hadron - hadron scattering parameters from energy levels calculated on the lattice . in section [ sec : calcdet ] we describe the details of our mixed - action lattice qcd calculation . we also discuss the relevant correlation functions and outline our fitting procedures . in section [ sec : matching ] we present the results of our lattice calculation , and the analysis of the lattice data with @xmath11-pt . in section [ sec : resdisc ] we conclude . the s - wave scattering amplitude for two particles below inelastic thresholds can be determined using lscher s method @xcite , which entails a measurement of one or more energy levels of the two - particle system in a finite volume . for two particles of identical mass , @xmath24 , in an s - wave , with zero total three momentum , and in a finite volume , the difference between the energy levels and those of two non - interacting particles can be related to the inverse scattering amplitude via the eigenvalue equation @xcite @xmath25 where @xmath26 is the elastic - scattering phase shift , and the regulated three - dimensional sum is @xmath27 the sum in eq . ( [ eq : sdefined ] ) is over all triplets of integers @xmath28 such that @xmath29 and the limit @xmath30 is implicit @xcite . this definition is equivalent to the analytic continuation of zeta - functions presented by lscher @xcite . in eq . ( [ eq : energies ] ) , @xmath31 is the length of the spatial dimension in a cubically - symmetric lattice . the energy eigenvalue @xmath32 and its deviation from twice the rest mass of the particle , @xmath33 , are related to the center - of - mass momentum @xmath34 , a solution of eq . ( [ eq : energies ] ) , by @xmath35 in the absence of interactions between the particles , @xmath36 , and the energy levels occur at momenta @xmath37 , corresponding to single - particle modes in a cubic cavity . expanding eq . ( [ eq : energies ] ) about zero momenta , @xmath38 , one obtains the familiar relation @xmath39 \ + \ { \cal o}\left({1\over l^6}\right ) \ \ , \label{luscher_a}\end{aligned}\ ] ] with @xmath40 and @xmath41 is the scattering length , defined by @xmath42 for the @xmath0 @xmath1 scattering length , @xmath43 , that we consider in this work , the difference between the exact solution to eq . ( [ eq : energies ] ) and the approximate solution in eq . ( [ luscher_a ] ) is much less than @xmath44 . however , in determining the phase - shift associated with the first excited state on the lattice , the full eigenvalue equation in eq . ( [ eq : energies ] ) is solved . our computation of the @xmath0 @xmath1 scattering amplitude consists of a hybrid lattice qcd calculation using staggered sea quarks and domain - wall valence quarks @xcite . the parameters of the three sets of @xmath45 asqtad - improved @xcite milc configurations generated with staggered sea quarks @xcite that we used in our calculations are shown in table [ table : configs ] . in the generation of the milc configurations , the strange - quark mass was fixed near its physical value , @xmath46 , ( where @xmath47 is the lattice spacing ) determined by the mass of hadrons containing strange quarks . the two light quarks in the three sets of configurations are degenerate , with masses @xmath48 and @xmath49 . these lattices were hyp - blocked @xcite in order to avoid large residual chiral symmetry breaking . we used the domain - wall valence propagators that had been previously generated by the lhp collaboration on each of these sets of lattices . the domain - wall height is @xmath50 and extent of the extra dimension is @xmath51 . as this is a mixed - action calculation , the parameters used to generate the light - quark propagators have been `` matched '' to those used to generate the milc configurations . this was achieved by requiring that the mass of the pion computed with the domain - wall propagators be equal ( to few - percent precision ) to that of the lightest staggered pion computed from staggered propagators generated with the same parameters as the given gauge configuration @xcite . the parameters used in the generation of the domain - wall propagators are shown in table [ table : configs ] . 0.1 in .the parameters of the milc gauge configurations and lhpc domain - wall propagators used in our calculations . for each propagator the extent of the fifth dimension is @xmath51 . [ cols="^,^,^,^,^,^,^",options="header " , ] 0.35 in .2 in the results of our calculation of the product @xmath52 are shown as a function of @xmath53 in fig . ( [ fig : extrapolate ] ) . in addition , we have shown the lowest pion mass datum from the dynamical calculations of the cp - pacs collaboration @xcite , and have not attempted to extrapolate their result to the continuum . this lattice spacing is comparable to the one used in this work . ] . the uncertainty in the cp - pacs measurement is significantly smaller than that of our calculation and the agreement is very encouraging . in order to extrapolate @xmath54 to the physical value of @xmath53 , we performed a weighted fit of eq . ( [ eq : ascattglwithchexp2 ] ) to the three data points in table [ table : summary ] and extracted a value of the counterterm @xmath55 . as both quantities , @xmath52 and @xmath53 , are dimensionless there is no systematic uncertainty arising from the scale setting ( @xmath56 ) . we determined that @xmath57 , where the first error is statistical and the second is an estimate of the systematic error ( see fig . ( [ fig : extrapolate ] ) ) . this fit of @xmath55 allows , through eq . ( [ eq : ascattglwithchexp2 ] ) , a prediction of the scattering length at the physical value of the light - quark masses , which we find to be @xmath58 the last uncertainty , @xmath59 , is the largest and is an estimate of the systematic error resulting from truncation of the chiral expansion of the scattering length -pt result multiplied by an arbitrary coefficient : @xmath60 . this yields @xmath61 where the first error is statistical and the second is an estimate of the systematic error . the scattering length at the physical value of the light - quark masses is , in this case , @xmath62 . ] . the two - loop expression for the scattering length @xcite is given by @xmath63 \nonumber \right . \\ & & \left . \qquad\qquad + \ { m_\pi^4\over 64\pi^4 f_\pi^4}\ ; \left[\ \frac{31}{6}\,\left(\log{\frac{m_\pi^2}{\mu^2}}\right)^2 \ + \ l^{(2)}_{\pi\pi}(\mu ) \ ; \log{\frac{m_\pi^2}{\mu^2}}\ + \ l^{(3)}_{\pi\pi}(\mu ) \ \right ] \right\ } , \label{eq : ascattglwithchexp2twoloop}\end{aligned}\ ] ] where @xmath64 and @xmath65 are linear combinations of undetermined constants that appear in the @xmath66 and @xmath67 chiral lagrangians @xcite ( see appendix [ app : twoloop ] ) . it is not possible to provide a meaningful fit of these three undetermined constants from the few data points we have calculated . while there are estimates of these low - energy constants from a variety of sources , using these estimates in our extrapolation to the physical point would amount to trading one unknown systematic error for another . in order to estimate the systematic error due to extrapolation , we first set @xmath68 and refit @xmath55 keeping only the double - log piece at two - loop order . we then set @xmath69 and do a two - parameter fit to @xmath55 and @xmath64 . the difference between the extrapolation of the one - loop expression and the extrapolation of the two - loop expression with these simplifications gives the estimate of the systematic error resulting from truncating the chiral expansion . in fig . ( [ fig : extrapolate ] ) we show the unique prediction of leading order @xmath11-pt for the scattering length ( the dashed line ) , which is seen to agree remarkably well with both the lattice data and the physical value . the gray band in fig . ( [ fig : extrapolate ] ) shows the statistical @xmath70 region . in order to better isolate the contributions from higher orders in @xmath11-pt it is convenient to define a scale - independent `` curvature '' function @xmath71 : @xmath72 where , by construction , @xmath73 at tree level . the values for @xmath74 that we have calculated are listed in table [ table : summary ] , and are shown in fig . ( [ fig : chiralshit ] ) as a function of @xmath53 together with the experimental point and the cp - pacs result . 0.55 in a weighted fit of eq . ( [ eq : curvefun ] ) to the results of the lattice calculation gives : @xmath75 , consistent with the fit to @xmath52 . the gray band in fig . ( [ fig : chiralshit ] ) shows the @xmath70 region for @xmath76 . the extrapolated value of the scattering length is obviously the same as in the direct fit to @xmath52 . the value of @xmath55 favored by the fits are such that there is an almost perfect cancellation between the counterterm and the logarithm in eq . ( [ eq : ascattglwithchexp2 ] ) in the range of @xmath13 considered . this cancellation may be an unfortunate coincidence in this channel . a more refined lattice qcd calculation is required in order to detect the predicted chiral curvature . the best experimental determination of the @xmath0 scattering length is obtained through an analysis of @xmath77 decays @xcite , which gives @xmath78 where the first error is statistical , the second is systematic and the third is theoretical . this data point is plotted in fig . ( [ fig : extrapolate ] ) and in fig . ( [ fig : chiralshit ] ) . the two - loop @xmath11-pt `` prediction '' @xcite is @xmath79 . our result is in very good agreement with these determinations . we are only able to extract the phase - shift at one non - zero value of the pion momentum , and only at the heaviest pion mass , despite having a relatively clean signal for the first excited state in the lattice volume for all three sets of milc configurations . the reason for this is that the pion masses are sufficiently light that the first excited state is at an energy very near the four - pion inelastic threshold on the two sets of configurations with the lightest pion masses . at @xmath80 , and for pion momentum @xmath81 , the phase - shift is found to be @xmath82 degrees . in this paper we have presented the results of a lattice qcd calculation of the @xmath0 @xmath1 scattering length performed with domain - wall valence quarks on asqtad - improved milc configurations with 2 + 1 dynamical staggered quarks . the calculations were performed at a single lattice spacing of @xmath83 and at a single lattice size of @xmath23 with three values of the light quark masses , corresponding to pion masses of @xmath84 , and @xmath4 . we have also presented the phase - shift at the heavier quark mass . we have used one - loop @xmath11-pt to fit the combination of counterterms contributing to @xmath1 scattering at next - to - leading order to the lattice data and extrapolated in the light - quark masses down to the physical point . at the one - loop level we are able to make a prediction for the value of the scattering length , @xmath85 , which agrees within errors with the experimental value . we thank r. edwards for help with the qdp++/chroma programming environment @xcite with which the calculations discussed here were performed . we are also indebted to the milc and the lhp collaborations for use of their configurations and propagators , respectively . mjs would like to thank the center for theoretical physics at mit and the high - energy and the nuclear - theory groups at caltech for kind hospitality during the completion of this work . the work of mjs is supported in part by the u.s . dept . of energy under grant de - fg03 - 97er4014 . the work of ko is supported in part by the u.s . dept . of energy under grant df - 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dependent low - energy constants that appear in the @xmath66 and @xmath67 chiral lagrangians @xcite , respectively .

0.5 cm we compute the @xmath0 @xmath1 scattering length at pion masses of @xmath2 , @xmath3 and @xmath4 in fully - dynamical lattice qcd using lscher s finite - volume method . the calculation is performed with domain - wall valence - quark propagators on asqtad - improved milc configurations with staggered sea quarks at a single lattice spacing , @xmath5 . chiral perturbation theory is used to perform the extrapolation of the scattering length from lattice quark masses down to the physical value , and we find @xmath6 , in good agreement with experiment . the @xmath0 @xmath1 scattering phase shift is calculated to be @xmath7 at @xmath8 for @xmath9 . -1.5 cm 0.8 cm

the approach we are going to explain is closely related to the dieks - vermaas version of the modal interpretation : the same type of rules are used to assign properties to physical systems . but instead of the usual treatment in which properties are supposed to correspond to monadic predicates , we will propose an analysis according to which properties have a relational character . physical systems of a given kind are described within a characteristic hilbert space ; we will allow arbitrary hilbert spaces . in our perspectival approach the state of a physical system @xmath0 ( corresponding to physical characteristics of @xmath0 ) needs the specification of a ` reference system ' @xmath1 with respect to which the state is defined . this reference system is a larger system , of which @xmath0 is a part . as already mentioned , we will allow that one and the same system , at one and the same instant of time , can have different states with respect to different reference systems . however , the system will have one single state with respect to any given reference system . this state of @xmath0 with respect to @xmath1 will be denoted by @xmath2 . it is a density matrix , i.e. , a hermitian operator acting on the hilbert space of @xmath0 that is positive semidefinite and has unit trace . in the special case in which @xmath1 coincides with @xmath0 the state is in general ( i.e. , if there is no degeneracy , see below ) a one - dimensional projector @xmath3 this state @xmath4 ( or equivalently @xmath5 ) , the ` state of @xmath0 with respect to itself ' , is the same as ` the physical state ' assigned to @xmath0 in the dieks - vermaas version of the modal interpretation ; i.e. it is one of the projectors occurring in the spectral decomposition of the reduced density operator of @xmath0 , and @xmath5 is one of the eigenvectors , if there is no degeneracy see @xcite for these ideas . the rules for determining all states , for arbitrary @xmath0 and @xmath1 , are as follows . if @xmath6 is the whole universe , then @xmath7 is taken as the quantum state assigned to @xmath6 by standard quantum theory . if system @xmath0 is contained in system @xmath8 , the state @xmath9 is defined as the density operator that can be derived from @xmath10 by taking the partial trace over the degrees of freedom in @xmath8 that do not pertain to @xmath0 : @xmath11 any relational state of a system with respect to a bigger system containing it can be derived by means of eq.([g4 ] ) . we already saw that for an arbitrary system @xmath0 , contained in the universe @xmath6 , @xmath4 is postulated to be one of the projectors contained in the spectral resolution of @xmath12 . if there is no degeneracy among the eigenvalues of @xmath13 these projectors are one - dimensional and the state can be represented by a vector @xmath5 , see eq.([g3 ] ) ; in the case of degeneracy the state of the system with respect to itself is a multi - dimensional projector . for simplicity we will in the following focus on the non - degenerate case and assume that the state of @xmath0 with respect to itself is given by one of the eigenvectors @xmath14 of @xmath13 . the state @xmath7 evolves unitarily in time . because there is no collapse of the wave function in our approach , this unitary evolution of the total quantum state is the main dynamical principle of the theory . furthermore , we assume that the state assigned to a _ closed _ system @xmath0 undergoes a unitary time evolution @xmath15\label{g5}\end{aligned}\ ] ] as always in the modal interpretation , the theory specifies only the probabilities of the various possibilities ( the interpretation is indeterministic ) : the probability that @xmath5 is the eigenvector @xmath14 is given by the corresponding eigenvalue of @xmath13 . if the systems @xmath16 , @xmath17 , ... @xmath18 are pair - wise disjoint and @xmath6 is the whole universe , then the joint probability that @xmath19 coincides with @xmath20 , @xmath21 coincides with @xmath22 , ... , @xmath23 coincides with @xmath24 , is given by @xmath25 and eq.([g6 ] ) shows that the states of @xmath26 and @xmath27 are uniquely correlated . ( this result played an important role in earlier versions of the modal interpretation . ) therefore , knowledge of @xmath27 is equivalent to knowledge of @xmath26 . this suggests that one may consider the state of @xmath0 with respect to the reference system @xmath8 , @xmath28 , alternatively as being defined _ from the perspective _ @xmath29 ( here @xmath8 is an arbitrary quantum reference system , while @xmath6 is again the whole universe ) . sometimes this concept of a ` perspective ' is intuitively more appealing than the concept of a quantum reference system ( cf . @xcite ) . nevertheless , there are some limitations inherent in this alternative . first , if @xmath8 itself is the whole universe , the concept of an external perspective can not be applied . moreover , the state of the system @xmath29 in itself does not contain sufficient information to determine the state of system @xmath8 ; one also needs the additional information provided by @xmath30 in order to compute @xmath26 . but @xmath26 does contain all the information needed to calculate @xmath28 ( cf . eq.([g4 ] ) ) . we will therefore relativize the states of @xmath0 to reference systems that contain @xmath0 , although we shall sometimes in cases in which this is equivalent also speak about the state of @xmath0 _ from the perspective _ of the complement of the reference system . of course , we must address the question of the physical meaning of the states @xmath28 . in our approach it is a fundamental assumption that basic descriptions of the physical world have a relational character , and therefore we can not explain the relational states by appealing to a definition in terms of more basic , and more familiar , non - relational states . but we should at the very least explain how these relational states connect to actual experience . minimally , the theory has to give an account of what observers observe . we postulate that experience in this sense is represented by the state of a part of the observer s perceptual apparatus ( the part characterized by a relevant indicator variable , like the display in our simple model ) _ with respect to itself_. more generally , the states of systems with respect to themselves correspond to the ( monadic)properties assigned by the earlier , non - perspectival , version of the modal interpretation . as we shall illustrate , the empirical meaning of many other states can be understood and explained - by using the rules of the interpretation - through their relation to these states of observers , measuring devices , and other systems , with respect to themselves . the simple model we are going to use is sketched in fig.1 . the right hand side part of the drawing represents a digital measuring device . it consists of several individual blocks that do not interact with each other ( as a representation of the situation in the nerve system this is a strong simplification , though part of the perceptual system can be modelled this way , especially in situations in which the incoming stimulus has a low intensity ) . the blocks are assumed to be very close to each other and their diameters are larger than but comparable to the wavelength of light . ( the diameters of the rods and cones in the human retina are 1 - 2 @xmath31 m ) . each block consists of a receptor ( drawn as a small rectangle ) and a display ( drawn as a circle ) . the operation of a receptor may be described roughly by means of two states , one corresponding to the receptor being excited , the other the ` ready - to - measure ' state . we shall denote these by @xmath32 and @xmath33 , respectively . any superposition of these states is also allowed . a basis in the hilbert space of the total measuring device consisting of @xmath34 receptors is given by the states @xmath35 , in which @xmath36 , referring to the @xmath37-th receptor , can be 0 or 1 . the displays have corresponding states that we denote by @xmath38 and @xmath39 for an individual display , and by @xmath40 for the whole set of displays . to make the ideas clear we shall assume that the interaction between the receptors and displays is such that if the @xmath37-th receptor is excited , and its state accordingly becomes @xmath32 , then the @xmath37-th display will with certainty end up in the corresponding indicator state @xmath38 . in other words , the receptor - display system accomplishes an ideal von neumann measurement . this is known to be approximately realizable . in the last section of this paper we will investigate a more refined form of the model , in which the degeneracy of the just - mentioned receptor and display states is taken into account . the just - mentioned assumption immediately implies that an arbitrary measurement leads to the entangled state @xmath41 of the whole model system , where the states @xmath42 of the measured object+environment system are normed but not necessarily mutually orthogonal . due to the one - to - one relationship between the orthonormal states @xmath35 and @xmath40 , the reduced density matrix of the display system is @xmath43 i.e. , it is diagonal in the ` definite display result ' basis . its eigenstates ( except in case of degeneracy , i.e. exact equality of the squared coefficients in ( [ e2 ] ) ) are the elements @xmath40 of the display result basis . a basic principle of our interpretation is that the eigenstates of the reduced density matrix are the ` states of the system with respect to itself ' and correspond to possible physical properties of the system ( in the same way as the states assigned in the earlier versions of the modal interpretation did ) . one of these possibilities will be actually realized , and the probability for any particular eigenstate of representing the actual state of affairs is given by the value of the corresponding eigenvalue of the reduced density matrix . it can thus be concluded from eq.([e2 ] ) that the observation has a definite result corresponding to one of the display states . as stressed before , in the perspectival approach this definite property is represented by the state of the display with respect to itself this leaves it open that the state and the corresponding properties may be different if relativized to another reference system . below we will indeed encounter an example of such a difference in state ascriptions corresponding to different perspectives . at this point we can already make a brief remark about the role of decoherence . in the model the environment of the display system is the receptor system , while coupling to the rest of the world is neglected . therefore , environment - induced decoherence in the usual sense does not play a role here , although entanglement between receptors and displays is essential . in other words , the display states are ` decohered ' by their correlation with the mutually orthogonal receptor states . that a measurement , performed with a given device , invariably leads to one of a number of alternatives that are determined by the nature of the device and are independent of the measured object ( the latter determines only the probability that a particular outcome occurs ) has always been a standard assumption in quantum measurement theory . in the present treatment we derive this assumption from the modal approach applied to our specific model of the measuring device . it should be noted that we did not presuppose classical features of the device ; the whole model is treated quantum mechanically . suppose that a single photon is scattered from the object . if the object possesses a large mass , we may neglect the back reaction ( recoil ) , so the total wave function of object and photon after the scattering is of the form @xmath44 in this equation @xmath45 stands for the position of the object , whereas @xmath46 and @xmath47 denote the degrees of freedom of the environment and the photon , respectively . after the photon has been absorbed in one of the receptors , the state of the complete system ( including the receptors and the displays ) is @xmath48 is the amplitude that the photon which has been scattered from @xmath45 , in the situation depicted in fig . 1 , is absorbed at a later time in the @xmath37-th receptor . the amplitude is negligible unless the position of the @xmath37-th receptor is near the geometrical optical image of the point @xmath45 . it follows that the reduced density matrix of the display system is given by @xmath49 here @xmath50 stands for the reduced density matrix of the object in coordinate representation , still before the object is exposed to the light . due to the recoil - free nature of the interaction , the diagonal elements @xmath51 are not influenced by the light scattering ( compare eq.([e4prime ] ) below ) . therefore , according to the modal scheme the actual physical condition of the display system ( its state with respect to itself ) is @xmath52 with probability @xmath53 note that eq . ( [ e5 ] ) is a special case of eq . ( [ e2 ] ) , thus eq . ( [ e7 ] ) actually follows from the previous general considerations . in a more general treatment , we should also take into account that the particle experiences a back - reaction because of its interaction with the photon . instead of eq.([e4 ] ) , we then have @xmath54 are transformed by the recoil are denoted by @xmath55 ; these states are not necessarily mutually orthogonal . the probability of the j - th display being excited now becomes @xmath56 since the states @xmath55 are in general not mutually orthogonal , non - diagonal elements of the reduced density matrix of the object enter the expression . if the recoil is negligible , one has @xmath57 and eq.([e8 ] ) is recovered . it is instructive to calculate the reduced density matrix @xmath58 of the object ( state of the object with respect to the whole system ) after it has been exposed to light , but still before the absorption of the photon in the receptors . using eq.([e3 ] ) one obtains @xmath59 after the absorption of the photon in the receptors one gets the same result because the time evolution during the absorption is unitary and the object degrees of freedom are not involved in the interaction . a calculation based on eq . ( [ e4 ] ) gives the alternative expression @xmath60 the equality of expressions ( [ e9 ] ) and ( [ e10 ] ) gives a condition the ` transfer functions ' @xmath61 must satisfy . according to the modal interpretations one of the eigenstates of @xmath62 represents the state of the object with respect to itself ( this is the physical state of the object according to the terminology of the dieks - vermaas modal interpretation ) . in general , these eigenstates are not localized @xcite , so they do not correlate with the result of the observation , which indicates a definite position . the question naturally arises what state the observation then corresponds to . according to the modal scheme the answer follows from the biorthogonal decomposition of the state of the whole system ( note that eq.([e4 ] ) is already of this form ) . thus , the observation is perfectly correlated to the state of the object+environment+receptor system with respect to itself . in other words , from the perspective of @xmath63 all information about the object state is contained in the relational state of the object with respect to object+environment+receptor . direct calculation based on eq.([e4 ] ) yields for this object state from the perspective of @xmath63 : @xmath64 in this equation the superscript @xmath65 refers to the object , and the subscript @xmath66 to the reference system , which is the complement of the system @xmath63 defining the perspective . owing to the properties of the amplitudes @xmath61 , for a given @xmath37 the expression ( [ e11 ] ) will be appreciable only if @xmath45 and @xmath67 are located in a small interval ( from which light will be scattered to the @xmath37-th receptor ) . this means that the object is indeed localized from the point of view of @xmath63 ( or , equivalently , with respect to considered as part of@xmath68 ) . it should be noted that the form of the state of @xmath68 is determined by the interaction constituting the observation that took place . without the observation we could not speak of localization as a relational property of the object . the example illustrates how different properties may be ascribable to an object from different perspectives . in this case , the state of the object with respect to itself is not localized . however , if the complement of the display system is chosen as the reference system , a description in terms of a localized state does apply . both descriptions are objective , but relational ; they involve the specification of different reference systems . an analogy may be helpful to see that the relational character of descriptions does not entail a lack of objectivity . according to the special theory of relativity one and the same object can be described as moving or as resting , without the implication that one of these descriptions is more fundamental than the other . both descriptions are entitled to be called objective , but make use of different ` perspectives ' . the ascription of properties in our interpretation of quantum mechanics has a similar relational character . we will now consider a situation in which two or more measurements are performed on the same object , by means of two different measuring devices . we know from everyday experience that the results of such measurements will ( more or less ) agree . our traditional notions concerning physical reality ( in particular , the idea that the properties of objects are independent of any perspective and can be represented by non - relational , monadic , predicates ) are to a large extent based on this and similar facts of experience . it is therefore important to show that our perspectival approach is in accordance with this agreement among observers ( or measuring devices ) . for simplicity we assume that the two devices are at the same position and do not disturb each other . in this way we ensure that both measurements take place under identical circumstances . analogously to eq.([e4 ] ) , the state of the compound system consisting of object , first device , and second device will after the measurement be given by @xmath69 with probability @xmath70 similarly , the state of the second display system with respect to itself is @xmath71 with probability @xmath72 according to the standard rules of the modal interpretation ( @xcite , see eq.([g6 ] ) ) the joint probability that the state of the first display system with respect to itself is given by eq.([f7 ] ) and the state of the second display system with respect to itself is given by eq.([f7a ] ) is @xmath73 this expression has the same form as the joint probability distribution predicted by classical theory for a situation in which independent measurements are made on each member of an ensemble of systems distributed in space with probability density @xmath51 . the properties of the coefficients @xmath61 imply that the expression ( [ f9 ] ) vanishes unless @xmath74 . indeed , @xmath61 is practically zero unless the @xmath37-th receptor block is situated near ( i.e. , in a distance of few wavelengths ) of the geometrical optical image of @xmath45 . therefore , our interpretational scheme predicts that two observers looking at the same macroscopic object , at the same time and under identical circumstances , will see it ( practically ) in the same place . we assumed in the calculation that the interaction between the photon and the object was recoil - free ; this is justified in the case of a macroscopic object . we have already seen that if recoil should be taken into account , off - diagonal elements @xmath75 from the narrow band in the matrix where @xmath61 and @xmath76 are both appreciably different from zero will enter the expression for the probability of the j - th receptor being excited . moreover , if there is substantial recoil the second measurement will show a different result than the first , because of the disturbance caused by the first measurement . this is not different from what would happen according to classical physics . in expression ( [ f9 ] ) it does not matter whether the system was in a pure or mixed state prior to the measurement . therefore , in our analysis of the observational mechanism macroscopic objects will be observed as localized quite independently of whether decoherence of the object state by interaction with its environment has taken place . our analysis indicates that this point generalizes to arbitrary observation mechanisms that possess the characteristic finiteness and determinism assumed in our model . this result brings to light an important difference between our approach , in which features of the measurement process take a central role , and approaches according to which the localization of macroscopic objects is due to environment - induced decoherence of the object state . as emphasized before , according to the modal interpretation environment - induced decoherence will in general _ not _ guarantee that a macroscopic object will be localized ( in the sense that its state with respect to itself is localized ) , because of the lack of localization of the eigenstates of the object s reduced density matrix . a similar analysis applies to the case in which the measurement is repeated , possibly many times and in rapid succession , by means of the same device . an adequate mathematical treatment of that case includes the description of a memory which stores the result of the first measurement ( this memory would be analogous to the first display system in the above situation ) and a mechanism which resets the measuring device and prepares it for the next measurement . without discussing the details , we just mention that the results ( especially the counterpart of eq.([f9 ] ) ) are completely analogous . although the agreement between different observers , which fits in naturally with the classical notion of physical reality and may even seem to imply it , was just found to be present in our interpretation of the quantum formalism as well , the overall picture of physical reality that emerges is very different from the usual one . a good starting - point for an explanation of the differences is a discussion of the applicability of the einstein - podolsky - rosen reality criterion . this criterion says that _ if , without in any way disturbing a system , we can predict with certainty ( i.e. , with probability equal to unity ) the value of a physical quantity , then there exists an element of physical reality corresponding to this physical quantity._@xcite . let us see how this applies to the just - discussed measurement situation , in which the object interacts with the impinging photons , which is followed by an interaction between the scattered photons and the receptors , after which there finally is an interaction between the displays and the receptors . the whole process is repeated in the second measurement . the result shown by the first display system allows an ( almost ) certain prediction of the result of the second measurement . the question is what we can say about the state of the object , after the light has been scattered for the first time , on the basis of the epr - criterion . the interactions between the photons and the receptors and between the receptors and the displays obviously do not disturb the object , which may find itself at a large distance . as soon as the display shows a result ( or if the result is read off , but we prefer to avoid the introduction of a conscious observer ) , there is a one - to - one relation with the result of the second measurement . in other words , the result of the second measurement is predicted , with certainty , by the result of the first measurement . as we have seen , the prediction is that the system will be found at a definite position . at first sight , the epr - criterion therefore seems to imply that the object system already possessed a definite position from the moment it interacted with the photons . however , in the approach that we are explaining things are not so straightforward . in our scheme , the physical quantity that is predicted corresponds to a _ relational _ state of the object , namely its state with respect to the object+environment+receptor system ( cf . eq.([e11 ] ) ) . now , the important point is that the first measurement _ has given rise to the perspective _ from which it is possible to make this prediction . therefore , although it is true that there was no physical interaction between the display and the object , the display nevertheless plays a part in determining the object state with respect to object+environment+receptor ( which is the complement of the display system itself ) . the physical interaction with the display affected the reference system , and therefore influenced the relational state . the fact that the relevant states have this relational ( perspectival ) character is responsible for the failure of ordinary counterfactual reasoning : from the fact that no physical disturbance has affected the object , it can not always be concluded that the object state has remained the same . one should also look at the reference system , with respect to which the state is defined , and see whether anything has changed _ there _ that is relevant . einstein , podolsky and rosen thought it very unlikely that the quantities of an object system would depend on whether or not a remote measurement is performed . within the conceptual framework of classical physics , in which properties attach to an object as monadic ( non - relational ) predicates , this skepticism is completely justified . however , in our present framework the possibility of the dependence in question naturally appears , not as an effect of physical disturbances acting on the object but as a consequence of a change in the conditions that define the perspective . this change comes about by local physical influences on the quantum reference system . this line of reasoning is in accordance with bohr s qualitative arguments that any reasonable definition of physical reality in the realm of quantum phenomena should also include the experimental setup @xcite . in the present relational approach states of a system are defined with respect to _ any _ larger physical system , so the concept of reality is not exclusively connected to the presence of instruments . nevertheless , in our scheme too , observed reality contains elements relating to us as observers in an essential way : we define the perspective . however , this does not imply any subjectivism . the various relational states follow unambiguously from the quantum formalism , and the way the world should be described depends accordingly on objective physical features ( whether or not the observing system is discrete , the nature of the interaction , whether the object has a large mass , and so on ) . these considerations show how the very concept of reality is modified in our interpretation of quantum mechanics . the essential new point is that quantum properties and quantum states possess a relational character . in general , one may expect that this quantum feature will not be noticeable on the level of observations , because of the agreement between different observers . yet , the modification persists even in the macroscopic domain . as we saw , the observed localization of macroscopic objects is absent from a different perspective . and on the observational level , bell - type experiments reveal the untenability of the traditional notions of reality ( monadic properties combined with locality ) . let us return to the epr reality criterion and draw conclusions about its status within our conceptual framework . as it stands , the criterion is ambiguous ( as observed by bohr in his reply to einstein , podolsky and rosen ) , since nothing is said about the perspective from which the physical quantity whose value can be predicted is defined . if the reference system _ is _ specified , the criterion is valid if neither the described system nor the quantum reference system is disturbed . that is , if it is possible to predict a relational state without any changes either in the reference system or the object , the state is there ( as a part of physical reality ) independently of whether or not the prediction is made . specifically , one could understand the epr - criterion as referring to the state of the system with respect to itself . in this case , the well - known ` no - signaling ' theorem becomes relevant : a system s density matrix ( found by partial tracing from @xmath7 ) , and therefore its eigenstates , will not change as a result of things happening elsewhere ( remember that we do not have collapses of the wave function in our scheme ) . so , if it is possible to predict the state of a far - away particle ( w.r.t . itself ) on the basis of measurements performed elsewhere , we surely should conclude that that state existed independently of those measurements , and the epr criterion therefore holds . let us now apply the epr reality criterion to the case for which it was devised , the case of distant correlated particles . we find that the state of the second particle that becomes precisely known after a measurement on the first particle , is the state of this second particle with respect to the two - particle system ( i.e. , the state from the perspective of the measuring device ) . however , it can not be concluded that this state was already present before the measurement , because the state of the reference system w.r.t.itself ( from which the state of particle 2 w.r.t . this reference system is derived ) was changed by the measurement . if one writes down the states explicitly , applying the given rules to the situations before and after the measurement , one easily establishes that the state of particle 2 w.r.t . the two - particle system indeed changes as a result of the measurement , in spite of the fact that there was no mechanical disturbance of particle 2 . by contrast , the state of particle 2 with respect to itself does not change and there is no influence on the reference system . so , if the state of particle 2 with respect to itself can be predicted from the result of the first measurement , application of the epr criterion is possible and yields a result which is in harmony with quantum mechanics ( in our interpretation ) : the state of particle 2 w.r.t . itself was indeed an element of physical reality already before , and independently of , the measurement on particle 1 . more generally , although the epr criterion can be upheld within our conceptual framework ( by specification of the missing reference system ) , its application does not lead to the conclusion that there are more elements of reality than the relational states admitted in our interpretation from the outset . as it appears , the modification of the reality concept proposed here makes the introduction of ` quantum nonlocality ' superfluous . indeed , the change in the relational state of particle 2 ( with respect to the 2-particle system ) can be understood as a consequence of the local change in the reference system , brought about by the measurement interaction . the local measurement interaction is responsible for the creation of a new perspective ( the state of the measuring device ) , and from this new perspective there is a new state of particle 2 . this agrees with a conclusion not infrequently drawn from the violation of bell s inequalities , namely that we should _ either _ abandon the usual realism concept ( something we do here ) _ or _ give up the principle of locality ; but not necessarily both . let us now consider the case in which the two measurements take place at different instants of time . as before , we shall assume that the measurements are performed by two different measuring devices , both situated at the same place . in this section we suppose that the interaction between the object and its environment is negligible and that the object initially has its own wave function . finally , we restrict our considerations to the case in which the object moves in one spatial dimension . after the first measurement the whole system evolves freely during a time interval @xmath77 . at the end of this interval the total state ( i.e. , the state of the compound system , object+first receptor+first display , with respect to itself ) can be written as @xmath78 is the propagator representing the free evolution between the measurements . after the second measurement has finished , the state of the total system reads @xmath79 according to the modal interpretation rules , the state of the object with respect to the object plus receptor system ( i.e. , the object state from the perspective of @xmath80 and @xmath81 ) is one of the states @xmath82 with the probability @xmath83 if the object system is macroscopic , by which we mean that the action is large compared to @xmath84 , the state @xmath85 is well localized in both coordinate and momentum space . the former follows directly from the narrowness of the function @xmath86 . note , however , that the width in coordinate space is comparable to the wavelength of the light , so it is still very wide on the scale of the de broglie wavelength of the object . as for the momentum , we can make use of the fact that for a macroscopic object the propagator has the approximate form @xmath87 where @xmath88 is the classical action as a function of the time and the initial and final coordinate of the orbit , and the function @xmath89 is smooth at the scale of the de broglie wavelength . inserting this into eq.([f10a ] ) and calculating the probability distribution of the momentum , one gets via saddle point integration @xmath90 here @xmath91 stands for the solution of the equation @xmath92 eqs.([f10e ] ) and ( [ f10f ] ) imply that , again due to the narrowness of @xmath93 and @xmath86 , the momentum is distributed in a narrow range around the classical value @xmath94 . in other words , the two measurements define a classical orbit . the question may be asked whether further measurements will confirm that the object will be near this orbit . in order to investigate this , let us consider a third measurement that takes place after the second one , after an elapsed time interval @xmath95 . the wave function of the whole system , including the object and the three measuring devices , is now @xmath96 if we now calculate the conditional probability of getting the @xmath97-th result at the third measurement ( the @xmath97-th display showing a result ) , given that the @xmath37-th and the @xmath98-th result had been obtained at the first and the second measurement , respectively , we get @xmath99 as discussed above , @xmath85 is a wave packet with fairly well defined coordinate and momentum . therefore , it evolves in time in such a way that the expectation value of the coordinate and the momentum obeys the classical equations of motion , as stated by ehrenfest s theorem . hence , the conditional probability ( [ f12 ] ) will be different from zero only if @xmath100 is nonzero near the classical trajectory at time @xmath95 . if @xmath97 does not correspond ( in the sense of optical imaging ) to the end point of this trajectory , the conditional probability ( [ f12 ] ) vanishes . this result demonstrates how the classical laws of motion emerge from a purely quantum mechanical description . note that the interaction of the object with its environment certainly influences the resulting classical equations ( for example , through the appearance of dissipative terms ) , but the emergence of classical motion itself is independent of whether or not there is environment - induced decoherence . in summary , what this section shows is that object systems that have a semi - classical propagator ( action large compared to @xmath84 ) follow classical paths . the mechanism of observation , and the fact that we consider object states defined from the perspective of the displays , is essential here . without these ingredients we would have no guarantee that the object wave packet is small , and no localization and definite path would therefore result . our considerations indeed demonstrate in the literal sense heisenberg s famous statement _ die `` bahn '' ensteht erst dadurch , da wir sie beobachten_.@xcite a central idea of our approach is not to _ assume _ the localization of physical objects , but to derive it as a result of the measurement interaction . we should therefore take into account the possibility that even the measuring device itself may be delocalized , in the sense that its wavefunction is not narrow in the position representation . in this section we consider what happens in this case . for simplicity we assume again that both the object and the measuring device move in one dimension , along parallel lines . we consider two simultaneous measurements taking place at the same spot . under these circumstances the total wave function is @xmath101 represents the center of mass coordinate of the measuring device . we have assumed in ( [ h1a ] ) that the interaction between the photons and the apparatus depends only on their mutual distance ( and not on absolute position ) ; in other words , that the interaction hamiltonian is translationally invariant . as before , we can conclude from the strict coupling between receptors and displays that the states of the display systems with respect to themselves are such that only one display block is excited . the joint probability that the @xmath37-th block of the first device and the @xmath98-th block of the second device are excited is given by @xmath102 as the functions @xmath61 are well localized in their arguments around a coordinate value that depends on their indices , eq.([h2 ] ) implies that @xmath74 . indeed , if @xmath37 and @xmath98 differ appreciably , for any value of @xmath103 at least one of the functions @xmath104 and @xmath105 is zero , so that the integral in eq.([h2 ] ) vanishes . therefore , both measurements find the object at the same place . the next question is what the state of the outside world is that corresponds to this well - defined position . in order to answer this question we should calculate the state of the system consisting of the object and the center of mass of the measuring device , with respect to the bigger system that also contains the receptors , because this state gives a description from the perspective of the displays . using the rules of our approach , we get latexmath:[\ ] ] in eq.([dx15 ] ) @xmath123 stands for a matrix whose @xmath124-th element is the operator @xmath125 . expression ( [ dx14 ] ) is the scalar product of the states @xmath126 and @xmath127 as the operators @xmath128 and @xmath129 are in general quite different , states ( [ dx16 ] ) and ( [ dx17 ] ) soon behave like two randomly chosen vectors in the @xmath34-dimensional hilbert space of the environment . the expectation value of the modulus square of the scalar product of two such random vectors is @xmath130 , thus latexmath:[\[\begin{aligned } therefore , if @xmath34 is large , @xmath132 becomes approximately block diagonal in the basis @xmath133 . suppose that there is a very large , but finite , number @xmath134 of such basis vectors . the constraints that the eigenvalues must be positive and that they add up to unity lead to a small level spacing if @xmath134 is large . assuming equidistant eigenvalues , the level spacing of @xmath135 is @xmath136 . the ensuing closeness of the eigenvalues implies that the eigenstates tend to be superpositions of the basis states @xmath133 . the elements of the off - diagonal block must be much smaller than the level spacing in order to avoid a mixing between the two subspaces ( compare @xcite ) . the requirement @xmath137 is easily satisfied if the dimensionality of the environment s hilbert space is large . if it is satisfied , the eigenstates of @xmath132 have the stable property of belonging to one or the other subspace ( ready or excited states , respectively ) . it is this property that corresponds to a definite outcome of a measurement . a generalization to the case of several ( @xmath138 ) displays is straightforward . in that case the system of interest contains all the displays , and one has to assume @xmath139 subspaces within the system s hilbert space that are invariant under time evolution . the above formalism applies with these amendments . the conclusion is again that interaction with an environment whose hilbert space has a sufficiently high number of dimensions leads to a block diagonal density matrix . this density matrix will have eigenstates which for each display belong stably to one or the other subspace . this corresponds to a well - defined excitation pattern of the display system . one of us ( g.b . ) acknowledges the support given by the nato science fellowship program , grant , and a jnos bolyai research fellowship . he also thanks for the hospitality extended by the institute for the history and foundations of science , faculty of physics and astronomy , utrecht university . 99 g. bene , quantum reference systems : a new framework for quantum mechanics , _ physica a _ * 242 * ( 1997 ) 529 - 565 . d. dieks and p.e . vermaas , _ the modal interpretation of quantum mechanics _ , kluwer academic publishers , dordrecht , 1998 . vermaas and d. dieks , the modal interpretation of quantum mechanics and its generalization to density operators , _ foundations of physics _ * 25 * ( 1995 ) 145 - 158 . g. bacciagaluppi , _ modal interpretations of quantum mechanics _ , cambridge university press , cambridge , 2000 . j. bub , _ interpreting the quantum world _ , cambridge university press , cambridge , 1997 . d. dieks , resolution of the measurement problem through decoherence of the quantum state , _ physics letters a _ * 142 * ( 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we study the process of observation ( measurement ) , within the framework of a ` perspectival ' ( ` relational ' , ` relative state ' ) version of the modal interpretation of quantum mechanics . we show that if we assume certain features of discreteness and determinism in the operation of the measuring device ( which could be a part of the observer s nerve system ) , this gives rise to classical characteristics of the observed properties , in the first place to spatial localization . we investigate to what extent semi - classical behavior of the object system itself ( as opposed to the observational system ) is needed for the emergence of classicality . decoherence is an essential element in the mechanism of observation that we assume , but it turns out that in our approach no environment - induced decoherence on the level of the object system is required for the emergence of classical properties . modal interpretations ( see , e.g. , ) aim at assigning _ properties _ ( or _ states _ that represent these properties in a one - to - one way ) to physical systems on the basis of the standard quantum mechanical formalism , though stripped from the postulates that attribute a special role to measurements . the motivation for introducing states that correspond to physical properties is the wish to give _ descriptions _ of systems , and thus to transcend the traditional interpretational framework in which systems are only discussed in terms of possible measurement results . the removal of the measurement postulates has the same background . we want to treat measurements as ordinary physical interactions , and measurement outcomes as properties of measuring devices or displays , and thus to remove any mysterious aspects of the concept of quantum measurement . as a first step towards this goal we assume all time evolution in hilbert space to be unitary , so that there is no collapse of the wave function . the modal approach based on these starting - points has proved to be appealing and successful when applied to situations in which the hilbert spaces are finite and the number of dimensions is not too large @xcite . however , in the case of continuous model systems , like freely moving particles or harmonic oscillators ( assumed to be interacting with the environment ) , the existing prescriptions are not guaranteed to lead to the expected classical properties . in particular , one such model study @xcite has failed to demonstrate more or less sharp localization of a particle in circumstances in which one would expect a classical description to be applicable . ( as for other kind of difficulties - not to be considered here - see @xcite . ) the essence of the failure to produce localization is not the continuity of the model , since the difficulty persists in models in which the hilbert space has a finite but large dimensionality . this can be clearly seen in computer simulations , in which one always works with finite hilbert spaces . at present it is not clear whether in more realistic models the situation will improve and classical properties will result . nevertheless , it seems to be worth considering - still within the modal scheme - another possibility , namely , that the observed classical properties are inevitable and generic consequences of the observation itself , but need not be present in the absence of observations . as already emphasized , we consider observations and measurements as ordinary physical interactions between object system and measuring device , and treat them quantum mechanically . it is indeed a - priori not implausible that applying the modal scheme to the perceptual system itself will lead to results that are in accordance with experience . the reason is that our nerve system has an inherent discreteness , both in its spatial structure ( cells ) and in its functionality ( a nerve cell either fires or does not fire ) . it is this kind of discreteness which seems to be needed to recover the expected classical alternatives within a quantum mechanical treatment @xcite . here , we will make a detailed investigation of the implications of such a discreteness in a simple model ( which can be conceived either as a model of a digital measuring device or as a very crude model of a part of the nerve system ) . the result is that according to this model an observer looking at an object will see the object localized : _ from the point of view of the observer the object is localized_. however , it turns out that the object is delocalized from a different perspective . the wider question addressed in this paper is whether these ( and similar ) results can be fitted into a consistent and satisfactory picture . an object can not be both localized and delocalized , so that it seems that inconsistency threatens . however , we will propose , and to some extent develop , an interpretational scheme ( a generalisation of the dieks - vermaas modal interpretation ) according to which it is _ not _ contradictory to assign such seemingly conflicting properties to an object . in this ` perspectival ' version of the modal interpretation properties of physical systems have a _ relational _ character and are defined with respect to another physical system that serves as a reference system @xcite . it is important to emphasize already now that the core idea of this new conceptual scheme is that the different descriptions , given from different perspectives , are equally objective and all correspond to physical reality . because of the relational character of the descriptions this involves no contradiction . a contradiction would only arise if different descriptions would be given from _ one _ perspective , or from compatible perspectives that can be combined into one . this will not happen in the interpretation that we will propose . furthermore , we will show that different observers observing the same object will agree about the results , just as in classical physics ( provided , of course , that the observations do not change the object ) . we shall also consider the time evolution of a macroscopic object and study when and why the classical description becomes applicable . finally , we discuss the role of environment induced decoherence .

generators using radiation from an electron beam in a periodic slow - wave circuit ( travelling wave tubes , backward wave oscillators , free electron lasers ) are now widespread @xcite . diffraction radiation @xcite in periodical structures is in the basis of operation of travelling wave tubes ( twt ) @xcite , backward wave oscillators ( bwo ) and such devices as smith - purcell lasers @xcite and volume fels using two- or three - dimensional distributed feedback @xcite . analysis shows that during operation of such devices electrons lose their energy for radiation , therefore , the electron beam slows down and gets out of synchronism with the radiating wave . these limits the efficiency of generator , which usually does not exceed @xmath0 . in the first years after creation of travelling wave tube it was demonstrated @xcite that to retain synchronism between the electron beam and electromagnetic wave in a twt change of the wave phase velocity should be provided . application of systems with variable parameters in microwave devices allows significant increase of efficiency of such devices @xcite . the same methods for efficiency increase are widely used for undulator fels @xcite . in the present paper we consider generation process in smith - purcell fels , volume fels , travelling wave tubes and backward wave oscillators using photonic crystal built from metal threads @xcite . it is shown that applying diffraction gratings ( photonic crystal ) with the variable period one can significantly increase radiation output . it is also shown that diffraction gratings ( photonic crystal ) can be used for creation of the dynamical wiggler with variable period in the system . this makes possible to develop double - cascaded fel with variable parameters changing , which efficiency can be significantly higher that of conventional system . in general case the equations , which describe lasing process , follow from the maxwell equations : @xmath1 here @xmath2 and @xmath3 are the electric and magnetic fields , @xmath4 and @xmath5 are the current and charge densities , the electromagnetic induction @xmath6 and , therefore , @xmath7 , the indices @xmath8 correspond to the axes @xmath9 , respectively . the current and charge densities are respectively defined as : @xmath10 where @xmath11 is the electron charge , @xmath12 is the velocity of the particle @xmath13 ( @xmath13 numerates the beam particles ) , @xmath14- \frac{\vec{v}_{\alpha}}{c^2}(\vec{v}_{\alpha}(t)\vec{e}(\vec{r}_{\alpha}(t),t ) ) \right\},\label{sys1}\ ] ] here @xmath15 is the lorentz - factor , @xmath16 ( @xmath17 ) is the electric ( magnetic ) field in the point of location @xmath18 of the particle @xmath13 . it should be reminded that the equation ( [ sys1 ] ) can also be written as @xcite : @xmath19 \right\},\label{sys11}\ ] ] where @xmath20 is the particle momentum . combining the equations in ( [ sys0 ] ) we obtain : @xmath21 the dielectric permittivity tensor can be expressed as @xmath22 , where @xmath23 is the dielectric susceptibility . when @xmath24 the equation ( [ sys01 ] ) can be rewritten as : @xmath25 when the grating is ideal @xmath26 , where @xmath27 is the reciprocal lattice vector . let the diffraction grating ( photonic crystal ) period is smoothly varied with distance , which is much greater then the diffraction grating ( ptotonic crystal lattice ) period . it is convenient in this case to present the susceptibility @xmath23 in the form , typical for theory of x - ray diffraction in crystals with lattice distortion @xcite : @xmath28 where @xmath29 , @xmath30 is the reciprocal lattice vector in the vicinity of the point @xmath31 . in contrast to the theory of x - rays diffraction , in the case under consideration @xmath32 depends on @xmath33 . it is to the fact that @xmath32 depends on the volume of the lattice unit cell @xmath34 , which can be significantly varied for diffraction gratings ( photonic crystals ) , as distinct from natural crystals . the volume of the unit cell @xmath35 depends on coordinate and , for example , for a cubic lattice it is determined as @xmath36 , where @xmath37 are the lattice periods . if @xmath38 does not depend on @xmath33 , the expression ( [ chi01 ] ) converts to that usually used for x - rays in crystals with lattice distortion @xcite . it should be reminded that for an ideal crystal without lattice distortions , the wave , which propagates in crystal can be presented as a superposition of the plane waves : @xmath39 where @xmath40 . let us use now that in the case under consideration the typical length for change of the lattice parameters significantly exceeds lattice period . this provides to express the field inside the crystal with lattice distortion similarly ( [ field ] ) , but with @xmath41 depending on @xmath33 and @xmath42 and noticeably changing at the distances much greater than the lattice period . similarly , the wave vector should be considered as a slowly changing function of coordinate . according to the above let us find the solution of ( [ 1 ] ) in the form : @xmath43 where @xmath44 , where @xmath45 can be found as solution of the dispersion equation in the vicinity of the point with the coordinate vector @xmath33 , integration is done over the quasiclassical trajectory , which describes motion of the wavepacket in the crystal with lattice distortion . let us consider now case when all the waves participating in the diffraction process lays in a plane ( coupled wave diffraction , multiple - wave diffraction ) i.e. all the reciprocal lattice vectors @xmath27 lie in one plane @xcite . suppose the wave polarization vector is orthogonal to the plane of diffraction . let us rewrite ( [ * ] ) in the form @xmath46 where @xmath47 @xmath48 then multiplying ( [ 1 ] ) by @xmath49 one can get : @xmath50 applying the equality @xmath51 and using ( [ * 1 ] ) we obtain @xmath52 , \label{3}\end{aligned}\ ] ] therefore , substitution the above to ( [ 2 ] ) gives the following system : @xmath53 + \nonumber \\ % & & + \textrm{~conjugated~terms~ } % = 4 \pi \vec{e } \left ( \frac{1}{c^2 } \frac{\partial \vec{j}(\vec{r},t)}{\partial t } + \vec{\nabla } \rho ( \vec{r},t ) \right ) , \nonumber \\ % & & \frac{1}{2 } e^{i(\phi_{2 } ( \vec{r})-\omega t ) } [ 2i \vec{k}_2(\vec{r } ) \vec{\nabla } a_2 + i \vec{\nabla } \vec{k}_2 ( \vec{r } ) a_2 - k_2 ^ 2(\vec{r } ) a_2 + \nonumber \\ % & & + \frac{\omega^2}{c^2 } \varepsilon_0(\omega,\vec{r } ) a_2 + i \frac{1}{c^2 } \frac{\partial \omega^2 \varepsilon_0(\omega,\vec{r})}{\partial \omega } \frac{\partial a_2}{\partial t } + \frac{\omega^2}{c^2 } \varepsilon_{\tau}(\omega,\vec{r } ) a_1 + i \frac{1}{c^2 } \frac{\partial \omega^2 \varepsilon_{\tau}(\omega,\vec{r})}{\partial \omega } \frac{\partial a_1}{\partial t } % ] + \nonumber \\ % & & + \textrm{~conjugated~terms~ } % = 4 \pi \vec{e } \left ( \frac{1}{c^2 } \frac{\partial \vec{j}(\vec{r},t)}{\partial t } + \vec{\nabla } \rho ( \vec{r},t ) \right ) , \label{3}\end{aligned}\ ] ] where the vector @xmath54 , @xmath55 , here notation @xmath56 is used , @xmath57 . note here that for numerical analysis of ( [ 3 ] ) , if @xmath58 , it is convenient to take the vector @xmath59 in the form @xmath60 . for better understanding let us suppose that the diffraction grating ( photonic crystal lattice ) period changes along one direction and define this direction as axis @xmath61 . thus , for one - dimensional case , when @xmath62 the system ( [ 3 ] ) converts to the following : @xmath63 + \nonumber \\ % & & + \textrm{~conjugated~terms~ } % = 4 \pi \vec{e } \left ( \frac{1}{c^2 } \frac{\partial \vec{j}(\vec{r},t)}{\partial t } + \vec{\nabla } \rho ( \vec{r},t ) \right ) , \nonumber \\ % & & \frac{1}{2 } e^{i(\vec{k}_{\perp } \vec{r}_{\perp } + \phi_{2z}(z)-\omega t ) } [ 2i { k}_{2z}(z ) \frac{\partial a_2}{\partial z } + i \frac{\partial { k}_{2z}(z)}{\partial z } a_2 - ( k_{\perp}^2+k_{2z}^2({z } ) ) a_2 + \nonumber \\ % & & + \frac{\omega^2}{c^2 } \varepsilon_0(\omega , z ) a_2 + i \frac{1}{c^2 } \frac{\partial \omega^2 \varepsilon_0(\omega , z)}{\partial \omega } \frac{\partial a_2}{\partial t } + \frac{\omega^2}{c^2 } \varepsilon_{\tau}(\omega , z ) a_1 + i \frac{1}{c^2 } \frac{\partial \omega^2 \varepsilon_{\tau}(\omega , z)}{\partial \omega } \frac{\partial a_1}{\partial t } % ] + \nonumber \\ % & & + \textrm{~conjugated~terms~ } % = 4 \pi \vec{e } \left ( \frac{1}{c^2 } \frac{\partial \vec{j}(\vec{r},t)}{\partial t } + \vec{\nabla } \rho ( \vec{r},t ) \right ) , \label{4}\end{aligned}\ ] ] let us multiply the first equation by @xmath64 and the second by @xmath65 . this procedure provides to neglect the conjugated terms , which appear fast oscillating ( when averaging over the oscillation period they become zero ) . considering the right part of ( [ 4 ] ) let us take into account that microscopic currents and densities are the sums of terms , containing delta - functions , therefore , the right part can be rewritten as : @xmath66 here @xmath67 is the time of entrance of particle @xmath13 to the resonator , @xmath68 is the time of particle leaving from the resonator , @xmath69functions in ( ref5 ) image the fact that for time moments preceding @xmath70 and following @xmath71 the particle @xmath72 does not contribute in process . let us suppose now that a strong magnetic field is applied for beam guiding though the generation area . thus , the problem appears one - dimensional ( components @xmath73 and @xmath74 are suppressed ) . averaging the right part of ( [ 5 ] ) over the particle positions inside the beam , points of particle entrance to the resonator @xmath75 and time of particle entrance to the resonator @xmath67 we can obtain : @xmath76 where @xmath5 is the electron beam density , @xmath77 is the mean electron beam velocity , which depends on time due to energy losses , @xmath78 , @xmath79 , @xmath80 indicates averaging over transversal coordinate of point of particle entrance to the resonator @xmath75 and time of particle entrance to the resonator @xmath67 . according to @xcite averaging procedure in ( [ 6 ] ) can be simplified , when consider that random phases , appearing due to random transversal coordinate and time of entrance , presents in ( [ 6 ] ) as differences . therefore , double integration over @xmath81 can be replaced by single integration @xcite . the system ( [ 4 ] ) in this case converts to : @xmath82 where the currents @xmath83 , @xmath84 are determined by the expression @xmath85 @xmath86 @xmath87 is the current density , @xmath88 , @xmath89 , @xmath90 , @xmath91 . the expressions for @xmath83 for @xmath92 independent on @xmath61 was obtained in @xcite . when more than two waves participate in diffraction process , the system ( [ sys2 ] ) should be supplemented with equations for waves @xmath93 , which are similar to those for @xmath94 and @xmath95 . now we can find the equation for phase . from the expressions ( [ phi1],[phi2 ] ) it follows that @xmath96 let us introduce new function @xmath97 az follows : @xmath98 therefore , @xmath99 in the one - dimensional case the equation ( [ sys11 ] ) can be written as : @xmath100 therefore , @xmath101 @xmath102 @xmath103 @xmath104 @xmath105,~ p \in [ -2 \pi , 2 \pi ] , ~l ~\textrm{is the length of the photonic crystal}.\ ] ] these equations should be supplied with the equations for @xmath106 . it is well - known that @xmath107 therefore , @xmath108 the above obtained equations ( [ sys2],[cz],[cz2],[gamma1 ] ) provide to describe generation process in fel with varied parameters of diffraction grating ( photonic crystal ) . analysis of the system ( [ cz2 ] ) can be simplified by replacement of the @xmath109 with its averaged by the initial phase value @xmath110 note that the law of parameters change can be both smooth and stair - step . use of photonic crystals provide to develop different vfel arrangements ( see fig.[volume ] ) . = 10 cm it should be noted that , for example , in the fel ( twt , bwo ) resonator with changing in space parameters of grating ( photonic crystal ) the electromagnetic wave with depending on @xmath61 spatial period is formed . this means that the dynamical undulator with depending on @xmath61 period appears along the whole resonator length i. e. tapering dynamical wiggler becomes settled . it is well known that tapering wiggler can significantly increase efficiency of the undulator fel . the dynamical wiggler with varied period , which is proposed , can be used for development of double - cascaded fel with parameters changing in space . the efficiency of such system can be significantly higher that of conventional system . moreover , the period of dynamical wiggler can be done much shorter than that available for wigglers using static magnetic fields . it should be also noted that , due to dependence of the phase velocity of the electromagnetic wave on time , compression of the radiation pulse is possible in such a system . the equations providing to describe generation process in fel with varied parameters of diffraction grating ( photonic crystal ) are obtained . it is shown that applying diffraction gratings ( photonic crystal ) with the variable period one can significantly increase radiation output . it is mentioned that diffraction gratings ( photonic crystal ) can be used for creation of the dynamical wiggler with variable period in the system . this makes possible to develop double - cascaded fel with variable parameters changing , which efficiency can be significantly higher that of conventional system .

the equations providing to describe generation process in fel with varied parameters of diffraction grating ( photonic crystal ) are obtained . it is shown that applying diffraction gratings ( photonic crystal ) with the variable period one can significantly increase radiation output . it is mentioned that diffraction gratings ( photonic crystal ) can be used for creation of the dynamical wiggler with variable period in the system . this makes possible to develop double - cascaded fel with variable parameters changing , which efficiency can be significantly higher that of conventional system . , free electron laser , travelling wave tube , backward wave oscillator , diffraction grating , smith - purcell radiation , diffraction radiation , photonic crystal 41.60.c , 41.75.f , h , 42.79.d

the main modern achievements in studying processes of double beta decay ( @xmath2-decay ) can be attributed to detection of its two - neutrino mode . this process has been discovered in as many as ten nuclei ( see reviews @xcite,@xcite,@xcite ) . the data obtained for the two - neutrino mode offer a chance to directly compare different models of the nuclear structure , which form the basis for calculations of nuclear matrix elements @xmath3 , and to select the optimal one . though direct correlation between the values of nuclear matrix elements for the two - neutrino and neutrinoless modes of @xmath4 decay is absent lacking , the methods for calculating @xmath3 and @xmath5 are very close , and a chance possibility to estimate their accuracy in calculating @xmath5 appears only when comparing experimental data and theoretical results calculations for the probability of @xmath6 decay . it can be expected that acquisition of experimental data on the other types of @xmath7 transitions ( @xmath8 , and @xmath9 processes ) will make it possible to considerably increase the quality of calculations for both @xmath10 and @xmath11 decays . much efforts have been currently made in searching for these processes ( @xcite,@xcite,@xcite ) in spite of the fact that the @xmath12 and @xmath13 modes are strongly suppressed relative to @xmath14 decay due to the coulomb barrier for positrons , and a substantially lower kinetic energy attainable in such transitions . positrons are absent in the final state of the @xmath15 decay , and the kinetic energy of the transition may be rather high ( up to 2.8 mev ) , which dictates determines an increased probability of a decay . however , this process is also difficult to detect , since it is only characteristic radiation that is detectable in it . the state - of - the - art experimental limit on the @xmath0kr half - life period with respect to the @xmath15 capture is @xmath16 yr ( 90% c.l . ) @xcite . the theoretical calculations based on different models predict the following @xmath0kr half - lives for this process : @xmath17 yr @xcite , @xmath18 yr @xcite , and @xmath19 yr @xcite . the last two values were obtained from the estimates of the @xmath0kr half - life with respect to the total number of @xmath20 captures including in view of the 78.6% fraction of @xmath15 capture events which makes 78.6% @xcite . from comparison of the experimental and theoretical results , it is apparent that the sensitivity of measurements has reached the lower limit of theoretical estimates . the @xmath0kr@xmath21se reaction produces a @xmath0se@xmath22 atom with two vacancies in its @xmath23-shell . the technique for seeking this reaction is based on the assumption that the values of energies of characteristic photons , and of the probability that they will be emitted when the double vacancy is being filled , coincide with the corresponding values when two separate single vacancies in the @xmath23 shells of two isolated singly ionized se@xmath24 atoms are being filled . in this case , the total measured energy is @xmath25 kev , where @xmath26 is the binding energy of a @xmath23 electron in a se atom ( 12.65 kev ) . the fluorescence yield upon filling of a single vacancy in the @xmath23-shell of se is 0.596 . the energies and relative intensities of the characteristic lines in the @xmath23 series are @xmath27 kev ( 100% ) , @xmath28 kev ( 52% ) , @xmath29 kev ( 21% ) , and @xmath30 kev ( 1% ) @xcite . there are three possible ways for deexcitation of a doubly ionized @xmath23-shell : 1 ) emission of auger electrons only ( @xmath31 , @xmath31 ) , 2 ) emission of a single characteristic quantum and an auger electron ( @xmath32 ) , and 3 ) emission of two characteristic quanta and low - energy auger electrons ( @xmath33 ) , with probabilities @xmath34 , @xmath35 , and @xmath36 , respectively . a characteristic quantum can travel a long enough distance in a gas medium between the points of its production and absorption . for example , 10% of characteristic quanta with energies of 11.2 a nd 12.5 kev is absorbed in krypton at a pressure of 4.35 atm ( @xmath37 g/@xmath38 ) on a path 1.83 and 2.42 mm long , respectively ( the values of absorption factors are taken from @xcite ) . the paths of photoelectrons with the same energies are 0.37 and 0.44 mm , respectively . they produce almost pointwise charge clusters of primary ionization in the gas . in case of the event with the escape of two characteristic quanta absorbed in the working gas and a single auger electron , the energy will be distributed among three pointwise charge clusters . it is these three - point ( or three - cluster ) events possessing a unique set of features that were the subject of the search in @xcite . a large proportional counter ( lpc ) with a casing made of m1-grade copper is used to detect the above considered processes . the lpc has a cylindrical shape with inner and outer diameters of 140 and 150 mm , respectively ; its section along the axis is schematically shown in fig.[lpc ] . a gold - plated tungsten wire of 10 @xmath39 m in diameter goes along the lpc axis and serves as the anode . the potential of + 2400 is applied to the wire , and the casing ( the cathode ) is grounded . both ends of the anode are lead to the appropriate end cap flanges via high - voltage pressure - sealed bushings - ceramic insulators with a central electrode taken from spark plugs . to reduce the influence of edge effects on the operating characteristics of the counter , the end segments of the wire are passed through the copper tubes with dimensions of @xmath40 mm , which are electrically connected to the anode . gas amplification is absent on these segments , and charges are collected in an ionization mode . with the fluoroplastic insulator , the distance from the working region to the flange is 70 mm . the working part of the lpc is 595 mm in length ( the distance between the end caps of the tubes ) ; therefore , the lpc s operating volume is 9,159 l. the total capacitance of the counter and the outlet insulator is @xmath41 pf . the total resistance of the anode and two output electrodes is @xmath42 om . all detachable joints are sealed with indium wire . all nipple joints are sealed with fluoroplastic gaskets . the internal insulators are made of fluoroplastic . their thickness was selected so as to be the smallest possible in order to improve the degassing conditions during vacuum treatment of the counter and stabilize its operating characteristics in the course of measurements . the lpc is filled with a pure kr sample to a total pressure of 4.51 att ; no quenching or accelerating gases are added . prior to filling , kr is purified of electronegative impurities in a ni / sio@xmath43 reactor . the lps s signals are read out by a charge - sensitive amplifier ( csa ) from one end of the anode wire . the csa parameters have been selected so that the signal is transmitted with minimum distortions , and information of the spatial distribution of primary - ionization charges in a projection onto a counter radius is fully represented by the pulse shape . when amplified in an auxiliary amplifier , the pulses arrive at the input of the digital oscilloscope la - n20 - 12pci , the output data of which ( the pulse waveform digitized with a frequency of 6.25 mhz ) are recorded with a personal computer . the length of the scanning frame with a resolution of 160 ns is 1024 points ( 163.8 @xmath39s ) , of which @xmath44 @xmath39s is the `` prehistory '' and @xmath45 @xmath39s is the `` history '' . the counter is calibrated through the wall of its casing by @xmath46 rays of a @xmath47cd source ( @xmath48 kev ; relative yield 0.036 photons / decay ) . figure [ pic2 ] presents ( _ 0 _ ) the total pulse amplitude spectrum and the energy spectra of ( _ 1 _ ) single- , ( _ 2 _ ) two- , and ( _ 3 _ ) three - point events from the source located in the middle of the lpc length . the procedure for obtaining them from digitized pulses is described in what follows . the following factors make their contribution to the low - energy part of the spectrum : characteristic radiation ag@xmath49 ( @xmath50 kev ) from this source , which `` survived '' after passing through a 5-mm - thick copper wall ; scattered radiation from the wall , which is in equilibrium with the characteristic radiation , and compton electrons from scattering of 88-kev photons in the gas with the escape of a compton photon beyond the counter . the 88-kev peak is wider on the low - energy side due to the contribution of 88-kev @xmath46 rays scattered from the wall . the energy resolution of this peak , determined by its right half , is 6.5% . peak _ 4 _ at an energy of 75.4 kev corresponds to the escape of kr characteristic radiation ( @xmath51 kev ) beyond the counter . the 88-kev full - energy peak contains events with different internal structures . quanta with this energy are absorbed in kr mostly by photoeffect in the @xmath23-shell ( 86.7% ) . the photoeffect in other shells makes 13.3% @xcite , @xcite . filling of the vacancy in the @xmath23-shell of kr is accompanied by emission of characteristic radiation in 66.0% of cases and auger electrons in 34.0% @xcite . the theoretical efficiency of characteristic radiation absorption in the counter s working volume is 86.9% . therefore , the photoeffect is responsible for 49.7% of two - point events @xmath52 and 42.8% of single - point events @xmath53 out of the total number of absorptions due to photoeffect in the full - energy peak . by single - point events , we mean all events in which only electrons escaping from the shell of a single atom ( photoelectrons @xmath54 auger electrons ) , including events of photoelectric absorption in the upper shells of kr . only single - point events in an amount of 7.5% @xmath55 $ ] of the total number of photoabsorption events will be presented in peak _ 4_. some primary quanta can be absorbed as a result of two - step process of `` compton scattering - photoeffect '' . a compton electron creates one ionization point . a compton photon absorbed by photoeffect participates in the above - described processes . therefore , the two - step process makes its contributions to the full - energy peak in the form of two- and three - point events and to peak _ 4 _ in the form of two - point events . upon normalization to the peak area , the estimated final composition of events for the full - energy peak contains 44.1% ( single - point events ) + 51.2% ( two - point events ) due to photoeffect + 2.2% ( two - point events ) + 2.5% ( three - point events ) due to the two - step process . in peak _ 4 _ , there are 95.3% single - point events + 4.7% two - point events . examples of pulses ( dark lines ) corresponding to events of two types are presented in fig.[exam_pulse ] . the pulse due to photoabsorption of a 88-kev photon with escape of electrons only ( a single - point event ) is shown in fig.[exam_pulse]a , and simultaneous escape of characteristic photon kr with an energy of 12.6 kev and a photoelectron @xmath56 kev - 12.6 kev=75.4 kev ( a two - point event ) is illustrated in fig.[exam_pulse]b . the maximum distance between pointwise charge clusters in projection onto the counter s radius is equal to the radius . for pure kr , the evaluated time it takes for ionization electrons to drift from the cathode to the anode is 53 @xmath39s . from figs.[exam_pulse]a and [ exam_pulse]b , it is apparent that the second pulse with a @xmath57 times smaller amplitude is produced in the counter in about @xmath58 @xmath39s after the first one . this pulse is generated by secondary photoelectrons knocked out of the cathode by photons produced during development of an avalanche from the primary ionization . the probability of photoeffect on the cathode is rather high , since the working gas contains no quenchers . the digitized output pulse from the measuring channel can be presented as direct convolution of sought signal @xmath59 with response pulse @xmath60 from the linear system , which is distorted by stochastic or determinate noise @xmath61 : @xmath62 the noise level determines the lower limit on the sensitivity of the measuring channel , while the instrumental function defines the resolution value . given the values of @xmath63 and @xmath60 , one can try to estimate @xmath64 in the presence of noise @xmath65 ; i.e. , in principle , it is possible to state the inverse problem ( deconvolution ) of determining the signal at the linear system output by the values of the output signal : @xmath66 where @xmath67 is the operator inverse . to determine the values of charges released in individual clusters of a multipoint event , one can differentiate the original charge pulse @xmath68 and represent the obtained shape by a set of gaussian curves . the evaluated area under an individual gaussian curve will correspond to the charge ( energy ) value in the relevant cluster . from figs.[exam_pulse]_c _ and [ exam_pulse]_d _ , it is apparent that direct differentiation provides an asymmetric bell shape ( dark line ) . such shape results from the nearly gaussian distribution of the current pulses due to electrons of primary ionization from a pointwise energy deposit , which arrive at the boundary of the gas amplification region near the anode wire . this shape is determined by the spatial distribution of the charge density in projection onto the radius . the parameters of this distribution depend on the time it takes for the primary charge cluster to drift to the anode . as it drifts , the charge cluster spreads out into a cloud due to electron diffusion . the pulse read out from the anode wire is mostly produced by a negative charge induced on the anode moving toward the cathode by positive ions produced near the wire in gas amplification process and moving toward the cathode i.e. , the ion component ( i.c . ) . the estimated total ion drift time is 0.447 s. the contribution of the equilibrium ( with ions ) electron component ( e.c . ) to the total induced charge is @xmath69% . the electron collection time is @xmath70 ns . the output pulse shape is defined by the superposition of induced charges from single electron avalanches distributed in time and intensity according to : the shape of the current pulse from primary ionization electrons , the shape of a pulse from an individual avalanche , and a finite time of the csa self - discharge . the last two parameters are responsible for the asymmetry of the output current pulse . the output current pulse can be transformed to a symmetric shape by taking into account the analytical dependence of the amplitude of the output voltage pulse generated by a point ( in projection onto the radius at the boundary of the gas amplification region ) group of primary ionization electrons as a function of time and discharge constant of the output storage capacitor @xcite : @xmath71 where @xmath72 is the amplitude of the voltage pulse from the @xmath73th group of electrons , @xmath74 is the number of primary electrons in the @xmath73th group , @xmath75 is the current time for the dependence of the voltage pulse amplitude from the @xmath73th group , @xmath76 is the time of origin of the total pulse , and @xmath77 is the time of origin of the pulse from the @xmath73th group , @xmath78 , @xmath79 is the gas amplification factor , @xmath80 is the total voltage pulse amplitude produced at output capacitor @xmath81 by a single ion generated in the gas discharge , @xmath82 is the electron charge , @xmath83 is the radius corresponding to the avalanche s center of gravity , @xmath84 is the anode radius , @xmath85 is the cathode radius , _ b _ is the time parameter associated with the motion of positive ions of the gas discharge in a particular gas ( for the lpc filled with kr at 4.51 at , @xmath86 ns ) , @xmath87 is the csa discharge constant , @xmath88 is the leakage resistance , @xmath89 , and @xmath90 is the total voltage pulse amplitude produced at output capacitor @xmath81 by a single electron generated in the gas discharge . in eq.(3 ) , the electron component is assumed to appear instantly . at @xmath91 , eq . ( 3 ) assumes the form @xmath92 in our case , @xmath93 @xmath39s . for a time interval satisfying condition @xmath94 , eq . ( 3 ) can be reduced to the first two expansion terms . if its is assumed that the gas discharge from @xmath95 of primary electrons happens at the beginning of the digitization interval , the pulse amplitude at the end of this interval can be described by eq . ( 3 ) for @xmath96 ns , since the influence of the output capacitor discharge over this time is negligible . if a contribution of the earlier discharges is absent in this time interval , the pulse amplitude at the upper bound of the interval can be used to determine the @xmath95 value . in this case , it is taken into account that , at the end of a 160-ns interval , the contributions of the terms to the total @xmath72 value in eq . ( 3 ) make 74 and 26% , respectively . these conditions are satisfied in the recorded actual pulse in the first time channel from the beginning of the pulse . the @xmath97 value obtained from the actual pulse is used in eq . ( 3 ) to calculate the total shape of the partial pulse in the entire time interval from the beginning to the end of the frame . the pulse obtained thereby is subtracted from the actual one . therefore , the above condition is now fulfilled for the first digitization interval of the residual pulse or for the second interval of the original pulse . this procedure is repeated until the last time channel in the frame . the sequence of @xmath95 values for a single - point event has a symmetrical distribution with a nearly gaussian shape . it is this distribution that is used for further analysis . the area under the gaussian curve or , in the case of a multiparticle event , the sum of the gaussian areas on a time interval of 53 @xmath39s from the beginning of the pulse yields the total number of primary ionization electrons . to plot the spectra in fig.2 , this sum is multiplied by the coefficient equal to the averaged ratio of areas of the actual current pulse and of the calculated gaussian curve for purely single - point events . the calculated , area - normalized current pulses of primary ionization electrons at the boundary of the gas amplification region are shown with light curves in figs.[exam_pulse]_c _ and [ exam_pulse]_d _ , and the corresponding voltage ( charge ) pulses obtained by integrating these current pulses are depicted with light lines in figs [ exam_pulse]_a _ and [ exam_pulse]_b_. from figs.[exam_pulse]_c _ and [ exam_pulse]_d _ , it is apparent that , at an energy deposit of 88 kev , the signal - to - noise ratio ( snr ) is rather high . in the energy range of @xmath98 kev , in which the @xmath99-capture in @xmath0kr is sought , the snr for individual components of the total energy deposit corresponding to the possible effect ( 25.3 kev ) is not so favorable . figures [ pic4]_a _ and [ pic4]_b _ present examples of recalculated current pulses for two types of two - point energy deposits due to the @xmath23-capture of @xmath100kr isotope . the total energy deposit corresponds to the binding energy of an electron in the @xmath23-shell of a daughter @xmath100br ( 13.5 kev ) . the energies of a characteristic quantum ( @xmath101 kev ) and concomitant auger electrons ( @xmath102 kev ) are close to the energies of individual components for events of the @xmath99-capture of @xmath0kr . more detailed information on the @xmath100kr source is given in what follows . from fig.[pic4 ] , it is apparent that these signals have a high noise level . noises and possible electric pickup may both mask the low - energy component and create a false one . the use of traditional methods of frequency filtering with different window functions , e.g. , in the form of the hamming @xcite , wiener @xcite , and savitzky - golay @xcite filters , sometimes fail to ensure reliable extraction of closely spaced ( fig . 4a ) and masking each other components of a compound event . mathematical studies carried out in late 1980s initiated intense development of a principally new class of orthogonal transforms based on the use of wavelet functions @xcite . wavelet transforms are distinguished by a high degree of locality of their base functions both in the time and frequency regions , which allows one to use them for processing of many nonstationary processes . `` in the preliminary processing of our lpc data it is expedient to use multiresolutional signal analysis @xcite based on the _ dyadic transform of discrete signals _ , and often called discrete wavelet transform ( dwt ) . in this case , the analyzed signal @xmath103 is presented as the decomposition @xmath104 where are the well - known orthonormal scaling ( scaling functions ) and wavelet functions- ' ' ancestor " wavelets ; @xmath105 are the empirical approximation coefficients and @xmath106 are the empirical detailing coefficients ; @xmath107 are the current values of the scale and the shift ; @xmath108 is the number of approximation ( detailing ) coefficients considered at the relevant levels of the decomposition ; @xmath109 is the initial scale value ; and @xmath110 is the final scale value . scaling functions @xmath114 and mother wavelet functions @xmath115 , which have @xmath99 nonzero coefficients , satisfy the so - called two - level relations @xcite : @xmath116 @xmath117 where @xmath118 and @xmath119 are the coefficients of low- and high - frequency filters of the wavelet transform , when @xmath120 . as distinct from other types of transforms , in which the base functions are explicitly specified , one succeeds in analytically obtaining the base functions in the wavelet analysis only in rare cases , and the basis is most frequently specified by coefficients @xmath118 and @xmath119 . in this paper , we use wavelets of the daubechies family . the scaling functions and the daubechies wavelets are continuous functions that are not identically equal to zero on a finite segment and are not differentiable anywhere on this segment . table [ tab1 ] contains the filter coefficients used in the daubechies scaling functions @xmath121 and @xmath122 ( numbers 4 and 6 denote the number of nonzero coefficients in the filters ) . they are rational numbers and fully define the daubechies wavelet transform ( dvt ) @xcite . danevich , f.a . , kobychev , v.v . , nagorny , s.s . , and tretyak , v.i . , nim phys . res , 2005 , vol . * a544 * , p.553 . barabash , a.s . , hubert , ph . , nachab , a. , et al , nucl . phys . a , 2008 , vol . * 807 * , p. 269 ; nucl . phys . a , 2007 , vol . * 785 * , p. 371 ; j. phys . g , 2007 , vol . * 34 * , p. 1721 . rukhadze , n.i . , benes , p. , briancon , ch . , et al . , bull . phys , 2008 , vol . 72 , p. 731 . gavriljuk , ju.m . , kuzminov , v.v . , osetrova n.ya . , and ratkevich s.s , phys . atomic nucl . , 2000 , vol . * 63 * , p. 2201 . aunola , m. and suchonen , j. , nucl . a. , 1996 , vol . * 602 * , p. 133 . hirsch , m. , muto , k. , oda , t. , and klapdor - kleingrothaus , h.v . , z. phys . a , 1994 , vol . * 347 * , p. 151 . rumyantsev , o.a . and urin , m. , phys . lett.b , 1998 , vol . * 443 * , p. 51 . doi , m. and kotani , t. , prog . theor . phys . , 1992 , vol . * 87 * , p. 1207 . blokhin , m.a . and shveitser , i.g . , rentgenospektralnyi spravochnik ( handbook on x - ray spectra ) , moscow : nauka , 1982 . storm , a. and israel , h. , secheniya vzaimodeistviya gamma - izlucheniya ( cross sections of the gamma - radiation ) , moscow : atomizdat , 1973 . blokhin , m.a . and shveitser , i.g . , rentgenospektral nyi spravochnik ( handbook on x - ray spectra ) , moscow : nauka , 1982 . abramov , a.i , kazanskii , yu.a . , and matusevich , e.s . , osnovy eksperimental nykh metodov yadernoi fiziki ( bases of the experimental methods of nuclear physics ) , moscow : energoatomizdat , 1985 . hamming , r.w . , digital filters . , ( englewood cliffs ) , nj : prentice - hall , 1983 . hillery , a.d . and chin , r.t . , ieee trans . signal proc . , 1991 , vol . 39 , p. 1892 . savitzky , a. and golay , m.j.e . , , 1966 , vol . * 36 * , p. 1627 . daubechies , i.,ten lectures on wavelets , philadelphia : siam , 1992 . wavelet representation , . ieee trans . pattern anal . and machine intel . , 1989 , vol . * 11 * , no . * 7 * , pp . 674 . mala , s. , veivlety v obrabotke signalov ( wavelets in signal processing ) , moscow : mir , 2005 . donoho , d. and johnstone , i. , biometrika , 1994 , vol . * 81 * , p. 425 . donoho , d.l . and johnstone , i.m . , j. amer . stat . assoc . , 1995 , vol . * 90 * , no.*432 * , p. 1200 ; neumann m. , j. time series analysis , 1996 , vol . 17 , p. 601 . abramovich , f. , bailey , t.c . , and sapatinas , t. , the statistician , 2000 , vol . * 49 * , p. 1 . neumann , m. and sachs , r. , annals of statistics , 1997 , vol . * 25 * , p. 38 . ratkevich , s.s . , vestnik kharkovskogo nats . univ . yadra , chastitsy , polya , 2006 , no . * 746 * , issue * 4 * , p. 23 sweldens , w. , siam j. math . anal , 1996 , vol . * 3 * , no . * 2 * , p. 186 . http://www-stat.stanford.edu/ wavelab/ ; http://www-dsp.rice.edu/software/rwt.shtml loosli , h.h . and oeschger , h. , earth planet lett . , 1968 , * 7 * , no . * 1 * , p. 67 . kuzminov , v.v . and pomansky , a.a . , radiocarbon , 1980 , vol . * 22 * , no . * 2 * , p. 311 . chew , w.m . , xenoulis , a.c . , fink , r.w . , et al . , nucl . a. , 1974 , vol . 1 , p. 79 . geant detector description and simulation tool . cern program library long write - up , * w5013 * , cern , 1994 .

a pulse shape analysis algorithm and a method for suppressing the noise component of signals from a large copper proportional counter in the experiment aimed at searching for 2k capture of @xmath0kr are described . these signals correspond to a compound event with different numbers of charge clusters due to from primary ionization is formed by these signals . a technique for separating single- and multipoint events and determining the charge in individual clusters is presented . using the daubechies wavelets in multiresolutional signal analysis , it is possible to increase the sensitivity and the resolution in extraction of multipoint events in the detector by a factor of @xmath1 .

the presence of ionized material in the interstellar medium ( hereafter , ism ) can be attributed to two distinctive mechanisms . first , photoionization of the surrounding neutral gas by the strong , energetic ultra - violet ( uv ) flux of nearby massive stars is largely responsible for the detection of the standard hydrogen balmer series , typically used for the morphological and kinematical description of hii regions . secondly , stellar winds with high terminal velocities and violent supernova blasts are commonly associated with the propagation of transonic and supersonic shock waves in the surrounding medium . the important increase of the post - shocked gas temperature favors its ionization through shock excitation . pioneering work by @xcite has compared specific flux ratios for a variety of ionic transitions in the ism optical gas . this allowed the authors to approximately separate standard hii regions and planetary nebulae ( hereafter , pne ) mostly governed by photoionization from shock - dominated supernova remnants ( hereafter , snrs ) . these diagnostic diagrams were often used , in literature , to classify large amount of ionized targets in large - scale objects , for example more - or - less distant galaxies ( e.g. , @xcite ) . this has led to emission - line ratio plots in which each object is usually statistically represented by a single point . obviously , intrinsic variations , within a given object , can be investigated by targeting galactic objects individually ( e.g. , @xcite ) . this allows to spatially resolve much smaller ( @xmath70.1 pc ) structures and artifacts characterized by peculiar line ratios that are , otherwise , unrevealed or statistically negligible in observations using poorer angular resolutions . the investigation of close ionized objects has already revealed that photoionization and shock excitation can both be found in individual regions . kinematically speaking , standard hii regions encompassing massive stars earlier than o6 have shown little indication of a strong impact on the surrounding gas attributed to stellar winds , referred earlier as a potential source for shock excitation in nebulae . this remains usually true even when using observations with high spectral resolutions that allow to measure velocity fluctuations down to a few km s@xmath8 . the main reason for this resides in the double - shock model said to accurately describe the dynamical evolution of wind - blown bubbles @xcite . the reverse shock quickly ( i.e. , often within less than a spatial element of resolution ) converts high - velocity ( @xmath91000 km s@xmath8 ) terminal winds into low - velocity , high - temperature gas . the thermal energy of this hot , pressurized material then initiates and fuels the expansion of the dense shell of post - shock ism material found at much greater distances with respect to the central star . the typical expansion velocity of the shocked shell is usually a few km s@xmath8 above 10 km s@xmath8 , roughly the speed of sound for warm ( @xmath1010000 k ) h@xmath11 gas . unfortunately , in complex tridimensional geometries , this kind of velocities could be easily confused with standard accelerated outflows in hii regions , such as champagne @xcite or photoevaporated @xcite flows , turbulent motions @xcite or fluid instabilities . this work mainly explores the possibility of detecting shock waves associated , in particular , to stellar winds using the emissive properties of the ionized gas rather than a more typical , and not always successful , approach based on the information retrieved from radial velocities and non - thermal line widths . the ic1805 nebula is located in the perseus arm of our galaxy . the most massive stars of the melotte 15 star cluster are currently responsible for the energetic support of the large hii region . the south - central portion of the nebula ( see figure 1 ) , in the direct vicinity of the star cluster , is gas - rich and will be used in this work to fill the @xcite diagrams . our goals are to ( 1 ) obtain reliable , high signal - to - noise ratios ( hereafter , s / n ) , spectra of the optical emission in the brightest portions of the ic1805 star - forming complex , ( 2 ) provide the corresponding series of line - ratio diagnostic diagrams , and ( 3 ) investigate the impact of photoionization and shock excitation in the targeted gas volume . we present , in @xmath12 2 , the galactic hii region ic1805 and its associated star cluster , melotte 15 . information related to our spectrometric observations and methods used for data reduction are detailed in @xmath12 3 . results of our study and diagnostic diagrams are provided in @xmath12 4 . interpretation and discussion follow in @xmath12 5 . summarized results and general conclusions will finally be provided in @xmath12 6 . introduced by @xcite as an irregular nebula of medium brightness , the galactic hii region ic1805 is often referred as the heart nebula due to the heart - shaped morphology of its southern hemisphere ( 60@xmath1328@xmath14@xmath15@xmath16@xmath1562@xmath1310@xmath14 ) . it is suggested that outflows from the ob association melotte 15 ( @xmath17=02@xmath1832@xmath1945@xmath20 , @xmath16=61@xmath1326@xmath1440@xmath21 ) are responsible for the actual expansion of the large hii region / superbubble @xcite . proper motion investigation has allowed the identification of 126 stars intrinsic to the star cluster @xcite . the authors confirmed , through spectroscopic observations , the presence of numerous massive stars ; about forty from spectral types o4 to b2 ( including 10 o - type stars ) have been identified with a high probability of membership . an important proportion of these massive stars are still found on the main - sequence branch indicating a very young cluster age evaluated at 2.5 myr @xcite . the dynamical age of the ic1805 nebula being estimated at 14 myr @xcite , it appears that the large hii region was formed and shaped by the mechanical deposit , in the ism , of energy ( i.e. , photon flux , stellar winds , supernovae ) attributed to a succession of different star clusters , melotte 15 being the most recently formed . no non - thermal emission being actually detected in the large star - forming complex , this suggests that the impact of old supernovae could be negligible in the nebula s current dynamics . ubv photometry , properly corrected for absorption , puts the melotte 15 complex at an heliocentric distance of 2.35 kpc @xcite . @xcite provide a high spectral - resolution ( @xmath2215 km s@xmath8 ) kinematical investigation of the h@xmath11 content on an appreciable fraction of the ic1805 ionized extent . the northernmost portions ( 63@xmath1320@xmath14@xmath15@xmath16@xmath1566@xmath1350@xmath14 ) of the large ovoid bubble have shown kinematical evidences for high - latitude gas venting and the early development of a galactic chimney . it appears that leaking uv photons , emanating from the melotte 15 star cluster and unabsorbed by the denser low - declination material , may participate in the sustainment of the high - temperature gas found in the galactic halo . in the southern half of the nebula , schematically represented in figure 1 , the last residual fragments of an old giant molecular cloud appear to be eroded by the uv flux and stellar winds of the nearby massive stars . as a result , series of criss - crossing accelerated flows are detected in agreement with the champagne phase ( e.g. , @xcite ) . ionization fronts , stellar winds , and champagne shocks are therefore expected to have developed in the vicinity of the melotte 15 star cluster . this could suggest that shock excitation may be responsible for the presence of a certain fraction of the observed ionized material although a well - defined kinematical signature attributed to shock waves has not been formally identified in our previous study . this will be largely addressed in the following sections . 0.3 cm [ cols="^,^,^,^,^ " , ] observations of the ic1805 optical gas complex were performed during the nights of 2008 september 24 - 25 , 25 - 26 using the ritchey - chrtien 1.6 m telescope of the observatoire du mont - mgantic ( omm ) . the data were gathered using the imaging fourier transform spectrometer spiomm ( spectromtre imageur de lobservatoire du mont mgantic ) . technical details regarding spiomm are provided in @xcite , @xcite and references therein . for each pixel of the detector , the instrument has the capacity to obtain the emission spectrum of the corresponding nebular - gas column in selected bandwidths of the optical regime between 3500 @xmath2 and 9000 @xmath2 . prior to entering the observation mode , the spectral resolution is fixed by the observer between r=1 and 25000 . for this work , the use of an interference filter enabled the acquisition of the spectral information in the red portion of the electromagnetic spectrum between 6480 @xmath2 and 6820 @xmath2 . this allowed to obtain , simultaneously , emission spectra displaying the h@xmath0@xmath16563 @xmath2 , @xmath3nii@xmath4$]@xmath1@xmath16548 , 6584 @xmath2 , and @xmath3sii@xmath4$]@xmath1@xmath16716 , 6731 @xmath2 ionic transitions . a spectral resolution of r=2100 was judged appropriate corresponding to a resolution of roughly 3 @xmath2 or 150 km s@xmath8 centered on the h@xmath0 rest frequency . the field - of - view ( fov ) is roughly 12@xmath14@xmath2312@xmath14 while data were spatially binned 2@xmath232 during acquisition to reduce readout times . the resulting raw data cube is formed , for each pixel , of a discrete interferogram . following classical data reduction for ccd observations ( i.e. , bias subtraction and flatfield correction ) , all panchromatic images were realigned to correct for guiding errors . according to the photometry of stars located in gas - depleted zones , all images were also corrected for variations of the sky transparency occurring during data acquisition . each interferogram is then fourier transformed and calibrated in velocities using a he - ne laser ( @xmath16328 @xmath2 ) as frequency of reference . the data were again spatially binned 2@xmath232 during reduction in order to account for the 2@xmath245 seeing measured at the telescope . the resulting spatial resolution is of 2@xmath242 pixel@xmath8 for 325@xmath23335 pixel@xmath25 . using the adopted distance to melotte 15 ( see @xmath12 2 ) , this corresponds to an angular resolution of 0.026 pc pixel@xmath8 as seen on the plane of the sky . two cubes were obtained in the vicinity of the melotte 15 star cluster . the observational parameters for both cubes , labeled east and west , are summarized in table 1 . the heliocentric correction being very similar for both cubes , the two were mosaicked to a new extended fov of 21@xmath265@xmath2312@xmath265 ( see figure 1 ) and over 200000 emission spectra . following subtraction of the continuum , a multi - component gaussian fit procedure , written in idl , was applied to the mosaicked cube . the fitting procedure returns , for each component , the peak intensity in analog - to - digital units ( adu ) , the line centroid ( or velocity ) , and the line dispersion ( @xmath27@xmath28@xmath29 ) . since this work is mostly based on line ratios , we followed the recommendation of @xcite by imposing a s / n greater than 6 for a given gaussian fit to be usable . for each spectrum , the noise level was estimated from continuum fluctuations measured in empty channels . figure 2 shows a high - quality spectrum obtained in the brightest parts of our mosaicked cube . this kind of spectrum is typical only for the regions of high emissivity of our fov . emission - line profiles corresponding to weaker areas are much noisier and complementary lines , found in the wavelength interval covered by the interference filter ( see above ) , such as the @xmath3ni@xmath4$]@xmath16500 @xmath2 and @xmath3hei@xmath4$]@xmath16678 @xmath2 transitions are not discernible from noise fluctuations . the weakest line ( i.e. , @xmath3nii@xmath4$]@xmath16548 @xmath2 ) in the example provided by figure 2 has a s / n of 24 . figures 3 to 7 provide the monochromatic , peak - intensity maps for all five emission lines investigated in this work . north is up and east is left in all figures . figure 8 spectacularly displays the @xmath6\,\lambda6584}{\textnormal{h}{\alpha}\,\lambda6563}$ ] ratio throughout the whole fov ( see caption ) . red emphasizes regions dominated by h@xmath0 while bluer shades pinpoint a stronger emission of the @xmath3nii@xmath4$]@xmath16584 @xmath2 transition ( with respect to the redder areas ) . three stars with strong h@xmath0 lines are detected ( the third being barely perceptible very close to the center of the figure ) . at the position of all other stars , bad pixels were removed and replaced by mean values statistically representing the surrounding nebular - gas content . this figure is particularly useful to describe ionized features in our fov . the five college radio astronomy observatory ( fcrao ) co(1 - 0 ) survey of the 2@xmath30 galactic quadrant @xcite has revealed very faint emission corresponding to tenuous molecular material at the position of the bright , central ionized structure in figures 3 to 7 . this suggests the presence of a large molecular fragment , surrounded by the most massive stars of the melotte 15 cluster , that has undergone almost full erosion by the uv flux and stellar winds of the nearby ionizing sources . figure 8 reveals the filamentary nature of the central structure . these filaments most likely trace out the spatial disposition , on the plane of the sky , of the eroded ( sometimes fully ionized ) molecular envelopes . the central structure is surrounded by diffuse ionized material likely associated to photoevaporated flows kinematically in agreement with the champagne phase @xcite . one of these flows ( displaying blue shades in figure 8) is particularly well - defined , propagating from the bottom - center of our fov approximately toward the upper - left corner of figure 8 . west of the central structure , the spatial arrangement of the nebular gases is extremely complex . ionized filaments are still perceptible although the contrast with the diffuse surroundings is not as clear . toward the western boundary of figure 8 , even the h@xmath0 emission appears darker , obscured by interstellar dust . shock excitation in the vicinity of the bright , central ionized structure , displayed in figures 3 to 8 , will be largely discussed and investigated in @xmath12 5.2.1 . toward the south - eastern portion of our fov , a small co fragment is clearly detected in the fcrao survey at the systemic velocity of the ic1805 region ( see co contours in figures 3 to 7 ) . its ionized counterpart is revealed by a thin , rounded ionization front especially visible in the lower - left corner of figure 8 ( see also figure 19__a _ _ ) . this specific shape of the ionization front results from the vast majority of the ionizing sources in melotte 15 being located behind the molecular clump . this was kinematically confirmed , in @xcite , by the detection of an accelerated ionized outflow moving away from the observer ( see @xmath12 5.2.2.1 ) . this flow eventually collides or simply coincides in lines - of - sight with the south - central / north - east flow mentioned in the previous paragraph . this results in particularly complicated kinematical motions in central ic1805 with h@xmath0 non - thermal line widths approaching the supersonic regime ( @xmath1010 km s@xmath8 ) according to high - resolution observations @xcite . in the same area of figure 8 ( see also figure 19__c _ _ ) , a cylindrical , cigar - like feature appears to be associated with an isolated star with strong h@xmath0 lines ( see above ) . the feature has bright h@xmath11 and n@xmath11 rims but is almost completely gas - deprived near its center . this structure as well as the rounded ionization front found in the lower - left corner of figure 8 will be discussed in @xmath12 5.2.2 . figure 9 was processed identically to figure 8 and displays the @xmath5\,\lambda\lambda6716 , 6731}{[\textnormal{n}\,\textsc{ii}]\,\lambda6584}$ ] ratio ( where the numerator is the sum of both lines of the @xmath3sii@xmath4 $ ] doublet ) . again , blue shades identify nitrogen - rich zones while red areas indicate that a sizeable role is played by sulfur in the overall gas emissivity . one immediately notices the complexity of the chemical properties in ic1805 . while @xmath3nii@xmath4 $ ] clearly dominates over @xmath3sii@xmath4 $ ] where h@xmath0 is the brightest , weaker zones ( in h@xmath0 integrated intensity ) show an important increase of the relative contribution of the sulfur material . in particular , the reader s attention is directed toward the north - eastern filament of the bright , central structure described above . series of small , quasi - circular blobs are found . the top three reveal two nitrogen - dominated features ( in blue ) and one with a stronger relative contribution in @xmath3sii@xmath4 $ ] ( in red ) . investigating their corresponding spectrum , the @xmath3sii@xmath4 $ ] intensity remains roughly constant from one blob to another while the @xmath3nii@xmath4 $ ] lines suffer a strong attenuation along the line - of - sight of the `` redder '' one ( which explains the relatively poor contribution of nitrogen towards it ) . this behavior is not generalized to our whole fov although similar features can be found here and there . the supernova remnant hb3 ( see figure 1 ) is located too far away from our fov to have had a sizeable impact on the chemical properties in central ic1805 . alternatively , we can argue that the old , large molecular cloud , that gave birth to melotte 15 roughly 2.5 myr ago , may have had an inhomogeneous distribution of its chemical compounds . old supernovae in ic1805 , whose non - thermal emission has vanished since , could also be held responsible . we reiterate that the ic1805 region was most likely formed by a succession of different star clusters ( see @xmath12 2 ) i.e. , although no indication for supernova remnants is currently found inside the large hii region , it is highly probable that the inner zones of ic1805 were , at some time in the past , disturbed by supernova events associated to previous generations of massive stars . these stars could have had an intrinsic inhomogeneity in their inner nitrogen distribution which eventually led to an anisotropic dispersion of these chemical compounds , products of the cno cycle , as each stellar object reached the end of its life ( e.g. , see the works by @xcite , @xcite , and @xcite on the crab nebula ) . line ratios , in the literature , usually correspond to ratios between two ( or more ) line fluxes . for a given emission line , the line flux is proportional to the product between its peak intensity and its width as returned by the gaussian fit . in this work , the relatively low spectral resolution used ( see @xmath12 3 ) is roughly a factor 10 to 20 greater than typical non - thermal velocity fluctuations , along the line - of - sight , found in ic1805 using high - resolution observations of the h@xmath11 kinematics @xcite . hence , in this work , the width of each emission line is entirely dominated by the instrumental response and the returned widths are very similar , from one ion to another , independently of the position in the fov . therefore , line ratios , in the following discussion , were strictly estimated using returned values for peak intensities while line widths were simply not considered . to assure the selection of the most reliable emission - line profiles in our sample of over 200000 spectra , a series of conditions are proposed . * all * of them need to be fulfilled otherwise the given spectrum is rejected from the discussion to follow . the conditions are summarized as follow : 1 . the profile must show all five lines ( i.e. , h@xmath0@xmath16563 @xmath2 , @xmath3nii@xmath4$]@xmath1@xmath16548 , 6584 @xmath2 , and @xmath3sii@xmath4$]@xmath1@xmath16716 , 6731 @xmath2 ) investigated in this work with reliable s / n above 6 . two conditions are proposed in order to confirm that the kinematical properties of a given profile are physically acceptable . more specifically , the @xmath3nii@xmath4$]@xmath16548 @xmath2 and @xmath3nii@xmath4$]@xmath16584 @xmath2 transitions must exhibit identical non - thermal motions since both lines are emitted by the same ions . the same argument also applies to the @xmath3sii@xmath4$]@xmath16716 @xmath2 and @xmath3sii@xmath4$]@xmath16731 @xmath2 lines . 1 . first , the difference , in centroid velocities , between both lines of a given doublet ( i.e. , @xmath3nii@xmath4 $ ] or @xmath3sii@xmath4 $ ] ) must not exceed 15 km s@xmath8 . this corresponds roughly to 1/10th the spectral resolution of our observations ( see @xmath12 3 ) and therefore roughly accounts for the uncertainties on the gaussian fits . centroid - velocity differences , along a given line - of - sight , between the h@xmath11 , n@xmath11 , and s@xmath11 material are allowed ( and expected ! ) since all three ions are not necessarily co - spatial in central ic1805 . 2 . secondly , the difference , in measured line widths , between all five components must not exceed 15 km s@xmath8 . this is expected from all lines being dominated by the instrumental response ( see @xmath12 4.2.1 ) . if observations with high spectral resolution ( e.g. , of the order of a few km s@xmath8 ) were used here , a condition similar to the previous one ( on centroid velocities ) would have been required ( i.e. , identical line widths , within the uncertainty bars of the gaussian fits , between both lines of a given doublet , @xmath3nii@xmath4 $ ] or @xmath3sii@xmath4 $ ] ) . finally , specific line ratios of the @xmath3nii@xmath4 $ ] and @xmath3sii@xmath4 $ ] transitions must agree with standard theoretical models of ionized nebulae . 1 . the computation of the @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] line ratio must lead to a finite electron density . the upper and lower uncertainties on the density measurement ( see @xmath12 4.2.3 ) must also be finite . the computation of the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio must not largely deviate from the theoretical value approaching 3 in low - density regimes typically found for hii regions . small deviations from 3 are expected ( see @xmath12 4.2.4 ) and explained ( see @xmath12 5.1 ) although values @xmath72 or @xmath314 would seem particularly hard to reconcile with the theory and would therefore demand to be rejected . all conditions taken in consideration , only 3057 emission - line profiles were retained , most of them being associated to the bright , central structure found in the overlapping region between the eastern and western field ( see @xmath12 5.2.1 ) . obviously , the summation of both cubes largely contributes to increase the data quality for duplicated pixels not only by increasing the peak signal for all lines but also by flattening noise fluctuations in empty channels ( see figure 2 ) . considering the very large number of emission - line profiles available in our initial data set ( more than 200000 ) , we are fully aware that the conditions listed above drastically reduce the size of the retained sample . on the other hand , these conditions assure that the retained profiles are undoubtedly the most reliable available . we reiterate that a s / n greater than 6 is required to appropriately conduct an investigation on line ratios . however , a `` physical detection '' [ of a given line ] is considered when s / n@xmath323 @xcite . this will be used in a later subsection to investigate notable , although not necessarily reliable , line ratios for particular structures in ic1805 ( see @xmath12 5.2.2.2 ) . the @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] line ratios were computed and used as input values for the fivel fortran procedure @xcite adapted , for convenience , in idl . assuming a constant electron temperature of 7400 k throughout ic1805 @xcite , mean electron densities ( @xmath33 ) , in the s@xmath11 ionic volume , were obtained for each nebular - gas column ( i.e. , line - of - sight ) . since uncertainties can not be recovered from the use of the fivel procedure , error bars on density measurements @xmath33 were computed as follows . for each line of the @xmath3sii@xmath4 $ ] doublet , the uncertainty on the peak intensity , as returned by the gaussian fit procedure , is provided by equation 4__a _ _ of @xcite . from there , the statistical error on each line ratio is simply provided by the standard propagation of uncertainty . for each ratio , minimal and maximal plausible values are then easily obtained by respectively subtracting and adding this calculated uncertainty . using these lower and upper limits on @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] , maximal ( @xmath34 ) and minimal ( @xmath35 ) densities , associated to @xmath33 , can be obtained via fivel . an asymmetrical error bar usually results where the upper ( lower ) uncertainty is provided by the difference between @xmath34 ( @xmath33 ) and @xmath33 ( @xmath35 ) . only @xmath33 values where both @xmath35 and @xmath34 are finite were retained ( see @xmath12 4.2.2 ) . according to fivel , this implies that @xmath35 and @xmath34 must be greater than 10 @xmath36 and lower than 15000 @xmath36 respectively ( these two values mostly depend on our choice of a constant electron temperature of 7400 k in ic1805 ) . our method considers that the statistical error on @xmath33 is entirely dominated by the data quality ( i.e. , s / n of the @xmath3sii@xmath4 $ ] lines ) and that the uncertainties on the atomic parameters , used by fivel , are negligible . however , using @xmath34 and @xmath35 to calculate the statistical error on @xmath33 probably overestimates the actual uncertainty on the electron density measurement . the distribution of the retained electron densities is provided by the black curve in figure 10__a _ _ ( the blue and red distributions will be discussed in @xmath12 5.2.1 ) . the histogram has a mean of 155@xmath37 @xmath36 where the error bars correspond to the mean upper ( @xmath38 ) and lower ( @xmath39 ) uncertainty retrieved from the 3057 electron - density points preserved . panel ( b ) provides the spatial distribution of these points , superimposed to a black - and - white reproduction of figure 4 . purple dots have the lowest density values while red dots , the largest ( see caption ) . figure 11 provides the distribution of the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio in the vicinity of the melotte 15 star cluster . the vertical long - dashed line indicates the theoretical value as suggested by the ratio of the transition probabilities of both ionic lines in a low - density regime ( * ? ? ? * chapter 5 ) . all 3057 points mentioned in @xmath12 4.2.3 formed the regular - line distribution . as shown by the thick - line histogram , the presence of large values i.e. , above @xmath103.5@xmath403.6 , is reduced by considering only data points of very high quality ( i.e. , requiring here , arbitrarily , that s / n@xmath920 in the first condition of @xmath12 4.2.2 ) . uncertainties on the mean and one - standard deviation of both histograms are statistical uncertainties retrieved assuming normal distributions ( * ? ? ? * chapter 5 ) . extreme values for @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] i.e. , below 2 or well - above 4 , were not measured . hence , condition 5 , listed in @xmath12 4.2.2 , was not used in this work to reject particular emission - line profiles although it could used by other authors for future , similar studies . considering the relatively high s / n used here ( above 20 for the thick - line histogram of figure 11 hence suggesting a very good signal quality ) , deviations from @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$]@xmath103 seem genuine and will be addressed in @xmath12 5.1 . panels ( a ) of figures 12 to 14 provide a series of line - ratio diagnostic diagrams commonly used in optical observations . each diagram contains the 3057 points already discussed in @xmath12@xmath12 4.2.3 and 4.2.4 . areas labeled `` hii regions '' , `` snrs '' , and `` pne '' were all reproduced according to figures 1 to 3 of @xcite . long - dashed lines , in figures 13__a _ _ and 14__a _ _ indicate the lower and upper limits for @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] that provide finite values of electron densities @xmath33 given a constant temperature of 7400 k in ic1805 ( see @xmath12 4.2.3 ) . in all three diagrams , crosses , colored in blue , are used to represent points found ( in majority ) within the `` hii regions '' area . the relatively high numbers of crosses is not surprising for ic1805 , already cataloged as an hii region . a certain fraction of the sample , however , falls outside this area . these points are symbolized as red filled circles and point approximately toward the `` snrs '' area , hence suggesting evidence for shock excitation in the targeted ism volume . this will be discussed in @xmath12 5.2 . out of the 3057 points retained for this study , respectively 134 ( 4.4% ) , 67 ( 2.2% ) , and 84 ( 2.7% ) are displayed as red filled circles in figures 12__a _ _ to 14__a__. panels ( b ) , in all three figures , provide the spatial distribution of all crosses ( shown as blue dots ) and circles ( shown as red dots ) displayed in the diagnostic diagram of their respective panel ( a ) . as in figure 10__b _ _ , a black - and - white reproduction of figure 4 was used . red dots in figures 12__b _ _ to 14__b _ _ all point at the same areas and reveal that the southern portions of the bright , central structure ( @xmath16@xmath2261@xmath1326@xmath1430@xmath21 ) may encompassed material subject to ionization by shocks . as demonstrated in @xmath12 4.2.4 , values for the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio above 3.5 and approaching 4 are partially attributed to poorer data quality especially affecting the lower - signal , noisy @xmath3nii@xmath4$]@xmath16548 @xmath2 line ( e.g. , see figure 2 ) . figure 11 indicates that the use of profiles whose emission lines are strictly characterized by `` very '' high s / n ( in our case , greater than 20 ) contributes to attenuate the right tail of the distribution . the asymmetry coefficient ( i.e. , skewness ) is reduced from roughly 0.4 to 0.2 . consequently , the width of the distribution is statistically narrower while the mean of the histogram slightly varied toward the expected value of @xmath103 for @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] . nonetheless , independently of the threshold in s / n used ( e.g. , 6 or 20 ) , figure 11 reveals that the mean @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] ratio exceeds , by a few tenths , the theoretical value . we propose that this could be attributed to a simple statistical effect by which the ratio of transition probabilities between the @xmath3nii@xmath4$]@xmath16548 @xmath2 and @xmath3nii@xmath4$]@xmath16584 @xmath2 transitions does not necessarily equal 3 given the physical properties governing the ic1805 nebula . note that the curve of transmission of the interference filter used for data acquisition has not revealed any significant difference between the transmission coefficients of both lines of the @xmath3nii@xmath4 $ ] doublet . table 3.15 of @xcite provides , for different ionic transitions , the critical electron density above which collisional de - excitations , caused by free electrons , become important . this density is estimated at 80 and 310 @xmath36 respectively for the @xmath3nii@xmath4$]@xmath16548 @xmath2 and @xmath3nii@xmath4$]@xmath16584 @xmath2 transitions . hence , for a given electron density between these two values and characterizing the n@xmath11 volume , the @xmath3nii@xmath4$]@xmath16548 @xmath2 line intensity will be less than expected , a fraction of its flux being `` lost '' in ( non - radiative ) collisional de - excitations . on the other hand , the @xmath3nii@xmath4$]@xmath16584 @xmath2 line intensity will not be affected until densities reach the 310 @xmath36 plateau . figure 10__a _ _ provides reasonable evidences for electron densities between 80 and 310 @xmath36 in ic1805 . hence , a mean @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] ratio of 3.3@xmath403.4 could be explained by a @xmath3nii@xmath4$]@xmath16548 @xmath2 line partially experiencing collisional de - excitations . from the results presented in @xmath12@xmath12 4.2.3 and 4.2.4 ( for s / n@xmath96 ) , figure 15 provides the electron densities ( as measured from the @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] line ratio ; see figure 10 ) vs. the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio ( see figure 11 ) relation . each @xmath33@xmath40@xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] pair is represented by a black cross . for each bin of 10 @xmath36 between 10 and 500 @xmath36 , the mean @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio is displayed as a red filled square . a standard linear regression ( * ? ? * chapter 4 ) was applied to the filled - square sample , in the 80@xmath40310 @xmath36 interval , and is displayed as the dashed green line . the value of each bin was adequately weighted in the regression i.e. , bins with large numbers of crosses were given a more important statistical weight . a clear , monotonic increase of the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio with increasing densities would be expected as the @xmath3nii@xmath4$]@xmath16548 @xmath2 line becomes more - and - more affected by collisional de - excitations ( i.e. , decreases in intensity ) . as expected , the linear relation shows a positive slope although its value , very close to 0 , largely suggests no correlation at all . indeed , the quality of the linear fit is relatively poor with a correlation coefficient of 0.35 . hence , we believe that no conclusion can be drawn from figure 15 . two possibilities are however provided in order to explain such behavior . first , the reader should note that the uncertainties are relatively large for both parameters investigated . the mean uncertainty for the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio is 0.17 while typical uncertainties for the electron - density measurements @xmath33 are provided in the last paragraph of @xmath12 4.2.3 . the poor correlation of the linear fit in figure 15 could be attributed to large error bars . moreover , the absence of a well - defined correlation between @xmath33 and @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] would not be surprising if the n@xmath11 and s@xmath11 material are not perfectly cospatial in the vicinity of melotte 15 . given a difference of more than 4 ev between the ionization potentials of neutral nitrogen and sulfur , it can be assumed that the electron densities , computed from the @xmath3sii@xmath4 $ ] doublet , may not reflect accurately the physical conditions prevailing in the n@xmath11 ionic volume . therefore , the @xmath33 vs. @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] diagram of figure 15 could be an interesting diagnostic tool although our data set may not provide the appropriate spectral information to reliably construct such a diagram . alternatively , given the first ionization potential of nitrogen ( 14.5 ev ) and oxygen ( 13.6 ev ) , the @xmath41\,\lambda3726}{[\textnormal{o}\,\textsc{ii}]\,\lambda3729}$ ] line ratio would likely give a more accurate indication of the density behavior inside the n@xmath11 volume . panels ( a ) of figures 12 to 14 reveal that a large fraction of the nebular - gas content in the vicinity of the melotte 15 star cluster appears to be dominated by photoionization , typical of standard hii regions . however , points identified outside the expected regime for hii regions and located near the `` snrs '' area demand to be closely investigated ( see @xmath12 4.2.5 ) . in all three figures , these are symbolized as red filled circles and potentially indicate the presence of shocks in the targeted nebular volume . these areas of our fov , suggesting shock excitation , are identified as red dots in all three panels ( b ) of the same figures . however , at this point , shock excitation in ic1805 remains highly hypothetical . this is somewhat surprising considering the presence of relatively massive stars in melotte 15 and their associated strong stellar winds ( see @xmath12 2 ) . in order to deeply investigate the impact of shock excitation in ic1805 , we assume that , if present , the shock - excited ionized material is relatively confined and localized along the line - of - sight i.e. , considering the imposing dimensions of the large hii region ( e.g. , @xcite ) , the interface between compressive shocks and the surrounding ism could be several times smaller than the size of the nebula itself . this said , the expected spectral signature of shock excitation could be strongly diluted by photoionized foreground / background material . a total of 2378 out of 3057 emission - line profiles retained for this study are directly associated to the bright , central structure clearly visible in figure 8 . the strong emission of its ionized component ( see peak intensities in figures 3 to 7 ) and therefore the quality of the gathered signal make this structure , surrounded by the most massive stars of the melotte 15 cluster , the ideal feature in our quest for shock excitation in central ic1805 . first , a series of four weaker portions [ in gas emission ] , specifically surrounding the central structure , were selected . each of these portions was spatially binned into a single 1@xmath231@xmath23249 ( spatial@xmath23spatial@xmath23channels ) emission - line profile . for each of the 2378 spectra selected and investigated in this subsection , the corresponding foreground / background emission was approximately recovered using a linear combination of these four 1@xmath231@xmath23249 profiles . each of the four linear coefficients was weighted via the inverse of the distance separating the targeted pixel from the center of the corresponding weaker portion ( e.g. , the larger this distance is , the smaller is the corresponding linear coefficient and , therefore , the smaller is the statistical impact of this weaker zone on the computation of the foreground / background spectral signature at the position of this given pixel ) . hence , a unique foreground / background spectrum is constructed for each of the 2378 points ( see below ) although neighbor pixels have [ as it should be ] foreground / background material with very similar spectral signatures ( i.e. , roughly identical linear coefficients ) . the subtraction of the foreground / background emission obviously had a certain incidence on the s / n of the resulting profiles , the peak intensity of all five lines being reduced in the subtracting process . still following all conditions listed in @xmath12 4.2.2 , 2229 emission - line profiles out of 2378 were said to `` survive '' the procedure . panels ( a ) and ( b ) of figures 16 present the log@xmath42)}\right]$ ] vs. log@xmath43)}\right]$ ] diagnostic diagram for these 2229 points respectively before and after the subtraction of the foreground / background material . figure 16__a _ _ is obviously a subset of figure 12__a _ _ and uses the same symbol definitions ( see @xmath12 4.2.5 ) . note that each point preserves its original symbol , shape ( cross / circle ) and color(blue / red ) , from figure 16__a _ _ to 16__b__. in figure 16__a _ _ , 2109 are displayed as blue crosses while 120 are symbolized as red filled circles , being located close but outside the lower - left corner of the `` hii regions '' area . note that 134 points were displayed as red circles in figure 12__a _ _ ( see @xmath12 4.2.5 ) : 14 of those points were not considered here , either not associated with the central structure or have simply failed to fulfill the conditions of @xmath12 4.2.2 following the subtracting process discussed above . in figure 16__b _ _ , a tail precisely directed toward the `` snrs '' area appears as numerous points migrate out of the `` hii regions '' area . recent surveys ( e.g. , @xcite ) have shown that the boundaries of the different areas circumscribed in the diagnostic diagrams used here could be slightly modified with respect to the first work carried on by @xcite . it appears that the lower and upper limits , in the parameter space , for each regime are still uncertain to this day . we therefore chose to arbitrarily redefine ( or expand ) the zone of shock - excitation for each relation investigated . for example , in the log@xmath42)}\right]$ ] vs. log@xmath43)}\right]$ ] relation , the hatched area in the lower - left corner of figure 16__b _ _ was , from now on , used to separate gas columns ( i.e. , pixels ) likely enclosing post - shocked material from those for which photoionization effects appear to primarily dominate . now , 509 points out of 2229 suggest evidence for potential shock excitation in the direct vicinity of the melotte 15 cluster . the tail is formed of both blue crosses and red circles indicating that figure 16__a _ _ alone would have not be sufficient in order to precisely pinpoint those sections of the central structure revealing ( or hiding ) a well - defined signature of ionization by shocks . panel ( a ) of figure 17 provides the post - subtraction spatial distribution of all points found in figure 16__b__. points found in the hatched area of the lower - left portion of the diagram are colored in red . others , still associated to the `` hii regions '' area hence suggesting photoionization effects , are in blue . similar exercises were carried on using the corresponding subsets for figures 13__a _ _ and 14__a _ _ although we will simply provide the results here . the zone of shock excitation was , respectively for both figures , redefined by log@xmath42)}\right]$]@xmath220.3 and log@xmath43)}\right]$]@xmath220.2 , and also bordered by the two long - dashed lines ( i.e. , lower and upper limits on @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] ) . for the log@xmath42)}\right]$ ] vs. @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] diagram , while only 54 out of 2229 points were being found initially outside the `` hii regions '' area prior to the subtraction of the foreground / background material , 509 points have shown log@xmath42)}\right]$]@xmath220.3 afterwards . again , as in figure 16__b _ _ , evidences for shocks appear for both blue crosses and red filled circles . panel ( b ) of figure 17 gives the spatial distribution of the post - subtraction points . one clearly sees the perfect correspondence between panels ( a ) and ( b ) i.e. , using two distinct diagrams the same points , that suggest shock excitation , are extracted . for the log@xmath43)}\right]$ ] vs. @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] relation , the situation is more complex as the `` snrs '' and `` hii regions '' areas overlap in the diagnostic tools ( see figure 14__a _ _ ) . the removal of the foreground / background material has led to 1419 points displaying log@xmath43)}\right]$ ] values below 0.2 although roughly half of these remain very close to the `` hii regions '' area . for the log@xmath43)}\right]$ ] vs. @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] relation , panel ( c ) of figure 17 gives the spatial distribution of the post - subtraction points using the same color code as in panels ( a ) and ( b ) . combining the post - subtraction results for all three diagrams allows us to spatially identify those sections on the plane of the sky that suggest shock excitation . panel ( d ) of figure 17 combines all results found in the first three panels . red dots that can be found in all three post - subtraction diagnostic diagrams are again colored in red . these points targeted regions with high probability of shock excitation in central ic1805 . those red dots that can only be found in panel ( c ) are colored in green in panel ( d ) . these can be referred as interesting candidates for shock excitation although 2 out of 3 diagrams have failed to identify them . blue dots in all three post - subtraction diagnostic diagrams are colored in blue and identify the zones , mostly located near the northern tip of the bright ionized feature , where photoionization clearly dominates . shock - dominated material seems to prevail along the southern portion of the central structure , directly exposed to the stellar winds and currently eroded by the ionizing flux of the two most massive stars identified in melotte 15 ( identified by black arrows in figure 17__d _ _ ) . these are cataloged by @xcite as hb15558 and hb15570 , two evolved o4iii(f ) ( @xmath17=02@xmath1832@xmath1943@xmath20 , @xmath16=61@xmath1327@xmath1422@xmath21 ) and o4if ( @xmath17=02@xmath1832@xmath1950@xmath20 , @xmath16=61@xmath1322@xmath1442@xmath21 ) stars respectively . ionization by shocks clearly dominates the bright ionized gas at the periphery of the weak molecular material found in the southernmost portion of our fov . our results reveal that shock excitation appears to be correlated with bright ionized material , suggesting high electron densities if we assume the nebula to be optically - thin to the ionic light . this would then be easily correlated to the compressive nature of the hypothetical shocks . extracting the electron densities @xmath33 for all 1419 points ( rather in red or green in figure 17__d _ _ ) yields the blue - line histogram of figure 10__a__. its mean is estimated at 125@xmath44 @xmath36 , slightly below the mean value previously derived for the whole sample of 3057 emission - line profiles retained for this work ( see @xmath12 4.2.3 ) . this hardly makes sense considering the compression expected from shock waves . however , keeping in mind that the subtraction of , what we referred to as , the foreground / background material was required in order to spectrally detect evidences for shock excitation in ic1805 , it seems logical that the same procedure should be applied in the extraction of the electron densities specifically associated to the shocked , ionized component . using the 1419 post - subtraction emission - line profiles that showed evidences for shock excitation , the corresponding electron densities ( @xmath28@xmath45 ) were computed following the method described in @xmath12 4.2.3 . the red histogram , in figure 10__a _ _ , reveals the distribution obtained for @xmath45 . the mean electron density along the line - of - sight at the position of shocks is estimated at 175 @xmath36 , a mean roughly 20 @xmath36 greater than what was found for the black histogram in figure 10__a__. this suggests , as expected , that the foreground / background material probably occupies , near melotte 15 , a large volume filled with a very diffuse ionized component . this tenuous material undoubtedly contributes to attenuate the ionic emission of high - density condensations along the line - of - sight . using the 1419 spectra of the foreground / background emission , we extracted to corresponding densities ( @xmath28@xmath46 ) . as expected , this tenuous component has a mean electron density , along the line - of - sight , of 25 @xmath36 . the distribution shows a narrow width of roughly 15 @xmath36 which reflects the homogeneity ( in density ) of the foreground / background material over the entire spatial extent of the central , ionized structure . figure 18 shows the distribution of the pixel - to - pixel compression factor i.e. , @xmath45/@xmath46 . all remaining 1419 points formed the black histogram . the mean compression is estimated at 8.5 although the histogram is clearly dominated by values of @xmath45/@xmath46 below 5 . as expected , the compression factor is always greater than 1 . sub - distributions , in green and red , gives the compression factor respectively for the green and red dots obtained in figure 17__d__. the green histogram , formed of points suggesting possible shock excitation , is very similar statistically to the black distribution with a mean of 7.5 . the red histogram , this time formed of points with a high probability of shock excitation , is slightly different , peaking at a greater @xmath45/@xmath46 value . its mean is estimated slightly above 10 i.e. , those points ( red in figure 17__d _ _ ) where , we believe , shocks may certainly dominate seem to be characterized by greater compression effects with respect to points ( green in figure 17__d _ _ ) where the presence of shocks appears less obvious . for abiadatic jump conditions , rankine - hugoniot relations indicate a compression factor of precisely 4 between pre- and post - shocked gas ( * ? ? ? * chapter 12 ) . the relatively large widths of the distributions displayed in figure 18 indicate that isothermal shocks may dominate . also , the fact that forbidden lines such as @xmath3nii@xmath4 $ ] and @xmath3sii@xmath4 $ ] are detected favors radiative cooling and therefore non - adiabatic effects [ although these lines most likely originate from ionized material trapped in the cooling , shocked outer shell and therefore may not be related to the wind - blown bubble s current state of expansion i.e. , energy- or momentum - driven ] . theoretically , for an isothermal shock ( * ? ? ? * chapter 6 ) , the compression factor corresponds to the square of the shock s mach number , @xmath47 ( see the abscissa in figure 18 ) . therefore , for a speed of sound of 10 km s@xmath8 in the h@xmath11 medium ( in agreement with our choice of 7400 k for the electron temperature in ic1805 ; see @xmath12 4.2.3 ) , shocks with typical velocities between 15 and 30 km s@xmath8 ( i.e. , @xmath48=2@xmath4010 ) may have led to shock excitation near melotte 15 . this is in agreement with typical expansion velocities found for wind - blown bubbles ( see @xmath12 1 ) . shock velocities up to 50 km s@xmath8 ( i.e. , @xmath48@xmath1025 ) are also found although such dynamics is certainly peculiar . note , however , that the compression measured here could be an upper limit since photoeroded gas located in the molecular cloud s envelope , rather than the diffuse foreground / background component , may act as pre - shocked material . the cloud s envelope is expected to be , at least , a few tens of particles per @xmath49 denser than typical values found for @xmath46 in this work @xcite . assuming a coplanar geometry , roughly 3@xmath404 pc separate the most massive star of the melotte 15 cluster and the central , ionized structure . using the standard model for expanding wind - blown bubbles @xcite , the distance reached by a given bubble s post - shocked shell can be estimated from the mechanical luminosity of the winds ( @xmath50=@xmath51@xmath52@xmath53 ) , the density of the pre - shocked medium ( @xmath46 ) and the total time during which winds have been blown by the central star ( @xmath54 ) . given @xmath54=2.5 myr , the age of the melotte 15 cluster ( see @xmath12 2 ) , we expect that two massive giant and supergiant o4 stars have left just very recently the main - sequence branch . therefore , for the calculations to follow , these were treated as o4v stars . from different studies proposing typical mass - loss rates and terminal velocities with respect to spectral types @xcite , @xmath52 and @xmath55 were respectively estimated at 5@xmath2310@xmath56 m@xmath57 yr@xmath8 and 3000 km s@xmath8 for standard o4v stars . given @xmath46@xmath1025 @xmath36 ( see above ) , nebular material found within 15@xmath4020 pc of melotte 15 is likely to have been previously disrupted by stellar winds emanating from the current cluster . this was calculated assuming radiative ( non - adiabatic ) shocks ( * ? ? ? * equation 13.5 ) . given that interstellar shocks are usually adiabatic at the earliest times of the expansion , the non - radiative model yields a distance of roughly 40 pc in the vicinity of melotte 15 ( * ? ? ? * equation 13.9 ) . both distances exceed the extent of our fov ( 15@xmath239 pc@xmath25 ) and support the fact that shocks , attributed to the current star cluster , have had sufficient time to reach the bright structure in central ic1805 . besides the bright , central structure observed near the most massive stars of the melotte 15 cluster , our fov mostly reveals diffuse ionized gas , partially obscured by interstellar dust . however , two structures are nonetheless detected , particularly well - defined in h@xmath0 ( see @xmath12 4.1 ) . in order to investigate the presence of shock excitation in other portions of our fov , a similar approach , to what was carried on in @xmath12 5.2.1 , is used here on these two features . [ [ south - east - co - fragment . ] ] south - east co fragment . + + + + + + + + + + + + + + + + + + + + + + + panel ( a ) of figure 19 reveals the shape of the ionization front observed at the periphery of the co fragment found in the south - east portion of our fov ( see figure 8) . as in figures 3 to 9 , north is up and east is left . the molecular feature is well - defined in the millimeter regime and its western side shows signs of erosion by the nearby star cluster . the ionized gas associated to the ionization front was first circumscribed while weaker areas near the bright h@xmath11 feature were selected and used to extract the foreground / background spectrum . in order to obtain a post - subtraction sample containing a sufficient number of emission - line profiles , only the first condition of @xmath12 4.2.2 therefore , results presented here should be cautiously interpreted . prior to the subtraction , all points were initially found inside the `` hii regions '' area of the log@xmath42)}\right]$ ] vs. log@xmath43)}\right]$ ] diagnostic diagram and black filled circles were used as symbols . panel ( b ) displays the diagram for post - subtraction spectra only . these results reveal , as in @xmath12 5.2.1 , that points are displaced toward the shock - dominated `` snrs '' area although the tail is much less developed when compared to figure 16__b__. no post - subtraction point actually enters the hatched area defined earlier . shock excitation hence appears to have played a much more minor role in the ionization of this cloud s envelope when compared to the central structure of our fov . the apparent shape of the ionization front indicates that the ionizing sources are located behind the molecular cloud ( see @xmath12 4.1 ) . using kinematical information retrieved from the photoevaporated material in the vicinity of the cloud , @xcite suggested that an appreciable distance could exist between the co fragment and the melotte 15 cluster ( see the authors figure 11 and their associated flow g ) . hence , large distances could signify here that stellar - wind shocks may have been partially dissipated before they could reach the molecular cloud . shock excitation is therefore likely measurable only within a certain distance of the shock sources i.e. , stars with strong stellar winds in our case . obviously , the stronger the shocks are , the greater this corresponding distance is . we reiterate that only the first 15 to 40 pc in the vicinity of melotte 15 may have been disrupted by stellar winds ( see @xmath12 5.2.1 ) . at this point , the strongest wind shocks ( fuelled up by the o4 stars ) would have receded to the subsonic regime with velocities between 4 and 9 km s@xmath8 depending on the model used ( radiative or non - radiative respectively ) . if , by any chance , the molecular fragment found near the south - eastern boundary of our fov is located relatively far from the ionizing sources ( @xmath3215 pc ) , this could explain the apparent absence of strong , compressive shocks at its periphery . [ [ cigar - like - structure . ] ] cigar - like structure . + + + + + + + + + + + + + + + + + + + + + panel ( c ) of figure 19 shows the very tenuous , but well - defined , ionized counterpart of a cigar - like structure pointing toward a nearby star with strong h@xmath0 emission line . again , north is up and east is left . the star is particularly visible on the left - hand side of figure 8 and has been cataloged by @xcite as hbh@xmath06211 - 05 with a v - band magnitude of 14.6 and no known spectral type . its coordinates are ( @xmath17=02@xmath1834@xmath1910@xmath20.06 , @xmath16=61@xmath1324@xmath1435@xmath21.7 ) . the elongated feature was briefly introduced in @xmath12 4.1 . emission in h@xmath0 and @xmath3nii@xmath4 $ ] is detected on the outskirts of the cigar - like feature while its center appears mostly gas depleted . this overall scheme is particularly similar to what is expected from elephant trunks in hii regions ( e.g. , @xcite ) . shadowing effects , caused by a dense neutral globule located in a radiation field , allow the warm gas behind it to recombine which could explain the absence of ionized material at the center of the elongated structure . the bright ionized rims are formed of photoevaporated material created by the erosion of the neutral globule . these flows move away from the ionizing star creating the cigar - like shape . both the fcrao co(1 - 0 ) survey and the canadian galactic plane survey at 21 cm @xcite reveal no indication for either molecular or atomic gas associated to the elongated feature . theoretically , h@xmath58 gas is expected to constitute the main component of the eroded globule while hi material should result from recombinations in the tail . the whole structure , however , has relatively small angular dimensions ( 45@xmath21@xmath237@xmath21 ) which suggests that emission from the neutral gas could be beam diluted in the low - resolution radio observations . theory has suggested that shocks will develop in elephant trunks @xcite although , in first approximation , the log@xmath42)}\right]$ ] vs. log@xmath43)}\right]$ ] diagram has shown nothing particular using the first set of gaussian fits i.e. , all points associated to the elongated structure are well - confined into the `` hii regions '' area . however , subtracting the foreground / background material yields the diagram displayed in panel ( d ) of figure 19 . none of the conditions listed in @xmath12 4.2.2 were considered . only a s / n greater than 3 ( sufficiently high to confirm a physical detection ; see @xmath12 4.2.2 ) was here required . therefore , the results are highly questionable although worth mentioning . the ionized material at the center of the cigar - like feature being very tenuous ( or simply non - existent ; see above ) , all points filling the diagram are found along the external bright rims . panel ( d ) reveals that the `` snrs '' area of shock - dominated material is highly favored , in agreement with the prediction of shock development proposed by the theory . the particular shape of the elongated structure presented in panel ( c ) could be easily mistaken for jets attributed to herbig - haro ( hh ) objects . however , figure 4__a _ _ of @xcite indicates the peculiar low intensities of the @xmath3nii@xmath4 $ ] lines in hh objects . this leads to a position , for hh objects in the log@xmath42)}\right]$ ] vs. log@xmath43)}\right]$ ] diagram , clearly above the `` snrs '' area . this said , the diagram of panel ( d ) most likely confirms that the cigar - like feature is not part of an hh object in ic1805 . rather , the elephant - trunk nature , suggested here , appears very plausible . the use of the imaging fourier transform spectrometer spiomm allowed us to obtain series of emission - line profiles of the optical gas in the brightest , central portions of the galactic ic1805 hii region . the bandwidth used at data acquisition allowed the simultaneous observations of the h@xmath0@xmath16563 @xmath2 , @xmath3nii@xmath4$]@xmath1@xmath16548 , 6584 @xmath2 , and @xmath3sii@xmath4$]@xmath1@xmath16716 , 6731 @xmath2 ionic lines ( see @xmath12 3 ) . the main goal of this work was to investigate on the presence of supersonic shock waves attributed to stellar winds in the vicinity of melotte 15 , the current star cluster actually fueling up the expansion of the ic1805 nebula . literature has long suggested that a kinematical detection of stellar winds , in hii region , might represent a difficult task since the typical expansion velocity of shocked shells could be commonly confused with other dynamical processes revealing similar kinematical behaviors ( see @xmath12 1 ) . on the other hand , specific line ratios retrieved from the optical gas may indicate if the presence of ionized material is attributed to standard photoionization or to shock excitation . these line ratios therefore provide a non - kinematical tool for identifying shocks in the nebular volume . our results are summarized as follow : 1 . the @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio genuinely deviates from the theoretical value of 3 ( see @xmath12 5.1 ) . values varying between 2.5 and 4 were commonly found . the distribution of the density measurements has not allowed to demonstrate that the @xmath3nii@xmath4$]@xmath16548 @xmath2 line could be affected by collisional de - excitations , hence reducing its peak intensity . this scenario remains nonetheless plausible and could be verified if densities , in the n@xmath11 volume , could be measured precisely ( see @xmath12 5.1 ) . densities extracted from the @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] ratio may not perfectly reflect the conditions prevailing in the n@xmath11 volume if both nitrogen and sulfur are not co - spatial in central ic1805 . 2 . the diagnostic diagrams , introduced by @xcite , indicate , in first approximation , that photoionization most likely dominate in ic1805 ( see @xmath12 4.2.5 ) . this initially holds even for the densest , most emissive structures in our field - of - view , directly exposed to the radiation fields and stellar winds of the nearby , most massive stars ( see @xmath12 5.2 ) . evidences for shock excitation appear only following the subtraction of the diffuse foreground / background material . this component likely occupies a very large fraction of the nebular volume in ic1805 ; a volume sufficiently large that , even though being tenuous in nature , the foreground / background material strongly dilutes the signal emanating from shocked condensations along the line - of - sight . its mean density is estimated at 25 @xmath36 ( see @xmath12 5.2.1 ) . 4 . oriented on a south - north axis and surrounded by numerous o - stars , a bright , large ionized feature occupies the central area of our field - of - view . the last , tenuous fragments of an old molecular cloud can be found near its southern portion . shocks may have contributed to ionize material found in the direct vicinity of the molecular clump while the northern parts of the ionized feature , deprived of molecular emission , appears to be largely dominated by photoionization ( see @xmath12 5.2.1 ) . the shock - excited ionized gas has a mean density of 175 @xmath36 and the compression factor , between pre- and post - shocked gas , is typically between 2 and 10 . for isothermal shocks , this suggests shock velocities between 15 and 30 km s@xmath8 , in agreement with models describing the expansion of wind - blown bubbles ( see @xmath12 5.2.1 ) . geometrically speaking , given the apparent proximity between the central , ionized structure and the most massive stars of melotte 15 , winds seem to have had sufficient time to reach the structure within a timescale corresponding to the age of the star cluster ( see @xmath12 5.2.1 ) . this gives credence to our assumption that shock excitation can be found in the south - central portions of our field - of - view . points identified with a high probability of shock excitation reveal compression factors typically greater than those points where shocks may be present although less certain ( see @xmath12 5.2.1 ) . 7 . shocks did not seem to have played a major role in the ionization of a molecular cloud s envelope located in the south - eastern portion of our field - of - view . this most likely results from the molecular fragment being located too far from the ionizing sources so that ( 1 ) shocks induced by stellar winds have not reached yet the cloud or ( 2 ) shocks have reached the cloud with velocities too low to initiate shock excitation ( see @xmath12 5.2.2.1 ) . shock development was clearly detected on the outskirts of an apparently weak , but well - defined elephant trunk located in the eastern portion of our field - of - view ( see @xmath12 5.2.2.2 ) . this is in agreement with theoretical works developed on such feature typically found in hii regions . the authors would like to thank the natural sciences and engineering research council of canada and the fonds qubcois de la recherche sur la nature et les technologies who provided funds for this research project . d. l. is grateful to m .- a . miville - deschnes and d. j. marshall who provided useful idl routines to carry out data reduction . d. l. and l. d. would like to thank b. malenfant , g. turcotte and p .- l . lvesque for technical support during numerous observing nights at the observatoire du mont - 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large hii regions , with angular dimensions exceeding 10 pc , usually enclose numerous massive o - stars . stellar winds from such stars are expected to play a sizeable role in the dynamical , morphological and chemical evolution of the targeted nebula . kinematically , stellar winds remain hardly observable i.e. , the typical expansion velocities of wind - blown bubbles being often confused with other dynamical processes also regularly found hii regions . however , supersonic shock waves , developed by stellar winds , should favor shock excitation and leave a well - defined spectral signature in the ionized nebular content . in this work , the presence of stellar winds , observed through shock excitation , is investigated in the brightest portions of the galactic ic1805 nebula , a giant hii region encompassing at least 10 o - stars from main - sequence o9 to giant and supergiant o4 . the use of the imaging fourier transform spectrometer spiomm enabled the simultaneous acquisition of the spectral information associated to the h@xmath0@xmath16563 @xmath2 , @xmath3nii@xmath4$]@xmath1@xmath16548 , 6584 @xmath2 , and @xmath3sii@xmath4$]@xmath1@xmath16716 , 6731 @xmath2 ionic lines . diagnostic diagrams , first introduced by sabbadin and collaborators , were used to circumscribe portions of the nebula likely subject to shock excitation from other areas dominated by photoionization . the gas compression , expected from supersonic shocks , is investigated by comparing the pre- and post - shocked material s densities computed from the @xmath5\,\lambda6716}{[\textnormal{s}\,\textsc{ii}]\,\lambda6731}$ ] line ratio . the typical @xmath6\,\lambda6584}{[\textnormal{n}\,\textsc{ii}]\,\lambda6548}$ ] line ratio slightly exceeds the theoretical value of 3 expected in low - density regimes . to explain such behavior , a scenario based on collisional de - excitations affecting the @xmath3nii@xmath4$]@xmath16548 @xmath2 line is proposed . [ firstpage ] techniques : spectroscopic ism : h regions ism : lines and bands ism : individual : ic1805

optical surveys sampling the sky over time scales of a few days significantly advanced our knowledge of astronomical transients of different origins , including super - luminous supernovae ( slsne ; @xcite ) , very fast - rising stellar explosions ( e.g. @xcite ) and stellar tidal disruptions by super - massive black holes ( tdes ; @xcite ) . occasionally , a transient is found with properties that seem to defy all previous classification schemes . the event asassn-15lh belongs to this category . asassn-15lh ( @xcite ) was discovered by the all - sky automated survey for supernovae ( asas - sn ) on 14 june 2015 at z=0.2326 ( @xmath1 mpc for standard _ cosmology ) . its extremely large peak luminosity @xmath2 and the blue , almost featureless spectrum with no apparent sign of h or he ( and some spectroscopic resemblance to the slsn 2010gx ) led @xcite and @xcite to suggest that asassn-15lh is the most luminous slsn ever detected . the very large energy radiated by assasn-15lh ( @xmath3 , @xcite ) , requires extreme properties of the progenitor star and sources of energy that are different from the standard radioactive decay of @xmath4ni that powers normal h - stripped sne in the local universe ( @xcite ) . in this context , the double - humped light - curve of asassn-15lh has been interpreted by @xcite as a signature of the interaction of massive sn ejecta @xmath5 with an h - poor circumstellar shell of @xmath6 , possibly supplemented by radiation from a newly - born rapidly - rotating magnetar @xcite . the old , massive @xmath7 host galaxy of asassn-15lh , with limited star formation ( @xmath8 ; @xcite ) , is however markedly different from those of core - collapse sne ( e.g. @xcite ) as well as of envelope - stripped slsne , which tend to be younger star forming systems with significantly lower stellar mass ( @xcite ) . this observation , together with the location of the transient , astrometrically consistent with the host galaxy nucleus , inspired a connection between asassn-15lh and the tidal disruption of a star by the host - galaxy super - massive black hole ( smbh , @xcite ) . in this context asassn-15lh would be the most luminous tde ever observed , associated to a smbh with mass @xmath9 @xcite significantly larger than any smbh presently associated to a tde ( e.g. @xcite ) . it is clear that the luminosity , the spectral properties and the double - humped ( i.e. , `` bactrian '' ) light - curve of asassn-15lh , as well as its host galaxy , are unprecedented both in the context of slsne and in the context of tdes . in this paper we present and discuss the following observational facts : [ i ] uncovering of persistent , soft x - ray emission at the location of asassn-15lh ( sec . [ subsec : cxo ] and sec . [ subsec : xrt ] ) ; [ ii ] detection of significant temporal variability at uv wavelengths during the re - brightening phase ( sec . [ subsec : uv ] ) ; [ iii ] detection of narrow nebular spectral features connected to the host - galaxy nucleus ( sec . [ subsec : spec ] ) . we propose a scenario where a single physical mechanism can naturally explain the double - humped light - curve of asassn-15lh and suggest that its location , very close or coincident with the nucleus of a galaxy that harbors a massive smbh , is likely the key to unlocking the mysterious nature of the transient ( sec . [ sec : int ] ) . conclusions are drawn in sec . [ sec : conclusions ] . in our analysis we assume the object s time of first light to be april 29 , 2015 , corresponding to a 30-day ( rest - frame ) rise - time to maximum v - band luminosity @xcite . our main conclusions do not depend on this assumption . -0.0 true cm we obtained four epochs of deep x - ray observations of asassn-15lh with the chandra x - ray observatory ( cxo ) on november 12 , 2015 ( exposure of 10 ks ) , december 13 , 2015 ( 10 ks ) , february 20 , 2016 ( 40 ks ) and august 19 , 2016 ( 30 ks , pi margutti ) , corresponding to @xmath10@xmath11@xmath12 days , @xmath10@xmath11@xmath13 days , @xmath10@xmath11@xmath14 and @xmath10@xmath11@xmath15 days rest frame since optical maximum light , which occurred on june 5 , 2015 @xcite . cxo data have been reduced with the ciao software package ( version 4.8 ) and corresponding calibration files . standard acis data filtering has been applied . asassn-15lh is not detected in our first epoch of observations ( i d 17879 ) , with a 3@xmath16 count - rate upper limit of @xmath17 ( 0.5 - 8 kev ) . the galactic column density in the direction of the transient is @xmath18 @xcite . for an assumed power - law spectrum with photon index @xmath19 and galactic absorption , the unabsorbed 0.3 - 10 kev flux limit is @xmath20 ( @xmath21 ) . our analysis below favors a soft x - ray spectrum with negligible absorption and @xmath22 or a thermal spectrum with @xmath23 . for these parameters , the unabsorbed 0.3 - 10 kev flux limit is @xmath24 ( non - thermal spectrum ) and @xmath25 ( thermal spectrum ) . in our second epoch of observation ( i d 17880 ) we find evidence for weak , soft x - ray emission at the location of asassn-15lh . we detect two photons with energy @xmath261 kev in a @xmath27 region around the transient , corresponding to a 4.3@xmath28 c.l . detection in the 0.5 - 1 kev energy range , and to a 3.3@xmath28 c.l . detection in the 0.5 - 8 kev energy range . we constrain the spectral parameters by using the observed background and the actual instrumental response to simulate the expected emission from a grid of thermal and non - thermal spectral models with a wide range of intrinsic absorption @xmath29 . the regions excluded at 3@xmath16 confidence are shaded in fig . [ fig : chandra ] . an x - ray source is clearly detected at the location of asassn-15lh at the time of our third cxo observation ( i d 17881 ) , with count - rate @xmath30 and significance of 5.2@xmath16 in the 0.5 - 8 kev range ( 4.7@xmath16 in the 0.5 - 1 kev energy range ) . in our fourth epoch ( i d 17882 ) the source is still detected with count - rate @xmath31 and significance of 4.9@xmath16 in the 0.5 - 8 kev range ( 3.6@xmath16 in the 0.5 - 1 kev energy range ) . we employ the cash statistics to fit the spectra ( we have a total of six and five photons in a 1@xmath32 region around the transient in the third and fourth epoch , respectively ) , and perform a series of mcmc simulations to constrain the spectral parameters . the analysis of each of the two epochs taken separately points to a soft x - ray spectrum , with limited absorption and no evidence for statistically significant evolution between the two epochs . we thus constrain the x - ray source spectral parameters with a joint spectral fit of the two cxo epochs of observation , where the spectral normalization is allowed to vary from one epoch to the other . the results are displayed in fig . [ fig : chandra ] . for an absorbed , non - thermal power - law spectrum , the best - fitting parameters are @xmath33 and @xmath34 . the inferred ( 0.3 - 10 kev ) unabsorbed flux for this model is @xmath35 corresponding to @xmath36 ( third epoch ) and @xmath37 ( @xmath38 , fourth epoch ) . the best - fitting parameters for an absorbed blackbody spectrum are : @xmath39 kev , @xmath40 . the inferred ( 0.3 - 10 kev ) unabsorbed flux for this model is @xmath41 , corresponding to @xmath42 ( third epoch ) and @xmath43 ( @xmath44 , fourth epoch ) . both spectral models point to a very limited amount of neutral hydrogen in the host galaxy along our line of sight , consistent with the very low n(hi ) inferred by @xcite from ly-@xmath45 and the very strong high ionization lines ( and ) . with reference to fig . [ fig : chandra ] we find that : ( i ) the x - ray source shows a soft spectrum ( most of the allowed parameter space is at @xmath46 and @xmath47 kev ) with limited intrinsic absorption ( of the order of a few @xmath48 at most ) . there is no evidence for strong temporal and/or spectral variability of the x - ray source . we first evaluate the possibility that the x - ray emission arises from a population of low - mass x - ray binaries ( lmxbs ) residing in the early - type host galaxy , using the @xmath49 and @xmath50 relations by @xcite . for the host - galaxy of asassn-15lh @xcite measure @xmath51 mag and @xmath52 mag , which imply @xmath53 ( 0.3 - 8 kev ) . this is a factor @xmath54 smaller then the measured x - ray emission at the location of asassn-15lh ( re - calibrated with the same spectral model as @xcite in the 0.3 - 8 kev band ) . we conclude that lmxbs are unlikely to be the source of the detected x - rays . we thus envision two possible scenarios : either the x - rays originate from weak agn activity from the host galaxy nucleus or they are physically connected to the optical / uv transient . in the first case we expect a somewhat stable x - ray emission over the time scale of years , while we anticipate fading if the x - ray emission is directly connected to asassn-15lh . future observations will clarify the origin of the detected high - energy emission . below we put our results into the context of x - ray emission from known transients ( i.e. sne and tdes ) . the detected emission is softer than the typical x - ray spectrum of sne associated with gamma - ray bursts ( @xmath55 , e.g. @xcite ) and normal h - stripped sne ( e.g. @xcite ) , which typically show @xmath55 and a decaying flux with time . a way to sustain luminous x - ray emission over a long time is to invoke the sn shock interaction with a thick medium ( see e.g. sn2014c , @xcite ) . however , the observed x - ray spectrum of h - stripped sne strongly interacting with the environment is even harder ( @xmath56 kev ) , and thus even more different from what we observe at the location of asassn-15lh ( e.g. @xcite ) . it is thus unlikely that a sn shock interaction with the medium is powering both the x - ray and optical / uv emission from asassn-15lh . finally , compared to the only other x - ray source associated to a slsn - i so far , the emission at the location of asassn-15lh is also softer and significantly longer lived ( fig . [ fig : xrayslc ] ) : for the slsn - i scp06f6 , @xcite reports @xmath57 ( or a thermal spectrum with @xmath58 kev ) . the x - ray properties of asassn-15lh are instead more reminiscent of the soft x - ray emission detected in non - jetted tdes . non - jetted tdes detected with _ _ xmm - newton _ , _ chandra _ and , more recently , with _ swift _ show peak luminosities of @xmath59 and very soft spectra that later harden with time on a time - scale of years and with initial temperatures @xmath60 kev ( e.g. @xcite for a recent review ) . as for the tdes asassn-15oi @xcite and asassn-14li @xcite , the x - ray emission is more luminous than what expected based on the extrapolation of the optical / uv blackbody model ( see sec . [ subsec : uv ] ) and a more complex model is needed . in this context asassn-15lh would show the most extreme ratio @xmath61 ( compared to @xmath62 for asassn-14li and @xmath63 for asassn-15oi ) . in fig . [ fig : xrayslc ] we put asassn-15lh on the x - ray luminosity plane of energetic envelope - stripped core - collapse sne ( i.e. grb - sne and slsne ) and tdes . asassn-15lh is @xmath64 times less luminous than the slsn - i scp06f6 and does not experience a similar drop in luminosity . at @xmath65100 days , the x - ray emission at the location of asassn-15lh is more luminous than grb - sne . however , observations obtained around the same epoch by the atca in fig . [ fig : radio ] put deep limits to the radio emission from asassn-15lh @xcite , and rule out the presence of powerful jets seen on axis ( most of the parameter space associated with off - axis grb - like jets is also ruled out ) . also in this case , the luminous and not strongly variable x - ray emission at the location of asassn-15lh , which lacks a luminous radio counterpart , seems to be more in line with observations of non - jetted tdes ( recent examples are asassn-14li , @xcite ; or asassn-15oi , @xcite ) . -0.0 true cm -0.0 true cm = @xmath66 propagating into a circumburst medium with density @xmath67 ( orange lines , @xcite ) . swiftj1644 and swiftj2058 are the two relativistic tdes known to date with radio observations . references : @xcite . , title="fig : " ] we re - processed all the x - ray data collected by the _ swift_-xrt @xcite between june 24 , 2015 until july 22 , 2016 ( total exposure time of @xmath65270 ks ) , following the prescriptions outlined in @xcite . a targeted search for x - ray emission at the location of asassn-15lh identifies the presence of a weak x - ray excess with significance of @xmath68 in the 0.3 - 5 kev range . the significance is reduced to @xmath69 in the 0.3 - 10 kev energy range , consistent with the soft x - ray spectrum suggested by the cxo observations . we infer a background subtracted count - rate of @xmath70 ( 0.3 - 5 kev ) , which corresponds to an unabsorbed 0.3 - 10 kev flux @xmath71 and @xmath72 for a blackbody and power - law spectral model , respectively , and the best - fitting spectral parameters derived from the cxo data . the average flux inferred from _ swift_-xrt observations is thus consistent with the results from the cxo analysis and suggests that the x - ray source at the location of asassn-15lh experienced at most mild temporal variability over the @xmath651 yr of _ swift _ monitoring . we note that flux variations of the order of a factor of a few are consistent with our findings , given the uncertainties affecting both the _ swift_-xrt and the cxo measurements . a delayed onset of the x - ray emission with respect to the optical emission is also clearly allowed , since _ swift_-xrt data started to be collected after optical maximum light . finally , xmm - newton observed asassn-15lh on november 18 , 2015 ( @xmath73 days rest - frame since maximum light ) , six days after our first cxo epoch , which yielded a non - detection . from the xmm - newton observations @xcite infer a @xmath74 confidence level flux limit @xmath75 ( 0.3 - 1 kev ) . we do not confirm the results from @xcite . adopting their inferred count - rate limit of 11 source counts in 9 ks of exposure time with epic - mos2 , their assumed blackbody spectrum with @xmath76 ev , and following the flux calibration procedure outlined in @xcite , we infer a flux limit which is @xmath77 times larger . we re - analyzed the xmm data using standard routines in the scientific analysis system ( sas version 15.0.0 ) and the relative calibration files . we employ a source region of 32@xmath32 radius and extract the background from a source - free region on the same chip . no x - ray source is detected at the location of asassn-15lh . our best constraints are derived from observations obtained with epic - mos2 , with total exposure time of 9 ks ( after removal of time windows contaminated by proton flaring ) and a 3@xmath28 count - rate upper limit of @xmath78 ( 0.3 - 10 kev ) . for the best fitting spectral models derived from cxo detections , we infer the following unabsorbed 0.3 - 10 kev flux limits : @xmath79 and @xmath80 for the blackbody and the power - law spectrum , respectively . xmm observations do not reach the necessary depth to probe the emission from the x - ray source that we detect with cxo and the stacking of _ swift_-xrt observations . a summary of the results from the x - ray observations of asassn-15lh can be found in table [ tab : xray ] . -0.0 true cm days across the _ swift_-uvot bands at the time of the re - brightening . vertical dotted lines mark the times of the cxo observations . , title="fig : " ] we re - analyzed all the _ swift_-uvot observations obtained from june 24 , 2015 until july 22 , 2016 following the prescriptions by @xcite and adopting the updated calibration files and revised zero points by @xcite . each individual frame has been visually inspected and quality flagged . observations with insufficient exposure time have been merged to obtain higher signal - to - noise ratio ( s / n ) images from which we extracted the final photometry reported in table [ tab : uvot ] . we corrected for galactic extinction in the direction of the transient ( @xmath81 mag , @xcite ) and subtracted the host galaxy flux component as constrained by @xcite . we performed a self - consistent flux calibration , and applied a dynamical count - to flux conversion that accounts for the spectral evolution of asassn-15lh , following the procedure outlined in @xcite . finally , we computed a bolometric light - curve of asassn-15lh by integrating the best - fitting blackbody spectra . a partial collection of the _ swift_-uvot photometry of asassn-15lh has already been presented by @xcite , @xcite , @xcite and @xcite . here we update the observations and focus on the presence of significant temporal variability that appears at the time of the re - brightening . figure [ fig : uvvar ] shows the presence of pronounced temporal variability across the uvot bands , and more pronounced at uv wavelengths as first noticed by @xcite . the short variability time scale @xmath82 days at @xmath83 days since first light argues against the interpretation of the sn shock interaction with the surroundings as the main source of energy powering the re - brightening @xcite . for a typical sn shock velocity @xmath84 c ( e.g. @xcite , their fig . 2 ) we do not expect significant temporal variability on @xmath85 days at @xmath86 days , contrary to what we observe in asassn-15lh . this observation motivates us to consider alternative explanations of the uv re - brightening ( sec . [ subsec : rep ] ) . -0.0 true cm we acquired deep multi - epoch optical spectroscopy of asassn-15lh , spanning the time range @xmath87 35350 rest - frame days after maximum light and sampling key points in the late evolution of the transient . a more detailed analysis will be presented in future work ( chornock et al . , in prep . ) . here we concentrate on an analysis of our highest s / n late - time spectrum , which was acquired well after the second re - brightening and when the underlying emission from the host galaxy stellar population is better revealed . we observed asassn-15lh on 2016 june 10 ( @xmath88 days rest - frame since maximum light ) using the low dispersion survey spectrograph ( ldss3c ; @xcite ) on the 6.5 m magellan clay telescope . we obtained three 1800 s exposures using the vph - all grism and a 1-wide slit near the center of the field of view oriented at a position angle of 1283 , which was close to the parallactic angle @xcite . this setup covered the range 380010500 with a resolution of 8.1 . standard iraf tasks were used to perform two - dimensional image processing . we used custom idl scripts to perform flux calibration and correction for telluric absorption using observations of eg131 obtained immediately prior to the object . we took particular care to mitigate the effects of second - order light contamination by combining observations of the standard star taken both with and without an order - blocking filter . however , small residual contamination at long wavelengths ( @xmath89>8000 ) is possible . the resulting spectrum is shown in black in figure [ fig : spec1 ] . numerous stellar absorption features from star light in the host galaxy are visible , as well as two emission peaks near h@xmath90 ( observed wavelengths @xmath658100 ) . several authors have fit the available pre - outburst host galaxy photometry @xcite and have found consistent results . however , the presence of spectral features from the host stellar population has the potential to improve the constraints on the stellar population synthesis , so we used an iterative procedure to incorporate this information while avoiding the flux from the transient . first , we estimated a best - fit blackbody temperature of @xmath91 k at the time of observations from the analysis of the uvot photometry described above . we then subtracted a scaled blackbody spectrum from the observed spectrum under the constraint that the blackbody - subtracted spectrum had to match the observed colors of the host galaxy to obtain an initial estimate of the host - only spectrum . we then used the fast code @xcite to fit the host - only spectrum combined with , and normalized by , the broadband @xmath92 host photometry @xcite . for simplicity , we fixed the metallicity to solar and assumed a @xcite initial mass function and zero internal extinction . we obtained a satisfactory fit using the @xcite stellar models and an exponentially - declining star formation law . the best - fit model has a total stellar mass of 1.2@xmath9310@xmath94 @xmath95 , a current stellar age of 10 gyr , and an @xmath96-folding timescale of 2 gyr , resulting in a current star - formation rate of @xmath650.8 @xmath97 . these numbers are in broad agreement with those reported previously ( e.g , @xcite ) . other choices for the stellar population model produced qualitatively similar results , although usually with smaller current star - formation rates . our best fit for the host is plotted in red in figure [ fig : spec1 ] . we then fitted our observed spectrum as a linear combination of the host galaxy model and a blackbody to find appropriate flux scaling factors . the scaled blackbody is plotted as blue in figure [ fig : spec1 ] and good agreement can be seen with the host - subtracted uvot @xmath98 photometry ( green squares ) interpolated to the date of observation . both the fitted host spectrum and the overplotted host photometry ( gray circles ) have been scaled by a factor of 0.40 from the values for the whole host , which presumably results from the smaller size of our spectroscopic aperture relative to the host as a whole . the @xcite models clearly have narrower features than those visible in our spectrum , so the host template was smoothed with a 10 boxcar function to mimic the combined effects of our spectral resolution and the internal velocity dispersion of the host galaxy . our results are not very sensitive to the width of this smoothing kernel . the sum of the scaled blackbody and the smoothed galaxy template is plotted in magenta in figure [ fig : spec1 ] and is a good match to the observed spectrum in black . @xcite noted two emission peaks near 4000 and 5200 in their late - time spectra of asassn-15lh . however , accurate modeling of the host galaxy stellar component from our late - time spectrum demonstrates that the most prominent broad spectral features detected in the observed ( host plus transient ) late - time spectra have to be attributed to the underlying continuum from the host galaxy star light ( figure [ fig : spec1 ] ) . we do not find unambiguous evidence for broad spectral features associated with the transient at this epoch . small , broad , low - amplitude discrepancies between the observed spectrum and combined fit ( black and magenta lines , respectively ) are present , but it is not yet clear if they represent true spectral features of the transient or limitations in the stellar population synthesis modeling . more observations of the host will be required after the optical transient fades further to more accurately constrain the presence of possible low amplitude broad spectral features in the transient spectrum at late times . without any correction for the host galaxy , the spectrum has the two obvious narrow emission features near 8085 and 8111 ( in air ) noted above , which can be clearly associated with h@xmath90 and [ ] @xmath896583 at @xmath99=0.2320 ( lower - right panel of figure [ fig : spec1])=0.2326 measured from narrow uv absorption lines @xcite . we do not discuss further the implications of this possible velocity offset for the uv absorbers in this work . ] . [ ] @xmath896548 is blended in the blue wing of h@xmath90 . h@xmath100 is only visible in emission after subtraction of the host model . weaker features also appear to be present in the difference spectrum near the [ ] doublet and [ ] @xmath895007 . we searched for [ ] @xmath893727 emission and none is visible , but the s / n of the spectrum is not as high at those wavelengths . inspection of our spectral sequence reveals that the h@xmath90/ [ ] lines are present in several of our higher s / n spectra throughout the evolution of the transient , consistent with a constant low - level contribution that is strongly diluted by light from the transient at earlier times . the peaks of h@xmath90 and [ ] @xmath896583 are of comparable height _ prior _ to subtraction of the host model . strong [ ] /h@xmath90 is a possible sign of ionization by an agn - like continuum . however , after correction for the underlying balmer absorption in our best - fit host model , the ratio decreases to @xmath650.5 . this line ratio , combined with weak [ ] /h@xmath100 and [ ] /h@xmath90 , is consistent with the nebular emission being powered by star formation instead of agn activity @xcite . we caution that these ratios are sensitive to systematic errors in the modeling of the underlying stellar absorption , and in particular the strength of the stellar balmer absorption . if all of the inferred h@xmath90 emission ( flux @xmath652.7@xmath9310@xmath101 erg @xmath102 s@xmath103 ) is powered by star formation , the inferred rate is @xmath650.4 @xmath95 yr@xmath103 @xcite , in rough agreement with that estimated from the host galaxy stellar population fit . @xcite reported h@xmath90 emission from asassn-15lh with a full width at half - maximum ( fwhm ) of 2500 km s@xmath103 , but in our data , it is clear that the reported emission feature is just the narrow nebular h@xmath90 and [ ] from the host blended together at low s / n or low resolution in their data . note that in their highest s / n spectra ( inset of their figure 1 ) , the putative h@xmath90 from the transient is flat - topped or double peaked , consistent with the two strong nebular emission lines of roughly equal height ( inset of our figure [ fig : spec1 ] ) being blended together . we also note that @xcite do not attempt to correct for the contribution from the underlying stellar continuum . therefore , we do not confirm their claim of h@xmath90 emission from the transient itself and the reported velocity fwhm likely reflects the spacing of the two [ ] lines , which are each offset by @xmath651000 km s@xmath103 from the central h@xmath90 emission . @xcite also report a `` bump '' near h@xmath90 at late times , but they do not report a fwhm , so it is not clear if they are are also possibly referring to a noisy detection of the narrow nebular lines . in addition , we re - evaluated the early optical spectra of asassn-15lh and were unable to confirm the likeness to slsne reported by @xcite . the ion , which is commonly observed in slsne @xcite , has a number of distinctive absorption features not observed in asassn-15lh ( fig . [ fig : spec - compare ] ) . the strongest two features centered near 4100 and 4400 are always observed to be of comparable strength and no reasonable values of temperature or density can change this ratio . asassn-15lh only shows the 4100 feature ( fig . [ fig : spec - compare ] , see also @xcite ) . without the accompanying 4400 feature , it is hard to reconcile the proposed association with , and thus the spectroscopic connection to slsne is not robust . the spectral features of asassn-15lh trend redward over time toward declining velocities . this is similar to the spectral evolution of supernovae where it is attributed to an expanding and cooling photosphere . however , unlike supernovae , the features of asassn-15lh do not show traditional p - cyg profiles and become increasingly inconspicuous . for example , the + 30 day spectrum of sn2010gx exhibits pronounced features , whereas the + 39 day asassn-15lh spectrum is nearly featureless ( fig . [ fig : spec - compare ] ) . we explored a variety of possible ions using the highly parameterized spectrum synthesis tool , ` syn++ ` @xcite , to determine whether blending of features could reproduce the spectral features and evolution of asassn-15lh , but were unsuccessful . to our knowledge the only previous examples of spectral features becoming increasingly inconspicuous in the early phases of a supernova involve interaction with dense csm . sn - csm interaction can rescale or `` mute '' the line profile relative to the continuum @xcite . most confirmed instances of sn - csm interaction involve h - rich material that can be readily identified by the presence of h balmer lines that may be narrow ( @xmath26 100 kms@xmath103 ) to broad ( @xmath104 kms@xmath103 ) in width , depending on their origin of formation . the hydrogen - poor slsn iptf13ehe exhibited h@xmath90 balmer emission with broad and narrow components + 251 days post maximum @xcite . however , no such lines are observed in asasn-15lh . interaction with h - poor csm ejected by rapidly rotating pulsational pair instability supernovae is possible @xcite , but the spectroscopic consequences of such interaction are poorly understood @xcite , and the timescales of variability observed in the uv strongly favor against this scenario ( section [ subsec : uv ] ) . a luminous central source over - ionizing expanding ejecta is a speculative , though attractive , scenario that may explain the spectroscopic evolution of asassn-15lh toward a featureless continuum . as there is no precedent for this scenario , the specific spectral signatures are unclear . certainly , the ionizing photons must be extremely energetic for no strong optical or uv lines to be observable . an analogous phenomenon may be variable uv absorption commonly seen in seyfert galaxies @xcite . in some cases variability in the form of absorption components appearing and disappearing , or decreasing outflow velocities @xcite , can result from changes in the ionizing flux @xcite . asassn-15lh may be an extreme version of these processes . , the -4 day spectrum of the slsn 2010gx can be reproduced . ( b ) by contrast , we can not reproduce + 13 day spectrum of asassn-15lh . it clearly misses an accompanying feature around 4400 . ( c ) evolution in the spectra are observed . most conspicuous is the 4100 feature that drifts to longer wavelengths . ( d ) the evolution toward increasingly inconspicuous spectral features is unlike slsne that exhibit increasingly stronger spectral features . here we show the + 30 day spectrum of sn2010gx , which is unlike the nearly featureless + 39 day spectrum of asassn-15lh . data have been retrieved from wiserep @xcite , normalized according to the procedure outlined in @xcite to aid in visual comparison , and were originally published in @xcite and @xcite . ] -0.0 true cm is shown for comparison with blue triangles . , title="fig : " ] although the mechanisms behind slsne powered by a stellar - mass compact object@xmath105such as a magnetar@xmath105 and the tidal disruption and accretion of a star by a smbh do differ significantly , the basic physical process driving the light curves of these events may be similar . a central source of uv / x - ray radiation ( an accreting smbh or the pulsar wind nebula of a rapidly spinning neutron star , ns ) is absorbed by a dense column of gas , and downgraded into optical radiation , where the lower opacity allows the radiation to more readily escape . such a reprocessing " picture has been applied to explain both tdes ( @xcite ) and slsne ( @xcite ) . consider the characteristic timescale of the central engine in a magnetar - powered slsn and tde scenarios . in a magnetar - powered slsn , the central engine lifetime is the magnetic dipole spin - down timescale of the magnetar : @xmath106 where @xmath107 , @xmath108 ms , and @xmath109 are , respectively , the mass , initial spin period , and dipole surface magnetic field strength of the magnetar ( e.g. @xcite ) . the maximum energy of the engine is limited to the rotational energy of the ns , @xmath110 which can vary from @xmath111 erg for the minimum value of the spin - period set by the mass shedding limit , depending on the mass of the ns ( @xcite ) . in order to simultaneously explain the large radiated energy and duration of asassn-15lh with a magnetar , we require a maximally spinning neutron star ( @xmath112 ) and a relatively weak magnetic field @xmath113 g ( e.g. , @xcite , see also @xcite ) . in the tde scenario , the engine lifetime is uncertain , but is commonly attributed to the fall - back time of the most tightly bound stellar debris following the disruption ( e.g. , @xcite ) , @xmath114 where @xmath115 and @xmath116 are the mass of the smbh and the star , respectively , and we have assumed a stellar mass - radius relationship @xmath117 appropriate to lower main - sequence stars . the maximum radiated energy is that liberated by the accretion of the half of the stellar mass which remains bound to the smbh , @xmath118 where the radiative efficiency for geometrically thin accretion varies from @xmath119 , depending on the spin of the smbh and its orientation relative to the angular momentum of the accreting gas . in the tde scenario , the energetics of asassn-15lh are reasonably accommodated by the accretion of a solar - mass star . however , the high mass smbh @xmath120 inferred from the host of asassn-15lh , would appear to predict a long duration of the transient @xmath121 2 year , inconsistent with the much shorter observed decay time of the first peak of a few weeks . this inconsistency could be solved by considering that a main - sequence star can only be disrupted by such a massive black hole if the smbh is _ spinning _ in a prograde direction with respect to the orbit of the disrupted star ( e.g. @xcite ) . precession of the star during the phase of tidal compression due to the bh spin may substantially enhance the spread in the energy distribution of the stellar debris as compared to the newtonian case , by partially aligning the direction of the hydrodynamic bounce with the velocity vector of the star ( @xcite ) . more tightly bound debris has a shorter orbital period , which could significantly speed - up the flare evolution timescale as compared to the newtonian gravity estimate in equation [ eq : tfb ] . though promising , general relativistic numerical simulations are needed to confirm this possibility . in addition to possibly speeding up the flare evolution , the high bh spin required to explain asassn-15lh as a tde would ( i ) naturally result in a large value of the accretion efficiency @xmath122 , accounting for its high luminosity ; ( ii ) possibly aid in the process of debris circularization by inducing precession of the stellar debris streams ( e.g. @xcite , @xcite , @xcite ) . precession of the streams out of the orbital plane due to misaligned bh spin could also help make the geometry of the reprocessing material relatively spherical ( e.g. , @xcite ) , consistent with the low measured optical polarization of asassn-15lh reported by @xcite . in both the magnetar slsn and tde scenarios , uv / x - ray radiation from the central source may ionize its way through the ejecta at late times . this process can result in the direct escape of uv / soft x - ray radiation while having an indirect influence on the observed optical light curve by changing the ejecta opacity ( sec . @xmath123 ) . approximating the ejecta as a homogeneously expanding sphere of mass @xmath124 , velocity @xmath125 cm s@xmath103 , and radius @xmath126 , the neutral column density is @xmath127 where @xmath128 is the neutral fraction . this is much higher than the inferred x - ray absorption column of @xmath129 @xmath102 towards asassn-15lh , requiring a very low neutral fraction if the x - ray source is related to the optical transient . this is consistent with the very low n(hi ) inferred by @xcite from ly-@xmath45 . the ejecta from tdes and slsne are expected to have markedly different chemical composition . in a tde the ejecta has nearly solar composition ( e.g. @xcite ) and the escape of soft x - rays is inhibited primarily by the bound - free opacity of neutral helium ( @xcite ) . by contrast , in a h - poor slsn , x - rays are blocked more severely by neutral oxygen and carbon ( @xcite ) . a central engine with an uv / x - ray luminosity @xmath130 releases an energy @xmath131 in ionizing radiation on a timescale @xmath132 . if the ejecta contains a mass fraction @xmath133 of elements with atomic number @xmath134 , the radiation ionizes its way through the ejecta on a timescale @xmath135 where @xmath136 and @xmath137 k is the temperature of electrons in the recombination layer and @xmath138 is the ratio of absorptive and scattering opacity in the ejecta @xcite . for typical parameters and an engine similar to asassn-15lh with @xmath139 ergs , we have @xmath140 month in the case of a he - rich composition ( @xmath141 ) of a tde - like scenario . by contrast , for a co - rich composition of an exploding massive star ( @xmath142 ) , we have @xmath140 months , making break - out harder to achieve . in the latter case , x - ray break - out is even less likely considering that the k - shell valence electrons of oxygen have a binding energy of @xmath143 kev , while the _ measured _ kev x - ray luminosity of asassn-15lh @xmath144 erg s@xmath103 is much less than the optical / uv luminosity ( in other words , the true value of @xmath145 to use in equation [ eq : tbo ] should be much lower than @xmath146 erg ) . we conclude that an ionization break - out could allow the escape of x - rays in the tde scenario , but is probably not sufficient to do so in the case of a h - poor supernova given the observed soft x - ray spectrum . the ionization of the ejecta reduces the bound - free opacity , allowing the escape to the observer of uv and x - ray radiation with energies _ above _ the ionization threshold . this process is unlikely to explain the observed uv re - brightening by itself , as even the highest frequency uv bands of _ swift_-uvot are below the first ionization energies of the most abundant elements ( h , he , c , o ) . however , an ionization break - out may have an indirect effect on the light - curve via the continuum opacity . at early times the ejecta is largely neutral and the opacity at optical frequencies is dominated by electron scattering , while the opacity at uv frequencies is dominated by line transitions of metals . however , once the ejecta becomes ionized by the central engine , the electron scattering opacity will increase , while the uv opacity will decrease as the ionized atoms have fewer bound - bound transitions . therefore , following ionization break - out we expect a shift of the peak of the spectral energy distribution from optical to uv frequencies . the appeal of this model is that a _ single _ central - engine timescale would naturally reproduce the double - peaked temporal structure of asassn-15lh , which has no analogue in previously observed tde or slsn light curves . as a comparison , the tde model invoked by @xcite combines two luminosity mechanisms , which result into two different timescales . while accurate modeling , beyond the scope of this paper , is necessary to understand if this effect alone can quantitatively explain the observations of asassn-15lh , here we consider a toy model to illustrate the basic principles . for illustrative purposes we use the spin down luminosity of a magnetar as the central source of ionizing photons . in particular , we consider a magnetar light - curve with parameters @xmath147 ms and @xmath148 g , similar to that described in @xcite , and a total ejecta mass of @xmath149 . however , we artificially change the grey opacity from @xmath150 @xmath151 g@xmath103 to @xmath152 @xmath151 g@xmath103 at a time corresponding to ionization break - out of about 50 days . as shown in fig . [ fig : opacity ] , this produces a minimum / flattening in the bolometric light curve , similar to that observed in asassn-15lh . although we have applied the model to a magnetar for concreteness ( and since the process of debris circularization in tdes remains uncertain ) , a similar result applies to the tde case if the central uv / x - ray accretion power smoothly rises on a timescale of a few weeks and then decays @xmath153 at later times . we also caution that a simple change in the grey opacity is unlikely to accurately predict the effect of wavelength - dependent opacity change created by an ionization break - out . we presented evidence for luminous , soft and persistent x - ray emission at the location of asassn-15lh , and discussed its origin in the context of multi - wavelength observations of the transient , which include constraints on its radio emission and early and late - time optical spectroscopy . our re - analysis of early - time spectra does not confirm the robust association of asassn-15lh with slsne claimed by previous studies , and invites us to be open - minded about the nature of asassn-15lh . late - time spectra reveal the emergence of _ narrow _ emission features from the host galaxy , while we associate the most prominent _ broad _ spectral features to the underlying stellar population . no clear evidence is found for broad spectral features associated with the transient at late times . we propose a model that explains the double - peaked temporal structure of asassn-15lh in the optical / uv band as originating from the temporal evolution of the ejecta opacity , which changes as a result of persistent ionizing flux from a long - lived central source ( either a magnetar or an accreting smbh ) . we speculate that the evolution of asassn-15lh towards a featureless spectrum also results from the presence of a persistent central source of ionizing photons . the exceptionally long active time - scale and high luminosity of the ionizing central source powering asassn-15lh ( i.e. months ) is most likely the key physical property that distinguishes asassn-15lh from all the tdes and slsne discovered so far . the optical / uv spectral evolution of asassn-15lh , its peculiar re - brightening and the presence of soft and persistent x - ray emission are indeed unprecedented among slsne and tdes and suggest two scenarios : ( i ) either asassn-15lh is the first member of a class of stellar explosions with extreme properties that are intrinsically rare or that have been overlooked because of the very close location to the host - galaxy nucleus or , alternatively , ( ii ) asassn-15lh results from refreshed nuclear activity of the host - galaxy smbh . in the first scenario the detected x - ray emission is physically unrelated to the transient and most likely originates from the host galaxy nucleus . we thus expect no fading of the x - ray source over the time scales of years . instead , if the x - ray emission is physically associated with the optical / uv transient , then asassn-15lh is unlikely to originate from a stellar explosion and an association with the activity of the host nucleus is favored . in this case , asassn-15lh would be a tde from the most massive spinning smbh observed to date . the fast initial decay timescale of the transient is challenging to understand based on the fall - back timescale of the disrupted star in newtonian gravity , possibly suggesting that bh spin plays a key role in enhancing the energy spread of the disrupted star . asassn-15lh and similar events discovered in the future would then constitute direct probes of matter under strong gravity around very massive , dormant , spinning smbh in galaxies . we emphasize that this scenario predicts significant temporal evolution of the x - ray emission over the next few years , as we expect a tde to have a non - negligible impact on the inner part of the accretion disk even in the case of a pre - existing weak agn . continued deep x - ray monitoring of asassn-15lh will constrain the temporal evolution of the x - ray source and its fading , revealing in this way if the x - ray source is indeed physically related to the optical / uv transient . future x - ray observations thus hold the keys to unveil the true nature of asassn-15lh . r. m. acknowledges partial support from the james arthur fellowship at nyu during the completion of this project and the research corporation for science advancement . bdm gratefully acknowledges support from the nsf ( ast-1410950 , ast-1615084 ) , nasa astrophysics theory program ( nnx16ab30 g ) , the alfred p. sloan foundation , and the research corporation for science advancement . g.m . acknowledges the financial support from the univearths labex programof sorbonne paris cite ( anr10labx0023 and anr11idex000502 ) . the scientific results reported in this article are based on observations made by the chandra x - ray observatory under program go 17500103 , pi margutti , observations ids 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, et al . 2016 , arxiv e - prints , r. m. , et al . 2011 , , 474 , 487 , m. j. 1988 , , 333 , 523 , n. , kasen , d. , guillochon , j. , & ramirez - ruiz , e. 2015 , arxiv e - prints , e. f. , & finkbeiner , d. p. 2011 , , 737 , 103 , a. 2006 , , 648 , l51 , n. , sari , r. , & loeb , a. 2013 , , 435 , 1809 , t. , & woosley , s. e. 2016 , , 820 , l38 , r. c. , nugent , p. e. , & meza , j. c. 2011 , , 123 , 237 , h. , zhang , w. , & macfadyen , a. 2010 , , 722 , 235 , s. , et al . 2016 , science , 351 , 62 , s. , frail , d. a. , krding , e. , & falcke , h. 2013 , , 552 , a5 , s. e. 2010 , , 719 , l204 , l. , et al . 2015 , arxiv e - prints , o. , & gal - yam , a. 2012 , , 124 , 668 , b. a. , berger , e. , margutti , r. , pooley , g. g. , sari , r. , soderberg , a. m. , brunthaler , a. , & bietenholz , m. f. 2013 , , 767 , 152 lllcccccc 57046 - 57591 & swift / xrt & 270 & @xmath154 & bb + & & & @xmath155 & pl + 57338 & cxo & 10 & @xmath156 & pl + & & & @xmath157 & bb + 57344 & xmm & 9 & @xmath158 & pl + & & & @xmath159 & bb + 57369 & cxo & 10 & @xmath160 & pl + & & & @xmath161 & bb + 57438 & cxo & 40 & @xmath162 & pl + & & & @xmath163 & bb + 57619 & cxo & 30 & @xmath164 & pl + & & & @xmath165 & bb + [ tab : xray ] lllcccccccccccc 197.10 & 16.86(0.07 ) & 197.09 & 16.76(0.04 ) & 197.09 & 15.39(0.04 ) & 197.09 & 15.27(0.04 ) & 197.10 & 15.63(0.04 ) & 197.10 & 15.25(0.04 ) + 199.79 & 16.85(0.07 ) & 199.79 & 16.82(0.04 ) & 199.79 & 15.43(0.04 ) & 199.79 & 15.33(0.04 ) & 199.79 & 15.67(0.04 ) & 199.79 & 15.35(0.04 ) + 201.82 & 16.93(0.08 ) & 201.82 & 16.87(0.04 ) & 201.82 & 15.48(0.04 ) & 201.82 & 15.40(0.04 ) & 201.82 & 15.83(0.04 ) & 201.82 & 15.48(0.05 ) + 205.64 & 16.98(0.08 ) & 205.64 & 16.94(0.05 ) & 205.64 & 15.54(0.04 ) & 205.63 & 15.56(0.04 ) & 205.64 & 15.88(0.04 ) & 205.64 & 15.58(0.04 ) + 208.66 & 17.05(0.08 ) & 208.66 & 17.03(0.05 ) & 208.66 & 15.72(0.04 ) & 208.66 & 15.74(0.04 ) & 208.66 & 16.07(0.04 ) & 208.67 & 15.76(0.04 ) + 211.66 & 17.07(0.09 ) & 211.66 & 17.05(0.05 ) & 211.66 & 15.77(0.04 ) & 211.66 & 15.83(0.05 ) & 211.66 & 16.18(0.05 ) & 211.66 & 15.94(0.04 ) + 221.69 & 17.27(0.13 ) & 214.68 & 17.09(0.05 ) & 214.68 & 15.85(0.04 ) & 214.67 & 15.95(0.05 ) & 214.68 & 16.35(0.04 ) & 214.68 & 16.06(0.05 ) + 223.52 & 17.32(0.10 ) & 221.68 & 17.29(0.07 ) & 216.56 & 15.90(0.03 ) & 219.79 & 16.26(0.06 ) & 217.01 & 16.48(0.04 ) & 221.69 & 16.52(0.05 ) + 231.07 & 17.51(0.08 ) & 223.51 & 17.27(0.06 ) & 221.68 & 16.03(0.06 ) & 221.68 & 16.31(0.06 ) & 220.69 & 16.71(0.05 ) & 223.52 & 16.68(0.05 ) + 244.89 & 17.54(0.11 ) & 229.27 & 17.41(0.06 ) & 220.52 & 16.08(0.04 ) & 223.51 & 16.45(0.05 ) & 223.51 & 16.90(0.05 ) & 229.27 & 16.99(0.05 ) + 255.12 & 17.58(0.07 ) & 232.89 & 17.45(0.06 ) & 223.51 & 16.18(0.05 ) & 229.27 & 16.74(0.06 ) & 229.27 & 17.25(0.06 ) & 232.90 & 17.26(0.06 ) + 268.79 & 17.59(0.12 ) & 244.88 & 17.71(0.07 ) & 229.27 & 16.39(0.05 ) & 232.89 & 16.99(0.06 ) & 232.89 & 17.50(0.06 ) & 244.89 & 17.70(0.06 ) + 310.35 & 17.80(0.08 ) & 250.70 & 17.49(0.09 ) & 232.89 & 16.48(0.05 ) & 244.88 & 17.43(0.07 ) & 244.88 & 17.99(0.07 ) & 250.71 & 17.81(0.10 ) + 383.29 & 18.18(0.32 ) & 253.70 & 17.66(0.10 ) & 244.88 & 16.76(0.06 ) & 250.70 & 17.31(0.10 ) & 250.71 & 18.07(0.11 ) & 253.63 & 17.76(0.13 ) + 214.68 & 17.16(0.09 ) & 259.40 & 17.87(0.08 ) & 250.70 & 16.83(0.09 ) & 253.67 & 17.43(0.08 ) & 253.70 & 18.11(0.12 ) & 259.41 & 18.09(0.08 ) + 226.52 & 17.39(0.07 ) & 265.56 & 17.82(0.07 ) & 253.70 & 16.68(0.08 ) & 259.40 & 17.55(0.08 ) & 259.40 & 18.46(0.10 ) & 268.79 & 18.08(0.09 ) + 241.13 & 17.57(0.09 ) & 268.79 & 17.89(0.08 ) & 259.40 & 16.97(0.07 ) & 265.56 & 17.51(0.07 ) & 267.22 & 18.29(0.08 ) & 283.28 & 17.45(0.06 ) + 248.47 & 17.51(0.09 ) & 309.34 & 18.06(0.06 ) & 265.56 & 17.05(0.06 ) & 268.78 & 17.51(0.07 ) & 283.88 & 17.82(0.04 ) & 290.87 & 17.37(0.06 ) + 262.73 & 17.67(0.10 ) & 313.24 & 17.95(0.09 ) & 268.79 & 17.00(0.07 ) & 282.88 & 17.18(0.06 ) & 290.88 & 17.71(0.06 ) & 293.56 & 17.03(0.06 ) + 272.67 & 17.68(0.10 ) & 383.28 & 18.59(0.16 ) & 282.88 & 16.89(0.05 ) & 283.41 & 17.14(0.06 ) & 293.57 & 17.56(0.06 ) & 296.62 & 17.06(0.06 ) + 277.36 & 17.55(0.07 ) & 226.52 & 17.35(0.04 ) & 283.41 & 16.93(0.05 ) & 290.87 & 17.16(0.06 ) & 296.63 & 17.50(0.05 ) & 299.51 & 17.14(0.06 ) + 284.57 & 17.77(0.07 ) & 241.12 & 17.58(0.05 ) & 290.87 & 16.93(0.05 ) & 293.56 & 17.03(0.06 ) & 299.52 & 17.51(0.05 ) & 302.57 & 17.06(0.06 ) + 298.65 & 17.84(0.11 ) & 248.47 & 17.60(0.05 ) & 293.56 & 16.99(0.05 ) & 296.62 & 16.90(0.05 ) & 302.57 & 17.55(0.05 ) & 307.46 & 16.97(0.05 ) + 305.49 & 17.90(0.10 ) & 262.73 & 17.88(0.06 ) & 296.62 & 17.02(0.05 ) & 299.51 & 16.96(0.06 ) & 307.45 & 17.28(0.06 ) & 309.68 & 16.95(0.06 ) + 317.77 & 17.93(0.14 ) & 272.67 & 17.83(0.05 ) & 299.52 & 16.96(0.05 ) & 302.57 & 17.00(0.05 ) & 309.68 & 17.30(0.07 ) & 311.31 & 16.81(0.05 ) + 330.19 & 17.93(0.11 ) & 277.36 & 17.72(0.05 ) & 302.57 & 16.90(0.05 ) & 307.45 & 16.93(0.06 ) & 311.31 & 17.20(0.06 ) & 313.24 & 16.93(0.05 ) + 346.20 & 17.92(0.10 ) & 284.57 & 17.86(0.05 ) & 307.45 & 16.91(0.06 ) & 309.67 & 16.81(0.07 ) & 313.24 & 17.27(0.06 ) & 353.56 & 16.96(0.07 ) + 371.42 & 18.19(0.17 ) & 298.64 & 18.09(0.07 ) & 309.68 & 16.84(0.08 ) & 311.30 & 16.85(0.06 ) & 353.63 & 17.30(0.10 ) & 356.22 & 17.12(0.07 ) + 393.66 & 18.10(0.15 ) & 255.69 & 17.74(0.05 ) & 311.30 & 16.92(0.07 ) & 313.23 & 16.83(0.06 ) & 356.23 & 17.21(0.05 ) & 359.31 & 17.04(0.07 ) + 447.41 & 18.49(0.29 ) & 305.49 & 17.98(0.06 ) & 313.23 & 16.93(0.07 ) & 353.56 & 16.91(0.07 ) & 359.32 & 17.29(0.06 ) & 362.84 & 17.21(0.08 ) + 398.03 & 18.02(0.11 ) & 317.77 & 17.91(0.07 ) & 353.57 & 16.94(0.09 ) & 356.23 & 16.96(0.06 ) & 362.84 & 17.23(0.06 ) & 365.09 & 17.19(0.07 ) + 432.60 & 18.59(0.19 ) & 330.18 & 18.08(0.08 ) & 356.23 & 17.01(0.06 ) & 359.32 & 17.05(0.06 ) & 365.73 & 17.26(0.06 ) & 374.40 & 17.14(0.07 ) + 452.20 & 18.33(0.17 ) & 346.19 & 18.10(0.06 ) & 359.32 & 16.89(0.06 ) & 362.84 & 16.96(0.07 ) & 374.41 & 17.51(0.06 ) & 377.19 & 17.23(0.07 ) + 456.92 & 18.24(0.17 ) & 366.43 & 18.15(0.11 ) & 362.84 & 17.22(0.08 ) & 365.10 & 17.13(0.07 ) & 377.20 & 17.42(0.05 ) & 380.91 & 17.16(0.07 ) + 472.23 & 18.35(0.39 ) & 371.41 & 18.36(0.10 ) & 365.10 & 17.16(0.07 ) & 368.95 & 16.99(0.07 ) & 380.92 & 17.53(0.06 ) & 384.52 & 17.31(0.09 ) + 477.55 & 18.22(0.26 ) & 393.66 & 18.35(0.11 ) & 368.95 & 17.04(0.07 ) & 374.40 & 17.17(0.07 ) & 384.51 & 17.35(0.06 ) & 226.52 & 16.89(0.05 ) + 533.61 & 18.74(0.51 ) & 427.75 & 18.64(0.13 ) & 374.41 & 17.23(0.07 ) & 377.19 & 17.29(0.08 ) & 226.52 & 17.16(0.05 ) & 241.13 & 17.61(0.05 ) + 552.75 & 18.26(0.25 ) & 447.41 & 19.10(0.22 ) & 377.20 & 17.20(0.08 ) & 380.92 & 17.25(0.07 ) & 241.12 & 17.77(0.06 ) & 248.48 & 17.83(0.06 ) + 554.88 & 18.02(0.17 ) & 396.75 & 18.56(0.13 ) & 380.92 & 17.22(0.08 ) & 384.51 & 17.18(0.08 ) & 248.47 & 18.12(0.06 ) & 262.73 & 18.16(0.07 ) + 557.37 & 18.52(0.17 ) & 399.25 & 18.62(0.14 ) & 384.51 & 17.22(0.09 ) & 226.52 & 16.62(0.05 ) & 262.73 & 18.38(0.07 ) & 272.68 & 17.88(0.07 ) + 560.22 & 18.57(0.19 ) & 433.43 & 18.86(0.10 ) & 226.52 & 16.30(0.04 ) & 241.12 & 17.25(0.05 ) & 272.67 & 18.22(0.06 ) & 277.33 & 17.62(0.07 ) + 563.21 & 18.82(0.40 ) & 452.19 & 18.91(0.14 ) & 241.12 & 16.63(0.05 ) & 248.47 & 17.43(0.06 ) & 277.36 & 17.87(0.05 ) & 284.57 & 17.32(0.05 ) + 569.44 & 18.42(0.27 ) & 456.91 & 18.91(0.13 ) & 248.47 & 16.78(0.05 ) & 262.73 & 17.55(0.06 ) & 298.64 & 17.54(0.05 ) & 298.65 & 17.11(0.05 ) + & & 472.22 & 19.14(0.30 ) & 262.73 & 17.08(0.06 ) & 272.67 & 17.39(0.05 ) & 255.70 & 18.29(0.07 ) & 255.66 & 18.02(0.09 ) + & & 477.55 & 19.03(0.21 ) & 272.88 & 17.01(0.07 ) & 277.35 & 17.15(0.05 ) & 305.49 & 17.34(0.05 ) & 305.50 & 17.03(0.06 ) + & & 533.61 & 18.79(0.23 ) & 277.36 & 16.95(0.05 ) & 284.37 & 17.11(0.05 ) & 317.77 & 17.39(0.06 ) & 317.77 & 16.83(0.09 ) + & & 552.75 & 19.01(0.19 ) & 284.37 & 16.97(0.05 ) & 298.64 & 17.01(0.05 ) & 330.19 & 17.28(0.05 ) & 330.19 & 16.97(0.05 ) + & & 554.88 & 19.13(0.16 ) & 298.64 & 16.88(0.05 ) & 255.69 & 17.51(0.06 ) & 346.20 & 17.35(0.05 ) & 346.20 & 16.98(0.06 ) + & & 557.36 & 19.09(0.12 ) & 255.69 & 16.91(0.05 ) & 284.77 & 17.12(0.06 ) & 371.42 & 17.41(0.05 ) & 371.43 & 17.25(0.07 ) + & & 560.22 & 19.12(0.13 ) & 272.60 & 16.96(0.05 ) & 305.48 & 16.87(0.05 ) & 316.24 & 17.27(0.05 ) & 316.23 & 16.98(0.07 ) + & & 563.21 & 19.04(0.21 ) & 284.77 & 16.95(0.05 ) & 317.73 & 17.02(0.06 ) & 319.77 & 17.43(0.06 ) & 319.76 & 17.13(0.07 ) + & & 569.43 & 19.41(0.24 ) & 305.49 & 16.87(0.05 ) & 330.18 & 16.92(0.05 ) & 325.34 & 17.52(0.06 ) & 322.75 & 17.19(0.07 ) + & & & & 317.77 & 16.94(0.06 ) & 346.19 & 17.02(0.05 ) & 328.53 & 17.30(0.05 ) & 325.33 & 17.22(0.07 ) + & & & & 330.18 & 17.00(0.06 ) & 366.39 & 17.08(0.07 ) & 331.06 & 17.24(0.05 ) & 328.52 & 16.96(0.06 ) + & & & & 346.19 & 16.97(0.05 ) & 371.41 & 17.16(0.06 ) & 334.58 & 17.32(0.06 ) & 331.05 & 17.03(0.07 ) + & & & & 366.43 & 17.18(0.08 ) & 316.23 & 16.95(0.06 ) & 343.44 & 17.34(0.06 ) & 334.58 & 17.02(0.07 ) + & & & & 371.41 & 17.17(0.07 ) & 321.23 & 16.98(0.06 ) & 349.22 & 17.27(0.06 ) & 343.43 & 17.06(0.06 ) + & & & & 316.24 & 16.87(0.06 ) & 325.34 & 16.98(0.06 ) & 393.66 & 17.55(0.05 ) & 349.21 & 16.94(0.07 ) + & & & & 319.76 & 17.01(0.06 ) & 328.53 & 16.78(0.06 ) & 427.75 & 18.16(0.07 ) & 393.67 & 17.37(0.07 ) + & & & & 325.34 & 16.92(0.06 ) & 331.05 & 16.82(0.06 ) & 447.41 & 18.06(0.07 ) & 427.75 & 17.83(0.08 ) + & & & & 328.53 & 16.86(0.05 ) & 334.58 & 16.80(0.06 ) & 396.76 & 17.55(0.06 ) & 447.41 & 17.87(0.08 ) + & & & & 331.05 & 16.86(0.06 ) & 343.43 & 16.93(0.06 ) & 399.25 & 17.41(0.06 ) & 396.76 & 17.31(0.07 ) + & & & & 334.58 & 16.97(0.06 ) & 349.21 & 16.88(0.06 ) & 429.65 & 18.06(0.09 ) & 399.25 & 17.29(0.07 ) + & & & & 343.43 & 17.03(0.06 ) & 393.66 & 17.27(0.06 ) & 433.51 & 18.15(0.09 ) & 429.65 & 17.83(0.10 ) + & & & & 349.21 & 16.98(0.06 ) & 427.74 & 17.59(0.07 ) & 437.32 & 18.16(0.08 ) & 435.30 & 17.84(0.08 ) + & & & & 393.66 & 17.37(0.07 ) & 447.41 & 17.73(0.09 ) & 450.04 & 18.09(0.08 ) & 450.04 & 18.00(0.09 ) + & & & & 427.75 & 17.73(0.09 ) & 396.75 & 17.20(0.07 ) & 454.22 & 18.17(0.08 ) & 454.22 & 17.90(0.09 ) + & & & & 447.41 & 18.00(0.12 ) & 399.25 & 17.27(0.07 ) & 457.64 & 18.19(0.10 ) & 456.09 & 17.89(0.09 ) + & & & & 396.75 & 17.36(0.08 ) & 429.64 & 17.74(0.10 ) & 456.08 & 18.03(0.07 ) & 465.55 & 18.01(0.08 ) + & & & & 399.25 & 17.33(0.07 ) & 433.50 & 17.76(0.10 ) & 465.56 & 18.17(0.07 ) & 470.33 & 17.92(0.07 ) + & & & & 431.64 & 17.85(0.10 ) & 437.32 & 17.71(0.09 ) & 470.34 & 18.19(0.06 ) & 472.23 & 18.05(0.16 ) + & & & & 437.32 & 17.75(0.11 ) & 450.03 & 17.82(0.12 ) & 472.23 & 18.18(0.13 ) & 475.67 & 17.94(0.08 ) + & & & & 450.03 & 17.85(0.13 ) & 454.21 & 17.78(0.09 ) & 475.68 & 18.29(0.07 ) & 477.55 & 18.06(0.11 ) + & & & & 454.21 & 17.80(0.11 ) & 457.64 & 17.92(0.12 ) & 477.55 & 18.26(0.10 ) & 480.30 & 18.20(0.09 ) + & & & & 456.91 & 17.81(0.08 ) & 456.07 & 17.83(0.08 ) & 480.30 & 18.39(0.07 ) & 487.48 & 18.29(0.15 ) + [ tab : uvot ]

we present the detection of persistent soft x - ray radiation with @xmath0 at the location of the extremely luminous , double - humped transient asassn-15lh as revealed by _ chandra _ and _ swift_. we interpret this finding in the context of observations from our multiwavelength campaign , which revealed the presence of weak narrow nebular emission features from the host - galaxy nucleus and clear differences with respect to superluminous supernova optical spectra . significant uv flux variability on short time - scales detected at the time of the re - brightening disfavors the shock interaction scenario as the source of energy powering the long - lived uv emission , while deep radio limits exclude the presence of relativistic jets propagating into a low - density environment . we propose a model where the extreme luminosity and double - peaked temporal structure of asassn-15lh is powered by a central source of ionizing radiation that produces a sudden change of the ejecta opacity at later times . as a result , uv radiation can more easily escape , producing the second bump in the light - curve . we discuss different interpretations for the intrinsic nature of the ionizing source.we conclude that , _ if _ the x - ray source is physically associated with the optical - uv transient , asassn-15lh most likely represents the tidal disruption of a main - sequence star by the most massive spinning black hole detected to date . in this case , asassn-15lh and similar events discovered in the future would constitute the most direct probes of very massive , dormant , spinning , supermassive black holes in galaxies . future monitoring of the x - rays may allow us to distinguish between the supernova and tde hypothesis .

the notion of completeness of a set of addition laws for an abelian variety @xmath0 in @xmath8 was introduced by lange and ruppert @xcite . we recall that an addition law is an @xmath9-tuple of bihomogeneous polynomials @xmath10 such that the map @xmath11 determines the group law @xmath12 on an open subset of @xmath13 , and a set of addition laws is complete if these open sets cover @xmath13 ( see definition [ def : complete ] ) . the bidegree @xmath14 of an addition law is the bidegree of the polynomials @xmath15 in @xmath16 and @xmath17 . lange and ruppert prove that the minimal bidegree of any addition law is @xmath18 and determine exact dimensions for the spaces of all addition laws of given bidegree . for an elliptic curve @xmath19 in @xmath3 in weierstrass form , the space of addition laws has dimension @xmath20 , and bosma and lenstra @xcite proved that two suffice for a complete set , determining @xmath21 on all of @xmath22 . in 2007 , edwards introduced a new normal form for elliptic curves @xmath23 with particularly simple rational expression for the group law . after a coordinate scaling , bernstein and lange @xcite descend this model to @xmath24 for @xmath25 , which admits the group law @xmath26 where @xmath27 for @xmath28 and @xmath29 . in addition to giving a precise analysis of the efficiency of this group law , bernstein and lange observe that the addition law is @xmath4-complete over any field @xmath4 in which @xmath30 is a nonsquare ( i.e. the addition law is well - defined on all pairs of @xmath4-rational points of @xmath19 ) . to interpret these rational expressions in terms of projective addition laws as analyzed by lange and ruppert , we note that @xmath31 forms a basis of global sections for the riemann roch space of the divisor at infinity for the pair of coordinate functions @xmath32 , and that this basis determines a projective embedding @xmath33 in @xmath34 which is projectively normal ( see section 2 for precise definitions ) . namely the image curve is of the form @xmath35 the edwards addition law can be interpreted as the bidegree @xmath18 addition law @xmath36 any elliptic curve specified by an affine model has a canonical embedding associated to the complete linear system . consequently , we refer only to such abelian varieties with projective embeddings . in terms of degree 3 models , bernstein , kohel and lange @xcite construct a @xmath4-complete addition law on the family of twisted hessian curves @xmath37 which admit the @xmath4-complete addition laws @xmath38 and @xmath39 over any field @xmath4 in which @xmath40 is not a cube . any such model is equivalent to a weierstrass model by a linear change of variables , which shows that the property of @xmath4-completeness is not special to quartic models in @xmath34 . both the edwards and twisted hessian models share the property that they require a level structure of rational torsion . in analogy with the quartic edwards model , bernstein and lange @xcite demonstrate by example that a general elliptic curve admits a quartic model with @xmath4-complete addition law ( subject to some coefficient being a nonsquare ) , while resorting to a rational expression for an addition law of high bidegree . the second author of the present article gives an elementary characterization of @xmath4-completeness of addition laws of bidegree @xmath18 in terms of the galois action on an associated divisor on the curve ( * ? ? ? * corollary 12 ) . in particular , the property of @xmath4-completeness on elliptic curves is not special . in this paper , we generalize the above results to abelian varieties . we determine new , tight bounds on the size of a complete set of addition laws under any embedding , a generalization of the result of bosma and lenstra @xcite for elliptic curves . moreover we prove that if @xmath4 is any field with infinite absolute galois group , then there exists , for every abelian variety @xmath5 , a projective embedding and an addition law defined for every pair of @xmath4-rational points ( see theorem [ thm : abvar ] ) . our work builds on the elegant paper of lange and ruppert @xcite , in which the authors interpret addition laws on an abelian variety @xmath5 in terms of sections of a certain line bundle @xmath41 on @xmath42 . our key idea is to observe that an addition law associated to a section @xmath43 of @xmath44 with zero divisor @xmath45 is defined on @xmath46 . we obtain a @xmath4-complete addition law by constructing a @xmath4-rational divisor @xmath47 without any @xmath4-rational point . this gives an exact analog of the elliptic curve case studied by the second author @xcite . in section 2 , we recall some definitions and concepts of @xcite , explain more explicitly the link between addition laws on a projective embedding of @xmath5 and sections of @xmath44 , and also deal with the geometric case @xmath48 . for any principally polarized abelian variety of dimension @xmath1 , we give bounds on the cardinality of any complete set of addition laws . in particular we show that its cardinality is at least @xmath2 . in section 3 , we consider the case of a field @xmath4 with infinite absolute galois group , and prove the aforementioned result on existence of a pair consisting of a projective embedding and a @xmath4-complete addition law . in section 4 , we specialize to elliptic curves and jacobians of genus 2 curves over a finite field @xmath4 , noting that the results also extend to other fields ( see remarks [ rem : hilbertianfields ] and [ rem : numberfields ] ) . we prove that there exists a @xmath4-complete addition law for their classical embeddings in @xmath3 and @xmath6 , respectively , as soon as @xmath49 for elliptic curves and @xmath50 for jacobian surfaces . in particular , we exhibit an explicit @xmath4-complete addition law on a weierstrass model of an elliptic curve @xmath19 over @xmath4 when @xmath19 has no nontrivial rational @xmath51-torsion point . let @xmath4 be a field and @xmath5 be an abelian variety of dimension @xmath1 . we assume that @xmath0 is embedded in some projective space @xmath8 over @xmath4 , by a very ample line bundle @xmath52 for @xmath53 an effective divisor , and we denote by @xmath54 the corresponding morphism . we also assume in the sequel that the embedding is projectively normal . recall that @xmath0 is said to be _ projectively normal _ in @xmath8 if for every @xmath55 the restriction map @xmath56 is surjective . this is the case in the classical settings where @xmath57 with @xmath58 an ample line bundle and @xmath59 @xcite . let @xmath60 and @xmath61 be the homogeneous defining ideal for @xmath0 in @xmath62 $ ] and @xmath63 $ ] , respectively . the _ group law _ @xmath64 defined by @xmath65 , can be locally described by bihomogenous polynomials . more precisely , an _ addition law _ @xmath66 of bidegree @xmath14 on @xmath67 is an @xmath9-tuple @xmath68 of elements @xmath69/i_1 \otimes k[y_0,\ldots , y_r]/i_2,\ ] ] which are bihomogeneous of degree @xmath70 and @xmath71 in @xmath72 and @xmath73 , respectively , and for which there exists a nonempty open subset @xmath74 of @xmath13 such that , for all @xmath75 , @xmath76 when @xmath0 is given with a fixed embedding in @xmath8 we may suppress the reference to the embedding @xmath77 and speak of addition laws on @xmath0 . [ def : complete ] a set @xmath78 of addition laws is said to be _ @xmath4-complete _ if for any @xmath4-rational point @xmath79 there is an addition law in @xmath78 defined on an open set @xmath74 containing @xmath80 . this set is said to be _ complete _ if the previous property is true over @xmath81 . if @xmath82 is a singleton , we say the addition law @xmath66 is _ @xmath4-complete _ and _ complete _ when @xmath48 . in ( * ? * lem.2.1 ) , lange and ruppert give the interpretation of the possible addition laws in terms of the sections of certain line bundles . [ prop : addition - law - sheaf ] let @xmath83 be the projection maps on the first and second factor . there is an addition law ( respectively a complete set of addition laws ) of bidegree @xmath14 on @xmath0 with respect to the embedding in @xmath8 determined by @xmath84 if and only if @xmath85 ( respectively the linear system @xmath86 is basepoint - free ) , where @xmath87 we explain how one associates an addition law to a nonzero section @xmath88 in @xmath89 . for @xmath90 , let @xmath91 be the basis given by @xmath92 where @xmath93 are the coordinate functions on @xmath8 . as shown in @xcite , @xmath94 , so @xmath95 is a basis of @xmath96 . for each @xmath97 and @xmath98 , we have @xmath99 now @xmath100 . as the embedding is projectively normal we have @xmath101 then there exists a bihomogeneous polynomial @xmath102 of bidegree @xmath14 such that for all points @xmath98 @xmath103 therefore , if @xmath104 , we have @xmath105 another natural requirement to ask is that @xmath106 be symmetric , i.e. @xmath107^*\mathcal{l } { \cong}\mathcal{l}$ ] , or equivalently @xmath108^*d$ ] , as we can see in the following lemmas . [ lem : inversion ] if @xmath5 is embedded in @xmath8 by a very ample symmetric line bundle @xmath84 ( projectively normal ) , then the inversion map @xmath107 $ ] on @xmath0 is induced by a linear automorphism of @xmath8 . moreover if @xmath109 there is a choice of coordinates such that the inversion acts by @xmath110 on each coordinate . the first statement is a direct consequence of the symmetry of @xmath84 . now fix a basis @xmath111 of @xmath112 and let @xmath113 be the matrix of the coordinates of @xmath107^*t_i$ ] in the basis @xmath111 . the morphism @xmath107 $ ] is induced by an involution of @xmath8 so there exists @xmath114 such that @xmath115 . the neutral element @xmath116 of @xmath117 is a fixed point for @xmath107 $ ] . hence , the vector @xmath118 is an eigenvector of the matrix @xmath113 with eigenvalue @xmath119 . this implies that @xmath120 and if @xmath109 then @xmath121 factors as @xmath122 . this proves that @xmath113 can be diagonalized over @xmath4 with eigenvalues in @xmath123 and the conclusion holds . before considering non - algebraically closed fields , it is natural to consider what happens over @xmath81 . we start by giving an upper bound on the cardinality of a complete set of addition laws . in what follows we define the difference map @xmath124 by @xmath125 , and use the product partial order on bidegree given by @xmath126 if and only if @xmath127 and @xmath128 . for bidegree @xmath129 we denote the line bundle @xmath130 of proposition [ prop : addition - law - sheaf ] by @xmath41 . we begin by recalling a fundamental lemma of lange and ruppert ( * ? ? ? * prop.2.2 , prop.2.3 ) . [ lem : image ] let @xmath84 be an ample line bundle on @xmath0 if @xmath84 is not symmetric then @xmath131 , and 2 . if @xmath84 is symmetric then @xmath41 is isomorphic to @xmath132 and is basepoint - free , and consequently @xmath133 . if @xmath134 then @xmath135 . for @xmath134 , the proof follows the case @xmath136 treated in ( * ? ? ? * prop.2.3 ) . for @xmath129 , lange and ruppert prove in ( * ? ? ? * proposition 2.2 ) that @xmath137 and that @xmath41 is basepoint - free . the equality @xmath133 is an easy consequence of the fact proved in _ loc . cit . _ that @xmath138 is trivial and of the fact that , as @xmath84 is ample , its index is zero . indeed , according to ( * ? ? * theorem 1(ii ) p.95 ) , one then has the isomorphism @xmath139 . the isomorphism of @xmath41 with @xmath140 allows us to consider line bundles on @xmath0 instead of @xmath42 . the following well - known lemma shows that we can always find a symmetric embedding of @xmath141 . let @xmath142 be a principally polarized abelian variety over @xmath81 . there exists a symmetric line bundle which induces the polarization @xmath143 on @xmath0 . suppose that @xmath144 is a line bundle attached to the polarization @xmath143 . we construct a symmetric line bundle @xmath84 algebraically equivalent to @xmath144 . since @xmath144 is algebraically equivalent to @xmath107^*\mathcal{l}'$ ] ( see @xcite ) , there exists @xmath145 such that the translation @xmath146 is algebraically equivalent to @xmath107^*\mathcal{l}'$ ] . let @xmath17 be an element of @xmath147 such that @xmath148 , and set @xmath149 . then @xmath84 is algebraically equivalent to @xmath144 and @xmath150^*\mathcal{l } ' = [ -1]^*\mathcal{l},\ ] ] hence symmetric . suppose that @xmath84 is a symmetric line bundle as in the preceeding lemma . by lemma [ lem : image ] the embedding defined by @xmath151 has a complete set of biquadratic addition laws of cardinality equal to @xmath152 . this gives an upper bound on the minimal size of a complete set of addition laws . we now determine a lower bound . assume @xmath0 is embedded in @xmath8 by a symmetric line bundle . if @xmath78 is a complete set of addition laws on @xmath0 then @xmath153 . suppose that @xmath78 is a complete set of addition laws of bidegree @xmath14 on @xmath0 , and let @xmath154 . by lemma [ lem : inversion ] , the isomorphism @xmath155 \times [ -1 ] : a \longrightarrow \nabla\ ] ] is linear , and so @xmath156 \times [ -1])^*s$ ] is a set of polynomial ( rational ) maps for @xmath157 . it follows that there exists a set @xmath158 of polynomials of degree @xmath159 such that @xmath160 \times [ -1])^*s = \left\ { \big(a_0 q(x_0,\dots , x_r),\dots , a_r q(x_0,\dots , x_r)\big ) \,:\ , q \in i \right\},\ ] ] where @xmath161 . since @xmath78 is complete , the subvariety @xmath162 is empty . on the other hand , its dimension is at least @xmath163 , hence the cardinality of @xmath78 must be at least @xmath2 . although the interval @xmath164 $ ] is quite large , the lower bound shows that there is no complete addition law on any abelian variety of any dimension . for @xmath165 , these bounds show that the minimal size of a complete set of addition laws is either @xmath51 or @xmath20 . an explicit set of cardinality @xmath20 was already given by lange and ruppert ( * ? ? ? * sec.3 ) if @xmath166 , and in @xcite for any characteristic , and bosma and lenstra @xcite proved that a set of minimal cardinality @xmath51 is in fact sufficient . let @xmath84 be a very ample symmetric line bundle defined by an effective @xmath4-rational divisor @xmath53 on @xmath5 . since @xmath167 there exists @xmath88 in @xmath44 such that @xmath168 . as we have seen in section [ sec : addition ] , @xmath88 defines a biquadratic addition law on the complement of @xmath169 . hence it is sufficient that @xmath53 has no @xmath4-rational point for the group law to be @xmath4-complete . note that this is also a necessary condition since a @xmath4-rational point @xmath16 on @xmath53 gives the @xmath4-rational point @xmath170 on @xmath171 . [ thm : abvar ] let @xmath5 be an abelian variety and @xmath172 be an embedding for some @xmath173 . assume that @xmath4 has infinite absolute galois group and let @xmath174 be such that there exists a separable extension @xmath175 of degree @xmath30 over @xmath4 . then there exists an embedding @xmath176 and a @xmath4-complete biquadratic addition law on @xmath177 , with @xmath178 . let @xmath179 be a separable extension , and denote by @xmath180 its distinct galois conjugates in the normal closure of @xmath175 . for @xmath181 , let @xmath182 be the hyperplane in @xmath183 @xmath184 since @xmath174 , the sets @xmath185 are linearly independent over @xmath4 for every @xmath186 and hence @xmath187 is empty . now @xmath188 is a @xmath4-rational divisor , so let @xmath189 and define the divisor @xmath190^*d_0 $ ] . then @xmath53 is a symmetric , effective , @xmath4-rational divisor without @xmath4-rational points . denote by @xmath58 the line bundle associated to the embedding @xmath191 . the line bundle @xmath52 is isomorphic to @xmath192 , so @xmath84 is very ample and provides a projectively normal embedding @xmath193 with a @xmath4-complete biquadratic addition law . by the riemann - roch theorem , the dimension @xmath194 is equal to @xmath195 . in the previous section , a @xmath4-complete ( biquadratic ) addition law is proved to exist , for an embedding of the abelian variety in a projective space of high dimension . when @xmath196 is a finite field and the abelian variety @xmath5 has dimension @xmath197 or @xmath51 , we will show that we can take the embedding to be the classical ones . in what follows , we let @xmath198 denote the frobenius automorphism of @xmath199 . the intersection of @xmath208 with @xmath209 is the group of @xmath210-rational @xmath20-torsion points of @xmath19 so @xmath211 . on the other hand , for all @xmath201 , we have @xmath212 so such a point @xmath213 exists in @xmath214 . [ thm : pointshorizontalline ] let @xmath4 be the finite field @xmath210 with @xmath218 and @xmath219 be an elliptic curve . there exists a @xmath4-complete biquadratic addition law on the weierstrass model of @xmath220 . let @xmath213 be a point as in lemma [ lem : alignedorbit ] and @xmath53 be the divisor given by the sum of the galois conjugates of @xmath213 . it is a @xmath4-rational divisor without @xmath4-rational points . it is not a symmetric divisor but @xmath52 is a symmetric line bundle as @xmath221^*d$ ] . another consequence of the relation @xmath222 is that the embedding associated to @xmath223 is projectively equivalent to the weierstrass model of @xmath19 . [ rem : hilbertianfields ] we use the fact that @xmath4 is a finite field only to prove the existence of the point @xmath213 . it is easy to see that when @xmath4 is a number field , such a point always exists and so the conclusion of theorem [ thm : pointshorizontalline ] still holds . indeed , if @xmath19 is defined by @xmath224 , then , since @xmath4 is hilbertian ( see @xcite ) , there exists @xmath225 such that @xmath226 is irreducible . we can take @xmath227 where @xmath228 is any root of @xmath229 in @xmath81 . in particular , for @xmath230 or @xmath20 , by means of a change of variables we may assume @xmath19 is of the form @xmath231 . moreover , if @xmath19 has no non trivial @xmath4-rational @xmath51-torsion point , then the polynomial @xmath232 is irreducible over @xmath4 and the sum @xmath233 is given by the addition law @xmath234 of bosma and lenstra @xcite : @xmath235 let @xmath238 be a genus @xmath51 curve over a finite field @xmath196 , with hyperelliptic involution @xmath239 . by ( * ? ? ? * proposition 2.3.21 , p.180 ) , there exists a ( not necessarily effective ) @xmath4-rational divisor @xmath240 of degree @xmath197 , such that @xmath241 is equivalent to the canonical divisor @xmath242 of @xmath238 . the divisor @xmath243 , defined as the image of @xmath238 in @xmath244 under the map @xmath245 , is then a @xmath4-rational , ample , symmetric divisor which defines the canonical principal polarization on @xmath244 . for any @xmath246 , we denote by @xmath247 its translation @xmath248 . + the following result can be found for instance in @xcite . let @xmath257 be the quotient by the hyperelliptic involution . note that @xmath213 is a point in @xmath258 such that @xmath259 is in @xmath260 . moreover , no such point exists if and only if @xmath261 , or equivalently if @xmath262 where @xmath263 is the number of ramification points of @xmath264 in @xmath265 . for @xmath266 , this equality contradicts the weil bound @xmath267 , and such a point exists . for each @xmath268 and @xmath269 , there exists at least one genus @xmath51 curve over @xmath210 with no such point @xmath213 . in particular , for @xmath270 , the bound is tight ( for @xmath271 ) : @xmath272 and is satisfied for the curve @xmath273 over @xmath274 . [ thm : genus2 ] let @xmath238 be a genus 2 curve over @xmath210 with @xmath254 . there exists a @xmath4-complete biquadratic addition law for the classical embedding of @xmath244 in @xmath6 determined by @xmath275 . for the canonical divisor @xmath242 and a point @xmath213 as in lemma [ lem : points ] , we define @xmath276 using proposition [ prop : decalage ] , we find @xmath277 @xmath278 by construction , the divisor @xmath279 is ample , symmetric and @xmath4-rational . moreover , since there exists a transitive action on the components @xmath280 , any @xmath4-rational point of @xmath53 must be a point of the intersection @xmath281 which is empty . finally , we have @xmath282 by construction , so @xmath283 and @xmath53 determines a @xmath4-complete addition law for the classical embedding of @xmath244 in @xmath6 determined by @xmath275 . [ rem : numberfields ] this construction can be generalized to other fields . for instance , following the same lines as remark [ rem : hilbertianfields ] , lemma [ lem : points ] has an analogue over number fields @xmath4 . however , a @xmath4-rational divisor @xmath284 of degree @xmath197 may no longer exist , but for the family of curves @xmath238 such as @xmath285 with @xmath286 , we can take @xmath284 to be the divisor with support the point at infinity . in this case , the analogue of theorem [ thm : genus2 ] holds over a number field . arene and cosset have developed an algorithm to construct such an addition law @xcite . the construction of theorem [ thm : genus2 ] uses differences of effective divisors of degree @xmath287 . in general such degree @xmath1 divisors are necessary , since if @xmath238 is a curve of genus @xmath1 and if we define @xmath288 , then by @xcite the intersection @xmath289 is nonempty for any @xmath290 and any @xmath291 . c. arene and r. cosset . construction of a @xmath4-complete addition law on abelian surfaces . in _ arithmetic , geometry , cryptography and coding theory 2011 _ , contemp . math . soc . , providence , ri , 2012 . to appear . daniel j. bernstein and tanja lange . faster addition and doubling on elliptic curves . in _ advances in cryptology asiacrypt 2007 _ , volume 4833 of _ lecture notes in comput . _ , pages 2950 . springer , berlin , 2007 .

we prove that under any projective embedding of an abelian variety @xmath0 of dimension @xmath1 , a complete set of addition laws has cardinality at least @xmath2 , generalizing of a result of bosma and lenstra for the weierstrass model of an elliptic curve in @xmath3 . in contrast , we moreover prove that if @xmath4 is any field with infinite absolute galois group , then there exists , for every abelian variety @xmath5 , a projective embedding and an addition law defined for every pair of @xmath4-rational points . for an abelian variety of dimension 1 or 2 , we show that this embedding can be the classical weierstrass model or the embedding in @xmath6 , respectively , up to a finite number of counterexamples for @xmath7 .

the circumgalactic medium ( cgm ) supplies the gas reservoirs required for galaxy star formation and is replenished via accretion / recycling and outflowing gas . with the likelihood that the cgm can make up about 50% of the baryonic mass bound to galaxies @xcite and could consist of about 50% of the baryons unaccounted for around galaxies @xcite , the cgm s importance in dictating galaxy evolution can not be overstated . determining how the cgm interacts with galaxies is critical to understanding how galaxies evolve . gas accretion from the intergalactic medium , along with previously ejected recycled gas , and large - scale galactic outflows are principal components of theoretical galaxy formation and evolution models @xcite . many studies have shown evidence for the existence of both multi - phase cold accretion / recycling ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and large - scale galactic outflows ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? both models and observations indicated that gas accretion should occur along filaments co - planar to the galaxy disk , whereas outflows are expected to extend along the galaxy projected minor axis . evidence for the geometric preference of inflowing and outflowing gas has already been observed for cool gas traced by absorption @xcite . it was originally reported that the equivalent width is dependent on galaxy inclination @xcite with the absorption kinematics consistent with being coupled to the galaxy angular momentum suggesting co - planar geometry @xcite . @xcite reported a bimodality in the azimuthal angle distribution of gas around galaxies , where cool ( @xmath710@xmath8 k ) dense cgm gas prefers to exist along the projected galaxy major and minor axes ( also see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) with the gas covering fraction being enhanced by as much as 20%30% along these axes . they found that blue star - forming galaxies drive the bimodality while red passive galaxies contain gas along their projected major axis . outflows likely contain more metal enriched gas and higher velocity width absorption profiles since higher equivalent width absorption tends to reside along the projected minor axes of galaxies @xcite . @xcite has shown that absorption profiles with the largest velocity dispersion are associated with blue , face - on galaxies probed along the projected minor axis while the cloud column densities are largest for edge - on galaxies and blue galaxies . @xcite have also shown that absorption detected along the projected minor axes is kinematically consistent with bi - conical outflows . these combined results are consistent with galaxy evolution scenarios where star - forming galaxies accrete new cool / warm co - planer gas within an half - opening angle of about 20@xmath5 , forming new stars and producing metal - enriched galactic scale outflows with half opening angles of 50@xmath5 , while red galaxies exist passively due to reduced gas reservoirs @xcite . the conclusions are drawn from observations conducted using absorption , however both outflowing and infalling gas are expected to be mulit - phased and should be explored using other gas - phase tracers such as , , etc . another standard tracer of the diffuse cgm gas is the o0.1emvi @xmath9 doublet . we know that there is a significant fraction of contained in the cgm @xcite that is typically gravitationally bound within the host galaxy s gravitational potential @xcite . compared to , the doublet can be more difficult to interpret since it can be commonly detected as photo - ionized or collisionally ionized gas . this implies that can trace warm / hot coronal regions surrounding galaxies , which may dictate the formation and destruction of the cool / warm cgm @xcite or trace other multi - phase gas structures . although absorption has been extensively studied in and around galaxies @xcite , the geometrical distribution of absorption around galaxies , which can yield improved understanding of the origins of such gas , has not been thoroughly observationally studied . @xcite attempted to address the azimuthal dependence of using 14 galaxies and found a mostly spatially uniform distribution of absorbing gas out to 300 kpc . there was small hint of a bimodality but was based on only five galaxies that were probed within one viral radius . we aim to further explore the multi - phase inflow and outflow azimuthal distribution using absorption for a large sample of spectroscopically confirmed galaxies . in section [ sec : data ] we present our sample and data reduction . in section [ sec : results ] we present the equivalent width ( ew ) and covering fraction dependence on impact paramater , d , and we compute the mean azimuthal angle probability distribution function for absorbing ( ew>0.1 ) and non - absorbing ( ew<0.1 ) galaxies along with the azimuthal angle covering fraction . we further compare the colors for absorbing and non - absorbing galaxies and show the azimuthal angle probability distribution function depends on galaxy color . in section [ sec : discussion ] , we discuss what can be inferred from the results and concluding remarks are offered in section [ sec : conclusion ] . throughout we adopt an h@xmath10 mpc@xmath11 , @xmath12 , @xmath13 cosmology . limits ( dashed red ) . we have adopted a detection threshold of @xmath140.1 given the distribution of equivalent widths . detections below this threshold are considered as non - absorbers such that our sample is treated with the same level of sensitivity . ( bottom ) the column density distribution is shown for the same sample . our equivalent width bifurcation at 0.1 translates to a column density bifurcation log @xmath15()=14.0 . ] we have constructed a sample of 53 candidate absorbing galaxies with spectroscopically confirmed redshifts ranging between 0.08@xmath1@xmath2@xmath10.67 ( @xmath16 ) within @xmath3 kpc ( 21@xmath1@xmath2@xmath1203 kpc ) of bright background quasars . these galaxies are selected to be isolated such that there are no neighbors within 100 kpc and have velocity separations less than 500 . all these galaxy absorber pairs were identified as part of our `` multiphase galaxy halos '' survey ( from pid 13398 plus from the literature ) . the quasars and quasar fields are selected to have _ hst _ imaging and medium resolution _ hst _ ultraviolet spectra . we discuss the data and analysis below . we have compiled 32 quasars from our survey that have medium resolution ( @xmath1720,000 ) spectra that cover the o0.1emvi @xmath9 doublet for the targeted galaxies . details of the _ hst_/cos and stis observations are contained in table [ tab : hst ] . the data were reduced using the calcos pipeline software . the pipeline reduced data ( ` x1d ' files ) were flux calibrated . in order to increase the spectral signal - to - noise ratio ( @xmath18 ) , individual grating integrations were aligned and co - added using the idl code ` coadd_x1d ' developed by @xcite . since the archival spectra come from different observing programs , our sample shows a range in @xmath18 from 525 per resolution element . as the cos fuv spectra are significantly over - sampled , i.e. , six raw pixels per resolution element , we binned the data by three pixels . binning of data improves @xmath18 per pixel by a factor of @xmath19 . all the measurements and analysis were performed on the binned spectra . continuum normalizations were done by fitting the line - free regions with smooth low - order polynomials . the line spread function ( lsf ) of the cos spectrograph is not a gaussian . for our voigt profile analysis , we adopt the non - gaussian lsf given by @xcite . the lsf was obtained by interpolating the lsf tables at the observed central wavelength for each absorption line and was convolved with the model voigt profile while fitting absorption lines using vpfit software . whenever possible , for each identified system , both the members of o0.1emvi @xmath9 doublet were fitted simultaneously in order to get best fitting component column densities . in cases for which one of the doublets is affected by a blend , we used the unblended transition to constrain the fit parameters . we then generate synthetic profile(s ) for the blended component(s ) for a consistency check . in all cases , we have used the minimum number of components required to get a satisfactory fit with reduced @xmath20 . the model profiles of @xmath21 1031 were used to compute the equivalent width ( ew ) . errors and 3@xmath22 limits on the ews were computed using the error spectrum . the ews and column densities are listed in table [ tab : morph ] . in figure [ fig : limits ] , we show the distribution of @xmath211031 rest - frame equivalent widths and 3@xmath22 equivalent width upper - limits along with the column density distribution . there is a clear transition between absorbing and non - absorbing galaxies at 0.1 with only four absorbers detected below this threshold . in order to treat our data / sample with uniform sensitivity limits , we consider only stronger systems with ew>0.1 as absorbers and all galaxies below this threshold as non - absorbers . the 0.1 bifurcation translates to a column density cut of log @xmath15()@xmath2314 . our final sample consists of 29 absorbing and 24 non - absorbing galaxies . llllclcllrr j012528.84@xmath24000555.9 & 1.075 & 01:25:28.84 & @xmath2400:05:55.93 & cos & g160 m & 13398 & wfpc2 & f702w & 700 & 6619 + j022815.17@xmath24405714.3 & 0.493 & 02:28:15.17 & @xmath2440:57:14.29 & cos & g130 m , g160 m & 11541 & acs & f814w & 1200 & 13024 + j035128.54@xmath24142908.7 & 0.616 & 03:51:28.54 & @xmath2414:29:08.71 & cos & g130 m , g160 m & 13398 & wfpc2 & f702w & 800 & 5949 + j040748.44@xmath24121136.8 & 0.573 & 04:07:48.43 & @xmath2412:11:36.66 & cos & g130 m , g160 m & 11541 & wfpc2 & f702w & 800 & 5949 + j045608.92@xmath24215909.4 & 0.534 & 04:56:08.92 & @xmath2421:59:09.40 & cos & g160 m & 12466,12252 & wfpc2 & f702w & 600 & 5098 + j072153.44@xmath25712036.3 & 0.300 & 07:21:53.45 & @xmath2571:20:36.36 & cos & g130 m , g160 m & 12025 & acs & f814w & 1200 & 13024 + j085334.25@xmath25434902.3 & 0.515 & 08:53:34.25 & @xmath2543:49:02.33 & cos & g130 m , g160 m & 13398 & wfpc2 & f702w & 800 & 5949 + j091440.39@xmath25282330.6 & 0.735 & 09:14:40.39 & @xmath2528:23:30.62 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j094331.61@xmath25053131.4 & 0.564 & 09:43:31.62 & @xmath2505:31:31.49 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j095000.73@xmath25483129.3 & 0.590 & 09:50:00.74 & @xmath2548:31:29.38 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j100402.61@xmath25285535.4 & 0.327 & 10:04:02.61 & @xmath2528:55:35.39 & cos & g130 m , g160 m & 12038 & wfpc2 & f702w & 800 & 5949 + j100902.07@xmath25071343.9 & 0.457 & 10:09:02.07 & @xmath2507:13:43.87 & cos & g130 m , g160 m & 11598 & wfc3 & f625w & 2256 & 11598 + j104116.16@xmath25061016.9 & 1.270 & 10:41:17.16 & @xmath2506:10:16.92 & cos & g160 m & 12252 & wfpc2 & f702w & 1300 & 5984 + j111908.67@xmath25211918.0 & 0.177 & 11:19:08.68 & @xmath2521:19:18.01 & cos & g130 m , g160 m & 12038 & wfpc2 & f606w & 2200 & 5849 + j113327.78@xmath25032719.1 & 0.525 & 11:33:27.79 & @xmath2503:27:19.17 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j113910.79@xmath24135043.6 & 0.557 & 11:39:10.70 & @xmath2413:50:43.64 & cos & g130 m & 12275 & acs & f814w & 520 & 9682 + j121920.93@xmath25063838.5 & 0.331 & 12:19:20.93 & @xmath2506:38:38.52 & cos & g130 m , g160 m & 12025 & wfpc2 & f702w & 600 & 5143 + j123304.05@xmath24003134.1 & 0.471 & 12:33:04.05 & @xmath2400:31:34.20 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j124154.02@xmath25572107.3 & 0.584 & 12:41:54.02 & @xmath2557:21:07.38 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j124410.82@xmath25172104.5 & 1.273 & 12:44:10.82 & @xmath2517:21:04.52 & cos & g160 m & 12466 & wfpc2 & f702w & 1300 & 6557 + j130112.93@xmath25590206.7 & 0.478 & 13:01:12.93 & @xmath2559:02:06.75 & cos & g130 m , g160 m & 11541 & wfpc2 & f702w & 700 & 6619 + j131956.23@xmath25272808.2 & 1.015 & 13:19:56.23 & @xmath2527:28:08.22 & cos & g160 m & 11667 & wfpc2 & f702w & 1300 & 5984 + j132222.46@xmath25464546.1 & 0.375 & 13:22:22.68 & @xmath2546:45:35.22 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j134251.60@xmath24005345.3 & 0.327 & 13:42:51.61 & @xmath2400:53:45.31 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j135704.43@xmath25191907.3 & 0.720 & 13:57:04.43 & @xmath2519:19:07.37 & cos & g160 m & 13398 & wfpc2 & f702w & 800 & 5949 + j154743.53@xmath25205216.6 & 0.264 & 15:47:43.53 & @xmath2520:52:16.61 & cos & g130 m , g160 m & 13398 & wfpc2 & f702w & 1100 & 5099 + j155504.39@xmath25362847.9 & 0.714 & 15:55:04.40 & @xmath2536:28:48.04 & cos & g130 m , g160 m & 11598 & acs & f814w & 1200 & 13024 + j170100.60@xmath25641209.3 & 2.741 & 17:01:00.62 & @xmath2564:12:09.12 & cos & g130 m & 13491 & acs & f814w & 12520 & 10581 + j170441.37@xmath25604430.5 & 0.372 & 17:04:41.38 & @xmath2560:44:30.50 & stis & e140 m & 8015 & wfpc2 & f702w & 600 & 5949 + j213135.26@xmath24120704.8 & 0.501 & 21:31:35.26 & @xmath2412:07:04.79 & cos & g160 m & 13398 & wfpc2 & f702w & 600 & 5143 + j213811.60@xmath24141838.0 & 1.900 & 21:38:11.60 & @xmath2414:18:38.0 & cos & g130 m , g160 m & 13398 & wfpc2 & f702w & 1400 & 5343 + j225357.74@xmath25160853.6 & 0.859 & 22:53:57.74 & @xmath2516:08:53.56 & cos & g130 m , g160 m & 13398 & wfpc2 & f702w & 700 & 6619 we further required that all 32 quasar fields were imaged with high - resolution spaced - based cameras such that we were able to adequately model the morphological parameters and orientations of all 53 galaxies . details of the _ hst _ imaging observations using acs , wfc3 , and wfpc2 for a range of filters are listed in table [ tab : hst ] . the wfpc2/_hst _ images were reduced using the wfpc2 associations science products pipeline ( waspp ) ( see * ? ? ? * ) . the acs and wfc3 data reduction was carried out using the drizzlepac software @xcite . if enough frames were present , cosmic rays were removed during the multidrizzle process otherwise we used lacosmic @xcite . galaxy photometry was performed using the source extractor package ( sextractor ; * ? ? ? * ) with a detection criterion of 1.5 @xmath22 above background . the @xmath26 magnitudes in each filter were measured using the wfpc2 vega zeropoints @xcite , which were then converted to ab magnitudes ( see * ? ? ? * ) , while the acs , and wfc3 zero points are based upon the ab system . the @xmath26 , and its corresponding filter , for each galaxy is listed in table [ tab : morph ] . galaxy morphological parameters and orientations were quantified by fitting a two - component ( bulge+disk ) model using gim2d ( galaxy image 2d ; * ? ? ? we fit the surface brightness of the disk component with an exponential profile and we fit the bulge component with a srsic profile @xcite where the srsic index may vary between @xmath27 . this technique has been successfully applied in previous works @xcite . the gim2d outputs were manually inspected to see if models were realistic representations of the observed galaxies . to model the galaxies , gim2d extracts `` portrait size '' images from parent _ hst _ images with an area 10 times larger than the @xmath28 galaxy isophotal area such that an accurate background can be computed . figure [ fig : absimage ] shows the portraits of the absorbing and non - absorbing galaxies . all galaxies appear to have similar sizes since they are shown for an area 10 times larger than the @xmath28 galaxy isophotal area . the orientations of the images are that of the parent _ hst _ and are arbitrary . note there is a wide range of galaxy morphological types with the dominant population being disk galaxies . during the gim2d process , the models are convolved with the point spread function ( psf ) . for wfpc-2 , we modeled the psf at the appropriate locations on the wfpc-2 chip using tiny tim @xcite as performed by @xcite . the psfs for wfc3 and acs depend both on time and position on the chip and , in addition , the images also contain significant geometrical distortions . so for wfc3 and acs , we used tiny tim to create the psfs and place them into blank frames every 500 pixels , with the same size and header parameters as those of the real flat - fielded individual exposures which were then reduced following the same procedure as for the data . the psf that was closest to the galaxy of interest was used for the gim2d modeling . the galaxy properties are listed in table [ tab : morph ] . we adopt the convention of the azimuthal angle @xmath29 to be along the galaxy major axis and @xmath30 to be along the galaxy minor axis . we have included in our analysis the full range of galaxy inclinations present in our sample . the sample contains only three galaxies with @xmath31 degrees . we find that the inclusion or exclusion of these three galaxies does not change our main results . we also note that , in the case of absorption , the geometric distribution of the low - ionization cgm is only weakly dependent upon galaxy inclination @xcite , but is strongly dependent upon azimuthal angle @xcite . the combined @xmath0 distribution of absorbers and non - absorbers is shown to be consistent with a uniform random distribution using a ks test with p(ks)=0.635 the impact parameters , @xmath32 , are computed using the galaxy and quasar isophotal centroids determined by sextractor . the uncertainty in @xmath32 is computed from the pixel offset of the galaxy sextractor isophotal center and the center of the gim2d galaxy model , which is typically @xmath33 pixels . an additional @xmath340.05 pixel uncertainty is included for centroiding error of the quasar , which is based upon centroiding errors of unresolved sources in the our images . 1031 equivalent width versus impact parameter with 3@xmath22 upper limits shown as red arrows . the horizontal dashed line shows bifurcation value of 0.1 separating absorbing and non - absorbing galaxies . there are four detections below this limit , however they are treated as non - detections for our analysis but shown for completeness . ( bottom ) the covering fraction as a function of impact parameter for an equivalent sensitivity of 0.1 . the horizontal error bars indicate the full range in impact parameter within each bin . the covering fraction 1@xmath22 errors are derived from binomial statistics @xcite . ] lllcccccrrrrr j035128 & 03:51:28.933 & @xmath2414:29:54.31 & 0.2617 & 1 & f702w & 21.0 & 2.3@xmath35 & @xmath36 & @xmath37 & @xmath38 & @xmath39 & @xmath40 + j040748 & 04:07:48.481 & @xmath2412:12:11.13 & 0.3422 & 2 & f702w & 21.5 & @xmath41 & @xmath42 & @xmath43 & @xmath44 & @xmath45 & @xmath46 + j040748 & 04:07:43.930 & @xmath2412:12:08.49 & 0.1534 & 2 & f702w & 18.5 & @xmath41 & @xmath47 & @xmath48 & @xmath49 & @xmath50 & @xmath51 + j072153 & 07:21:51.403 & @xmath2571:20:10.80 & 0.2640 & 3 & f814w & 19.0 & @xmath41 & @xmath52 & @xmath53 & @xmath54 & @xmath55 & @xmath56 + j072153 & 07:21:54.962 & @xmath2571:20:11.20 & 0.2490 & 3 & f814w & 22.1 & @xmath41 & @xmath57 & @xmath58 & @xmath59 & @xmath60 & @xmath61 + j085334 & 08:53:35.160 & @xmath2543:48:59.81 & 0.4402 & 1 & f702w & 20.6 & 1.8@xmath35 & @xmath62 & @xmath63 & @xmath64 & @xmath65 & @xmath66 + j085334 & 08:53:33.384 & @xmath2543:49:03.97 & 0.1635 & 1 & f702w & 18.8 & 1.8@xmath35 & @xmath67 & @xmath68 & @xmath69 & @xmath70 & @xmath71 + j085334 & 08:53:36.881 & @xmath2543:49:33.32 & 0.2766 & 1 & f702w & 20.9 & @xmath41 & @xmath72 & @xmath73 & @xmath74 & @xmath75 & @xmath76 + j085334 & 08:53:34.481 & @xmath2543:49:37.51 & 0.0872 & 1 & f702w & 17.5 & 2.1@xmath35 & @xmath77 & @xmath78 & @xmath79 & @xmath80 & @xmath81 + j094331 & 09:43:29.210 & @xmath2505:30:41.75 & 0.1431 & 4 & f814w & 17.5 & 2.82@xmath82 & @xmath83 & @xmath84 & @xmath85 & @xmath86 & @xmath87 + j094331 & 09:43:33.789 & @xmath2505:31:22.26 & 0.2284 & 4 & f814w & 18.6 & 2.24@xmath82 & @xmath88 & @xmath89 & @xmath90 & @xmath91 & @xmath92 + j111908 & 11:19:06.675 & @xmath2521:18:29.56 & 0.1383 & 5 & f606w & 17.7 & @xmath41 & @xmath93 & @xmath94 & @xmath95 & @xmath96 & @xmath97 + j113910 & 11:39:08.330 & @xmath2413:50:45.64 & 0.2198 & 1 & f702w & 22.4 & 2.1@xmath35 & @xmath98 & @xmath99 & @xmath100 & @xmath101 & @xmath102 + j130112 & 13:01:20.123 & @xmath2559:01:35.72 & 0.1967 & 1 & f702w & 20.9 & 1.6@xmath35 & @xmath103 & @xmath104 & @xmath105 & @xmath106 & @xmath107 + j131956 & 13:19:55.729 & @xmath2527:28:12.88 & 0.6719 & 6 & f702w & 21.3 & 1.5@xmath35 & @xmath108 & @xmath109 & @xmath110 & @xmath111 & @xmath112 + j134251 & 13:42:52.235 & @xmath2400:53:43.10 & 0.2013 & 4 & f814w & 19.7 & 2.45@xmath82 & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath117 + j135704 & 13:57:03.290 & @xmath2519:18:44.41 & 0.4295 & 1 & f702w & 21.9 & @xmath41 & @xmath118 & @xmath119 & @xmath120 & @xmath121 & @xmath122 + j135704 & 13:57:02.914 & @xmath2519:18:55.51 & 0.4406 & 1 & f702w & 21.3 & 1.8@xmath35 & @xmath123 & @xmath124 & @xmath125 & @xmath126 & @xmath127 + j154743 & 15:47:45.561 & @xmath2520:51:41.37 & 0.0949 & 1 & f702w & 21.4 & 1.21@xmath82 & @xmath128 & @xmath129 & @xmath130 & @xmath131 & @xmath132 + j154743 & 15:47:41.642 & @xmath2520:52:39.39 & 0.1343 & 1 & f702w & 19.7 & 3.04@xmath82 & @xmath133 & @xmath134 & @xmath135 & @xmath136 & @xmath137 + j170100 & 17:01:03.261 & @xmath2564:11:59.89 & 0.1900 & 7 & f814w & 22.4 & @xmath41 & @xmath138 & @xmath139 & @xmath140 & @xmath141 & @xmath142 + j170441 & 17:04:37.106 & @xmath2560:44:20.35 & 0.3380 & 1 & f702w & 21.6 & 2.0@xmath35 & @xmath143 & @xmath144 & @xmath145 & @xmath146 & @xmath147 + j213811 & 21:37:48.702 & @xmath2414:33:16.51 & 0.1857 & 1 & f702w & 19.5 & @xmath41 & @xmath148 & @xmath149 & @xmath150 & @xmath151 & @xmath152 + j213811 & 21:37:45.083 & @xmath2414:32:06.27 & 0.0752 & 1 & f702w & 18.5 & @xmath41 & @xmath153 & @xmath154 & @xmath155 & @xmath151 & @xmath152 + j012528 & 01:25:28.257 & @xmath2400:06:08.20 & 0.3787 & 8 & f702w & 20.7 & 1.3@xmath35 & @xmath156 & @xmath157 & @xmath158 & @xmath159 & @xmath160 + j012528 & 01:25:27.671 & @xmath2400:05:31.39 & 0.3985 & 8 & f702w & 19.7 & 1.8@xmath35 & @xmath161 & @xmath162 & @xmath163 & @xmath164 & @xmath165 + j035128 & 03:51:27.892 & @xmath2414:28:57.88 & 0.3567 & 1 & f702w & 20.7 & 0.28@xmath35 & @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 + j040748 & 04:07:49.020 & @xmath2412:11:20.76 & 0.4942 & 2 & f702w & 22.6 & @xmath41 & @xmath171 & @xmath172 & @xmath173 & @xmath174 & @xmath175 + j045608 & 04:56:08.913 & @xmath2421:59:29.00 & 0.4838 & 9 & f702w & 20.4 & 1.78@xmath35 & @xmath176 & @xmath177 & @xmath178 & @xmath179 & @xmath180 + j045608 & 04:56:08.820 & @xmath2421:59:27.40 & 0.3818 & 1 & f702w & 20.7 & 1.66@xmath35 & @xmath181 & @xmath182 & @xmath183 & @xmath184 & @xmath185 + j091440 & 09:14:41.759 & @xmath2528:23:51.18 & 0.2443 & 4 & f814w & 19.6 & 1.24@xmath82 & @xmath186 & @xmath187 & @xmath188 & @xmath189 & @xmath190 + j094331 & 09:43:30.671 & @xmath2505:31:18.08 & 0.3530 & 4 & f814w & 21.2 & 1.17@xmath82 & @xmath191 & @xmath192 & @xmath193 & @xmath194 & @xmath195 + j094331 & 09:43:32.376 & @xmath2505:31:52.15 & 0.5480 & 4 & f814w & 21.0 & 0.96@xmath82 & @xmath196 & @xmath197 & @xmath198 & @xmath199 & @xmath200 + j095000 & 09:50:00.863 & @xmath2548:31:02.59 & 0.2119 & 4 & f814w & 18.0 & 2.74@xmath82 & @xmath201 & @xmath202 & @xmath203 & @xmath204 & @xmath205 + j100402 & 10:04:02.353 & @xmath2528:55:12.50 & 0.1380 & 1 & f702w & 21.9 & @xmath41 & @xmath206 & @xmath207 & @xmath208 & @xmath209 & @xmath210 + j100902 & 10:09:01.579 & @xmath2507:13:28.00 & 0.2278 & 4 & f625w & 20.1 & 1.39@xmath35 & @xmath211 & @xmath212 & @xmath213 & @xmath214 & @xmath215 + j104116 & 10:41:17.801 & @xmath2506:10:18.97 & 0.4432 & 10 & f702w & 20.9 & 2.81@xmath35 & @xmath216 & @xmath217 & @xmath218 & @xmath219 & @xmath220 + j113327 & 11:33:28.218 & @xmath2503:26:59.00 & 0.1545 & 4 & f814w & 19.2 & 1.29@xmath82 & @xmath221 & @xmath222 & @xmath223 & @xmath224 & @xmath225 + j113910 & 11:39:11.520 & @xmath2413:51:08.69 & 0.2044 & 1 & f702w & 20.0 & 2.30@xmath35 & @xmath226 & @xmath227 & @xmath228 & @xmath229 & @xmath230 + j113910 & 11:39:09.801 & @xmath2413:50:53.08 & 0.3191 & 1 & f702w & 20.6 & 1.60@xmath35 & @xmath231 & @xmath232 & @xmath233 & @xmath234 & @xmath235 + j113910 & 11:39:09.533 & @xmath2413:51:31.46 & 0.2123 & 1 & f702w & 20.0 & 2.10@xmath35 & @xmath236 & @xmath237 & @xmath238 & @xmath239 & @xmath240 + j121920 & 12:19:23.469 & @xmath2506:38:19.84 & 0.1241 & 5 & f702w & 18.2 & 1.20@xmath35 & @xmath241 & @xmath242 & @xmath243 & @xmath244 & @xmath245 + j123304 & 12:33:04.084 & @xmath2400:31:40.20 & 0.3185 & 4 & f814w & 20.2 & 1.38@xmath82 & @xmath246 & @xmath247 & @xmath248 & @xmath249 & @xmath250 + j124154 & 12:41:53.731 & @xmath2557:21:00.94 & 0.2053 & 4 & f814w & 19.9 & 1.42@xmath82 & @xmath251 & @xmath252 & @xmath253 & @xmath254 & @xmath255 + j124154 & 12:41:52.410 & @xmath2557:20:43.28 & 0.2178 & 4 & f814w & 20.1 & 1.53@xmath82 & @xmath256 & @xmath257 & @xmath258 & @xmath259 & @xmath260 + j124410 & 12:44:11.045 & @xmath2517:21:05.05 & 0.5504 & 1 & f702w & 21.7 & 1.34@xmath35 & @xmath261 & @xmath262 & @xmath263 & @xmath264 & @xmath265 + j131956 & 13:19:55.773 & @xmath2527:27:54.84 & 0.6610 & 11 & f702w & 21.6 & 1.45@xmath35 & @xmath266 & @xmath267 & @xmath268 & @xmath269 & @xmath270 + j132222 & 13:22:22.470 & @xmath2546:45:45.98 & 0.2142 & 4 & f814w & 18.6 & 2.02@xmath82 & @xmath271 & @xmath272 & @xmath273 & @xmath274 & @xmath275 + j134251 & 13:42:51.866 & @xmath2400:53:54.07 & 0.2270 & 4 & f814w & 18.2 & 1.59@xmath82 & @xmath276 & @xmath277 & @xmath278 & @xmath279 & @xmath280 + j135704 & 13:57:04.539 & @xmath2519:19:15.15 & 0.4592 & 1 & f702w & 21.4 & 1.40@xmath35 & @xmath281 & @xmath282 & @xmath283 & @xmath284 & @xmath285 + j155504 & 15:55:05.295 & @xmath2536:28:48.46 & 0.1893 & 4 & f814w & 18.5 & 1.43@xmath82 & @xmath286 & @xmath287 & @xmath288 & @xmath289 & @xmath290 + j213135 & 21:31:35.635 & @xmath2412:06:58.56 & 0.4300 & 12 & f702w & 20.7 & 2.06@xmath35 & @xmath291 & @xmath292 & @xmath293 & @xmath294 & @xmath295 + j225357 & 22:54:00.417 & @xmath2516:09:06.82 & 0.3526 & 1 & f702w & 20.6 & 1.30@xmath35 & @xmath296 & @xmath297 & @xmath298 & @xmath299 & @xmath300 in figure [ fig : d ] we show the ew distribution as a function of @xmath32 for our sample . note that the dashed horizontal line at 0.1 is our bifurcation between absorbing and non - absorbing galaxies , although we have shown four detections that reside below this threshold for completeness only . there is an anti - correlation with ew and @xmath32 , which is consistent with previous studies that typically show this anti - correlation between @xmath15 ( ) and @xmath32 @xcite . a kendall-@xmath301 rank correlation test shows that the ew and @xmath32 are anti - correlated at the 2.7@xmath22 level for absorbers with ew>0.1 . if limits are included , then the rank correlation test shows no correlation ( 0.9@xmath22 ) . the lack of a statistically significant correlation for the full sample suggests that non - absorbers exist at all impact parameters and that gaseous halos have a non - unity covering fraction , even at low @xmath32 . also shown in figure [ fig : d ] is the covering fraction as a function of @xmath32 . the covering fraction is defined as the ratio of the number of absorbers to the sum of absorbers and non - absorbers in each impact paramater bin . the covering fraction 1@xmath22 errors are derived from binomial statistics @xcite . the covering fraction starts high at 80% within @xmath302 kpc and decreasing to 33% at @xmath303200 kpc , which is consistent with previous studies ( e.g. , * ? ? ? * ; * ? ? ? we follow the method of @xcite to study the relationship between absorption and azimuthal angle ( @xmath0 ) . they discuss that direct binning of the azimuthal angles , an approach taken by some authors , applies only when the measured uncertainties are smaller than the bin size and each galaxy can be quantized in single @xmath0 bin , which is not always the case . additional complications occur when the uncertainties are asymmetric , as they are for gim2d model output parameters . we model the measured azimuthal angles and their uncertainties as asymmetric univariate gaussian probability distribution functions ( see * ? ? ? * ) , thus creating an azimuthal angle probability distribution function ( pdf ) for each galaxy . from the continuous azimuthal pdfs , we then compute the mean pdf combining all galaxies as function of @xmath0 . the mean pdf represents the probability of detecting absorption at a given @xmath0 . this technique provides higher weight per azimuthal angle bin for galaxies with well determined @xmath0 . however , even the less robustly modeled galaxies provide useful information ; the method is similar to stacking low signal - to - noise spectra or images to search for a coherent signal . in figure [ fig : main ] , we present the binned mean azimuthal angle pdf for the 29 absorbing ( ew@xmath304 ) and 24 non - absorbing ( ew@xmath305 ) galaxies . the binned pdfs are normalized such that the total area is equal to unity : this provides an observed frequency for each azimuthal bin . the shaded regions about each bin are the @xmath306 deviations computed from a bootstrap analysis produced by resampling the galaxies in the @xmath0 distribution with replacement 10,000 times . the top panel of figure [ fig : main ] shows that the pdf for absorbing galaxies is highest along the projected major axis ( @xmath307@xmath5 ) within 20@xmath5 , drops dramatically , and then slowly rises towards another maximum along projected minor axis ( @xmath308@xmath5 ) . the middle panel of figure [ fig : main ] shows that the non - absorbers almost have the opposite trend as the absorbers . the frequency of non - absorbers occur in a narrow window along the projected major axes within 10@xmath5 and then drops dramatically . they have the highest representation at intermediate @xmath0 between 2060@xmath5 where absorbers actually are at a minimum and have little - to - no representation along the projected minor axis . following the methods of @xcite , we used the chi - squared statistic on the @xmath309 binned @xmath0 distributions to test the null hypothesis that the distribution of azimuthal angles for non - absorbers and absorbers were drawn from the same population . the null - hypothesis was ruled out at the 3.7@xmath22 significance level . to examine sensitivity to binning , we also tested bin sizes of @xmath310 , @xmath311 , and @xmath312 . the significance levels were all greater than 3.1@xmath22 . the data show that the presence or absence of gas is highly driven by galaxy orientation ; absorption is more common along the major and minor axes while the lack of absorption is common at intermediate @xmath0 . we discuss how these results are consistent with inflow and outflow models and how they compare to the azimuthal dependence seen by @xcite in the next section ( also see * ? ? ? * ; * ? ? ? * ; * ? ? ? distribution for absorbing galaxies . the binned pdfs are normalized such that the total area is equal to unity : this provides an observed frequency in each azimuthal bin . absorption is detected with increased frequency towards the major and minor axes . shaded regions are 1@xmath22 errors produced by bootstrapping the sample . ( middle ) @xmath0 distribution for non - absorbing galaxies . note the opposite effect , where non - absorbers peak where absorbers are near a minimum . ( bottom ) the gas covering fraction and 1@xmath22 errors . ] we assume that the width of the peaks of the pdf provides constraints to the geometry of outflowing and inflowing gas . the peak at @xmath29 suggests that accreting gas is found within @xmath313 of the projected galaxy major axis plane . one may also interpret that the deficit of absorption and the surplus of non - absorbers within @xmath410@xmath5 of the major axis as physical and could highlight that may be much colder gas may reside in this area ( no such deficiency of absorbers and surplus of non - absorbers was observed for the cool / dense gas ) while more diffuse gas surrounds the cool dense as indicated by the peak at 20@xmath5 . the slow rise of the absorber pdf and the sharp decline of the non - absorber pdf along the minor axis may suggest that outflowing gas could occur within a half - opening angle as small as 30@xmath5 ( from 6090@xmath5 ) or even larger to 50@xmath5 since the covering fraction in outflows likely decreases with increasing @xmath0 . in the bottom of figure [ fig : main ] , we further show the gas covering fraction as a function of azimuthal bin . as defined in the literature , we define the covering fraction to be @xmath314 , where @xmath315 is the number of absorbers and @xmath316 is the number of non - absorbers . we compute the covering fractions and their 1@xmath22 errors by bootstrapping the aforementioned ratio for each azimuthal bin 10,000 times . this first presentation of the covering fraction as a function of azimuthal angle shows , as previously mentioned , a deficit of absorption with a covering fraction of 35% within 10@xmath5 of the projected major axis . the covering fraction then peaks at 85% at 20@xmath5 before dropping back to 35% . the covering fraction of is again at a maximum above 80% within 30@xmath5 of the projected minor axis . with our current sample , the covering fraction is highly dependent on @xmath0 and is the highest along the projected major and minor axes . the azimuthal angle averaged covering fraction of our sample is 52% . we explore how the incidence of absorbers and non - absorbers depend on galaxy inclination and how the @xmath0 pdfs depend on inclination . in figure [ fig : i ] , we show the distribution of absorbers ( black ) and non - absorbers ( red ) as a function of @xmath0 and @xmath317 . this figure shows that regardless of galaxy inclination , the absorbers have a preference for existing along the projected major and minor axes . the same is true for the non - absorbers ; they continue to have a preference to exist at intermediate @xmath0 . we do note that the distribution of non - absorbers has a larger @xmath0 range at high galaxy inclination angles . to further explore this , we show the @xmath317 pdfs for absorbers and non - absorbers in figure [ fig : i ] . the galaxy inclination pdfs for absorbers and non - absorbers have very similar / consistent distributions except for the most edge - on systems where a non - absorbers are more frequent that absorbers . a chi - squared statistic on the @xmath309 binned @xmath317 distributions shows the null - hypothesis was ruled out at the 3.0@xmath22 significance level . however , the chi - squared statistic on the pdfs below @xmath318 shows that the null - hypothesis was ruled out at the 1.0@xmath22 significance level . therefore , the largest statistical difference is observed at high galaxy inclination . this is further reflected in the covering fractions as a function of galaxy inclination shown in figure [ fig : i ] . the only significant change in covering fraction as a function of @xmath317 is shown as a drastic decrease at high inclinations . ) and galaxy inclination ( @xmath319 . note that the bi - modality in @xmath0 is present for all values of @xmath317 . however , non - absorbers exist over a broader @xmath0 range for edge - on galaxies ( @xmath320 ) . ( middle ) @xmath317 distribution for absorbing and non - absorbing galaxies . the binned pdfs are normalized such that the total area is equal to unity : this provides an observed frequency in each inclination bin . shaded regions are 1@xmath22 errors produced by bootstrapping the sample . absorbers and non - absorbers have consistent distributions except above ( @xmath320 ) . ( bottom ) the gas covering fraction and 1@xmath22 errors . the drop in covering fraction is likely due to a minimized outflow / inflow cross - section geometry at high galaxy inclination . ] ) and along the projected minor ( @xmath321 ) axes . the mean value and its standard deviation is also shown . the mean equivalent widths , with the standard error in the mean , along the major and minor axes are @xmath322 and @xmath323 , respectively , are shown . note that the highest equivalent width systems are found along the projected minor axis . ] we further explore how the strength of the absorption is dependent on the projected location around the galaxy . in figure [ fig : diff ] , we show the distribution of ew separated into projected major and minor axis bins bifurcated at 45@xmath5 . the ew distributions for the projected major axis ( for 14 galaxies ) and minor axis ( for 15 galaxies ) differ slightly with stronger equivalent width systems found along the projected minor axis . the mean equivalent widths , with the standard error in the mean , along the major and minor axes are @xmath322 ( standard deviation of 0.1 ) and @xmath323 ( standard deviation of 0.17 ) , respectively . our results suggest that high equivalent width systems ( recall equivalent width is a measure of both kinematic spread and column density ) tend to originate along the projected minor axis from outflows . this is consistent with previous results found along the minor axis for absorption @xcite . the same is true for the column densities , with the logarithmic mean column densities along the major and minor axes being @xmath324 and @xmath325 , respectively . we also explore how the strength of the absorption is dependent on galaxy inclination . the mean equivalent widths , with the standard error in the mean , for low inclination ( @xmath326 ) and high inclination ( @xmath327 ) galaxies are 0.35@xmath40.01 ( standard deviation of 0.1 ) and 0.36@xmath40.01 ( standard deviation of 0.16 ) , respectively . thus , unlike for high and low @xmath0 , we find no significant difference in the mean equivalent widths as a function of galaxy inclination . color distribution for a subset of our sample of absorbing and non - absorbing galaxies . note non - absorbing galaxies tend to be redder than absorbing galaxies . ( middle ) @xmath328 colors for a subset of our sample of absorbing and non - absorbing galaxies , again , non - absorbing galaxies are red . ( bottom ) combined @xmath329 and @xmath328 colors for 27/29 absorbers and 14/24 non - absorbers . note that the bifurcation value between red and blue sequences inferred by a color magnitude diagram in @xmath329 and @xmath328 are both around 2 , which makes for roughly an equal comparison . non - absorbing galaxies are redder than absorbing galaxies suggesting link between star - formation and halo gas cross - section / abundance . ] distribution for absorbing galaxies from the top panel of figure [ fig : main ] . we show the relative contributions of blue and red galaxies bifurcated at a @xmath329 and @xmath328 color of 1.5 ( see figure [ fig : color ] ) . red galaxies show a preference along the projected major axis while blue galaxies prefer the project minor axis . this difference in orientation preference may explain why blue galaxies are more frequently detected as absorbers given the difference in opening angle between inflowing and outflowing gas . ] here we assess whether absorbing and non - absorbing galaxies have different colors . @xcite has shown that the majority of absorbers are produced by star - forming galaxies while non - absorbers are mostly associated with quiescent galaxies . we do not have star - formation rates for the majority of our sample , however we do have @xmath329 or @xmath328 colors for 27/29 absorbers and 14/24 non - absorbers . the @xmath329 colors were either obtained from @xcite or from magiicat @xcite while the @xmath328 colors were obtained from @xcite . in figure [ fig : color ] we show the @xmath329 and @xmath328 color distributions for absorbing and non - absorbing galaxies . consistent with @xcite , we also find the absorbers tend to have bluer colors on average than non - absorbing galaxies . we combine the @xmath329 and @xmath328 colors in the bottom panel of figure [ fig : color ] . we validate doing this since the bifurcation value between red and blue sequences inferred by color magnitude diagrams in @xmath329 and @xmath328 are both at @xmath330 ( e.g. , * ? ? ? * ; * ? ? ? * ) , thus making for an equal comparison . the difference in colors between absorbers and non - absorbers is more accentuated in this combined figure . a ks test shows that absorbers and non - absorbers differ in their color distribution by 2.5@xmath22 ( cl=0.98793 ) . this is consistent with the idea that star - forming galaxies make up a significant fraction of absorbers and a smaller fraction of non - absorbing galaxies , which are dominated by red galaxies . this may suggest that blue star - forming galaxies have an overall higher covering fraction since they may be both accreting and outflowing gas , while quiescent galaxies have much less cgm activity . to test our previous statement , we further explore how the different colors of absorbing galaxies contribute to the distribution of @xmath0 . we divide 27 absorbers with measured colors into red and blue galaxies using a color cut at 1.5 , which is where the population of absorbers and non - absorbers split as seen the bottom panel of figure [ fig : color ] . this color - cut separates blue star - forming galaxies from redder dusty star - formers , green valley , and red sequence galaxies . in figure [ fig : oricolor ] , we show the relative contribution of absorbers for blue ( 15 galaxies ) and red ( 12 galaxies ) galaxies . blue galaxies show a low contribution along the projected major axis with a gradual increase to peak in frequency along the minor axis . the red galaxies show the strongest contribution along the projected major axis with little - to - no contribution along the minor axis . these results suggest that in fact blue galaxies are dominated by outflowing gas with wide opening angles while redder galaxies appear to have along the projected major axis , which has a much smaller opening angle . this difference in orientation and opening angle between blue and red galaxies may explain why red galaxies are less frequently detected due to their low gas sky cross - section while blue galaxies have a much higher gas cross - section . the abundance of contained in the halos of galaxies is significant @xcite and mostly bound to the galaxy s gravitational potential @xcite . we are just beginning to understand how this diffuse gas is associated with its host galaxies and what role it plays in galaxy evolution . we and others have shown that the covering fraction and equivalent width are dependent on the distance away from the galaxy @xcite . @xcite has shown that the majority of absorbers are produced by star - forming galaxies while non - absorbers are mostly associated with quiescent galaxies . this suggests that either the quantity of absorbing gas and/or the geometric distribution of this gas is different between star - forming and quiescent galaxies . for the first time , we have demonstrated that the distribution of absorption within 200 kpc of isolated galaxies exhibits an azimuthal angle bimodality . @xcite presented the first hint of a possible bimodality with only five galaxies detected within one viral radius with the remaining sample probing larger impact parameters . our significant increase in sample size to 29 absorbers and 24 non - absorbers was key to our findings . here we have shown that the highest frequency of absorption is found within 1020@xmath5 of the projected major axis and within 30@xmath5 of the projected minor axis . the non - absorbing galaxies occur at the highest frequencies at intermediate @xmath0 between 20 - 60@xmath5 . this is further reflected in the azimuthal covering fraction distribution , which is typically around 35% except along the projected major axis between 10 - 20@xmath5 and near the projected minor axis between 60 - 90@xmath5 . these results are consistent with the idea that gas inflows along the major axis and outflows along the minor axis of galaxies . this bimodality is also consistent with bimodality found for absorption @xcite , suggesting the infalling and outflowing gas is multi - phased . we constrain the half opening angle of inflowing gas to be around 20@xmath5 , which is a factor of two larger than found for @xcite , while the half opening angle of outflowing gas is at least 30@xmath5 , which is consistent with previous results determined using different tracers for gas ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? unlike , the covering fraction of within 10@xmath5 of the projected major axis is very low while the covering fraction of is above 80% within 10@xmath5 . if this result remains with a larger sample size , then it may suggest a detection of cool gas accretion surrounded by an envelope of a diffuse gas phase traced by . this could have interesting implications for how gas is fed into galaxies . @xcite and @xcite reported that projected minor axis absorption originated from outflows , traced by absorption , and has higher equivalent widths on average than absorption found along projected major axis . here we find the same result : higher equivalent width systems occur along the projected minor axis . the mean equivalent widths , with the standard error in the mean , along the major and minor axes are @xmath322 and @xmath323 , respectively . these results suggest that either the column density , metallicity , and/or kinetic spread of outflowing gas is higher along the minor axis . @xcite has shown that absorption profiles with the largest velocity dispersion are associated with blue , face - on galaxies probed along the projected minor axis while the column densities are largest for edge - on galaxies and blue galaxies . these results are consistent with bi - conical outflows along the minor axis for star - forming galaxies causing an increase in kinematic spread and column density . there should also be an increase in metallicity as well due to enriched supernovae - driven winds but this has yet to be confirmed observationally . the equivalent width distribution along the projected major axis still contains moderately high equivalent width systems . this may suggest that we are not observing purely pristine accretion onto galaxies and it is likely associated with the recycling of previously ejected gas ( e.g. , * ? ? ? * ; * ? ? ? * ) . we have shown that the dependence on galaxy inclination is much weaker than the galaxy position angle and this is consistent with previous results shown for absorption @xcite . we do show that there appears to be a broader @xmath0 distribution of non - absorbers for edge - on galaxies with @xmath320 , which translates to a lower covering fraction for @xmath320 . these results are consistent with the picture that gas inflows along the major axis and outflows along the minor axis of galaxies . both the cross - section of the co - planer flows and the cross - section of conical outflows will be minimized for edge - on galaxies , thus leading to a higher frequency of non - absorbers . consistent with @xcite , we find bluer galaxies are preferentially selected to be absorbers while redder galaxies tend to be non - absorbers . this suggests that star - formation is a key driver in the distribution and covering fraction of around galaxies . however our results shown in figure [ fig : oricolor ] also suggest that blue galaxies have detected preferentially along their minor axes while red galaxies have absorption preferentially detected along their major axes . this result could arise from our selected color bifurcation between blue and red galaxies at 1.5 in @xmath329 and @xmath328 , where the `` red '' galaxies could also contain red star - forming galaxies however , even if we changed our color cut to 2 ( selecting almost only quiescent galaxies ) , then out of seven red galaxies six still have @xmath331@xmath5 . thus , the cross - section of red galaxies could be dominated by small opening angle accretion of recycled material , while blue galaxies have a cross - section dominated by large - opening angle outflows . it is possible then , that the feedback mechanisms of the cgm also drive the predictions of gas cross - section and not just the presence of star - formation alone . we have constructed a sample of 29 absorbing ( ew>0.1 ) and 24 non - absorbing ( ew<0.1 ) galaxies with spectroscopically confirmed redshifts ranging between 0.08@xmath1@xmath2@xmath10.67 within @xmath3 kpc of bright background quasars . these galaxies are selected to be isolated such that there are no neighbors within 100 kpc and have velocity separations less than 500 . all these galaxy - absorber pairs were identified as part of our `` multiphase galaxy halos '' survey and from the literature . the background quasars all have medium resolution _ hst_/cos and stis ultraviolet spectra covering the o0.1emvi @xmath9 doublet . the quasar fields have high resolution _ hst _ wfpc-2 , acs , and wfc3 imaging . we used gim2d to model the galaxy morphological properties and the azimuthal angle relative to the galaxy project major ( @xmath307@xmath5 ) and projected minor ( @xmath308@xmath5 ) axes and the quasar sight - line . we have analyzed the dependence of absorption on @xmath32 , azimuthal angle , and @xmath329 and @xmath328 color . our results are summarized as follows : 1 . we have shown that , for ew@xmath304 , the ovi rest - frame equivalent width is anti - correlated with impact parameter , @xmath32 ( 2.7@xmath22 ) . non - absorbers ( ew@xmath305 ) are found at all impact parameters , such that the covering fraction is 80% within 50 kpc and decreases monotonically to 33% at 200 kpc . the presence of absorption ( ew>0.1 ) is azimuthally dependent and primarily occurs along the projected major axis within a half opening angle of 20@xmath5 and along the project minor axis within a half opening angle of at least 30@xmath5 . these results are consistent with what is expected for major axis - fed inflows and minor axis - driven outflows as traced by and consistent with previous results found for absorption . 3 . the frequency of non - detected absorption ( ew<0.1 ) as a function of azimuthal angle is greatest in the range 3060@xmath5 . thus , there is a paucity of detectable ovi absorbing gas at intermediate azimuthal angles . this suggests that absorbing gas is not mixed throughout the cgm , but remains confined primarily within the outflowing winds and near to the planar disk region . non - absorbers also exist within @xmath410@xmath5 of the projected major axis . we show that the covering fraction of is 35% within 0 - 10@xmath5 , then peaks at 85% within 10 - 20@xmath5 , drops back to 35% again between 20 - 60@xmath5 and peaks up at 80% from 60 - 90@xmath5 . the lack of absorption within 10@xmath5 , increased presence of ovi absorption at 2030@xmath5 , and then sudden decrease beyond 30@xmath5 may suggest that cool gas resides in a narrow planar geometry surrounded by warm / hot gas . this could indicate accreting gas in which the cool material is narrowly confined to the disk plane and is surrounded by an also accreting warm / hot envelope , or indicate an extended disk with a pressure supported corona , or some combination of both scenarios . the covering fraction within 60 - 90@xmath5 suggests that outflows have very high covering fractions . the dependence of absorption on galaxy inclination is much weaker than the galaxy position angle and this is consistent with previous results shown for absorption . the covering fraction is constant at 60% at all @xmath317 except it drops by a factor of two at @xmath320 . this is interpreted as geometric minimization in the cross - section of co - planer flows and conical outflows occurring for edge - on systems . we determine that the equivalent width distributions for projected major axis gas for @xmath332@xmath5 and projected minor axis gas for @xmath333@xmath5 have mean equivalent widths , with the standard error in the mean , of @xmath322 and @xmath323 , respectively . therefore , higher equivalent width systems , including the highest ew systems , are found along the projected minor axis , which is consistent with an outflow scenario . consistent with previous results , we show that absorbers tend to have bluer colors while non - absorbers tend to have redder colors on average . this suggests that star - formation is a key driver in the detection rate of absorption . we show the relative dependence of blue ( 15 galaxies ) and red ( 12 galaxies ) absorbing galaxies on the distribution of @xmath0 for a color cut of 1.5 in @xmath329 and @xmath328 . blue galaxies show a low detection rate along the projected major axis with a gradual increase to its peak along the minor axis . the red galaxies show the strongest detection rate of absorption along the projected major axis with little - to - no detections along the minor axis . these results suggest that in the halos of blue galaxies are dominated by outflowing gas with wide opening angles while red galaxies appear to have gas along the projected major axis , which have smaller opening angles . this difference in orientation and opening angle between blue and red galaxies may explain why red galaxies less frequently produce absorption due to their low gas sky cross - section while blue galaxies have a much higher gas cross - section . our results are consistent with current models of gaseous multi - phase outflows and accretion / recycling . it is clear that high opening angle outflows are ubiquitous at all redshifts for star - forming galaxies , however , it is unclear whether we are still observing small opening angle cold accretion filaments at redshifts @xmath334 or , the more likely scenario , we are detecting recycling of previously ejected gas , which can be seen for most red and a few blue galaxies . placing constraints of the metallicity of these absorption systems hold the key in constraining the exact feedback processes that are occurring in and around galaxies . we would like to thank the referee for his / her thorough read of the manuscript . we thank , and are grateful to , roberto avila ( stsci ) for his help and advice with modeling psfs with acs and wfc3 . ggk acknowledges the support of the australian research council through the award of a future fellowship ( ft140100933 ) . cwc , jcc , nmn , and sm are supported by nasa through grants hst go-13398 from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 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we report a bimodality in the azimuthal angle ( @xmath0 ) distribution of gas around galaxies traced by absorption . we present the mean @xmath0 probability distribution function of 29 _ hst_-imaged absorbing ( ew>0.1 ) and 24 non - absorbing ( ew<0.1 ) isolated galaxies ( 0.08@xmath1@xmath2@xmath10.67 ) within @xmath3 kpc of background quasars . we show that ew is anti - correlated with impact parameter and covering fraction decreases from 80% within 50 kpc to 33% at 200 kpc . the presence of absorption is azimuthally dependent and occurs between @xmath41020@xmath5 of the galaxy projected major axis and within @xmath430@xmath5 of the projected minor axis . we find higher ews along the projected minor axis with weaker ews along the project major axis . highly inclined galaxies have the lowest covering fractions due to minimized outflow / inflow cross - section geometry . absorbing galaxies also have bluer colors while non - absorbers have redder colors , suggesting that star - formation is a key driver in the detection rate . surrounding blue galaxies exists primarily along the projected minor axis with wide opening angles while surrounding red galaxies exists primarily along the projected major axis with smaller opening angles , which may explain why absorption around red galaxies is less frequently detected . our results are consistent with cgm originating from major axis - fed inflows / recycled gas and from minor axis - driven outflows . non - detected occurs between @xmath0=2060@xmath5 , suggesting that is not mixed throughout the cgm and remains confined within the outflows and the disk - plane . we find low covering fractions within @xmath6@xmath5 of the projected major axis , suggesting that cool dense gas resides in a narrow planer geometry surrounded by diffuse gas .

recent long - baseline neutrino experiments , t2k @xcite and minos @xcite , gave rise to the first indications to a non - zero @xmath6 . following reactor neutrino experiments successfully presented certain values of @xmath7 . double chooz reported their first result @xmath8 at the 68% cl @xcite . daya bay narrowed down the range to @xmath9 at 5.2@xmath10 @xcite , and reno reported a definitive result with a value of @xmath11 at 4.9@xmath10 @xcite . the current bound on other angles , determined from neutrino oscillation experiments , are @xmath12 and @xmath13 at the @xmath14 cl . the current data also include the mass squared differences that are accompanied by solar and atmospheric neutrino oscillations , @xmath15 and @xmath16 , respectively @xcite . non - abelian discrete symmetries have provided theoretical frameworks for neutrino masses with tri - bi - maximal(tbm ) mixing @xcite with @xmath17 and @xmath18 @xcite . due to the signals from recent measurements of @xmath0 , its non - zero value , which is still small relative to other two angles , is considered as being generated by a mechanism based on the symmetrical background rather than being a perturbation effect . two models with non - zero @xmath6 are introduced using @xmath1 flavor symmetry . both models have the same field contents with the same flavor charges , except for two righthanded neutrinos . whether the @xmath19 representations of the two fields are double @xmath20s or a single @xmath21 , the non - zero @xmath6 is obtained in charged lepton masses or in neutrino masses . besides the standard model(sm ) higgs , a few scalar multiplets are added . the @xmath22 which commutes with @xmath19 divides the fields by their parity , in the sense that all sm fields have even parity and so their couplings are not affected by the @xmath22 symmetry . the @xmath22-odd righthanded neutrinos couple with only @xmath22-odd scalar fields to make 5-dimensional majorana masses in an effective theory . using a simple assumption of yukawa coupling constants , predictions of the mass ratios and mixing angles are presented . this paper is organized as follows : section ii introduces the representations of flavor symmetry @xmath19 , and contains the construction of yukawa interactions of sm charged leptons with @xmath22-even higgs contents . in section iii , two models with non - zero @xmath6 are presented . the first model obtains the pmns angle by a simple relation of tbm angles and the mixing angle @xmath23 of charged leptons . in the second model , the transformation of neutrino mass matrix becomes the pmns matrix to the leading order . the predictions are examined in the figures . the conclusion section contains a summary and mentions some exclusion regions as the prediction , and an appendix is attached to describe the interactions of higgs fields and their vevs to make the potential minimum . the minimal non - abelian discrete symmetry @xmath19 is the group of the permutation of the three sides of an equilateral triangle . there are six elements of the group in three classes , and their irreducible representations are @xmath24 and @xmath25 its character table is mentioned in many models @xcite . the clebsch - gordon coefficients in the real representations are given by the following product rules @xcite , @xmath26 .group representation of sm particles [ cols="^,^,^,^,^",options="header " , ] here , an abelian discrete symmetry @xmath22 is also adopted , which is the parity that distinguishes extra particles from the sm contents . the sm fields are assigned to representations of @xmath1 as listed in table i , where the su(2 ) representation and hypercharge of a field are denoted by the subscription ` @xmath27 ' of the gauge symmetry , and the @xmath19 representation and @xmath22 charge of the field are denoted by the subscription ` @xmath28 ' . although the @xmath22 charges of fields are presented in table i , @xmath22 symmetry does not affect the interactions among @xmath22-even sm fields . then , the lagrangian of yukawa couplings of the charged leptons and the higgs scalar doublet @xmath29 is @xmath30 where the su(2 ) fermion doublet @xmath31 is a flavor singlet but @xmath32 and @xmath33 belong to a doublet such as @xmath34 under @xmath19 . also , the righthanded charged lepton singlet @xmath35 is a flavor singlet while @xmath36 and @xmath37 belong to a doublet such as @xmath38 . the higgs scalar doublet @xmath29 of sm is involved in the above interactions as a flavor singlet . the higgs self potential is @xmath39 so that , after spontaneous su(2 ) symmetry breaking , three dirac mass matrices of the charged leptons from the yukawa couplings become @xmath40 it is shown that for the flavor model we can build a basis where the matrix of charged lepton masses is diagonal and @xmath41 , if only sm fields contribute to generate the dirac masses . it follows that the right mass hierarchy is obtained from additional yukawa couplings with additional scalar fields beyond the sm . there is an additional higgs scalar particle that couple with sm leptons , which is represented by , under @xmath1 , @xmath42 the interactions of only @xmath22-even higgs particles , @xmath43 and @xmath44 , among themselves are @xmath45 where the term @xmath46 include such three contributions as @xmath47 since the product @xmath48 can be any of the following representations , @xmath49 or @xmath50 of @xmath1 . according to the processes in appendix in order to make the potential minimum , the vevs are obtained such that @xmath51 , @xmath52 , and @xmath53 . if the masses of charged leptons were obtained by using the yukawa couplings in eq.([smlepton ] ) , the muon and tau lepton could have the same mass , @xmath54 . here , we introduce an additional contribution to the masses derived from the yukawa couplings with another higgs @xmath44 . @xmath55 while the yukawa couplings of quarks are protected from the non - sm additional higgs @xmath44 , since the flavor symmetry is leptonic so that all quarks are @xmath19 singlets and @xmath22-even . the dirac mass matrix of charged leptons derived from both eq.([sm ] ) and eq.([sm_extended ] ) has the following form @xmath56 the masses of leptons are obtained from @xmath57 . the transformation matrix @xmath58 is required for pontecorvo - maki - nakagawa - sakata(pmns ) matrix along with the transformation of neutrino masses @xmath59 such that @xmath60 . we denote the 1 - 3 block of the matrix @xmath61 by @xmath62 such as @xmath63 which is plausible by the relation in terms of the masses and the mixing angle as in @xmath64 , where the 1 - 3 block rotation @xmath65 is given by @xmath66 the elements of the matrix @xmath62 in eq.([mdaggerm ] ) are described by physical parameters , @xmath67 in opposite way , the mixing angle @xmath23 and the phase @xmath68 are obtained from the elements in eq.([elementk ] ) as @xmath69 or from the matrix in eq.([mdaggerm ] ) @xmath70}{|c_2 v + c_3 v_1|^2-|c_1|^2 v^2 - |c_5|^2 v_1 ^ 2}.\label{theta } \end{aligned}\ ] ] in general , the squared masses can be expressed in the following way , @xmath71 where @xmath72 . for @xmath73 , the squared masses are approximated to @xmath74 where @xmath75 and @xmath76 are derived from the independent yukawa couplings from each other , and @xmath77 is close to a diagonal matrix due to the suppressed numerator in eq.([theta ] ) . for @xmath78 , the squared masses in eq.([msquare ] ) reduce to @xmath79 the strong hierarchy between @xmath75 and @xmath76 and the large mixing angle require a careful fine tuning in eq.([largetan ] ) . two models are presented to explain neutrino masses in the normal hierarchy and the mixing matrix . both models contain @xmath22-odd additional particles , which can be distinguished from @xmath22-even sm fields . the righthanded neutrinos are characterized by @xmath22 charge -1 and coupled with @xmath22-odd higgs fields , while the sm righthanded leptons are all assigned to the charge + 1 and coupled with @xmath22-even higgs fields . those additional higgs contents are @xmath80 the higgs potential which include the interactions of @xmath81 and @xmath82 is presented in eq.([potential_sigma ] ) . its minimum is obtained by the real vevs of the higgs fields , which are denoted as @xmath83 and @xmath84 . as far as the mixing of neutrino mass matrix is concerned , one model generates tbm , and the other generates nonzero @xmath0 in the pmns matrix . both types of mixing matrices are derived in terms of the @xmath19 symmetry and its breaking mechanism by the vacuum expectation value(vev ) of an @xmath19-doublet scalar field . the field contents and their charge assignments in the two models are identical to each other except the flavor charges of the righthanded neutrinos . the operators for majorana masses have four external lines with an internal line , as shown in fig.1 . if the internal line is a heavy righthanded neutrino which has its ends coupled in a yukawa interaction , the process giving rise to low energy effective masses is equivalent to the seesaw mechanism . here , we describe the generation of neutrino masses while comparing the two models , the difference between which originated from the choice of group representations for the internal righthanded neutrinos . all additional fields beyond the sm , including righthanded neutrinos , are distinguished from the sm particles by @xmath22 parity . all the sm fields are @xmath22 even , so that the parity does not affect any interaction of sm particles . additional higgs scalars , @xmath85 and @xmath81 , and right - handed neutrinos , @xmath86 , all have @xmath22-odd quantum number . their representation under the gauge symmetry is @xmath87 , and their representations under the flavor symmetry are @xmath88 their yukawa interactions are as follows : @xmath89 where @xmath90 is redundant so that the same result can be obtained by just two righthanded neutrinos . but we keep the two identical singlets , @xmath91 and @xmath90 , for the comparison with other representation for them in another model . the couplings of @xmath92 and @xmath93 can be rephrased as @xmath94 , @xmath95 , and @xmath96 . the gauge singlets @xmath97 have majorana masses as in @xmath98 , while the @xmath91 and @xmath90 have a cross term such as @xmath99 . if majorana neutrinos @xmath97 are very heavy , any two yukawa couplings could be linked to each other as shown in fig.1 . the vertex @xmath100 or @xmath101 in figure 1 can correspond to any term in eq.([yukawatbm ] ) if the @xmath97 that is substituted into @xmath102 is the same for both vertices . then , an effective lagrangian is obtained by integrating out the internal heavy fermion @xmath102 . @xmath103 here , @xmath104 and @xmath105 are the elements of mass matrix of singlet majorana neutrinos @xmath97 for @xmath106 . since @xmath107 and @xmath90 all belong to different representations , their yukawa coupling constants @xmath108 can be different and so can be their masses , @xmath104 . when the scalar fields obtain vevs by spontaneous breaking of @xmath19 symmetry , the above 5-dimensional interactions reduce to low - energy effective mass terms of light neutrinos @xmath109 . the symmetric matrix is @xmath110 and @xmath111 where @xmath112 and @xmath113 . for simplicity , it is assumed that all @xmath108 are 1 ; then the mass matrix in eq.([tbm ] ) has a simple pattern as follows : @xmath114 where @xmath115 . the above matrix has a vanishing determinant , implying that one mass should be zero . the ratio of non - zero masses @xmath4 is @xmath116 . the type of mass hierarchy depends on the relative sizes of @xmath117 and @xmath118 and those of @xmath119 and @xmath120 . if @xmath121 , furthermore , the matrix in eq.([tbm_simple ] ) is exactly of the form that results from the tri - bi - maximal mixing : @xmath122 . even when the yukawa coupling constants , @xmath108 , are allowed to be different from each other , @xmath0 and one of the masses vanish . only the mass ratio @xmath4 is shifted to @xmath123 . the specific pattern described above was discussed further in a previous work @xcite . the pmns matrix @xmath124 is obtained by @xmath125 where @xmath58 is the transformation of @xmath77 in eq.([masslepton ] ) that imbeds @xmath126 in eq.([tanslepton ] ) into the 1 - 3 block . on the other hand , if @xmath124 is expressed in the standard parametrization , @xmath127 where @xmath128 and @xmath129 are @xmath130 and @xmath131 , respectively . from the comparison of two expressions in eq.([pmnsproduct ] ) and eq.([standard ] ) , the three angles in pmns matrix are obtained by the following simple relations , @xmath132 where @xmath133 in eq.([pmnsproduct ] ) and @xmath134 in eq.([standard ] ) . the prediction of the model can be estimated with respect to the @xmath135 in eq.([theta ] ) . the @xmath136 in eq.([pmns ] ) matched to @xmath137 can predict the range of @xmath0 measured in recent neutrino oscillation experiments @xcite . another predicted result is that @xmath138 belongs to the second octant . however , the range of @xmath139 predicted from the given @xmath23 is barely overlapped with the @xmath140 range of @xmath139 in the global analysis . a narrow range of @xmath135 , @xmath141 , can generate simultaneously the angles @xmath142 and @xmath143 marginally allowed in experiment , of which the eligibility is about to be tested by a higher precision data of the current oscillation experiments @xcite . when the @xmath144 is very small as considered in eq.([diagmm ] ) , i.e. , when @xmath58 is almost the unit , the pmns matrix can be simply the transformation of neutrino mass matrix . the two righthanded neutrinos are in a doublet so they are not distinguishable in terms of flavor symmetry . the majorana neutrinos @xmath91 and @xmath90 are tied into @xmath145 , a two - dimensional representation of @xmath19 , so that @xmath146 . the scalar particles and @xmath147 do not change their charges , and all non - sm particles have @xmath22-odd quantum number . the group representations of particles are summarized as follows : @xmath148 the higgs potential is the same as the one for model a in eq.([potentialnh ] ) . the yukawa interactions are @xmath149 the couplings of @xmath150 and @xmath93 can be rephrased as @xmath151 , @xmath152 , @xmath153 , and @xmath154 . the majorana masses of the righthanded neutrinos are given by @xmath155 where @xmath156 becomes @xmath157 according to the product rule in eq.([prod3 ] ) , and breaking of @xmath19 gives rise to a common mass @xmath158 for @xmath91 and @xmath90 . for the same reason , the neutrinos have common yukawa coupling constants @xmath159 and @xmath160 . for very heavy majorana neutrinos , any two yukawa couplings can link to each other , as shown in fig.1 . the interactions in figure 1 can be considered as non - renomalizable 5 dimensional couplings obtained by integrating out the internal fermions @xmath97 . then the effective lagrangian is given by @xmath161 where @xmath104 are the heavy masses of singlet majorana neutrinos @xmath97 for @xmath106 . when the scalar fields obtain vevs by spontaneous breaking of @xmath19 symmetry as shown in eq.([first_der])-eq.([second_der ] ) , the above 5-dimensional interactions reduce to low - energy effective mass terms of light neutrinos @xmath109 . the matrix is @xmath162 where @xmath163 and @xmath164 . for the simplest analysis , it is assumed that all @xmath165 s are one except @xmath166 . the @xmath166 is smaller than 1 to make @xmath167 smaller than @xmath168 . a rather tedious estimation of masses and mixing angles from the above mass matrix is presented using pictorial analysis . fig.2 presents the possible parametric plots that can arise from eq.([t2kmass ] ) in spaces of @xmath169 , @xmath170 , and @xmath171 , respectively . the value of @xmath166 is chosen as 0.7 in the following example . five curves in each figure represent the choices of @xmath172 as 0.0001 , 0.1 , 1.0 , 2.0 , and 5.0 by thick solid , dashed , dotted , dot - dashed , and thin solid lines , respectively . four circles on each curve represent the choices of @xmath173 as 0.07 , 0.17 , 0.27 , and 0.37 as @xmath5 increases . the horizontal shadow in each figure represents the allowed region at the 90% cl for the physical parameters according to the current experimental results : @xmath174 , @xmath175 , and @xmath176 . as for the mass ratio , the region above 0.201 is not ruled out since @xmath177 can be larger than @xmath178 . for @xmath5 , two bounded regions are presented . one is @xmath179 from reno @xcite , which is indicated by brown stripes . the other is @xmath180 from daya bay @xcite , which is indicated by gray shadows . the bundle of curves in figure 2 shows the area that the model can cover as @xmath181 increases . different curves in a figure come from the relative ratio of the vevs of non - sm higgs fields . the construction of mass matrix does not derive coefficients of elements , leaving them free parameters , when the symmetry builds up a pattern of a mass matrix based on the charge assignments of a flavor symmetry . here , we examine the prediction from the model , while the effect of yukawa couplings , @xmath108 or @xmath165 , is suppressed by setting them one . whatever the value of @xmath181 is and whatever the ratio @xmath182 is , there is some area in @xmath183 , which is larger than 0.707 , excluded by the prediction . as for @xmath4 , likewise , the mass type of degeneracy or quasi - degeneracy is ruled out . the thin solid line in each figure describes the fit for @xmath184 , and figures 2(b ) and 2(c ) show that the resulted curves miss the allowed range , if @xmath185 . so the mass ratio range is confined within @xmath186 at most . on the other hand , the ranges of @xmath181 and @xmath182 are also trimmed off by the experimental bounds of mixing angles . for instance , if @xmath187 , any value of @xmath181 smaller than 0.17 and that larger than 0.40 are excluded by reno bound on @xmath0 . the recent measurements of @xmath188 motivate an idea which is that non - zero @xmath5 is generated by a mechanism based on the symmetrical background rather than being a perturbation effect from tbm with @xmath18 . two lepton models were introduced in terms of @xmath1 flavor symmetry . one provides tbm from @xmath59 and non - zero @xmath0 from @xmath58 to the pmns matrix , while the other provides all leading orders of mixing angles from the neutrino mixing matrix , @xmath59 . the difference between two models is caused by the only difference between the flavor charge assignments for righthanded neutrinos . other group theoretical properties are all the same for both models . the @xmath22 symmetry splits the field contents into the particle fields beyond the sm and the fields in the sm , whether the charge is -1 or + 1 . since the sm fields are @xmath22-even , the yukawa couplings of the sm fermions are protected from the coupling with a @xmath22-odd scalar field . on the other hand , a @xmath22-odd righthanded neutrino makes a vertex with a @xmath22-odd scalar field . if a righthanded neutrino as an internal line is integrated out and the effective 5-dimensional coupling is suppressed by the mass scale of the righthanded neutrino , the majorana masses of lefthanded neutrinos become then light . the @xmath22-even scalar higgs fields , @xmath29 and @xmath44 , in yukawa couplings contribute to the masses of charged leptons , and @xmath22-odd scalar higgs fields , @xmath81 and @xmath82 , in effective 5-dimensional couplings contribute to the masses of neutrinos . depending on the flavor charges of @xmath91 and @xmath90 among three generations , the type of neutrino mixing was determined . when they belong to separate @xmath189 representations , the model gave rise to the exact @xmath190 , as shown in eq.([tbm_simple ] ) . when they belong to a single @xmath191 representation , the model gave rise to @xmath192 unless @xmath193 . the prediction of the model , neglecting the contributions from most @xmath165 , was studied in fig.2 . the results obtained for various ranges of the relative ratio , @xmath194 , of majorana masses and for those of the relative scales of vevs of non - sm higgs , @xmath195 , rule out the area of @xmath183 larger than 0.707 and the mass pattern of ( quasi- ) degeneracy . thus , the survival of model b can be determined , depending on whether @xmath196 and whether the mass type is hierarchical . this work was supported by the basic science research program through nrf(2011 - 0014686 ) . the contents of higgs scalar particles and their representations under @xmath1 are @xmath197 which commonly belong to @xmath198 under @xmath199 gauge group . the full invariant higgs potential can be organized into three parts as follows : @xmath200 where @xmath201 and @xmath202 are the interactions of only @xmath22-even particles and those of only @xmath22-odd particles , respectively , while @xmath203 is the cross interactions of @xmath22-even and @xmath22-odd particles . each contribution to the potential @xmath204 is given as ; the subscript ` 1 ' or ` 2 ' in each term indicates that the product of two fields belongs to the representation @xmath206 or @xmath21 in @xmath19 . each term with a subscript ` @xmath207 ' consists of three types of products , @xmath208 and @xmath21 representations as in eq.([3reps ] ) . according to the product rules in eqs . ( [ prod1 ] ) - ( [ prod3 ] ) , @xmath209 , and @xmath210 . the higgs potential in eq.([potentialnh ] ) can be rephrased in terms of component fields @xmath211 with @xmath212 and 2 , and @xmath213 : where @xmath224 and @xmath225 . the @xmath226 are rather complicated polynomials of @xmath227 and @xmath228 in eq.([potential_ext ] ) , such that @xmath229 and @xmath230 . their details are not necessary for the following examination of the minimal condition . the first derivatives of the full potential given in eq.([3potential ] ) are where each @xmath62 denotes the part that corresponds to the coefficients of linear terms . the vevs , @xmath232 and @xmath233 , can make the potential minimum , when the following conditions are satisfied . @xmath234 @xmath235 it is clear that any of @xmath236 and @xmath237 should not be zero to satisfy the above conditions . according to the symmetry of the potential under the interchange of @xmath238 and @xmath239 , vevs can be taken as @xmath240 . thus , in summary , the following vevs of the fields in eq.([4higgs ] ) can be adopted for the masses of leptons : @xmath241 then , the derivatives in eqs.([6min ] ) reduces to the following conditions : according to eq.([first_der ] ) and eq.([second_der ] ) , @xmath243 and @xmath244 are necessary . the mass matrices are examined upon the assumptions of @xmath245 and @xmath246 with weak hierarchy . the assumptions do not show any conflicts with either the minimum conditions in eq.([4min ] ) or the phenomenological constraints , @xmath247 , since the constraints on masses and vevs contain a sufficient number of independent parameters . k. abe _ et al . _ [ t2k collaboration ] , phys . lett . * 107 * , 041801 ( 2011 ) p. adamson _ et al . _ [ minos collaboration ] , phys . lett . * 107 * , 181802 ( 2011 ) y. abe _ et al . _ [ double - chooz collaboration ] , phys . lett . * 108 * , 131801 ( 2012 ) f. p. an _ [ daya - bay collaboration ] , phys . lett . * 108 * , 171803 ( 2012 ) j. k. ahn _ [ reno collaboration ] , phys . lett . * 108 * , 191802 ( 2012 ) k. nakamura _ et al . _ [ particle data group ] , j. phys . g * 37 * , 075021 ( 2010 ) . m. c. gonzalez - garcia and m. maltoni , phys . rept . * 460 * , 1 ( 2008 ) p. f. harrison , d. h. perkins and w. g. scott , phys . b * 530 * , 167 ( 2002 ) p. f. harrison and w. g. scott , phys . b * 557 * , 76 ( 2003 ) a. zee , phys . b * 630 * , 58 ( 2005 ) e. ma , phys . rev . d * 72 * , 037301 ( 2005 ) k. s. babu and x. g. he , arxiv : hep - ph/0507217 . e. ma , mod . a * 20 * , 2601 ( 2005 ) g. altarelli and f. feruglio , nucl . b * 741 * , 215 ( 2006 ) x. g. he , y. y. keum and r. r. volkas , jhep * 0604 * , 039 ( 2006 ) x. g. he and a. zee , phys . b * 645 * , 427 ( 2007 ) r. r. volkas , arxiv : hep - ph/0612296 . f. feruglio , c. hagedorn , y. lin and l. merlo , nucl . b * 775 * , 120 ( 2007 ) [ erratum - ibid . * 836 * , 127 ( 2010 ) ] w. grimus , l. lavoura and p. o. ludl , j. phys . g * 36 * , 115007 ( 2009 ) t. araki , j. mei and z. z. xing , phys . b * 695 * , 165 ( 2011 ) n. haba , a. watanabe and k. yoshioka , phys . lett . * 97 * , 041601 ( 2006 ) c. s. lam , phys . * 101 * , 121602 ( 2008 ) c. hagedorn , m. lindner and r. n. mohapatra , jhep * 0606 * , 042 ( 2006 ) h. zhang , phys . b * 655 * , 132 ( 2007 ) f. bazzocchi , l. merlo and s. morisi , nucl . b * 816 * , 204 ( 2009 ) f. bazzocchi , l. merlo and s. morisi , phys . d * 80 * , 053003 ( 2009 ) r. z. yang and h. zhang , phys . lett . b * 700 * , 316 ( 2011 ) s. morisi and e. peinado , phys . b * 701 * , 451 ( 2011 ) n. w. park , k. h. nam and k. siyeon , phys . d * 83 * , 056013 ( 2011 ) s. morisi , k. m. patel and e. peinado , phys . d * 84 * , 053002 ( 2011 ) d. meloni , s. morisi and e. peinado , j. phys . g * 38 * , 015003 ( 2011 ) p. v. dong , h. n. long , c. h. nam and v. v. vien , phys . d * 85 * , 053001 ( 2012 ) x. chu , m. dhen and t. hambye , jhep * 1111 * , 106 ( 2011 ) e. ma , arxiv : hep - ph/0409075 . s. l. chen , m. frigerio and e. ma , phys . d * 70 * , 073008 ( 2004 ) [ erratum - ibid . d * 70 * , 079905 ( 2004 ) ] j. kubo , h. okada and f. sakamaki , phys . d * 70 * , 036007 ( 2004 )

this work proposes two models of neutrino masses that predict non - zero @xmath0 under the non - abelian discrete flavor symmetry @xmath1 . we advocate that the size of @xmath0 is understood as a group theoretical consequence rather than a perturbed effect from the tri - bi - maximal mixing . so , the difference of two models is designed only in terms of the flavor symmetry , by changing the charge assignment of righthanded neutrinos . the pmns matrix in the first model is obtained from both mass matrices , charged leptons giving rise to non - zero @xmath2 and neutrino masses giving rise to tri - bi - maximal mixing . the physical mixing angles are expressed by a simple relation between @xmath2 and tri - bi - maximal angles to fit the recent experimental results . the other model generates pmns matrix with non - zero @xmath0 , only from the neutrino mass transformation . the 5 dimensional effective theory of majorana neutrinos obtained in this framework is tested with phenomenological bounds in the parametric spaces @xmath3 and @xmath4 vs. @xmath5 .

hii regions are astronomical sources that represent early stages of deeply embedded high mass ( o or early b ) stars . their study can provide vital information about high mass star formation as well as their interaction with the parent molecular cloud . being deeply embedded in interstellar cloud including the dust component , almost all of their energy is absorbed and re - emitted in the infrared wavebands . ( g282.0 - 1.2 ) and ( g281.6 - 1.0 ) are galactic star forming regions in the southern sky , which are generally less - studied . radio measurements indicate that is an extended hii region ( hill , 1968 ; manchester , 1969 ) . a number of distance estimates to can be found in the literature , ranging from 5.1 to 7.1 kpc . here we use the distance of 6.3 kpc estimated by caswell & haynes ( 1987 ) based on radio recombination line measurements ( for r@xmath6 = 8.5 kpc and galactic rotation velocity = 220 km s@xmath7 at r@xmath6 ) . the distance to is estimated to be 3.7 kpc ( caswell & haynes , 1987 ) . recently , it has been concluded that complex harbours a very massive ob star cluster ( bik et al . 2005 ; hanson et al . , 2003 ) . bik et al . ( 2005 ) have carried out near - infrared k - band spectroscopy of few members of this cluster and find two very massive ( o3-o4 ) stars here . has been studied as a part of surveys for the search of emission lines including masers . formaldehyde absorption has been detected towards this source at 4.8 ghz ( whiteoak and gardner , 1974 ) and 14.5 ghz ( gardner and whiteoak , 1984 ) . whiteoak et al ( 1982 ) detected co ( 1 - 0 ) line emission from this source using the 4-m radio telescope of csiro . search for methanol transition ( peng and whiteoak , 1992 ) , methanol maser ( schutte et al , 1993 ) and oh maser ( cohen et al , 1995 ) close to this source have led to negative results . has been imaged by puchalla et al . ( 2002 ) at 42 ghz using the mobile anisotropy telescope on cerro toco ( mat / toco ) and an integrated flux of @xmath8 jy within a 0.3@xmath9 circular beam has been obtained by them . both these regions , and have been studied as a part of the parkes - mit - nrao ( pmn ) survey ( kuchar and clark , 1997 ) . they find the peak radio flux density of and at 5 ghz to be 25.7 jy / beam and 1.0 jy / beam , respectively , where the beam is @xmath10 . cs ( 2 - 1 ) line emission has been observed close to both these regions ( bronfman et al . , 1996 ) using the sest telescope . the iras - lrs ( low resolution spectrometer ) spectrum of shows strong [ ne ii ] emission line at 12.8 @xmath2 m ( de muizon et al . , 1990 ; simpson & rubin , 1990 ) , a relatively weak [ s iii ] emission line at 18.7 @xmath2 m as well as emission in the unidentified infrared bands ( uibs ) at 7.7 , 8.6 and 11.3 @xmath11 ( zavagno et al . , 1992 ; de muizon et al . , 1990 ) water vapor maser ( braz et al . 1989 ) as well as methanol maser ( schutte et al . 1993 ) have not been found close to . the less known southern galactic massive star forming regions are being studied under a long - term program which involves observing these sources in far infrared ( vig et al . 2007 , ojha et al . 2002 , karnik et al . 2001 , ghosh et al . 2000 , verma et al . 1994 ) . in this paper , we present a systematic study of the star forming regions associated with and . the star forming region associated with is believed to harbour a cluster of very massive stars . is , on the other hand , a young star forming region with few members belonging to the cluster . we have carried out an infrared study of these southern galactic hii regions in detail with the aim of understanding the energetics , physical sizes , the spatial distribution of interstellar dust and its temperature as well as the associated young clusters . section 2 describes observations and other data sets used . section 3 describes the results and in section 4 , radiative transfer modelling of these sources is presented . a comprehensive discussion of these sources is carried out in section 5 and a brief summary is presented in section 6 . the galactic star forming regions associated with and have been observed using the two - band far infrared ( fir ) photometer system at the cassegrain focus of the tifr 100 cm ( f/8 ) balloon borne telescope . the observations were carried out during the balloon flight from the tifr balloon facility , hyderabad in india ( latitude @xmath12 north , longitude @xmath13 east ) on feb 20 , 1994 . details of the telescope and the observational procedure are given by ghosh et al ( 1988 ) . the two fir bands use a pair of 2@xmath143 composite silicon bolometer arrays , cooled to 0.3k by liquid @xmath15he which view identical parts of the sky simultaneously . the field of view of each detector is @xmath16 . the absolute positions were established from the detections of catalogued stars with an optical photometer located at the focal plane of the telescope , simultaneously during the fir observations . the planet jupiter was observed at the beginning as well as at the end of the flight . the observations of jupiter were used for the absolute flux calibration of the two band fir photometer ( 12 channels ) as well as for the determination of the point spread function ( psf ) . the spectral response of each band of the fir photometer was determined in the laboratory using a michelson interferometer and a golay cell as a comparison detector . the two fir wavebands will be referred to as 150 and 210 @xmath11 bands corresponding to the @xmath0 for 30 k source with @xmath17 emissivity law . the region around ( @xmath18 ) and ( @xmath19 ) was mapped by raster scanning the region of the sky in cross - elevation with steps in elevation at the end of each scan . the fir signals were gridded into a matrix with a pixel size of @xmath20 . the deconvolution of the observed chopped signal matrix was carried out using the maximum entropy method similar to that of gull & daniell ( 1978 ) ( for details see ghosh et al , 1988 ) . an angular resolution of @xmath21 has been achieved in the fir maps using this method . the data from the infrared astronomical satellite ( iras ) survey in the four bands ( 12 , 25 , 60 and 100 @xmath11 ) for and were hires ( high resolution processing using maximum correlation method ; aumann et al . , 1990 ) processed at the infrared processing and analysis center ( ipac , caltech ) to obtain high angular resolution maps . the flux densities of the sources within a circular region of diameter @xmath22 ( centered on the peak ) have been extracted from these images . appears in iras - lrs catalog ( iras science team , 1986 ) while the lrs spectrum of is presented by volk & cohen ( 1989 ) . these spectra , in the wavelength range 8 - 22 @xmath11 , and the flux densities have been used for constructing the spectral energy distributions ( seds ) . the midcourse space experiment ( msx ) surveyed the entire galactic plane within @xmath23b@xmath24 in four mid infrared wavebands : 8.3 , 12.1 , 14.7 and 21.3 @xmath11 at a spatial resolution of @xmath25 ( price et al . , 2001 ) . the panoramic images of the galactic plane survey of msx were taken from ipac . the images of and were used to extract the sources and obtain the flux densities within a circular region of diameter @xmath22 in order to construct the seds . point sources close to these star forming regions have been selected from msx point source catalog version 2.3 ( egan et al . , 2003 ) and cross - correlated with 2mass sources ( see sect . 2.2.3 ) . the point sources around the regions and were selected from the two micron all sky survey ( 2mass ) point source catalog ( psc ) . the 2mass psc is complete down to j @xmath26 , h @xmath27 and k@xmath28 mag for s / n@xmath29 , in the absence of confusion . the 2mass sources used in this study are those with good photometric quality ( rdflg @xmath30 ) . the j , h and k@xmath31 magnitudes of the selected sources have been used to construct colour - magnitude ( cm ) and colour - colour ( cc ) diagrams which have been used to study the embedded clusters in these regions . the jhk@xmath31 magnitudes and images were taken from ipac . the sydney university molonglo sky survey ( sumss ) is a radio imaging survey of the sky south of declination @xmath32 ( bock et al . , 1999 ) . this survey uses the molonglo observatory synthesis telescope ( most ) , operating at 843 mhz with a bandwidth of 3 mhz . the synthesised beam size is @xmath33cosec@xmath34 . the sumss radio images of and have been used to study the distribution of ionised gas around this region . in this section , we present the results obtained from the observations as well as from the available data . as our study includes a wide wavelength range ( near infrared to radio ) , we classify our results according to those for the dust and gas components ( interstellar medium ) versus the stellar component for both the regions , and . the deconvolved maps of at 150 and 210 @xmath2 m are presented in fig . [ firmap_a ] . figure [ firmap_a ] ( left ) shows emission from the region at 150 @xmath2 m while fig . [ firmap_a ] ( right ) shows the emission at 210 @xmath2 m , from the complete region ( @xmath35 ) mapped by the telescope . the angular size ( 50% contour level ) of at 150 and 210 @xmath2 m is @xmath36 . the fir emission in these maps samples the cold dust around these regions . is well resolved in both the bands . the fir emission in this region shows extension towards north - east in both the bands . this emission towards the north - east extends to a larger scale in the 210 @xmath2 m map in fig . [ firmap_a ] ( right ) . taking advantage of the simultaneous observations in the two bands , with almost identical field of view , we have generated maps of the dust temperature ( @xmath37 ) and optical depth at 200 @xmath2 m ( @xmath38 ) . the dust temperature t(150/210 ) and optical depth ( @xmath39 ) maps are shown in fig . [ ttaubal_a ] . for these maps , we have assumed dust emissivity of the form @xmath40 . the temperature distribution shows plateaus of maximum ( 60 k ) towards north - west and south - east of the fir peak . we have detected dust as cold as 20 k. the peak optical depth at 200 @xmath2 m is determined to be @xmath41 and is located at the position of the peak of fir emission . another peak is seen to the west of the maximum optical depth with an extension corresponding to north - east . the iras - hires maps of the region around at 12 , 25 , 60 and 100 @xmath2 m are shown in fig . [ hires_a ] . the achieved angular resolutions are @xmath42 at 12 and 25 , @xmath43 at 60 @xmath11 , @xmath44 at 100 @xmath11 . the flux densities of integrated over @xmath22 diameter , centered on the peak , from the tifr and iras - hires maps are listed in table [ fluxes ] . we have modelled thermal continuum emission from interstellar dust alongwith the emission in unidentified infrared bands ( uibs ) , using mid - ir data from the msx galactic plane survey in the 8.3 , 12.1 , 14.7 and 21.3 @xmath11 bands . this has been carried out using the scheme developed by ghosh & ojha ( 2002 ) . in this scheme , the emission from each ( @xmath45 ) pixel in the msx images is modelled to be a combination of two components : ( i ) thermal continuum from the warm dust grains ( gray body ) and ( ii ) the emission from the uib features falling within the msx band . the scheme assumes that dust emissivity follows the power law of the form @xmath46 and the total radiance due to uibs in the 12 @xmath2 m band is proportional to that in the 8 @xmath2 m band . the spatial distribution of uib emission predicted by this scheme is presented in fig . [ 1004_pah ] . the morphology of uib emission shows extensions towards the south - west near the peak as well as due north - west at the fainter levels . the peak strength of the modelled uib emission is @xmath47 w m@xmath48 sr@xmath7 and is close ( @xmath49 ) to the iras position . table [ fluxes ] also lists the flux densities of from msx maps integrated within a circular region of diameter @xmath22 . the sumss radio continuum emission from the region around at 843 mhz is shown in fig . [ sumss_a ] . a dynamic range of @xmath50 is achieved in this region ( peak flux is 4.3 jy / beam ; rms noise is @xmath51 mjy / beam ) . the radio emission peaks at ( @xmath52 = @xmath53 @xmath54 @xmath55 , @xmath56 = -57@xmath9 12@xmath57 40.0@xmath58 ) . the integrated radio flux density is @xmath59 jy over 43.6 arcmin@xmath60 . the distribution of near - infrared sources ( selected from 2mass psc ) in the region around was investigated . a higher density of sources close to the iras region compared to neighbouring regions implied an embedded cluster . we have used 2mass sources to study the nature of embedded cluster . we first estimate the cluster radius . for this , we select a large region of radius @xmath61 around the iras peak . to account for the contribution from the field stars ( for background determination ) , we select a control field which is 20@xmath62 to the east of . the centre of the cluster was estimated by convolving a gaussian with the stellar distribution and taking the point of maximum density as the centre . to determine the radial profile , the cluster region was divided into a number of concentric annuli with respect to the cluster centre . the surface number density of stars were obtained by counting them in each 7@xmath63 annulus and dividing by the annulus area . the king s model , @xmath64 , and the inverse radius model , @xmath65 , of the following functional forms are fitted to the surface density radial profile . @xmath66 @xmath67 here , @xmath68 and @xmath69 are the fitted background constants for the king s and inverse radius profile , respectively . for the king s profile , @xmath70 represents the core concentration at radius zero and @xmath71 is the core radius . @xmath72 represents the stellar density of the inverse profile at the core . the radial profile of the observed star density as well as the fits are shown in fig . [ clust_prof ] . a clear gradient in surface number density distribution confirms the existence of clustering ( crowdiness ) at centre of the region . the radial profile of the cluster merges with the background field at @xmath73 , yielding the extent of the cluster to be @xmath4 pc . the background level as estimated from the control field is @xmath74 stars pc@xmath48 , which agrees well with the values of @xmath75 and @xmath76 stars pc@xmath48 yielded by the inverse radius and king s profile fitting , respectively . note that these estimates of the stellar density towards the hii region when compared with the adjacent fields are unlikely to be affected by extinction ( i.e. background stars ) since the cluster is fairly distant . within a cluster radius of @xmath77 around the cluster centre ( @xmath52 = @xmath53 @xmath54 @xmath78 , @xmath56 = -57@xmath9 12@xmath57 25.2@xmath58 ) , 89 sources were found . of these , 45 were detected in all the three jhk@xmath31 bands of 2mass . forty - four sources have been detected in either h and k@xmath31 bands or only k@xmath31 band . we have discussed the nature of stellar populations in this region using the cm ( j - h vs j ) and the cc ( j - h vs h - k ) diagrams which are shown in fig . [ cmcc_a ] . we have assumed extinction values of a@xmath79/a@xmath80 = 0.282 , a@xmath81/a@xmath80 = 0.175 and a@xmath82/a@xmath80 = 0.112 from rieke & lebofsky ( 1985 ) . all the 2mass magnitudes as well as the curves are in the bessel & brett ( 1988 ) system . in the cm diagram , the nearly vertical solid lines from left to right ( with increasing j - h ) represent the zams curves ( for a distance of 6.3 kpc ) reddened by a@xmath80 = 0 , 15 and 30 mag , respectively . the slanting lines joining them trace the reddening vectors of these zams stars . in the cc diagram , the locii of the main sequence and giant branches are shown by the solid and dotted lines , respectively . the short - dash line represents the locus of classical t - tauri stars ( meyer et al . , 1997 ) . the parallel dot - dash straight lines follow their reddening vectors . the long - dash line represents the locus of herbig ae / be stars ( lada and adams , 1992 ) . we classify the cc diagram into three regions , as shown in the figure ( see ojha et al . , the ` f ' region is considered to be the region where field stars ( main sequence stars , giants ) , class iii and class ii objects with small infrared excess are located . the ` t ' sources are located redward of region ` f ' but blueward of the reddening line projected from the red end of the t - tauri locus of meyer et al . these sources may be considered to be mostly class ii with large nir excess ( classical t tauri like - stars ) and/or extremely reddened early type zams stars , having excess emission in k - band ( e.g. , blum , damineli & conti , 2001 and references therein ) . the ` p ' region is where the protostar - like class i objects and herbig ae / be stars are mostly located . in fig . [ cmcc_a ] ( left ) cm diagram , the 10 sources lying above the zams curve of spectral type o9 are shown as asterisk symbols while 13 sources with infrared excess are shown as open circles . it is likely that few of these objects are foreground objects or bright background giants . the source lying above the zams curve of o9 and lying in the t region is shown as a solid triangle . the remaining sources are represented by plus - symbols . it is important to note that the cm and cc diagrams are useful tools for estimating the approximate nature of the stellar populations within the cluster in a statistical sense . the intensity maps of the region around in the 150 and 210 @xmath2 m fir bands are shown in fig . [ firmap_b ] . the emission at 210 @xmath2 m from the entire mapped region ( @xmath83 ) is shown in fig . [ firmap_b ] ( right ) . the isophotes in both the maps have been displayed upto 1% level of the peak intensities . a detailed investigation of these maps reveals that is unresolved at 150 @xmath2 m but is extended at 210 @xmath2 m map . at fainter levels , we observe large - scale extended emission , particularly towards the south - east . the hires - processed iras maps for the corresponding region around are shown in fig . [ hires_b ] . the angular resolution achieved in these maps are @xmath84 at 12 and 25 , @xmath43 at 60 @xmath11 , @xmath44 at 100 @xmath11 . the peak position and flux density details of are given in table [ fluxes ] . the emission in the mid infrared bands of msx has been used to model the the peak uib emission which is @xmath85 w m@xmath48 sr@xmath7 . the sumss radio emission in region around at 843 mhz is shown in fig . [ sumss_b ] . the dynamic range of the radio map is @xmath86 ( peak flux is 652 mjy / beam ; rms noise is @xmath4 mjy / beam ) . the peak of radio emission is at @xmath52 = @xmath53 @xmath87 @xmath88 , @xmath56 = -56@xmath9 46@xmath57 38.0@xmath58 . the integrated flux density is @xmath89 jy over 4.9 arcmin@xmath60 . we have used the 2mass sources in the vicinity of to study the stellar populations here . in a circular region of radius @xmath90 around the iras peak , 27 sources have been detected in jhk@xmath31 bands while 14 sources are detected in either h and k@xmath31 bands or only k@xmath31 band . the cm ( h - k vs k ) and cc ( j - h vs h - k ) diagrams of the sources detected in all the three bands are shown in fig . [ cmcc_b ] . the zams curves in the cm diagram are for a distance of 3.7 kpc reddened by a@xmath80 = 0 , 15 and 30 mag , respectively . as in the case of , all the 2mass magnitudes as well as the curves are in the bessel & brett ( 1988 ) system . the source lying above the zams curve of spectral type o9 is represented by an asterisk while the filled triangle represents a source lying above the zams curve of spectral type o9 and having an infrared excess . the 7 sources detected only in hk@xmath31 bands are shown as open squares in the figure . the remaining sources are denoted by crosses . an investigation into the distribution of 2mass psc sources in the region around shows a few sources grouped together near the iras peak . for this cluster , we have been unable to determine the cluster radius owing to the low surface density of stars ( very few excess stars close to the iras peak ) . in an attempt to obtain a self - consistent picture of these star forming regions using all the available data as well as to extract important physical parameters , we have carried out radiative transfer modelling of and . the symmetric morphology of contours near the centrally located peak in the fir maps supports the one - dimensional treatment of the radiative transfer modelling . each star forming region is modelled as a spherically symmetric cloud of gas ( hydrogen ) and dust , powered by a centrally embedded source ( single or cluster of zams stars ) . the cloud is immersed in an average interstellar radiation field . the radiative transfer equations have been solved assuming a two - point boundary condition for the spherical cloud . the gas exists throughout the cloud , i.e. from the stellar surface to the edge of the cloud . the dust exists in a spherical shell with a cavity at the centre . this is because close to the exciting source(s ) , the dust grains are destroyed when exposed to excessive radiative heating . two commonly used types of interstellar dust are explored ; the dl type ( draine & lee , 1984 ) and the mmp type ( mezger , mathis & panagia , 1982 ) . in the spherical shell where gas and dust co - exist , the gas - to - dust ratio is held constant . other details of this self - consistent scheme are given in mookerjea & ghosh ( 1999 ) . the parameters explored are as follows : the geometrical dimensions of the cloud ( primarily the radius of the dust - free cavity ; the outer diameter is guided by the observed angular extent and the distance ) , radial density distribution law ( three power law exponents : n(r ) @xmath1 r@xmath91 , r@xmath7 , or r@xmath48 ) , radial optical depth due to dust , the nature of embedded source/(s ) ( single zams star or a cluster , consistent with the total observed luminosity ) , relative abundances of different grain types ( silicate , graphite ) , and the gas - to - dust ratio . the observational constraints include the sed due to thermal emission from the dust component , angular sizes at different wavelengths and radio continuum emission from the h ii region . the parameters corresponding to the best fit models for the two sources are presented in table 2 . while no claim is made about the uniqueness of these parameter - sets , the following comments in support of the robustness of the results are in order : - a constant density radial profile provides the best fit for both regions and the r@xmath7 or r@xmath48 density profiles are conclusively ruled out by the observed sed ; - inner ( dust - free ) cavity radii lower than those derived result in short wavelength fluxes significantly higher than those observed and are thus ruled out ; - radio continuum emission is quite sensitive to the nature of the central energy source , viz . , single star versus a cluster . we have also modelled the infrared nebular / ionic fine structure line emission from gas around and , using a sophisticated scheme which uses more details of the interstellar gas component . this scheme developed by mookerjea & ghosh ( 1999 ) , uses the photoionisation code cloudy ( ferland , 1996 ) , which solves for statistical and thermal equilibrium by balancing various ionization - neutralization processes as well as heating - cooling processes . the scheme additionally includes : ( a ) the exact structure of the cloud / h ii region ( viz . , central dust - free cavity ) and ( b ) absorption effects of the dust component on the emergent line intensities . typical h ii region abundance of the gas component has been taken into consideration . the overall structure of the cloud is defined by the parameters that provide the best fit to the continuum spectral energy distribution ( see subsection 4.1 ) . the first part of the calculations involve the pure gas inner shell and the emerging spectrum comprises of continuum as well as line emission . this emergent continuum from the inner shell provides the inner surface boundary condition for the second shell comprising of gas and dust . the line emission from the inner shell is transported outwards through extinction by the dust column in the second shell . similarly , line emission originating from the second shell is transported considering the absorption effects of the dust grains lying between the emission zone and the outer surface of the second shell . the finally predicted emergent line luminosity includes both these components . a total of 27 spectral lines in the wavelength range @xmath92 @xmath2 m have been considered . to compare the predictions of the model with the spectral lines detected in the iras - lrs spectra of and , we convolve the model predicted spectrum with iras - lrs instrument profile in the wavelength range 8 - 22 @xmath2 m . however , to show the complete spectrum for the wavelength range 2.5 - 200 @xmath2 m , the spectral lines have been convolved with iso - sws and lws typical spectral resolutions of 1000 for @xmath93 @xmath2 m , 20000 for @xmath94 @xmath2 m , 8100 for @xmath95 @xmath2 m and 6800 for @xmath96 @xmath2 m while predicting the expected emergent spectrum . the sed for is constructed using the flux densities in the two tifr bands , the four iras bands ( from hires maps ) , the iras - lrs data and the four msx bands . the flux densities used in the sed are the fluxes integrated over a circular region of diameter @xmath22 around the peak . the total luminosity is @xmath97 l@xmath6 for a distance of 6.3 kpc . [ radtran_1004 ] ( left ) shows the observed sed and the predicted spectrum from the best fit model . this best fit model implies a uniform dust and gas density distribution with the embedded energy source as a single zams star of type o5-o4 ( see table 2 ) . the inner cloud dust radius is 0.008 pc while the outer cloud radius is 4.2 pc . the optical depth at 100 @xmath2 m is 0.002 . the radius of the ionised gas from the model is 2.8 pc . the observed angular sizes are explained by this model . the measured radio flux density , @xmath98 jy at 843 mhz is obtained by integrating within a circular region of 2.8 pc around the radio peak . this flux is quite high and can not be explained by the model as the predicted radio flux from the model is 4 jy for gas - to - dust ratio of 100:1 by mass . the dust composition for si : gr is 11:89 for the dl type of dust . the total dust mass is determined to be 12 m@xmath99 . the modelling of line emission for has been carried out using the physical sizes obtained from the modelling of continuum emission . in all , 21 nebular / ionic lines satisfy our detectability criterion ( power in the line be at least 1% of the power in the neighbouring continuum ) . the wavelengths and luminosities of these lines are presented in table [ model_linea ] . the modelled spectrum , including the lines from the 10 elements considered as well as the continuum predicted by this model is shown in fig . [ radtran_1004 ] ( right ) . we have looked for signatures of the ionic lines generally found in hii regions in the iras - lrs spectrum using a simple method . this method assumes the line signal to be present within data points , d@xmath100 , corresponding to one resolution element of lrs , centered at the expected line position . the local spectral baseline is estimated by interpolating ( power law ) data points in one resolution element each on both the _ red _ & _ blue _ sides of d@xmath100 . the estimated baseline is then subtracted from d@xmath100 to obtain the line emission . for , the ` brighter ' lines : [ ne ii ] at 12.8 @xmath2 m , [ ne iii ] at 15.5 @xmath2 m , [ s iii ] at 18.7 @xmath2 m and [ ar iii ] at 21.8 @xmath2 m are detected . the luminosities of these lines as well as the ratios of line - to - continuum are listed in table [ model_linea ] . we compare the luminosities as well as the ratios of line - to - continuum of the predicted lines from the model ( convolved with iras - lrs instrument profile ) with respect to those observed . the line - to - continuum ratios for the ` lrs - convolved ' model are listed in table [ model_linea ] . we find that the ratios of luminosities ( model / observations ) for the lines [ ne ii ] , [ ne iii ] and [ s iii ] agree within a factor of 4 . for the [ ar iii ] line , we find that the observed value is @xmath101 times larger than the modelled value . however , it is to be noted that [ ar iii ] line is close to the edge of the wavelength range of the spectrometer and could have instrumental uncertainities . the lines [ ar iii ] at 9.0 @xmath2 m and [ s iv ] at 10.5 @xmath2 m are barely detected although the model predicts them to be bright . the sed for has been constructed using flux densities ( integrated over a circular region of @xmath22 diameter centred on the peak ) from the msx , iras - hires as well as the tifr maps . the iras - lrs spectrum has also been used in the construction of the sed . the spectrum shows a strong silicate feature . for , the total luminosity is @xmath102 l@xmath6 ( d@xmath103 kpc ) . we have used a single zams star of spectral type o9 as the centrally exciting source . the best fit radiative transfer model along with the observed sed is shown in fig . [ radtran_1003 ] ( left ) , and the parameters of the best fit model are given in table [ radtran_parm ] . this model implies a uniform density distribution of gas and dust . the optical depth at 100 @xmath2 m ( from the model ) is 0.07 . the ratio of silicates and graphite dust grains is estimated to be 62:38 . the radio flux predicted by the model at 843 mhz is 0.07 jy for a gas - to - dust ratio of 100 by mass and radius of ionised region is 0.03 pc . this is comparable to the measured radio flux density of 0.04 jy at 843 mhz ( obtained by integrating the flux within a circular region of radius 0.03 pc centred on the radio peak ) . a dust mass of 24 @xmath104 is obtained from the model . for , 17 nebular / ionic lines satisfy the detectability criterion . the luminosities of the lines as well as the ratio of luminosities of each line with respect to the continuum ( for the model convolved with the lrs spectral resolution ) are also listed in table [ model_lineb ] . figure [ radtran_1003 ] ( right ) shows the emerging spectrum from the model . in addition , following the method in sect . 4.3 , we have searched for lines in the lrs spectrum of . the [ ne ii ] line at 12.8 @xmath2 m is clearly detected with a luminosity of 30 l@xmath6 . among the other lines in the range of lrs , we find that [ ar iii ] at 9.0 @xmath2 m and [ s iii ] at 18.7 @xmath2 m also show detections . a comparison of the luminosites of the detected lines with the predicted values shows that the observed luminosities are much higher than the modelled ones . table [ model_lineb ] also lists the ratio of line - to - continuum for these lines . a comparison of these ratios shows that the model ratios are a factor of @xmath105 lower than those observed . this can be attributed to the elevated level of the continuum from the model . a comparison of the model sed with the observed one in fig . [ radtran_1003 ] ( left ) shows that the radiative transfer model in the mid infrared overestimates the observed lrs spectrum . much larger discrepancies between the observed and predicted line emissions for , may be understood as follows . our scheme for predicting the line emission utilizes the description of the cloud , which is obtained from modelling the continuum emission from the dust component distributed throughout the cloud . in contrast , emission of the specific ionic lines under discussion , viz . , [ s iii ] , [ ne ii ] and [ ar iii ] are expected to originate from the innermost part of the h ii region ( due to high ionization potentials ) . in addition , the lower excitation of the central source ( zams o9 ) for this source , makes these line emissions more sensitive to the precise physical details ( e.g. inhomogeneities like clumpiness ) in the immediate vicinity of the star . since the emission of forbidden lines depends on square of the local density , denser clumps in an inhomogeneous medium will show enhanced emission compared to an equivalent uniform medium . hence , a clumpy medium around the exciting star in , could help explain the observed higher nebular line luminosities . this scenario is also consistent with the higher radio continuum emission observed . using the fir map of ( fig . [ firmap_a ] ) , we see an extension of the dust emission towards north - east . this is also seen in the 60 and 100 @xmath11 emission iras - hires maps of this region , shown in fig . [ hires_a ] . the iras - hires mid infrared emission from warm dust , however , shows an extension towards the west . this western extension is also clearly seen in the msx maps . the flux densities from the tifr maps at 150 and 210 @xmath11 have been used to compute the mass of dust using the formulation of hildebrand ( 1983 ) and sandell ( 2000 ) . for a temperature of 30 k , the dust mass obtained is @xmath106 @xmath104 . as can be seen from fig . [ sumss_a ] , a north - east extension seen also in the cold dust emission is observed for the ionised gas also . the total flux of 42.4 jy at 843 mhz can be compared with the flux density of 40 jy ( beam @xmath107 ) at 1410 mhz ( manchester , 1969 ) . from the cm ( j - h vs j ) diagram of the 2mass sources ( fig . [ cmcc_a ] left ) within the cluster radius ( @xmath108 ) , we see that there are 11 sources lying above the zams curve of o9 . these 11 sources are designated as ira-1 , ... , ira-11 and a list of their positions and magnitudes is given in table [ 2mass_parm_a ] . however , it is important to note that the spectral types inferred from cm diagram are the earliest possible spectral type ( upper limits ) when the stars have infrared excess . sources with infrared excess can be found from the cc ( j - h vs h - k ) diagram . in the cc diagram ( see fig . [ cmcc_a ] ) , 8 ` asterisks ' lie in the band occupied by reddened zams stars . one source , ira-7 ( shown as a triangle lying in the ` t ' region ) , shows an infrared excess while ira-10 lies to the left of the reddening band of the zams objects ( drawn from the top of the main sequence branch ) in the cc diagram . the upper left part of the cc diagram is not an allowed region for young stellar objects ( lada & adams , 1992 ) . ira-10 is faint in j band ( @xmath109 mag ) with larger errors though it is relatively brighter in the h ( @xmath110 mag ) and k@xmath31 ( @xmath111 ) bands as compared to the j band . one possibility to explain the near infrared colours of ira-10 as well as its position in the cc diagram is that ira-10 could comprise two or more unresolved sources . the sources lying near the unreddened main sequence in the cc diagram are possibly foreground sources not associated with the cluster . a small but significant fraction of cluster stars lie outside and to the right of the reddening band of the zams stars . these are mostly young stellar objects ( ysos ) with intrinsic colour excess . fourteen objects lie in this infrared excess zone , i.e. in the ` t ' and ` p ' ) regions . by dereddening the stars ( on the cc diagram ) that fall within the reddening vectors encompassing the main sequence stars , we find the visual extinction ( a@xmath80 ) towards each star . the individual extinction values range from @xmath112 magnitudes . from the cm diagram , we find that the extinction values of most ` asterisks ' lie at about @xmath113 magnitudes . figure [ 2mass_a ] shows the 2mass k@xmath31-band image of the region around in grayscale . the 2mass image of this region shows diffuse emission apart from the sources ( earlier than spectral type o9 ) clustered together . it is interesting to note the clustering of these sources close to the iras peak and the distribution of the other such stars in the ne - sw direction which is the direction of extension of the ionised gas as well as the cold dust . the morphology of emission in the uib as well as the emission from warm dust in msx bands also shows extension along the south - west direction . the brightest infrared source among the selected 2mass sources is ira-11 . ira-7 is closest ( @xmath114 ) to the iras peak , is of spectral type earlier than o5 and has an infrared excess . this is consistent with the spectral type determined by bik et al . ( 2005 ) for ira-7 ( referred to as 10049nr411 in their paper ) using high resolution k - band spectra . they find that it is of spectral type o3-o4 . recent multi - epoch radial velocity measurements by apai et al . ( 2007 ) show large amplitude variations in radial velocities pointing towards ira-7 being a massive binary system ( 50 and 20 m@xmath6 ) . bik et al . ( 2005 ) have also carried out spectroscopy of ira-3 ( 10049nr324 ) and find it to be of spectral type o3-o4/o5-o6 . this is consistent with its zams spectral type , o5-o6 , obtained from the cm diagram . it is interesting to note that the ` asterisk ' closest to the radio peak is ira-8 . the cm diagram indicates ira-8 to be of zams spectral type o9-o6 . we have also cross - correlated the msx psc sources with those from 2mass psc which lie within the cluster radius . there are 2 such msx psc sources namely g282.0341 - 01.1810 and g282.0176 - 01.1793 ( hereafter m1 and m2 , respectively ; listed in table [ msx ] ) . m1 coincides ( @xmath115 ) with an infrared excess source , 2mass - j10064115 - 5712377 ( j@xmath116 , h@xmath117 , k@xmath118 ) . m2 coincides ( @xmath119 ) with a source detected only in k@xmath31 band , 2mass - j10063592 - 5711563 ( k@xmath120 mag ) . from the 2mass - msx colour - colour diagram ( @xmath121 vs @xmath122 ; lumsden et al . 2002 , their fig . 9 ) , we find that these sources lie in the region generally covered by compact h ii regions and massive young stellar objects . we can , therefore , conclude that these are young stars associated with the cluster . the dust emission around is compact ( unresolved ) at 150 @xmath2 m , as well as at the iras - hires wavebands . is not resolved in the sumss radio map also . however , is barely resolved in the 210 @xmath2 m map . deconvolving the beam from 210 @xmath2 m image gives us @xmath123 as an estimate of its size . the dust mass obtained using the flux density at 210 @xmath11 from the tifr map is @xmath124 @xmath104 for a temperature of 30 k. however , from the radiative transfer modelling , we obtain a larger dust mass of 24 @xmath104 . the total radio flux density is 1.1 jy . considering that the radio flux is from a single ionising zams star , we use the formulation of schraml & mezger ( 1969 ) and panagia ( 1973 ) to estimate its spectral type . we find it to be of zams spectral type o9-o8.5 . this can be compared with the zams spectral type of o9 obtained from the fir luminosity . unlike the case of , we do not see a rich cluster with a large number of stars around . rather a small group comprising of a few stars is seen close to this region . there are seven sources ( labelled as irb-1 to irb-7 ) clustered close to the fir ( iras ) peak ; of which two , irb-1 and irb-5 , are detected in all the three ( j , h and k@xmath31 ) bands . the 2mass designation and flux details of these sources are given table [ 2mass_parm_b ] . from the cm diagram ( h - k vs k ; fig . [ cmcc_b ] left ) , it is observed that irb-1 is of zams spectral type earlier than o5 . it is to be noted that this spectral type of irb-1 obtained from nir study is the earlier possible spectral type ( upper limit ) as it shows an infrared excess . irb-5 is however of later spectral type , b3-b2 . it lies among a well - defined group of stellar sources which have lower extinction values ( a@xmath125 mag ) . from the cc diagram ( j - h vs h - k ) , it is found to lie near the main - sequence curve . it is therefore possible that irb-5 is not associated with the cluster but is a foreground source . the other five sources are detected in h and k@xmath31 or only in k@xmath31 bands . the source closest ( @xmath126 ) to the iras peak , irb-3 , is detected in h ( @xmath127 mag ) and k@xmath31 ( @xmath128 mag ) bands but not detected in the j band . from the cm diagram , it is found that irb-3 is a heavily extincted early spectral type ( o9-o6 ) zams star . it is located at the peak of 843 mhz radio emission . the other four sources irb-2 , irb-4 , irb-6 and irb-7 are detected only in k@xmath31 band . this implies that these are deeply embedded objects . thus , among the small number of cluster members , the majority of them are deeply embedded indicating that the star formation here is in an early stage . the 2mass image of the region around is shown in fig . [ 2mass_b ] . the sources lying above the zams curve of spectral type o9 are shown as asterisk symbols while the ` asterisk ' having an infrared excess is shown as a solid triangle . the open squares represent sources detected either in h and k@xmath31 bands or only in k@xmath31 band . the other sources are represented with plus - symbols . the seven sources discussed above are labelled as irb " in the figure . it is interesting to note the presence of young objects ( detected in h and k@xmath31 or only k@xmath31 bands ) clustered near the iras peak . in the region , we find that there is 1 msx psc source within @xmath90 radius centered on the iras peak , g281.5857 - 00.9706 ( designated as m3 and listed in table [ msx ] ) . m3 could possibly be associated with either irb-3 ( @xmath129 ) or irb-2 ( @xmath129 ) . in both the cases , we obtain the 2mass - msx colours ( @xmath121 and @xmath122 , lumsden et al . 2002 ) and observe that m3 lies in the general region covered by compact h ii regions . the massive star forming regions associated with and have been studied using the infrared ( near , mid and far ) wavebands . the dust and gas environments as well as the stellar sources have been probed using data from tifr balloon - borne telescope , msx , sumss and 2mass . the spatial distributions of far infrared emission from cold dust at 150 and 210 @xmath2 m have been obtained alongwith maps of optical depth ( @xmath39 ) and colour temperature , t(150/210 ) , using msx data , the emission from warm dust and uibs in the region has been studied . region shows the presence of a rich cluster of ob stars which gives rise to strong radio continuum ( @xmath130 jy at 843 mhz ) . the cluster radius is estimated to be @xmath108 and within the cluster radius , there are 11 2mass sources lying above the zams curve of spectral type o9 , designated ira-1 to ira-11 . one source , ira-7 also has an infrared excess and is located closest to the iras peak . it is a massive star of early spectral type , earlier than o5 consistent with earlier studies . unlike , the star forming region associated with , comprises of a very compact group of stars . the sumss radio flux at 843 mhz predicts a radio spectral type of o9-o8.5 . the sources clustered near the iras peak comprise of young sources detected in h and k@xmath31 or only k@xmath31 bands . the 2mass source detected in all the three bands , designated irb-1 , has an infrared excess and is found to be of zams spectral type earlier than o5 from the 2mass colour - magnitude diagram . self - consistent radiative transfer modelling constrained by observations have been carried out for both these sources . the geometric details of the clouds , the dust - composition and optical depths , etc have been obtained from the best fit models . we have also carried out modelling of line emission from and using a scheme based on cloudy . the predictions of the model for emission in the ionic lines are closer to lrs detections for than for . we speculate that for , some of the basic assumptions of the modelling scheme are not valid . we thank the anonymous referee for useful suggestions that improved the paper . it is a pleasure to thank the members of the infrared astronomy group at tifr for their support during laboratory tests and balloon flight campaigns . all members of the balloon group and control instrumentation group of the tifr balloon facility , hyderabad , are thanked for their technical support during the flight . we thank ipac , caltech , for providing us the hires - 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we investigate the physical conditions of the interstellar medium and stellar components in the regions of the southern galactic star forming complexes associated with and . these regions have been mapped simultaneously in two far infrared bands ( @xmath0 @xmath1 150 & 210 @xmath2 m ) , with @xmath3 angular resolution using the tifr 1-m balloon borne telescope . spatial distribution of the temperature of cool dust and optical depth at 200 @xmath2 m have been obtained taking advantage of the similar beams in the two bands . the hires processed iras maps at 12 , 25 , 60 and 100 @xmath2 m have been used for comparison . using the 2mass near infrared sources , we find the stellar populations of the embedded young clusters . a rich cluster of ob stars is seen in the region . the fits to the stellar density radial profile of the cluster associated with has been explored with inverse radius profile as well as the king s profile ; the cluster radius is obtained to be @xmath4 pc . the source in the cluster closest to the iras peak is ira-7 which lies above the zero age main sequence curve of spectral type o5 in the colour - magnitude diagram . unlike , a small cluster comprising of a few deeply embedded sources is seen at the location of . self consistent radiative transfer modelling aimed at extracting important physical and geometrical details of the two iras sources show that the best fit models are in good agreement with the observed spectral energy distributions . the geometric details of the associated cloud and optical depths ( @xmath5 ) have been estimated . a uniform density distribution of dust and gas is implied for both the sources . in addition , the infrared ionic fine - structure line emission from gas has been modelled for both these regions and compared with data from iras - low resolution spectrometer . for , the observed and modelled luminosities for most lines agree to within a factor of four while for , we find a discrepancy of a factor of 100 and it is likely that some basic assumptions of the model are not valid in this case .

in this paper we study the moduli spaces of tropical curves with marked points from the topological point of view . these spaces were recently introduced by mikhalkin in @xcite as important gadgets in tropical geometry , see also the work of sturmfels , e.g. , see @xcite for a nice introduction . mikhalkin s investigation centers on the tropical geometry of these spaces , going in particular depth in the case of genus @xmath4 ; here we complement his pioneering work by focusing exclusively on the topological properties . accordingly , we define the moduli spaces as imbedded in a larger space which we call _ moduli space of metric graphs with @xmath1 marked points , _ in particular , inheriting the natural topology from that larger space . we then prove that a simultaneous contraction of all the bridges is a strong deformation retraction . the rigorous proof of this fact is somewhat technical and requires corresponding rigorosity in the definition of the topology on the set of the isometry classes of the metric graphs with @xmath1 marked points . the main technical problem is to take care of the symmetries arising from the action of the automorhism group of the graph . all of this is done in section 3 . since all the edges in a tree are bridges , the shrinking bridges strong deformation retraction contracts the entire moduli space of tropical curves of genus @xmath4 with @xmath1 marked points to a point . therefore , the corresponding space is not of much interest from the topological point of view , as far as our current study is concerned . the first topologically interesting case , that of genus @xmath2 , is dealt with in section 4 . in this framework the shrinking bridges strong deformation retraction simplifies the analysis of the space dramatically , reducing it to the quotient of the torus by the conjugation action of @xmath3 . this can be done because a bridge - free graph of genus @xmath2 is isomorphic to a cycle . at present it seems difficult to describe the homotopy type of the obtained space in simple terms . however , it is possible to view it as a homotopy colimit of a simple diagram ( actually just a gluing of two identical mapping cylinders ) . we use that presentation to compute the betti numbers of the space with coefficients in @xmath3 , as well as to make a conjecture about the homology groups with integer coefficients . we refer to @xcite for the concepts and tools of combinatorial algebraic topology which are used throughout this paper . the few notions of the category theory that we use here can also be found in ( * ? ? ? * chapters 4 and 15 ) or in @xcite . @xmath5 all the graphs considered in this paper will be finite and undirected , however loops and multiple edges are allowed , and will , in fact , be essential for our investigation . let us now fix our notations . [ df : graph ] a * graph * @xmath6 is a pair of finite sets @xmath7 and @xmath8 equipped with set maps @xmath9 and @xmath10 , such that * @xmath11 , * the map @xmath12 has no fixed points . when more precise specification is needed , we also write @xmath13 instead of @xmath12 . we shall call @xmath7 the _ set of vertices_. we think of the elements of @xmath8 as _ directed edges _ , where the map @xmath12 changes orientation of such a directed edge to the opposite one , and the map @xmath14 takes a directed edge to its source vertex . accordingly , we introduce notation @xmath15 for the target vertex map . one can think of a graph as a diagram of sets over the category with two objects and three non - identity morphisms , see figure [ fig : graph ] , with the compositions of the morphisms given by the rules @xmath16 and @xmath17 . the condition that @xmath12 has no fixed points implies that elements of @xmath8 come in pairs @xmath18 . these pairs , or equivalence classes , are the _ ( undirected ) edges _ of @xmath6 and we denote the corresponding set by @xmath19 . elements @xmath20 , such that @xmath21 are called _ directed loops _ , and the corresponding equivalence classes in @xmath22 are called _ loops_. for arbitrary vertices @xmath23 , we let @xmath24 denote the set of edges directed from @xmath25 to @xmath26 . clearly @xmath27 , and the union is disjoint . the map @xmath12 is a bijection between @xmath28 and @xmath29 , and we set @xmath30 we have @xmath31 , for all @xmath23 . in particular @xmath32 denotes the set of directed loops at @xmath25 , and @xmath33 denotes the set of loops at @xmath25 . as an example , for a graph @xmath6 with one vertex and one loop we have @xmath34 , @xmath35 , with @xmath36 , @xmath37 , @xmath38 , and @xmath39 , @xmath40 . for two graphs @xmath6 and @xmath41 , a * graph homomorphism * from @xmath6 to @xmath41 is simply a map between corresponding diagrams of sets . in concrete terms , it consists of two set maps @xmath42 and @xmath43 , such that @xmath44 , and @xmath45 . a graph homomorphism is called a * graph isomorphism * if the involved set maps @xmath46 and @xmath47 are bijections . since @xmath44 we obtain induced map @xmath48 . for example , the graph with one vertex and one loop described above has two automorphisms , i.e. , invertible graph homomorphisms to itself . both are identity maps on the sets @xmath7 and @xmath22 . however , on the set @xmath8 , one is the identity map , and the other one swaps the directed edges @xmath49 and @xmath50 . @xmath5 we shall now associate a topological space @xmath51 to a graph @xmath6 . to avoid making noncanonical choices , and to aid our further considerations , we would like to think of the space @xmath51 as obtained by gluing together closed intervals corresponding to elements of @xmath8 . for this , let @xmath52 denote the closed interval , a copy of @xmath53\subset{{\mathbb r}}$ ] , corresponding to the element @xmath54 , and let @xmath55 be the union of all disjoint closed intervals @xmath52 . let furthermore @xmath7 also denote the discrete set of points indexed by elements of @xmath7 , and let @xmath56 be the discrete set of points indexed by the union @xmath57 consider the following maps : * a map @xmath58 , defined by @xmath59 ; * a map @xmath60 , which takes @xmath61 to the point in @xmath52 corresponding to @xmath4 , and takes @xmath62 to the point in @xmath52 corresponding to @xmath2 ; * a map @xmath63 which takes a point in @xmath52 corresponding to @xmath64 $ ] to the point in @xmath65 corresponding to @xmath66 , for all @xmath54 , and all @xmath64 $ ] . together with spaces @xmath56 , @xmath7 , and @xmath55 these maps form a diagram shown in figure [ fig : wvod ] , which one can think of as a gluing data for @xmath51 . this is made precise by the following definition . [ df : dag ] for an arbitrary graph @xmath6 , we let the topological space @xmath51 be the colimit of the diagram shown in figure [ fig : wvod ] . we let @xmath67 denote the map induced by the structural maps from the spaces in the diagram to the colimit of that diagram . the map @xmath68 , and hence also the map @xmath69 , is surjective . clearly , @xmath51 has a structure of a @xmath2-dimensional cw complex , whose @xmath4-cells are indexed by the vertices of @xmath6 , @xmath2-cells are indexed by the edges of @xmath6 , i.e. , by @xmath12-invariant pairs of elements from @xmath8 , and the attachment maps are given by the vertex - edge incidences . when appropriate we shall identify vertices of @xmath6 with corresponding @xmath4-cells of @xmath51 , and edges of @xmath6 with corresponding open @xmath2-cells of @xmath51 . a graph homomorphism from @xmath6 to @xmath41 induces a natural diagram map from the gluing data of @xmath6 to the gluing data of @xmath41 , and therefore it also induces a natural cw map from @xmath51 to @xmath70 , which is in fact a homeomorphism when restricted to any open @xmath2-cell of @xmath51 . we denote both maps by @xmath71 . when @xmath72 is a graph isomorphism , the map @xmath71 is a cw isomorphism . as the last piece of terminology here , the first betti number of @xmath51 will be called the _ genus _ of @xmath6 , and denoted by @xmath73 . clearly @xmath74 . @xmath5 [ ssect : mgraphs ] to introduce more structure we now vary the lengths of the edges . let @xmath6 be a graph . we say that @xmath6 is a * metric graph * when we are given a function @xmath75 , called the * edge - length function*. the index @xmath6 will be skipped in @xmath76 whenever it is clear which graph is considered . we shall also use @xmath76 to denote the corresponding @xmath12-invariant function @xmath77 . given a metric graph @xmath78 , there is a standard way to use the function @xmath76 to turn the topological space @xmath51 into a metric space , which we now describe . we identify each @xmath52 with the metric space @xmath79 $ ] , instead of the topological space @xmath53 $ ] , with the standard distance function given by @xmath80 , for @xmath81 $ ] . we can now define a distance function @xmath82 on @xmath51 as follows : @xmath83 , for all @xmath84 , and for @xmath85 , @xmath86 we set @xmath87 where the minimum is taken over all @xmath88-tuples @xmath89 of points from @xmath55 , such that points @xmath90 and @xmath91 belong to the same closed interval @xmath52 , for all @xmath92 , so the distance @xmath93 is taken in this interval , and furthermore @xmath94 , @xmath95 , and @xmath96 for all @xmath97 . from now on , whenever @xmath6 is a metric graph , we think of @xmath51 as a metric space with the standard metric which we just described . given a graph homomorphism from @xmath6 to @xmath41 , we can adjust the induced diagram map from the gluing data of @xmath6 to the gluing data of @xmath41 to the metric setting , by taking the map from @xmath52 to @xmath98 to be the dilation with the scaling factor @xmath99 . in the colimit we get the induced map from @xmath51 to @xmath70 , which we also denote by @xmath71 . let @xmath6 be a metric graph , and let @xmath1 be a nonnegative integer . we say that @xmath6 is a * metric graph with @xmath1 marked points * , when we are given a function @xmath100{\rightarrow}{\delta}(g)$ ] called the * marking function*. here we use the convention @xmath101:=\{1,\dots , n\}$ ] for natural numbers @xmath1 , and @xmath102:=\emptyset$ ] . formally , metric graphs with @xmath1 marked points are given by triples @xmath103 . clearly , these generalize metric graphs , which we can recover by setting @xmath104 . for @xmath84 , we say that _ @xmath25 is marked with @xmath105 _ , or simply that @xmath25 _ is marked _ , in case that subset of @xmath101 $ ] is not empty . we call a point @xmath84 _ special _ if it is vertex or a marked point ( or both ) . two metric graphs @xmath6 and @xmath41 with @xmath1 marked points are said to be * isometric * if there exists a graph isomorphism consisting of the maps @xmath42 and @xmath106 , such that we have @xmath107 , and the marked points are mapped appropriately by the corresponding isometries of the edges , i.e. , @xmath108 . a graph isomorphism @xmath109 from @xmath6 to @xmath41 , which is also an isometry of metric graphs , induces an isometry @xmath71 of the corresponding metric spaces . isometry of metric graphs with @xmath1 marked points is clearly an equivalence relation . when @xmath6 is a metric graph with @xmath1 marked points we let @xmath110 $ ] denote the corresponding equivalence class ( that is the set of all metric graphs with @xmath1 marked points which are isometric to @xmath6 ) . likewise , for a set @xmath111 of metric graphs with @xmath1 marked points we set @xmath112:=\{[g]\,|\,g\in s\}$ ] . @xmath5 [ ssect : shrink ] given a metric graph @xmath6 with @xmath1 marked points and @xmath113 , which is not a loop , we can define a new metric graph @xmath114 as follows . let @xmath115 be the set of the endpoints of @xmath116 , @xmath117 , and let @xmath118 be a label which is not in @xmath7 ; for reasons which will become clear shortly , we let the label @xmath118 be the set @xmath115 itself . let furthermore @xmath119 be the directed edges corresponding to @xmath116 . we set @xmath120 and @xmath121 . in particular , we see that @xmath122 . let @xmath123 be the map defined by @xmath124 , and @xmath125 for @xmath126 . we now set @xmath127 and @xmath128 . in concrete terms , we have @xmath129 the function @xmath130 is set to be the restriction of @xmath76 to @xmath131 . clearly , we have a surjective map @xmath132 which `` shrinks '' the edge @xmath116 , and the marking function @xmath133 is taken to be the composition @xmath134 . more generally , let @xmath135 be a set of edges which forms a subforest of @xmath6 , i.e. , the induced graph _ contains no cycles _ ( in particular , the set @xmath111 contains no loops ) , and let @xmath136 be the set of corresponding directed edges . one can then shrink the set @xmath111 just like we shrunk a single edge . more precisely , let @xmath137 be the graph whose set of vertices is @xmath7 and whose set of edges is @xmath111 . the new graph @xmath138 is now obtained by taking the connected components of @xmath137 as vertices , and setting @xmath139 , hence @xmath140 . let @xmath123 be the map taking every vertex of @xmath6 to the connected component of @xmath137 which contains it . we can then define @xmath127 and @xmath128 ; just like for the case when @xmath111 consists of a single edge . for @xmath141 we now have @xmath142 we again let @xmath130 be the restriction of @xmath76 to @xmath143 , we have a surjective shrinking map of topological spaces @xmath132 , and we set @xmath144 . we want to point out a subtlety related to the edge shrinking . given two disjoint sets of edges @xmath145 and @xmath146 we could shrink all these edges provided that @xmath147 forms a subforest . if we shrink first @xmath145 and then @xmath146 we get a different graph from the one obtained by shrinking the set @xmath147 right away . this is because the labels of the vertices will be different . for example , for a graph with 3 vertices and 2 edges g given by @xmath148 , @xmath149 , @xmath150 , shrinking first @xmath116 and then @xmath151 yields a graph with a single vertex labelled @xmath152 and no edges , while shrinking the entire set @xmath153 right away yields a graph with a single vertex labelled @xmath154 and no edges . however , it is easy to see that the isometry class @xmath155 $ ] of the obtained graph does not depend on the order in which we do the shrinking . we will implicitly use this fact in the future arguments . @xmath5 let @xmath1 be a nonnegative integer , and let @xmath156 denote the set of all isometry classes of finite metric graphs with @xmath1 marked points . we would like to turn this set into a topological space . for this we need to say when two isometry classes of metric graphs with @xmath1 marked points `` are close . '' let @xmath6 be a metric graph with @xmath1 marked points . we set @xmath158 , where the minimum is taken over all pairs of special points @xmath85 . note that since the number of special points is necessarily finite , the minimum is well - defined . we shall refer to the open interval @xmath159 as the _ admissible range _ of @xmath6 , this is the range from which the sizes of the neighborhoods of @xmath110 $ ] in @xmath156 shall be sampled , and depends only on the isometry class @xmath110 $ ] , not on the choice of @xmath6 . let @xmath160 be a number from the admissible range of @xmath6 . we now define a set @xmath161 as follows : a metric graph with @xmath1 marked points @xmath41 lies in @xmath161 if and only if 1 . the edges of @xmath41 of length less than @xmath160 form a subforest ; 2 . the metric graph @xmath6 can be obtained from @xmath41 by shrinking all the edges of lengths less than @xmath160 , as described in subsection [ ssect : shrink ] , and by subsequently varying lengths of remaining edges and positions of marked points by up to @xmath160 . the latter can be formalized as follows . we say that a metric graph @xmath6 with @xmath1 marked points can be obtained from another metric graph @xmath41 with @xmath1 marked points by varying lengths of edges and the positions of marked points by up to @xmath160 if there exists a graph isomorphism @xmath72 from @xmath6 to @xmath41 such that 1 . for all @xmath113 we have @xmath162 ; 2 . for all @xmath163 $ ] , we have @xmath164 . finally , we set @xmath165):=[n_\varepsilon(g)]\subseteq{mg_n}$ ] . choosing different representatives of @xmath110 $ ] means simply changing labels of vertices and edges , therefore the set @xmath165)$ ] is independent on the particular choice of @xmath6 . it is easy to see that when @xmath166 lies in the admissible range of @xmath6 , and @xmath167 , we have @xmath168)\subseteq n_{\varepsilon_1}([g])$ ] . this is because for any @xmath169 , and for any edge @xmath116 of @xmath41 we can not have @xmath170 , as otherwise @xmath116 would correspond to an edge of @xmath6 , such that @xmath171 , which is impossible since the inequalities @xmath172 and @xmath173 imply @xmath174 we are now ready to topologize the set of all isometry classes of metric graphs with @xmath1 marked points . let @xmath1 be a nonnegative integer . the * moduli space of metric graphs with @xmath1 marked points * is the topological space whose set of points is given by @xmath156 , and whose topology is generated by the sets @xmath165)$ ] as follows : a subset @xmath175 is open if and only if for every @xmath110\in x$ ] there exists @xmath176 , such that @xmath177)\subseteq x$ ] . we leave to the reader the verification of the fact that the spaces @xmath177)$ ] are themselves open . we extend the usage of @xmath156 to denote the corresponding topological space as well . there are various natural modifications of @xmath156 . for example , one could require the metric graphs to be connected . we denote the corresponding subspace of @xmath156 by @xmath178 . another , independent possibility is to require that the marked points are vertices of the graph . we denote the corresponding subspace of @xmath156 by @xmath179 . combining , we let @xmath180 denote the subspace of @xmath156 consisting of the isometry classes of connected metric graphs with @xmath1 marks on vertices ( as we constructed it , multiple marks are allowed ) . given a metric graph @xmath6 with @xmath1 marked points , we let @xmath181 denote the metric graph obtained from @xmath6 by turning all marked points into vertices ( of course , in case they were not vertices already ) . clearly , the isometry class @xmath182 $ ] depends on the isometry class @xmath110 $ ] only , hence the map @xmath183 given by @xmath184{\rightarrow}[g^{{\mathfrak{v}}}]$ ] is well - defined . the map @xmath183 making all marked points into vertices is a retraction . * by construction , we have @xmath185 . furthermore , we see that the map @xmath72 is continuous . indeed , for any metric graph @xmath6 with @xmath1 marked points the neighborhood @xmath186)$ ] is mapped inside the neighborhood @xmath187)$ ] . this is because allowed ( for staying in the neighborhood @xmath186)$ ] ) deformations of the graph @xmath6 : edge contraction of edges shorter than @xmath188 , changing the edge lengths by up to @xmath188 , shifting a marked point by at most @xmath188 , can all be realized by edge contractions of edges shorter than @xmath160 and changing edge lengths by at most @xmath160 in the graph @xmath181 , when @xmath160 is in the admissible range of @xmath6 . we therefore conclude that @xmath72 is a retraction . @xmath5 let us now describe the connected components of @xmath156 . let @xmath6 be a metric graph with @xmath1 marked points , and let @xmath189 be its connected components . assume that @xmath190 has genus @xmath191 , for @xmath192 , and let @xmath193 be disjoint , possibly empty sets whose union is @xmath101 $ ] ( this is like a set partition , but with empty sets allowed ) . the set @xmath194 is now the data which we associate to the graph @xmath6 . it is not difficult to see that any graph with the same data lies in the connected component which contains @xmath110 $ ] . furthermore , the data of this type , meaning a set of tuples @xmath195 , such that @xmath196 and @xmath197 is a set partition of @xmath101 $ ] , possibly involving empty sets , index connected components of @xmath156 . consider now a connected component @xmath198 indexed by the set @xmath199 where the sets @xmath200 are non - empty . denote the number of appearances of the tuple @xmath201 in that set by @xmath202 , for @xmath203 . by our assumptions , only finitely many of these are different from @xmath4 . then we have a homeomorphism @xmath204 where @xmath205 denotes the @xmath206-fold symmetric product , i.e. , the quotient space @xmath207 , where the symmetric group @xmath208 acts on the direct product by permutation of its factors . we shall use @xmath209 to denote the topological space whose points are the isomorphism classes of connected graphs of genus @xmath0 with @xmath1 marked points , which is the same as the connected component @xmath210)}$ ] . @xmath5 the moduli space @xmath156 has a natural stratification . to produce a stratum , fix a graph @xmath6 , and for each @xmath163 $ ] , fix @xmath211 , which is either a vertex or an edge of @xmath6 . now , consider the set of all isometry classes of metric graphs with @xmath1 marked points , which have a representative @xmath212 , such that there exists a graph isomorphism @xmath72 from @xmath6 to @xmath41 , for which the point @xmath213 is equal to @xmath214 , if @xmath211 is a vertex , or belongs the open edge @xmath215 , if @xmath211 is an edge with endpoints @xmath25 and @xmath26 , for all @xmath163 $ ] . this stratum is indexed by the graph @xmath6 together with the @xmath1-tuple @xmath216 ; we denote it by @xmath217 . we shall call this stratification _ standard _ , and we shall its strata the _ standard strata_. the stratum does not change if we replace @xmath6 with an isomorphic graph , and change the @xmath1-tuple @xmath218 accordingly . we shall implicitly use this fact in our discussion . we let @xmath219 denote the closure of the stratum @xmath217 , and we let @xmath220 denote @xmath221 , which we shall call the _ boundary _ of the stratum . one can see that the boundary of an arbitrary stratum @xmath217 is a union of other strata . the indexing data of these strata can be obtained from @xmath222 by a combination of the steps of the following two kinds 1 . replacing an edge by one of its endpoints in the @xmath1-tuple @xmath218 ; 2 . shrinking a non - loop edge in @xmath6 and replacing this edge and its endpoints by the label of the thus obtained vertex in the @xmath1-tuple @xmath218 . in general the strata do not have to be manifolds . consider , for example , the stratum @xmath217 , where @xmath6 is a graph with one vertex @xmath223 and one edge @xmath116 ( which hence must be a loop ) , @xmath224 , and @xmath225 . we can vary the length of the edge in the open interval @xmath226 , and we can slide the marked point along the edge . thus potentially the point moves in the interval @xmath227 , however , because of the symmetry which flips the loop , we need to identify coordinates @xmath25 and @xmath228 , so we can choose a representative from the half - closed interval @xmath229 $ ] . these considerations show that the stratum @xmath217 is homeomorphic to the space @xmath230 , which in turn is homeomorphic to the space @xmath231 . on the positive side , as easily seen , the generic points of every stratum @xmath217 form an open manifold whose dimension is equal to the number of edges of @xmath6 plus the number of labels which are edges in the @xmath1-tuple @xmath218 . it follows , that there are infinitely many strata of dimension @xmath4 ; these are indexed by graphs @xmath6 with no edges , whose vertices are labeled by disjoint subsets of @xmath101 $ ] , so that the union of all labels is @xmath101 $ ] , which is the same as to index them by sets @xmath232 , where we might have @xmath233 , such that @xmath101=\cup_{i=1}^t a_i$ ] , and the union is disjoint . in general , the space @xmath156 is somewhat technical to handle directly : it is infinite dimensional , and an arbitrarily small neighborhood of each point intersects infinitely many strata ; for example , in the case with no marked points , an arbitrarily small neighborhood of the graph with one vertex and no edges intersects all strata indexed by trees . we shall therefore start by performing the shrinking bridges strong deformation retraction , in order to replace @xmath156 by a more manageable space . @xmath5 recall , that an edge @xmath116 of a graph @xmath6 is called a _ bridge _ if deleting it from the graph @xmath6 increases the number of connected components . equivalently , @xmath116 is a bridge if the endpoints of @xmath116 belong to different connected components of @xmath234 ( cf . clearly , shrinking a bridge will neither change the number of connected components of @xmath6 , nor will it change the genuses of these connected components . for example , every edge of a forest is a bridge . we shall call a graph which has no bridges _ bridge - free _ , and we shall denote the set of bridges of @xmath6 by @xmath235 . let us now define a homotopy @xmath236{\rightarrow}{mg_n}$ ] . let @xmath6 be a metric graph with @xmath1 marked points , and let @xmath237 $ ] . let @xmath238 denote the metric graph with @xmath1 marked points obtained from @xmath6 by scaling down all the edges in @xmath235 by the factor @xmath206 , and adjusting the marking function accordingly . for @xmath239 we set @xmath240 to be the graph obtained from @xmath6 by shrinking all the bridges , this is allowed since the set of all bridges forms a subforest of @xmath6 . up to isomorphism , the metric graph @xmath238 with @xmath1 marked points is uniquely determined by the isomorphism class of @xmath6 , and by the parameter @xmath206 , hence the assignment @xmath241,t):=[\beta(g , t)]$ ] is well - defined . let @xmath242 denote the subspace of @xmath156 consisting of all the isometry classes of metric bridge - free graphs @xmath6 with @xmath1 marked points . clearly , the space @xmath242 is a union of standard strata , and the map @xmath243 surjectivity takes @xmath156 to @xmath242 . [ thm : bbh ] the space @xmath242 is a strong deformation retract of the space @xmath156 . the map @xmath236{\rightarrow}{mg_n}$ ] provides a corresponding strong deformation retraction . * proof . * as already mentioned , we have @xmath241,0)\in{mg_n}^{{\mathfrak b}}$ ] , for any metric graph @xmath6 with @xmath1 marked points . furthermore , it follows directly from our definition of the map @xmath244 , that @xmath241,t)=[g]$ ] , for all @xmath110\in{mg_n}^{{\mathfrak b}}$ ] , and that @xmath245 . therefore , to prove that @xmath244 is an appropriate strong deformation retraction , it is enough to show that it is continuous . we shall now provide a direct , albeit somewhat tedious verification . we take a point @xmath246 and a point @xmath247 $ ] , such that @xmath248 , and then show that for every sufficiently small @xmath160 ( how small it needs to be shall depend on @xmath25 and @xmath26 ) , there exists a neighborhood @xmath249 of @xmath26 , which is taken by @xmath244 inside of @xmath250 : @xmath251 . we use the notations @xmath252,t)\in{mg_n}\times[0,1]$ ] , where @xmath6 is a metric graph with @xmath1 marked points , and accordingly @xmath253,t)=[\beta(g , t)]$ ] . it is technically easier to divide the argument into considering two separate cases . * _ we assume that @xmath254 . _ this is the easier one of the two cases . for sufficiently small @xmath176 , we look for @xmath255 , such that @xmath244 maps the @xmath26-neighborhood @xmath256)\times[t-\delta_2,t+\delta_2]$ ] inside of @xmath250 . since @xmath256)=[n_{\delta_1}(g)]$ ] , and @xmath257,t))= n_\varepsilon([\beta(g , t)])= [ n_\varepsilon(\beta(g , t))]$ ] , it is enough to find @xmath160 , @xmath258 , and @xmath259 , such that @xmath244 maps @xmath260 $ ] inside of @xmath261 , as long as the conditions on @xmath160 , @xmath258 , and @xmath259 depend only on the isomorphism class of @xmath6 , not on the specific representative . in any case , we assume that @xmath160 is sampled from the admissible range of @xmath6 ( which only depends on @xmath110 $ ] ) . let now @xmath41 be a metric graph with @xmath1 marked points in @xmath262 , and let @xmath137 denote the set of edges of @xmath41 of length less than @xmath258 . by our construction , these must form a subforest . let us fix some @xmath263 in the interval @xmath264 $ ] . the graph @xmath265 is obtained from @xmath41 by shrinking all the bridges by a factor @xmath266 , @xmath267 . therefore , requesting that @xmath268 will ensure that the images of the edges from @xmath137 will have length less than @xmath160 . on the other hand , we want the images of the edges from @xmath269 , i.e. , the original edges of @xmath6 , to have lengths larger than @xmath160 . this can be ensured by requesting that @xmath270 therefore , the graph @xmath271 can be obtained from the graph @xmath265 by shrinking all the edges of length less than @xmath160 , and then varying the lengths of the remaining edges , as well as positions of marked points , by up to an @xmath160 . the latter follows from the fact that @xmath268 , and shrinking some of the edges only decreases the edge length variation . passing on to the whole interval @xmath264 $ ] , we choose @xmath160 so that @xmath272 and then take @xmath268 and @xmath273 . this choice of parameters verifies the continuity of the map @xmath244 at the point @xmath26 . * _ we assume that @xmath239 . _ we have @xmath253,0)\in{mg_n}^{{\mathfrak b}}$ ] . this time , for sufficiently small @xmath176 , we need to find @xmath255 , such that @xmath244 maps the @xmath26-neighborhood @xmath256)\times[0,\delta_2]$ ] inside of @xmath250 . just like in the first case , we can drop the isomorphism brackets , and search for @xmath160 , @xmath258 , and @xmath259 , such that @xmath244 maps @xmath274 $ ] inside of @xmath275 . let again @xmath41 be a metric graph with @xmath1 marked points in @xmath262 , and let @xmath137 denote the set of edges of @xmath41 of length less than @xmath258 . it important to note , that the set of edges @xmath276 forms a subforest . this is because @xmath137 is a forest , and adding bridges to any forest will not create cycles , since bridges can not be a part of any cycle . also , as noted before , we have @xmath277 . the graph @xmath278 is obtained from @xmath41 by shrinking all the bridges . this means shrinking all the bridges of @xmath6 , and possibly some of the edges from @xmath137 . choosing @xmath160 smaller than @xmath279 , and then choosing @xmath280 ensures that the edges of @xmath278 whose lengths are less than @xmath160 are precisely the non - bridges from the set @xmath137 . this means that the graph @xmath240 can be obtained from the graph @xmath278 by shrinking the edges of length less than @xmath160 , and varying the lengths of other edges , as well as positions of the marked points , by up to an @xmath160 . assume now @xmath259 is chosen so that @xmath281 and choose @xmath282 . the graph @xmath283 is obtained from @xmath41 by scaling all the bridges down by the factor @xmath206 . the way @xmath259 is chosen , this will scale down all the bridges of @xmath6 , so that they become shorter than @xmath160 . furthermore , the conditions @xmath284 imply that also all the edges from @xmath137 will be shorter than @xmath160 ( since they were shorter than @xmath160 to start with , and some were additionally shrunk ) , and that all the non - bridges of @xmath6 will not be shorter than @xmath160 . hence again the graph @xmath240 can be obtained from the graph @xmath283 by shrinking the edges of length less than @xmath160 , and varying the lengths of other edges , as well as positions of the marked points , by up to an @xmath160 . this finishes the verification of the fact that the homotopy @xmath244 is continuous at the point @xmath26 in this case . @xmath5 we now define a subspace of @xmath285 which is of special interest in tropical geometry and has been the starting point of the current investigation . let @xmath1 be a nonnegative integer , and let @xmath82 be a positive number . we define @xmath286 to be the subspace of @xmath285 consisting of the isomorphism classes of all metric graphs @xmath6 with @xmath1 marked points , such that 1 . @xmath6 has no vertices of valency 2 ; 2 . @xmath6 has exactly @xmath1 leaves , cf . @xcite . ] , and these are marked @xmath2 through @xmath1 ; 3 . the lengths of the edges leading to leaves are equal to @xmath82 . simultaneous dilation of the edges leading to leaves gives a homeomorphism between spaces @xmath287 and @xmath288 , for arbitrary positive @xmath289 and @xmath290 . letting @xmath82 go to @xmath4 we obtain yet another homeomorphic space , which it would be natural to denote by @xmath291 , but for simplicity we just call it @xmath157 . this space can also be described directly , as is done in the next definition . [ df : tmn ] let @xmath1 be a nonnegative integer . we define @xmath157 to be the subspace of @xmath285 consisting of the isomorphism classes of all metric graphs @xmath6 with @xmath1 marked points , such that for every vertex of @xmath6 the sum of its valency with the number of times it is marked should be at least 3 . the condition in definition [ df : tmn ] just means that every vertex of valency 2 should be marked , and that every leaf should be marked at least twice . it is not a difficult exercise to see that every two points of @xmath157 corresponding to metric graphs of the _ same _ genus , can be connected by a path inside @xmath157 , whereas obviously , every two points of @xmath157 corresponding to metric graphs of _ different _ genus , can not be connected by such a path , not even inside of @xmath156 . hence the connected components of @xmath157 are indexed by nonnegative integers @xmath0 , corresponding to the genuses of the involved graphs , and we call them @xmath292 . is a strong deformation retract of the graph moduli space @xmath293 , see @xcite . ] it is important to not that , unlike the spaces @xmath286 , the space @xmath157 is a union of the standard strata of @xmath156 . [ crl : tmnb ] let @xmath294 . then the space @xmath295 is a strong deformation retract of @xmath157 . * proof . * the space @xmath157 is obviously closed under the shrinking bridges strong deformation retraction . therefore , the theorem [ thm : bbh ] implies that @xmath157 strongly deformation retracts to @xmath295 . the space @xmath295 can be also described directly : it is the subspace of @xmath285 consisting of all isomorphism classes of metric graphs @xmath6 with @xmath1 marked points , such that 1 . every vertex of @xmath6 of valency 2 is marked ; 2 . the graph @xmath6 has no bridges . the connected components of @xmath295 are again indexed by genuses of the constituting graphs , and we shall use the notation @xmath296 . since all the edges of a tree are bridges , we see that @xmath297 and @xmath298 are just single points . theorem [ thm : bbh ] and corollary [ crl : tmnb ] imply that the spaces @xmath299 and @xmath300 are contractible . in this section we shall focus on the next interesting case : namely the spaces of connected metric graphs of genus @xmath2 with @xmath1 marked points . let us start by analysing the space @xmath301 . the bridge - free graphs of genus @xmath2 are simply cycles , hence , since all their vertices have valency equal to @xmath302 , they should all be marked , and @xmath1 should be at least @xmath2 . reversely , all the marked points are vertices . we can therefore forget about the vertices and just record the marked points . the points of the space @xmath303 can thus be indexed by @xmath1-tuples of points on a circle , whose radius is an arbitrary positive real number , divided by the action of the orthogonal group on the circle . factoring out the radius length of the circle , we have a homeomorphism @xmath304 for @xmath305 , where @xmath306 denotes the unit circle , and the division is done with respect to the diagonal action of the orthogonal group @xmath307 , which acts on each term in the natural way . note , that the described action of @xmath307 on the direct product @xmath308 is transitive on the last coordinate . let us fix the last coordinate to be @xmath309 . then the space @xmath310 can be rewritten as @xmath311 where @xmath3 is the subgroup of @xmath307 which fixes the point @xmath312 . clearly , the group @xmath3 consists of two elements , and the non - identity element is an involution which acts diagonally on the direct product of @xmath313 copies of @xmath306 by a reflection about the @xmath314-axis . one way to think of this action is to view the points of @xmath306 as complex numbers with absolute values equal to @xmath2 , in which case the action is just a simultaneous conjugation : @xmath315 consider the cw structure on a unit circle @xmath306 consisting of two @xmath4-cells @xmath312 and @xmath316 , and two @xmath2-cells corresponding to upper and lower semicircles . this induces a cw structure on the direct product of @xmath1 copies of @xmath306 ( which is an @xmath1-torus ) , where the cells are simply direct products of cells of the factors . these are indexed by the ordered @xmath1-tuples of the cells of the factors and we shall use the following encoding : we write `` @xmath317 '' for the @xmath4-cell @xmath312 , `` @xmath318 '' for the @xmath4-cell @xmath316 , `` @xmath319 '' for the upper semicircle , and `` @xmath320 '' for the lower semicircle , as shown on figure [ fig : circle ] . so , for example , @xmath321 would denote a @xmath302-cell in the @xmath322-torus . the open cells are actually cubes , so we can think of this decomposition as some sort of a cubical structure on @xmath308 . in this encoding , the boundary of a cell is obtained by replacing symbols @xmath319 and @xmath320 with symbols @xmath317 and @xmath318 . the number of @xmath82-cells is @xmath323 , as we can put a plus or a minus in every coordinate , and then distribute @xmath82 symbols `` @xmath324 '' arbitrarily . clearly , this cubical structure on @xmath325 is invariant under the @xmath3-action of simultaneous conjugation . in our symbolic notations , the action changes the signs assigned to @xmath324 s and fixes the other coordinates . in particular , all @xmath4-cells are fixed , and all higher dimensional cells come in pairs , the cells in each pair are being swapped by the involution . this means that in the quotient @xmath310 we get the induced cubical structure , consisting of the orbits of the cells of the @xmath326-torus . these are also indexed by the @xmath1-tuples of the symbols from the set @xmath327 , with an additional constraint that the first symbol @xmath319 comes before the first symbol @xmath320 ( if at all ) . hence the number of vertices in this cubical structure on @xmath310 is @xmath328 , whereas the number of @xmath82-cubes , for @xmath329 , is @xmath330 . the presentation can be used to analyze what happens for the small values of @xmath1 . the space @xmath331 is just a point : the empty direct product is to be interpreted as a point here . the space @xmath332 is homeomorphic to a closed interval , e.g. , @xmath333 $ ] if we take the projection of @xmath306 onto the @xmath314-axis . let us consider @xmath334 . we have a described a cubical structure on this space which has @xmath322 vertices , @xmath322 edges and @xmath302 squares . the two squares are indexed with @xmath335 and @xmath336 and it is easy to see that @xmath337 is obtained by gluing these @xmath302 squares together along their entire boundaries . hence we conclude that @xmath337 is homeomorphic to a @xmath302-dimensional sphere . as the last case , let us consider @xmath338 . this is a space glued together from @xmath322 cubes ; it has @xmath339 vertices , @xmath340 edges and @xmath340 squares . all of the cubes share the same set of vertices , and it is easy to see that all points , except for these @xmath339 vertices , have neighborhoods which are homeomorphic to open balls in @xmath341 . however , the space @xmath342 is not a manifold . to see this , let us take a look at the neighborhoods of these vertices . clearly , it does not matter which one we take , so let us take the vertex @xmath343 . as an open neighborhood of @xmath223 we can take a cone with an apex in @xmath223 over the link of @xmath223 . the link of a vertex in a cubical complex is always a simplicial complex . the one we have here has @xmath344 vertices , @xmath345 edges , and @xmath322 triangles . a presentation of this simplicial complex is given on figure [ fig : link ] , from which it is clear that this link is homeomorphic to @xmath346 . this shows that @xmath342 fails to be a manifold at these points . let us now generalize our description of @xmath342 to the general case . in particular , we shall see that for @xmath347 the vertices in the cubical structure on @xmath310 are singularities , and if they are removed , we are left with an @xmath326-dimensional manifold . we start with by viewing the @xmath1-torus @xmath348 in the standard way as the quotient space of @xmath349 divided by the group action of @xmath350 , given by @xmath351 since @xmath352 , as described above , we have @xmath353 , where @xmath354 is a group defined by @xmath355\rangle,\ ] ] with the action of @xmath356 on @xmath349 given by @xmath357 it follows from the presentation that the group @xmath354 is the semidirect product @xmath358 , with the group homomorphism @xmath359 given by @xmath360 , where @xmath356 is the nontrivial element of @xmath3 and @xmath361 is arbitrary . in particular , the elements of @xmath354 can be uniquely presented either as @xmath362 or as @xmath363 , for @xmath364 . the action of the element @xmath363 is fixed - point - free , whereas the action of the element @xmath365 has a unique fixed point , whose coordinates are @xmath366 . we see that the action of @xmath354 on @xmath367 is free , and that accordingly @xmath368 is an @xmath1-dimensional manifold . for the points in @xmath369 we see that the stabilizers are of cardinality @xmath302 , and that given such a point @xmath223 , the action of its stabilizer on an @xmath326-dimensional sphere centered at @xmath223 is the standard antipodal action , hence the quotient of this sphere by that action is the projective space @xmath370 . we conclude that @xmath371 can be obtained from an @xmath1-dimensional manifold , whose boundary consists of @xmath372 @xmath326-dimensional projective space @xmath370 , by attaching @xmath372 cones - one at each boundary projective space . we would like to present the space @xmath310 yet in another way , the one which will also introduce the terminology for dealing with the case of higher genus . the following concept is an important construction in combinatorial algebraic topology , see ( * ? ? ? * chapter 15 ) for relevant background . [ hocolimdf ] the * homotopy colimit * , denoted @xmath373 , of a diagram @xmath374 of topological spaces over a triangulated space @xmath375 , is the quotient space @xmath376 where the disjoint union is taken over all simplices in @xmath375 . the equivalence relation @xmath377 is generated by : for @xmath378 , @xmath379 , let @xmath380 be the inclusion map , then for @xmath391 , the space @xmath392 can be considered as a homotopy colimit of a diagram @xmath374 over an interval @xmath53 $ ] , where the latter is viewed as a triangulated space with two vertices and one edge . namely , set @xmath393 , @xmath394 , and let both diagram maps @xmath395 and @xmath396 be the quotient maps @xmath397 . see figure [ fig : hc1 ] . we see that the base projection map @xmath398 $ ] is induced by the projection of the first coordinate of @xmath399 to the real axis : @xmath400 by definition , this homotopy colimit is homeomorphic to the space obtained by taking the cylinder with the base @xmath348 and then quotioning both ends using the map @xmath401 . the reader is welcome to compare this to the definition of whitehead group , see @xcite . [ [ the - homology - groups - of - x_n - with - coefficients - in - mathbb - z_2 ] ] the homology groups of @xmath310 with coefficients in @xmath3 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ it follows from the presentation of @xmath392 as a homotopy colimit that it can be seen as a union of two disjoint copies of a mapping cylinder of the map @xmath401 , glued together along the topmost copies of @xmath348 . let us slightly modify this picture . let @xmath402 $ ] be the base projection map corresponding to this diagram . we set @xmath403 and @xmath404)$ ] , where @xmath160 is a small positive number , say @xmath405 . then we have @xmath406 . since @xmath407 , and the subspaces @xmath408 , @xmath409 , and @xmath410 are open in @xmath392 , we can use the mayer - vietoris sequence to compute the homology groups of @xmath392 : @xmath411 clearly , both @xmath408 and @xmath409 are homotopy equivalent to @xmath412 , whereas @xmath410 is homotopy equivalent to @xmath348 . the inclusion maps @xmath413 , and @xmath414 induce the same maps on the homology as the quotient map @xmath397 . it follows that the mayer - vietoris sequence translates to the long exact sequence @xmath415 this long exact sequence splits into short exact sequences as follows from the next proposition . * proof . * we know that @xmath418 . choose a subset @xmath111 of @xmath101 $ ] , such that @xmath419 . let @xmath420 to be the sum of all @xmath324-cells , indexed by @xmath1-tuples @xmath421 , such that @xmath422 for all @xmath423 . there are @xmath424 cells like that , and their sum is a cycle . if we let @xmath111 run over all cardinality @xmath324 subsets of @xmath101 $ ] , we shall obtain precisely the representatives of @xmath425 generators of the group @xmath426 . all the cycles @xmath420 are @xmath3-invariant , with all the @xmath324-cells in each sum coming in @xmath3-invariant pairs . in every pair , both cells are mapped to the same cell in @xmath412 , hence their contributions under the map @xmath427 cancel out each other , when we work with @xmath3-coefficients . it follows that @xmath428 already on the level of chains . since @xmath429 s generate @xmath426 , we conclude that @xmath427 is a @xmath4-map on the homology groups . certainly , if @xmath427 is a @xmath4-map , then so is the map @xmath430 in the long exact sequence . hence , the latter splits into the short ones of the type @xmath431 since we are working over field coefficients we conclude that we have an isomorphism @xmath432 for all @xmath433 and @xmath391 . accordingly , we get @xmath434 using that formula we arrive at the following statement . for the integer coefficients the situation is somewhat more complicated , as the map @xmath427 is not necessarilly a @xmath4-map anymore . based on the examples for the small values of @xmath1 and the general intuition for what this map is , we make the following conjecture . * acknowledgments . * the author would like to thank bernd sturmfels for introducing him into the world of tropical geometry and grisha mikhalkin for explaining the importance of the tropical moduli spaces . he would also like to thank alek vainshtein and alex suciu for discussions of torus quotients , as well as eva - maria feichtner for debating various formal approaches to the topology of the space of metric graphs . finally , the author expresses his gratitude to the mathematical institute at oberwolfach for the hospitality during the time a part of this research was done . j. richter - gebert , b. sturmfels , t. theobald , _ first steps in tropical geometry _ , idempotent mathematics and mathematical physics , contemp . * 377 * , amer . soc . , providence , ri , 2005 , 289317 .

in this paper we study topology of moduli spaces of tropical curves of genus @xmath0 with @xmath1 marked points . we view the moduli spaces as being imbedded in a larger space , which we call the _ moduli space of metric graphs with @xmath1 marked points . _ we describe the shrinking bridges strong deformation retraction , which leads to a substantial simplification of all these moduli spaces . in the rest of the paper , that reduction is used to analyze the case of genus @xmath2 . the corresponding moduli space is presented as a quotient space of a torus with respect to the conjugation @xmath3-action ; and furthermore , as a homotopy colimit over a simple diagram . the latter allows us to compute all betti numbers of that moduli space with coefficients in @xmath3 .

in recent years many experimental papers on enhanced faraday rotation in the systems with nanoscale inhomogeneities have appeared @xcite-@xcite . this increased research interest is largely motivated by the fact that the faraday effects is widely used in optical isolators , phase modulators @xcite , spin dynamics @xcite and etc . experiments found that the origin of the strong enhancement in the classical regime is intimately connected with the different plasmon @xcite resonances . the papers @xcite deal with the three - dimensional ( 3d ) random systems consisted of a solution with the embedded in it metallic nanoparticles . in such systems , an enhancement of faraday rotation at the frequencies close to the nanoparticle surface plasmon resonance frequency is observed in modest magnetic fields . other experimental papers @xcite were devoted to the plasmon induced enhancement of faraday rotation when an electromagnetic wave passes through a subwavelength thin metallic film with nanostructured surface profile . the surface profile can be in the form of a periodical grating ( as in the experiment @xcite where inhomogeneity is created by the periodically placed nanowires on the surface ) as well as random , as in case of randomly embedded nanoparticles on the surface @xcite . the theory , outlined in ref.@xcite for faraday rotation in 3d disordered media , could correctly predict many of the peculiar features of the experiments @xcite . according to ref . @xcite the faraday rotation angle in 3d disordered system is inversely proportional to the photon elastic mean free path , depends on the frequency , and has a minimum at the frequency of nanoparticle local plasmon resonance , due to a large scattering cross section . however , most of the experiments on enhanced faraday rotation are carried out with the subwavelength thin metallic films that can be considered as 2d systems . a consistent theory of faraday rotation in 2d disordered systems is absent . in the present paper , we theoretically consider the faraday rotation of light passing through a thin metallic film with structured surface profile . within a common approach we study both periodical and random surface profiles . we show that the plasmon scatterings on the inhomogeneities of surface profile lead to rotation angle enhancement . let a p - polarized wave impinge on the interface between two media ( see fig.1 ) . . the external magnetic field is directed on @xmath0 . after passing through a thin film , the incoming beam is rotated by the faraday angle @xmath1.,width=317 ] the incident wave magnetic field @xmath2 is directed on @xmath3 . after passing the magnetooptical medium , it rotates in the plane @xmath4 . the plane of incidence of wave vector is @xmath5 . the dielectric permittivity tensor of the system has the form @xmath6 where @xmath7 if @xmath8 and @xmath9 and @xmath10 if @xmath11 . the term @xmath12 describes the smooth surface and @xmath13 describes the surface roughness . here @xmath14 is the surface profile ( @xmath15 is a two dimensional vector on the plane @xmath4 ) that can be random as well as periodical and @xmath16 is the antisymmetric tensor . @xmath17 describes the magneto - optical properties of the medium , and we assume that the external magnetic field is directed on @xmath18 . the above geometry is more frequently used in the experiments . faraday rotation angle is determined as @xmath19 where @xmath20 is the thickness of the film . assuming that the profile function @xmath21 is smooth and neglecting its derivatives , from maxwell equations one obtains a helmholtz equation for the magnetic field @xmath22 substituting eq.([tensor ] ) into eq.([helmholtz ] ) , it is easy see that equations for right - hand and left - hand polarized photons are separated @xmath23 where @xmath24 , @xmath25 and @xmath26 . we neglect the difference between left - hand and right - hand polarizations in @xmath27 because it is already proportional to a small parameter @xmath28 . the faraday angle is determined through @xmath29 as @xmath30 because @xmath31 are continuous at @xmath32 , the same is correct for @xmath29 . from the maxwell equations and continuity of @xmath33 follow the continuity of @xmath34 at @xmath32 . when the surface roughness is absent ( @xmath35 ) one can solve the helmholtz equation , eq.([helmholtz ] ) , for @xmath2 with the above mentioned boundary conditions and find @xmath36 where the reflection and transmission amplitudes are determined as follows ( see , e.g. , ref . @xcite ) @xmath37 here @xmath38 and analogous expressions for @xmath39 and @xmath40 can be written substituting @xmath41 by @xmath42 . substituting eqs.([backsol ] ) and ( [ reftrans ] ) into eq.([detangle ] ) , for the thick films @xmath43 one finds the well known result for the faraday angle @xcite @xmath44 in the thin film limit @xmath45 similarly , we find @xmath46 where we assume that @xmath47 . in analogous manner one can find the polarization rotation angle for the reflected wave @xmath48 note that in both limits the rotation angle does not depend on @xmath20 . the solution of the eq . ( [ leftright ] ) consists of two contributions : one is caused by the smooth surface and the second one is caused by the scattering from the inhomogeneities @xmath49 , where the background field obeys a homogeneous equation @xmath50 the scattered field @xmath51 is determined through the green s function @xmath52 where the green s function obeys the equation @xmath53 below , we separately consider the case when the surface of metal film has a periodical grating and when the surface profile is random . in this case , @xmath21 is a two - dimensional periodic function . one can expand the profile function into discrete fourier series @xmath54 where @xmath55 are two dimensional discrete vectors on the inverse lattice ; @xmath56 are the profile periods in @xmath57 directions respectively and @xmath58 . when one of the periods tends to infinity , one recovders a one dimensional periodical profile , considered in the experiment @xcite . substituting eq.([perprof ] ) into eq.([scafield ] ) , taking its 2d fourier transforms , and integrating over @xmath0 using the explicit form of @xmath59 , one finds @xmath60 where @xmath61 is the two dimensional fourier transform : @xmath62 it is worth noticing , that the presence of the @xmath63-function in the expression of @xmath27 will lead to the different values of any physical quantity at @xmath64 , while evaluating the integral over @xmath0 . to avoid the problem with discontinuous physical quantities at @xmath65 in our further calculations , we will take their value at @xmath66 . one has an analogous expression for the left - hand polarized component . note that the background field @xmath67 which is the solution of homogeneous equation eq.([back ] ) depends only on @xmath0 . in order to obtain the faraday rotation angle , see eq.([detangle ] ) and fig.1 , we need to evaluate the coherent part of the scattered field , eq.([scafield2 ] ) , that is the part with wave vector directed on @xmath0 @xmath68 as it is seen from eq.([cohpart ] ) , the scattered field includes a green s function that has a plasmon pole which plays a crucial role in our study of the magneto - optic effects in 2d disordered systems . more precisely , when one of the wave numbers of the inverse lattice @xmath69 coincides with the plasmon wave number then the scattered field resonantly enhances ( see below ) . in order to analyze the faraday angle , taking into account the scattered field , let us represent it in the form @xmath70 for the thin films @xmath71 . at the resonance @xmath72 can be essentially larger than @xmath73 . at the same time , @xmath74 is proportional to the roughness height @xmath75 and is significantly smaller than @xmath76 . the latter is proportional to unity provided that @xmath77 . correspondingly the faraday angle will resonantly enhance provided that resonance condition is fulfilled . such an experimental enhancement of faraday rotation is observed in the recent experiment @xcite . it is worth noting that for the reflected wave the above mentioned effect is absent due to the fact that the denominator and numerator of eq . ( [ faranglesc ] ) at the resonance are of the same order , i.e. , @xmath78 . for the thick films , @xmath43 , the resonance effect is possible . in order to estimate the plasmon contribution to the faraday rotation angle for thin film , we assume that grating height is small @xmath79 and substitute the green s function in eq.([cohpart ] ) by the bare one @xmath80 . for a given @xmath81 , the green s function has a sense of magnetic field of a point source . therefore it satisfies the same boundary conditions as magnetic field , namely continuity of @xmath82 and @xmath83 . solving eq.([grfunc ] ) for @xmath35 with the above mentioned boundary conditions at @xmath32 , one obtains @xmath84 it is easy to find the green s function for other values of @xmath85 also . however for our purposes the above mentioned one is enough . note also that here we consider only @xmath32 plasmon contribution believing that it is more important due to the roughness at @xmath32 and not at @xmath86 . one can also be convinced that the green s function eq.([magplasmon ] ) has a pole . to find the pole , we equate the denominator of eq.([magplasmon])to @xmath87 . getting free from the square roots near the pole values , the green s function can be represented in the form @xmath88 where @xmath89 here @xmath90 describes damping of right - hand magnetoplasmon due to electromagnetic losses and we assume that @xmath91 and @xmath92 . it follows from eqs.([magpole ] ) and ( [ cohpart ] ) that the scattered field and corresponding faraday angle will resonantly increase provided that one of the inverse lattice wave numbers @xmath69 coincides with the magnetoplasmon wave number @xmath93 , see also @xcite . using eqs.([magpole]),([magpoledet ] ) and ( [ cohpart ] ) and keeping only the resonance term in the sum of eq.([cohpart ] ) , from eq.([faranglesc ] ) one has @xmath94 , \label{farangleper}\ ] ] where @xmath95 characterizes the height of periodical grating . to get the plasmon resonance contribution into faraday rotation angle in the periodical grating case , one has to substitute the parameters @xmath96 and @xmath97 from eq.([magpoledet ] ) into eq.([farangleper ] ) . the final result , in the limit @xmath98 , reads as follows @xmath99 equation ( [ farperfin ] ) with eq . ( [ ranfin ] ) ( see below ) represent the central results of this paper . the main difference of the plasmon resonance contribution @xmath100 compared to smooth surface metallic film contribution , eq.([thin ] ) , is that the former dependences on the imaginary part of the dielectric permittivity @xmath101 . comparing eqs.([thin ] ) and ( [ farperfin ] ) , we have @xmath102 for nanoscale metallic films , usually @xmath103 . taking into account that for noble metals in the optical region @xmath104 , one has @xmath105 . to apply the obtained results to the experiment @xcite , one can model the composite system consisting of garnet substrate with gold surface profile by an effective metallic film with dielectric permittivity tensor . the diagonal part of the latter is mainly determined by gold ( at optical wavelengths its absolute value is much larger than that of garnet ) and non - diagonal part determined by bismuth substituted yttrium iron garnet value . taking at @xmath106 @xcite @xmath107 , @xmath108 @xcite and @xmath103 , from eq.([ratioper ] ) we obtain that the plasmon enhancement factor is of order @xmath109 . that agrees well with the experimental value @xcite @xmath110 . for the profile periods @xmath111 and @xmath112 the resonant number is @xmath113 . note that faraday rotation angle for smooth garnet film , follows from eqs.([angleflat],[thin ] ) . taking @xmath114 , @xmath115 , @xmath106 , @xmath116 @xcite , one has @xmath117 . for the large @xmath118 one can not separate out a single resonance term from the sum over the inverse lattice wave numbers , eq.([cohpart ] ) . in this case the summation can be replaced by integration ( @xmath119 , where @xmath120 is the area of the system ) . carrying out the integration over @xmath121 and making use of eqs.([backsol],[reftrans],[faranglesc ] ) , one finds for the real part of faraday angle , in the limit @xmath77 , the following expression ( @xmath122 ) @xmath123}{2}. \label{percont}\ ] ] this is a general expression , independent of the surface periodic profile model . for simplicity we discuss only constant harmonic grating case , i.e. , @xmath124 . the quantities @xmath125 are the local density of states of right hand and left hand polarized magnetoplasmons . because of translational invariance , they depend only on the difference of the arguments and therefore are independent of local point @xmath126 . expanding @xmath125 on @xmath17 , one gets that @xmath127 . thus , the measurement of the rotation angle gives information on the density of states @xcite . more as a consequence of the periodicity of @xmath21 , the plasmon spectrum consists of energetic bands and gaps . the above mentioned derivative gets its maximal values at the edges of these bands . similar behavior for the faraday rotation was found in 1d periodical systems @xcite . now consider the case when the surface profile is random . we assume that @xmath21 is a gaussian distributed random function @xmath128 where @xmath129 denotes the ensemble average and @xmath130 and @xmath131 are the root - mean - square roughness and correlation length , respectively . in order to average the faraday angle over the realizations of random roughness and to separate its real part , it is convenient to multiply the numerator and denominator of eq.([detangle ] ) by @xmath132 , see also @xcite @xmath133 like in the periodical grating case , we decompose the magnetic field into background and scattered parts @xmath134 . in the denominator of eq.([multiply ] ) for small roughness @xmath135 , one can keep only the terms containing background fields @xmath136 . in respect to the numerator , one should keep only the terms containing the scattered fields because the terms associated with the background field are small for thin films : @xmath137 . for the real part of plasmon diffusional contribution to the faraday angle , one finds from eq.([multiply ] ) @xmath138 recall that the scattered field is determined by eq.([scafield ] ) and that the terms @xmath139 do not contribute to the real part of faraday angle because the denominator of eq.([realdif ] ) is real . to find the averages in eq.([realdif ] ) one needs the averaged over random roughness green s function , taking into account the plasmon multiple scattering effects on the surface roughness ( see refs . random roughness leads to damping of surface magnetoplasmon due to elastic scattering . the final answer for the magnetoplasmon green s function , averaged over the randomness , reads @xmath140 where magnetoplasmon elastic mean free path is determined as @xmath141 and @xmath142 . in eq.([avgrfun ] ) we neglect @xmath143 compared to @xmath144 . we will take the contribution of the former into account in the diffusional propagator ( see below ) . in the weak scattering limit @xmath145 , the main contribution to the average quantities in eq.([realdif ] ) gives the magnetoplasmons diffusion . using eq.([scafield ] ) one can represent the diffusional contribution in the form @xmath146 where @xmath147 , @xmath148 magnetoplasmon diffusion propagator which is determined by ladder diagrams presented in fig.2 ( see for example @xcite ) , width=377 ] summing the ladder diagrams in the limit @xmath98 , one has @xmath149 expression ( [ difprop ] ) was derived in the limits @xmath150 and @xmath151 . the propagator @xmath152 is obtained from @xmath153 by changing the sign of @xmath17 . calculating the integrals in eq.([difcontr ] ) in the limits @xmath98 and @xmath154 , we arrive at @xmath155 finally , using eqs.([realdif ] ) , ( [ difprop ] ) and ( [ difav ] ) , for diffusional contribution to the faraday angle , we obtain @xmath156 note , that the ratio @xmath157 has is the average number of scatterings of plasmon . a similar result for faraday rotation in 3d disordered medium is obtained in @xcite . however in 2d systems , faraday rotation is more sensitive to the number of scatterings ( square dependence against the linear in 3d case ) as well as to dielectric permittivity of thin film . near the surface plasmon resonance @xmath158 , the faraday angle enhances because @xmath159 . comparing eq . ( [ thin ] ) with the flat surface contribution , eq.([angleflat ] ) , we have @xmath160 if the diffusion of magnetoplasmon is realized on the film surface , i.e. , the inequality @xmath151 is met , then the condition @xmath161 should hold . as a consequence , the diffusion contribution to the faraday rotation angle can be the dominant one . now let us make some numerical estimates to clarify whether or not the mentioned inequality takes place . assuming that the roughness is created by the randomly adsorbed on the surface nanoparticles , we have : @xmath162 , where @xmath163 is the radius of a nanoparticle . for gold nanoparticles with radius @xmath164 at @xmath165 @xcite , @xmath166 , @xmath167 @xcite and @xmath168 . the surface plasmon wave number @xmath169 and the constant @xmath170 in eq.([magpoledet ] ) are estimated as @xmath171 and @xmath172 , respectively . the surface plasmon elastic mean free path @xmath173 is found from eq.([elmean ] ) to be approximately @xmath174 . if the losses are caused by the gold substrate , then the inelastic mean free path can be estimated using eq.([magpoledet ] ) and the above mentioned numbers , @xmath175 . so plasmon diffusion inequalities @xmath151 are not realized in ordinary conditions . however , close to the nanoparticle surface plasmon resonance the physical situation is completely different , because the expression eq.([elmean ] ) , for the determination of the plasmon elastic mean free path is not valid any more . in this case the elastic mean free path can be estimated as @xmath176 , where @xmath177 is the surface concentration of nanoparticles and @xmath131 is the cross section of the interaction of the surface plasmon with the nanoparticle . when the surface plasmon wave number coincides with the nanoparticle surface plasmon resonance wave number , the plasmon elastic cross section resonantly enhances up to several orders compared to ordinary situation , see for example @xcite . therefore , close to the resonance , the magnetoplasmon elastic mean free path becomes essentially smaller and the condition of its diffusion can be easily fulfilled . we have considered the faraday rotation of light passing through a thin metallic film with nanostructured surface grating . in the periodical grating case , the enhancement of the faraday angle happens when the surface plasmon wave number coincides with one of the wave vectors of inverse lattice , characterizing the grating periods on the surface . in the random surface profile case , the dominant contribution to the faraday angle gives the diffusion of magnetoplasmons . if the random roughness is created by the randomly embedded nanoparticles on the surface , then the maximum faraday rotation angle of the transmitted wave is achieved when surface plasmon wave number coincides with the nanoparticle surface plasmon resonance wave number . experimental manifestations of the obtained results are discussed . * acknowledgments . * we are grateful to o. del barco for preparing the figures . we thank w. whitaker for a critical reading of the mansucript . v.g . acknowledges partial support by feder and the spanish dgi under project no . fis2010 - 16430 . 99 h. uchida , y. masuda , r.fujikawa , a.v.baryshev , m.inoue , journal of magnetism and magnetic materials * 321 * 843 ( 2009 ) . prashant k.jain , yanhong xiao , ronald walswort , and adam e.cohen , nano lett.,*9(4 ) * 1644 ( 2009 ) . s.tkachuk , g.lang , c.krafft and i.mayergoyz , journal of applied physics , * 109 * 07b717 ( 2011 ) . raj kumar dani , hongwang wang , stefan h.bossmann , gary wysin and viktor chikan , the journal of chemical physics , * 135 * 224502 ( 2011 ) . i. crassee , j. levallois , a. l. walter , m. ostler , a. bostwick , e. rotenberg , t. seyller , d. van der marel , and a. b.kuzmenko , nat . phys . 7 , 48 ( 2011 ) . g. m. wysin , viktor chikan , nathan young , and raj kumar dani , j. phys . : condens . matter * 25 * 325302 ( 2013 ) . jessie yao chin , tobias steinle , thomas wehlus , daniel dregely , thomas weiss , vladimir i. belotelov , bernd stritzker and harald giessen , nature communications,*doi : 10.1038/ncomms2609 * ( 2013 ) 1 - 6 . k.hayashi , r.fujikawa , w.sakamoto , m.inoue and t.yogo , j.phys.chem.c*112 * 14255 ( 2008 ) . h.c.y.yu , m.a.eijkelenborg , s.g.leon-saval , a.argyros and g.w.barton , appl.opt . * 47 * , 6497 ( 2008 ) . f.liu , t.makino , t.yamasaki , k.ueno , a.tsukazaki , t.fukumura , y.kong and m.kawasaki , phys.rev.lett . * 108 * , 257401 ( 2012 ) . h.raiether , _ surface plasmons on smooth and rough surfaces and on gratings _ ( springer tracts in modern physics vol.111 ) 1988 berlin , springer . v. gasparian and zh . s. gevorkian , phys.rev . a * 87 * , 053807(2013 ) . aronov , v.m . gasparian , and ute gummich , j. phys . condens . matter ; 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we have analyzed analytically the faraday rotation of an electromagnetic wave for magnetoactive thin metallic film with a nanostructured surface profile . periodical as well as random surface profiles were considered . the plasmon contribution to the faraday angle was studied . for periodical grating case , we have shown that the maximum of rotation angle is achieved when surface plasmon wave number coincides with one of the wave numbers of the inverse lattice . enhancement of the faraday angle at plasmonic band edges is predicted . in the case of random surface profile , it is shown that the diffusion of surface magnetoplasmons gives a dominant contribution to faraday rotation . comparison with the experiments is carried out .

amartya sen @xcite argues that famine and other catastrophes are easily avoided in a democracy . this argument relies on the fact that where information can freely diffuse , decision makers can form an unbiased picture of the state of a society , and take proper measures . biases due to individual opinions are expected to be washed out in the information aggregation process , a phenomenon often referred to as the `` wisdom of crowds '' @xcite . still , cases of information aggregation failure abound even in democratic societies . for example , in the aftermath of the 2008 lehman brothers bankruptcy , former federal reserve chairman alan greenspan expressed his state of `` shocked disbelief '' during his hearing before the us committee of government oversight and reform , leaving the public opinion to wonder where did he get the information the policy of the fed was based on . al gore @xcite argues that shortcuts between decision makers and the media are often such that the former are not in the best position to be informed about what is going on . a number of models on social dynamics have addressed the issue of information aggregation ( see e.g. @xcite for an excellent review ) . the simplest is probably the voter model , which entails agents taking the same opinion of a randomly chosen node amongst their neighbors . this allows for sharp predictions @xcite where generally the information aggregation process converges to the incorrect outcome with finite probability . other contributions have instead proposed different opinion dynamics mechanisms , such as majority rules @xcite or social impact models @xcite , which support different conclusions . these models , however , come short in their micro - economic foundation , as the interaction mechanism is somewhat arbitrary . more detailed micro - economic models of social learning have been proposed in the economics literature . it is well known that information aggregation may fail when agents free ride on the information gathered by others , without seeking independent sources . this phenomenon , called rational herding @xcite , is also supported by experimental evidence @xcite . a sequel of papers have focused on bayesian learning schemes @xcite , coming to the generic conclusion that when agents update their beliefs following bayes rule society correctly aggregates information ( still see @xcite ) . some authors have focused on the impact of dominant groups of individuals on the aggregation of information . for example , bala and goyal @xcite introduced the notion of `` royal family '' as a group of agents whose behavior is observed by anyone else . alternatively , golub and jackson @xcite defined @xmath0-step `` prominent groups '' as those groups whose behavior eventually influences all other agents within time @xmath0 . regardless of the specific definitions , these and other studies unanimously highlighted the negative role that exceedingly influential groups have on the information aggregation process . in this paper we focus on an extremely stylized model of a society and we address the issue of whether information distributed across the population is able to diffuse to an uniformed well connected clique of decision makers . our model assumes bayesian learning , but differently from @xcite , who study a continuum of agents , we study a finite but large population of agents connected by a social network . on a finite network , when agents talk repeatedly with their peers , they may not be able to disentangle what in their peer s opinion is new information and what reflects information exchanged in previous interactions , including the one provided by themselves to them . this phenomenon , called `` persuasion bias '' in @xcite , introduces a non - trivial positive feedback and leads to information aggregation failure , at odds with the conclusions of @xcite . the main conclusions of our paper can be summarized in two points : _ i ) _ information aggregation crucially depends on the synchronicity of the information updates of different agents : in the extreme case of a parallel update dynamics , where we can derive analytic results , information diffusion leads to the correct outcome in the limit of a very large society or for very informative initial signals . when the fraction of agents who update their beliefs at each time step is lower than a critical threshold , the society converges with finite probability to the wrong outcome , no matter how large the society is . _ ii ) _ in the case of parallel dynamics , information aggregation _ degrades _ as the size of the clique of uninformed agents gets smaller . in particular , the limit of a vanishingly small clique of uninformed agents , behaves markedly differently from the case of a homogeneous society ( with no clique ) . both results suggests that it might not be wise to rely on crowds in situations which are reminiscent of those prevailing in our societies , where update is sequential and the social network is characterized by highly connected cliques ( news corporations , political parties ) . the paper is organized as follows . in section [ dyn_topology ] we detail the network structure and the information update rules , as well as our quantitative measure of a social network s ability to correctly aggregate information . in section [ prob_computing ] we provide analytic results for the case of parallel information update . in section [ diff_dyn ] we numerically compare these results with those obtained in sequential update schemes . we conclude the paper with a few conclusive remarks in section [ conclusions ] . following ref . @xcite , let us consider a population of agents who need to decide whether a certain event @xmath1 will occur ( @xmath2 ) or not ( @xmath3 ) . let us denote the prior probability of @xmath1 as @xmath4 , where @xmath5 . as already stated , our interest is mainly focused on the information aggregation process as performed by societies where a fraction of individuals matters much more than the vast majority of the population . in the language of networks , the most obvious measure of the importance of a node is its degree , i.e. the number of neighbors . for this very reason , throughout the rest of the paper we shall focus on a highly stylized society structure , where only few nodes have a large degree , which we build starting from a connected regular graph where all @xmath6 nodes have degree @xmath7 . then , a randomly chosen set @xmath8 made of @xmath9 non - neighboring sites are connected among themselves , thus forming a clique of nodes with degree @xmath10 ( this construction is such that each hub has exactly @xmath11 links connecting it to nodes outside @xmath8 ) . in the following , we shall be mostly interested in the case @xmath12 , i.e. when @xmath8 becomes a group of mutually connected hubs . at time @xmath13 , each agent @xmath14 receives signals about event @xmath1 which are independently drawn from a probability distribution @xmath15 . we assume these signals to be informative @xcite , i.e. @xmath16 @xmath17 , and we focus on the particular case @xmath18 on the other hand , the agents @xmath19 the leaders are assumed to be initially uninformed . this means that their signals are independently drawn from a probability @xmath20 for @xmath21 . in our model , agents repeatedly exchange information with their neighbors . in this exchange , the generic agent @xmath22 collects a certain number @xmath23 of signals that we denote by @xmath24 , where @xmath25 are the initial signals discussed above . given this information set @xmath26 , by bayes theorem @xcite , the agent s state of knowledge about @xmath1 is quantified by the conditional probability @xmath27 where @xmath28 is the probability of the signals @xmath26 . notice that the likelihood ratio of @xmath29 and @xmath30 does not depend on @xmath28 . if the agent believes that the different signals are independent , then @xmath31 and the logarithm of the likelihood ratio , which embodies the state of information of agent @xmath22 , can be described by a single variable @xmath32 : @xmath33 at @xmath34 , agents have just one signal . then we have @xmath35 and the above expression reduces to the very compact form @xmath36 when two agents , say @xmath22 and @xmath37 with signals @xmath26 and @xmath38 respectively , meet , they communicate by exchanging signals and , as a result , their state of knowledge changes . indeed , if @xmath39 , then @xmath40 . likewise , if @xmath41 , then @xmath42 . starting from an initial state of knowledge @xmath43 , for @xmath44 , one can think of different types of information update . our assumption will be that at each time step @xmath45 , a certain fraction @xmath46 ( where @xmath47 ) of randomly selected agents update their state of knowledge by _ listening _ to their neighbors . so , assuming that agents in the set @xmath48 are the ones to update their information at time @xmath0 , one has : @xmath49 where @xmath50 is the @xmath51 element of the adjacency matrix @xmath52 , i.e. @xmath53 if agents @xmath22 and @xmath37 are connected and @xmath54 if they are not . clearly , the above dynamics has two limiting cases : @xmath55 and @xmath56 . the former describes cases where agents update their information one at a time , and we shall refer to this particular situation as random node sequential ( rns ) dynamics . the latter case , instead , describes a parallel dynamics where all agents simultaneously update their state of knowledge . this information update rule was initially proposed in @xcite , and , due to its analytical tractability , represents the most frequent choice in social learning models . in the following , we also shall investigate this type of dynamics , and then explore other cases in section [ diff_dyn ] . the dynamics in eq . ( [ general_dyn ] ) is unbounded , i.e. each @xmath32 will either diverge to @xmath57 or @xmath58 . thus , information aggregation properties can be assessed simply by looking at the signs of the @xmath32s in the long run . thus , a good measure of information aggregation is given by the `` magnetization '' of the system : @xmath59 the quantity @xmath60 tells what is the fraction of the population holding the right information on event @xmath1 at time @xmath0 . a quantitative measure of information aggregation is given by the probability @xmath61 that the majority will converge to the true outcome , in an ensemble of repeated trials . according to the parallel dynamics prescription , all agents in a social network listen to their neighbors at any time @xmath62 , and update their state of knowledge accordingly : @xmath63 by collecting all @xmath64s into a column vector @xmath65 the column vector with components @xmath66 , and by @xmath67 the corresponding row vector . ] , the dynamics described in equation can be rewritten as @xmath68 play a crucial role in the time evolution of the state of knowledge vector @xmath65 . being symmetric , the adjacency matrix @xmath69 yields @xmath6 real eigenvalues @xmath70 , whose corresponding eigenvectors @xmath71 ( @xmath44 ) form an orthogonal set in @xmath72 . by decomposing the adjacency matrix as @xmath73 , one can see that , for large enough times , equation becomes @xmath74 , corresponding to the largest eigenvalue of the adjacency matrix @xmath69 , share the same sign , which we shall assume to be positive from now on . thus , in the light of the relation in , two main points become apparent : * for large enough times @xmath65 is proportional to @xmath75 , meaning that all agents on the network either learn the correct value of @xmath1 or they all get it wrong . * the sign of the components in @xmath75 is completely determined by the sign of the overlap @xmath76 , so that the probability of the whole network learning the right information reads @xmath77 in the following we shall compute the probability for the simple network topology discussed above . for the sake of simplicity , let us assume @xmath2 , so that the probability in equation is equivalent to the probability of the scalar product @xmath78 being positive , and that each agent is initially given one signal @xmath79 at time @xmath13 . assuming that hubs , i.e. nodes in the clique @xmath8 , have no initial information ( @xmath80 for @xmath81 ) , such a scalar product can be written as a sum over the @xmath82 sites not belonging to @xmath8 : @xmath83 where @xmath84 denotes the @xmath22-th component of the first eigenvector , and @xmath85 ( see equation ) . a good approximation scheme to estimate the probability of the quantity in equation being positive is via the central limit theorem : as a matter of fact the scalar product in is the sum of @xmath82 random variables , each given by the product of two random variables : @xmath86 . thus , the probability of @xmath87 in equation being positive is approximately given by @xmath88 where @xmath89 and @xmath90 denote the mean and standard deviation , respectively , of the random variable @xmath87 . given the independence of the @xmath32s and the eigenvector components @xmath84s , such two quantities are given by @xmath91 where @xmath92 and @xmath93 denote the mean and standard deviation of the random variables @xmath32 , whereas @xmath94 and @xmath95 denote the mean and standard deviation of the eigenvector components @xmath84 for @xmath14 . computing @xmath92 and @xmath93 is easy . recalling that signals must be informative ( see equation ) , one has @xmath96 . let us rewrite such probability as @xmath97 with @xmath98 . then , one can immediately verify that @xmath99 as regards @xmath94 and @xmath95 , good approximate expressions for them can be computed by employing standard perturbation theory up to second order ( see appendix [ perturb ] for the details ) . to leading order in @xmath6 one gets : @xmath100 where @xmath101 denotes the fraction of hubs in the network . as can be seen from the inset in fig . [ ls ] , the above approximations are in excellent agreement with results obtained from numerical diagonalization of adjacency matrices , especially for large network sizes . plugging equations and into equation , one can eventually compute the probability of converging to the right value of event @xmath1 as in equation : @xmath102 as already stated , we are mostly interested in cases where only a few nodes in the network play the role of hubs , i.e. @xmath103 : in this case the probability in equation further simplifies to the following remarkably simple expression : @xmath104 in fig . [ ls ] the prediction by the above equation is compared with the results of numerical simulations for @xmath105 , @xmath106 , and for several different system sizes @xmath6 : all results are in very good agreement with equation ( all data points are rescaled in order to collapse on the function @xmath107 ) . a few comments are in order on the approximate result of equation . since @xmath108 , according to equation for each system size @xmath6 correct information aggregation happens with probability that for all practical purposes can be considered equal to 1 when initial signals informativeness is @xmath109 , where @xmath110 . this point essentially means that for _ any _ population size @xmath6 correct information aggregation is possible , for informative enough initial signals , despite the presence of a fraction @xmath111 of dominant nodes . such a result shows that the presence of a group of individuals with large influence does not necessarily jeopardize correct information aggregation . moreover , the threshold value @xmath112 is inversely proportional to @xmath113 , meaning that large populations will be able to aggregate information correctly as soon as signals are informative , i.e. as soon as @xmath114 is slightly larger than @xmath115 . this is essentially a stronger statement of previous results obtained for infinite networks ( see for example @xcite ) , where the presence of signals with arbitrarily large informativeness , combined with the lack of individuals with unbounded influence , is identified as a sufficient condition for correct information aggregation . on the other hand , for @xmath116 the population reaches consensus on the wrong value of @xmath1 with non - zero probability . a very interesting role in the information aggregation process is played by the fraction of hubs @xmath111 . in fig . [ clt ] , one can see how , for a fixed system size @xmath6 , the probability of correct information aggregation behaves when increasing the fraction @xmath111 of hubs in the network . also , it is rather interesting to compare such results with the information aggregation capabilities of a regular graph where all nodes have the same degree @xmath11 . in such a case , one can immediately verify that the first eigenvector of the adjacency matrix is uniform with all components equal to @xmath117 , and the probability of the scalar product in equation being positive simply reduces to the probability of the sum @xmath118 being positive . therefore , one can compute the probability of correct information aggregation of a regular network with easy central limit theorem considerations , analogous to those already presented in this section . such a probability does not depend on @xmath11 and reads : @xmath119 where the last approximation holds for large values of @xmath6 . as one can see , equation reduces to the above expression for @xmath120 ( though numerically one does not find perfect agreement between the two , since equations and represent good approximations only for very low values of @xmath111 ) . so , the lesson to be learned from the plots in fig . [ clt ] is twofold . first , one can see that as soon as a very small clique of uninformed hubs is introduced in a regular graph the overall population s ability to correctly aggregate information decreases sharply . this can be also understood by observing that the probability in equation does not recover the regular network ( rn ) result when considering vanishingly small fractions of hubs , i.e. : @xmath121 on the other hand , whenever a clique of hubs is present in the network , then information aggregation can actually be improved by increasing the size of the clique itself , up to the point ( for @xmath122 ) where the aggregation ability of the original regular graph can almost be reproduced . intuitively , the above findings can be altogether understood in the following terms . according to our setting , all hubs in the clique @xmath8 are mutually connected and have a degree equal to @xmath123 . this means that each hub has exactly @xmath11 neighbors outside @xmath8 , so that one can expect roughly @xmath124 nodes to fall within the clique s neighborhood @xmath125 . so , for very low values of @xmath111 , @xmath125 contains a negligibly small number of nodes , which , however , will largely influence the initially uninformed hubs whenever they communicate for the first time . given the small size of @xmath125 , its initial state of knowledge will be much more sensitive to fluctuations in the initial signals distribution among agents . on the other hand , when @xmath122 , the number of nodes in the neighborhood of @xmath8 becomes of order @xmath6 , hence much more robust with respect to fluctuations . in summary , the role of hubs in our model is subtle , as a handful of them is enough to heavily damage the good information aggregation properties of a population of equals ( as modeled by a regular graph ) , whereas increasing their number also has `` healing '' effects which can restore such good properties . so far , we only have considered the most popular and widely used evolution rule for the information propagation on a network , i.e. the parallel dynamics introduced in equation . however , as already discussed in section [ dyn_topology ] , parallel dynamics represents one of the two extreme cases of the general dynamics , according to which a fraction @xmath126 of agents listens to their neighbors at each time step @xmath0 , i.e. the case @xmath56 . the other extreme case is the already mentioned rns dynamics ( @xmath127 ) , according to which agents update their state of knowledge one at a time . numerical simulations highlight significant differences in a social network s ability to aggregate information correctly under parallel or rns dynamics , the latter performing much worse than the former : as shown in the left panel of fig . [ transition ] , the probability of correct information aggregation under parallel dynamics outperforms the one obtained under rns dynamics over a wide range of signal informativeness levels a link is randomly selected and the two nodes that share it exchange information . however , none of the simulations we performed highlighted any significant difference between such a dynamics and the parallel one . ] . moreover , results obtained via rns dynamics show no relevant dependence on the system size @xmath6 . the above findings suggest to look for a transition in information aggregation as a function of the number of agents that update their state of knowledge at a given time step by letting the parameter @xmath126 take values over the whole interval @xmath128 $ ] . in the right panel of fig . [ transition ] we plot the probability @xmath129 of correct information aggregation as a function of @xmath126 for different system sizes and a fixed informativeness level of the signals initially distributed to agents ( the qualitative overall appearance of the results is not changed when considering different levels of informativeness ) . as can be seen , for increasing values of @xmath126 a transition is observed towards better information aggregation capabilities for all system sizes . this can essentially be interpreted in terms of the speed of information update . as one could expect , rns dynamics is extremely slow compared to parallel dynamics ( depending on the system size , we find on average that rns dynamics reaches consensus in times that are 3 - 4 orders or magnitude larger than the ones required by parallel update ) , hence more prone to allow the spreading of misleading signals in the agents initial distribution . on the other hand , parallel dynamics is fast , in such a way that in a few time steps each agent receives through his / her neighbors aggregated information coming from the whole network . in summary , we have presented a stylized dynamic network model of the information diffusion throughout a large society featuring a small fraction of uninformed leaders . the model s simplicity allows , in some cases , to make analytical considerations . namely , when assuming all agents to simultaneously update their state of knowledge on a given issue , we are able to provide a closed - form expression for the probability of correct information aggregation as a function of the system size , i.e. the number of agents in the society , and the fraction of individuals playing the role of hubs . our results partially overlap with previous works from the social learning literature in economics , as we show that larger populations are better , on average , at aggregating information . on the other hand , we provide interesting novel results on the role played by the size of an uninformed lite , portrayed in our model by a clique of nodes that do not own any prior information on the issue being discussed by the population . first , we show a rather counterintuitive result , i.e. that increasing the relative size ( compared to the overall population ) of such uninformed lites actually helps the information aggregation process . moreover , we show that letting the fraction of hubs go to zero does not recover the results obtained for the corresponding hub - free regular network . rather interestingly , we also show our model to be sensitive to the information update speed , as defined by the fraction of agents who simultaneously revise their information at each time step , by showing the existence of a transition towards better information aggregation capabilities when moving from the low speed towards the high speed regime . when assuming hubs to be identified by nodes @xmath130 , the network adjacency matrix @xmath69 takes the following block form : @xmath131 in the above equation , @xmath132 is an @xmath133 block such that @xmath134 for @xmath135 and @xmath136 . the off - diagonal block @xmath137 is of size @xmath138 , and it accounts for neighbors of the clique @xmath8 , i.e. @xmath139 for @xmath81 and @xmath140 , or vice versa , and zero otherwise . lastly , the block @xmath141 is of size @xmath142 , and it accounts for links between nodes that do not belong to @xmath8 . spectral properties of the adjacency matrix @xmath69 , expressed in block form as in equation , can be deduced from standard perturbation theory . as a matter of fact , such a matrix can be decomposed as @xmath143 , where @xmath144 for small values of @xmath11 ( i.e. the degree of nodes outside of @xmath8 ) , the matrix @xmath145 above is sparse and can be interpreted as a perturbation to the matrix @xmath146 describing the fully connected clique @xmath8 plus a sea of @xmath147 disconnected nodes . * the largest eigenvalue reads @xmath150 , and its normalized eigenvector @xmath149 has the first @xmath9 components equal to @xmath151 and the remaining @xmath147 ones equal to zero . * @xmath152 for @xmath153 , with eigenvectors having non - zero components only in the first @xmath147 sites . * @xmath154 for @xmath155 , with eigenvectors that can simply be chosen as having all components equal to zero except for the @xmath22-th component being equal to one . let us then approximate the first eigenvector of the full adjacency matrix @xmath69 as @xmath156 and @xmath157 denote the first and second order corrections , respectively , to the unperturbed eigenvector @xmath158 . the first order correction only involves neighbors of the clique @xmath8 , and it reads @xmath159 where @xmath160 represents the number of neighbors that node @xmath22 has within the clique @xmath8 . the second order correction components of @xmath161 . however , such corrections are irrelevant to our analysis , so we can safely neglect them . ] involves neighbors of the nearest neighbors of the clique @xmath8 : @xmath162 where @xmath163 denotes the set of next to nearest neighbors of the clique @xmath8 , whereas @xmath164 is the number of neighbors that node @xmath22 has amongst neighbors of the clique @xmath8 . in order to perform exact calculations up to second order , one should in principle compute the expected number of nodes belonging to @xmath125 and @xmath163 , and the expected values of the quantities @xmath165 in and @xmath166 in by averaging over all possible network configurations built as explained in section [ dyn_topology ] for given @xmath6 , @xmath9 and @xmath11 . however , in order to keep things simple , let us just assume that each node in @xmath125 has just one neighbor in the clique @xmath8 , and , in a similar fashion , that each node in @xmath163 has just one neighbor in @xmath125 , which amounts to posing @xmath167 , @xmath168 , and @xmath169 , @xmath170 . clearly , both such approximations work well as long as the number of nodes in @xmath8 is small compared to @xmath6 , i.e. for @xmath103 where @xmath101 . * @xmath171 components ( i.e. the number of nodes in @xmath125 ) equal to @xmath172 * @xmath173 components ( i.e. the number of nodes in @xmath163 ) equal to @xmath174 * @xmath175 components equal to zero . therefore , the mean @xmath94 and standard deviation @xmath95 ( see equation ) can be computed as follows : @xmath176 and the approximations in equation can be immediately derived as leading order results in @xmath6 of the above expressions . 99 sen , a. _ development as freedom _ ; oxford university press ( oxford ) , 1999 ( p. 182 ) . surowiecki , j. _ the wisdom of crowds : why the many are smarter than the few and how collective wisdom shapes business , economies , societies , and nations _ ; doubleday books ( new york ) , 2004 . shafak , e. the view from taksim square : why is turkey now in turmoil ? + _ http://www.guardian.co.uk/world/2013/jun/03/taksim-square-istanbul-turkey-protest _ gore , a. _ the assault on reason _ ; penguin press ( london ) , 2007 . castellano , c. ; fortunato , s. ; loreto , v. statistical physics of social dynamics . phys . _ * 2009 * , _ 81 _ , 591 - 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the ability of a society to make the right decisions on relevant matters relies on its capability to properly aggregate the noisy information spread across the individuals it is made of . in this paper we study the information aggregation performance of a stylized model of a society whose most influential individuals the leaders are highly connected among themselves and uninformed . agents update their state of knowledge in a bayesian manner by listening to their neighbors . we find analytical and numerical evidence of a transition , as a function of the noise level in the information initially available to agents , from a regime where information is correctly aggregated to one where the population reaches consensus on the wrong outcome with finite probability . furthermore , information aggregation depends in a non - trivial manner on the relative size of the clique of leaders , with the limit of a vanishingly small clique being singular . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the chinese famines of 1958 - 1961 killed , it is now estimated , close to thirty million people [ ... ] . the so called great leap forward initiated in the late 1950s had been a massive failure , but the chinese government refused to admit that and continued to pursue dogmatically much of the same disastrous policies for three more years [ ... ] . in 1962 , just after the famine had killed so many millions , mao made the following observation , to a gathering of seven thousand cadres : _ `` without democracy , you have no understanding of what is happening down below ; the situation will be unclear ; you will be unable to collect sufficient opinions from all sides ; there can be no communication between top and bottom ; top - level organs of leadership will depend on one sided and incorrect material to decide issues [ ... ] '' . _ + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

the origin of the ultra - high energy cosmic rays ( uhecrs ) is still an unsolved problem in astroparticle physics . a model for acceleration of cosmic rays , firstly devised by hillas@xcite predicts that cosmic rays are accelerated to the highest energies ( above 10@xmath0 ev ) by electromagnetic fields of astrophysical objects . therefore , the identication of possible astrophysical sources of uhecrs is possible by analysing the arrival directions of the cosmic rays . the correlation of the positions of point - like astrophysical objects ( point sources ) with the arrival directions of cosmic rays defines a small scale anisotropy . in the search of point - like sources it is a common procedure to convolve the sky maps containing arrival directions of cosmic rays with mathematical functions ( the kernel of the convolution operation ) aiming to optimize the signal to noise ratio . in this work it is studied the performance of some kernels of the mexican hat wavelet family ( mhwf ) to identify point sources of cosmic rays , and disentangle genuine signals from the background . wavelets are defined as mathematical functions belonging to the @xmath1 space . they can be thought as localized wave - like oscillating functions which can be operated with a given signal and provide information about it . the continuous wavelet transform ( cwt ) in two dimensions may be formally written as @xmath2 where @xmath3 ( @xmath4 , @xmath5 ) is the scaling factor and @xmath6 and @xmath7 ( @xmath8 ) are the translation parameters . so , the cwt decomposes a function @xmath9 in a basis of wavelet @xmath10 . one can scale and translate a mother - wavelet " @xmath11 and obtain a wavelet @xmath10 , as follows : @xmath12 the mexican hat wavelet family ( mhwf ) , introduced by gonzlez - nuevo _ et al_.@xcite , and its extension on the sphere have been widely used to detect point sources in maps of cosmic microwave background radiation@xcite , due to the amplification of the signal - to - noise ratio ( snr ) when transiting from real to wavelet space . the mhwf is obtained by successive application of the laplacian operator to the two - dimensional gaussian @xmath13 . a generic member of this family , of order @xmath14 , is : @xmath15 celestial maps are pixelations of the celestial sphere taking into account the angular resolution of the experiment . the events map is a celestial map representing the arrival directions of cosmic rays in a suitable coordinate system . due to intrinsic limitations of detector , every event detected is convolved with a probability related to the angular resolution of the detector , which means that there is a point spreading function ( psf ) associated to the detector . the convolution of celestial maps with filters is given by : @xmath16 where @xmath17 is the number of cosmic rays within the pixel of index @xmath18 , in the direction @xmath19 . @xmath20 is the used filter and @xmath21 is the position vector representing the point where the integral is being calculated . this work is an extension of previous ones@xcite . the simulated detector has two sites , one located in the southern hemisphere ( 36@xmath22 s and 65@xmath22 w ) , and the other in the northern hemisphere ( 38@xmath22 n and 102@xmath22 w ) , seven times larger than the one in the south , implying on a flux of cosmic rays seven times greater . the acceptance law has the form @xmath23 , where @xmath24 is the zenith angle . it was also considered the case of an ideal detector with uniform exposure and full sky coverage . it was simulated a point source located at ( l , b)=(320@xmath22,30@xmath22 ) ( galactic coordinates ) . this source was modeled by a gaussian : @xmath25 and was embedded in a background consisting on a superposition of four different patterns of arrival directions of cosmic rays . the simulated patterns were : ( i ) an isotropic distribution of events ; ( ii),(iii ) dipoles , modeled according to @xmath26 , where @xmath27 is the flux of cosmic rays , @xmath28 is related to the isotropic flux , @xmath29 is the amplitude of the dipole ( @xmath30 for ( ii ) and @xmath31 for ( iii ) ) , @xmath32 is the vector which points towards the dipole ( @xmath33 = @xmath34 ) for ( ii ) and @xmath33 = @xmath35 ) for ( iii ) ) and @xmath36 is a unit vector pointing in an arbitrary direction@xcite ; ( iv ) several sources modeled according to equation [ eq : source ] , in the directions @xmath33 : @xmath34 [ @xmath37 , @xmath38 , @xmath39 [ @xmath40 , @xmath41 , @xmath42 [ @xmath43 , @xmath44 , @xmath45 [ @xmath46 , @xmath41 , @xmath47 [ @xmath48 , @xmath49 , @xmath50 [ @xmath51 , @xmath41 , @xmath52 [ @xmath53 , @xmath54 , @xmath55 [ @xmath56 , @xmath57 , @xmath58 [ @xmath59 , @xmath60 and @xmath61 [ @xmath46 , @xmath62 . this last background pattern was included in the simulation because unknown sources might be present during an analysis tagged on a given source , and their effects must be evaluated . from this combination of source ( signal ) and background patterns ( noise ) , it was obtained the events map . the events map resulting from the sum of the background patterns and the simulated source was convolved with gaussian and mhwf ( orders 1 , 2 and 3 ) filters . the amplification of the snr ( @xmath63 ) is calculated by the expression @xmath64 where @xmath65 is the value of the central pixel associated to the source in the non - filtered source map , @xmath66 is the corresponding value in the filtered source map , @xmath67 is the root mean square ( rms ) of the non - filtered background map and @xmath68 is the rms of the filtered background map . aiming to study the impact of the number of events from the source and its intensity upon the filter , different number of events ( @xmath69 ) were simulated in the direction of the source , ranging from 10 events up to 1000 . the amplitude of the gaussian source ( @xmath70 ) was also varied , from @xmath71 to @xmath72 . in figures [ fig : evt ] and [ fig : amp ] it is shown the maximum amplification of the snr , as a function of the corresponding scale . in the case of figure [ fig : evt ] , @xmath69 is fixed and @xmath70 is varied . for figure [ fig : amp ] , @xmath70 is fixed and @xmath69 from the source is varied . from figure [ fig : evt ] it can be noted that the uniform exposure acts like a constraint for @xmath63 , and that the gaussian has a slightly better performance than the mhwf filters . the maximum amplification for nonuniform exposure is clearly achieved by using mhwf filters , whereas the gaussian filter has amplification close to 1 , which means no amplification . comparing the two graphs in figure [ fig : evt ] , in the bottom graph the amplification is greater , which seems reasonable since the number of events in this case is 100 times greater . in figure [ fig : amp ] it can be seen the behavior of the filters when @xmath70 is varied . for the uniform exposure the amplification of both the gaussian and the mhwf filters are low , but the gaussian has a slightly better performance . comparing the top and bottom graphs in figure [ fig : amp ] , it is clear that the amplification achieved by the filters is proportional to @xmath69 . also , when there is an acceptance , the gaussian filter does not provide a good amplification of the snr , which can be achieved by using the mhwf . in this work it was analyzed the performance of the gaussian and the mhwf filters to detect point sources of cosmic rays embedded in a non uniform background , whose features are modulated both by the acceptance of the detector and the background patterns imposed to the incoming particles . some parameters from the source such as the amplitude @xmath70 and the number of events @xmath69 were varied , and the effects of theses changes on amplification of the snr was studied . the trivial conclusion is that the amplification achieved by these kernels is proportional to the source intensity parameters ( @xmath70 and @xmath69 ) . it is interesting to notice that for a realistic case , i. e. , a source with low amplitude and only a few events coming from its direction , the amplifications achieved are low for both filters . the mhwf filters are more robust to these parameters and can provide a greater amplification of the snr even if @xmath69 is small . regarding the contribution of the acceptance of the experiment for the detection , it can be clearly seen that for uniform exposure the amplifications are smaller . in this case , the gaussian filter provides a slightly better amplification compared to the mhwf . however , for a more realistic case taking into account the nonuniform exposure of the detector , the mhwf filters always achieve a greater amplification . also , they are more robust to low statistic of events , which makes them particularly useful for cosmic ray studies .

an analysis of the sensitivity of gaussian and mexican hat wavelet family filters to the detection of point sources of ultra - high energy cosmic rays was performed . a source embedded in a background was simulated and the number of events and amplitude of this source was varied aiming to check the sensitivity of the method to detect faint sources with low statistic of events .

the hamiltonian for a classical scalar field with potential @xmath2 and an environmental temperature @xmath3 is , ( with @xmath4 ) @xmath5}\over { t}}={1\over { t}}\int dx\left [ { 1\over 2 } \left ( { \partial\phi\over\partial t}\right ) ^2 + { 1\over 2 } \left ( { \partial\phi\over\partial x}\right ) ^2 + v_0(\phi)\right ] \ , .\ ] ] even though 1-dimensional field theories are free of ultra - violet divergences , the ultra - violet cutoff imposed by the lattice spacing will generate a _ finite _ contribution to the effective potential which must be taken into account if we are to obtain a proper match between the theory formulated in eq . ( [ e : hamiltonian ] ) and its numerical simulation on a discrete lattice . if neglected , this contribution will compromise the measurement of physical quantities such as the density of kink - antikink pairs or the effective kink mass . however , before investigating the particular example of kink - antikink production , we present the method in its most general form . for classical , 1-dimensional finite - temperature field theories , the one - loop corrected effective potential is given by the momentum integral @xcite @xmath6 = v_0(\phi ) + { t\over 4}\sqrt{v_0''(\phi ) } \ ; .\ ] ] as mentioned before , the lattice spacing @xmath1 and the lattice size @xmath0 introduce long and short momentum cutoffs @xmath7 and @xmath8 , respectively . lattice simulations are characterized by one dimensionless parameter , the number of degrees of freedom @xmath9 . for sufficiently large @xmath0 one can neglect the effect of @xmath10 and integrate from @xmath11 to @xmath12 . for @xmath13 ( satisfied for sufficiently large @xmath12 ) , the result can be expanded into @xmath14 as is to be expected for a 1-dimensional system , the limit @xmath15 exists and is well - behaved ; there is no need for renormalization of ultra - violet divergences . however , the effective one - loop potential is lattice - spacing dependent through the explicit appearance of @xmath12 , and so are the corresponding numerical simulations . in order to remove this dependence on @xmath1 , we follow the renormalization procedure given by bg @xcite ; it is irrelevant if the @xmath12-dependent terms are ultra - violet finite ( @xmath16 ) or infinite ( @xmath17 ) . in the lattice formulation of the theory , we add a ( finite ) counterterm to the tree - level potential @xmath18 to remove the lattice - spacing dependence of the results , @xmath19 there is an additional , @xmath12-independent , counterterm which was set to zero by an appropriate choice of renormalization scale . the lattice simulation then uses the corrected potential @xmath20 as we will show later in the context of kink - antikink pair production , this lattice formulation simulates the continuum limit to one loop as given by eq . ( [ e.oneloopdef ] ) . note that the above treatment yields two novel results . first , that the use of @xmath21 instead of @xmath18 gets rid of the dependence of simulations on lattice spacing . [ of course , as @xmath22 , @xmath23 . however , this limit is often not computationally efficient . ] previous works @xcite , have explored the influence of a counterterm quadratic on lattice spacing . however , we note that for small enough @xmath1 , the limit of interest here , our linear correction is dominant . second , that the effective interactions that are simulated must be compared to the one - loop corrected potential @xmath24 of eq . ( [ e.oneloopdef ] ) ; once the lattice formulation is made independent of lattice spacing by the addition of the proper counterterm(s ) , it simulates , within its domain of validity , the thermally corrected one - loop effective potential . as an application of the method discussed above we consider the symmetric double - well potential @xmath25 . the excitations of the associated quantum theory have a mass @xmath26 . thus , in order for the system to remain in the classical regime , the condition @xmath27 must hold . this constrains the dimensionless temperature @xmath28 to be larger than @xmath29 . for @xmath30 , where @xmath31 is the dimensionless kink mass corresponding to the tree - level potential @xmath18 @xcite , we can expect to have only a dilute gas of kink - antikinks at thermal equilibrium . with these two conditions jointly satisfied , the system will also obey @xmath32 , indicating weak coupling . the corrected lattice potential is @xmath33 simulations using @xmath21 will , in principle , match the continuum theory @xmath34 which has ( shifted ) minima at @xmath35 , with @xmath36 . for the numerical simulations we introduce the dimensionless variables @xmath37 , @xmath38 , and @xmath39 . to keep the notation simple we will subsequently suppress the tilde . the field is prepared as @xmath40 , and evolved in time according to a langevin equation with white noise that incorporates the environmental temperature @xmath3 through the fluctuation - dissipation theorem . the details of this and of the numerical implementation are laid out in @xcite . the viscosity coefficient @xmath41 has been set to unity throughout this study . the time step is @xmath42 , and @xmath43 . the heat bath takes a time @xmath44 to achieve equipartition so that the energy per degree of freedom is @xmath45 . * ensemble average of field . * for sufficiently low temperatures the simulated field will remain in the vicinity of the minimum @xmath46 for a very long time ( compared to typical fluctuation time - scales ) , until large - amplitude fluctuations drive portions of the space over the barrier at @xmath47 and beyond . the subsequent evolution is then the formation of the first kink - antikink pair . true thermal equilibrium consists of reaching the final equilibrium kink - antikink density together with zero mean field . in a lose sense , this situation corresponds to symmetry restoration , although in one spatial dimension `` symmetry restoration '' will occur for any nonzero temperature ; it is all a matter of time . as a first test of our procedure , we investigate the mean field value @xmath48 _ before _ the nucleation of a kink - antikink pair , _ i.e. _ , while the field is still well localized in the bottom of the well . in fig . [ f.phiave ] we show the ensemble average of @xmath49 ( after 100 experiments ) for different values of @xmath1 , ranging from 1 down to 0.1 , at @xmath50 . the simulations leading to the left graphs use the `` bare '' potential @xmath18 , whereas the right graphs are produced employing @xmath21 ( eq . [ e.vlatt ] ) . apart from a discrepancy for very coarse grids ( @xmath51 ) , where the resolution nears the correlation length , the average field value is clearly lattice - spacing independent when using @xmath21 , in contrast to the use of @xmath18 . as discussed before , the average mean field value should correspond to the minimum @xmath52 of the effective potential . however , since we are only using a one - loop approximation , this agreement will get progressively worse as the temperature increases . for example , for @xmath53 , the discrepancy between the theoretical value , @xmath54 , and the numerical result is 10% . for higher temperatures , we should not trust the one - loop approximation ; other nonperturbative effects , such as subcritical fluctuations , too small in width and amplitude to emerge as a kink - antikink pair but still large enough to bring the average value of the field away from its one - loop value , will become important@xcite . thus , we restrict our investigation to temperatures safely within the limits of validity of the one - loop approximation . in a subsequent study , we intend to investigate the role of these nonperturbative effects . * density of kink - antikink pairs . * perhaps the most difficult task when counting the number of kink - antikink pairs that emerge during a simulation is the identification of what precisely is a kink - antikink pair at different temperatures . typically , we can identify three `` types '' of fluctuations : i ) small amplitude , perturbative fluctuations about one of the two minima of the potential ; ii ) full - blown kink - antikink pairs interpolating between the two minima of the potential ; iii ) nonperturbative fluctuations which have large amplitude but not quite large enough to satisfy the boundary conditions required for a kink - antikink pair . these latter fluctuations are usually dealt with by a smearing of the field over a certain length scale . basically , one chooses a given smearing length @xmath55 which will be large enough to `` iron out '' these `` undesirable '' fluctuations but not too large that actual kink - antikink pairs are also ironed - out . in this study , a similar smoothing was implemented as a four - pole butterworth low - pass filter of the field with a filter cutoff length @xmath55 . the filter removes fluctuations with wavelengths smaller than @xmath55 . the choice of @xmath55 is , in a sense , more an art than a science , given our ignorance of how to handle these nonperturbative fluctuations . in table 1 we show the number of pairs for different choices of filter cutoff length and for different temperatures . we counted pairs by identifying the zeros of the filtered field . from table 1 it is clear that as the temperature increases , the discrepancies in the count of pairs also increase . for this reason we only trust our data for fairly low temperatures . the problem is agravated by the fact that the `` size '' of the kink - antikink pair , _ i.e. _ , the minimal separation between the two , not only changes due to dynamical effects , but also changes with temperature . thus , choosing the filter cutoff length to be too large may actually undercount the number of pairs . choosing it too low may include nonpertubative fluctuations as pairs . we chose @xmath56 in the present work , as this is the smallest `` size '' for a kink - antikink pair . in contrast , in the works by alexander et al . a different method was adopted , that looked for zero - crossings for eight lattice units ( they used @xmath57 ) to the left and right of a zero crossing @xcite . we have checked that our simulations reproduce the results of alexander et al . if we : i ) use the bare potential in the lattice simulations and ii ) use a large filter cutoo length @xmath55 . specifically , the number of pairs found with the bare potential for @xmath58 are : @xmath59 , for @xmath60 , respectively . alexander et al . found ( for our lattice length ) @xmath61 . comparing these with table 1 , it is clear that the differences between our results and those of alexander et al . come from using a different potential in the simulations , _ viz . _ a corrected vs. an uncorrected potential . we believe that at this point it is fair to say that the `` smearing issue '' remains unresolved , at least for temperatures @xmath62 or so . we intend to address the issue of how to deal with these nonperturbative effects in a forthcoming publication . in any case , the focus of the present work is mostly on how to achieve a lattice - independent count , irrespective of the particular method used for identifying the kink - antikink pairs . [ f.kinkdens ] compares measurements of the kink - antikink pair density ( half the number of zeros of the filtered field ) , ensemble - averaged over 100 experiments , for different lattice spacings . again it is clear from the graphs on the left that using the tree - level potential @xmath18 in the simulations causes the results to be dependent on @xmath1 , whereas the addition of the finite counterterm removes this problem quite efficiently ; both diagrams of fig . [ f.kinkdens ] contain four graphs each , although the graphs on the right are almost indistinguishable . unless the properly corrected potential is used in the lattice simulations , the measured number density of topological defects is sensitive to the lattice spacing . one must be careful when counting kinks , especially for large lattice spacings , say @xmath63 or larger . the next step is to extract the correct continuum theory from the lattice simulations . what theory is the lattice simulating ? most previous simulations of thermal nucleation of kink - antikink pairs have overlooked this problem . although a temperature - dependent kink mass was conjectured in the works of reference @xcite , not much has been done to understand its origin or its value . one way of addressing it is by comparing the numerically measured kink mass with its theoretical prediction . it has been found that the measured mass was smaller than the theoretical prediction by a factor ranging from 25% to 45%@xcite , a disturbing result . this has been attributed to several effects , such as the finite size of the lattice , the finite size of the kinks , and phonon dressing effects due to the lattice discretization@xcite . we will show that this problem is rooted in the incorrect matching between theory and numerical simulations . in the works by alexander et al . a beautiful agreement between the low temperature limit and a @xmath64 wkb approximation was obtained , as well as between high temperatures and a double gaussian nonperturbative method @xcite . our method is effective precisely between these two regimes , and could be interpreted as a @xmath3-dependent wkb approximation obtained naturally from the inclusion of counterterms . one should expect the equilibrium kink - antikink pair density to follow the proportionality @xcite @xmath65 where @xmath66 is the kink mass , given by @xmath67 \ , , \ ] ] and @xmath68 is the kink solution to the equation of motion . note that we left the potential @xmath69 unspecified . if we use the tree - level potential , @xmath70 , we obtain the well - known result @xmath71 . or , in dimensionless variables , @xmath72 . one can extract the numerical value of @xmath66 by measuring the pair density and plotting the results in a logarithmic scale , as in ref . the result should be a straight line with negative slope @xmath73 . however , as mentioned above , the measured slope was found to be about @xmath74 . the reason for the discrepancy is that the potential which should be used when comparing theory and simulation is not the tree - level potential @xmath75 but the effective potential @xmath24 . thus , one must compute the effective kink mass @xmath76 using the corrected potential @xmath24 and _ then _ compare the results with the numerical simulations . the effective kink mass can be found using the equation of motion and the real part of @xmath77 @xcite , @xmath78 = 2\int_0^{\phi_{\rm min } } \sqrt{2 { \rm re}(v_{\rm 1l}(\phi ) ) } \ , d\phi \ ; .\ ] ] this integration can easily be carried out numerically . in fig . [ f.kinktheory ] we plot the ratio @xmath79 vs. the dimensionless temperature , @xmath80 . of course , for @xmath64 , @xmath81 . as the temperature increases , the effective kink mass decreases . the points represent the kink mass extracted from the numerical simulations , while the error bars were obtained by propagating the standard deviation of the ensemble average . it is quite clear that the effective kink mass tracks the numerical values quite well . in fact , within the validity of our approximations , the `` averaged '' value for the effective kink mass is @xmath82 . also , since the mass extracted from the simulations depends on the filter cutoff length @xmath55 , the reasonable agreement between theory and numerical experiment offers indirect support for our choice of @xmath56 . for very small and very large temperatures the theory fails to track the numerical data . at large temperatures @xmath83 , the one - loop approximation breaks down , while for low temperatures @xmath84 , the large pair nucleation time - scale precludes a proper statistical analysis ( not enough experiments ) . however , the conclusion is quite clear : by controlling the dependence on lattice spacing of the simulations we were able , within the validity of our approximations , to obtain the correct effective potential that should be used when comparing theory and numerical experiment . mg was partially supported by the national science foundation through a presidential faculty fellows award no . phy-9453431 and by the national aeronautics and space administration grant no . hrm was supported by a national science foundation grant no . phy-9453431 and by the national aeronautics and space administration grant no .

we investigate the matching between ( 1 + 1)-dimensional nonlinear field theories coupled to an external stochastic environment and their lattice simulations . in particular , we focus on how to obtain numerical results which are lattice - spacing independent , and on how to extract the correct effective potential which emerges from the simulations . as an application , we study the thermal production of kink - antikink pairs , obtaining a number density of pairs which is lattice - spacing independent and the effective barrier for pair production , _ i.e. _ , the effective kink mass . psfig key words : field theory , lattice simulations , solitons the possibility that the universe underwent a series of symmetry breaking phase transitions during the earliest stages of its evolution has triggered a great deal of interest in the application of nonequilibrium statistical mechanics to cosmology . of particular interest is the potential role that coherent field configurations , which arise from the interplay between nonlinearities and out - of - equilibrium conditions , could have played in shaping the earlier evolution and present - day structure of the universe . examples range from the nucleation of bubbles in the context of inflation and electroweak baryogenesis to the formation of topological defects@xcite . given the relevance of the topic , and the obvious difficulties in performing experiments in a cosmological context , attempts to investigate the emergence of coherent field structures rely heavily on numerical simulations and possible analogies with condensed matter experiments@xcite . here we would like to focus on the former , namely , on numerical simulations designed to investigate the emergence of coherent structures in thermal field theories . an obvious limitation of such an approach is that , although field theories are continuous and usually formulated in an infinite volume , lattice simulations are discrete and finite , imposing both a maximum ( `` size of the box '' @xmath0 ) and a minimum ( lattice spacing @xmath1 ) wavelength that can be probed by the simulation . when the system is coupled to an external thermal ( or quantum ) bath , fluctuations will be constrained within the allowed window of wavelengths , leading to discrepancies between the continuum formulation of the theory and its lattice simulations ; the results will be dependent on the choice of lattice spacing . parisi suggested that if proper counterterms were used , this depedence on lattice spacing could be attenuated@xcite . this technique was implemented in a study of 2-dimensional nucleation by alford and gleiser@xcite . however , these studies still left open the question of how to match the lattice results to the correct continuum field theory . this is a crucial step if we want to test numerically certain predictions from field theories of relevance not only for cosmology but also for condensed matter physics , such as decay rates of metastable states and the production of topological defects . recently , borrill and gleiser ( bg ) have examined this question within the context of 2-dimensional critical phenomena@xcite . they have computed the counterterms needed to render the simulations indepedent of lattice spacing and have obtained a match between the simulations and the continuum field theory , valid within the one - loop approximation used in their approach . inspired by their results , we decided to investigate the validity of this method within the context of topological defects . the results presented here should be relevant to numerical studies of the formation of topological defects and their comparison with experiments@xcite , as well as to elucidating the general nature of the effective potential which emerges from coupling nonlinear field theories to a stochastic thermal ( or quantum ) background .

lipid bilayers are the most elementary and indispensable structural component of biological membranes , which form the boundary of all cells in living systems . in biological membranes , the bilayer consists of many different lipids and other amphiphiles , which can separate into coexisting liquid phases , or domains , with distinct compositions . two types of phases are typically observed : a liquid phase with short - range order and a liquid disordered phase ( see , e.g. , @xcite , @xcite ) , which we label phase @xmath0 and @xmath1 in the sequel . their configurations , however , are fundamentally distinct from other interfaces , since they are not determined by a surface tension but rather by a bending elasticity , as introduced independently by @xcite , @xcite , and @xcite . as in , e.g. , @xcite and @xcite , the canham - helfrich - evans energy functional of a surface @xmath2 and a phase @xmath3 is given by @xmath4 where @xmath5 denotes the integration with respect to the ordinary two - dimensional area measure , @xmath6 is the sum of the principal curvatures of @xmath2 , i.e. , twice the mean curvature and @xmath7 is the gaussian curvature . in a state where the phases are completely separated by a sharp interface , we let @xmath8 denote the characteristic function of the phase @xmath0 , @xmath9 are the phase - dependent bending rigidities , and @xmath10 is the phase - dependent spontaneous curvature . in the last term of , @xmath11 is the ( constant ) line tension coefficient , @xmath12 is the interface between the phases @xmath0 and @xmath1 , and @xmath13 is the one - dimensional hausdorff measure . @xmath2 at 280 250 @xmath14 at 185 350 we prove the existence of a minimizer of in the set of families of axisymmetric surface - phase couples @xmath15 such that the total area of the surface @xmath2 , the area ratio between the phases , and the volume enclosed by the surface @xmath2 are fixed . in the last forty years , homogeneous membranes have been extensively studied from the experimental , theoretical , and numerical points of view ( see , e.g. , @xcite ) . investigation of inhomogeneous systems started more recently ( @xcite , @xcite ) and is nowadays at the center of an increasing focus . multicomponent vesicles have recently been considered in numerical studies aiming at understanding the dynamics of the phase separation , the stability of nanodomains , and the complex morphology of the membranes ; see the review paper @xcite . we refer in particular to the paper @xcite concerning the dynamics of one - dimensional curves in viscous fluids , while two - dimensional surfaces where investigated with the phase field method in @xcite and @xcite , by means of surface finite elements in @xcite , and by adaptive finite elements in @xcite . our result provides a theoretical basis , at least in the axisymmetric case , for the existence of the shapes approximated by these numerical studies . more advanced models , with respect to , which couple the chemical and mechanical properties of lipid bilayers , are subject of current research , see , e.g. , @xcite , @xcite . we note that in the case where the spontaneous curvature vanishes , the canham - helfrich functional becomes the famous willmore functional ( see , e.g. , ( * ? ? ? * chapter 7 ) ) . while there has recently been tremendous amount of research associated with minimizing the willmore functional , this research does not directly carry over to the doubly - constrained canham - helfrich functional ( cf . @xcite ) . we detail now the assumptions on the variables , the parameters , and the constraints that compose our problem and we state the main result . [ [ i - surfaces . ] ] i ) surfaces . + + + + + + + + + + + + axisymmetric surfaces , i.e. , surfaces of revolution , can be obtained by rotating a curve about a line . let @xmath16 be a half - plane in @xmath17 ; we first consider a smooth curve @xmath18\to { \mathbb{r}}^2_+$ ] , @xmath19 . by rotating @xmath20 around the axis @xmath21 , we obtain the surface @xmath2 parametrized by : @xmath22,\qquad ( t,\theta)\in [ 0,1]\times [ 0,2\pi].\ ] ] if a surface @xmath2 admits the parametrization , then we say that @xmath2 _ is generated by _ a standard computation ( see section [ sec : surrev ] below ) shows that if @xmath20 generates a smooth surface @xmath2 without boundary , then the 2-dimensional surface area @xmath23 , the enclosed volume @xmath24 , and the principal curvatures @xmath25 of the generated surface are given by @xmath26 denoting by @xmath27 the one - dimensional lebesgue measure , let @xmath28}$ ] be the area measure induced by the curve @xmath20 . it is not difficult to see , at least in the case of only one phase ( see @xcite ) , that a bound on @xmath29 provides an a priori estimate on @xmath30 in the space @xmath31 , which translates into a bound for @xmath20 in the space @xmath32 on any stretch of curve such that @xmath33 . precisely , by proposition [ prop : compcurv ] , it is not restrictive to assume that @xmath34 moreover , in order for @xmath29 to be finite , @xmath20 is forced to meet the @xmath35axis orthogonally , so if @xmath36 , then @xmath37 is the @xmath38 immersion of a closed surface . if @xmath37 is not an embedding , as , for example , in the case of the surface generated by the limit curve in figure [ fig1 ] , we _ define _ the _ generalized _ two - dimensional surface area , enclosed volume and principal curvatures of the generated surface by the quantities in - . [ [ ii - phases . ] ] ii ) phases . + + + + + + + + + + + if @xmath39 is the characteristic function of phase @xmath0 , then in order for @xmath40 to be bounded , and in particular in order for @xmath41 to be bounded , we need to impose some kind of regularity on the class of admissible phases @xmath42 , for example by restricting to characteristic functions of finite perimeter sets on the surface @xmath2 . under the simplifying assumption of axisymmetry for the phases , as well as for the surfaces , a useful approach is then to follow the parametrization @xmath43\to \{0,1\}$ ] . let @xmath44 be the set of points where @xmath3 has a jump discontinuity . owing to axisymmetry , the ( measure - theoretical ) interface @xmath12 between the two phases on the surface @xmath2 is a union of circles @xmath45 \right\}\ ] ] and @xmath46 where @xmath47 is the counting measure restricted to the jump set of @xmath3 ( see section [ ssec : bv ] below ) . since @xmath48)$ ] and @xmath49 for @xmath50 , then according to , @xmath51 if and only if @xmath52 . in the sequel , it will be convenient to deal with the weaker request that @xmath53 extending @xmath54 to @xmath55 if the quantity in is not bounded . the area measure of phase @xmath0 can then be expressed as @xmath56 it would also be possible to choose , as ambient space for the phases , the space of _ special _ functions of ( locally ) bounded variation @xmath57 . since we are dealing with two - valued functions , and the cantorian part of the measure will not appear in any case , we prefer to keep the setting as simple as possible and use @xmath58 functions . [ [ iii - parameters . ] ] iii ) parameters . + + + + + + + + + + + + + + + + let the bending rigidities for phase @xmath59 be given by @xmath60 , and the spontaneous curvature be @xmath61 , for @xmath62 . in order for the the functional @xmath54 to be coercive , we require ( see lemma [ lemma : fundest ] below ) that @xmath63 we note that the physical range in which the parameters @xmath64 and @xmath65 are typically found is contained in the one we impose in , see e.g. @xcite and @xcite ( note that the latter cites the former , but inverting numerator and denominator , by mistake ) . we define the coefficient functions on the interval @xmath66 $ ] in such a way that @xmath64 is the linear interpolation of @xmath67 , @xmath68 , and the same holds for @xmath65 and @xmath69 , i.e. , @xmath70 according to the parametrization , the contributions in depending on the mean curvature , the gaussian curvature , and the line tension of the interface between the phases , can then be written as @xmath71 we say that a couple of surface and phase @xmath72 is _ admissible _ if the surface @xmath2 is generated by a curve @xmath20 satisfying and the phase @xmath3 satisfies . finally , for fixed area and volume , a configuration made of several connected components may have a lower energy than a one - component configuration . this could also be favored by a relatively high value of @xmath73 , since separation of phases in different components would have no interface between phases and thus @xmath74 . from a dynamical point of view , in certain conditions , shape transformations involving topological changes , like budding and fission ( see , e.g. , ( * ? ? ? * section 3 ) ) , could be expected . we take this possibility into account by studying families of admissible surface - phase couples @xmath75 , and defining the total energy of such a system as the sum of the helfrich energies of the single components : @xmath76 . [ [ iv - main - result . ] ] iv ) main result . + + + + + + + + + + + + + + + + we prove the following . [ th : main ] let @xmath77 , @xmath62 be given such that is satisfied . let @xmath78 @xmath79 , @xmath62 be given . let @xmath80 be given such that @xmath81 let @xmath82 denote the set of finite families @xmath75 of admissible couples of surfaces and phases such that the _ generalized _ area , volume , and phase area constraints @xmath83 are satisfied ( see , ) . let @xmath84 . then the problem @xmath85 has a solution . the proof of theorem [ th : main ] follows the direct method of the calculus of variations : given a minimizing sequence of ( systems of ) surfaces and phases @xmath86 satisfying the area and volume constraints , by compactness we obtain a subsequence converging to a system @xmath87 , and by lower semicontinuity of the functional @xmath88 we prove that @xmath87 is a global minimizer . regarding compactness , it is fundamental that the phase - dependent parameters @xmath64 , @xmath65 , @xmath69 are chosen in such a way that the functional @xmath54 is an upper bound for the @xmath89-norm of the second fundamental form of the surface @xmath2 ( lemma [ lemma : fundest ] ) . in this way we can exploit the compactness result obtained in @xcite in the case of homogeneous membranes . the main idea in modeling the phases ( see also section [ ssec : discussion ] ) is to follow the parametrizations @xmath20 instead of their images , so that the sequence of parametrized phases is defined on a fixed interval rather than a sequence of surfaces . the drawback with this approach is that the phases are only of _ locally _ bounded variation ; where the curves touch the axis of revolution , the area measure vanishes . in other words , where the horizontal component @xmath90 of the curve @xmath20 becomes infinitesimal , the line tension part of the functional may allow for an infinite number of discontinuities in the phases @xmath3 . however , this will not constitute a problem in the proof of lower semicontinuity , since the combined phases and curves are well - behaved ( lemma [ lemma : hlsc ] ) . the proof of the lower semicontinuity for the curvature terms and requires a special care , since we have to pass to the limit simultaneously in the surfaces and in the phases defined on the surfaces . a useful tool can be found in the function - measure pairs introduced in @xcite . in section [ sec : geoineq ] we describe the notation , we derive the geometrical quantities involved in helfrich s functional , and we recall the basic definitions and the main results regarding functions of bounded variation and measure - function couples . in section [ sec : existence ] we study the compactness and lower semicontinuity of a bounded admissible sequence , and end with the proof of theorem [ th : main ] . [ [ modeling - the - phases . ] ] modeling the phases . + + + + + + + + + + + + + + + + + + + + the choice of modeling the phases by following the parametrization @xmath43\to \{0,1\}$ ] , instead that by functions with support on the surface @xmath91 , yields first of all a much simpler setting , since the domain becomes a fixed real interval , instead of surface . it is also a way to solve a more intrinsic question related to the possible ill - posedness of the problem . consider , as in the example in figure [ fig1 ] , a sequence of curves @xmath92 and phases @xmath93 such that , in the limit , two stretches of curve carrying different phases overlap . how should the limit phase be defined in this case ? at 251 176 at 232 154 at 235 154 at 213 131 @xmath20 at 310 135 @xmath94 at 163 5 @xmath95 at 236 5 @xmath96 at 355 5 @xmath97 at 385 165 @xmath98 at 385 50 any @xmath99-valued phase @xmath39 defined on the support of @xmath20 would not be a limit ( not even in the sense of distributions ) of the sequence @xmath100 . instead , defining the parametrized phases on the interval @xmath66 $ ] , the situation in figure [ fig1 ] could be described , e.g. , by choosing a constant sequence @xmath93 equal to the characteristic function of the interval @xmath101 , which clearly converges to the same characteristic function . another way to approach the problem of overlapping curves could be , for example , to consider the varifolds @xcite associated with the curves and to describe the segment @xmath102\times \{0\}$ ] by a stretch with double density . in the case of a homogeneous curve in two dimensions , this approach was followed , for example , in @xcite , but in the case of multiple phases it is not evident which approach to follow . a possibility could be by multivalued functions on varifolds , but we deem more natural to follow the parametrization of the curves . in higher dimensions this could be generalized by following the immersion of a riemann surface as opposed to studying its image . experimental evidence of higher - genus membranes is known for homogeneous membranes ( see , e.g. , @xcite ) , but we are not aware of any experiment or simulation showing multiphase biological or artificial membrane of genus higher than zero . nonetheless , from a mathematical viewpoint , it is straightforward to allow for genus-1 axisymmetric membranes ( see @xcite ) by considering closed generating curves with strictly positive first component . since the @xmath32-norm of such curves is bounded from above by , they have the same compactness properties of the curves that we consider , and theorem [ th : main ] could be directly extended to include also genus-1 generators . what is not clear to us , is whether the genus of the minimizer could be prescribed , since a priori a sequence of genus-1 minimizer may degenerate to a genus-0 surface in the limit . even though , according to experiments and simulations , we expect the phases of an axisymmetric surface to be axisymmetric as well , one could pose the problem of studying general phases on surfaces of revolution . we believe that this step could be performed following the same steps as in the symmetric case , as the theorems used for compactness and semicontinuity for functions of bounded variation do not depend on the dimension . actually , the functional for a two - dimensional phase would provide a better bound , namely in @xmath103 instead of @xmath58 . generalizations to non axisymmetric surfaces and phases would require a completely different approach : our method relies on ascoli - arzel compactness for equicontinuous curves , which in one dimension can be applied owing to the compact immersion @xmath104 , which fails in higher dimensions . [ [ necessity - of - constraints . ] ] necessity of constraints . + + + + + + + + + + + + + + + + + + + + + + + + + in order to model realistic configurations of multiphase membranes , we considered classes of surfaces with fixed total area , phases area , and enclosed volume . we notice , though , that the only constraint which is necessary in order to obtain compactness of a minimizing sequence is that on the total area , which is needed in order to bound from above the full second fundamental form of the surface ( lemma [ lemma : fundest ] ) and to ensure that the limit of a minimizing sequence does not vanish . the area of each phase is then bounded by the total area , and the enclosed volume results bounded by isoperimetric inequality . theorem [ th : main ] could then be restated by saying that if the total area is bounded from above and away from zero , then there is a minimizer of , and , in particular , there is one for any admissible choice of volume and phase area constraints . we refer to @xcite for the definitions of the following objects . let @xmath105 and @xmath106 be the @xmath107-dimensional lebesgue and hausdorff measures , let @xmath108 and @xmath109 be the restrictions to the set @xmath0 of the @xmath107-dimensional lebesgue and hausdorff measures , respectively . for an open set @xmath110 , let @xmath111 be the space of continuous functions with compact support in @xmath112 , let @xmath113 be the closure of @xmath111 with respect to uniform convergence , and let @xmath114 be the space of radon measures on @xmath112 , which can be identified with the dual of @xmath111 . if @xmath115 and @xmath116 we say that @xmath117 is a finite radon measure . for a curve @xmath18\to { \mathbb{r}}^2 $ ] , @xmath118 , we use the shorthand notation @xmath119 to denote the set @xmath120:{\gamma}_1(t ) \geq 0\}$ ] , we denote the first and second derivatives by @xmath121 , and we define the measure @xmath122 by @xmath28}$ ] . the derivation of the principal curvatures for a surface of revolution can be found , e.g. , in ( * ? ? ? * section 3 - 3 , example 4 ) or in @xcite . recalling the parametrization introduced in , we define the tangents to the surface along the coordinate lines as @xmath123 , \\ r_\theta:=\frac{\partial}{\partial \theta}r(t,\theta)&= \big[-{\gamma}_1(t)\sin \theta,\ { \gamma}_1(t)\cos \theta,\ 0\big],\end{aligned}\ ] ] and notice that @xmath124 , i.e. , they are always orthogonal to each other . the first fundamental form is given by @xmath125= \left [ \begin{array}{cc } r_t \cdot r_t & r_t \cdot r_\theta\\ r_\theta \cdot r_t & r_\theta \cdot r_\theta \end{array } \right ] = \left [ \begin{array}{cc } |\dot { \gamma}(t)|^2 & 0 \\ 0 & { \gamma}_1(t)^2 \end{array } \right],\ ] ] @xmath126 the normal vector to the surface can be oriented either inwards or outwards , depending on the direction of @xmath20 . for curves parametrized in counterclockwise direction , it is inwards : @xmath127\\ & = \frac{1}{|\dot { \gamma}(t)|}\big [ -\dot { \gamma}_2(t ) \cos \theta,-\dot { \gamma}_2(t ) \sin \theta,\ \dot { \gamma}_1(t)\big].\end{aligned}\ ] ] now we start to compute the second fundamental form , using a constant - speed parametrization . @xmath128\\ n_\theta:= \frac{\partial}{\partial \theta}n(t,\theta ) & = \frac{1}{|\dot { \gamma}(t)| } \big [ \dot { \gamma}_2(t ) \sin \theta,-\dot { \gamma}_2(t ) \cos \theta,\ 0\big].\end{aligned}\ ] ] the second fundamental form is given by @xmath129=-\left [ \begin{array}{cc } n_t \cdot r_t & n_t \cdot r_\theta\\ n_\theta \cdot r_t & n_\theta \cdot r_\theta \end{array } \right ] = \frac{1}{|\dot { \gamma}(t)| } \left [ \begin{array}{cc } \ddot { \gamma}_2 \dot { \gamma}_1 -\ddot { \gamma}_1 \dot { \gamma}_2 & 0 \\ 0 & { \gamma}_1 \dot { \gamma}_2 \end{array } \right].\ ] ] the _ signed _ curvature of @xmath20 is defined as @xmath130 the gaussian curvature is given by @xmath131 the mean curvature is ( for sake of notation , we define as mean curvature the double of what is often defined as mean curvature ) @xmath132 the principal curvatures are @xmath133 the area is @xmath134 and the volume enclosed by the surface is @xmath135 it is then straightforward to check that helfrich energy for axisymmetric surface and phase is given by expressions . let @xmath136 denote an open subset of @xmath137 ; following @xcite we say that a function @xmath138 has bounded variation in @xmath136 , and write @xmath139 , if @xmath140 we say that a function @xmath141 has locally bounded variation in @xmath136 , and write @xmath142 , if for each open set @xmath143 @xmath144 if @xmath142 there exists a finite radon measure @xmath117 and a @xmath117-measurable function @xmath145 such that @xmath146 for all @xmath147 . we write @xmath148 and @xmath149=\eta\|df\|$ ] . if @xmath139 , then for each @xmath150 @xmath151 ( since @xmath152 is scalar , we should have written @xmath153 instead of @xmath154 , but @xmath153 and @xmath149 $ ] look too similar ) . for example , denoting by @xmath155 the dirac distribution centered at @xmath96 , if @xmath139 is the characteristic function of an interval @xmath156 , then @xmath149=\delta_a-\delta_b$ ] , @xmath157 , @xmath158=\psi(b)-\psi(a),\qquad \int_u \psi\,d\|df\|=\psi(b)+\psi(a)\ ] ] and @xmath159 . in the sequel we will rely on the two fundamental results ( ( * ? * section 5.2 ) ) [ th : thlsc ] let @xmath160 be such that @xmath161 in @xmath162 . then @xmath163 [ th : thcpt ] let @xmath164 be such that @xmath165 . then there exist a subsequence @xmath166 and a function @xmath139 such that @xmath167 in @xmath168 . we recall that a sequence of radon measures @xmath169 is said to converge weakly-@xmath170 to @xmath171 if @xmath172 for every @xmath173 . we define the space of @xmath174-summable functions with respect to a positive radon measure @xmath117 as @xmath175 [ def : weakags ] following ( * ? ? ? * definition 5.4.3 ) , given a sequence of measures @xmath176 converging weakly-@xmath170 to @xmath117 , we say that a sequence of ( vector ) functions @xmath177 converges weakly to a function @xmath178 , and we write @xmath179 in @xmath180 , provided @xmath181 for every @xmath182 . for @xmath183 , we say that a sequence of ( vector ) functions @xmath184 converges _ weakly _ to a function @xmath185 , and we write @xmath179 in @xmath186 , provided @xmath187 for @xmath183 , we say that a sequence of ( vector ) functions @xmath184 converges _ strongly _ to a function @xmath185 , and we write @xmath188 in @xmath186 , if holds and @xmath189 [ lemma : moser ] let @xmath190 such that @xmath191 . suppose that @xmath169 and @xmath117 are radon measures on @xmath137 and that @xmath192 , @xmath193 , @xmath194 , @xmath195 be such that @xmath196 then @xmath197 [ th : ags ] let @xmath183 , let @xmath184 be a sequence converging _ weakly _ to a function @xmath185 in the sense of definition [ def : weakags ] , then @xmath198 for every convex and lower semicontinuous function @xmath199.$ ] the proof of existence relies on the following fundamental estimate , which shows that , for properly chosen coefficients , the functional @xmath200 bounds from above the @xmath89-norm of the second fundamental form of @xmath2 , independently of the phase @xmath3 . whereas the estimate follows from the phase - independent case ( see ( * ? ? ? * lemma 2.1 ) ) , for sake of completeness we include the details . let @xmath204 , note that @xmath205 and the last term is positive if and only if @xmath206 . for all @xmath207 it holds @xmath208 and thus @xmath209 for all @xmath210 , choosing @xmath211 and @xmath207 such that @xmath212 , we get @xmath213 where @xmath214 and @xmath215 . denoting by @xmath216 ( @xmath62 ) the constants obtained for @xmath217 , @xmath218 , etc . , we integrate on @xmath2 to obtain @xmath219 let the total area , the @xmath0-phase area and the volume constraints @xmath220 be given . let @xmath221 where @xmath222 is the cardinality of the @xmath223-th family , be a minimizing sequence for @xmath54 , i.e. , @xmath224 since for any @xmath220 satisfying it is possible to construct a spheroid with area @xmath225 , volume @xmath226 , and divide its surface in two domains such that one has area @xmath227 , the infimum above is finite , and there exists @xmath228 such that @xmath229 for all @xmath230 . in @xcite we studied the compactness properties of a sequence of surfaces for which the bound @xmath231 holds uniformly . owing to lemma [ lemma : fundest ] , we can apply the results in ( * ? ? ? * lemma 2.4 , lemma 2.5 , lemma 3.6 , proposition 3.7 ) to the sequence of surfaces @xmath232 . we summarize these results in the following proposition . regarding notation , we denote the first and the second component of a curve @xmath233 by @xmath234 and @xmath235 . [ prop : compcurv ] let @xmath86 be a sequence of finite systems of admissible surfaces and phases satisfying the bound for some fixed constant @xmath202 . denote the generating curves by @xmath236 . then , there exists a subsequence @xmath237 such that * the cardinality @xmath238 of the system is uniformly bounded . therefore , it is not restrictive to assume that @xmath239 , for some constant @xmath240 . * there exists a system of curves @xmath241 such that for all @xmath242 ( @xmath243 ) @xmath244 and , up to a permutation of the indices , @xmath245 * for @xmath246 , the curves @xmath247 converge to a point lying on the @xmath14-axis , i.e. @xmath248 for some @xmath249 . * for @xmath242 , it holds @xmath250 and @xmath251 * for @xmath242 , each curve @xmath233 meets the @xmath14-axis orthogonally , so , in particular , the generated surface @xmath252 is the union of a finite number of @xmath253-regular surfaces . * the area and volume constraints pass to the limit , i.e. , @xmath254 since singularities for @xmath255 can occur only on the @xmath14-axis , i.e. where @xmath256 , points ( iv ) and ( v ) imply that the limit system can be reparametrized as a finite family of admissible curves ( * ? ? ? * corollary 2.9 ) . we turn now to the question of compactness for the phases . since the system @xmath86 has fixed cardinality , it is enough to study the behavior of a single couple . let @xmath257 be a sequence of admissible surface - phase couples , where @xmath258 is generated by @xmath92 , and let @xmath20 be a curve such that @xmath259 as in proposition [ prop : compcurv ] ( ii ) , ( iv ) . first of all , since @xmath93 takes values in @xmath99 , we can find a subsequence ( not relabeled ) and a function @xmath260 such that @xmath261 since convergence implies that @xmath262 uniformly in @xmath263)$ ] , for every compact @xmath264 there exists @xmath207 such that @xmath265 for all @xmath266 more precisely , since by proposition [ prop : compcurv]-(iv ) the set @xmath267 is finite , for every @xmath268 there exists a compact set @xmath269 and a positive number @xmath270 such that @xmath271\backslash k)\leq \frac{\delta}2\qquad \text{and}\qquad { \gamma}_1^n(t)\geq { \varepsilon}\quad \forall\,t\in k.\ ] ] owing to lemma [ lemma : fundest ] @xmath272 recalling we obtain @xmath273 let @xmath274 be an open set such that @xmath275 , we have that @xmath276 and @xmath277 by classical compactness and lower semicontinuity for @xmath103 functions ( see theorems [ th : thlsc ] and [ th : thcpt ] ) , there exists a subsequence @xmath278 and a function @xmath279 such that @xmath280 by , we also have that @xmath281 a.e . in @xmath136 and thus @xmath3 is the strong limit for the whole sequence @xmath93 . by and , we found that for every @xmath282 there exists an open set @xmath283 and a function @xmath284 such that @xmath285 \backslash u } |{\varphi}^n-{\varphi}|\ , dt \right)\\ & \leq \lim_{n \to \infty}\left(\int_u |{\varphi}^n-\bar { \varphi}|\ , dt + 2\delta \right)\\ & \leq 2\delta.\end{aligned}\ ] ] since @xmath286 is arbitrary , we obtain strong convergence in @xmath287 . recalling that strong convergence , up to subsequences , implies convergence almost everywhere , we conclude that convergence is improved to @xmath288 or , actually , noting that @xmath289 , to @xmath290 for all @xmath291 note that @xmath292-convergence for the phases , combined with convergence for the curves , implies that the phase area constraint passes to the limit : if @xmath293 , then @xmath294 and since @xmath295 for @xmath296 , @xmath297 [ prop : comp ] let the area , @xmath0-phase area and volume constraints @xmath220 be given . let @xmath298 be a sequence of systems of admissible surfaces - phases such that @xmath299 and @xmath300 for some constant @xmath228 . then there exists a subsequence @xmath237 and an admissible system @xmath301 such that for all @xmath242 @xmath244 @xmath302 and , up to a permutation of the indices , @xmath303 and @xmath304 we first address the lower semicontinuity of the term @xmath41 . we need to prove that if the couple @xmath306 converges to @xmath307 as in , then @xmath308 this result would be straightforward , _ if _ we had @xmath309 in @xmath310 . since @xmath93 is only @xmath311 , we need one remark : [ lemma : hlsc ] let @xmath312 be open and bounded . let @xmath313 be nonnegative , and @xmath314 be such that @xmath315 if there exists @xmath202 such that @xmath316 then @xmath317 is a finite radon measure on @xmath318 and @xmath319 let @xmath320 , with @xmath321 , @xmath322 since @xmath323 , there exists @xmath324 such that @xmath325 by , for all @xmath207 there exists @xmath326 such that @xmath327 for @xmath328 . we have @xmath329 since @xmath330 was arbitrary , we obtain @xmath331 let now @xmath332 be a sequence such that @xmath333 if @xmath334 , @xmath335 . since @xmath336 , there exists a continuous and monotone increasing function @xmath337 such that @xmath338 and @xmath339 i.e. , @xmath340 uniformly in @xmath341 . since the mapping @xmath342 is continuous with respect to uniform convergence , we can extend @xmath317 to a finite radon measure on @xmath318 and conclude that regarding the remaining part of the functional , define @xmath344 we need to show that if the couple @xmath306 converges to @xmath307 as in , then @xmath345 define the radon measures @xmath346}\ , , & & \mu_{\gamma}:=2\pi { \gamma}_1\ , |\dot { \gamma}|\ , { \mathcal l}^1\llcorner_{[0,1]}\,,\\ & \lambda^n:= \pi { \kappa_h}({\varphi}^n ) { \gamma}_1^n|\dot { \gamma}^n|{\mathcal l}^1 \llcorner_{[0,1]}\ , , & & \psi^n:= 2\pi{\kappa_g}({\varphi}^n ) { \gamma}_1^n|\dot { \gamma}^n|{\mathcal l}^1 \llcorner_{[0,1]}\,.\end{aligned}\ ] ] by and by the linearity of @xmath347 , @xmath69 , @xmath348 strongly in @xmath349 for every @xmath350 , while by , @xmath351),\ ] ] so , in particular , @xmath352}$ ] and @xmath353}$ ] as measures . recalling also , it is straightforward to check that weak ( resp . strong ) convergence in @xmath354 is equivalent to weak ( resp . strong ) convergence in @xmath355 , or in @xmath356 , in the sense of definition [ def : weakags ] . by convergence and ( * ? ? ? * lemma 3.4 ) , @xmath357 by theorem [ th : ags ] and lemma [ lemma : moser ] we conclude that @xmath358 this proves , which together with concludes the proof of proposition [ prop : lsc ] . let the total area , @xmath0-phase area and volume constraints @xmath220 be given , such that the isoperimetric inequality and the bound on the phase are satisfied . let the parameters @xmath60 satisfy and @xmath79 , for @xmath62 . let the set @xmath82 and the functional @xmath88 be given as in the statement of theorem [ th : main ] . let @xmath359 be a sequence of admissible couples of surfaces and phases such that @xmath360 since @xmath86 satisfies @xmath229 , for a suitable @xmath228 , by lemma [ lemma : fundest ] there is a constant @xmath202 such that @xmath361 we can therefore apply proposition [ prop : comp ] , and find admissible couples @xmath362 and a subsequence ( not relabeled ) of constant cardinality @xmath363 , such that for @xmath364 , as @xmath365 @xmath366 in the sense of convergence , , while @xmath367 shrink to points , thus not contributing to the total area of the system . the system @xmath87 of surfaces - phases couples generated by @xmath368 satisfies the total area , phase area and enclosed volume constraints . by the lower semicontinuity proposition [ prop : lsc ] @xmath369 so that , by , @xmath370 . the proof of theorem [ th : main ] is thus complete .

we consider a canham - helfrich - type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area . the problem models the shape of multiphase biomembranes . it consists of minimizing the sum of the canham - helfrich energy , in which the bending rigidities and spontaneous curvatures are now phase - dependent , and a line tension penalization for the phase interfaces . by restricting attention to axisymmetric surfaces and phase distributions , we extend our previous results for a single phase @xcite and prove existence of a global minimizer . * keywords : * helfrich functional , biomembranes , global minimizers , axisymmetric surfaces , multicomponent vesicle . * ams subject classification * : 49q10 , 49j45 ( 58e99 , 53c80 , 92c10 ) .

the semi - direct product of the group of diffeomorphisms and the abelian group of smooth functions on @xmath0 may be regarded as a _ generalized symmetry group _ on @xmath0 . from the representation theory of this group , h .- doebner and g.a . goldin derived a family of nonlinear schrdinger equations on @xmath0 @xcite , which have been called the doebner goldin(dg)equation @xcite ( for recent progress in the study of these equations see the contributions in @xcite . ) : @xmath1\psi + \hbar d'\sum_{j=1}^5 c_j r_j[\psi]\psi , \label{gdge}\ ] ] where @xmath2 $ ] , @xmath3 are real - valued functionals of the density @xmath4 and the current @xmath5 , @xmath6 : = \frac{\grad\cdot\vec{j}}{\rho } = \im\frac{\delta\psi}{\psi},\quad r_2[\psi ] : = \frac{\delta\rho}{\rho } = \frac{\delta(|\psi|^2)}{|\psi|^2 } , \quad r_3[\psi ] : = \frac{\vec{j}^2}{\rho^2 } = \left(\im\frac{\grad\psi}{\psi}\right)^2 , \\[.5ex ] \ds r_4[\psi ] : = \frac{\vec{j}\cdot\grad\rho}{\rho^2 } = \im\left(\frac{\grad\psi}{\psi}\right)^2,\quad r_5[\psi ] : = \frac{(\grad\rho)^2}{\rho^2 } = \left(\frac{\grad(|\psi|^2)}{|\psi|^2}\right)^2 . \end{array } \label{rj}\ ] ] here the real number @xmath7 ( with the physical dimension of a diffusion constant ) labels unitarily inequivalent representations of the generalized symmetry group involved in the derivation of the nonlinear equations ( [ gdge ] ) . it has been interpreted as a quantum number describing dissipative quantum systems @xcite . the real number @xmath8 ( also with the physical dimension of a diffusion constant ) describes the magnitude of the real non - linearity and the dimension - less constants @xmath9 are model parameters . for the purpose of this paper it is more convenient to use the parameterization that is obtained by rewriting equation in terms of the real functionals @xmath2 $ ] only , following the notation of @xcite : @xmath10\psi + \sum_{j=1}^5 \mu_j r_j[\psi]\psi + \mu_0 v\psi,\quad \nu_1\neq 0 . \label{nse}\ ] ] particular homogeneous equations of this type have also been considered in the context of quantum mechanics by other authors , e.g. @xcite . one of the interesting features of the family of dg equations ( [ gdge ] ) is its invariance under a certain group of transformations @xcite @xmath11 i.e. if @xmath12 is a solution of @xmath13 , then @xmath14 is a solution of @xmath15 , where the change of parameters under @xmath16 is @xmath17 \ds \mu_1^{\,\prime } = -\frac{\gamma}{\lambda}\nu_1 + \mu_1,\quad \mu_2^{\,\prime } = \frac{\gamma^2}{2\lambda}\nu_1-\gamma \nu_2 - \frac{\gamma}{2}\mu_1+\lambda \mu_2\,,\quad \mu_3^{\,\prime } = \frac{\mu_3}{\lambda},\\[.5ex ] \ds \mu_4^{\,\prime}= -\frac{\gamma}{\lambda}\mu_3 + \mu_4,\quad \mu_5^{\,\prime } = \frac{\gamma^2}{4\lambda}\mu_3 - \frac{\gamma}{2}\mu_4 + \lambda\mu_5,\quad \mu_0^{\,\prime } = \lambda \mu_0 . \end{array } \label{pt}\ ] ] thus , without loss of generality we can restrict our calculations to the particular choice of parameters ( a particular _ gauge _ , see below ) @xmath18 since the transformations leave the position probability invariant , i.e. @xmath19 , they have been called _ nonlinear gauge transformations _ @xcite . this notion is physically motivated by the fact , that in ( non - relativistic ) quantum mechanics we basically measure positions at different times . furthermore , the transformations have been used to construct a consistent notion of observables in a nonlinear quantum theory @xcite . it turned out that besides such important properties of the dg equation as homogeneity , separability , and euclidean invariance , which were input by construction , equations ( [ gdge ] ) possess a number of other attractive properties . among them one should emphasize the possibility of constructing explicit square integrable solutions , which is important for a physical interpretation . in particular , some stationary and non - stationary ( gaussian and traveling wave ) solutions have been obtained @xcite . the well - known connection between exact solutions of partial differential equations ( pdes ) and their symmetry properties @xcite as well as the necessity of classifying equations ( [ gdge ] ) in a unified way , motivated a systematic study of their lie symmetry in @xcite . as a result , one has to distinguish nine sub - families ( characterized by conditions on the parameters @xmath20 in the chosen gauge ) with different maximal lie symmetry algebras @xmath21 . the relationship between these sub - families and their symmetries is indicated in fig.1 ( using the notation of @xcite ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ fig . 1 : lie symmetries of the dg sub - families are characterized by their parameters and arrows indicate the subfamily structure . the equations dealt with in this paper are in bold frames . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ five of these symmetry algebras are finite dimensional , @xmath22 @xmath23 . among them we find the direct sum of the ( centrally extended ) schrdinger algebra and the real numbers ( due to real homogeneity of the equations ) . these equations thus fit into the classes of schrdinger invariant nonlinear evolution equations determined in @xcite . the four remaining symmetry algebras are infinite dimensional . @xmath24 and @xmath25 contain in addition to the elements of @xmath26 infinite dimensional algebras @xmath27 and @xmath28 , that depend on a pair of ( real ) solutions of a linear forward and backward heat equations and a ( complex ) solution of a linear schrdinger equation , respectively . actually these symmetries correspond to linearizations of these sub - families , the first to a pair of forward and backward heat equations , the latter to a schrdinger equation @xcite . on the contrary , the symmetry algebras @xmath29 and @xmath30 contain an infinite dimensional algebra @xmath31 that depends only on one real - valued function . as a consequence , there is no local transformation ( i.e. a transformation that does not involve integrals or derivatives of the dependent variables ) linearizing the corresponding dg equations . nevertheless , these equations as well as the one admitting the finite dimensional symmetry algebra @xmath32 are shown in the present paper to be integrable by a _ non - local _ transformation of dependent and independent variables in the case of one spatial variable ( @xmath33 ) . thus , _ all _ dg equations with exceptional symmetries ( bottom row of fig . 1 ) @xmath34 , @xmath35,@xmath36 , @xmath37 , @xmath38 are _ integrable _ , i.e.they can be reduced to an equation which is either linear or integrable by quadratures . the principal object of study in the present paper are dg equation in 1 + 1 dimensions with parameters @xmath39 i.e. the following coupled two - dimensional pdes : @xmath40 i\psi_t&=&\left\{(\mu_1-i ) \im\ , { \psi_{xx}\over\psi } + 2\left(\im { \psi_x\over\psi}\right)^2 - \mu_1\im\left({\psi_x\over\psi}\right)^2 + \mu_0v(x)\right\}\psi.\label{e2}\end{aligned}\ ] ] one of these dg - equations is contained in the so - called ehrenfest sub - family @xcite fulfilling the second ehrenfest relation : equation with @xmath41 , i.e. the dg equation with maximal lie symmetry @xmath32 . this equation is furthermore the only schrdinger and therefore galilei invariant equation among and . nevertheless , in the free case ( @xmath42 ) all of these dg - equations admit traveling ( solitary ) wave solutions with arbitrary shape @xcite . these solutions are rediscovered as a particular case of the general solutions in this paper . using a polar decomposition @xmath43 we rewrite the above equations in the following way : @xmath44 \ds s_t + \mu_3 s_x^2 & = & -\mu_0 v , \end{array}\right.\\[1ex ] \label{f2 } & f_2:&\left\ { \begin{array}{rrcl } \ds r_t + s_{xx } + 2 r_x s_x & = & 0 , \\[.5ex ] \ds s_t + \mu_1 s_{xx } + 2 s_x^2 & = & -\mu_0 v. \end{array}\right.\end{aligned}\ ] ] the paper is organized as follows . in the section [ 2:sec ] we integrate the free equations , i.e. equations ( [ f1 ] ) , ( [ f2 ] ) with a vanishing potential ( @xmath42 ) . in order to integrate @xmath45 we have to distinguish between the cases @xmath46 and @xmath47 , the latter corresponding to the subfamily with the larger lie symmetry algebra @xmath48 . section [ 3:sec ] contains some remarks on the integration of the equations with potential and two particular examples where the integration is carried out . the methods of integration of @xmath45 and @xmath49 in sections [ 2:sec ] and [ 3:sec ] yield their general solutions containing implicitly determined function . consequently , these solutions are , generally speaking , implicit . therefore , in section [ 4:sec ] we give some explicit solutions for the free equation as well as for linear and quadratic potentials by specifying one of the arbitrary functions of the general solutions obtained in the preceding sections . putting in ( [ f1 ] ) , ( [ f2 ] ) @xmath50 we obtain the following pdes : @xmath51 \ds s_t + \mu_3 s_x^2 & = & 0 , \end{array}\right.\\[1ex ] \label{1.2 } & \widetilde f_2:&\left\ { \begin{array}{rrcl } \ds r_t + s_{xx } + 2 r_x s_x & = & 0 , \\[.5ex ] \ds s_t + \mu_1 s_{xx } + 2 s_x^2 & = & 0 . \end{array}\right.\end{aligned}\ ] ] henceforth we suppose that in ( [ 1.2 ] ) @xmath52 , since otherwise system ( [ 1.2 ] ) is a particular case of ( [ 1.1 ] ) with @xmath53 . first , we turn to the integration of the system of nonlinear pdes ( [ 1.1 ] ) . as this system admits only a finite - dimensional lie symmetry group , there is no local transformation which linearizes it . so the only possibility to transform the system in question into an integrable form is to utilize a non - local transformation of dependent and independent variables . the choice of a desired change of variables is implied by the form of the second equation of the system ( [ 1.1 ] ) ; it is nothing but the one - dimensional hamilton jacobi equation , which is known to be linearizable by the following contact transformation ( called in the literature the euler - ampre transformation ) : @xmath55 let us recall that a transformation @xmath56 , @xmath57 where @xmath58 , is called _ contact _ , if it preserves the first - order tangency condition @xmath59 the above condition ensures that the functions @xmath60 determined by the last two formulae from ( [ 1.4 ] ) are really derivatives of a function @xmath61 determined by the third formula with respect to @xmath62 and @xmath63 determined by the first and second formula , correspondingly . it is readily seen that formulae ( [ 1.3 ] ) determine a contact transformation preserving the tangency condition . but before applying the euler - ampre transformation to the system under study we must ensure its invertibility , as we may loose some solutions otherwise . it is known that transformation ( [ 1.3 ] ) is invertible in a domain where @xmath64 . consequently , we have to consider the cases @xmath65 and @xmath66 separately . : : @xmath65 + let us apply the transformation ( [ 1.3 ] ) to the system ( [ 1.1 ] ) having prolonged it to the second derivatives @xmath67 as a result , we get @xmath68 now , the first equation becomes linear and is easily integrated to give the following expression for @xmath69 : @xmath70 inserting the result into the second equation of the system ( [ 1.6 ] ) we get a first - order linear pde with non - constant coefficients , @xmath71 when integrating the above equation we have to distinguish two sub - cases @xmath47 and @xmath72 . let us recall that the dg equation with parameters ( [ p1 ] ) under @xmath47 satisfies the ehrenfest relation and , what is more , admits an additional symmetry operator ( see fig.1 ) . sub - case 1.1 . : : @xmath47 + in this case equation ( [ 1.8 ] ) takes the form @xmath73 and its general solution reads @xmath74 to rewrite the result obtained in the initial variables @xmath75 we have to invert the transformation ( [ 1.3 ] ) . from the second and third relations it follows that @xmath76 to determine the function @xmath77 we make use of the last relation from ( [ 1.3 ] ) . substituting into it the formula ( [ 1.7 ] ) with @xmath47 yields @xmath78 hence @xmath79 the above relation determines the function @xmath80 in an implicit way . since @xmath81 , we can always solve ( [ 1.10 ] ) ( at least locally ) with respect to @xmath63 thus getting an explicit form of the function @xmath63 . + summarizing the results we conclude that the general solution of the system of pdes ( [ 1.1 ] ) with @xmath82 is of the form @xmath83 s(x , t ) & = & \ds -tz_1 ^ 2 + xz_1 - f(z_1 ) , \end{array}\right . \label{1.11}\ ] ] where @xmath84 are arbitrary functions and @xmath77 is determined by the relation ( [ 1.10 ] ) . sub - case 1.2 . : : @xmath85 + using in ( [ 1.8 ] ) the transformation @xmath86 we get @xmath87 from the theory of the first - order pdes it is well - known ( see , e.g. @xcite ) that a general solution of the above equation has the form @xmath88 , where @xmath89 is an integral of the euler - lagrange system @xmath90 + the above system is rewritten as a linear first - order ordinary differential equation for a function @xmath91 , @xmath92 the general solution of which can be represented in the form @xmath93 with an arbitrary constant @xmath94 . hence we conclude that the general solution of equation ( [ 1.8 ] ) is given by the following formula : @xmath95 where @xmath84 are arbitrary functions . + returning to the initial variables @xmath96 we obtain the general solution of dg equation ( [ 1.1 ] ) for the case @xmath97 @xmath98 s(x , t ) & = & - \mu_3 t z_1 ^ 2 + x z_1 - f(z_1 ) , \end{array}\right . \label{1.13}\ ] ] where @xmath84 are arbitrary functions and the @xmath77 is determined implicitly @xmath99 case 2 . : : @xmath66 + in this case @xmath100 with arbitrary smooth functions @xmath101 , @xmath102 . substituting this expression into the second equation from ( [ 1.1 ] ) we arrive at the relations : @xmath103 an integration of these equations gives rise to the following expressions for @xmath104 and @xmath105 : @xmath106 s(x , t ) & = & c_1x - \mu_3c_1 ^ 2 t + c_2 , \label{1.14 } \end{array}\right.\ ] ] where @xmath107 are arbitrary real constants . thus we have rediscovered the traveling ( solitary ) wave solutions with arbitrary shape @xcite as a particular case of the general solution . we have established that any smooth solution is contained ( at least locally ) in one of the classes given by equations ( [ 1.11 ] ) , ( [ 1.13 ] ) , and ( [ 1.14 ] ) . summarizing we arrive at the conclusion that the general solution of the free dg equation ( [ e1 ] ) splits into two inequivalent classes : 1 . @xmath108 @xmath109 where @xmath110 are arbitrary sufficiently smooth functions , @xmath107 are arbitrary real parameters , and @xmath77 is determined implicitly by formula ( [ 1.10 ] ) ; 2 . @xmath111 @xmath112 where @xmath110 are arbitrary sufficiently smooth functions , @xmath113 are arbitrary real parameters , and @xmath77 is determined implicitly by formula ( [ 1.14 * ] ) . although formulae ( [ 1.17 ] ) and ( [ 1.18 ] ) give the general solution of the corresponding dg equation for all @xmath72 , the case @xmath114 deserves a special consideration , as the system of pdes with @xmath114 is easily integrated without applying the contact transformation ( [ 1.3 ] ) , ( [ 1.5 ] ) . the second equation of ( [ 1.1 ] ) yields that @xmath105 does not depend on time , so @xmath115 inserting this into the first equation we get a first order pde for @xmath104 , @xmath116 the case @xmath117 leads to a traveling wave solution ( [ 1.17 ] ) with @xmath118 ; if @xmath119 , then the general solution reads @xmath120 where @xmath121 is again an arbitrary function . consequently , the general solution of the dg equation ( [ e1 ] ) with @xmath114 is either given by a traveling wave solution ( [ 1.17 ] ) with @xmath118 , or by @xmath122 let us turn to the integration of the dg equation ( [ 1.2 ] ) . first , we note that the second equation is the potential burgers equation , which is is linearized by the logarithmic substitution . furthermore , we reduce the order of spatial derivatives in the first equation by a linear transformation of the dependent variables . thus , the transformation @xmath124 reduces system ( [ 1.2 ] ) to the form @xmath125 the second equation may be taken as the integrability condition of the vector - field @xmath126 on space time , @xmath127 , so that it is the gradient of a smooth function @xmath128 , @xmath129 with this remark the first equation of the system ( [ 1.21 ] ) is rewritten to be @xmath130 and is easily integrated @xmath131 , where @xmath132 is arbitrary , sufficiently smooth function . solving ( [ 1.22 ] ) with respect to @xmath133 we get @xmath134 where @xmath94 is an arbitrary constant . returning to the initial variables @xmath135 we get the general solution of the dg equation ( [ e2 ] ) @xmath136 where @xmath137 is an arbitrary solution of the heat equation @xmath138 and @xmath132 is an arbitrary smooth function . finally , we note that the traveling wave solutions of @xcite are reobtained using the particular solution @xmath139 of the heat equation . surprisingly enough , dg equations ( [ f1 ] ) , ( [ f2 ] ) are integrated in quadratures even in the case when @xmath140 ( i.e. in the presence of a non - vanishing potential ) . unfortunately , for the family @xmath45 the corresponding formulae are implicit and cumbersome . that is why we restrict ourselves to considering in detail system ( [ f1 ] ) with an additional constraint @xmath41 ( i.e. the ehrenfest subfamily of ( [ f1 ] ) is studied ) ; this system was also considered in @xcite . choosing in ( [ f1 ] ) @xmath41 , we obtain the following system of pdes : @xmath141 to linearize this system we make use of the following trick : instead of system of pdes ( [ 1.26 ] ) one of its differential consequences is considered @xmath142 substituting in ( [ 1.27 ] ) @xmath143 we arrive at the system of first - order pdes @xmath144 thus , using the substitution ( [ 1.27a ] ) enables us to reduce the order of system of pdes under study . next , we apply to this system the hodograph transformation @xmath145 @xmath146 r_t = \frac{v_{z_0}}{u_{z_0}},\quad r_x = v_{z_1 } - \frac{u_{z_1}}{u_{z_0}}v_{z_0},\quad s_t = { 1\over u_{z_0}},\quad s_x = -\frac{u_{z_1}}{u_{z_0}}. \end{array } \label{1.29}\ ] ] this hodograph transformation is defined in an arbitrary domain where @xmath147 . so again we have to distinguish two cases , @xmath148 and @xmath149 . * case 1 . * : : @xmath147 + performing in ( [ 1.28 ] ) the change of variables ( [ 1.29 ] ) we get @xmath150 a further change of variables @xmath151 transforms the system to @xmath152 thus , combining local and non - local transformations of the dependent and independent variables we _ linearized _ and decoupled the differential consequence of ( [ 1.26 ] ) . integrating these equations yields @xmath153 \tilde v(z_0,z_1)&=&g\bigl(z_0 ^ 2 + \mu_0v(z_1)\bigr ) , \end{array}\ ] ] where @xmath84 are arbitrary functions . + returning to the variables @xmath154 we obtain the general solution of system ( [ 1.28 ] ) in an implicit form @xmath155 to rewrite these equations in the initial dependent variables @xmath135 we have to invert the transformation ( [ 1.27a ] ) . as a result , we get @xmath156 where @xmath157 is an arbitrary function . the ambiguity arising is connected to the fact that we are not solving the initial system ( [ 1.26 ] ) but its differential consequence ( [ 1.27 ] ) . the extra function @xmath158 is used to choose from the set of solutions of system ( [ 1.28 ] ) those ones which satisfy ( [ 1.26 ] ) . indeed , substituting formulae ( [ 1.33 ] ) into ( [ 1.26 ] ) and taking into account that the functions @xmath159 satisfy system of pdes ( [ 1.27 ] ) we arrive at the following ordinary differential equation for a function @xmath158 : @xmath160 so @xmath161 where @xmath94 is an arbitrary real constant . + summing up , we conclude that the general solution of the initial dg equation reads @xmath162 s(x , t)&=&\ds \int_0^x s(\tau , t)d\tau - \int_0^t s^2(0,\tau)d\tau -\mu_0v(0)t + c , \end{array } \right . \label{1.34}\ ] ] where @xmath163 is a smooth function determined implicitly by and @xmath110 are arbitrary sufficiently smooth functions . * case 2 . * : : @xmath149 + with this condition the system of pdes ( [ 1.28 ] ) is easily integrated to yield @xmath164 where @xmath165 is an arbitrary sufficiently smooth function and @xmath166 is an arbitrary real constant . + rewriting the above expressions in the initial variables @xmath135 we get @xmath167 s(x , t)&= & \ds \int_0^x \sqrt{c_1-\mu_0(\xi)}d\xi -c_1t+c_2\ , , \end{array } \right . \label{1.37}\ ] ] where @xmath168 is an arbitrary constant . thus , we have established that the general solution of dg equation ( [ e1 ] ) with @xmath41 splits into the following two classes : 1 . @xmath147 @xmath169 where @xmath121 , @xmath170 , and @xmath163 is determined implicitly by . 2 . @xmath149 @xmath171 where @xmath172 and @xmath173 . integrating system ( [ f2 ] ) with potentials is similar to integrating the free system ( @xmath174 ) in section [ 2b : sec ] . using the change of variables ( [ 1.19 ] ) for ( [ f2 ] ) we arrive at the following system of pdes for new functions @xmath137 and @xmath175 : @xmath176 now , given a potential @xmath177 and an arbitrary solution @xmath137 of the first equation of this system , one can construct a general solution @xmath175 of the second equation which leads to a general solution of the initial system : @xmath178 as mentioned before some of the solutions of dg equation obtained are _ local _ in a sense that they are not determined on the whole plane @xmath179 . but for physical applications one needs _ global _ solutions , and what is more , they should be square integrable , i.e. the integral @xmath180 is to be finite . if it is , the quantity @xmath181 is treated as a probability density of a distribution of the wave function @xmath12 in space at a given time . evidently , the traveling wave solutions of dg equation ( [ e1 ] ) given by ( [ 1.15 ] ) , ( [ 1.17 ] ) are defined on the whole plane and , consequently , are global . to ensure square integrability of these solutions one has to restrict the choice of the arbitrary function @xmath132 to square integrable ones , @xmath182 thus , the traveling wave solutions are square integrable provided @xmath132 is . solutions ( [ 1.16 ] ) , ( [ 1.18 ] ) are , generally speaking , local , since the function @xmath80 contained in these solutions is determined implicitly by formulae ( [ 1.10 ] ) and ( [ 1.14 * ] ) , correspondingly , and the existence of solution is only guaranteed locally by the implicit function theorem . in order to obtain explicit expressions for global and strictly local solutions we consider solutions of ( [ 1.16 ] ) with quadratic and cubic functions @xmath132 , respectively . for quadratic functions @xmath132 , @xmath183 the implicit equation ( resp . ) for @xmath63 can be solved globally and we get @xmath184 . thus , we arrive at the following class of explicit solutions of the dg equation ( [ e1 ] ) containing an arbitrary smooth function @xmath165 : 1 . @xmath185 2 . @xmath186 these solutions are square integrable , provided @xmath165 is , and are well defined on the whole plane @xmath179 with a possible exception of the line @xmath187 , where they converge to a @xmath188-function . in particular for @xmath189 solutions coincide with the gaussian wave solutions of @xcite . cubic functions @xmath132 , @xmath190 give rise to strictly local solutions of the dg equation . indeed , inserting them into ( [ 1.10 ] ) we obtain a quadratic equation with respect to @xmath63 @xmath191 it has real solutions in the case @xmath192 only , i.e.@xmath80 is not defined inside the parabola @xmath193 . solving ( [ 2.5 ] ) yields @xmath194 ; according to the general solutions and we have to choose the positive sign since in @xmath195 and in @xmath63 have to be positive . hence , we arrive at the following class of strictly local explicit solutions of the family @xmath45 containing an arbitrary smooth function @xmath165 : 1 . @xmath196 2 . @xmath197 the domain of definition of these solutions is the set @xmath198 . furthermore , as the function @xmath199 is not defined for @xmath200 at @xmath201 at any given time @xmath202 , @xmath203 , the solution ( [ 2.6 ] ) is not square integrable . in this context let us remark that in general solutions ( [ 1.16 ] ) are square integrable at a given time @xmath202 provided * the ( possibly infinite ) limits @xmath204 exist and * @xmath165 is square integrable on the interval @xmath205 $ ] . this statement follows from a change of the integration variable @xmath206 . before turning to dg equations with non - vanishing potentials we examine solution of the particular case @xmath114 of the family @xmath45 . if the first derivative of @xmath132 has no zeros , then the solution given by ( [ 1.18d ] ) is certainly global . again , a change of the integration variable shows that the solution is square integrable , provided that * the ( possibly infinite ) limits @xmath207 exist and * @xmath165 is square integrable on the interval @xmath208 $ ] . in case of non - vanishing potentials we concentrate on the following specific potentials : 1 . the linear potential @xmath209 2 . the harmonic oscillator potential @xmath210 3 . the anti - harmonic oscillator potential @xmath211 first we consider ehrenfest case @xmath41 , the integration of which has been studied in detail in section [ 3a : sec ] . 1 . for linear potentials the implicit equation ( [ 1.32a ] ) reads @xmath212 if we choose @xmath213 , then @xmath214 . thus , we get a class of explicit solutions from : @xmath215 these solutions are defined on the whole plane @xmath179 and square integrable , provided @xmath165 is . + another class of explicit solutions is obtained directly by means of formula ( [ 1.39 ] ) : @xmath216 where @xmath165 is an arbitrary twice continuously differentiable function , @xmath113 are arbitrary parameters . these solutions are defined on the half - plane @xmath217 . analogously we construct explicit solutions for the ( anti-)harmonic oscillator potentials . we give these without derivation . ( @xmath132 is an arbitrary sufficiently smooth function , @xmath113 are arbitrary parameters . ) . harmonic oscillator potential @xmath218 \psi(x , t)&= & \begin{array}[t]{l } \ds(c_1 ^ 2-a^2x^2)^{-\frac{1}{4}}f\left ( \sqrt{c_1 ^ 2-a^2x^2}\sin 2at - ax \cos 2at\right)\\ \ds\times\exp\left\{i\bigl(\frac{x}{2}\sqrt{c_1 ^ 2-a^2x^2 } + \frac{c_1 ^ 2}{2a } \arcsin { ax\over c_1 } -c_1 t + c_2\bigr)\right\ } ; \end{array}\label{2.16 } \end{aligned}\ ] ] 2 . anti - harmonic oscillator potential : @xmath219 \psi(x , t)&=&\begin{array}[t]{l } \ds(c_1+a^2x^2)^{-\frac{1}{4}}f\left(\left(ax + \sqrt{c_1+a^2x^2}\right ) { \rm e}^{-2at } \right ) \\ \ds \times\exp\left\{i\bigl(\frac{x}{2}\sqrt{c_1+a^2x^2 } + \frac{c_1}{2a } \ln \left|ax + \sqrt{c_1+a^2x^2}\;\right| -c_1 t + c_2\bigr)\right\}. \end{array}\label{2.18 } \end{aligned}\ ] ] as mentioned in section [ 3a : sec ] , general solutions of the family @xmath45 ( [ e1 ] ) with @xmath72 are given by cumbersome implicit formulae . but with the particular choice of the potentials above it has explicit solutions containing one arbitrary function : * @xmath114 1 . linear potential : @xmath220 2 . harmonic oscillator potential : @xmath221 3 . anti - harmonic oscillator potential : @xmath222 * @xmath223 1 . linear potential : @xmath224 2 . harmonic oscillator potential : @xmath225 3 . anti - harmonic oscillator potential : @xmath226 * @xmath227 1 . linear potential : @xmath228 2 . harmonic oscillator potential : @xmath229 3 . anti - harmonic oscillator potential : @xmath230 in all these cases @xmath231 and @xmath170 . clearly , if the function @xmath137 is a global solution of the heat equation ( [ 1.24 ] ) , then the formula ( [ 1.25 ] ) gives a global solution of dg equation ( [ e2 ] ) . and what is more , it is square integrable provided * the ( possibly infinite ) limits @xmath232 exist and * @xmath132 is square integrable on the interval @xmath208 $ ] . in order to construct explicit solutions of the family @xmath49 for linear and quadratic potentials ( [ 2.8a])([2.8c ] ) we have to solve equations . after some tedious calculations we obtain the following solutions : 1 . linear potential : @xmath233 2 . harmonic oscillator potential : @xmath234 3 . anti - harmonic oscillator potential : @xmath235 here @xmath132 is an arbitrary twice continuously differentiable function and @xmath94 is an arbitrary constant . as mentioned above lie symmetries of dg equations considered are not extensive enough to provide their linearizability by means of local transformations . pdes ( [ e1 ] ) , ( [ e2 ] ) prove to be integrable because of infinite _ non - local _ symmetries admitted . take , as an example , system ( [ 1.26 ] ) . it has been decoupled into a system of two linear first - order pdes ( [ 1.30 ] ) by means of non - local transformations of dependent and independent variables ( [ 1.27a ] ) , ( [ 1.29 ] ) , ( [ 1.29a ] ) . it is well - known ( see , e.g. @xcite ) that any linear first - order pde admits an infinite parameter lie transformation group . consequently , system ( [ 1.30 ] ) possesses an infinite local symmetry . but after being rewritten in the initial variables @xmath75 it becomes non - local and can not be found by using the infinitesimal lie algorithm . in the case involved an existence of non - local symmetry was indicated by a change of local symmetry of the dg equation when the parameters were specified to be ( [ p1 ] ) , ( [ p2 ] ) . these additional local symmetries form the top of the iceberg , the main part of which consists of non - local symmetries enabling us to integrate the corresponding dg equations . since we have the formulae for general solutions of systems of pdes ( [ e1 ] ) , ( [ e2 ] ) , it is not but natural to apply these to analyze the initial value problem for these systems , which is important for a physical interpretation of the equations . for example , using formula ( [ 1.25 ] ) it is not difficult to prove that the initial value problem @xmath236 \psi(x,0)&=&r_0(x)\exp\{is_0(x)\},\end{aligned}\ ] ] where @xmath237 are arbitrary functions such that @xmath238 , has a unique solution given by the formula ( [ 1.25 ] ) , where @xmath137 is a solution of the initial value problem for the heat equation @xmath239 and the function @xmath240 reads @xmath241 here @xmath242 is determined implicitly by the relation @xmath243 but for the dg equation ( [ e1 ] ) an analysis of the initial value problem is complicated due to the complex structure of its general solution . the method of integration of dg equations developed in the present paper for the case of one space variable can be extended to a physically more interesting case of three spatial dimensions . a principal idea of such an extension is a utilization of generalized euler - ampre transformations of the space @xmath244 suggested in @xcite . the above transformations were used to study compatibility and to construct a general solution of the four - dimensional nonlinear dalembert eikonal system . this problem is under investigation now and will be a topic of our future publications . we appreciate useful discussions with h .- d . doebner , w.i . fushchych , g.a . goldin , and w. lcke and we are grateful for their remarks and suggestions . one of us ( rz ) would like to thank the `` alexander von humboldt stiftung '' for financial support . and goldin : `` manifolds , general symmetries , quantization and nonlinear quantum mechanics '' . in _ proceedings of the first german - polish symposium on particles and fields , rydzyna castle , 1992 _ , p. 115 world scientific , singapore , 1993 . and g.a . goldin : `` group theoretical foundations of nonlinear quantum mechanics '' . in _ annales de fisica , monografias , vol . ii , proceedings of the 17th international conference on group theoretical methods in physics , salamanca , 1992 _ , p. 442445 ciemat , madrid , 1993 . , g.a . goldin , and p. nattermann : `` a family of nonlinear schrdinger equations : linearizing transformations and resulting structure '' . clausthal - preprint asi - tpa/8/94 , quant - ph/9502014 , to appear in the proceedings of the xiiith workshop on mathematical methods in physics , bialowieza , july 915 , 1994 , 1995 . : `` particular solutions of a non - linear schrdinger equation carrying particle - like singularities represent possible models of de broglie s double solution theory '' . a * 135 * , 99105 ( 1989 ) . : `` solutions of the general doebner - goldin equation via nonlinear transformations '' . in _ proceedings of the xxvi symposium on mathematical physics , toru , december 710 , 1993 _ , p. 4754 nicolas copernicus university press , toru , 1994 .

we suggest a method for integrating sub - families of a family of nonlinear schrdinger equations proposed by h .- d . doebner and g.a . goldin in the 1 + 1 dimensional case which have exceptional lie symmetries . since the method of integration involves non - local transformations of dependent and independent variables , general solutions obtained include implicitly determined functions . by properly specifying one of the arbitrary functions contained in these solutions , we obtain broad classes of explicit square integrable solutions . the physical significance and some analytical properties of the solutions obtained are briefly discussed . # 1#2 # 1 # 1([#1 ] )

since their discovery of the 21-@xmath0 line emitted by proto - planetary nebulae ( ppne ) in the iras data bank , kwok and coll . @xcite have added many observational contributions to the knowledge of ppn spectra : see the bibliography in zhang et al . we shall dwell here on this paper , which reports on their spitzer / irs study of 21 and 30 @xmath0 m by several galactic ppne . successive observational improvements have led to pinpoint the former s peak wavelength at 20.1 @xmath0 m ( hrivnak et al . @xcite ) . ppne have a relatively short lifetime between the stellar states of agb ( asymptotic giant branch ) and pn ( planetary nebula ) , and the number of available sources is correspondingly small . nevertheless , the position of the 21-@xmath0 m band is very accurately determined , as is its shape , slightly skewed , with a red wing more extended than the blue one . on the other hand , the amplitude of the 30-@xmath0 m band relative to the former is widely variable and its shape does not seem to be stable ; it extends from about 23 to about 40 @xmath0 m , and peaks between 26 and 35 @xmath0 m . the present work dwells on possible carriers of these two bands . over the years , several candidates were put forth . for the 30-@xmath0 m band , goebel @xcite initially proposed mgs , and was followed by szczerba et al . @xcite and hony et al . omont et al . @xcite concluded that this choice is disputable . papoular @xcite later suggested a strong contribution from hydroxyl groups attached to various carbonaceous structures . for the 21-@xmath0 m band , goebel @xcite , for instance , suggested solid sis@xmath1 , and solid particles of sic in various shapes and sizes were advocated by speck and hofmeister @xcite ) . posch et al . @xcite published experimental and theoretical spectra of sio@xmath1-coated sic and feo , which exhibit very convincing strong features near 20 @xmath0 m . von helden et al . @xcite advanced small clusters of titanium carbide with a special composition , ti@xmath2c@xmath3 , one of several others experimentaly studied by v. heijsnbergen , v. helden et al . each of these candidates has , of course , its pros and cons , which were discussed in the same literature ; see in particular the extensive and unbiased discussion by posch et al . however , tic attracted special attention in view of the excellent agreement , in position and width , of the published laboratory feature with the ppn feature . for the same reason , several authors looked further into this candidate . henning and mutschke @xcite showed that the feature is not present in the reflection spectrum of bulk tic , and deduced from their measurements that it could not be exhibited by small tic particles of various shapes . li @xcite ruled out the tic model based on the excessive amounts of tic required by kramers - kroenig relations between total dust mass and integrated extinction cross - section . the same conclusion was arrived at by chigai et al . zhang et al . @xcite also critically analyzed several of the published proposals . thiourea was already proposed as a carrier of the 21-@xmath0 m feature by sourisseau et al . @xcite . it is a common molecule on earth and in the industry , and was described in detail by stewart @xcite , kutzelnigg and meck @xcite , lin - vien @xcite and alia et al . @xcite , and in ullman s encyclopedia of industrial chemistry @xcite among others ( see further relevant bibliography in sourisseau et al . its chemical formula is sc(nh@xmath4 . as an aside , it differs from urea only by the presence of sulphur , s , instead of oxygen , o. its characteristic subgroup , cs , was observed along several sightlines in the galaxy ( e.g. turner @xcite ) . also note that the cosmic abundance of sulphur is slightly higher than that of silicon , which is known to associate readily with carbon , both being chemically active . sulphur associated with silicon was also detected in the form of sis molecules ( e.g. turner @xcite ) . according to stewart @xcite , thiourea exhibits an absorption band which is reproduced in fig.1 ; it peaks at 20.4 @xmath0 m , but is distinctly wider than the ppne feature , and much wider than molecular features , for that matter . the thiourea molecule , being made of 8 atoms , has 18 vibrational modes ; theoretical analysis indicates that only 3 of them are notably ir - active between 10 and 40 @xmath0 m , and the strongest falls indeed at 486 @xmath5 ( 20.6 @xmath0 m ) . but this alone can not be responsible for the band width in fig.1 . obviously , this is mainly due to its preparation , as solid state micrograins , embedded in a nujol mull ( which is expected to add a small , uniform absorbance ) . it is well known that the spectrum of small grains is different from that of the same material in bulk , and that particle size and shape have an enormous influence on the position , shape and width of spectral lines ( see smith @xcite ) . a popular example , of course , is sic , which has a very discrete mid - ir spectrum with a single lorentz oscillator at @xmath6=795 @xmath5 . this single oscillator is responsible for a great variety of measured spectra , depending on particle specifics ( see bohren and huffman @xcite , chap . therefore , if thiourea is to be considered , it could not be in the solid state . now , the single , unbroadened line of the gaseous molecule near 20 @xmath0 m can not mimic the 21-@xmath0 m band , whose width is 2.4 @xmath0 m . however , broadening of the line may also occur due to the staightforward formation of h - bonds and complexes with impurity ions ( coordination ) or to the presence of thiourea derivatives ( see stewart @xcite , masunov and dannenberg @xcite bryantsev and hay @xcite , brennan @xcite ) . these are of the form sc(nr1r2)(nr3r4 ) , where the r s represent radicals attached to the scn@xmath1 root . the present work , precisely , explores the possibility of taking advantage of this propensity of thiourea in order to control the position and shape of its 21-@xmath0 m feature so as to better fit the observations . indeed , attachment to a chemically different structure slightly shifts the thiourea lines and may also enhance the ir activity of some lines nearby . combining the spectra of many associations of this type will hopefully fill the window occupied by the astronomical band . in chosing the structures to be associated with the thiourea group , consideration of cosmic abundances shows that simple hydrocarbons may be considered as useful radicals for our purpose . the astronomical 21-@xmath0 m feature extends redward to merge with another prominent band peaking between 25 and 30 @xmath0 m , also known as the 30-@xmath0 m band . following the same line of thought , it is found that this ubiquitous band can be modelled by the combined spectra of a large number of aliphatic chains , made of ch@xmath1 groups , oxygen bridges and oh groups . the experimental literature on thiourea and its derivatives in the gas phase is scant , despite a recent surge of interest spured by possible chemical applications ( see lesarri @xcite ) . it is therefore necessary to resort to theory and computational chemistry ( see alia et al.@xcite , masunov et al . @xcite , bryantsev et al . @xcite , etc . ) . fortunately , the latter has progressed considerably in the last decade , especially due to the greatly increased speed of computers . the procedure followed in the present work is the same as that which was described in detail in papoular ( @xcite ) , except that the adopted chemical software was updated to version 7 of hyperchem ( hypercube , inc . ) . this software delivers , for each mode , the ir intensity and graphic illustration of the movement of each atom in the structure , together with the frequency of vibration . this is of great help in selecting chemical elements and molecular structures of interest , and later estimating their relative abundances in the model . the semi - empirical , pm3/rhf computation method was preferred for the aliphatic structures because it was specifically optimized for hydrocarbon structures and gives sufficiently accurate ir freqencies ( better than about 5 @xmath7 ) within reasonable computation times . for the thiourea family , the semi - empirical am1 method was chosen for its more accurate treatment of n and s atoms . sections 2 and 3 respectively deal with the 2 generic classes of molecules defined above . in each case , several examples of structures are illustrated , together with the corresponding ir spectra , and , whenever possible , the type of molecular vibration is described . in section 4 , we synthesize all the above line spectra and exhibit typical emission spectra , differing in the temperatures and abundances of the emitters , to be compared with typical observations . the required relative abundances of the various atomic species are indicated in sec . the theoretical fundamental vibrational spectrum of thiourea is obtained by first optimizing the structure of the molecule in its neutral ground state , then performing a normal mode analysis . this delivers the vibrational frequencies and geometries in the limit of weak excitation or low temperatures . the optimized thiourea molecule ( delivered by our computations ) is drawn in fig.2 , in its most stable state , where the molecule is shown to be ( only ) nearly plane . according to lesarri @xcite , the experimental values for the cs bond length , the cn bond length and the scn angle are , respectively , 1.645 , 1.368 , and 123 @xmath8 , all for the ground state . the corresponding values obtained in the present work are 1.63 , 1.38 and 120.5 respectively . the vibrational mode of interest here is illustrated by dashed arrows in fig . the line frequency computed by the software is 492 @xmath5 ; the corresponding value obtained by stewart @xcite and yamagushi et al . @xcite are 486 and 498 , respectively . appendix a shows the extent of aggreement over the ir spectrum , between stewart s , yamagushi s and the present results . considering thiourea as a functional group , it is reasonable to conjecture that its characteristic lines should survive attachment to different chemical structures , designated by r s in its general formula , sc(nr1r2)(nr3r4 ) . such associations are expected to slightly shift the thiourea lines to and fro , which could help filling the spectral window covered by the observed 21 @xmath0 m band . in space , of course , the most likely candidates for r s are hydrocarbons . as examples of possible associates to thiourea , fig . 3 collects mainly associations with compact aromatic clusters , while fig . 5 assembles cases of linear clusters . in each derivative , the thiourea root , with both n s or only one n left , is attached to a more or less complicated hydrocarbon structure . the spectral lines of the two groups of structures are represented in fig . 4 and 6 , respectively , by vertical lines of different colors , and lengths proportional to their computed integrated band intensities , @xmath9 , in units of km.mol@xmath10 , given by @xmath11 where @xmath12 is the corresponding absorbance in @xmath5 , @xmath13 is the band width in @xmath14 , and @xmath15 is the molecular density in mol.l@xmath10 of the sample used for measuring @xmath12 . graphic analysis of the modes shows that the thiourea family of fig . 4 contributes lines essentially from 18 to 23 @xmath0 m . by usual standards of ir activity , the first group is particularly strong . the lines close to 20 @xmath0 m are due to symmetric out - of - plane ( oop ) excursions of the upper nh s ( fig . 1 ) and little oop excursions of s. the lines beyond are contributed by molecules carrying 3 nh bonds in asymmetric oop excursions of the nh s and ip or oop excursions of s. the spectra of thiourea appended to linear aromatics are represented in fig . 6 in different colours . as compared to those of fig . 4 , they are roughly half as strong , but they extend more uniformly from 20 to 22.5 @xmath0 m , which will help building the extended red wing characteristic of the astronomical 21-@xmath0 m band , as will be seen shortly . note that the lines of the same , but bare , aromatics , flock around 22 @xmath0 m . they are due to orderly bulk oop vibrations , akin to string vibrations and their frequency depends on the relative phases of the carbon atoms excursions . as a rule vibrations of the straight leg of the structure contribute to the spectrum near 22 @xmath0 m , while the other leg contributes near 26 @xmath0 m . the association with the thiourea chemical group activates and shifts some of the thiourea lines . hence the increased line crowding near 20 and 22 @xmath0 m . apart from helping shift the 20-@xmath0 m line , the main contributions of the attached hydrocarbon sub - structures are in the ranges of the near- and mid - ir uibs ( unidentified ir bands ) observed in the sky , to which , of course , the thiourea functional group also contributes ( essentially with nh stretching and ip ( in - plane ) and oop motions ) . even when concatenated , the spectra of fig . 4 and 6 are obviously still too sparse to make a continuous band . the structures illustrated in fig . 3 and 5 are only intended as examples of generic molecules from which to generate other carriers of the 21-@xmath0 m line . this may be done by varying the number of attached phenyl rings , or by changing the position of the thiourea group around the periphery . leaving this for further work , we are content here with a smoothing operation consisting of fast fourier transform smoothing over 30 points ( this procedure is more effective than the usual adjacent averaging device ) . the central peak is at 20.7 @xmath0 m , to be compared with 20.4 @xmath0 m , for the experimental absorbance in fig . the much broader width of the latter may be explained along the lines suggested in the introduction . the points at half - height in fig . 7 are at 19.6 and 21.9 @xmath0 m , to be compared with 19.4 and 21.8 , on average , as determined by hrivnak et al . @xcite for the 21-@xmath0 m line . however , the peak wavelength is shifted redward from 20.1 by 0.6 @xmath0 m , closer to 21 @xmath0 m . this discrepancy remains to be resolved . the 30-@xmath0 m feature roughly extends from 23 to 40 @xmath0 m . while the thiourea family of molecules makes some contribution to this range , this is only a minor one . it is therefore necessary to look for another plausible carrier . it was shown in papoular @xcite that the oh radical ( hydroxyl ) considerably increases the ir activity of the structures to which it is attached , especially in the range 30 to 40 @xmath0 m . this , again , is because of the strong electric charge of the oxygen atom . the oop , wagging motion of the oh bond , which has a strong ir intensity , is the main contributor to this range . chain geometry is especially favourable to oh attachment . note that , in the kerogen model of uib carriers , aliphatic chains are expected to link other , compact structures , and contribute essentially to the 8.6-@xmath0 m band ( papoular @xcite ) . the spectral effect of oh attachment is best illustrated in the case of linear aliphatic chains , essentially made of ch@xmath1 groups , connected by single cc bonds . these have intrinsically only a very weak ir activity . however , when oh groups are attached , as in fig . 8 and 9 , strong lines arise , mainly in the range 20 to 40 @xmath0 m , as shown in fig 10 . apart from the lines of interest between 22 and 27 @xmath0 m , the ch stretching , ip and oop vibrations , as might be expected , contribute several intense lines within the uibs . again , this series of structures can be continued along the same line to obtain a denser concatenated spectrum . * we are now in a position to try and simulate typical ppn spectra by combining the 2 thiourea sub - groups and the aliphatic group of structures defined above . in particular , we wish to obtain a spectrum with nearly equal emission at 20 and 30 @xmath0 m ( e.g. iras source 23304 + 6147 ) , and a spectrum where the latter is much stronger than the former ( e.g. iras 22574 + 6609 ) . * note that zhang et al . @xcite , in fitting their latest spitzer / irs ppn spectra , also needed 3 lorentz features to account for the same fir spectral window . they also added , as a background , 2 dust continua at different temperatures . following suit , let us seek fitting combinations of molecular and background dust temperatures and intensities . the temperatures of our 3 molecular families will be assumed equal ( @xmath16 ) , but not necessarily equal to those of the dust ( @xmath17 and @xmath18 ) . consider , first , iras 22574 + 6609 . based on the line densities and intensities of our 3 families of structures ( fig . 4 , 6 and 7 ) , we shall assume that thiourea derivatives ( with compact and linear aromatics ) and the aliphatic chains ( fig . 2 , 3 , 5 , 8 and 9 ) are in the ratios 1:1:1 . thus , in fig.11 , we concatenate the spectra of fig . 4 ( red ) and 6 ( green ) together with the spectrum of fig . 10 ( blue ) . because we have not considered enough individual structures in each family , the concatenated spectrum of all the selected structures is still not nearly continuous . we therefore need some sort of interpolation and/or smoothing . accordingly , the concatenated line spectrum was fast fourier transform smoothed over 50 points , resulting in the continuous black line in fig . 11 . the 20-@xmath0 m feature peaks at 20.7@xmath0 m ; its apparent fwhm is 2.3 @xmath0 m . since the integrated band intensity , @xmath9 , is proportional to the absorbance , @xmath12 , of the material ( eq . 1 ) , its radiative emission , @xmath19 , will be proportional to the product of @xmath9 and the black body emission at the temperature , @xmath16 , of the molecules . this will be multiplied by @xmath20 , for comparison with the @xmath21 spectra of zhang et al . for the same reason , we adopt dust emissivities scaling like @xmath22 . the fitting procedure delivered the spectrum of fig . 12 ( continuous curve ) : @xmath16=150 k , @xmath17= 170 k , @xmath18=67 k ; to be compared with the spectrum of iras 22574 + 6609 ( overlaid , dots ) . no attempt was made in the model to account for the uibs , which are observed to be prominent between 11 and 13 @xmath0 m . although the thiourea family is seen to make a contribution to this range ( fig . 11 ) , its intensity and temperature are too low for it to emerge in fig . now to iras 23304 + 6147 . here , we found it necessary to reduce the abundances of the two thiourea families relative to that of the aliphatic chains , so the ratios became 0.67:0.67:1 . the second dust temperature was also changed into @xmath18=78 k , and the abundance of molecules relative to dust was increased by a factor @xmath23 . the measured and model spectra are drawn in fig . 13 , and the same comment as for fig . 12 can be made as to the intensities in the uib range . note that , in this second case , the 16-@xmath0 m band appears in both the original and the model . there is clearly a possible trade - off between temperatures and relative abundances of the 3 families of molecules , as well as the dust temperatures and abundances , so the choices made above are not unique . the resolution of this ambiguity requires a knowledge of the excitation process of the molecules , and of their interactions with local dust . an obvious defect of these model spectra is the intensity dip near 27 @xmath0 m ( fig . 11 ) , which distorts the band top and shifts the peak to @xmath24 m in fig . 12 and 13 . correction of such defects requires a more elaborate apportion of the various selected structures , or the addition of new ones . an example for the latter could be structure ( e ) in fig . 6 , where 2 oh groups happen to be nearly parallel and vibrate in phase , giving the strongest line in fig . 10 , at 27.9 @xmath0 m . * another defect is the shift of the model peaks redward from 20.1 to 20.7 @xmath0 m . this makes a relative mismatch of 3@xmath7 , well within the wavelength error margin of our chemical software . however , as noted earlier , the respective wavelength intervals occupied by the model fir bands comply with observations , and the wavelengths at half - maximum of the bare 21-@xmath0 m band ( fig . 7 ) nearly coincide with those of the normalized band determined by hrivnak et al . @xcite . rather than the wavelengths , one is therefore led to question the adopted relative structure abundances , or intensity errors ( due to intrinsic inaccuracy of the software , and larger than wavenumber errors ) which are more likely to affect the band profile and , hence , the peak position . thus , the lines of thiourea and thiourea derivative ( e ) in fig . 3 , fall at 20.3 and 20.1 @xmath0 m , respectively ; if their relative abundances or their intrinsic ir intensities were underestimated in the models , any increase in their values would shift the band peak in the right direction . it should also be worth looking into the details of extracting the 21-@xmath0 m band from the full spectra , and inquiring if these may affect the peak position . * table 1 lists the various structures with their elemental compositions for each family . * the last two lines pertain to case 1 ( fig . 12 ) and give the global composition of the model carriers . the global ratio c / h is characteristic of predominently aliphatic structures . the roles of n and s , on the one hand , and o ( mainly in oh ) on the other hand , in building the model spectra , are made clear by fig . 11 : in this model , they are responsible for the 21- and 30-@xmath0 m bands , respectively . these heavy elements are among the most abundant in the circumstellar medium ; they have been detected in a large number of molecules ( e.g. co , no , cs hncs , ocs , etc . ) . we recall that the cosmic abundances of o , n and s relative to c are of order 1 , 0.1 and 0.1 , respectively . by comparison , the small relative amounts required by our model are expected to be easily available in the ppn environment . * on the other hand , large variations of these amounts are expected according to the particular history and properties of each object and the consequent changes in circumstellar conditions and chemistry . hence the great variety of the relative intensities of the two fir bands in observed ppn spectra . changing the relative number of the structures selected in this work , or new ones for that matter ( thus adding more free parameters ) , allows one to tailor the overall spectrum at will . while sulphur may be found in a huge variety of molecules , it seems to carry the 21-@xmath0 m band only when in the form of thiourea . the latter has a quite specific geometry , and , therefore , can not be expected to occur under any circumstances . thus , turbulent mixing in the envelope is expected to enhance the population of nh@xmath25 and sulphur - bearing species ( see for instance heinzeller et al . @xcite ) . by contrast , the 30-@xmath0 m band is carried by structures which have much less specific geometries ( aliphatic chains ) and may therefore be present under less stringent conditions . hence the rarity of the 21-@xmath0 m band , and its absence in cases where the other one is observed . clearly , a simpler molecular structure or a solid carrier ( such as tic ) would be less elusive . the prerequisites for the existence of thiourea have probably more to do with atmospheric conditions and complicated chemical reaction paths rather than with gross relative abundances . this is typical food for molecular chemists . [ cols="<,<,<,<,<,<,<",options="header " , ] we have shown that the thiourea functional group , associated with various carbonaceous structures , has one or two strong emission lines in a spectral range of @xmath264 @xmath0 m , within the 21-@xmath0 m band emitted by a number of pre - planetary nebulae . the combination of nitrogen and sulphur in thiourea is the essential source of emission in this model : the band disappears if these species are replaced by carbon . these two elements are part of the ubiquitous chons family because of their high chemical activity . thiourea may therefore readily form in space , and be found as an independent molecule or as a peripheral group attached to carbonaceous structures believed to be abundant in space . in all cases , it carries a strong ir line near the molecular thiourea line , which is the strongest and thus determines the peak of the band . obviously , no single structure can exhibit the required spectrum , for each only contributes discrete lines which can not be broadened enough by usual broadening mechanisms . twelve structures have been selected here , but their list is far from being exhaustive ; they are only intended as examples of the generic thiourea class . the chemical software used here also allows to determine the types of modal vibrations which cary the lines of interest ; this helps designing new structures to fill the wide bands observed in the sky . using interpolation and smoothing between the concatenated discrete lines of the selected structures , we produced synthetic spectra which exhibit a prominent , asymmetric , feature between 18 and 25 @xmath0 m , with half - maximum points at 19.6 and 21.9 @xmath0 m , very near the observed values . however , the peak is 0.6 @xmath0 m redward of the observed average . the astronomical 21-@xmath0 m feature extends redward to merge with the other , prominent 30-@xmath0 m band . it is found that the main characters of this band can be modelled by the combined spectra of : a ) aliphatic chains , made of ch@xmath1 groups , oxygen bridges and oh groups , which provide the 30-@xmath0 m emission ; b ) small , mostly linear , aromatic structures , which contribute to raise the red wing of the 21-@xmath0 m band and fill the space between the two main features . the concatenated spectral lines of ten of these structures form a strong band between 23 and 38 @xmath0 m . the omission of oxygen in such structures all but extinguishes the 30-@xmath0 m emission . the fact that these carriers do not involve , and are likely more abundant than , thiourea derivatives ensures that the 30-@xmath0 m feature can still be present in the absence of the 21-@xmath0 m feature , as observed . combining the discrete lines of the 22 selected structures in different proportions , interpolating and smoothing , we produced 2 synthetic spectra which purport to mimic , respectively , 2 typical ppn spectra . the main defect of these model spectra is insufficient intensity near 27 @xmath0 m , resulting in a small redshift of the 30-@xmath0 m " band peak . as expected , the selected structures also contribute lines in the near- and mid- ir bands ( uibs ) , due to the various vibrations of their cc and ch bonds . to my knowledge , none of the lines displayed in the figures above has been shown to be absent in ppne . some of them could help confirming the presence of thiourea or derivatives thereof : e.g. the nh stretching lines near 3 @xmath0 m ( see also stewart @xcite ) , * although this might prove to be difficult in emission because of the low temperatures indicated by the fitting procedure . * finally , the mechanism of the commonly observed disappearance of the 21-@xmath0 m line is straightforward , if this line is due to thiourea : indeed the corresponding vibration modes mainly involve the s , c , n and h atoms in a very special conformation , and the latter are easily removed from the structure . thus , thiourea , by contrast with solid carriers or strongly bonded diatomic molecules , may be muted by radiation or h atom encounters , for instance . while the scn@xmath1 skeleton of thiourea is robustly framed into a plane , the two amine ( nh@xmath1 ) planes easily rotate about the cn axes or depart slightly from the skeletal plane . hence the existence of conformeres studied theoretically by several authors ( e.g. bryantsev and hay @xcite ) . a conformational search with the present chemical software also delivered four other conformeres : one of them is planar ( c@xmath27 symmetry ) , one is slightly skewed out of plane like an s , as in fig . 2 , one is slightly skewed in the other sense ( like a z , also of symmetry c@xmath1 ) , and one of symmetry c@xmath28 or ts1 ) , with one pair of nh s in the plane ncn and the other oop . other chemical algorithms yield similar results ( see bryantsev and hay @xcite , who provide clear representations of the geometrical differences between structures ) . the structural and energetic differences between the 4 structures are so small that they do not impede transitions from one to the other , and cause only minor changes in the vibrational modes . the corresponding frequencies for our conformeres are about 496 , 492 , 496 and 492 @xmath5 , covering the range 20.1 to 20.4 @xmath0 m . we therefore retained only one structure ( fig . 2 ) . figure 14 displays the lines calculated by stewart @xcite and yamagushi et al . @xcite , together with those obtained in this work . * it is a pleasure to acknowledge the useful suggestions of the reviewer , dr th . posch , and the help of prof . sun kwok in providing the data files for the ppne . * alia j. edwards h. and stoev m. 1999 , spectrochimica acta a 55 , 2423 bohren c. and huffman d. 1983 , absorption and scattering of light by small particles , wiley and sons , new york brennan n. 2006 , thesis , univ . pretoria , upeted.up . za / thesis . bryantsev v. and hay b. 2006 , j. phys . chem . a , 110 , 4678 chigai t. , yamamoto , kaito c. and kimura y. 2003 , apj 587 , 771 goebel j. and moseley s. 1985,apj 290 , l35 goebel j. a&a , 278 , 226 v. heinjsbergen 1999 , phys . 83 , 4986 heinzeller d. , nomura h. , walsh c. and millar t. 2011 , arxiv:1102.3972v1 henning th . and mutschke h. 2001 , spectrochimica acta a 57 , 815 hony s. , waters l. and tielens a. 2002 , a&a 390 , 533 hrivnak b. , volk k. and kwok s. 2009 , ap.j . 694 , 1147 kutzelnigg w. and mecke r. 1961 , spectrochimica acta 17 , 530 kwok s. , volk k. and hrivnak b. 1989 , apj 535 , 275 lesarri a. , mata s. , blanco s. , lopez j and alonso j. 2004 , j. chem . 120 , 6191 li a. 2003 , apj lett . lin - vien d. , colthup n. , fateley w. and grasselli j. 1991 , the handbook of infrared and raman characteristic frequencies of organic molecules , academic press , new york . masunov a. and dannenberg j. 2000 , j. phys . b 104 , 806 merck s ft - ir atlas 1988 , vch , germany omont a. , moseley s. , cox p. et al . 1995 , apj 454 , 819 papoular r. 2000 , a&a lett . 362 , l9 papoular r. 2001 , spectrochimica acta a 57 , 947 papoular r. 2010 , arxiv 1008.5136 posch th . , mutschke h. and andersen a. 2004 , apj 616 , 1167 smith b. 1999 , infrared spectral interpretation , crc press , new york . speck a. and hofmeister a. 2004 , apj 600 , 986 stewart j. 1957 , j. chem . 26 , 248 sourrisseau c. , coddens g. and papoular r. 1992 , a&a 254 , l1 szczerba r. , henning th . , volk k. , kwok s. and cox p. 1999 , a&a 99 , 345 , l39 turner b. 1987 , a&a , 183 , l23 turner b. 1987 , a&a , 183 , l15 ullman s encyclopedia of industrial chemistry 2010 , wiley - vch von helden g. , tielens a. , van heijnsbergen d. et al . 2000 , science , 288 , 313 yamagushi a. et al . 1958 , j. am . 80 , 527 zhang k. , jiang b. and li a. 2009 , mnras zhang y. , kwok s. and hrivnak b. 2010 , apj 725 , 990 after completion of this work , it was realized that the thiourea group in space could also be attached to aliphatic chains . the families of thiourea derivatives illustrated in sec . 2 were therefore complemented with several structures comprizing one thiourea group attached to one of the carbon atoms along an aliphatic chain , including the ends . again , the thiourea feature emerged in the vicinity of 20 @xmath0 m ( 19.6 to 20.4 @xmath0 m ) with considerable intensity . further , investigation of various more complex and larger structures showed that the specific thiourea feature again appeared within the right wavelength range , but with decreasing intensity as the size of the structure increased .

computational chemistry is used here to determine the vibrational line spectrum of several candidate molecules . it is shown that the thiourea functional group , associated with various carbonaceous structures ( mainly compact and linear aromatic clusters ) , is able to mimic the 21-@xmath0 m band emitted by a number of proto - planetary nebulae . the combination of nitrogen and sulphur in thiourea is the essential source of emission in this model : the band disappears if these species are replaced by carbon . the astronomical 21-@xmath0 m feature extends redward to merge with another prominent band peaking between 25 and 30 @xmath0 m , also known as the 30-@xmath0 m band . it is found that the latter can be modelled by the combined spectra of aliphatic chains , made of ch@xmath1 groups , oxygen bridges and oh groups , which provide the 30-@xmath0 m emission . the absence of oxygen all but extinguishes the 30-@xmath0 m emission . the emission between the 21- and 30-@xmath0 m bands is provided mainly by thiourea attached to linear aromatic clusters . the chemical software reveals that the essential role of the heteroatoms n , s and o stems from their large electronic charge . it also allows to determine the type of atomic vibration responsible for the different lines of each structure , which helps selecting the most relevant structures . obviously , no single structure can exhibit the required spectrum , for each only contributes discrete lines which can not be broadened enough by usual mechanisms . a total of 22 structures have been selected here , but their list is far from being exhaustive ; they are only intended as examples of 3 generic classes . however the concatenation , interpolation and smoothing of the computed line spectra deliver continuous spectra of the overall emission of the selected candidates . * when background dust emission is added , model spectra are obtained , which are able to satisfactorily reproduce recent observations of proto - planetary nebulae . * the relative numbers of atomic species used in this model are typically h : c : o : n : s=53:36:8:2:1 . [ firstpage ]

the technologically important lead zirconate titanate alloy ( i.e. , pbzr@xmath3ti@xmath4o@xmath5 usually denoted as pzt ) has an interesting phase diagram @xcite . increasing the ti @xmath6 composition yields progressively the following _ ground state _ phases : an antiferroelectric orthorhombic phase for @xmath6 @xmath7 0.1 , a ferroelectric rhombohedral fe@xmath8 phase for 0.1 @xmath7 @xmath6 @xmath7 0.4 , another ferroelectric rhombohedral fe@xmath9 phase for 0.4 @xmath7 @xmath6 @xmath7 0.5 , and finally a tetragonal ferroelectric phase for @xmath6 larger than 50% . the high - temperature phase of this alloy at all compositions is the cubic perovskite structure . previous theoretical studies @xcite focused on the long - range b - site ordering effects in pzt for a ti composition equal to 0.5 , i.e. , close to the morphotropic phase boundary between rhombohedral and tetragonal ferroelectric phases . the subject of the present theoretical study is rather different : we will investigate alloying and ferroelectric effects on structural , chemical and dielectric properties in the tetragonal phase of the pzt alloy . in other words , we would like to know what are the effects of alloying and ferroelectricity on bond lengths , chemical bonding and born effective charges in pzt with high ti content . 1.48truein we focus on the ordered structure shown in fig . 1 and exhibiting a ti composition equal to 2/3 . the b - site ordering of this supercell consists of one zr plane alternating with two ti planes along the [ 001 ] direction . we perform local - density approximation ( lda ) calculations on this supercell using the vanderbilt ultrasoft - pseudopotential scheme @xcite , and including the semicore shells for _ all _ the metals considered . specifically , the pb @xmath10 , @xmath2 and @xmath11 , the zr @xmath12 , @xmath13 , @xmath14 and @xmath15 , the ti @xmath16 , @xmath17 , @xmath0 and @xmath12 , and the o @xmath18 and @xmath1 electrons are treated as valence electrons . we choose the plane - wave cutoff to be 25 ry and use the ceperley - alder exchange and correlation @xcite as parameterized by perdew and zunger @xcite . the first - principles calculations throughout this work are performed using a ( 6,6,2 ) monkhorst - pack mesh @xcite . further technical details of the procedure used in the present study can be found in ref . @xcite . in fact , we perform two different calculations corresponding to two different symmetries of the structure shown in fig . 1 : ( 1 ) using a centrosymmetric cell ( i.e. , exhibiting an inversion symmetry about the central pb atom ) ; and ( 2 ) using a ferroelectric cell ( i.e. , relaxing the inversion symmetry constraint ) . results of calculation ( 1 ) identify the alloying effects on various physical properties , while comparison of ( 1 ) with ( 2 ) allows us to isolate the ferroelectric effects on those properties . the lattice parameter , the axial ratio c / a and the atomic positions along the [ 001 ] ( compositional ) direction are optimized in calculation ( 1 ) by minimizing the total energy and the hellmann - feynman forces , the latter being converged to within 0.02 ev / . the ferroelectric experimental ground state of pb(zr@xmath19ti@xmath20)o@xmath5 has the tetragonal p4 mm point group and does not present any evidence of ( long - range ) b - site ordering . the material is thus composed of a succession of equivalent planes of composition zr@xmath19ti@xmath20 stacked along the tetragonal direction . to mimic this situation , we have chosen the ferroelectric direction in the supercell of fig . 1 to lie along the [ 100 ] direction , rather than along the compositionally - modulated [ 001 ] direction . in our ferroelectric supercell , two axial ratios thus exist . these are the `` ferroelectric - related '' @xmath21 and the `` ordered - related '' @xmath22 , where @xmath23 , @xmath24 , and @xmath25 are the lengths of the supercell lattice vectors along the [ 100 ] , [ 010 ] , and [ 001 ] directions , respectively . we are thus dealing with a p2 mm orthorhombic ferroelectric supercell rather than with a p4 mm tetragonal ferroelectric supercell . however , in order to be as close as possible to the experimental situation , we will keep the @xmath22 ratio as equal to the ideal value of 3 . in this case , we shall refer to our ferroelectric cell as `` quasi - tetragonal '' along [ 100 ] ( i.e. , tetragonal as regards the axial ratios , although true tetragonal symmetry is broken down to orthorhombic by the b - plane ordering in the [ 001 ] direction ) . the lattice parameter @xmath24 , the axial ratio @xmath21 , and the atomic positions along the [ 100 ] and [ 001 ] directions are then optimized in calculation ( 2 ) by minimizing the total energy and the hellmann - feynman forces ( again to within a tolerance of 0.02 ev / for the forces ) . the determination of the electronic ground state in calculations ( 1 ) and ( 2 ) is used to investigate the ferroelectric effects on the bond length distribution and on the chemical bonding in pzt with high ti content . the effective charges in each case ( i.e. , in both non - centrosymmetric and centrosymmetric cells ) will then be calculated from the polarization differences between the ground state and slightly distorted structures , following the procedure introduced in ref . @xcite and intensively used in ref . optimizing each degree of freedom in the centrosymmetric supercell leads to the lattice vectors @xmath26 $ ] , @xmath27 $ ] , and @xmath28 $ ] , where @xmath29=7.498 a.u . is the lattice parameter . the renormalized @xmath22 ratio defined as the actual ratio ( i.e. , 2.99 ) divided by the ideal one ( i.e. , 3.00 ) is equal to 0.997 and is thus very close to unity . for this reason , our centrosymmetric supercell can be referred as `` quasi - cubic '' , which is consistent with the fact that the experimental paraelectric phase of pzt is cubic . lccccccc & & displacements & + atoms & @xmath6 ( a.u . ) & @xmath30 ( a.u . ) & @xmath31 ( a.u . ) & @xmath32 ( a.u . ) & @xmath33 & @xmath34 + pb1 & 3.749 & 3.749 & -7.194 & 0.279 & 3.90 & 4.04 + pb2 & 3.749 & 3.749 & 0.000 & 0.000 & 3.88 & 3.53 + pb3 & 3.749 & 3.749 & 7.194 & -0.279 & 3.90 & 4.04 + ti1 & 0.000 & 0.000 & -3.643 & 0.093 & 6.77 & 6.65 + ti2 & 0.000 & 0.000 & 3.643 & -0.093 & 6.77 & 6.65 + zr1 & 0.000 & 0.000 & 11.210 & 0.000 & 6.33 & 6.69 + o1 & 0.000 & 0.000 & -7.222 & 0.251 & -2.58 & -5.39 + o2 & 3.749 & 0.000 & -3.638 & 0.099 & -5.58 & -2.34 + o3 & 0.000 & 3.749 & -3.638 & 0.099 & -2.72 & -2.34 + o4 & 0.000 & 0.000 & 0.000 & 0.000 & -2.53 & -5.57 + o5 & 3.749 & 0.000 & 3.638 & -0.099 & -5.58 & -2.34 + o6 & 0.000 & 3.749 & 3.638 & -0.099 & -2.72 & -2.34 + o7 & 0.000 & 0.000 & 7.222 & -0.251 & -2.58 & -5.39 + o8 & 3.749 & 0.000 & 11.210 & 0.000 & -5.17 & -2.94 + o9 & 0.000 & 3.749 & 11.210 & 0.000 & -2.33 & -2.94 + the relaxed atomic positions and the effective charges in this non - polar structure are given in table i. it can be seen from table i that ( i ) the pb and o atoms lying between the zr and ti planes ( i.e. , pb1 , o1 , pb3 , and o7 ) move significantly towards the ti planes ; and ( ii ) the ti and o atoms belonging to the ti planes ( i.e. , ti1 , o2 , o3 , ti2 , o5 , and o6 ) move very slightly towards the central mirror ( pbo ) plane . these atomic motions lead to shortened ti o bonds and lengthened zr o bonds . for example , the ti1o1 bond length shrinks to 1.89 , while zr1o7 enlarges to 2.11 , to be compared with the unrelaxed b o bond length of 1.98 in the ideal structure . as a matter of fact , the appearance of several different bond lengths associated with the mixed sublattice seems to be a general feature of alloying , and has also been observed and predicted in zinc - blende , wurtzite and rocksalt alloys @xcite . alloying effects in the present ordered [ 001 ] structure also lead to a change in the lengths of the pb for example , the pb3o bonds can be decomposed into three different groups : shorter pb3o bonds ( e.g. , pb3o5 equal to 2.73 ) , roughly unrelaxed pb3o bonds ( e.g. , pb3o7 equal to 2.80 ) , and long pb3o bonds ( e.g. , pb3-o8 equal to 2.91 ) . the three groups are populated in the ratio 4:4:4 . thus , alloying has some significant effects on the b o bonds ( @xmath354.5% change in bond lengths ) , and to a smaller extent , on the pb o bonds ( @xmath352.5% change ) . lccccccccc & & & + atoms & @xmath6 & @xmath30 & @xmath31 & @xmath36 & @xmath37 & @xmath32 & @xmath33 & @xmath34 + pb1 & 3.547 & 3.731 & -7.250 & -0.334 & 0.000 & 0.211 & 3.17 & 3.92 + pb2 & 3.551 & 3.731 & 0.000 & -0.331 & 0.000 & 0.000 & 3.37 & 3.56 + pb3 & 3.547 & 3.731 & 7.250 & -0.334 & 0.000 & -0.211 & 3.17 & 3.92 + ti1 & 0.015 & 0.000 & -3.630 & 0.015 & 0.000 & 0.101 & 5.38 & 5.81 + ti2 & 0.019 & 0.000 & 3.630 & 0.015 & 0.000 & -0.101 & 5.38 & 5.81 + zr1 & 0.118 & 0.000 & 11.192 & 0.118 & 0.000 & 0.000 & 6.06 & 6.06 + o1 & 0.628 & 0.000 & -7.223 & 0.628 & 0.000 & 0.238 & -2.15 & -4.80 + o2 & 4.432 & 0.000 & -3.637 & 0.551 & 0.000 & 0.094 & -4.56 & -1.95 + o3 & 0.605 & 3.731 & -3.634 & 0.605 & 0.000 & 0.097 & -2.16 & -2.59 + o4 & 0.656 & 0.000 & 0.000 & 0.656 & 0.000 & 0.000 & -2.10 & -4.92 + o5 & 4.432 & 0.000 & 3.637 & 0.551 & 0.000 & -0.094 & -4.56 & -1.95 + o6 & 0.605 & 3.731 & 3.634 & 0.605 & 0.000 & 0.097 & -2.16 & -2.59 + o7 & 0.628 & 0.000 & 7.223 & 0.628 & 0.000 & -0.238 & -2.15 & -4.80 + o8 & 4.201 & 0.000 & 11.192 & 0.320 & 0.000 & 0.000 & -4.62 & -2.63 + o9 & 0.779 & 3.731 & 11.192 & 0.779 & 0.000 & 0.000 & -2.07 & -2.90 + the born effective charges for our pzt supercell are detailed in table i. they exhibit the same trends as in cubic bulk pbtio@xmath5 and pbzro@xmath5 compounds @xcite : large values of about + 4.0 for pb atoms ; large values around + 6.5 for the b atoms ; and two sets of values for the oxygen atoms , either close to -5.5 for oxygen atoms moving parallel to the b o b chain , or close to -2.5 for oxygen atoms moving perpendicular to these chains . the large values of the effective charges for b and o atoms are due to a ( weak ) hybridization between the b @xmath38 and o @xmath1 orbitals @xcite . it is interesting to note that along the [ 001 ] axis , the effective charge of ti is very similar to that of zr , while the difference between these two effective charges in the bulk parent compounds is larger than 1.0 ( i.e. , 7.06 _ vs. _ 5.85 for ti and zr respectively , according to ref . one can also point out that the effective charge along [ 001 ] for atom o7 sitting between the ti and zr atoms is -5.39 , i.e. , very close to the average value -5.32 of the corresponding oxygen effective charges in the bulk parents ( -5.83 and -4.81 for pbtio@xmath39 and pbzro@xmath39 respectively , according to ref . @xcite ) . we now turn to a consideration of the _ ferroelectric _ effects on bond - length distributions and effective charges . optimizing each degree of freedom in the non - centrosymmetric cell previously described yields the following lattice vectors in atomic units : @xmath40 $ ] , @xmath41 $ ] , and @xmath42 $ ] , where the lattice parameter is @xmath43=7.461 a.u . , i.e. , 0.5% smaller than for the centrosymmetric cell . a similar decrease of the lattice constant of around 0.7% has also been theoretically predicted when going from the paraelectric cubic phase to the tetragonal ferroelectric phase of the bulk pbtio@xmath5 compound @xcite . by looking at @xmath44 , we also notice that our calculation predicts a `` ferroelectric - related '' axial ratio @xmath21 of 1.040 . this prediction must be very close to the true value in pb(zr@xmath19ti@xmath20 ) , since recent measurements performed on pb(zr@xmath3ti@xmath4 ) films @xcite for @xmath6=0.6 found a value of 1.035 for this ratio , to be compared with values of 1.064 and 1.02 for @xmath45 and @xmath46 respectively @xcite . the optimized non - centrosymmetric cell has an energy that is 0.15 ev/5-atom - cell lower than that of the optimized centrosymmetric cell . this is consistent with the fact that the experimental ground state of pb(zr@xmath19ti@xmath20 ) is ferroelectric and tetragonal rather than paraelectric and cubic . the relaxed atomic positions and the effective charges in the non - centrosymmetric structure are shown in table ii . the quantities @xmath36 , @xmath37 , and @xmath32 are the [ 100 ] , [ 010 ] and [ 001 ] atomic displacements of the ferroelectric structure with respect to the ideal ordered structure associated with @xmath44 , @xmath47 and @xmath48 . @xmath33 and @xmath34 are the effective charges along the [ 100 ] and [ 001 ] directions , respectively . a comparison of tables i and ii leads to the following observations . ( i ) the atomic displacements along the [ 001 ] direction are quite comparable between the centrosymmetric and the non - centrosymmetric phases . ( ii ) the ferroelectricity in tetragonal pzt is mainly characterized by the very large displacement of oxygen atoms along the tetragonal [ 100 ] direction ( as in tetragonal pbtio@xmath5 bulk ) , the large displacement of pb atoms along the [ @xmath4900 ] direction , and by the slight displacement of zr atoms along the [ 100 ] direction . this ferroelectric atomic relaxation yields two different ti o bond lengths along the [ 100 ] direction : a very long bond of length 2.33 , which is even longer than the longest zr o bond ( 2.16 ) ; and a very short bond of length 1.55 . this very short ti o bond is much shorter than the shortest zr o bond of 1.95 , and is even much shorter than the shortest ti o bond of 1.78 occurring in tetragonal ferroelectric pbtio@xmath5 . ferroelectricity also leads to a drastic change in the pb o bonds . there are now some very short pb o bonds with an average length of 2.51 , `` normal '' pb o bonds with an average length of 2.84 , and very long pb o bonds with an average length of 3.26 . as in the centrosymmetric cell , the population ratio between these three groups is again 4:4:4 . however , the oxygens exhibiting the shortest pb o bonds now share a common ( 100 ) plane , while they share a common ( 001 ) plane in the non - polar structure . pair - distribution function analysis of recent pulsed neutron powder diffraction measurements on ferroelectric pzt alloys clearly confirms the existence of these three different groups @xcite . the experimental average value of the three different pb o bond lengths is @xmath352.5 , @xmath352.9 , and @xmath353.4 , i.e. , in excellent agreement with our predictions . comparing table i and table ii also indicates that ferroelectricity leads to a striking decrease of the born effective charges . the most spectacular decrease occurs for the atoms exhibiting a large change in their bond lengths . as a matter of fact , the effective charges along the [ 100 ] direction for atoms ti1 , pb3 and o6 all decrease by @xmath3520% with respect to the centrosymmetric case . in consequence , the effective charges of the ti atoms and pb atoms are reduced by 1.4 and 0.7 , respectively , relative to their non - polar values . previous theory has shown that a change of the effective charge by more than one unit of ` e ' is indeed not unusual in going from the cubic to tetragonal ferroelectric phase in perovskite compounds @xcite . the most extreme example appears to be for nb in knbo@xmath39 , where the effective charges change from 9.67 to 7.05 in the direction parallel to the tetragonal axis . to further understand the ferroelectric effects in pzt , fig . 2 compares the electronic charge density in the ( b , o ) planes for the [ t ! ] 3.6truein centrosymmetric and non - centrosymmetric cases . figure 3 shows a similar comparison but in the ( pb , o ) planes . figure 2 indicates that ferroelectricity in pzt leads ( i ) to a chemical breaking of some ti o bonds which generates the long ti o bonds of 2.33 , and ( ii ) to the formation of strong covalency between ti and o , which is the cause of the very short ti o bonds of 1.55 . in fact , we also found similar behavior for the electronic charge density in the ( ti , o ) plane for cubic and tetragonal lead titanate ( pt ) . thus , as in bulk pt @xcite , the formation of ferroelectricity in tetragonal pzt leads to an enhancement of hybridization between ti @xmath0 and o @xmath1 orbitals . interestingly , we do nt observe in fig . 2 any breaking of zr o bonds nor the formation of strong covalent zr o bonds . the different chemical behavior between zr and ti may perhaps be the cause of the difference in ground states exhibited by the corresponding bulk parents ( antiferroelectric and orthorhombic for pbzro@xmath5 _ vs. _ ferroelectric and tetragonal for pbtio@xmath5 ) . the striking feature of fig . 3 is the formation of covalent chains between pb and o atoms , which is the cause of the very short pb - o bonds of 2.5 . we also found similar trends in the ( pb , o ) planes of paraelectric and ferroelectric pt . thus , as in bulk pt @xcite , the hybridization between pb @xmath2 and o @xmath1 orbitals plays an important role in the ferroelectric behavior of tetragonal pzt . [ ! using 15-atom supercells and vanderbilt ultrasoft pseudopotentials within the local - density approximation , we investigated alloying and ferroelectric effects on the bond lengths , chemical bonding and effective charges in lead zirconate titanate alloys ( pzt ) . our principal findings are as follows . \(i ) the centrosymmetric pzt alloy is mainly characterized by two sets of b o bonds ( shorter ti o bonds _ vs. _ longer zr o bonds ) , while the pb o bonds differ only slightly ( by @xmath352.5% ) from the ideal structure . \(ii ) allowing ferroelectricity in pzt alloys has two striking chemical effects : enhancement of hybridization between ti @xmath0 and o @xmath1 orbitals , and hybridization between pb @xmath2 and o @xmath1 orbitals . \(iii ) these chemical and ferroelectric effects lead to the formation of very short covalent ti o bonds while breaking other ti o bonds , and give rise to the formation of covalent chains of very short pb o bonds . \(iv ) the atoms engaged in covalent bonding exhibit a striking decrease of their effective charges by @xmath3520% relative to the paraelectric phase . this work is supported by the office of naval research grant n00014 - 97 - 1 - 0048 . we thank professor t. egami for helpful discussions and for communicating his results with us .

first - principles calculations are performed to investigate alloying and ferroelectric effects in lead zirconate titanate ( pzt ) with high ti composition . we find that the main effect of alloying in the paraelectric phase of pzt is the existence of two sets of b o bonds , i.e. , shorter ti o bonds _ vs. _ longer zr o bonds . on the other hand , ferroelectricity leads to the formation of very short covalent ti o bonds and to the formation of covalent chains of pb o bonds . the covalency in the ferroelectric phase is mainly induced by an enhancement of hybridization between ti @xmath0 and o @xmath1 , and between pb @xmath2 and o @xmath1 . these hybridizations induce a striking decrease of the effective charges when going from the paraelectric to the ferroelectric phase of pzt .

lccccccccccl mkn 421 & 0.030 & 129 & 68 & 1 & 3.32 & @xmath4 & 45.6 & 3.15 & @xmath5 & h & @xcite + 1es 1959 + 650 & 0.047 & 201 & 78 & 1 & 3.18 & @xmath6 & 45.9 & 2.90 & 85 & h & @xcite + 1es 2344 + 514 & 0.044 & 190 & 120 & 0.5 & 2.95 & @xmath7 & 45.0 & 2.82 & @xmath8 & h & @xcite + mkn 501 & 0.034 & 150 & 8.7 & 1 & 2.58 & 85 & 44.4 & 2.39 & @xmath9 & h & @xcite + 3c 279 & 0.536 & 2000 & 520 & 0.2 & 4.11 & 68 & 46.9 & 2.53 & 2.0 & q & @xcite + pks 2155 - 304 & 0.116 & 490 & 1.81 & 1 & 3.53 & 64 & 45.4 & 2.75 & @xmath10 & h & @xcite + pg 1553 + 113 & @xmath11 & @xmath12 & 46.8 & 0.3 & 4.46 & 41 & @xmath13 & @xmath14 & @xmath15 & h & @xcite + w comae & 0.102 & 430 & 20 & 0.4 & 3.68 & 31 & 44.9 & 3.41 & @xmath16 & i & @xcite + 3c 66a & 0.444 & 1700 & 40 & 0.3 & 4.1 & 28 & 46.3 & 2.43 & 13 & i & @xcite + 1es 1011 + 496 & 0.212 & 870 & 200 & 0.2 & 4 & 26 & 45.5 & 3.66 & @xmath17 & h & @xcite + 1es 1218 + 304 & 0.182 & 750 & 11.5 & 0.5 & 3.07 & 24 & 45.4 & 2.37 & @xmath18 & h & @xcite + mkn 180 & 0.045 & 190 & 45 & 0.3 & 3.25 & 20 & 44.0 & 3.17 & @xmath19 & h & @xcite + 1h 1426 + 428 & 0.129 & 540 & 2 & 1 & 2.6 & 20 & 45.0 & 1.71 & @xmath20 & h & @xcite + rgb j0710 + 591 & 0.125 & 520 & 1.36 & 1 & 2.69 & 15 & 44.8 & 1.83 & @xmath21 & h & @xcite + 1es 0806 + 524 & 0.138 & 580 & 6.8 & 0.4 & 3.6 & 10 & 44.7 & 3.21 & @xmath22 & h & @xcite + rgb j0152 + 017 & 0.080 & 340 & 0.57 & 1 & 2.95 & 8.5 & 44.1 & 2.45 & @xmath23 & h & @xcite + 1es 1101 - 232 & 0.186 & 770 & 0.56 & 1 & 2.94 & 8.2 & 44.9 & 1.50 & @xmath24 & h & @xcite + 1es 0347 - 121 & 0.185 & 770 & 0.45 & 1 & 3.1 & 8.2 & 44.9 & 1.67 & @xmath25 & h & @xcite + ic 310 & 0.019 & 83 & 1.1 & 1 & 2.0 & 8.1 & 42.8 & 1.90 & @xmath26 & h & @xcite + pks 2005 - 489 & 0.071 & 300 & 0.1 & 1 & 4.0 & 8.0 & 44.0 & 3.56 & @xmath27 & h & @xcite + magic j0223 + 430 & & & 17.4 & 0.3 & 3.1 & 7.6 & & @xmath28 & & r & @xcite + 1es 0229 + 200 & 0.140 & 590 & 0.7 & 1 & 2.5 & 6.4 & 44.5 & 1.51 & @xmath29 & h & @xcite + pks 1424 + 240 & @xmath30 & @xmath31 & 51 & 0.2 & 3.8 & 6.3 & @xmath32 & @xmath33 & 4.0 & i & @xcite + m87 & 0.0044 & 19 & 0.74 & 1 & 2.31 & 5.9 & 41.4 & 2.29 & @xmath34 & r & @xcite + bl lacertae & 0.069 & 290 & 0.3 & 1 & 3.09 & 5.4 & 43.8 & 2.67 & @xmath35 & l & @xcite + h 2356 - 309 & 0.165 & 690 & 0.3 & 1 & 3.09 & 5.4 & 44.6 & 1.86 & @xmath36 & h & @xcite + pks 0548 - 322 & 0.069 & 290 & 0.3 & 1 & 2.86 & 4.0 & 43.7 & 2.44 & @xmath35 & h & @xcite + centaurus a & 0.0028 & 12 & 0.245 & 1 & 2.73 & 2.8 & 40.7 & 2.72 & @xmath37 & r & @xcite + imaging atmospheric cerenkov telescopes , such as h.e.s.s . , veritas , and magic , have opened the very - high energy gamma - ray ( vhegr , @xmath38 ) sky , finding a universe populated by a variety of energetic , vhegr sources . while the majority of observed vhegr sources are galactic in origin ( e.g. , supernova remnants , etc . ) , the extragalactic contribution is dominated by a subset of blazars . there are presently 46 extragalactic tev sources knownrwagner / sources/ for an up - to - date list . ] , of which 28 have well defined spectral energy distributions ( seds ) , and are collected in table [ tab : tevsources ] . of these 28 well - studied objects , 24 are blazars , implying that blazars make up an overwhelming majority of the bright vhegr sources . all of the extragalactic vhegr emitters are relatively nearby , with @xmath39 generally , and @xmath2 typical . this is a result of the large opacity of the universe to tev photons , which annihilate upon soft photons in the extragalactic background light ( ebl ) , producing pairs ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? typical mean free paths of vhegrs range from @xmath40 to @xmath41 depending upon gamma - ray energy and source redshift , and thus the absence of high - redshift vhegr sources is not unexpected . the pairs produced by vhegr annihilation are necessarily ultrarelativistic , with typical lorentz factors of @xmath42@xmath43 . the standard assumption is that these pairs lose energy almost exclusively through inverse - compton scattering the cosmic microwave background ( cmb ) and ebl . when the up - scattered gamma - ray is itself a vhegr the process repeats , creating a second generation of pairs and up - scattering additional photons . the result is an inverse - compton cascade ( icc ) depositing the energy of the original vhegr in gamma - rays with energies @xmath44 . this places the icc gamma rays in the lat bands of , and thus has played a central role in constraining the vhegr emission of high - redshift blazars . based upon observations of @xmath45 tev sources , a number of authors have now published estimated lower bounds upon the intergalactic magnetic field ( igmf ; see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? typical numbers range from @xmath46 to @xmath47 , with the latter values being of astrophysical interest in the context of the formation of galactic fields . ] . these limits on the igmf arise from the _ lack _ of the gev bump associated with the icc of the blazar tev emission , presumably due to the resulting pairs being deflected significantly under the action of the igmf itself . the wide range in the estimates upon the minimum igmf is due primarily to different assumptions about the tev blazar duty cycle . has also provided the most precise estimate of the unresolved extragalactic gamma - ray background ( egrb ) for energies between @xmath48 and @xmath49 . since iccs reprocess the vhegr emission of distant sources into this band , this has been used to constrain the evolution of the luminosity density of vhegr sources ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? generally , it has been found that these can not have exhibited the dramatic rise in numbers by @xmath50@xmath51 seen in the quasar distribution . that is , the comoving number of blazars must have remained essentially fixed , at odds with both the physical picture underlying these systems and with the observed evolution of similarly accreting systems , i.e. , quasars . both of these conclusions depend critically upon iccs dominating the evolution of the ultra - relativistic pairs . however , as we will show , the pairs constitute a cold , highly collimated plasma beam moving through a dense , stationary background , both of which are susceptible to collective plasma phenomena . such beams are notoriously unstable ; for instance , equal density beams typically lose a significant fraction of their energy after propagating distances measured in plasma skin depths of the background plasma . if the vhegr - generated pairs suffer a similar fate while propagating through the intergalactic medium ( igm ) , the cooling of the pairs would be dominated by plasma instabilities , thereby quenching the iccs . here we present a plausible alternative mechanism by which the energy in the ultra - relativistic pairs can be extracted . while a variety of potential plasma beam instabilities exist , we find that the most relevant for the vhegr - produced pair beams is the `` oblique '' instability @xcite . this is a more virulent cousin of the commonly discussed weibel and two stream instabilities . in section [ sec : pbs ] we discuss the formation and properties of the ultra - relativistic pair beam , including limits upon its temperature and density . section [ sec : pbis ] presents growth rates for a variety of plasma instabilities , including the oblique instability . particular attention is paid to if and when plasma instabilities dominate inverse compton as a means to dissipate the kinetic energy of the pairs . the resulting implications for studies of the igmf are described in section [ sec : implications igmf ] , including how such efforts might mitigate the systematic uncertainties arising from plasma cooling . the evolution of the luminosity function of vhegr - emitting blazars is discussed in section [ sec : blf ] , in which we construct a blazar luminosity function based upon that of quasars which is consistent with current tev source populations and the estimates of the egrb . finally , conclusions are contained in section [ sec : c ] . this is the first in a series of three papers that discuss the potential cosmological impact of the tev emission from blazars . here we provide a plausible mechanism for the local dissipation of the vhegr luminosity of bright gamma - ray sources . in addition to the particular consequences this has for studies of high - energy gamma - ray phenomenology , it also provides an novel heating process within the igm . * hereafter paper ii ) estimates the magnitude of the new heating term , describes the associated modifications to the thermal history of the igm , and shows how this can explain some recent observations of the forest . * hereafter paper iii ) considers the impact the new heating term has upon the structure and statistics of galaxy clusters and groups , and upon the ages and properties of dwarf galaxies throughout the universe , generally finding that blazar heating can help explain outstanding questions in both cases . an additional follow - up paper by @xcite shows that when combined with most recent estimates of the evolving photoionizing background and hydrodynamic simulations of cosmological structure formation , the heating by blazars results in excellent quantitative agreement with observations of the mean transmission , one- and two - point statistics , and line width distribution of high - redshift forest spectra . in particular , these successes depends upon the peculiar properties of the blazar heating via the dissipation of plasma instabilities . for all of the calculations presented below we assume the wmap7 cosmology with @xmath52 , @xmath53 , @xmath54 , and @xmath55 @xcite . originating at redshift @xmath56 as it propagates to the earth , @xmath57 . bottom : pair - production optical depth of a gamma ray of energy @xmath58 propagating across a hubble distance at redshift @xmath56 , @xmath59 . in both cases the thick black line shows @xmath60 of unity , blue and red lines show optically thick and thin regions of the parameter space , respectively . the sudden break in the contours at @xmath61 is due to the assumed ebl evolution ( see text ) . ] sources of vhegrs are necessarily attenuated by the production of pairs upon interacting with the ebl . namely , when the energies of the gamma ray ( @xmath58 ) and the ebl photon ( @xmath62 ) exceed the rest mass energy of the @xmath63 pair in the center of mass frame , i.e. , @xmath64 , where @xmath65 is the relative angle of propagation in the lab frame , an @xmath63 pair can be produced with lorentz factor @xmath66 @xcite . the optical depth the universe presents to high - energy gamma rays depends solely upon the energy of the gamma ray and the evolving spectrum of the ebl . detailed estimates for the ebl spectrum and its evolution have produced an estimated mean free path of tev photons of @xmath67 where the redshift evolution is due to that of the ebl alone , is dependent predominately upon the star formation history , and @xmath68 for @xmath69 and @xmath70 for @xmath71 @xcite and declines rapidly afterward , galaxies and type 1 active galactic nuclei compensate for the lost flux until @xmath61 . see , for example , figure 3 from @xcite . ] . the resulting optical depth to vhegrs emitted with energy @xmath58 from an object located at @xmath56 is @xmath72 h(z')(1+z ' ) } \propto e\,,\ ] ] ( in which @xmath73 is the hubble factor ) , is shown in the top panel of figure [ fig : taus ] . is given by @xmath74 $ ] . ] from this it is evident that above @xmath49 the universe is optically thick to sources at @xmath75 ( cf . a related , and perhaps more appropriate measure of the optical depth is that across a hubble length , @xmath76 , shown in the bottom panel of figure [ fig : taus ] , providing a sense of how opaque the universe is as a function of redshift . in all cases it is clear that at @xmath0 photons will pair - produce on the ebl for @xmath77 . in the future this may not be the case , since for sufficiently small @xmath56 , @xmath59 decreases with decreasing @xmath56 due to the dramatic decrease in the ebl photon density associated with both the hubble expansion and the slowing of star formation . the pairs produced by the tev gamma rays are ultra - relativistic , with typical lorentz factors of @xmath78 . they also necessarily constitute a cold , highly anisotropic , dilute beam propagating through the igm . this follows immediately from the intergalactic distances traversed by the gamma rays and the comparatively small ebl photon energies ( as seen in the igm frame ) . here we estimate the properties of this plasma beam . the momentum dispersion of the resulting pairs is set by the gamma - ray spectrum , geometry of the tev source , distribution of the ebl photons , and heating due to pair production . of these , only the last plays a significant role in setting the transverse momentum dispersion . , is @xmath79 . with @xmath80 , implied by the x - ray variability timescales @xcite , this gives @xmath81^\zeta \left(e/\tev\right)^2 $ ] . similarly , since the creation of pairs is dominated by ebl photons near the pair - production threshold , the typical energy of the relevant ebl photons is roughly @xmath82 ( i.e. , twice the threshold value for transverse ebl photons ) , and thus @xmath83 . ] the center - of - mass frame , i.e. , the `` beam frame '' , momentum dispersion resulting from pair production is roughly @xmath84 . this results in an igm - frame transverse momentum dispersion of @xmath85 . with @xmath86 , the temperature associated with this transverse momentum dispersion is @xmath87 since the transverse momentum dispersion of the pairs is much smaller than that associated with the bulk motion of the beam ( i.e. , since @xmath88 ) , we may safely assume that the beam is transversely kinematically cold . ) at a number of redshifts ( @xmath56 ) . in all cases @xmath89 , corresponding to a mean - density region , and the isotropic - equivalent luminosity of the source at energy @xmath58 , @xmath90 , is @xmath91 , similar to the brightest tev blazars seen from earth . finally , we list the initial pair lorentz factor , @xmath92 , and cooling lengthscale along the top and right axes , respectively . ] the density of the pair beam at a given point within the igm is set by the rate at which pairs are produced , duration of the tev emission , advection of the pairs through the igm , and the processes by which they lose their kinetic energy . that is , the evolution of the density of pairs per unit lorentz factor , @xmath93 , is governed by the boltzmann equation : @xmath94 where the left - hand side assumes all the pairs are moving away from the tev source relativistically ( @xmath95 and @xmath96 ) , and the right - hand side corresponds to pair production . generally , we may neglect advection , which alters @xmath93 over timescales of @xmath97 , much longer than any relevant timescale of interest here ( i.e. , @xmath98 may be neglected ) . furthermore , for most of the potential sources we will consider ( primarily tev blazars ) we will assume that the duration of the tev emission is sufficiently long that @xmath93 reaches a steady state ( i.e. , @xmath99 ) . in this case , we have @xmath100 . making further progress requires us to define the spectrum of the pairs , which itself depends upon the spectrum of the gamma rays and the energy dependence of the cooling processes . nevertheless , we may obtain an estimate of the beam density in the vicinity of a given lorentz factor , @xmath101 , by setting @xmath102 . the source term is given by twice ( since each gamma - ray produces two leptons ) the rate at which high - energy gamma rays with energy @xmath103 annihilate within the region of interest , i.e. , @xmath104 , where @xmath105 is the gamma - ray number flux , with units of @xmath106 . thus , upon defining a cooling rate @xmath107 , we have @xmath108 i.e. , the density of pairs of a given energy is determined by balancing cooling and pair creation . generally , @xmath109 is a function of energy and beam density , as well as external factors ( e.g. , seed photon density , igm density , etc . ) . thus this gives a non - linear algebraic equation to solve for @xmath110 , the particulars of which depend upon the various mechanisms responsible for extracting the bulk energy of the beam . in practice , given expressions for @xmath109 , associated with the processes discussed in following section , we solve equation ( [ eq : nb ] ) numerically to obtain @xmath111 . which mechanism dominates the cooling of the beam depends upon a variety of environmental factors and the properties of the pair beam itself . nevertheless , inverse - compton cooling via the cosmic microwave background ( cmb ) provides a convenient lower limit upon @xmath109 , and thus an upper limit upon @xmath110 . this is a function of @xmath56 and @xmath92 alone , given by @xmath112 where @xmath113 denotes the thompson cross section . the strong redshift dependence arises from the rapid increase in the cmb energy density with @xmath56 ( @xmath114 ) . furthermore , since it is @xmath115 , inverse - compton cooling is substantially more efficient at higher energies . the associated cooling rate is shown as a function of @xmath58 for a number of redshifts in figure [ fig : gs ] . when we set @xmath116 , we obtain the following upper limit upon the beam density : @xmath117 where we have defined @xmath118 to be the isotropic - equivalent luminosity ( per unit energy ) of a source located a distance @xmath119 from the region in question . setting @xmath90 to a typical value ( @xmath91 ) gives an idea of the typical pair - beam densities . note that despite the large blazar luminosities we consider , the associated beams are exceedingly dilute , a point that is of critical importance in the following section . since @xmath120 is independent of @xmath110 , this has no implication for inverse - compton cooling itself . plasma beams are notoriously unstable , with the instabilities driven by the anisotropy of the lepton distribution function . here we consider the implications of these instabilities upon the ultimate fate of the kinetic energy in the tev - blazar driven pair beams . for collective phenomena to be relevant , it is necessary for many pairs to be present within each wavelength of the unstably growing modes . as we shall see , the relevant scale for the beam plasma instabilities we describe below is the plasma skin depth of the igm , @xmath121 where @xmath122 is the igm plasma frequency and @xmath123 is the igm free - electron number density . generally , the growing mode must be uniform on scales considerably larger than @xmath124 , both longitudinally and transversely ( otherwise it is not well - represented as a single fourier component ) , and thus the volume in which many particles must be present is much larger than that defined by a sphere of diameter @xmath124 . nevertheless , this gives us a conservative constraint , i.e. , we require @xmath125 with equation ( [ eq : nb ] ) this gives a _ maximum _ plasma cooling rate : @xmath126 plasma processes with cooling rates that exceed this limit necessarily saturate near this cooling rate . such a super - critical process can potentially operate only until the beam density is driven below the value at which pairs can support collective phenomena , at which point they necessarily quench . however , after plasma cooling ceases , the plasma beam density rises again ( since @xmath120 is always less than @xmath127 in practice ) , and thus the super - critical plasma cooling may resume . since the efficiency of the plasma cooling decreases smoothly to zero at the critical density , this sequence stabilizes near @xmath127 . the associated excluded region is shown by the grey region in the upper - left corner of figure [ fig : gs ] . region of the igm . grey lines show the limit defining the applicability of the plasma prescription , below which less than a single beam lepton is found in a volume @xmath128 . below these collective phenomena are unimportant . black lines show the luminosity at which linear plasma cooling rate begins to dominate inverse - compton cooling . both limits are sensitive functions of the igm density , scaling as @xmath129 and @xmath130 , respectively . thus in low / high density regions these limits become moderately more / less permissive . for reference , the sources listed in table [ tab : tevsources ] are also plotted ( at @xmath131 ) , with hbl , ibl , radio galaxies , and quasars shown by the the blue triangles , green squares , red hexagons and magenta circles , respectively . filled points indicate sources that have been used to estimate the igmf ( see section [ sec : implications igmf ] ) . the only objects that fail to meet the plasma - cooling luminosity criterion are the radio galaxies m87 and cen a , both of which are detectable only due to their close proximity . ] the upper limit upon @xmath110 obtained in equation ( [ eq : nbic ] ) implies an analogous limit upon the relevant source isotropic - equivalent luminosities . again , this arises from requiring a sufficient number of beam pairs to be present to support plasma processes . since the beam density is linearly dependent upon the source luminosity , this gives a constraint upon the latter : @xmath132 effectively defining a luminosity cut - off , below which collective plasma phenomena may be ignored . this limiting luminosity shown as a function redshift by the grey lines in figure [ fig : nblimits ] for various gamma - ray energies . while the luminosity limit does depend upon @xmath56 , it is clear that all of the observed tev blazars ( listed in table [ tab : tevsources ] ) are sufficiently luminous for their resulting pair beams to support collective phenomena at the redshifts of interest for active galactic nuclei ( agns ) . the only source to fall marginally below the limiting luminosity at @xmath133 is the radio galaxy cen a ( the lower - most point in figure [ fig : nblimits ] ) , the closest and dimmest object in table [ tab : tevsources ] . within astrophysical contexts there are at least two well known beam instabilities : two - stream and weibel , the latter having been suggested as a mechanism for magnetizing strong shocks @xcite , and both implicated in the coupling at collisionless shocks @xcite . these are , however , simply different limiting examples of the same underlying filamentary instability , for which the maximum growth rate exceeds either the so - called `` oblique '' mode , which we discuss below @xcite . note that since the pair beam is neutral , it contains its own return current and thus beam instabilities that arise due to the electron return currents within the background plasma ( e.g. , the bell and buneman instabilities , see * ? ? ? * ) are not relevant . here we discuss the nature of these instabilities , and their growth rates in the cold - plasma limit ( i.e. , mono - energetic beams ) . while the low - temperature approximation is unlikely to be applicable in practice for the beams of interest here , it provides a convenient limit in which to present the relevant processes within their broader context . for a similar reason , and because we have not found analogous derivations elsewhere in the astrophysical literature , we present the ultra - relativistic pair beam instability growth rates for the two - stream and weibel instabilities in appendixes [ sec : ts ] and [ sec : w ] , only summarizing the results here . we defer a discussion of the more directly relevant warm - plasma oblique instability to the following section . generally , we find that the plasma instabilities are capable of dominating inverse compton cooling as a means to dissipate the bulk kinetic energy of the pair beams from tev blazars . the pair two - stream instability arises due to the interaction of the anisotropic electron and positron distribution functions with the comoving background electrostatic wave ( i.e. , @xmath134 , where @xmath135 is the electromagnetic mode wavevector and @xmath136 is the beam momentum ) with wavelength @xmath137 for @xmath138 , which is generally the case of interest here . the associated cooling rate in the cold - plasma limit is @xmath139 which depends only weakly upon the igm density and the pair beam density , though decreases rapidly as @xmath92 becomes large . the weibel instability is associated with coupling to a secularly growing , anharmonic transverse magnetic perturbation ( i.e. , @xmath140 ) . the most rapidly growing wavelength is again that associated with the background plasma skin depth , @xmath137 , with associated cooling rate in the cold - plasma limit @xmath141 which depends only upon the pair beam density and the pair lorentz factor . at large @xmath92 this is suppressed more weakly than the two - stream instability , though is a moderately stronger function of @xmath110 . the computation of the growth rates of the two - stream and weibel instabilities are greatly simplified by the particular geometries of the coupled electromagnetic waves and beam momenta . however , a generalized oblique treatment has shown the presence of continuum of unstable modes @xcite , characterized by the orientation of their wave - vector relative to the bulk beam velocity . of these neither the two - stream nor weibel are generally the most unstable . rather , generically the most robust and fastest growing mode occurs at oblique wave - vectors , and thus referred to as the oblique instability by @xcite . the cold - plasma cooling rate of this maximally - growing mode is @xmath142 this can be much larger than the two - stream and weibel growth rates when @xmath138 and @xmath143 . the cold - plasma oblique instability cooling rates dominate inverse - compton cooling by orders of magnitude over the region of interest . however , at the very dilute beam densities of relevance here , the cold - plasma approximation requires exceedingly small beam temperatures . above an igm - frame temperature of roughly @xmath144 pairs can traverse many wavelengths of the unstable modes over the cold - instability growth timescale , a situation commonly referred to as the kinetic regime . as a consequence , significant phase mixing can occur , substantially reducing the effective growth rate @xcite . the way in which finite beam temperatures limit the growth rate depends sensitively upon the nature of the velocity dispersion and the modes of interest ( see , e.g. , * ? ? ? for example , the two - stream instability , associated with wave vectors parallel to the beam , enters the strongly suppressed `` quasi - linear '' regime when the parallel momentum dispersion is large , though is insensitive to even large transverse dispersions . conversely , the weibel instability , associated with wave vectors orthogonal to the beam , is sensitive to even small transverse velocity dispersions but unaffected by large parallel velocity dispersions . this is simply because a given mode can tolerate large velocity dispersions within but not across the phase fronts of the unstable electromagnetic modes . for the situation of interest here , dilute beams and cool igm ( i.e. , @xmath145 ) , the oblique modes are nearly transverse , and thus sensitive primarily to large transverse velocity dispersions . nevertheless , even for the small temperatures we have inferred for the pair beams , we find ourselves in the kinetic regime . in this case the oblique instability cooling rate has been numerically measured to be @xmath146 where we have set the beam temperature in the beam frame to @xmath147 , and thus @xmath148 @xcite . both the cold and hot growth rates have been verified explicitly using particle - in - cell ( pic ) simulations , though for somewhat less dilute beams than those we discuss here @xcite . when the kinetic oblique mode dominates the beam cooling the beam density is given by @xmath149 the associated cooling rate is then @xmath150 this is a stronger function of gamma - ray energy than inverse - compton cooling , implying that it will eventually dominate at sufficiently high energies , assuming a flat tev spectrum . in addition it is a very weak function of @xmath151 , being only marginally faster in lower - density regions , and thus the cooling of the pairs is largely independent of the properties of the background igm . the rates obtained by numerically solving equation ( [ eq : nb ] ) for @xmath110 , with @xmath152 , are shown for a number of redshifts in figure [ fig : gs ] . for the luminosity shown ( @xmath153 , typical of the bright tev blazars ) at @xmath154 plasma cooling dominates inverse compton above a tev . in the present epoch , @xmath155 is roughly two orders of magnitude larger than @xmath120 for bright tev blazars . the luminosity dependence of @xmath155 implies a luminosity limit below which inverse compton does dominate the linear evolution of the pair beam , @xmath156 ( note that at this luminosity @xmath157 , and thus @xmath110 is half the value shown in equation ( [ eq : nbgm ] ) ) . this limit is shown as a function of redshift for a number of different energies by the black lines in figure [ fig : nblimits ] . note that these lie above the corresponding limits associated with the applicability of the plasma prescription , suggesting that in practice the beam does support collective phenomena . the critical luminosity ranges from @xmath158 to @xmath91 , depending upon redshift and energy of interest . it also depends upon the over - density as roughly @xmath159 , and thus at low densities the critical luminosity is moderately smaller . the only two sources in table [ tab : tevsources ] which fall below the plasma - cooling luminosity limit are the radio galaxies m87 and cen a , both of which are detectable only as a result of their close proximity . at @xmath133 , all but a handful of the remaining sources lie more than an order of magnitude above this limit , and in the case of the two sources that dominate the tev flux at earth , more than two orders of magnitude above this limit . + given the technical nature of our discussion above , it is useful to have a qualitative understanding of these instabilities . we caution , however , that intuitive pictures of plasma processes frequently fail to capture all the relevant physics . hence , generalization of these intuitive pictures beyond their limited range of applicability is potentially misleading . this is explicitly illustrated by our examples here : all of the instabilities discussed in this work belong to the same family , i.e. , instabilities that arise from interpenetrating plasmas , but the underlying qualitative pictures for each differ substantially . we begin with the weibel ( or filamentation ) instability @xcite , for which a mechanical viewpoint ( an initial magnetic perturbation deflects particles into opposing current streams that reinforce the perturbed field ) can be found in @xcite , to which we refer the interested reader . here we present picture that although having the virtue of being simpler is not entirely correct : because like currents attract , small - scale current perturbations arising out of the fluctuations within the interpenetrating plasmas will coalesce preferentially to produce increasingly larger - scale currents . these induce stronger magnetic fields , and thus larger attractive lorentz forces between neighboring currents ; a positive feedback loop develops leading to instability . this process continues until the associated magnetic field strengths become sufficiently large to disrupt the currents ( in the case of equal density beams ) or until the transverse velocity of the constituent particles is large enough to efficiently migrate between current structures on the linear growth timescale , i.e. , enter the kinetic regime . we should note that the aggregation of currents in the weibel instability does not rely upon _ oscillatory _ waves . hence , there is no oscillatory component to this instability it is a purely growing mode . in contrast , the two - stream instability is an overstable mode , where the oscillatory components are langmuir ( or plasma ) waves with a wavevector parallel to the beam velocity . these are longitudinal waves , associated with local charge oscillations , and are completely described by a propagating perturbation in the electrical potential . as particles in the beam traverse langmuir waves in the background plasma , they experience successive periods of acceleration and deceleration , with electrons ( positrons ) collecting in minima ( maxima ) of the electric potential , where the particle speeds are at their smallest . this charge - separated bunching of the beam plasma enhances the background electric perturbation , potentially growing the background langmuir wave . the bunching within the beam is simply an excitation of langmuir waves within the beam plasma itself . thus the growth of the charge perturbations in the background and beam plasmas corresponds to the resonant coupling between langmuir waves in the background and beam . when the beam density is much less than the background density , as is the case here , background and beam langmuir waves only overlap in frequency , and therefore satisfy the conditions for resonance , when the wavevector of the latter is parallel to the beam velocity ( see the discussion above equation ( [ eq : displin ] ) ) . this is always satisfied for the family of comoving beam langmuir waves ( i.e. , waves which in the beam frame move counter to the background plasma ) . however , of particular importance for the two - stream instability are the counter - propagating beam langmuir waves ( i.e. , waves which in the beam frame move parallel to the background plasma ) . if the beam velocity exceeds the phase velocity of these waves ( as seen in the beam frame ) , the counter - propagating wave will be dragged in the direction of the beam ( as seen in the background frame ) , and therefore has a wavevector which satisfies the resonant condition . nevertheless , the counter - propagating langmuir wave still carries momentum in the direction opposite to the beam . thus , as the counter - propagating wave grows , the momentum , and therefore energy , of the beam - wave system necessarily decreases . this implies that the counter - propagating langmuir wave is also a _ negative energy mode _ as seen in the background frame . as a consequence , the resonant coupling can transfer energy to the positive - energy background wave from the negative - energy beam wave , while growing the amplitudes of both , and thereby leading to instability . note that if the phase velocity of the counter - propagating langmuir wave is larger than the beam velocity , it no longer is dragged in the direction of the beam and no longer satisfies the necessary resonant condition . for a distribution of particles , this constraint upon the velocities within the beam corresponds to the familiar penrose criterion @xcite , and is satisfied in the pair beams resulting from vhegrs . the oblique instability encompasses the two - stream and weibel instabilities , though the most unstable mode is most similar to the former in that the intuitive picture focus solely on the electrostatic forces , ignoring electromagnetic forces . in practice , this is a relatively good approximation and can be used to calculate the growth of these modes in idealized situations ( e.g. * ? ? ? * ; * ? ? ? the qualitative picture proceeds similarly to that for the two - stream instability described above , with the minor modification that the perturbing background langmuir waves now move at an angle @xmath65 relative to the _ relativistic _ beam . as a consequence , the resonant langmuir waves have a phase velocity such that @xmath160 . from the simple intuitive picture above , if @xmath65 is selected such that the projected beam velocity is slightly faster than the nearly resonant langmuir wave , an instability develops . finally , to understand why the growth rates between the two - stream instability above and the oblique instability differ , it is useful to make the approximation that the electric fields generated are in the direction of the k - vector . in the two - stream case , the electric field must slow down ( or speed up ) particles . this gets progressively harder for more relativistic particles . in the oblique case , the electric field deflects the particles , changing their _ projected _ velocity . while this is also more difficult for more relativistic particles , this is not nearly as hard as changing the particles parallel ( along the beam ) velocity . hence , the oblique instability more easily drives charge density enhancements ( and therefore instabilities ) at large @xmath65 , i.e. , easier deflection , than the two - stream instability . we have thus far only treated the linear development of the relativistic two - stream , weibel , and oblique instabilities . however , the impact pair beams have upon the igm , gamma - ray cascade emission , and measures of the igmf will ultimately depend upon their nonlinear development . to address this , however , we are presently forced to appeal to analytical and numerical calculations of systems in somewhat different ( and less extreme ) parameter regimes . motivated by the applicability of the weibel instability in the context of grbs , the nonlinear saturation of the relativistic weibel instability for equal density plasma beams is well understood analytically and numerically . initially , the weibel instability rapidly grows until the energy density of the generated magnetic field becomes of order the kinetic energy of the two beams @xcite . analytically , @xcite argued that the weibel instability would saturate when the generated magnetic fields become so large that the larmor radius of the beam particles is of order the skin depth , i.e. , when the energy of generated magnetic fields is equal to the kinetic energy of two equal density relativistic beams ( see also * ? ? ? * ) . the particles rapidly isotropize with a maxwellian distribution @xcite , i.e. , heat , and the magnetic energy then rapidly decays within an order of a few tens of skin depths @xcite . hence for two equal density , relativistic , interpenetrating beams , the weibel instability converts anisotropic kinetic energy into heat . however , as we have already noted , for the pair beams of interest here the weibel instability is completely suppressed for tiny transverse beam temperatures . hence , while this instability may initially operate , it may quickly become suppressed if it results in significant transverse heating of the beam . unlike the weibel instability , the two - stream and kinetic oblique instabilities continue to operate , though more slowly than implied by their cold - plasma limits , in the presence of substantial beam temperatures . due to the geometry of the coupled modes , the oblique instability is primarily sensitive to transverse beam - velocity dispersions , though shares the resistance to beam heating with its two - stream cousin @xcite . unlike the two - stream instability , the oblique instability in the parameter range of interest here is also largely insensitive to longitudinal heating . thus , generally , it appears that the oblique instability is substantially more robust than its more commonly discussed brethren . what is less clear a priori is if these instabilities primarily heat the beam or primarily heat the background plasma . here we appeal to the numerical simulations of @xcite , where a mildly relativistic beam ( @xmath161 ) penetrating into a hot , dense background plasma ( beam - to - background density ratio of @xmath162 ) was studied . in these the oblique instability resulted in a significant fraction ( @xmath163 ) of the beam energy heating the background plasma before the heating of the beam suppressed the oblique instability in favor of the two - stream instability . the relative effectiveness with which the beam heats the background plasma is due to the efficiency with which the longitudinal electrostatic modes are dissipated in the background plasma ; unlike electromagnetic modes ( e.g. , those generated by the weibel instability ) , electrostatic modes are rapidly dissipated via landau damping . we note that @xcite found that the heating of the beam by the oblique instability eventually led to its suppression , allowing the two - stream instability to grow , continuing the dissipation of the beam kinetic energy . as a result , in their simulations a total of 30% of the energy was deposited into the background plasma via a _ combination _ of the oblique and two - stream instabilities , i.e. , an additional 10% of the beam energy was thermalized via the two - stream instability during and after the suppression of the oblique instability . in our case , we expect that much more beam energy ( more than the 20% , and possibly up to @xmath164 ) will be deposited into the background igm because we are much deeper into the regime in which the oblique instability dominates , i.e. , @xmath143 and @xmath138 . in particular , for the case of interest here , @xmath165 and @xmath166 the extremely large lorentz factor and tiny density ratio make it computationally prohibitive to assess the beam evolution with numerical pic directly . nevertheless , because we find ourselves in a regime in which the oblique instability is much more strongly dominant than that simulated in @xcite , we expect the linear growth of the kinetic oblique instability to continue for much longer before the beam changes character and moves out of the oblique - dominated regime potentially once @xmath167 of the beam energy has been dissipated . however , this remains to be studied in future work . the effect of nonlinear processes might also effect the evolution of the linear instability . for instance , @xcite argued that for the relativistic electrostatic two - stream instability , nonlinear coupling to daughter modes arrest the growth of the linearly unstable mode at a very low mode energy . hence , they claim that the electrostatic two - stream instability can only bleed energy from the beam at a slow rate . utilizing the order of magnitude estimates in @xcite or the expression for nonlinear landau damping in @xcite , we have found that damping of the pair beam via the relativistic two - stream instability could be highly suppressed . the oblique instability may be similarly suppressed , however , this effect is much more marginal due to the instabilities much larger growth rate . nevertheless , determining its behavior for the parameters relevant here is an important unanswered question that is left for future work . for the purposes of this paper , we will rely upon the intuition developed from numerical studies of the oblique instability and argue that the beam ends up heating the igm primarily . in what follows , we will presume the nonlinear evolution of the `` oblique '' or related plasma instabilities lead to the heating of the background igm , i.e. , beam cooling . however , we note that the beam itself may be the primary recipient of this kinetic energy , i.e. , beam disruption . most of our results depend critically on beam cooling and not beam disruption . this includes the effect of blazar heating on the igm temperature - density relation studied in paper ii , the effects on structure formation studied in paper iii , the excellent reproduction of the statistical properties of the high - redshift forest found in @xcite , and the implications on the egrb and the redshift evolution of tev blazars studied in section [ sec : blf ] . however , our conclusions on the inapplicability of igmf constraints determined from the non - observation of gev emission from blazars still remains in the presence of beam disruption . this is because the self - scattering of pairs in the beam would suppress this gev emission in similar manner to an igmf . + the beam instabilities we have discussed have been analyzed primarily within the context of unmagnetized plasmas . however , for a variety of theoretical reasons a weak igmf is not unexpected . for example , a field strength of @xmath47 is sufficient to explain the observed @xmath168 galactic fields via compression and winding alone . here we consider the implications that an igmf has for the instability growth rates we have described above . a strong igmf causes the ultra - relativistic pairs to gyrate , and therefore to isotropize , suppressing the growth of instabilities that feed upon the beam anisotropy ( e.g. , those we have described above ) . however , for this to efficiently quench the growth of the plasma beam instabilities , this isotropization must occur on a timescale comparable to the instability growth time , i.e. , the larmor frequency must be comparable to the cooling rate , @xmath109 . this condition gives a lower - limit upon igmf strengths sufficient to appreciably suppress the growth of plasma beam instabilities of @xmath169 considerably larger than those typical of both , primordial formation mechanisms @xcite and implied by galactic magnetic field estimates , assuming galactic fields are produced by contraction and winding alone . because the vhegrs emitted by the tev blazars travel cosmological distances prior to producing pairs , an igmf capable of suppressing the plasma beam instabilities must necessarily have a volume filling fraction close to unity . in particular , it must permeate the low - density regions , where most of the cooling occurs . however , the alfvn velocity within those areas is extraordinarily small , roughly @xmath170 the igm sound speed , @xmath171 is considerably larger , implying convection is much more efficient . nevertheless , even after substantial heating via the thermalization of the tev blazar emission ( paper ii ) is taken into account , magnetic fields will have propagated @xmath172 over a hubble time via convection and much less via diffusion , implying that a pervasive , sufficiently strong magnetic field can not be produced via ejection from galactic dynamos . while galactic winds can produce much faster outflows , @xmath173 , unless they inject a mass comparable to that contained in the low - density regions over a hubble time , they are rapidly slowed via dissipation at shocks in the igm , again limiting the spread of galactic fields . moreover , we note that since the magnetic field must be volume filling to suppress the plasma beam cooling substantially it is insufficient to produce pockets of strong fields , and thus any galactic origin powered by winds requires a nearly complete reprocessing of the low - density regions . however , the ultimate thermalization of such fast , dense winds would raise the igm temperature to @xmath174 , in conflict with the forest data and exceeding the entire bolometric output of quasars by at least a factor of two . for these reasons we conclude that a volume - filling strong igmf would demand a primordial origin . the existence of plasma processes that can cool the pair beams associated with tev blazars has profound consequences for efforts to constrain the igmf using the spectra of tev blazars . here we describe how the reported igmf limits have been obtained , the consequences of plasma cooling for these , and potential strategies for overcoming the constraints it imposes . the general argument made in efforts to constrain the igmf based upon the gev emission from blazars proceeds as follows : the beamed tev blazar emission pair - creates off of the ebl . the resulting pairs subsequently up - scatter cmb photons to gev energies . in principle , this should produce an observable gev excess , or bump , in the spectra of these objects . however , in the presence of a large - scale igmf , the ultra - relativistic pairs can be deflected significantly , directing the beamed gev emission away from earth . thus , it is argued , the lack of a discernible gev bump in a number of tev blazars implies a lower limit upon the igm field strength ( as a function of tev jet opening angle and variability timescale ) . the crucial components of the argument are 1 . the tev emission is beamed with the typical opening angles inferred from radio observations of agn jets . 2 . the variability timescale within the tev source is long in comparison to the geometric time delays between the original tev and inverse - compton produced gev gamma rays due to the orbit of the pairs through some angle @xmath175 , @xmath176 ( see * ? ? ? 3 . the pairs produced by tev absorption on the ebl cool primarily via inverse - compton scattering the cmb ( and therefore evolve only due to inverse - compton cooling and orbiting within the large - scale igmf ) . the first of these is supported indirectly by the lack of tev emission from non - blazars . the implied tev source stability required by the second is at odds with the variability observed in blazars at longer wavelengths . however , since the tev - gev delay is a strong function of deflection angle , given an empirical limit upon the tev blazar variability timescale ( presently @xmath177 , * ? ? ? * ) it is possible to produce a substantially weaker constraint upon the igmf ( @xmath178 ) @xcite . ) is @xmath179 , resulting in @xmath180 up - scattered cmb photons , the energy at which the /lat instrument is most sensitive . for reference , the sources listed in table [ tab : tevsources ] are also plotted ( at @xmath131 ) , with hbl , ibl , radio galaxies , and quasars shown by the the blue triangles , green squares , red hexagons and magenta circles , respectively . filled points indicate sources that have been used to estimate the igmf , corresponding to ( in increasing flux ) pks 0548 - 322 , 1es 0347 - 121 , 1es 1101 - 232 , rgb j0152 + 017 , 1es 0229 + 200 , 1es 1218 + 304 , rgb j0710 + 591 , and mkn 501 . for reference , the inferred isotropic luminosity is shown in the top axis for sources located at @xmath181 . ] plasma cooling provides a fundamental limitation for these methods , however , by violating the third condition explicitly . figures [ fig : nblimits ] and [ fig : fic ] imply that for all but the dimmest and highest - redshift ( @xmath182 ) gamma - ray blazars only a small fraction of the pair energy is lost to inverse - compton on the cmb ( @xmath183 ) . in particular , the black lines in figure [ fig : nblimits ] shows where @xmath184 , i.e. , roughly 50% of the tev - photon power is ultimately converted into heat via plasma instabilities . as a result , the putative gev component is typically much less luminous than otherwise expected , reducing the significance of non - detections substantially . this may be seen explicitly in figure [ fig : fic ] , which shows @xmath185 as a function of gamma - ray flux for @xmath179 ( corresponding to a comptonized - cmb photon energy of approximately @xmath180 ) , at number of source redshifts . typical values for the tev blazars collected in table [ tab : tevsources ] ( including those employed by * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) lie in the range @xmath186 to @xmath187 , implying correspondingly small gev comptonization signals . more importantly , when the beam evolution is dominated by plasma instabilities , over the inverse - compton cooling timescale the pair distribution necessarily becomes isotropized . as a consequence , the angular distribution of the resulting gev gamma rays ( i.e. , the orientation of the gev `` beam '' ) is no longer indicative of the beam propagation through a large - scale magnetic field . that is , one expects large - angle deviations regardless of the igmf strength . unfortunately , subject to the caveats of the preceding section , plasma instabilities appear to be the dominant cooling mechanism for the pair beams associated with the tev blazars that have been used to constrain the igmf thus far . the isotropic - equivalent luminosities of these sources range from @xmath188 to @xmath91 , placing them well within the plasma - instability dominated regime . as a result , the reported igmf limits inferred from tev blazars are presently unreliable . nevertheless , it may be possible to avoid the limitations imposed by plasma instabilities . at sufficiently low pair densities the cooling rates associated with the beam instabilities described in section [ sec : pbis ] fall below that due to inverse compton . moreover , at some point the plasma prescription breaks down altogether , suggesting that any plasma instabilities that operate on the skin - depth of the igm are strongly suppressed . there are two distinct ways in which low pair densities can arise : low intrinsic luminosities and very short timescale events . below isotropic - equivalent luminosities of roughly @xmath189 inverse - compton cooling dominates the beam instabilities we describe in section [ sec : pbis ] at @xmath190 . below @xmath191 the plasma prescription itself breaks down altogether . while no tev blazars with isotropic - luminosities below @xmath191 are known , there are a handful of very nearby sources below @xmath192 . the two dimmest sources , the radio galaxies m87 and cen a , however , are observable only due to their proximity ( i.e. , they have proper distances @xmath193 ) , intrinsically preventing them from providing a significant constraint on the imgf . the highest source in figure [ fig : fic ] , and thus presumably the source with the largest fractional inverse - compton signal , is pks 2005 - 489 , a dim , soft tev blazar at @xmath194 . even in this case , however , the inverse - compton cooling timescale is roughly 4 times longer than the plasma cooling timescale . hence , probing the igmf with tev blazars will require observing considerably dimmer objects than those used thus far . since doing so will likely require substantial increases in detector sensitivities , the incompleteness of the tev observations are well - described by a single flux limit , corresponding to an isotropic - equivalent luminosity limit at @xmath195 of roughly @xmath196 . the flux limit of is estimated to be roughly @xmath197 at @xmath45 ( see figure 23 of * ? ? ? ] we will not consider this possibility any further here . low pair densities may also be produced by limiting the duration of the vhegr emission . equation ( [ eq : nb ] ) was derived assuming that the pairs had reached a steady state between their formation via vhegrs annihilating upon the ebl and their cooling via inverse - compton and plasma processes . however , it takes roughly a cooling timescale for the pair beam densities to saturate at this level , which can be as long as @xmath198 at some @xmath58 and @xmath56 . therefore , generally there is a lag between the onset of the vhegr emission and the time at which plasma processes begin to dominate the cooling of the beam . we can estimate this by setting @xmath199 where @xmath200 is the duration of the emission . this is the case when the pair density is initially zero ( i.e. , it has been many cooling timescales since the last period of vhegr emission ) and @xmath201 . inserting this into the various cooling rates and setting @xmath202 gives an estimate for the maximum duration for which the pair density remains sufficiently low that inverse - compton cooling dominates the cooling : @xmath203 which may be found for the observed tev sources in table [ tab : tevsources ] . typically @xmath204 for the blazars that have been used to constrain the igm , though @xmath200 ranges from @xmath205 to @xmath206 for tev blazars generally . note that it is not sufficient for emission to vary on @xmath200 ; such variations will be temporally smoothed on the much larger cooling timescale . rather , it is necessary for the source to have been quiescent ( i.e. isotropic - equivalent luminosity considerably less than @xmath192 ) for periods long in comparison to @xmath207 and have turned on less than a time @xmath208 ago . as a consequence , only igmf estimates using new tev blazars can potentially avoid plasma instabilities in this way . a natural example of a class of transient gamma - ray sources is gamma - ray bursts ( grbs ) . for grbs an isotropic - equivalent energy of roughly @xmath209 is required for the plasma instabilities to grow efficiently , comparable to the total energetic output of the brightest events . using grbs to probe the igmf has been suggested previously , and limits based upon the lack of a delayed gev component in some grbs already exist , finding field strengths above @xmath210 ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? future efforts should be able to probe field strengths of @xmath47 for @xmath211 ( see figure 4 of * ? ? ? this is predicated , however , upon the existence of a significant vhegr component in the prompt or afterglow emission of grbs . @xcite have attempted to directly detect the gev signal in the vicinity of agns that do not exhibit tev emission . that is , since an igmf can in principle induce large - angle deviations in the propagation direction of the up - scattered gev photon relative to the original vhegr , one may search for a gev excess at large distances from non - blazars . this is made much more difficult by the extraordinarily low surface gamma - ray brightness due to the large @xmath119 , random source orientations , and potential dilution associated with the beam diffusion . nevertheless , @xcite claim a marginal detection of this component in stacked images , though the point - spread function appears to be capable of completely explaining their result @xcite . if such a gev - halo were found around tev - bright sources with the anticipated luminosity , it would argue strongly against plasma instabilities dominating the beam evolution . however , we note that for many tev - dim sources inverse - compton cooling still dominates the beam evolution , and thus in stacked images of these sources gev - halos may still be present . note that , as discussed in section [ sotoibai ] , a strong igmf ( @xmath212 ) could , in principle , suppress the growth of the plasma beam instabilities significantly . as we argue there , a sufficiently pervasive and strong igmf would necessarily be primordial in origin . however , few constraints upon such a primordial field exist , especially since we have argued that the lack of an inverse compton gev bump can not be used as a constraint on the igmf . hence , we are left with constraints from big bang nucleosynthesis ( @xmath213 , * ? ? ? * and references therein ) , from the cmb ( @xmath214 , * ? ? ? * ) , and limits on the faraday rotation of distance radio sources ( @xmath214 , * ? ? ? * and references therein ) . a stronger limit can be obtained from limits upon the deflection angle ( @xmath215 ) of ultra - high energy cosmic rays ( @xmath216 ) from distant sources at a distance @xmath217 , which upon assuming a magnetic correlation length , @xmath218 , gives @xmath219 ( see for instance * ? ? ? * ; * ? ? ? * ) , though this depends strongly upon the assumed structure of the igmf and small - scale fields may be significantly stronger than this @xcite . proposed mechanisms by which a primordial igmf can be theoretically generated are essentially unconstrained , yielding b - field strengths between @xmath220@xmath221 @xcite . however , in the remainder of this work and in paper ii and paper iii , we show that the effect of the `` oblique '' ( or similar ) instability may potentially explain many different observed phenomena in cosmology and high energy astrophysics . taking the beam instabilities we present here at face value , that so many observational puzzles can be simultaneously explained by the local dissipation of tev blazar emission would imply the strongest upper limit upon primordial magnetism to date . namely , the apparent impact of tev blazars upon the large - scale cosmological environment places a constraint on the igmf of @xmath222 . the egrb corresponds to the diffuse gamma - ray background ( @xmath223 ) not associated with galactic sources . this has been measured at high energies by egret @xcite and more recently strongly constrained between @xmath48 and @xmath49 by @xcite . the most recent estimate of the egrb is a featureless power - law , somewhat softer than that found by egret , and does not exhibit the localized high - energy excess claimed by a reanalysis of the egret data @xcite . while bright , nearby blazars are resolved , and therefore excluded from the egrb , the remainder is thought to arise nearly exclusively from distant gamma - ray emitting blazars , upon which the egrb places severe constraints @xcite . beyond @xmath224 can not detect blazars with isotropic - equivalent luminosities comparable to the objects listed in table [ tab : tevsources ] , and thus the vast majority of such objects contribute to the egrb . a significant egrb above @xmath49 is not expected due to annihilation upon the ebl . however , if they operate efficiently , iccs can reprocesses the vhegr emission of distant sources into the egrb energy range ( i.e. , @xmath44 ) . thus , a number of efforts to constrain the vhegr emission from extragalactic sources based upon the egrb can be found in the literature @xcite . these have typically found that the comoving number density of vhegr - emitting blazars could not have been much higher at high-@xmath56 than it is today . even with moderate evolutions ( e.g. , that in * ? ? ? * ; * ? ? ? * ) consistent with the egret egrb , substantially over - produce the egrb , and are therefore believed to be excluded @xcite . however , here we show that this conclusion is predicated upon the high - efficiency of the icc . we have already shown that for bright vhegr sources plasma beam instabilities extract the kinetic energy of the first generation of pairs much more rapidly than inverse - compton scattering . as a consequence , the icc is typically quenched , substantially limiting the contributions of these sources to the egrb . thus , even with a dramatically evolving blazar population , e.g. , similar to that of quasars , it is possible for these objects to be consistent with the egrb . while our discussion of the egrb and the evolution of blazars is predicated on the extraction of the pair - beam kinetic energy by plasma beam instabilities , our conclusions are not . rather they will continue to hold as long as the pair beams are locally dissipated via plasma beam instabilities or some other equally powerful mechanism . blazars dominate the extragalactic gamma - ray sky , and thus represent the best studied vhegr source class . since we are interested exclusively in those objects responsible for the bulk of the vhegr emission , we necessarily concentrate upon the subset of blazars that are luminous vhegr emitters . of the 28 tev sources listed in table [ tab : tevsources ] , 23 are peaked at very - high energies ( the hbl and hard ibl sources , for a full definition see below ) , and comprise what we will call collectively tev blazars . this necessarily is limited to @xmath45 as a consequence of the annihilation of vhegrs upon the ebl . nevertheless , we will find that this is remarkably similar to the local quasar luminosity function ( @xmath225 ) , and thus attempt to construct a blazar luminosity function by analogy : @xmath226 in which @xmath227 is the comoving number density of blazars at redshifts less than @xmath56 with isotropic - equivalent luminosities below @xmath228 . this is different from previous efforts to empirically constrain @xmath229 from the egrb ( see , e.g. , * ? ? ? * ; * ? ? ? * ) in at least two ways . first , we begin with an empirically determined local @xmath229 and attempt to extend this by placing the tev blazars within the broader context of accreting supermassive black holes , rather than beginning with the egrb and working backwards to infer an acceptable @xmath229 . second , we are primarily concerned with @xmath229 of tev blazars , and specifically do not consider the contributions from other kinds of objects . while this does not represent a significant oversight in terms of the high - energy contributions to the egrb , which almost certainly arises from the vhegr - emitting blazars , it does mean that our conclusions regarding the luminosity function of the tev blazars do not necessarily apply to all agn ( e.g. , the fsrqs , see below ) . the tev blazars presumably fit within the broader context of the blazars observed by specifically , and agns generally . for this reason , here we briefly review the physical classification scheme based on the widely accepted agn standard paradigm that provides a unified picture of the emission emission properties of these objects ( e.g. , * ? ? ? * ) . specifically , we summarize the classes of objects believed to be capable of producing significant tev luminosities and the potential physical processes responsible for the observed emission . based upon these we then assess the implications for the number , variability , and redshift evolution of the tev blazars . in general there exist two main classes of agns that differ in their accretion mode and in the physical processes that dominate the emission . 1 . _ thermal / disk - dominated agns : _ infalling matter assembles in a thin disk and radiates thermal emission with a range of temperatures . the distributed black - body emission is then comptonized by a hot corona above the disk that produces power - law x - ray emission . hence the emission is a measure of the accretion power of the central object . this class of objects are called qsos or seyfert galaxies and make up about 90% of agns . they preferentially emit in the optical or x - rays and do not show significant nuclear radio emission . none of these sources have so far been unambiguously detected by or imaging atmospheric cherenkov telescopes because the comptonizing electron population is not highly relativistic and emits isotropically , i.e. there is no beaming effect that boosts the emission . _ non - thermal / jet - dominated agns : _ the non - thermal emission from the radio to x - ray is synchrotron emission in a magnetic field by highly energetic electrons that have been accelerated in a jet of material ejected from the nucleus at relativistic speed . the same population of electrons can also compton up - scatter any seed photon population either provided by the synchrotron emission itself or from some other external radiation field such as uv radiation from the accretion disk . hence the sed of these objects shows two distinct peaks . the luminosity of these non - thermal emission components probes the jet power of these objects . observationally , this leads to the class of radio - loud agns which can furthermore be subdivided into blazars and non - aligned non - thermal dominated agns depending on the orientation of their jets with respect to the line of sight . there are no known sources above @xmath230 that correspond to agns with jets pointed at large angles ( @xmath231 , see @xcite ) with respect to the line of sight ( for an example of a non - aligned agn , ngc1275 , that shows a very steep high - energy spectrum , emitting a negligible number of vhegrs , see * ? ? ? hence we turn our attention to blazars , which can be powerful tev sources . blazars can further be subdivided into two main subclasses depending upon their optical spectral properties : flat spectrum radio quasars ( fsrq ) and bl lacs . fsrqs , defined by broad optical emission lines , have seds that peak at energies below @xmath232 , implying a maximum particle energy within the jet and limiting the inverse - compton scattered photons mostly to the soft gamma - ray band . it is presumably for this reason that no continuous tev component has been detected in an fsrq ( note , however , that tev flares from fsrqs have been detected in two cases @xcite ) . in contrast , bl lacs or blazars of the bl lac type @xcite can be copious tev emitters . these are very compact radio sources and have a broadband sed similar to that of strong lined blazars , though lack the broad emission lines that define those . depending upon the peak energy in the synchrotron spectrum , which approximately reflects the maximum particle energy within the jet , they are classified as intermediate- , or high - energy peaked bl lacs , respectively called lbl , ibl , and hbl @xcite . while lbls peak in the far - ir or ir band , they exhibit a flat or inverted x - ray spectrum due to the dominance of the inverse - compton component ( see fig 15 of * ? ? ? * for a visualization of the sed of bl lacs ) . the synchrotron component of ibls peaks in the optical which moves their inverse - compton peak into the gamma - ray band of . hbls are much more powerful particle accelerators , with the synchrotron peak reaching into the uv or , in some cases , the soft x - ray bands . the inverse - compton peak can then reach tev energies @xcite . in the gamma - ray band , the subclass of ibls that emit vhegrs are almost indistinguishable from the hbls , suggesting that the location of the synchrotron peak does not uniquely characterize the vhegr emission from these sources ( e.g. , due to variations among individual blazars in the magnetic field strength within the synchrotron emitting region and the origins and properties of the seed photons that are ultimately comptonized ) . hence we identify hbls and vhegr - emitting ibls with the single source class of tev blazars . we note that there is presently no evidence for the hypothetical class of ultra - hbls that were proposed to have a very energetic synchrotron component extending to @xmath92-rays @xcite . if such a population of bright and numerous sources exists , should have seen it @xcite . the ultra - hbls may have escaped detection from thus far by being either intrinsically dim @xmath92-ray sources or very rare objects @xcite . tev blazars have a redshift distribution that is peaked at low redshifts extending only up to @xmath233 . this is most likely entirely a flux selection effect ; tev blazars are intrinsically less luminous than lbls and fsrqs , with an observed isotropic - equivalent luminosity range of @xmath234 , with the highest redshift tev blazars also being among the most luminous objects ( see figures 23 and 24 in * ? ? ? that tev blazars should be intrinsically less luminous than fsrqs is not entirely unexpected , however . @xcite have argued that the physical distinction between fsrqs and tev blazars has its origin in the the different accretion regimes of the two classes of objects . using the gamma - ray luminosity as a proxy for the bolometric luminosity , the boundary between the two subclasses of blazars can be associated with the accretion rate threshold ( nearly 1% of the eddington rate ) separating optically thick accretion disks with nearly eddington accretion rates from radiatively inefficient accretion flows . the spectral separation in hard ( bl lacs ) and soft ( fsrqs ) objects then results from the different radiative cooling suffered by the relativistic electrons in jets propagating into different surrounding media @xcite . hence in this model , tev blazars can not reach higher luminosities than approximately @xmath235 since they are limited by the nature of inefficient accretion flows that power these jets and by the maximum black hole mass , @xmath236 . and @xmath237 , respectively ) . the solid lines show the absolute @xmath238 ( in comoving mpc ) , while the dashed lines show @xmath238 rescaled in magnitude by @xmath239 and shifted to lower luminosities by a factor of @xmath240 . in both cases the black , orange , and red lines correspond to @xmath241 , @xmath242 , and @xmath243 , respectively . the points and upper - limits show @xmath229 of the hbl and ibl sources listed in table [ tab : tevsources ] , assuming the relevant limits for pks 1553 + 113 and pks 1424 + 240 . presented in the inset is the tev source luminosity distance as a function of source luminosity for all of the sources in table [ tab : tevsources ] with redshift estimates ( including limits ) . the dotted line shows the distance - dependence of the flux limit we employ in the completeness correction . ] while we have attempted to place the observed tev blazars , which we have identified with the hbls and vhegr - emitting ibls , into the broader context of agns using the unified model , due to the substantial distinctions in accretion rate , emission properties , object morphology and geometry , it is not obvious that any of the properties of tev blazars should be similar to those of agns more generally . nevertheless , evidence for a simple connection between the two populations can be found in the similarity between their the luminosity functions ( a fact we will exploit later in estimating the redshift evolution of the tev blazars ) . here we define the luminosity for the purposes of defining @xmath229 to be the isotropic - equivalent value associated with emission between @xmath49 and @xmath244 . while this may be considered to be a vhegr luminosity , because most tev blazars are peaked within this band , this corresponds to the majority of the emission from these sources . the objects listed in table [ tab : tevsources ] were chosen because they have well defined seds , based upon a combination of veritas , h.e.s.s . , and magic observations . these 28 sources have vhegr spectra that are well fit by the form , @xmath245 where @xmath246 is the normalization in units of @xmath247 . the gamma - ray energy flux is trivially related to @xmath248 by @xmath249 , from which we obtain a vhegr flux , @xmath250 and for sources with a measured redshift a corresponding isotropic - equivalent luminosity , @xmath251 , where @xmath252 is the luminosity distance . , this overestimates the luminosity by a factor of order unity . ] the resulting @xmath246 , @xmath253 , @xmath254 , @xmath255 , and @xmath228 are collected in table [ tab : tevsources ] . in addition we list the redshift , inferred comoving distance , and absorption - corrected intrinsic spectral index , defined at @xmath253 , obtained via @xmath256}}{d\ln e}\right|_{e_0 } \simeq \alpha - \tau_{e}\left[e_0(1+z),z\right]\,.\ ] ] for high - redshift sources @xmath257 can be substantially less than @xmath51 , implying that an intrinsic spectral upper - cutoff must exist . to produce @xmath258 , we must account for a variety of selection effects inherent in the sample listed in table [ tab : tevsources ] . the objects in table [ tab : tevsources ] were originally selected for study for a variety of source - specific reasons , e.g. , existing well known sources , extremely high x - ray to radio flux ratio in the sedentary high energy peaked bl lac catalog , hard spectrum sources in the point source catalog , and flagged as promising by the -lat collaboration . in addition , the source selection suffered from the usual problems associated with surveys ( e.g. , scheduling conflicts with other targets , moon , bad weather , etc . ) . as a consequence , this sample is somewhat inhomogeneous . nevertheless , in lieu of a less - biased sample , we will treat it as homogeneous and correct for the selection effects were possible , focusing upon those due to the sky coverage and duty cycle of tev observations , and those due to sensitivity limits of current imaging atmospheric cerenkov telescopes . to estimate the sky completeness and duty cycle of this set of objects we rely upon the all - sky @xmath259 gamma - ray observations of hbl and ibl sources by @xcite . outside of the galactic plane , observes 118 high - synchrotron peaked ( hsp ) blazars and a total of 46 intermediate - synchrotron peaked ( isp ) blazars . roughly half of the latter are likely to emit vhegrs as indicated by their flat spectral index between 0.1 and 100 gev ( @xmath260 ; see the spectral index distribution of figure 14 in @xcite ) . of these potential 141 tev blazars , only 22 have also been coincidentally identified as tev sources , whereas there are a total of 33 known tev blazars ( 29 hbl , 4 ibl ) . if these 141 sources are all @xmath261 emitters , but have not been detected due to incomplete sky coverage of current tev instruments , then the selection factor is @xmath262 . in addition , the duty cycle of coincident @xmath259 and @xmath261 emission is @xmath263 .. ] finally , by excluding the galactic plane for galactic latitudes @xmath264 , this is an underestimate by roughly @xmath265 . we make a rough attempt to correct for the sensitivity limit of the tev instruments , inferring a flux limit by fitting the upper - envelope of the tev - blazar flux luminosity distance distribution . somewhat surprisingly , despite the heterogeneous nature of the tev observations , are remarkably well described by a single flux limit : @xmath266 . this process and the associated limit are shown explicitly in the inset of figure [ fig : blf ] . from this we obtain a maximum redshift , and therefore comoving volume , associated with each luminosity . we then construct @xmath229 by counting the number of tev blazars per unit @xmath267 in each logarithmic luminosity bin and dividing by the comoving volume . the resulting @xmath229 , weighted by luminosity ( and therefore showing the luminosity density in comoving units ) , is shown in figure [ fig : blf ] . it peaks at @xmath268 , implying that , as expected , these objects are systematically dimmer than most other agn , and exhibits the broken - power law shape typical of agn luminosity functions . more importantly , despite a handful of sources with redshifts @xmath269 , the objects in table [ tab : tevsources ] are all nearby , and therefore our @xmath229 corresponds to that in the local universe , i.e. , @xmath2 . based upon this , the inferred present - day tev - blazar luminosity density is roughly @xmath270 . also shown in figure [ fig : blf ] is the @xmath238 obtained by @xcite . after rescaling @xmath238 to lower luminosities ( @xmath240 ) and lower luminosity densities ( @xmath239 ) , it provides a remarkably good fit to our @xmath271 ( reduced-@xmath272 of 0.13 with 3 degrees of freedom ) , i.e. , we find @xmath273 where an explicit expression for @xmath225 is given in appendix [ app : qlf ] . while there is considerable uncertainty in @xmath271 , especially at low luminosities , it clearly does not fit @xmath238 at higher @xmath56 . this suggests two immediate conclusions : 1 . the bolometric output of tev blazars and quasars are regulated by similar mechanisms , presumably accretion , despite the large difference in luminosity and the details of the emission processes between the two populations . tev blazars and quasars are contemporaneous elements in a single agn distribution ; specifically , tev - blazar activity does not lag that of quasars . based upon the strong similarities between @xmath229 and @xmath225 , and the associated implications , we make the conservative _ theoretical _ assumption that the redshift evolution of the tev blazars follows that of quasars . that is , we suppose that equation ( [ eq : blfqlf ] ) holds at all @xmath56 . this implies that the integrated comoving tev - blazar isotropic - equivalent luminosity density is given by @xmath275 where the constant of proportionality , @xmath276 , is then set by the comparison between @xmath229 and @xmath225 at @xmath181 . as a consequence , in comoving units , the tev - blazar luminosity density would be roughly an order of magnitude larger at @xmath277 , @xmath278 . , though neither varies significantly with @xmath254 . the total number of objects is sensitive to the spectral index , and shown as a function of @xmath254 in the inset . different colors correspond to different normalizations between the @xmath49@xmath244 and @xmath279@xmath49 luminosities : @xmath280 ( red ) , 1.6 ( green ) , and 3.1 ( blue ) , i.e. , the former is the latter luminosity multiplied by @xmath281 ) . for reference , the normalized number of hard gamma - ray blazars ( hsps and half of the isps ) observed by are shown by the black circles , and the black square in the inset shows the average spectral index and total number observed , with horizontal error bars giving the @xmath282-@xmath283 range . ] the redshift distribution of the bl lac sample ( i.e. , the first lat catalog , 1lac ; * ? ? ? * ) is peaked at @xmath284 and falls rapidly thereafter . inasmuch as the hsp and isp counts directly probes the low-@xmath56 @xmath229 , it may appear that they are inconsistent with the rapidly evolving @xmath229 described in the previous section . indeed , an analysis of the first three months of observations , suggested that the bl lac population does not grow substantially with redshift @xcite . however , as seen in figure [ fig : fhblc ] , this is not necessarily the case . assuming , as we have , that the number of tev blazars is equal to the the number of hard gamma - ray blazars , corresponding to isps with @xmath260 ( comprising roughly half of that class ) and hsps , the expected number observed inside of a given redshift is : @xmath285 where @xmath286 $ ] and @xmath287 are the proper and angular diameter distances , respectively , @xmath288 is the intrinsic upper - cutoff of the tev blazar isotropic - equivalent luminosity function , and @xmath289^{2 } \,\erg\,\s^{-1 } \end{aligned}\ ] ] is lower - limit set by the flux limit ( see figure 23 of * ? ? ? * and surrounding discussion ) . the factor @xmath281 is a correction relating the @xmath49@xmath244 isotropic - equivalent luminosities we employ to define @xmath229 to the @xmath279@xmath49 luminosities used by to define the flux limit . the dependence upon @xmath254 arises from the limited spectral coverage of both definitions , though here we fix @xmath290 for all objects based upon the sources that dominate the tev flux at earth . note that since the @xmath225 from @xcite diverges at small @xmath228 , the lower - luminosity cutoff is critical to getting both the total number of tev blazars and the shape of their redshift distribution correct . the resulting @xmath291 is shown in figure [ fig : fhblc ] for three different choices of @xmath281 : @xmath292 , @xmath293 , and @xmath294 , of which @xmath295 is most similar to the 1lac hard gamma - ray blazars . generally , our @xmath229 does an excellent job of reproducing the overall number of 1lac hard gamma - ray blazars and the dominance of nearby objects in their redshift distribution . this is despite the strong redshift evolution of @xmath229 implied by its relationship with @xmath225 . the reason for this is the flat distribution at low @xmath228 , the steep drop - off at high @xmath228 ( a result of which is that the shape is only marginally sensitive to the cutoff at @xmath296 ) and the rapidly growing @xmath297 due to the fixed flux limit . however , we note that the comparison between @xmath229 and the 1lac hard gamma - ray blazar statistics assumes that those with measured redshifts , comprising roughly half of the 1lac hard gamma - ray blazar sample , are representative of the hard gamma - ray blazar population as a whole . this appears not to be the case ; as we discuss in detail in appendix [ app : fhspzs ] , it is clear that the 1lac hsps with and without measured redshifts are not drawn from the same underlying @xmath254-distribution . based upon a similar analysis for 1lac bl lacs generally , @xcite argued that the objects without redshifts are more consistent with @xmath298 population . however , this conclusion does not extend to hsps , for which there are only three sources in the clean 1lac hsp sample with @xmath298 . rather , in appendix [ app : fhspzs ] , we show that the 1lac hsps without redshifts are distributed both in spectral index and flux much more similarly with nearby hsps ( @xmath299 ) than more distant hsps ( @xmath300 ) . if this is because these sources are intrinsically under - luminous , nearby objects , it could marginally improve the already remarkable comparison between the number of objects implied by our @xmath229 and those observed . however , because the number of predicted 1lac hard gamma - ray blazars is strongly dependent upon the flux limit , even a marginal increase in either @xmath229 at low luminosities or the effective flux limit results in a @xmath301 that is more strongly weighted at low @xmath56 , potentially allowing even more dramatically evolving luminosity functions . for this reason we conclude that our @xmath229 is broadly consistent with the 1lac hard gamma - ray blazar distribution , and that does not , at present , exclude a rapidly evolving @xmath229 in the recent past . , roughly the set for which can no - longer resolve them . the dashed lines show the contribution from all tev blazars , including those which should have been able to detect individually and to remove . the dotted lines show the contribution from all tev blazars when the annihilation upon the ebl is neglected . different curves correspond to different parameters of the intrinsic spectrum assumed for the blazar populations . top : variations in the low - energy spectral index ( top to bottom , @xmath302 , @xmath303 , @xmath304 ) . middle : variations in the break energy ( top to bottom , @xmath305 , @xmath133 , @xmath306 ) . bottom : variations in the high - energy spectral index ( top to bottom , @xmath307 , @xmath308 , @xmath309 ) . unless otherwise specified , the remaining parameters are @xmath310 , @xmath311 , and @xmath290 . finally , the measurement of the egrb reported in @xcite is shown by the blue points . note that in our scenario , the integral of the difference between the unabsorbed ( dotted ) and absorbed ( dashed ) energy fluxes , which dominates the total energy budget of these sources , has been dissipated into heat within the igm . , title="fig : " ] + , roughly the set for which can no - longer resolve them . the dashed lines show the contribution from all tev blazars , including those which should have been able to detect individually and to remove . the dotted lines show the contribution from all tev blazars when the annihilation upon the ebl is neglected . different curves correspond to different parameters of the intrinsic spectrum assumed for the blazar populations . top : variations in the low - energy spectral index ( top to bottom , @xmath302 , @xmath303 , @xmath304 ) . middle : variations in the break energy ( top to bottom , @xmath305 , @xmath133 , @xmath306 ) . bottom : variations in the high - energy spectral index ( top to bottom , @xmath307 , @xmath308 , @xmath309 ) . unless otherwise specified , the remaining parameters are @xmath310 , @xmath311 , and @xmath290 . finally , the measurement of the egrb reported in @xcite is shown by the blue points . note that in our scenario , the integral of the difference between the unabsorbed ( dotted ) and absorbed ( dashed ) energy fluxes , which dominates the total energy budget of these sources , has been dissipated into heat within the igm . , title="fig : " ] + , roughly the set for which can no - longer resolve them . the dashed lines show the contribution from all tev blazars , including those which should have been able to detect individually and to remove . the dotted lines show the contribution from all tev blazars when the annihilation upon the ebl is neglected . different curves correspond to different parameters of the intrinsic spectrum assumed for the blazar populations . top : variations in the low - energy spectral index ( top to bottom , @xmath302 , @xmath303 , @xmath304 ) . middle : variations in the break energy ( top to bottom , @xmath305 , @xmath133 , @xmath306 ) . bottom : variations in the high - energy spectral index ( top to bottom , @xmath307 , @xmath308 , @xmath309 ) . unless otherwise specified , the remaining parameters are @xmath310 , @xmath311 , and @xmath290 . finally , the measurement of the egrb reported in @xcite is shown by the blue points . note that in our scenario , the integral of the difference between the unabsorbed ( dotted ) and absorbed ( dashed ) energy fluxes , which dominates the total energy budget of these sources , has been dissipated into heat within the igm . , title="fig : " ] while the statistics of the 1lac hard gamma - ray blazar sample probes the evolution of @xmath229 at @xmath39 , the egrb is sensitive to tev blazars at high @xmath56 . the contribution to the egrb from tev blazars given our @xmath229 is shown in figure [ fig : egrb ] . because iccs are suppressed , computing the egrb flux requires only summing the individual intrinsic @xmath259 spectra of the tev blazars . for simplicity , to estimate the tev blazar contribution to the egrb we assume all tev blazars have identical intrinsic spectra , given by a broken power law , @xmath312 where the normalization is set such that the @xmath313 ( i.e. , @xmath314 , corresponding to the luminosity used to define @xmath229 , and thus determine @xmath315 ) , @xmath316 is the energy of the spectral break , and @xmath317 are the low and high - energy spectral indexes , respectively . was determined assuming an unbroken spectrum , and thus in this case would _ over estimate _ the total flux from the tev blazars . nevertheless , here this is not accounted for , i.e. , we renormalize the broken power - law spectrum to our previous power - law estimate of the tev blazar luminosity density . as a consequence , our estimates of the egrb are also _ over estimates_. ] with this , after performing the integral over @xmath228 , the egrb flux is then @xmath318}\\ & = \frac{\eta_b}{4\pi } \int_z^\infty\ ! dz ' \frac{c\tilde{\lambda}_q(z')}{h(z')(1+z')^5 } \,\hat{f}_{e'}\,{{\rm e}}^{-\tau_e[e',z']}\ , , \end{aligned}\ ] ] where @xmath319 , and @xmath320 is the physical quasar luminosity density . we choose fiducial values for the spectral parameters of @xmath321 , @xmath311 , and @xmath290 , typical of the tev blazars ( see table [ tab : tevsources ] and * ? ? ? finally , since resolves individual gamma - ray blazars with isotropic - equivalent luminosities @xmath322 , roughly the location of the peak in @xmath229 , for @xmath323 , we construct the egrb from sources at larger redshift . if @xmath316 is sufficiently high ( @xmath324 ) the egrb is remarkably insensitive to the particular values of @xmath316 and @xmath254 ( see the bottom two panels of figure [ fig : egrb ] ) . this is a direct result of the annihilation of the vhegrs upon the ebl , effectively removing the relevant portion of the blazar sed from the egrb . more important is the low - energy spectral index . however , for values of @xmath325 that are consistent with the comptonization models for the vhegr emission , the anticipated egrb is consistent with the result . thus , we find that despite our dramatically evolving @xmath229 , we are are able to satisfy the limits imposed by the egrb . moreover , for reasonable spectral - parameter values it is possible to accurately reproduce both the magnitude and shape of the high - energy egrb ( i.e. , at energies @xmath326 ) . at first this may appear in conflict with other studies that have performed more sophisticated analyses of the gamma - ray emission from blazars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) that have typically found that such a strongly evolving @xmath229 over - produces the egrb . however , this is not the case for three reasons . first , the agn populations typically considered in computations of the egrb include the fsrqs and lbls , which dominate at low energies . the tev blazars of interest here are only the high - energy tail of the blazar distribution , and it is these alone that we fix to the quasar luminosity function ( indeed , it is for only these objects that the arguments surrounding figure [ fig : blf ] have been made ) . moreover , the tev blazars themselves are generally dimmer , both bolometrically and within the lat energy band ( @xmath48@xmath49 ) , than the lbls and fsrqs , which have typical isotropic - equivalent luminosities of @xmath327 @xcite . second , and perhaps most importantly , the suppression of the iccs means that the vhegr emission is lost to heat in the igm , not reprocessed to below @xmath49 . since the tev blazar flux is presumed to be dominated by emission above @xmath49 , the non - radiative dissipation of this emission substantially reduces the impact of the tev blazars upon the egrb . for all of the spectral parameters we considered the un - absorbed fluxes exceed the @xmath328 egrb by a significant margin , implying that the suppression of the iccs is crucial to bringing the tev blazar contributions in line with the egrb . thus , the lack of the iccs generally appears necessary to reconcile the blazar and quasar luminosity functions . third , the vhegr spectra from the most luminous tev blazars necessarily peaks near or above tev energies . for a number of the sources in table [ tab : tevsources ] , the intrinsic vhegr spectrum ( adjusted for annihilation on the ebl ) is inverted , indicating that this break is _ above _ a tev . the brightest tev blazar in table [ tab : tevsources ] , mkn 421 , has an inverted spectrum below @xmath49 @xcite , implying @xmath329 . the second brightest , 1es 1959 + 650 , is consistent with being flat below @xmath49 , and appears to peak near @xmath133 when in the high - state @xcite . similarly , 1es 2344 + 514 peaks near @xmath330 @xcite and mkn 501 is inverted below @xmath49 , implying a peak above that value @xcite . thus the sources likely to dominate the vhegr background , and by extension the egrb , all turn over near @xmath49@xmath133 . this is expected if the the vhegr emission arises from inverse - compton scattering the synchrotron bump ( see , e.g. , * ? ? ? * and references therein ) . the consequence of the spectral break is the suppression of the contribution from the tev blazars to the egrb below @xmath316 . were the tev blazar contribution to the egrb dominated by sources with @xmath331 it would exceed the egrb at @xmath332 . a more complete analysis of the egrb should include a variety of spectra , including a distribution of break energies , smoothly connecting the hbl , ibl and lbl populations . however , this is beyond the scope of this paper , which is primarily concerned with the fate of the vhegr emission , absent in the lbls and fsrqs . nevertheless a recent comprehensive model , which includes the contributions of the fsrqs and bl lacs observed by , as well as starburst galaxies , has had considerable success fitting the egrb with an even more extremely evolving @xmath229 ; fixing it to the luminosity function of radio galaxies @xcite . of particular relevance here is that @xcite and @xcite explicitly ignored the potential contributions of iccs . unlike their analysis , however , we find that the tev blazars are capable of reaching the highest bands despite the annihilation upon the ebl . the cold , highly anisotropic beams of ultra - relativistic @xmath63 pairs produced by the annihilation of vhegrs upon the ebl are unstable to plasma beam instabilities . more importantly , for a wide range of parameters relevant for the observed tev blazars these instabilities may be capable of isotropizing , and potentially extracting the kinetic energy of , the pairs at a rate orders of magnitude faster than inverse - compton scattering . this has far reaching consequences for efforts to constrain the igmf using empirical limits upon the gev emission from known tev sources . typically , @xmath333 after the onset of tev emission the pair beam density has grown sufficiently for plasma beam instabilities to dominate its evolution , randomize the beam , and potentially suppress the inverse - compton signal upon which the igmf limits are based . note that due to the beam disruption by the instabilities , this occurs even if the plasma instabilities do not ultimately cool the pairs . as a consequence , the present constraints upon the igmf , obtained by the non - observation of an inverse - compton gev bump in the spectra of bright tev blazars are inherently unreliable . nevertheless , the sudden appearance of a tev - bright blazar or intrinsically transient sources ( e.g. , grbs ) provide a means to temporarily avoid the consequences of plasma beam instabilities during the growth of the pair beam . alternatively , observing particularly dim sources , @xmath334 , limits the beam density directly , again avoiding the complications imposed by plasma processes . finally , the presence of these plasma instabilities in the pair beams of tev blazars , which manifest themselves through their impact on the igm ( see paper ii , paper iii , and @xcite ) , implies the most stringent upper limit to date on the igmf : @xmath222 . if the plasma instabilities can efficiently convert the pair beam kinetic energy into heat in the igm , as we anticipate based upon existing numerical simulations and the arguments in section [ sec : nls ] , they would necessarily suppress the development of iccs , and thus prevent the reprocessing of the vhegr emission from bright sources to gev energies . the lack of iccs , independent of the mechanism that facilitates the local dissipation of the pair kinetic energy , would greatly weaken the constraints upon the evolution of the blazar population derived from the unresolved egrb measured by . by introducing a spectral break near @xmath133 and eliminating the reprocessed vhegr emission , we find that the egrb is consistent with a tev blazar ( and therefore presumably hard gamma - ray blazar ) luminosity function fixed to that of quasars , normalized by comparing objects in the local universe ( @xmath45 ) , and motivated by the remarkable similarity between them in the local universe . this conclusion is relatively insensitive to the particular parameters governing the vhegr spectra of the tev blazars , requiring only that the vhegr emission is produced via inverse - compton scattering . for a wide range of spectral parameters , we are able to match the magnitude and shape of the egrb at high energies , with the low - energy component presumably arising from the fsrqs and lbls . this @xmath229 , and perhaps an even more rapidly evolving luminosity function , is also consistent with the observed redshift distribution of the 1lac hsp and hard - isp sample . matching the high - energy egrb ( above @xmath335 ) requires more tev blazars than are currently observed , though a comparable number to those inferred once the sky - coverage and gev duty - cycle completeness corrections are included . based upon these factors , and our rough estimate of the egrb , we predict that upcoming surveys performed with next - generation cerenkov telescope arrays ( see , e.g. , * ? ? ? * ) should find roughly @xmath336 sources above @xmath337 ( our estimate of the effective flux limit of current imaging cerenkov telescopes ) , and a handful of additional sources comparable to the brightest tev blazars observed . in addition , based upon the current number of known tev blazars and our estimate of @xmath274 , the improved anticipated sensitivities of these instruments , @xmath338@xmath339 time larger than current arrays , should result in the detection of @xmath340@xmath341 additional tev blazars , with median luminosities @xmath342 . these should allow more precise estimates for their gamma - ray seds and a better characterization of @xmath274 , especially for low - luminosity objects . unlike inverse - compton cooling , the plasma beam instabilities deposit the energy locally , heating the igm . moreover , the homogeneity of the ebl , and the weak dependence of the plasma cooling rates upon the igm density , result in a uniform _ volumetric _ heating , in clear contrast to either photoionization heating or mechanical feedback from agn . while we shall defer a detailed discussion of the consequences of this heating to papers ii , iii , and @xcite here we note that this unusual heating prescription naturally explains a number of heretofore outstanding questions , including the inverted equation - of - state ( temperature - density relation ) for low density regions in the igm ( paper ii ) , the suppression of dwarf galaxies and their histories , the segregation of galaxy clusters and groups into cool core and non - cool core populations ( paper iii ) , and the quantitative properties of the high - redshift forest @xcite . as a consequence , despite the fact that our estimates of the plasma cooling rates are limited to the linear regime ( though with some numerical support ) , there are a variety of observational reasons to believe that plasma cooling , or an analogous mechanism , does in fact dominate the evolution of the ultra - relativistic pair beams . we thank tom abel , marco ajello , marcelo alvarez , arif babul , roger blandford , james bolton , mike boylan - kolchin , luigi costamante , andrei gruzinov , peter goldreich , martin haehnelt , andrey kravtsov , ue - li pen , ewald puchwein , volker springel , chris thompson , matteo viel , marc voit , and risa wechsler for useful discussions . we are indebted to peng oh for his encouragement and useful suggestions . we thank steve furlanetto for kindly providing technical expertise . these computations were performed on the sunnyvale cluster at cita . a.e.b . and p.c . are supported by cita . gratefully acknowledges the support of the beatrice d. tremaine fellowship . c.p . gratefully acknowledges financial support of the klaus tschira foundation and would furthermore like to thank kitp for their hospitality during the galaxy cluster workshop . this research was supported in part by the national science foundation under grant no . nsf phy05 - 51164 . here we compute the growth rates for the various plasma instabilities discussed in the text within the context of relativistically moving pair plasmas . in all cases we make use of the kinetic theory description of the underlying plasmas . this necessarily assumes that the plasma , and in particular the beam plasma , is sufficiently dense that it is well described by a distribution function on the relevant scales . this is equivalent to requiring that many particles within a characteristic energy range be present on the plasma scale of the igm , i.e. , the conditions laid out in section [ sec : aflupcr ] . in our analysis the vlasov equation will play a central role , which owing to the relativistic nature of the calculation , we express in terms of the canonical pair , @xmath343 and @xmath344 , making use of the lorentz invariance of the electron / positron distribution functions @xmath345 and the phase - space volume element @xmath346 . in what follows we set @xmath347 unless otherwise specified . the two - stream instability arises due to the excitation of negative - energy electrostatic waves in the beam and target plasmas . these waves carry away both the energy and momentum of the beam . specifically , we compute the growth rate of the electrostatic wave moving in the direction of the beam in the absence of a background magnetic field . in this case , the vlasov equations for the electrons and positrons are : @xmath348 where @xmath349 is the stokes derivative and @xmath350 is the net electric field . linearizing these and fourier transforming in @xmath351 and @xmath343 gives @xmath352 where @xmath353 are the perturbations to the electron and positron distribution functions , @xmath354 is the electrostatic wave field and we have assumed that the background distributions , @xmath355 , are isotropic . as a result we have @xmath356 at this point we may compute the dielectric tensor associated with the plasma response , however it will suffice to consider gauss s law . thus we now compute the perturbed charge density : @xmath357 it is now necessary to specify the @xmath355 . we idealize the target ionic plasma as cold and the pair beam plasma as mono - energetic , yielding @xmath358 where @xmath359 and @xmath110 are the lepton number densities in the target and beam , respectively . after performing the trivial integrals we then obtain @xmath360 were in the final equality we used the fact that @xmath361 . from gauss s law we have @xmath362 , and therefore @xmath363 where @xmath364 and @xmath365 are the plasma frequencies associated with the target and beam plasmas . this explicitly provides the dispersion relation , quadratic in @xmath366 ( one electrostatic wave traveling in each direction for each plasma ) . when @xmath367 , @xmath368 . when @xmath369 , as is the case of interest here , we may solve the dispersion relation perturbatively . we do this by setting @xmath370 , with @xmath371 , which gives : @xmath372 which is no longer independent of @xmath373 . when @xmath374 , @xmath375 is real and thus there is no instability . on the other hand , where @xmath376 , we have @xmath377 and therefore @xmath378 . this has three solutions : @xmath379 the first of which is oscillatory , the second is decaying and the third is growing with timescale @xmath380 this differs from that associated with non - relativistic , ionic beams only by the factor of @xmath381 , arising due to time - dilation within the beam . note that since the energy within the electrostatic wave is proportional to @xmath382 , the rate at which energy is removed from the beam is @xmath383 . the relativistic version of the weibel instability has been discussed in some detail in the literature for the case of anisotropic , though symmetric beams @xcite . here we consider a the case of a dilute pair beam incident upon a much denser target plasma . this is essentially identical to the two - stream instability discussed previously , though coupling instead to an electromagnetic mode with @xmath384 . again we begin with the linearized vlasov equations for the electrons and positrons : @xmath385 where we have @xmath386 from faraday s law . fourier transforming in @xmath351 and @xmath343 and solving for @xmath353 gives @xmath387}{\omega-\k\cdot\v } \cdot\frac{\partial f^\mp_0}{\partial\p}\\ & = \pm \frac{ie}{\omega } \left(\e_1 + \frac{\v\cdot\e_1}{\omega-\k\cdot\v}\k\right ) \cdot\frac{\partial f^\mp_0}{\partial\p}\ , . \end{aligned}\ ] ] the associated perturbation to the current is @xmath388 \cdot \e_1 d^3\!p\ , , \end{aligned } \label{eq : wj1}\ ] ] in which we ve defined @xmath389 . choosing the @xmath390 given by equation ( [ eq : tsfpm0 ] ) and assuming @xmath391 we recover the standard two - stream instability . for our purposes here , however , the computations may be substantially simplified by boosting into a frame in which the target plasma is not at rest . in this frame we have @xmath246 given by , @xmath392 with @xmath393 , where we will take care at the end to properly relate all of these quantities to their target - frame analogs . in particular , we choose this frame such that @xmath394 , where @xmath395 and @xmath396 are the proper densities of the target and beam plasmas , respectively . if @xmath397 is the beam plasma velocity in the target ( or lab ) frame , this gives @xmath398 where @xmath399 and is in our case much less than unity . note that this is only the center of momentum frame if @xmath400 ( for which @xmath401 , the case most commonly discussed ) . if we choose @xmath402 this choice of frame causes the terms linear in @xmath403 in equation ( [ eq : wj1 ] ) to vanish identically , yielding @xmath404\cdot\e_1'\,.\ ] ] using the inhomogeneous wave equation , obtained from faraday s and ampere s law , @xmath405 and choosing @xmath406 produces the dispersion relation @xmath407 at this point we make use of the fact that in our case @xmath408 , which allows a perturbative solution of equation ( [ eq : vtp ] ) : @xmath409 and therefore @xmath410 . furthermore , @xmath411 and @xmath412 for the geometry under consideration . as consequence , @xmath413 and @xmath414 , which implies that @xmath415 this has a purely imaginary solution : @xmath416^{1/2 } - 1 } \,.\ ] ] for @xmath417 this rises linearly with @xmath373 , saturating for @xmath418 , giving the growth rate @xmath419 given that for our application @xmath408 , we will drop terms first order in @xmath420 as well . note that the plasma frequency that enters into the growth rate is that of the _ beam _ , which is much lower than that associated with the target plasma . nevertheless , the scale of the rapidly growing perturbations is limited to that associated with the _ target _ plasma , which are general much smaller than that associated with the perturbations in the beam alone . again , the rate at which energy is sapped from the beam is then @xmath421 . in the interests of completeness , here we reproduce the @xmath225 from @xcite , corresponding to the `` full '' case in that paper , that we employ . see @xcite for how this @xmath225 was obtained , and caveats regarding its application . cccccccc @xmath422 & @xmath423 & @xmath424 & @xmath425 & @xmath426 & @xmath427 & @xmath428 & @xmath429 + & & @xmath430 & @xmath431 & @xmath432 & @xmath433 & @xmath434 & @xmath435 + & & @xmath436 & @xmath437 & & & @xmath438 & @xmath439 + & & @xmath440 & @xmath441 & & & & + the form of @xmath225 is assumed to be a broken power law : @xmath442^{\gamma_1(z)}+[l / l_*(z)]^{\gamma_2(z)}}\,,\ ] ] where the location of the break ( @xmath443 ) and the power laws ( @xmath444 and @xmath445 ) are functions of redshift . these are given by , @xmath446 where @xmath447 the values of the relevant parameters are given in table [ tab : qlf ] , and where any estimate of the uncertainty is made we assume these are independently , normally distributed . ( red ) , hsps with measured redshifts @xmath448 ( blue ) , and hsps without redshift measurements ( black ) . in the bottom panel , error bars denote the poisson uncertainty only . ] and different line colors correspond to different subpopulations of the fermi hsp sample : all hsps with measured redshifts ( green ) , hsps with measured redshifts @xmath449 ( red ) , hsps with measured redshifts @xmath448 ( blue ) , and hsps without redshift measurements ( black ) . in the bottom panel , error bars denote the poisson uncertainty only . ] ( ranging between @xmath450 and @xmath451 , i.e. , large @xmath56 is denoted by large points and small @xmath56 by small points ) . hsps without redshifts are shown by the blue triangles . for reference , the dotted black lines show lines of constant redshift for a source with luminosity @xmath235 between @xmath279 and @xmath49 ( the definition of @xmath452 in * ? ? ? * note the difference with the definition of @xmath453 ) . ] lcccc all hsps with @xmath56 s & @xmath454 & @xmath455 & @xmath456 & @xmath457 + hsps with @xmath300 & @xmath458 & @xmath459 & @xmath460 & @xmath461 + hsps with @xmath299 & @xmath462 & @xmath455 & @xmath463 & @xmath464 + hsps without @xmath56 s & & @xmath465 & & @xmath466 + of the 118 hsps detected by fermi and reported in the first lat agn catalog ( 1lac , * ? ? ? * , there called hsps ) , 113 are members of the clean sample ( meaning that there are no ambiguities surrounding their detection ) , and of these only 65 have measured redshifts . this is a somewhat larger fraction that for 1lac bl lacs generally , though this leaves nearly half of the hsp population with their redshifts undetermined . based upon comparisons between the spectral index and flux distributions between the hsps without redshifts and those at various redshift ranges , here we argue that these are likely to be located nearby . already , based upon comparisons between the spectral index distributions ( sids ) of the 1lac bl lacs at large , it is clear that the objects with and without measured redshifts are not drawn from the same underlying population @xcite . similarities between the sids of the unknown-@xmath56 objects and the @xmath298 subset of those with measured redshifts , @xcite has suggested that the bl lacs without redshifts may be biased towards higher redshifts . however , we note that there are only three hsps with @xmath298 , and thus this conclusion is relevant for isps and lsps only . the sids of the unknown-@xmath56 hsps is shown in figure [ fig : hspalpha ] , compared with the sids of hsps with redshifts ( cf . figure 22 of * ? ? ? the kolmogorov - smirnov ( ks ) probability that these are drawn from the same parent population is @xmath454 , indicating that this is unlikely at the 2-@xmath283 level . in addition , we show the sids of hsps with @xmath299 ( blue line ) and @xmath300 ( red line ) , with corresponding ks probabilities of @xmath462 and @xmath458 , respectively ( these are collected in table [ tab : kshsp ] ) . thus , in contrast to the 1lac bl lac sample at large , the hsps without redshifts appear to have sids that are strongly inconsistent with the population observed to have high redshifts , and indistinguishable from those at low redshifts . nevertheless , the hsps with unknown-@xmath56s still tend to be softer than those with measured redshifts in either range . a similar analysis may be performed upon the reported flux measure , @xmath453 , corresponding to the flux between @xmath467 and @xmath49 . figure [ fig : hspflux ] shows the flux distributions ( fds ) of the unknown-@xmath56 , all measured @xmath56 , @xmath300 , and @xmath299 hsp populations , and is analogous to figure [ fig : hspalpha ] . as before , the hsps without redshifts have fds that are marginally inconsistent with being drawn from the same parent population as the complete set of hsps with measured redshifts , having a ks probability of @xmath456 . the inconsistency with the high-@xmath56 hsp fd is even more striking than for the sids , with a ks probability less than @xmath460 , i.e. , that the high-@xmath56 fd of the unknown-@xmath56 and @xmath300 hsps are from the same distribution is excluded . however , again , we find that the fds of the low-@xmath56 and unknown-@xmath56 populations are indistinguishable , having a ks probability of @xmath463 . that the sids and fds both favor a relationship between the unknown-@xmath56 and low-@xmath56 hsp populations provides some confidence that this may , in fact , be the case . this is supported by the redshift distribution in the @xmath254-@xmath453 plane , depicted in figure [ fig : hspfaz ] . hsps with high redshifts are clustered at low fluxes and hard @xmath254 . this is not unexpected given an upper limit upon the luminosity of hsps . the dotted lines in figure [ fig : hspfaz ] show constant redshift curves in the @xmath453@xmath254 plane for a @xmath279@xmath49 luminosity of @xmath235 ( the definition of @xmath452 in * ? ? ? * note the distinction with the definition of @xmath453 ) . if all hsps have intrinsic luminosities less than @xmath235 , no sources at a given @xmath56 should be found to the right of the associated line . since the volume of the visible universe is dominated by @xmath50 , the majority of high-@xmath56 objects should then be found up against the instrumental flux limit , i.e. , at low fluxes and nearly flat spectra . sources with harder spectra are likely to have higher bolometric gamma - ray luminosities since they are likely to be more below the inverse - compton peak , biasing high-@xmath56 sources towards harder spectra . however , the hsps without redshifts are noticeably absent within this high-@xmath56 dominated region . this is responsible for the fact that they more closely share their sid and fd with the low-@xmath56 hsps . the presence of hsps without redshifts at a variety of @xmath254s and @xmath453s suggests that this is not a result of a selection effect upon either . it is nonetheless possible that there is some instrumental effect which prevents the measurement of high redshifts in _ intrinsically _ bright , soft objects , in which case a population of high - luminosity hsps with high ( and therefore unmeasured ) @xmath56s would necessarily be located in regions with predominantly low measured @xmath56s . however , this is belied by the broad distribution of luminosities and spectral indexes among the hsps with measured redshifts . thus , we adopt the simpler explanation : the hsps with unknown redshifts have and sid and fd similar to low-@xmath56 objects because they are intrinsically dim , nearby objects . , m. , et al . 2010 , in astronomical society of the pacific conference series , vol . 427 , astronomical society of the pacific conference series , ed . l. maraschi , g. ghisellini , r. della ceca , & f. tavecchio , 302

inverse - compton cascades initiated by energetic gamma rays ( @xmath0 ) enhance the gev emission from bright , extragalactic tev sources . the absence of this emission from bright tev blazars has been used to constrain the intergalactic magnetic field ( igmf ) , and the stringent limits placed upon the unresolved extragalactic gamma - ray background ( egrb ) by has been used to argue against a large number of such objects at high redshifts . however , these are predicated upon the assumption that inverse - compton scattering is the primary energy - loss mechanism for the ultra - relativistic pairs produced by the annihilation of the energetic gamma rays on extragalactic background light photons . here we show that for sufficiently bright tev sources ( isotropic - equivalent luminosities @xmath1 ) plasma beam instabilities , specifically the `` oblique '' instability , present a plausible mechanism by which the energy of these pairs can be dissipated locally , heating the intergalactic medium . since these instabilities typically grow on timescales short in comparison to the inverse - compton cooling rate , they necessarily suppress the inverse - compton cascades . as a consequence , this places a severe constraint upon efforts to limit the igmf from the lack of a discernible gev bump in tev sources . similarly , it considerably weakens the limits upon the evolution of blazar populations . specifically , we construct a tev - blazar luminosity function from those objects presently observed and find that it is very well described by the quasar luminosity function at @xmath2 , shifted to lower luminosities and number densities , suggesting that both classes of sources are regulated by similar processes . extending this relationship to higher redshifts , we show that the magnitude and shape of the egrb above @xmath3 is naturally reproduced with this particular example of a rapidly evolving tev - blazar luminosity function .

we say a geodesic metric space @xmath1 is _ gromov hyperbolic _ , or _ @xmath2-hyperbolic _ , if there is a number @xmath3 for which every geodesic triangle in @xmath1 satisfies the _ @xmath2-slim triangle _ condition , i.e. any side is contained in a @xmath2-neighbourhood of the other two sides . throughout this paper we will assume that the space @xmath1 is separable , i.e. it contains a countable dense set , but we will not assume that @xmath1 is proper or locally compact . for example , any countable simplicial complex satisfies these conditions . let @xmath0 be a countable group which acts by isometries on @xmath1 . we say the action of @xmath0 on @xmath1 is _ non - elementary _ if @xmath0 contains a pair of hyperbolic isometries with disjoint fixed points in the gromov boundary . we say @xmath0 is _ weakly hyperbolic _ if it admits a non - elementary action by isometries on some gromov hyperbolic space @xmath1 . in this case , a natural boundary for the group is given by the gromov boundary @xmath4 of @xmath1 , which however need not be compact . several widely studied group actions are weakly hyperbolic in this sense , in particular : * hyperbolic and relatively hyperbolic groups ; * mapping class groups , acting on the curve complex ; * out(@xmath5 ) acts on various gromov hyperbolic simplicial complexes , for example the complex of free factors or the complex of free splittings ; * right - angled artin groups acting on their extension graphs ; * finitely generated subgroups of the cremona group . in particular , all acylindrically hyperbolic groups are weakly hyperbolic , see section [ section : discussion ] for further discussion and more examples . in this paper , we shall consider random walks on weakly hyperbolic groups , constructed by choosing products of random group elements . a probability distribution @xmath6 on @xmath0 determines a random walk on @xmath0 , by taking the product @xmath7 where the @xmath8 are independent identically distributed elements of @xmath0 , with distribution @xmath6 . a choice of basepoint @xmath9 determines an orbit map sending @xmath10 , and we can project the random walk on @xmath0 to @xmath1 by considering the sequence @xmath11 , which we call a _ sample path_. we say a measure @xmath6 on @xmath0 is _ non - elementary _ if the semi - group generated by its support is a non - elementary subgroup of @xmath0 . the first result we establish is that sample paths converge almost surely in the gromov boundary : [ theorem : convergence ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 . then , for any basepoint @xmath9 , almost every sample path @xmath12 converges to a point @xmath13 . the resulting hitting measure @xmath14 is non - atomic , and is the unique @xmath6-stationary measure on @xmath4 . we use the convergence to the boundary result to show the following linear progress , or positive drift , result . recall that a measure @xmath6 has finite first moment if @xmath15 . [ theorem : linear progress ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 , and @xmath9 a basepoint . then there is a constant @xmath16 such that for almost every sample path we have @xmath17 furthermore , if @xmath6 has finite first moment , then the limit @xmath18 exists almost surely . finally , if the support of @xmath6 is bounded in @xmath1 , then there are constants @xmath19 and @xmath16 such that @xmath20 for all @xmath21 . note that the constants @xmath22 , @xmath23 , and @xmath24 depend on the choice of the measure @xmath6 . moreover , the first statement also implies that @xmath25 ( for a possibly different constant @xmath22 ) . if we assume that @xmath6 has finite first moment with respect to the distance function @xmath26 , then we obtain the following geodesic tracking result . [ theorem : sublinear ] let @xmath0 be a countable group which acts by isometries on a separable gromov hyperbolic space @xmath1 with basepoint @xmath27 , and let @xmath6 be non - elementary probability distribution on @xmath0 , with finite first moment . then for almost every sample path @xmath28 there is a quasigeodesic ray @xmath29 which tracks the sample path sublinearly , i.e. @xmath30 if the support of @xmath6 is bounded in @xmath1 , then in fact the tracking is logarithmic , i.e. @xmath31 finally , we investigate the growth rate of translation length of group elements arising from the sample paths . [ theorem : translation ] let @xmath0 be a countable group which acts by isometries on a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 . then the translation length @xmath32 of the group element @xmath33 grows at least linearly in @xmath21 , i.e. @xmath34 for some constant @xmath22 strictly greater than zero . if the support of @xmath6 is bounded in @xmath1 , then there are constants @xmath19 and @xmath16 , such that @xmath35 for all @xmath21 . recall that the translation length @xmath36 of an isometry @xmath37 of @xmath1 is defined as @xmath38 as an element with non - zero translation length is a hyperbolic (= loxodromic ) isometry , this shows that the probability that a random walk of length @xmath21 gives rise to a hyperbolic isometry tends to one as @xmath21 tends to infinity . _ the poisson boundary of acylindrically hyperbolic groups . _ a special class of weakly hyperbolic groups are the _ acylindrically hyperbolic _ groups . in this case , we show that we may identify the gromov boundary @xmath39 with the poisson boundary . recall that a group @xmath0 acts acylindrically on a gromov hyperbolic space @xmath1 , if for every @xmath40 there are numbers @xmath41 and @xmath42 , which both depend on @xmath23 , such that for any pair of points @xmath43 and @xmath44 in @xmath1 , with @xmath45 , there are at most @xmath42 group elements @xmath37 in @xmath0 such that @xmath46 and @xmath47 . [ theorem : poisson ] let @xmath0 be a countable group of isometries which acts acylindrically on a separable gromov hyperbolic space @xmath1 , let @xmath6 be a non - elementary probability distribution on @xmath0 with finite entropy and finite logarithmic moment , and let @xmath14 be the hitting measure on @xmath48 . then @xmath49 is the poisson boundary of @xmath50 . _ word hyperbolic groups_. the simplest example of a weakly hyperbolic group is a ( finitely generated ) gromov hyperbolic group acting on its cayley graph , which by definition is a @xmath2-hyperbolic space , and so any non - elementary gromov hyperbolic group is weakly hyperbolic . convergence of sample paths to the gromov boundary in this case is due to kaimanovich @xcite , who also shows that the gromov boundary may be identified with the poisson boundary and the hitting measure is the unique @xmath6-stationary measure . note that in this case the space is locally compact , and the boundary is compact . _ relatively hyperbolic groups . _ the cayley graph of a relatively hyperbolic group is @xmath2-hyperbolic with respect to an infinite generating set , and so these groups are also weakly hyperbolic , but in this case the space on which the group acts need not be proper . in this case , convergence to the boundary was shown by gautero and mathus @xcite , who also covered the case of groups acting on @xmath51-trees . more recently , group actions on ( locally infinite ) trees have been considered by malyutin and svetlov @xcite . there are then groups which are weakly hyperbolic , but not relatively hyperbolic , the two most important examples being the mapping class groups of surfaces , and out(@xmath5 ) . _ mapping class groups . _ the mapping class group mod(@xmath52 ) of a surface @xmath52 of genus @xmath37 with @xmath53 punctures acts on the curve complex @xmath54 , which is a locally infinite simplicial complex . as shown by masur and minsky @xcite , the curve complex is @xmath2-hyperbolic , and moreover , the action is acylindrical , by work of bowditch @xcite , so we can apply our techniques to get convergence and the poisson boundary . convergence to the boundary of the curve complex also follows from work of kaimanovich and masur @xcite and klarreich @xcite , using the action of mod(@xmath52 ) on teichmller space ( which is locally compact , but not hyperbolic ) . indeed , kaimanovich and masur show that random walks on the mapping class group converge to points in thurston s compactification of teichmller space @xmath55 , and then klarreich ( see also hamenstdt @xcite ) shows the relation between @xmath55 and the boundary of the curve complex . our approach does not use fine properties of teichmller geometry . a third approach is to consider the action of mod(@xmath56 on teichmller space with the weil - petersson metric , which is a non - proper cat(0 ) space . by work of bestvina , bromberg and fujiwara @xcite , this space has finite telescopic dimension , and one may then apply the results of bader , duchesne and lcureux @xcite . we remark that @xmath54 does not possess a cat(0 ) metric , since it is homotopic to a wedge of spheres ( harer @xcite ) ; in fact , kapovich and leeb @xcite showed that the mapping class group ( of genus at least @xmath57 ) does not act freely cocompactly on a cat(0 ) space , though it is still open as to whether there is a proper cat(0 ) space on which the mapping class group acts by isometries ; bridson @xcite showed that any such action must have elliptic or parabolic dehn twists . _ out(@xmath5 ) . _ the outer automorphism group of a non - abelian free group , out(@xmath5 ) , acts on a number of distinct gromov hyperbolic spaces , as shown by bestvina and feighn @xcite and handel and mosher @xcite , and so is weakly hyperbolic . similarly to the case of mod(@xmath52 ) , convergence to the boundary also follows by considering the action of out(@xmath5 ) on the ( locally compact ) outer space , as shown by horbez @xcite . _ right - angled artin groups . _ a right - angled artin group acts by simplicial isometries on its extension graph , which has infinite diameter as long as the group does not split as a non - trivial direct product , and is not quasi - isometric to @xmath58 . kim and koberda showed that the extension graph is a ( non - locally compact ) quasi - tree @xcite , and in fact the action is acylindrical @xcite . _ finitely generated subgroups of the cremona group . _ manin @xcite showed that the cremona group acts faithfully by isometries on an infinite - dimensional hyperbolic space , known as the picard - manin space , which is not separable . however , any finite - generated subgroup preserves a totally geodesic closed subspace , which is separable , see for example delzant and py @xcite . _ acylindrically hyperbolic groups . _ the definition of an acylindrical group action is due to sela @xcite for trees , and bowditch @xcite for general metric spaces , see osin @xcite for a discussion and several examples of acylindrical actions on hyperbolic spaces . as every acylindrically hyperbolic group is also weakly hyperbolic , this gives a number of additional examples of weakly hyperbolic groups which are not necessarily relatively hyperbolic ; for example , all one relator groups with at least three generators , see @xcite for many other examples . _ isometries of cat(0 ) spaces . _ even though not all cat(0 ) spaces are hyperbolic , the two theories overlap in many cases . for isometries of general cat(0 ) spaces , karlsson and margulis @xcite proved boundary convergence and identified the poisson boundary . more recently , boundaries of cat(0 ) cube complexes have been studied by nevo and sageev @xcite , and ( not necessarily proper ) cat(0 ) spaces of finite telescopic dimension by bader , duchesne and lcureux @xcite . once we have proved convergence to the boundary , we apply this to show positive drift . in the locally compact case , positive drift results go back to guivarch @xcite . in particular , when the space on which @xmath0 acts is proper , positive drift follows from non - amenability of the group , but this need not be the case for non - proper spaces . in the curve complex case , linear progress is due to maher @xcite . we then show the sublinear tracking results , using work of tiozzo @xcite . sublinear tracking can be thought of as a generalization of oseledec s multiplicative ergodic theorem @xcite . in our context , these results go back to guivarch @xcite , and are known for groups of isometries of cat(0 ) spaces by karlsson and margulis @xcite , and for teichmller space by duchin @xcite . sublinear tracking on hyperbolic groups is due to kaimanovich @xcite ; moreover , karlsson and ledrappier @xcite proved a law of large numbers on general ( proper ) metric spaces using horofunctions . note that these results can be used to prove convergence to the boundary once one knows that the drift is positive . in the above - mentioned cases , the space is meant to be proper , so positive drift follows from non - amenability of the group , while a new argument is needed in general . in this paper we give an argument for the non - proper weakly hyperbolic case , where sublinear tracking and positive drift _ follow _ from convergence to the boundary . note that in our approach we use horofunctions , and indeed @xcite can be used to simplify our proofs if one assumes positive drift . recently ( after the first version of this paper appeared ) , mathieu and sisto @xcite provided a different argument for positive drift in the acylindrical case . logarithmic tracking was previously known for random walks on trees , due to ledrappier @xcite , on hyperbolic groups , due to blachre , hassinsky and mathieu @xcite , and on relatively hyperbolic groups , due to sisto @xcite . finally we show that the translation length grows linearly , which in particular shows that the probability that a random walk gives rise to a hyperbolic element tends to one . this generalizes earlier work of rivin @xcite , kowalski @xcite , maher @xcite and sisto @xcite . the methods in this paper build on previous work of calegari and maher @xcite , which showed convergence results with stronger conditions on @xmath1 and @xmath6 . to explain the argument in the proof of theorem [ theorem : convergence ] , we briefly remind the reader of the standard argument for convergence to the boundary for a random walk on a group @xmath0 acting on a locally compact @xmath2-hyperbolic space @xmath1 . the argument ultimately goes back to furstenberg @xcite , who developed it for lie groups . _ measures on the gromov boundary . _ let @xmath6 be the probability distribution on @xmath0 generating the random walk . the first step is to find a @xmath6-stationary measure @xmath14 on the gromov boundary @xmath48 , and then apply the martingale convergence theorem to show that for almost every sample path @xmath59 , the sequence of measures @xmath60 converges to some measure @xmath61 in @xmath62 , the space of probability measures on @xmath48 . one then uses geometric properties of the action of @xmath0 on @xmath1 to argue that @xmath61 is a @xmath2-measure @xmath63 for some point @xmath64 , almost surely , and that the image of the sample path under the orbit map @xmath28 converges to @xmath65 . this argument uses local compactness in an essential way in the first step . for a locally compact hyperbolic space @xmath1 , the gromov boundary @xmath48 is compact , as is @xmath66 . the space of probability measures on @xmath48 is also compact , and so the existence of a @xmath6-invariant measure on @xmath1 just follows from taking weak limits . in the non - locally compact case , the gromov boundary @xmath48 , and @xmath67 , need not be compact , as seen in the following example . [ ex : wedge ] a ray is a half line @xmath68 , with basepoint @xmath69 . let @xmath1 be the wedge product of countably many rays . this space is a tree , and so is @xmath2-hyperbolic , and is not locally compact at the basepoint . the gromov boundary is homeomorphic to @xmath70 with the discrete metric , and is not compact . _ the horofunction boundary . _ in order to address this issue , we shall consider the horofunction boundary of @xmath1 , which was also initially developed by gromov @xcite , and has proved a useful tool in studying random walks , see for example karlsson - ledrappier @xcite and bjorklund @xcite . we now give a brief description of this construction , giving full details in section [ section : horofunction ] . let @xmath1 be a metric space , and @xmath27 a basepoint . for each point @xmath43 in @xmath1 , one defines the horofunction @xmath71 determined by @xmath43 to be the function @xmath72 @xmath73 this gives an embedding of @xmath1 in the space @xmath74 of ( lipschitz- ) continuous functions on @xmath1 , which we shall consider with the compact - open topology ( we emphasize that we use uniform convergence on compact sets , not uniform convergence on bounded sets ) . with this topology , the closure of @xmath75 in @xmath74 is compact , even if @xmath1 is not locally compact ; it is called the _ horofunction compactification _ of @xmath1 and denoted by @xmath76 . in particular , there is a @xmath6-stationary measure @xmath14 on @xmath76 . we now consider a basic but fundamental example in detail . [ ex : r ] consider @xmath77 , with the usual metric . in this case the horofunction boundary @xmath76 consists of @xmath75 together with precisely two additional functions , namely @xmath78 , and @xmath79 . this example turns out to be very important in our case ; indeed , if @xmath1 is gromov hyperbolic , then the restriction of an arbitrary horofunction to a geodesic is equal ( up to a bounded additive error ) to one of the horofunctions described above , i.e. @xmath71 or @xmath80 . in example [ ex : wedge ] , the horofunction boundary equals the gromov boundary as a set , but the topology is different : namely , any sequence of horofunctions @xmath81 corresponding to a sequence of points @xmath82 which leaves every compact set converges to the horofunction @xmath83 associated to the basepoint @xmath27 . the gromov boundary may be recovered from the horofunction boundary by identifying functions which differ by a bounded amount , but in general the horofunction boundary may be larger than the gromov boundary , and is not a quasi - isometry invariant of the space . consider @xmath84 , with the @xmath85-metric , @xmath86 . then the sequences @xmath87 and @xmath88 have different values on @xmath89 , and so converge to distinct horofunctions , and in fact in this case the horofunction boundary consists of the product of the gromov boundary with @xmath90 . we shall distinguish two different types of horofunctions . we say a horofunction @xmath91 is _ finite _ if @xmath92 , and is _ infinite _ if @xmath93 . this partitions @xmath76 into two subsets : we shall write @xmath94 for the set of finite horofunctions , and @xmath95 for the set of infinite horofunctions . _ the local minimum map . _ we shall now construct a map from the horofunction boundary to the gromov boundary . recall that the restriction of a horofunction @xmath91 to a geodesic @xmath29 in @xmath1 is coarsely equal to one of the standard horofunctions on @xmath51 : in particular , it has ( coarsely ) at most one local minimum on @xmath29 . thus , if @xmath91 is bounded below on @xmath29 , we can map @xmath91 to the location where it attains its minimum , getting a map @xmath96 . on the other hand , if the horofunction is not bounded below , then we can pick a sequence @xmath97 of points for which the value of the horofunction tends to @xmath98 ; it turns out that such a sequence converges to a unique point in the gromov boundary , and the limit is independent of the choice of @xmath97 . thus , we can extend @xmath99 to a map @xmath100 . we show that this map is continuous on @xmath95 and @xmath0-equivariant , and that the stationary measure @xmath14 is supported on the infinite horofunctions @xmath95 . therefore the stationary probability measure @xmath14 on @xmath76 restricts to a probability measure on @xmath95 , and pushes forward to a @xmath6-stationary probability measure @xmath101 on @xmath48 . we may then complete the argument using with the geometric properties of the action of @xmath0 on @xmath48 . _ plan of the paper . _ in section [ section : background ] , we review some useful material about gromov hyperbolic spaces , and fix notation . in section [ section : horofunction ] , we develop the properties of the horofunction boundary that we will use , including the local minimum map , and the behaviour of shadows . in these initial sections we give complete proofs in the non - proper case of certain statements that are already known in the proper case . in section [ section : convergence ] , we use the horofunction boundary to show that almost every sample path converges to the gromov boundary . in section [ section : applications ] , we use the convergence to the boundary result to show results on positive drift , sublinear tracking , and the growth rate of translation distance , and then finally in section [ section : poisson ] we show that if the action of @xmath0 is acylindrical , and @xmath6 has finite entropy , then the gromov boundary with the hitting measure is the poisson boundary for the random walk . we would like to thank jason behrstock , danny calegari , romain dujardin , camille horbez , vadim kaimanovich , anders karlsson , andrei malyutin and samuel taylor for helpful conversations . the first author gratefully acknowledges the support of the simons foundation and psc - cuny . let @xmath1 be a gromov hyperbolic space , i.e. a geodesic metric space which satisfies the @xmath2-slim triangles condition . we will not assume that @xmath1 is _ proper _ , i.e. that closed balls are compact , but we will always assume that it is _ separable _ , i.e. that it contains a dense countable subset . we shall write @xmath26 for the metric on @xmath1 , and @xmath102 for the closed ball of radius @xmath103 about the point @xmath43 in @xmath1 . we shall now recall a few facts on the geometry of @xmath1 . we shall write @xmath104 to mean that the absolute value of the function @xmath105 is bounded by a number which only depends on @xmath2 , though this need not be a linear multiple of @xmath2 . similarly , we shall write @xmath106 to mean that the difference between @xmath107 and @xmath108 is bounded by a constant , which depends only on @xmath2 . recall that the gromov product in a metric space is defined to be @xmath109 in a @xmath2-hyperbolic space , for all points @xmath110 and @xmath44 , the gromov product @xmath111 is equal to the distance from @xmath27 to a geodesic from @xmath43 to @xmath44 , up to an additive error of at most @xmath2 : if we write @xmath112 $ ] for a choice of geodesic from @xmath43 to @xmath44 , then @xmath113 ) = { { ( x\cdoty)_{x_0 } } } + o(\delta),\ ] ] see e.g. @xcite*iii.h 1.19 . moreover , for any three points @xmath114 one has the following inequality @xmath115 which we shall refer to as the triangle inequality for the gromov product . we now recall the definition of the gromov boundary of @xmath1 , which we shall write as @xmath48 . we say that a sequence @xmath116 is a _ gromov sequence _ if @xmath117 tends to infinity as @xmath118 tends to infinity . we say that two gromov sequences @xmath82 and @xmath119 are equivalent if @xmath120 tends to infinity as @xmath21 tends to infinity . the _ gromov boundary _ @xmath4 is defined as the set of equivalence classes of gromov sequences . we can extend the gromov product to the boundary by @xmath121 where the supremum is taken over all sequences @xmath122 and @xmath123 . with this definition , the triangle inequality also holds for any three points @xmath124 in @xmath66 , but with a larger additive constant @xmath125 instead of @xmath2 ( see e.g. @xcite*iii.h remark 3.17(4 ) ) . the gromov product on the boundary may be used to define a complete metric on @xmath48 , see bridson and haefliger @xcite*iii.h.3 for the proper case , and visl @xcite for the non - proper case . moreover , the space @xmath126 can be equipped with a topology such that the relative topologies on both @xmath1 and @xmath48 are equal respectively to the usual topology on @xmath1 , and the above - mentioned metric topology on @xmath4 . if @xmath127 we shall write @xmath128 for the closure of @xmath129 in @xmath130 . if @xmath1 is proper , then @xmath48 is compact , but it need not be compact if @xmath1 is not proper . however , a bounded set does not have limit points in the gromov boundary , i.e. for @xmath131 we have @xmath132 . let @xmath133 be a connected subset of @xmath51 , and let @xmath1 be a metric space . a @xmath134-quasigeodesic is a ( not necessarily continuous ) map @xmath135 such that for all @xmath136 and @xmath137 in @xmath133 , @xmath138 if @xmath139 , then we will call the quasigeodesic @xmath29 a _ bi - infinite quasigeodesic_. if @xmath140 , then we shall call @xmath29 a _ quasigeodesic ray based at @xmath141_. if the metric space is @xmath2-hyperbolic , then quasigeodesics have the following stability property , which is often referred to as the morse lemma . [ prop : morse ] let @xmath1 be a @xmath2-hyperbolic space . given numbers @xmath142 and @xmath24 , there is a number @xmath22 such that for any two points @xmath43 and @xmath44 in @xmath143 , any two @xmath134-quasigeodesics connecting @xmath43 and @xmath44 are contained in @xmath22-neighbourhoods of each other . we shall refer to a choice of constant @xmath22 in proposition [ prop : morse ] above as a morse constant for the quasi - geodesic constants @xmath142 and @xmath24 . for any choice of basepoint @xmath27 and every point @xmath43 in the boundary there is a quasigeodesic ray based at @xmath27 which converges to the point @xmath43 , and any two points in the boundary are connected by a bi - infinite quasigeodesic . in fact , the quasigeodesics may be chosen to have quasigeodesic constants @xmath142 and @xmath24 bounded above in terms of the hyperbolicity constant @xmath2 , independently of the choice of basepoint or boundary points , see e.g. kapovich and benakli @xcite . by choosing @xmath142 and @xmath24 sufficiently large , we may assume that we have chosen these constants so that at least one of the @xmath134-quasigeodesics is continuous , see e.g. bridson and haefliger @xcite*iii.h.1 . we will use the fact that in a @xmath2-hyperbolic space nearest point projection onto a geodesic @xmath29 is coarsely well defined , i.e. there is a constant @xmath144 , which only depends on @xmath2 , such that if @xmath53 and @xmath145 are nearest points on @xmath29 to @xmath43 , then @xmath146 . we will make use of the following _ reverse triangle inequality_. [ prop : reverse triangle ] let @xmath29 be a geodesic in @xmath1 , @xmath147 a point , and @xmath53 a nearest point projection of @xmath44 to @xmath29 . then for any @xmath148 we have @xmath149 and furthermore , any geodesic from @xmath150 to @xmath44 passes within distance @xmath125 of @xmath53 . the upper bound for @xmath151 is immediate from the usual triangle inequality . to prove the lower bound , by the definition of nearest point projection , @xmath152 ) . \\ \intertext{recall that by \eqref{eq : gp estimate } , } d_x(y , p ) & \leqslant ( p \cdot z)_y + \delta , \\ \intertext{and writing out the gromov product , we get } d_x(y , p ) & = \frac{1}{2 } \big ( d_x(y , p ) + d_x(y , z ) - d_x(p , z ) \big ) + \delta , \ ] ] which yields @xmath153 as required . this implies that a path consisting of @xmath154 \cup [ p , z]$ ] is a @xmath155-quasigeodesic , and so by stability of quasigeodesics in a @xmath2-hyperbolic space , this path is contained in an @xmath125-neighbourhood of any geodesic @xmath156 $ ] from @xmath44 to @xmath150 , so in particular , the distance from @xmath53 to a @xmath156 $ ] is at most @xmath125 . finally we show that if two points @xmath43 and @xmath44 in @xmath1 have nearest point projections @xmath157 and @xmath158 to a geodesic @xmath29 , and @xmath157 and @xmath158 are sufficiently far apart , then the path @xmath159 \cup [ p_x , p_y ] \cup [ p_y , y]$ ] is a quasigeodesic , and in fact has the same length as a geodesic from @xmath43 to @xmath44 , up to an additive error depending only on @xmath2 . [ prop : npp2 ] let @xmath29 be a geodesic in @xmath1 , and let @xmath43 and @xmath44 be two points in @xmath1 with nearest points @xmath157 and @xmath158 respectively on @xmath29 . then if @xmath160 , then @xmath161 and furthermore , any geodesic from @xmath43 to @xmath44 passes within an @xmath125-neighbourhood of both @xmath157 and @xmath158 . this is well known , see e.g. @xcite*proposition 3.4 . given a point @xmath162 and a number @xmath163 the _ shadow _ @xmath164 is defined to be @xmath165 there are a number of similar definitions in the literature , and we emphasize that we define shadows to be subsets of @xmath1 , rather than subsets of say @xmath66 or @xmath48 . we will refer to the quantity @xmath166 as the _ distance parameter _ of the shadow , and it is equal to the distance from @xmath27 to the shadow , up to an additive error depending only on @xmath2 . by the triangle inequality for the gromov product , for any two points @xmath44 and @xmath150 in the closure of a shadow @xmath167 , there is a lower bound on their gromov product @xmath168 we now show that the nearest point projection of a shadow @xmath169 to a geodesic @xmath170 $ ] is contained in a bounded neighbourhood of the intersection of the shadow with @xmath170 $ ] , and the same result holds for the complement of the shadow . [ prop : npp shadow ] let @xmath150 be a point in the shadow @xmath171 , let @xmath29 be a geodesic from @xmath43 to @xmath44 , and let @xmath53 be a nearest point to @xmath150 on @xmath29 . then @xmath172 if @xmath173 , then @xmath174 if @xmath150 lies in the shadow @xmath171 , @xmath175 using the definition of the gromov product , we may rewrite this as @xmath176 and then using , and the fact that @xmath177 and @xmath44 lie in that order on a common geodesic , gives @xmath178 as required . if @xmath150 does not lie in @xmath179 , then the same argument works , with the opposite inequality . as a consequence , the complement of a shadow is almost a shadow : [ c : shadowcompl ] the complement of the shadow @xmath179 is contained in the shadow @xmath180 , where @xmath181 . let @xmath182 be a metric space . a function @xmath183 is called _ @xmath184-lipschitz _ if for each @xmath185 we have @xmath186 clearly , @xmath184-lipschitz functions are uniformly continuous . for each @xmath9 , let us define @xmath187 the space of @xmath184-lipschitz functions which vanish at @xmath27 . we shall endow the space @xmath188 with the topology of pointwise convergence . note that , since all elements of @xmath188 are uniformly continuous with the same modulus of continuity , this topology is equivalent to the topology of uniform convergence on compact sets , which is also equivalent to the compact - open topology as @xmath189 is a metric space . let @xmath1 be a separable metric space . then for each @xmath9 , the space @xmath188 is compact , hausdorff and second countable ( hence metrizable ) . note that for any function @xmath190 and each @xmath191 we have @xmath192 hence the space @xmath188 is a closed subspace of an infinite product of compact spaces , hence it is compact by tychonoff s theorem . let @xmath74 be the space of real - valued continuous functions on @xmath1 , with the compact - open topology . as @xmath51 is hausdorff , @xmath74 is also hausdorff , hence so is @xmath188 . since @xmath1 is a separable metric space , it is second countable ; moreover , as @xmath189 is also second countable , @xmath74 is second countable and so is @xmath188 . let @xmath9 be a basepoint . we define the _ horofunction _ map @xmath193 to be @xmath194 where we write @xmath195 for @xmath196 . note that the horofunction map @xmath193 depends on the choice of basepoint , but as we shall usually consider horofunctions defined from some fixed basepoint we omit this from the notation . in the few cases we consider horofunctions with different basepoints we will change notation on an ad hoc basis . for each @xmath44 , the horofunction @xmath195 is @xmath184-lipschitz , as @xmath197 and moreover @xmath198 for all @xmath44 , hence @xmath193 maps @xmath1 into @xmath199 . the map @xmath200 defined above is continuous and injective . for any @xmath147 , the function @xmath201 achieves a unique minimum at @xmath202 , so the map @xmath193 is injective on @xmath1 . the map @xmath193 is continuous , as for any @xmath114 we have @xmath203 and so if @xmath123 then @xmath204 uniformly on compact sets , in fact uniformly on all of @xmath1 . let us now define the fundamental object we are going to work with . let @xmath1 be a separable metric space with basepoint @xmath9 . we define the _ horofunction compactification _ @xmath205 to be the closure of @xmath75 in @xmath188 . we shall call the set @xmath206 the _ horofunction boundary _ of @xmath1 . elements of @xmath76 will be called _ note that in the proper case , the space @xmath76 contains @xmath1 as an open , dense set . in the non - proper case , although the map @xmath193 is injective on @xmath1 , the image @xmath75 need not be open in @xmath76 , so although @xmath76 is compact , it is not a compactification of @xmath1 in the standard sense . let @xmath0 be a group of isometries of @xmath1 . then the action of @xmath0 on @xmath1 extends to a continuous action by homeomorphisms on @xmath76 , defined as @xmath207 for each @xmath208 and @xmath209 . the action of @xmath208 on @xmath1 translates into an action on @xmath75 , by defining @xmath210 for each @xmath208 and @xmath147 . let us observe that @xmath211 thus we can define the action of @xmath208 on each @xmath209 as in . it is immediate from the definition that if @xmath212 pointwise then @xmath213 , hence @xmath37 acts continuously on @xmath76 , and so acts by homeomorphisms , as @xmath0 is a group . we shall write @xmath214 for the closure of @xmath129 in @xmath76 . we remark that as @xmath215 need not be compact , a sequence of points contained in a bounded set may have images under @xmath193 which converge to the horofunction boundary , i.e. @xmath216 may contain points in the horofunction boundary @xmath217 . we now record some basic observations about the behaviour of horofunctions . we start by describing the restriction of a horofunction @xmath195 to a geodesic @xmath29 in @xmath1 . as we shall see , for any geodesic @xmath29 , the restriction of @xmath195 to @xmath29 has a coarsely well defined local minimum a bounded distance away from the nearest point projection @xmath53 of @xmath44 to @xmath29 . moreover , for any point @xmath218 , the value of @xmath219 is equal to @xmath220 , up to bounded error depending only on @xmath2 . we now make this precise . let @xmath29 be a geodesic in @xmath1 , @xmath147 , and let @xmath53 be a nearest point projection of @xmath44 to @xmath29 . then the restriction of @xmath195 to @xmath29 is given by @xmath221 for all @xmath218 . this follows from the reverse triangle inequality , proposition [ prop : reverse triangle ] , by adding @xmath222 to both sides . we now describe the restriction of an arbitrary horofunction @xmath91 to a geodesic @xmath29 in @xmath1 . an _ orientation _ for a geodesic @xmath29 is a strict total order on the points of @xmath29 , induced by a choice of unit speed parameterization ( thus , each geodesic has exactly two orientations ) . we may then define the _ signed distance function _ along @xmath29 to be @xmath223 [ prop : horofunction geodesic ] let @xmath91 be a horofunction in @xmath76 , and let @xmath29 be a geodesic in @xmath1 . then there is a point @xmath53 on @xmath29 such that the restriction of @xmath91 to @xmath29 is equal to exactly one of the following two functions , up to bounded additive error : 1 . either @xmath224 2 . or @xmath225 for some choice of orientation on @xmath29 . so for example , for geodesic rays starting at the basepoint @xmath27 , the graphs of these functions are equal to one of the two graphs shown below , up to an error of @xmath125 . ( 12,-13 ) ( 12,-13.25 ) node [ below right ] @xmath53 ; ( 4.75,-13 ) node [ left ] @xmath69 ( 21,-13 ) ; ( 5,-13 ) ( 12,-20 ) ( 21,-11 ) ; ( 25,-8 ) ( 25,-22 ) ; ( 24.75,-13 ) node [ left ] @xmath69 ( 35,-13 ) ; ( 5,-8 ) ( 5,-22 ) ; ( 5,-20 ) ( 4.75,-20 ) node [ left ] @xmath226 ; ( 25,-13 ) ( 34,-22 ) ; ( 7.5,-9 ) node[anchor = base west ] @xmath227 ; ( 12.5,-12.5 ) ; ( 26,-9 ) node[anchor = base west ] @xmath228 ; let @xmath91 be a horofunction in @xmath76 , let @xmath229 be sequence of horofunctions which converge to @xmath91 , and let @xmath230 be the nearest point projection of @xmath231 to @xmath29 . first , suppose that there is a subsequence of the @xmath232 for which the projections @xmath230 to @xmath29 are bounded , i.e. contained in a subinterval @xmath133 of @xmath29 of finite length . we may pass to a further subsequence , which by abuse of notation we shall also call @xmath229 , such that the projections @xmath230 converge to a point @xmath233 , and in fact are all within distance @xmath184 of @xmath53 . therefore , for this subsequence , @xmath234 , by . as @xmath235 pointwise , this implies that @xmath236 for each @xmath237 , as required . now consider the case in which the nearest point projections @xmath230 eventually exit every compact subinterval of @xmath29 . in this case it will be convenient to choose @xmath53 to be a nearest point on @xmath29 to the basepoint @xmath27 , and we may pass to a subsequence such that @xmath238 for all @xmath21 , for some choice of orientation on @xmath29 . for any @xmath237 all but finitely many @xmath230 satisfy @xmath239 . as @xmath230 is a nearest point projection of @xmath231 to @xmath29 , and @xmath53 is a nearest point projection of @xmath27 to @xmath29 , we may use the reverse triangle inequality to rewrite @xmath240 in terms of @xmath241 , @xmath242 , and distances between points on the geodesic @xmath29 . there are two cases , depending on whether @xmath243 , or @xmath244 , illustrated below in figure [ pic : horofunction geodesic ] . the sign on each line segment in figure [ pic : horofunction geodesic ] indicates the sign of the corresponding line segment in the approximation for @xmath245 . this shows that @xmath246 as @xmath229 converges to @xmath91 , this implies the same coarse equalities for @xmath247 . as @xmath248 , and using the definition of signed distance along @xmath29 , this gives above . = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( 0,0 ) ( 10,0 ) node [ below ] @xmath29 ; ( 2,0 ) node [ midway , right ] @xmath249 ( 2,4 ) ; ( 8,0 ) node [ midway , left ] @xmath250 node [ midway , right ] @xmath249 ( 8,-4 ) ; ( 2,0 ) node [ midway , above ] @xmath249 ( 5,0 ) ; ( 5,0 ) node [ midway , above ] @xmath249 node [ midway , below ] @xmath250 ( 8,0 ) ; ( 2,4 ) node [ point , label = right:@xmath27 ] ; ( 2,0 ) node [ point , label = below:@xmath53 ] ; ( 8,0 ) node [ point , label = above:@xmath230 ] ; ( 8,-4 ) node [ point , label = right:@xmath231 ] ; ( 5,0 ) node [ point , label = below:@xmath43 ] ; ( -4,0 ) ( 10,0 ) node [ below ] @xmath29 ; ( 2,0 ) ( 2,4 ) ; ( 8,0 ) ( 8,-4 ) ; ( 2,4 ) node [ point , label = right:@xmath27 ] ; ( 2,0 ) node [ point , label = below:@xmath53 ] ; ( 8,0 ) node [ point , label = above:@xmath230 ] ; ( 8,-4 ) node [ point , label = right:@xmath231 ] ; ( -2,0 ) node [ point , label = below:@xmath43 ] ; ( 2,0 ) node [ midway , right ] @xmath249 ( 2 , 4 ) ; ( -2,0 ) node [ midway , below ] @xmath250 ( 2,0 ) node [ midway , below ] @xmath250 node [ midway , above ] @xmath249 ( 8,0 ) node [ midway , left ] @xmath250 node [ midway , right ] @xmath249 ( 8,-4 ) ; we say a function @xmath256 has no coarse local maxima if @xmath257 , where @xmath258 has no local maxima . similarly we say @xmath256 has at most one coarse local minima if @xmath259 , where @xmath258 has at most one local minima . so we have shown that any horofunction @xmath91 restricted to a geodesic @xmath29 has no coarse local maxima on @xmath29 , and at most one coarse local minimum on @xmath29 . for any horofunction @xmath209 we may consider @xmath260 which takes values in @xmath261 $ ] . we may partition @xmath76 into two sets depending on whether or not @xmath262 . we shall denote @xmath263 the set of _ finite horofunctions _ @xmath264 and @xmath265 the set of _ infinite horofunctions _ @xmath266 clearly , both @xmath263 and @xmath265 are invariant for the action of @xmath0 . note moreover that if a horofunction @xmath91 is contained in @xmath75 , i.e @xmath267 for some @xmath147 , then @xmath268 and the infimum is achieved at the unique point @xmath147 , hence @xmath75 is contained in the set @xmath94 of finite horofunctions . more generally , by the same proof , if @xmath108 is a bounded subset of @xmath1 , then @xmath269 note however that the subset @xmath270 need not be compact , so there may be sequences of elements of @xmath95 which do not have subsequences which converge in @xmath95 . if a sequence of horofunctions @xmath271 converges to @xmath91 , this does not in general imply that @xmath272 converges to @xmath273 . for example , in example [ ex : wedge ] , consider the sequence @xmath81 , where @xmath274 is the point distance @xmath21 from @xmath69 in the @xmath21-th ray . then @xmath275 , but @xmath81 converges to @xmath83 , for which @xmath276 . in fact , in this example there are sequences in @xmath95 which converge to horofunctions in @xmath94 . if we set @xmath277 to be equal to the limit of @xmath278 , where the @xmath279 are points distance @xmath280 from @xmath69 along the @xmath21-th ray , then @xmath281 , but @xmath282 converges to @xmath83 . [ l : horoineq ] for each basepoint @xmath9 , each horofunction @xmath209 and each pair of points @xmath185 the following inequality holds : @xmath283 let @xmath191 . then one has , by the triangle inequality @xmath284 now , by @xmath2-hyperbolicity , combined with the previous estimate , one has @xmath285 since every horofunction is the pointwise limit of functions of type @xmath286 , the claim follows . we define a sequence @xmath116 to be _ minimizing _ for a horofunction @xmath91 if @xmath287 as @xmath288 . we shall now prove that every minimizing sequence is a gromov sequence , hence it has a limit in the gromov boundary . [ l : converges ] let @xmath289 an infinite horofunction , and @xmath290 a sequence of points in @xmath1 such that @xmath291 . then the sequence @xmath290 converges to a point in the gromov boundary of @xmath1 . moreover , two minimizing sequences for the same horofunction converge to the same point in the gromov boundary . by lemma [ l : horoineq ] one has @xmath292 as @xmath293 , proving the first claim . to prove uniqueness of the limit , suppose that there are two sequences @xmath97 and @xmath290 such that @xmath287 and @xmath291 . then , by lemma [ l : horoineq ] , the gromov product @xmath294 , hence by definition the sequences @xmath97 and @xmath290 converge to the same point in the gromov boundary @xmath48 . note finally that if @xmath289 , then there is actually a quasigeodesic sequence @xmath290 in @xmath1 such that @xmath295 . furthermore , we can recover the gromov boundary from the horofunction boundary as follows . define an equivalence relation on @xmath76 by @xmath296 if @xmath297 is finite . this collapses @xmath94 to a single point , and the equivalence classes in @xmath95 are precisely the point pre - images of the local minimum map @xmath298 , so the gromov boundary is homeomorphic to @xmath299 . however , we will not use this result so we omit the proof . we now define a map @xmath300 , which may be thought of as the `` local minimum '' map , which sends a horofunction @xmath91 to the location at which it attains its minimum . if the horofunction @xmath91 does not attain a minimum in @xmath1 , it turns out that it makes sense to think of the minimum value as lying in the gromov boundary . we now make this precise . the _ local minimum _ map @xmath300 is defined as follows . * if @xmath301 , i.e. @xmath302 , then define @xmath303 the set of points of @xmath1 where the value of @xmath91 is close to its infimum ; * if @xmath289 , i.e. @xmath262 , then choose a sequence @xmath119 with @xmath304 , and set @xmath305 to be the limit point of @xmath290 in the gromov boundary . the local minimum map @xmath306 is well - defined and @xmath0-equivariant . by lemma [ l : converges ] , the map is well - defined on @xmath95 : indeed , every minimizing sequence for @xmath91 converges in the gromov boundary , and any two minimizing sequences yield the same limit . to prove equivariance , let us first pick @xmath301 , and @xmath307 . then for each @xmath147 we have @xmath308 , thus @xmath309 for each @xmath147 , hence the value of @xmath310 at @xmath311 is close to its infimum hence @xmath312 . if instead @xmath289 , then let @xmath290 a minimizing sequence for @xmath91 . then by definition of the action one gets @xmath313 hence @xmath314 is a minimizing sequence for @xmath310 , so @xmath315 as required . [ lemma : bdd diameter ] there exists @xmath23 , which depends only on @xmath2 , such that for each finite horofunction @xmath301 we have @xmath316 let @xmath317 , for some @xmath301 , and consider the restriction of @xmath91 along a geodesic segment from @xmath43 to @xmath44 . by proposition [ prop : horofunction geodesic ] , the restriction has at most one coarse local minimum : hence , since @xmath43 and @xmath44 are coarse local minima of @xmath91 , the distance between @xmath43 and @xmath44 is universally bounded in terms of @xmath2 . [ prop : phi cts on r ] let @xmath271 a sequence of horofunctions which converges to some infinite horofunction @xmath289 . then @xmath318 converges to @xmath319 in the gromov boundary . as a consequence , the local minimum map @xmath320 is continuous . ( if @xmath277 is a finite horofunction , in the above statement we mean that @xmath321 for any choice of @xmath322 . ) let @xmath323 , and let @xmath324 a sequence of horofunctions which converge to @xmath289 . let us pick a minimizing sequence @xmath325 for @xmath91 , and for each @xmath21 a sequence @xmath326 , such that @xmath327 as @xmath328 , so that @xmath329 $ ] and @xmath330 $ ] . the goal is to prove that @xmath331 as @xmath288 . since @xmath332 is a gromov sequence and it is minimizing for @xmath91 , there exists @xmath333 such that @xmath334 and for each @xmath335 one has @xmath336 since @xmath212 pointwise , there exists @xmath337 such that @xmath338 for each @xmath339 . now , since @xmath340 is minimizing for @xmath277 , there exists @xmath341 such that @xmath342 and @xmath343 hence , by lemma [ l : horoineq ] we have @xmath344 and by property , for all @xmath345 @xmath346 thus @xmath347 for @xmath339 , as claimed . [ cor : surjective ] the local minimum map @xmath298 is surjective . pick @xmath348 . by construction , there exists a sequence @xmath349 of points of @xmath1 which converge to @xmath65 . by compactness , the sequence of horofunctions @xmath350 has a subsequence @xmath351 which converges to some @xmath209 ; since @xmath352 , we have that @xmath91 belongs to @xmath95 ; thus , by the proposition , @xmath353 , hence by uniqueness of the limit @xmath354 , as required . however , we emphasize that if @xmath324 converges to @xmath91 in @xmath94 , then @xmath355 need not converge to @xmath319 . for instance , in the countable wedge of rays of example [ ex : wedge ] , if @xmath274 is the point on the branch @xmath356 at distance @xmath21 from the base point @xmath27 , then @xmath357 in the horofunction compactification , but the sequence @xmath82 does not converge in the gromov boundary . we define the _ depth _ of the shadow @xmath358 to be the quantity @xmath359 we now show that we may characterize points in a shadow in terms of the value of the corresponding horofunction at the basepoint and the depth . [ l : shadowhoro ] if @xmath358 is a shadow , then @xmath360 if and only if @xmath361 by definition of shadow one has @xmath362 hence by writing out the gromov product @xmath363 and by simplifying we get @xmath364 which proves the claim . for any shadow @xmath358 , the closure of @xmath52 in @xmath76 is @xmath365 where @xmath366 . it will also be useful to know how shadows are related to the topology of the gromov boundary @xmath48 . we shall use the following property of shadows : there is a constant @xmath367 , which only depends on the action of @xmath0 on @xmath1 , such that for any @xmath208 the closure of the shadow @xmath368 in @xmath4 contains a non - empty open set . this follows from the proposition below . [ prop : shadow open ] let @xmath0 be a group acting by isometries on a separable gromov hyperbolic space @xmath1 , such that @xmath0 contains at least one hyperbolic isometry . then there is a number @xmath369 such that for any @xmath208 the set @xmath370 contains a limit point of @xmath371 in its interior . this follows from the following result from blachre , hassinsky and mathieu @xcite . @xcite*proposition 2.1 [ prop : bhm ] for any @xmath372 sufficiently small , and any @xmath373 , there are positive numbers @xmath374 and @xmath367 , such that for any @xmath375 , and any @xmath376 with @xmath377 , @xmath378 where @xmath379 is the ball of radius @xmath103 about @xmath44 in the metric @xmath380 on @xmath48 . since @xmath0 contains at least one hyperbolic isometry , then the limit set @xmath381 contains at least two points in @xmath4 , which we call @xmath382 and @xmath383 . let @xmath384 be a quasigeodesic from @xmath382 to @xmath383 , and let @xmath53 be a closest point on @xmath384 to the basepoint @xmath27 . given a group element @xmath208 , consider the translate @xmath385 , as illustrated in figure [ pic : quasi - axis ] below . = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( 0,0 ) circle ( 4 ) ; ( 4 , 0 ) node [ point , label = right:@xmath386 .. controls ( 2 , 0 ) and ( 0 , 2 ) .. ( 0 , 4 ) node [ point , label = above:@xmath387 ; ( 1 , 2.3 ) node @xmath385 ; ( 3.5 , -3.5 ) node @xmath48 ; ( 0 , 0 ) node [ point , label = below:@xmath27 ] ( 1.25 , 1.25 ) node [ point , label = below:@xmath145 ] ; ( 2.2 , -1 ) node [ point , label = below:@xmath389 ( 2.7 , 0.25 ) node [ point , label = above:@xmath390 ; for any element @xmath208 , let @xmath145 be a nearest point on @xmath385 to @xmath27 . any quasigeodesic from @xmath27 to either @xmath391 or @xmath392 passes within distance @xmath125 of @xmath145 , so @xmath393 lies within distance @xmath394 of at least one of these quasigeodesics , which we may assume has endpoint @xmath395 , up to relabeling . therefore , the product @xmath396 is bounded above independently of @xmath208 , hence by proposition [ prop : bhm ] , there is a number @xmath367 , ( depending only on @xmath2 and the choice of @xmath384 ) such that the closure @xmath397 contains an open set containing @xmath398 , for any @xmath399 , as required . _ a priori _ , shadows need not be convex , or even quasi - convex . however , we now show various results about horofunctions and nested shadows which we can think of as weak versions of convexity . for example , for any two points contained in a shadow @xmath164 , the geodesic connecting them is contained in @xmath400 . we start by showing that the value of a horofunction along a geodesic is bounded by its values on the endpoints , up to an additive error of @xmath125 . [ l : qc ] let @xmath1 a @xmath2-hyperbolic , geodesic metric space . then there exists a constant @xmath374 , which depends only on @xmath2 , such that given any geodesic segment @xmath401 $ ] in @xmath1 , with @xmath402 , the following holds : @xmath403 for any @xmath404 $ ] and any @xmath162 . let us first assume that @xmath405 belong to @xmath401 $ ] . up to swapping @xmath406 and @xmath407 , we can assume @xmath408 $ ] ; then we have the bound @xmath409 which yields the claim . now , in the general case let @xmath157 be the closest point projection of @xmath43 to @xmath401 $ ] , and @xmath410 the projection of @xmath27 . by the definition of @xmath286 , @xmath411 hence for each @xmath412 @xmath413 where @xmath414 denotes the horofunction based at @xmath410 . since now @xmath410 and @xmath157 lie on @xmath401 $ ] , the claim follows by the previous case . lemma [ l : qc ] implies the following weak convexity property of both shadows and their complements . [ cor : weak convexity ] there exists a constant @xmath374 , which depends only on @xmath2 , such that for each shadow @xmath358 the following hold : 1 . if @xmath406 and @xmath407 belong to @xmath52 , then the geodesic segment @xmath401 $ ] lies in @xmath415 ; 2 . if @xmath406 and @xmath407 do not belong to @xmath52 , then the geodesic segment @xmath401 $ ] does not intersect @xmath416 . moreover , for each @xmath134 which satisfy proposition [ prop : morse ] , the above statements still hold with geodesic " replaced by @xmath417-quasi - geodesic " , where this time @xmath374 depends on @xmath418 , and @xmath24 . if @xmath419 belong to @xmath52 , then @xmath420 for each @xmath412 by lemma [ l : shadowhoro ] , hence lemma [ l : qc ] implies @xmath421 where @xmath422 , thus @xmath423 once again by lemma [ l : shadowhoro ] . the proof of ( 2 ) is similar , using the left - hand side of equation . the extension to quasi - geodesic is immediate by the fellow - traveling property of proposition [ prop : morse ] . in this section we prove the following theorem . [ t : conv ] let @xmath0 be a countable group of isometries of a geodesic , separable , @xmath2-hyperbolic space @xmath1 ( not necessarily proper ) , and let @xmath6 be a non - elementary probability measure on @xmath0 . then for each @xmath9 , almost every sample path @xmath12 converges in @xmath126 to a point of the gromov boundary @xmath4 . the proof of the theorem takes several steps , and it exploits the action of @xmath0 on the space of probability measures both on the horofunction compactification @xmath76 and on the gromov boundary @xmath4 . the strategy of the proof is the following : 1 . by compactness , there is a stationary measure @xmath14 on @xmath76 ( lemma [ l : existstat ] ) . 2 . the measure @xmath14 does not charge the finite part of the boundary : @xmath424 ( proposition [ prop : qzero ] ) . 3 . by the martingale convergence theorem ( proposition [ p : martingale ] ) , for almost every sample path the sequence of measures @xmath425 converges to some measure @xmath61 in @xmath426 . 4 . by pushing the sequence forward to @xmath4 , using the local minimum map @xmath99 , almost every sequence @xmath427 converges to some measure @xmath428 in @xmath429 ( lemma [ lemma : mart2 ] ) . almost every sample path @xmath28 has a subsequence which converges to a point @xmath65 in the gromov boundary @xmath4 ( proposition [ p : convsub ] ) . thus , by lemma [ lemma : delta ] , almost every sample path @xmath430 has a subsequence @xmath431 such that @xmath432 converges to a delta - measure @xmath63 , for some @xmath64 . since the limit exists , almost every sequence @xmath427 converges to a delta - measure @xmath63 ( proposition [ prop : delta ] ) . we prove that the fact that @xmath427 converges to @xmath63 implies that the sample path @xmath433 converges to @xmath348 ( proposition [ p : topconv ] ) . in the rest of the section we shall work out the details of the proof . we remark that sample paths do not in general converge to points in the horofunction compactification . consider the nearest neighbour random walk on the cayley graph of @xmath434 , with respect to the standard generating set @xmath435 , [ b , c ] , c^2 \rangle$ ] , and with basepoint @xmath27 corresponding to the identity element . if @xmath436 , then @xmath437 , and @xmath438 . as almost every sample path @xmath33 hits each coset of @xmath439 infinitely often , sample paths do not converge in @xmath76 , almost surely . we briefly review some background material on random walks and fix notation . let @xmath6 be a probability distribution on @xmath0 ; the _ step space _ of the random walk generated by @xmath6 is the measure space @xmath440 , which is the countable infinite product of the measure spaces @xmath441 . each element of @xmath442 is a sequence @xmath443 , whose entries are the increments of our ( bi - infinite ) random walk . the shift map @xmath444 sends @xmath443 to @xmath445 , and is measure preserving and ergodic . we define the _ location _ of the random walk at time @xmath21 , which we shall denote @xmath33 to be @xmath446 this gives a map @xmath447 , defined by @xmath448 . we shall denote the range of the location map as @xmath449 , to distinguish it from the step space , and call @xmath450 the pushforward to @xmath449 of the product measure @xmath451 on the step space @xmath452 . we shall refer to @xmath453 as the _ path space _ , and elements @xmath454 as _ sample paths _ of the random walk . the shift map @xmath455 acts on @xmath449 by @xmath456 , and is measure preserving and ergodic . let @xmath457 be a metrizable topological space , and denote @xmath458 the space of borel probability measures on @xmath457 . the space @xmath458 is endowed with the weak- * topology , which is defined by saying @xmath459 if for each continuous bounded function @xmath105 on @xmath457 one has @xmath460 . if now @xmath0 is a countable group which acts on @xmath457 by homeomorphisms , we denote @xmath461 the pushforward of @xmath462 under the action of @xmath208 , i.e. @xmath463 , and define the _ convolution operator _ @xmath464 as the average of the pushforwards : @xmath465 we say that a probability distribution @xmath14 on @xmath457 is _ @xmath6-stationary _ if @xmath466 , i.e. for each borel set @xmath129 we have @xmath467 a space @xmath457 equipped with a @xmath6-stationary measure @xmath14 is called a _ @xmath441-space_. now , if we fix a base point @xmath468 , we shall write @xmath469 for the pushforward of @xmath6 under the orbit map , i.e. if @xmath470 then @xmath471 . we shall write @xmath472 for the @xmath21-fold convolution of @xmath6 with itself on @xmath0 , we shall write @xmath473 for the pushforward of @xmath472 to @xmath457 , and finally , we shall write @xmath474 for the cesro averages of the pushforward measures , @xmath475 . classical compactness arguments yield the following : [ l : existstat ] let @xmath0 be a countable group which acts by homeomorphisms on a compact metric space @xmath457 , and let @xmath6 be a probability distribution on @xmath0 . then there exists a @xmath6-stationary borel probability measure @xmath14 on @xmath457 . since @xmath457 is a compact metrizable space , then @xmath458 is compact in the weak-@xmath476 topology . then any weak-@xmath476 limit point of the sequence @xmath477 of the cesro averages is @xmath6-stationary . an alternate proof follows from the schauder - tychonoff fixed point theorem . applying the above arguments to @xmath76 implies that there exists a @xmath6-stationary measure @xmath14 on @xmath76 , i.e. @xmath478 is a @xmath441-space . we now show that the measure @xmath14 is supported on @xmath95 . [ prop : qzero ] let @xmath0 be a non - elementary countable group of isometries of a separable gromov hyperbolic space @xmath1 . let @xmath6 be a non - elementary probability distribution on @xmath0 , and let @xmath14 be a @xmath6-stationary measure on @xmath76 . then @xmath479 in order to show that some set @xmath480 has measure zero , the basic idea is to consider the translates @xmath481 of the set @xmath480 , and to consider the supremum of the measures of these sets . if we choose a translate @xmath481 with measure very close to the supremum , then by @xmath6-stationarity , if @xmath482 is another translate with @xmath483 , then @xmath484 will also be close to the supremum . if there are enough disjoint translates with @xmath14-measures close to the supremum , then the total measure of @xmath14 is strictly greater than one , which contradicts the fact that @xmath14 is a probability measure . we now make this precise . [ lemma : measure zero ] let @xmath0 a countable group acting by homeomorphisms on a metric space @xmath457 , @xmath6 a probability distribution on @xmath0 whose support generates @xmath0 as a semigroup , and @xmath14 a @xmath6-stationary probability measure on @xmath457 . moreover , let us suppose that @xmath485 has the property that there is a sequence of positive numbers @xmath486 such that for any translate @xmath487 of @xmath480 there is a sequence @xmath488 of group elements ( which may depend on @xmath105 ) , such that the translates @xmath489 are all disjoint , and for each @xmath490 , there is an @xmath491 , such that @xmath492 . then @xmath493 . the proof of this is a variation on @xcite*lemma 3.5 , but we provide a proof for the convenience of the reader . suppose that @xmath494 . choose @xmath495 , let @xmath496 , and let @xmath497 . finally , choose @xmath105 such that the harmonic measure of @xmath487 is within @xmath498 of the supremum , i.e. @xmath499 . by hypothesis , there is a sequence of group elements @xmath500 such that the @xmath42 translates @xmath501 , are all disjoint , and for each @xmath490 there is an @xmath502 such that with @xmath503 . the harmonic measure @xmath14 is @xmath6-stationary , and hence @xmath504-stationary for any @xmath502 , which implies @xmath505 in particular , such an estimate holds for each of the @xmath42 disjoint translates @xmath506 , and furthermore , @xmath507 for each @xmath508 . this implies that @xmath509 as we chose @xmath510 , and @xmath511 , this implies that the total measure @xmath512 is greater than one , a contradiction . we now complete the proof of proposition [ prop : qzero ] . recall that the translation length @xmath36 of an isometry @xmath37 of @xmath1 is defined to be @xmath38 this definition is independent of the base point @xmath27 , and @xmath513 if and only if @xmath37 is a hyperbolic isometry , and furthermore @xmath514 . we shall apply lemma [ lemma : measure zero ] taking as @xmath480 the set of horofunctions whose local minimum lies in a given ball around the base - point : precisely , @xmath515 , where @xmath99 is the local minimum map , and @xmath516 is a ball of radius @xmath103 in @xmath1 . as @xmath0 is non - elementary it contains hyperbolic isometries of arbitrarily large translation length . choose a hyperbolic isometry @xmath37 with translation length @xmath36 greater than @xmath517 , where @xmath23 is the bound on the diameter of @xmath319 from lemma [ lemma : bdd diameter ] . now , the translates @xmath518 are all at least distance @xmath519 apart , hence no @xmath319 can intersect two of them , so the sets @xmath520 are all disjoint . as the semi - group generated by the support of @xmath6 is equal to @xmath0 , for each @xmath21 there is an @xmath502 such that @xmath521 . set @xmath522 , for some such @xmath502 , then lemma [ lemma : measure zero ] implies that @xmath493 . as this holds for every @xmath103 , this implies that @xmath523 , as required . the measure @xmath14 is therefore supported on @xmath95 , and as @xmath99 is continuous on @xmath95 , the measure @xmath14 pushes forward to a borel probability measure @xmath524 on the gromov boundary @xmath48 . the measure @xmath101 is a @xmath6-stationary probability measure on @xmath48 , so @xmath525 is a @xmath441-space . we now show that @xmath101 is non - atomic , which implies that @xmath14 is non - atomic as well . recall that if the action of @xmath0 on @xmath1 is non - elementary , then @xmath0 does not preserve any finite subset of the boundary @xmath48 . [ lemma : non - atomic ] let @xmath0 be a countable group which acts by isometries on a separable gromov hyperbolic space @xmath1 . let @xmath6 be a non - elementary probability distribution on @xmath0 , and let @xmath14 be a @xmath6-stationary measure on @xmath76 , with pushforward @xmath101 on @xmath48 under the local minimum map @xmath99 . then the measure @xmath101 is non - atomic ( hence so is @xmath14 ) . furthermore , any @xmath6-stationary measure on @xmath48 is the pushforward of a @xmath6-stationary measure on @xmath95 . we first observe that if there are atoms , then there must be an atom of maximal weight , as an infinite sequence of atoms @xmath526 of increasing weights has total measure greater than one . let @xmath502 be the maximal weight of any atom , and let @xmath527 be the collection of atoms of weight @xmath502 , which is a finite set . as @xmath101 is @xmath6-stationary , if @xmath528 , then @xmath529 as no atom has weight greater than @xmath502 , all elements of the orbit of @xmath530 under @xmath0 must have the same weight @xmath502 , so @xmath527 is a finite @xmath0-invariant set , which contradicts the fact that @xmath0 is non - elementary . finally , as the local minimum map @xmath298 is surjective , the pushforward map @xmath531 is also surjective , see e.g. @xcite*theorem 15.14 . if @xmath65 is a @xmath6-stationary measure on @xmath48 , then @xmath532 is a non - empty , convex subspace of the space of measures on @xmath95 , which can also be seen as a subspace of the space @xmath426 of probability measures on @xmath76 . thus , the closure @xmath533 of @xmath532 in @xmath426 is compact , convex , and invariant under convolution with @xmath6 , so by the schauder - tychonoff fixed point theorem there is a @xmath6-stationary measure @xmath14 in @xmath534 . however , by proposition [ prop : qzero ] , @xmath14 vanishes on the set of finite horofunctions , hence it belongs to @xmath535 , and since @xmath14 is a limit of elements in @xmath532 and @xmath536 is continuous , we also have @xmath537 , as required . the goal of this section is to prove the following step in the proof of theorem [ t : conv ] : [ p : convsub ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 . then , for @xmath450-almost every sample path @xmath430 there is a subsequence of @xmath538 which converges to a horofunction in @xmath95 . as a corollary , @xmath450-almost every sample path @xmath430 has a subsequence @xmath539 such that @xmath540 converges to a point in the gromov boundary @xmath48 . given a shadow @xmath358 , we define the _ open shadow _ @xmath541 to be the subset of @xmath76 given by @xmath542 as @xmath543 is compact and @xmath544 is open in @xmath51 , the set @xmath545 is an open subset of @xmath76 contained in the interior of @xmath546 . [ l : cover ] for each @xmath547 , the set @xmath95 is contained in a countable collection of open shadows of depth @xmath548 . we have immediately from the definition @xmath549 now , by picking a countable dense subset @xmath550 of @xmath1 and an enumeration @xmath551 of @xmath552 , we have @xmath553 as required . we shall now define a _ descending shadow sequence _ to be a sequence @xmath554 , where each @xmath555 is a finite collection of shadows , and each shadow @xmath556 has depth @xmath557 . given a descending shadow sequence @xmath558 , we shall introduce ( for convenience of notation ) an indexing of all its shadows , i.e. @xmath559 . we say a @xmath457-tuple @xmath560 of positive integers is _ an index set of depth @xmath457 _ for @xmath558 if each @xmath561 , the shadow @xmath562 is an element of @xmath563 ( hence , it has depth @xmath564 ) . given a descending shadow sequence @xmath558 , and an index set @xmath560 , we define the _ cylinder _ @xmath565 to be the intersection of the open shadows @xmath566 corresponding to the indexed shadows @xmath562 , i.e. @xmath567 so @xmath565 is an open set in @xmath76 . given a number @xmath568 , let @xmath569 be @xmath570 where the union is taken over all index sets of depth @xmath457 . the sets @xmath569 form a nested sequence of open sets in @xmath76 , i.e. @xmath571 finally , we observe that by lemma [ l : shadowhoro ] , if @xmath572 then @xmath573 , so @xmath574 [ l : conv ] let @xmath119 be a sequence of points of @xmath1 , and let @xmath575 be a descending shadow sequence . if @xmath576 intersects @xmath569 for each @xmath457 , then there exists a subsequence @xmath577 such that @xmath578 converges to a horofunction in @xmath95 . suppose the sequence @xmath576 intersects each @xmath569 . so for each @xmath568 there is an @xmath579 such that @xmath580 . as @xmath581 , each @xmath582 may lie in only finitely many @xmath569 , and so @xmath583 as @xmath584 . the horofunction @xmath585 lies in @xmath586 , which is a union of cylinders , so there is an index set @xmath587 of depth @xmath280 such that @xmath588 there are only finitely many choices for the first entry @xmath589 in each index set @xmath590 , so we may pass to a further subsequence in which @xmath591 is constant . choose @xmath592 to be the first element of this subsequence , and relabel the remaining elements as @xmath593 for @xmath594 . again , as there are only finitely many choices for the second entry @xmath595 in the index set @xmath590 , we may pass to a further subsequence in which the indices @xmath595 are constant for all @xmath594 . proceeding by induction , we may construct a subsequence @xmath577 , and a sequence of indices @xmath596 , such that @xmath597 for each @xmath280 . by compactness , the sequence @xmath578 has a limit point @xmath209 . then by lemma [ l : shadowhoro ] , for each @xmath598 and each @xmath599 , if @xmath600 , we have @xmath601 , hence @xmath602 thus by passing to the limit as @xmath603 we have @xmath604 which gives @xmath289 . [ l : largeshadow ] let @xmath605 , and let @xmath14 be a @xmath6-stationary measure on @xmath76 . then there exists a finite descending shadow sequence @xmath606 such that for each @xmath457 , @xmath607 recall by lemma [ l : cover ] that for any @xmath568 there exists a countable collection of shadows of depth @xmath608 which covers @xmath95 , and @xmath95 has full measure by proposition [ prop : qzero ] . thus , since probabilities are countably additive one can find a finite set @xmath609 of shadows of depth @xmath608 such that the union @xmath610 satisfies @xmath611 we may now set @xmath558 to be the sequence @xmath612 , which is a descending shadow sequence . we now observe that @xmath613 as required . we may now complete the proof of proposition [ p : convsub ] . we shall show that the set @xmath614 of sequences @xmath430 in the path space @xmath453 for which @xmath615 does not have limit points in @xmath95 has measure at most @xmath616 , for each @xmath617 . fix @xmath605 , and let @xmath618 be a descending shadow sequence constructed according to lemma [ l : largeshadow ] , using a measure @xmath14 which is a @xmath6-stationary weak limit of the cesro averages @xmath619 . now , suppose that the sequence @xmath615 does not have limit points in @xmath95 : then by lemma [ l : conv ] there exists an index @xmath457 such that @xmath620 does not belong to @xmath569 for any @xmath21 . thus we have the inclusion @xmath621 so if we set @xmath622 , then @xmath623 then , by definition of the cesro averages , @xmath624 , so this implies that @xmath625 furthermore as @xmath14 is the weak limit points of the @xmath626 of the cesro averages , and @xmath627 is closed , we have for each @xmath457 , @xmath628 hence by lemma [ l : largeshadow ] @xmath629 and the claim is proven . the corollary follows from proposition [ prop : phi cts on r ] . we now prove theorem [ theorem : convergence ] , convergence to the boundary . we start by showing that the action of @xmath0 on @xmath130 satisfies the following property ( which need not hold for the action of @xmath0 on @xmath76 ) : if the sequence @xmath630 converges to a point in @xmath48 , then the sequence @xmath631 converges to the same point , for any @xmath147 . [ lemma : insideconv ] let @xmath1 be a gromov hyperbolic space . if @xmath9 and the sequence @xmath632 converges to a point @xmath65 in @xmath48 , then the sequence @xmath633 converges to the same point @xmath65 , for any @xmath147 . consider the gromov product @xmath634 by the triangle inequality @xmath635 , and as @xmath490 is an isometry , @xmath636 . this implies @xmath637 which tends to infinity as @xmath21 tends to infinity , so @xmath633 converges to the same limit point as @xmath632 . the following is a version of kaimanovich @xcite*lemma 2.2 in the non - proper case . [ lemma : action ] let @xmath0 be a group of isometries of a gromov hyperbolic space @xmath1 . let @xmath638 be a sequence in @xmath0 such that @xmath639 . then there is a subsequence @xmath640 such that @xmath641 for all but at most one point of @xmath66 . we show that there is a subsequence @xmath640 for which there is at most one point @xmath642 such that @xmath643 . suppose there is a point @xmath644 in @xmath48 such that @xmath645 . this implies there is an open set @xmath646 containing @xmath65 such that infinitely many @xmath647 do not lie in @xmath646 . therefore , we may pass to a subsequence @xmath640 such that @xmath648 for all @xmath280 . now suppose there is another point @xmath649 such that @xmath650 . this implies there is an open set @xmath651 containing @xmath65 such that infinitely many @xmath652 do not lie in @xmath651 . as before , we may pass to a subsequence , which by abuse of notation we shall continue to call @xmath653 , such that @xmath654 for all @xmath21 . as @xmath655 is also an open neighbourhood of @xmath65 , it contains a shadow set of the form @xmath656 , with @xmath65 contained in the interior of the slightly smaller shadow @xmath657 , where @xmath374 will be chosen as the weak convexity constant of corollary [ cor : weak convexity ] for @xmath417-quasi - geodesics . let now @xmath29 be a @xmath134-quasigeodesic from @xmath644 to @xmath658 , and @xmath147 a point on @xmath29 . by corollary [ cor : weak convexity ] , since the endpoints @xmath659 and @xmath660 do not belong to @xmath661 , then the point @xmath662 does not belong to @xmath663 . however , this is a contradiction , as by lemma [ lemma : insideconv ] we know that @xmath664 , hence it must be eventually lie inside @xmath663 . we will use the following result , which goes back to furstenberg @xcite*corollary 3.1 , see also margulis @xcite*chapter vi . [ p : martingale ] let @xmath457 be a compact metric space on which the countable group @xmath0 acts continuously , and @xmath14 a @xmath6-stationary borel probability measure on @xmath457 . then for @xmath450-almost all sequences @xmath430 the limit @xmath665 exists in the space @xmath458 of probability measures on @xmath457 . to give a brief overview of the argument , one proves that , since the measure is stationary , for each continuous function @xmath666 the process @xmath667 is a bounded martingale , hence converges almost surely ; this defines a positive linear functional on the space @xmath668 , which is thus represented by a borel measure . furthermore , if @xmath669 a.s . , then we get the integral formula @xmath670 [ lemma : mart2 ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , let @xmath6 be a non - elementary probability distribution on @xmath0 , and let @xmath101 be a @xmath6-stationary borel measure on @xmath48 . then for almost every sample path @xmath671 , the sequence of measures @xmath427 converges to a measure @xmath672 . by lemma [ lemma : non - atomic ] , there is a @xmath6-stationary probability measure @xmath14 on @xmath95 such that @xmath101 is the pushforward of @xmath14 . applying proposition [ p : martingale ] to the action of @xmath0 on @xmath76 , the sequence @xmath425 converges to a measure @xmath673 , almost surely . moreover , since @xmath14 vanishes on @xmath94 and @xmath94 is @xmath0-invariant , the measures @xmath674 also vanish on @xmath94 for each @xmath33 ; furthermore , by equation the limit @xmath61 also vanishes on @xmath94 for almost every @xmath675 . now , note that @xmath95 is a countable intersection of open subsets of @xmath76 , hence it is a borel subset of @xmath76 , so the weak-@xmath476 topology on @xmath535 ( arising from @xmath676 ) is the relativization of the weak-@xmath476 topology on @xmath426 , see e.g. @xcite*theorem 15.4 . since @xmath669 a.s . in @xmath426 and both @xmath674 and @xmath61 belong to @xmath535 , this implies that a.s . @xmath669 in the weak-@xmath476 topology of @xmath535 . finally , since @xmath99 is continuous , the pushforward map @xmath677 is continuous , hence @xmath678 as claimed . we wish to show that for @xmath450-almost all sequences @xmath430 , the measure @xmath679 is a @xmath2-measure , and as the limit exists , it suffices to show this for any subsequence @xmath539 . we have already shown , by proposition [ p : convsub ] , that almost every sequence @xmath430 in @xmath453 has a subsequence @xmath431 such that @xmath540 converges to a point in @xmath48 , so it suffices to show that if a sequence @xmath638 has the property that @xmath632 converges to @xmath64 , then the measures @xmath680 converge to @xmath63 . [ lemma : delta ] let @xmath0 be a non - elementary countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath638 be a sequence in @xmath0 such that @xmath681 . then for any non - atomic probability measure @xmath101 on @xmath48 there is a subsequence @xmath640 such that the translations @xmath682 converge in the weak-@xmath476 topology to a delta - measure @xmath63 , supported on @xmath65 . by lemma [ lemma : action ] , there exists a subsequence @xmath683 and a point @xmath684 such that for all @xmath685 , one has @xmath686 . since the measure @xmath101 is non - atomic , @xmath687 and the claim follows by the dominated convergence theorem . [ prop : delta ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , let @xmath6 be a non - elementary probability distribution on @xmath0 , and let @xmath101 be a @xmath6-stationary probability measure on @xmath48 . then for almost every sample path @xmath671 , there is a boundary point @xmath688 such that the sequence of measures @xmath427 converges in @xmath62 to a delta - measure @xmath689 . by proposition [ p : convsub ] , almost every sample path @xmath690 has a subsequence which converges to some point @xmath688 . thus , by lemma [ lemma : delta ] , there exists a subsequence @xmath539 such that @xmath691 . by lemma [ lemma : mart2 ] , the sequence @xmath692 has a limit , hence the limit must coincide with @xmath689 . we have shown that @xmath450-almost every sequence @xmath693 converges to a @xmath2-measure on @xmath48 . finally , we now show that this implies that @xmath450-almost every sequence @xmath28 converges to some point in the gromov boundary @xmath48 . we start by showing that there are two shadows with disjoint closures and positive measure . [ l : disjshadows ] let @xmath1 be a separable , gromov hyperbolic space , and @xmath101 a non - atomic probability measure on @xmath4 . then there exist two shadows @xmath661 , @xmath663 in @xmath1 such that their closures @xmath694 in @xmath4 are disjoint , and both have positive @xmath101-measure . as @xmath4 is a separable metric space , the support of @xmath101 is a non - empty closed set , and , since @xmath101 is non - atomic , it contains at least two points @xmath695 and @xmath696 . now , for each @xmath697 we can choose a shadow @xmath698 in @xmath1 such that @xmath697 is contained in the interior of the closure @xmath699 , hence @xmath700 for each @xmath412 , and such that the distance parameter of each @xmath698 is much larger than @xmath701 , so that @xmath646 and @xmath651 are disjoint . the next proposition completes the proof of theorem [ t : conv ] . [ p : topconv ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , let @xmath6 be a non - elementary probability distribution on @xmath0 , and let @xmath101 be a @xmath6-stationary measure on @xmath48 . suppose that the sequence @xmath693 converges to a delta - measure @xmath63 on @xmath48 , for some @xmath702 . then @xmath703 converges to @xmath65 in @xmath66 . let @xmath704 and @xmath705 as in lemma [ l : disjshadows ] , applied to the stationary measure @xmath706 . set @xmath707 ; as @xmath646 and @xmath651 both have positive @xmath706-measure , @xmath708 is strictly greater than zero . as the sequence of measures @xmath709 converges to the delta - measure @xmath63 , for any shadow set @xmath710 containing @xmath65 in its interior , there is an @xmath42 such that @xmath711 as @xmath712 , the set @xmath713 intersects @xmath714 , hence @xmath715 intersects the ( slightly larger ) shadow @xmath716 , and similarly @xmath717 intersects @xmath718 . for each @xmath412 , let us pick @xmath719 and denote @xmath720 . by disjointness of @xmath646 and @xmath651 , there exists a constant @xmath374 such that @xmath721 for each @xmath722 and @xmath723 . moreover , since @xmath0 acts by isometries , we get @xmath724 hence we can bound the distance from @xmath725 to the geodesic @xmath726 $ ] as @xmath727 ) = { { ( x_1\cdotx_2)_{w_n x_0 } } } + o(\delta ) \leqslant c + o(\delta);\ ] ] note that the constant on the right - hand side depends only on @xmath646 , @xmath651 and @xmath2 , and not on @xmath21 . as @xmath728 and @xmath729 both lie in @xmath718 , by weak convexity ( corollary [ cor : weak convexity ] ) , a geodesic @xmath726 $ ] connecting them lies in a shadow @xmath730 , and as @xmath725 is a bounded distance from @xmath726 $ ] , this implies that @xmath725 lies in the slightly larger shadow @xmath731 , for all @xmath732 . as this holds for all shadow sets @xmath714 containing @xmath65 in their interiors , this implies that @xmath733 converges to @xmath65 , as required . theorem [ t : conv ] implies the convergence statement in theorem [ theorem : convergence ] , and so it remains to show that the hitting measure @xmath101 is the unique @xmath6-stationary measure on @xmath48 , and the convolution measures @xmath734 converge weakly to @xmath101 . by proposition [ prop : delta ] , for any @xmath6-stationary measure @xmath101 on @xmath48 , for @xmath450-almost every sample path @xmath690 , the sequence of measures @xmath735 converges to @xmath689 , where @xmath736 is the limit point of the sample path @xmath12 , hence it only depends on @xmath690 , not on @xmath101 . thus , uniqueness follows from the integral formula . furthermore , @xmath737 we may take the limit as @xmath21 tends to infinity , and by the integral formula , the distribution of the limit points is given by @xmath14 , so @xmath473 weakly converges to @xmath101 . in this section we use convergence to the boundary to show the results on positive drift , sublinear tracking and translation length . we will no longer use measures on the horofunction boundary , and so we shall from now on simply denote by @xmath14 the hitting measure on the gromov boundary @xmath48 . moreover , given @xmath738 , the symbol @xmath739 from now on will always mean the closure of @xmath52 in the space @xmath126 . we start by showing that the measure of a shadow tends to zero as the distance parameter of the shadow tends to infinity . in order to simplify notation , we shall denote @xmath740 the set of shadows based at @xmath27 , with centers on the orbit @xmath741 and with distance parameter @xmath742 . [ prop : shadow bound ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 . let @xmath6 be a non - elementary probability distribution on @xmath0 , and let @xmath14 be the hitting measure on @xmath48 . then we have @xmath743 this result also holds for the reflected measures @xmath744 and @xmath745 determined by @xmath746 , as if @xmath6 satisfies the hypotheses of proposition [ prop : shadow bound ] , then so does @xmath747 . the result follows from propositions [ prop : shadow open ] and [ prop : bhm ] , which says that a shadow centered at @xmath393 of distance parameter @xmath103 is contained in a ball of radius @xmath748 in the metric @xmath380 on @xmath48 , where @xmath374 is independent of @xmath103 , and as @xmath14 is non - atomic , the measure of a such a ball tends to zero as the radius of the ball tends to zero . if @xmath6 has bounded range in @xmath1 then the argument from @xcite*lemma 2.10 shows that the @xmath14- and @xmath472-measures of a shadow @xmath749 decay exponentially in the distance parameter , i.e. there are positive constants @xmath23 and @xmath750 such that @xmath751 for @xmath129 a subset of @xmath1 , let @xmath752 denote the probability that a random walk starting at @xmath43 ever hits @xmath129 in forward time , i.e. @xmath753 similarly , let @xmath754 be the probability that a random walk starting at @xmath43 ever hits @xmath129 in reverse time , i.e. the probability that @xmath755 lies in @xmath129 for some @xmath756 . [ prop : hitting ] let @xmath0 be a countable group which acts by isometries on a separable gromov hyperbolic space @xmath1 , and @xmath6 a non - elementary probability distribution on @xmath0 . then @xmath757 this immediately implies the same result with @xmath758 replaced by @xmath759 , by replacing @xmath6 with the reflected measure @xmath760 . suppose a sample path starting at @xmath27 hits a shadow @xmath761 with @xmath762 in forward time , at @xmath393 say . let @xmath29 be a geodesic from @xmath27 to @xmath43 , and let @xmath53 be a nearest point on @xmath29 to @xmath393 , as illustrated below in figure [ pic : hitting shadows2 ] . = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( -1 , 0 ) node [ point , label = below:@xmath27 ] ( 7 , 0 ) node [ point , label = below:@xmath43 ] ; ( 6 , 1.5 ) node [ point , label = right:@xmath389 ( 6,0 ) node [ point , label = below:@xmath53 ] ; ( 0 , 1.5 ) node [ point , label = right:@xmath44 ] ( 0,0 ) node [ point , label = below:@xmath145 ] ; ( 5 , 2 ) ( 5 , -2 ) node [ right ] @xmath764 ; ( 1 , 2 ) ( 1 , -2 ) node [ right ] @xmath765 ; consider the shadow @xmath765 , for some fixed @xmath373 which we will choose later . the main idea is that if the random walk ever hits @xmath661 , it will likely converge inside the closure of @xmath663 , and the probability of that happening is small if the distance parameter of @xmath663 is large . to make the idea precise , let @xmath44 be a point in the complement of @xmath663 , and let @xmath145 be the nearest point projection of @xmath44 to @xmath29 . by proposition [ prop : npp shadow ] , @xmath53 is within distance @xmath125 of @xmath661 , and @xmath145 is within distance @xmath125 of the complement of @xmath663 , so the distance between @xmath53 and @xmath145 is at least @xmath766 . now using proposition [ prop : npp2 ] , if @xmath767 , then the gromov product satisfies @xmath768 therefore , the complement of @xmath663 is contained in a shadow @xmath769 ( where @xmath770 ) of distance parameter @xmath766 . fix some positive @xmath771 ; then , since the measure of shadows tends to zero as the distance parameter tends to infinity ( proposition [ prop : shadow bound ] ) , there is a number @xmath772 sufficiently large such that @xmath773 for all shadows @xmath774 with distance parameter larger than @xmath772 . as a consequence , if we choose @xmath107 such that @xmath775 in the above construction , we have @xmath776 hence , since we proved @xmath777 , @xmath778 now , by the markov property of the random walk , the conditional probability of ending up in @xmath779 after hitting an element @xmath780 at time @xmath280 is given by @xmath781for each @xmath280 and @xmath37 , and such probability is large by equation . this implies the following lower bound on the probability of ending up in @xmath779 : @xmath782 which , by recalling the definitions of @xmath14 and @xmath758 , becomes @xmath783 now , as the distance parameter of @xmath661 tends to @xmath784 , so does the distance parameter of @xmath663 , hence @xmath785 by proposition [ prop : shadow bound ] , and by the above equation @xmath786 tends to @xmath69 , as required . as @xmath786 is an upper bound for @xmath787 for any @xmath21 , equation implies the following corollary . [ cor : mun estimate ] let @xmath0 be a countable group which acts by isometries on a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 . then there is a function @xmath788 , with @xmath789 as @xmath790 such that for all @xmath21 one has @xmath791 as the reflected random walk also satisfies the hypotheses of corollary [ cor : mun estimate ] , we obtain a similar result for @xmath744 , though possibly for a different function @xmath105 . [ prop : positive ] let @xmath0 be a non - elementary , countable group acting by isometries on a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a probability distribution on @xmath0 , whose support generates @xmath0 as a semigroup . then there is a number @xmath367 such that for any @xmath792 the closure of the shadow @xmath793 has positive hitting measure for the random walk determined by @xmath6 . let us first assume @xmath794 . by proposition [ prop : shadow open ] , there is a constant @xmath367 such that every shadow @xmath795 contains a limit point @xmath65 of @xmath371 in the interior of its closure . we may now follow the same argument as in proposition [ prop : hitting ] . choose a shadow @xmath796 containing @xmath65 such that @xmath797 is at least @xmath766 ( note that here the roles of @xmath661 and @xmath663 are reversed with respect to proposition [ prop : hitting ] , as @xmath798 ) . in particular , this implies that for any point @xmath799 , the complement of @xmath661 is contained in a shadow @xmath800 with distance parameter at least @xmath107 . by proposition [ prop : shadow bound ] the measure of shadows tends to zero as the distance parameter tends to infinity , so given a positive number @xmath771 we may choose @xmath107 sufficiently large so that @xmath773 for all shadows @xmath801 . let @xmath638 be a sequence in @xmath0 such that @xmath632 converges to @xmath65 , and let @xmath490 be an element of the sequence with @xmath802 , and let @xmath280 be such that @xmath803 . now using the markov property of the random walk , the conditional probability of converging to the closure of @xmath661 , having hit @xmath663 , is at least @xmath804 , and so @xmath805 which is positive , as required . now , the case @xmath806 can be reduced to the previous one ; indeed , given @xmath807 , there exists by hypothesis an @xmath21 such that @xmath808 , which implies @xmath809 which by the markov property of the walk equals @xmath810 and this is positive by the previous case . in this section we prove theorem [ theorem : linear progress ] , i.e. that the sample paths @xmath28 of the random walk have positive drift in @xmath1 . it will be convenient to consider the @xmath280-step random walk @xmath811 , and introduce the notation @xmath812 for each @xmath508 . let @xmath813 be the random variable given by the distance in @xmath1 traveled by the sample path from time @xmath814 to time @xmath815 , i.e. @xmath816 for fixed @xmath280 , the @xmath817 are independent identically distributed random variables with common distribution @xmath818 . given a number @xmath41 , we say a subsegment @xmath819 $ ] of the sample path is _ persistent _ if the following three conditions are satisfied : @xmath820 the constant @xmath374 in is the weak convexity constant from corollary [ cor : weak convexity ] , while @xmath821 will be depend on @xmath2 and will be chosen later . a persistent subsegment is illustrated in figure [ pic : persistent ] below . = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( 0 , 0 ) node [ point , label = above:@xmath822 ( 8 , 0 ) node [ point , label = below:@xmath823 ; ( 6 , 2 ) ( 6 , -2 ) node [ right ] @xmath824 ; ( 2 , 2 ) ( 2 , -2 ) node [ left ] @xmath825 ; ( -1 , -1 ) ( 0,0 ) ( 1.5,-1 ) ( 0.5 , -1 ) ( 4.5 , 2 ) ( 3 , 1.5 ) ( 8 , 1 ) ( 7 , 1.5 ) ( 8,0 ) ( 9 , 1 ) ; choose an @xmath708 , with @xmath826 . we now show that given such a choice of @xmath708 , we may choose both @xmath41 and @xmath280 sufficiently large such that for any @xmath508 each of the three conditions holds with probability at least @xmath804 . the probability that holds is the same as the probability that @xmath827 never hits the complement of the shadow @xmath828 for any @xmath829 . as the complement of this shadow is contained in a shadow @xmath830 where @xmath831 , the probability that holds is at least @xmath832 which equals by the markov property @xmath833 the distance parameter of @xmath834 , which equals the distance parameter of @xmath698 , is @xmath835 ; hence , by proposition [ prop : hitting ] , we may choose @xmath41 sufficiently large such that is at least @xmath804 . a similar argument show that the probability that holds is at least @xmath836 and again we may choose @xmath41 sufficiently large such that is at least @xmath804 . finally , the probability that holds is @xmath837 , since @xmath838 and @xmath839 have the same law . we have shown that almost every sample path converges to a point in the gromov boundary , so in particular , sample paths are transient on bounded sets . this implies that for any @xmath41 and @xmath708 , there is a sufficiently large @xmath280 , depending on @xmath41 and @xmath708 , such that @xmath840 as required . therefore , for the choice of @xmath841 and @xmath280 described above , the probability that each condition holds individually is at least @xmath842 . the three conditions need not be independent , but the probability that all three hold simultaneously is at least @xmath843 , which is positive as @xmath844 . thus , if we define for each @xmath508 the random variable @xmath845 @xmath846 \text { is persistent } \\ 0 \text { otherwise . } \\ \end{array } \right.\ ] ] we get that the @xmath847 are identically distributed ( but not independent ) , with finite expectation since they are bounded ; moreover , for each @xmath508 @xmath848 we now show that the number of persistent segments lying between @xmath27 and @xmath827 gives a lower bound on the distance @xmath849 . let @xmath29 be a geodesic from @xmath27 to @xmath850 , and suppose that @xmath819 $ ] is a persistent subsegment of the sample path . by , @xmath27 lies in @xmath825 for @xmath851 , and by , @xmath274 lies in @xmath824 for @xmath852 . as the two shadows are at least distance @xmath853 apart , the geodesic @xmath29 has a subsegment @xmath854 of length at least @xmath855 which fellow travels with @xmath856 $ ] , and which is disjoint from both @xmath857 and @xmath858 . now let @xmath859 $ ] be a different persistent subsegment . the same argument as above shows that there is a subsegment @xmath860 of @xmath29 of length at least @xmath861 which fellow travels with @xmath862 $ ] . we now show that @xmath854 and @xmath860 are disjoint subsegments of @xmath29 . up to relabeling , we may assume that @xmath863 . then both @xmath864 and @xmath865 lie in @xmath824 , and so by weak convexity , corollary [ cor : weak convexity ] , any geodesic connecting them lies in @xmath866 , and so in particular @xmath854 and @xmath860 are disjoint subsegments of @xmath29 . therefore the distance @xmath849 is at least @xmath861 times the number of persistent subsegments between @xmath27 and @xmath827 . we will now apply kingman s subadditive ergodic theorem , @xcite , using the following version from @xcite*theorem 8.10 : [ theorem : kingman ] let @xmath867 be a probability space and @xmath868 a measure preserving transformation . if @xmath869 is a subadditive sequence of non - negative real - valued random variables on @xmath449 , that is , @xmath870 for all @xmath871 , and @xmath872 has finite first moment , then there is a @xmath129-invariant random variable @xmath873 such that @xmath874 @xmath450-almost surely , and in @xmath875 . in order to apply the theorem , let us define for each @xmath21 the variable @xmath876 \text { is persistent } \}\ ] ] which gives the number of persistent subsegments along a given sample path from @xmath27 to @xmath850 . the random variables @xmath877 are non - negative and have finite expectation , since @xmath878 for each @xmath21 , and the sequence is subadditive by the markov property . moreover , as expectation is additive , we get from equation @xmath879 with @xmath880 . we now apply theorem [ theorem : kingman ] taking as @xmath449 the step space of the @xmath881-step random walk , @xmath129 the shift map , and the @xmath882 as random variables ( for fixed @xmath280 ) ; we get that the sequence @xmath883 converges almost surely and in @xmath85 to some random variable @xmath884 ; moreover , since @xmath129 is ergodic , @xmath884 must be constant almost everywhere , thus there exists a constant @xmath885 such that @xmath886 in @xmath85 ; finally , since @xmath887 , we have that @xmath373 . thus , since @xmath882 is a lower bound for the distance @xmath888 , we get almost surely for the @xmath881-step random walk @xmath889 which proves the first part of theorem [ theorem : linear progress ] , where we make no assumptions on the moments of @xmath6 . for the second part of theorem [ theorem : linear progress ] , we assume that @xmath6 has finite first moment with respect to the distance function @xmath26 . in this case , we can apply kingman s theorem directly to @xmath888 , and we know that the limiting value @xmath22 is positive , by the previous case . finally , if the support of @xmath6 is bounded in @xmath1 , then the arguments from @xcite apply directly . we will now prove theorem [ theorem : sublinear ] , using the following sublinearity result from tiozzo @xcite . [ lemma : tiozzo ] let @xmath890 be a non - negative measurable function , @xmath891 an ergodic , measure preserving transformation , and suppose that @xmath892 then @xmath893 for almost all @xmath894 . in order to apply it in this case , let us note that there are constants @xmath142 and @xmath24 , depending only on @xmath2 , such that any two distinct points in @xmath48 are connected by a @xmath895-quasigeodesic . we shall write @xmath896 for the set of @xmath897-quasigeodesics connecting @xmath43 and @xmath44 . we then define @xmath898 as @xmath899 as @xmath14 and @xmath745 are non - atomic , @xmath900 gives measure zero to the diagonal in @xmath901 , so @xmath902 is non - empty @xmath903-almost surely , and the function @xmath904 is well - defined @xmath450-almost surely . then by the triangle inequality , @xmath905 , and so @xmath906 lies in @xmath875 . thus it follows from lemma [ lemma : tiozzo ] ( where @xmath455 is the shift map on the step space ) that sample paths track quasigeodesics sublinearly , i.e. @xmath907 proving the first part of theorem [ theorem : sublinear ] . to prove the second part , we need to show that , if @xmath6 has bounded support in @xmath1 , then the tracking is in fact logarithmic . we apply the argument from blachre , hassinsky and mathieu @xcite*section 3 , combined with our exponential decay of shadows . we now give the details for the convenience of the reader . we first show that the distribution of distances from the locations of the sample path to the quasigeodesic satisfies an exponential decay property . let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 . there are positive constants @xmath23 and @xmath750 , which depend on @xmath6 , such that @xmath908 where @xmath29 is a quasigeodesic ray from @xmath27 to the limit point in @xmath48 of @xmath28 . by the definition of @xmath909 , @xmath910 ) , \ ] ] where @xmath736 is the limit point of @xmath911 in @xmath48 , which exists for @xmath450-almost every @xmath675 . applying the isometry @xmath912 gives @xmath913 ) .\ ] ] recall from , that the gromov product @xmath111 may be estimated up to an error of @xmath125 in terms of the distance @xmath914)$ ] , and a similar estimate holds if one of @xmath43 or @xmath44 is a point in @xmath48 , and @xmath170 $ ] is a quasigeodesic connecting them . this implies that @xmath915 so by the definition of a shadow , the condition @xmath916 is equivalent to @xmath917 where the parameter @xmath41 is given by @xmath918 the boundary point @xmath919 only depends on the increments of the random walk of index greater than @xmath21 , so @xmath920 and @xmath33 are independent . furthermore , the distribution of @xmath919 is equal to @xmath14 . therefore @xmath921 and as @xmath6 has bounded range in @xmath1 , we may use the exponential decay estimate for shadows , which gives @xmath908 as required . it follows immediately from the proposition above that there is a constant @xmath922 such that @xmath923 the logarithmic tracking result , @xmath924 then follows from the borel - cantelli lemma . this completes the proof of theorem [ theorem : sublinear ] . we briefly review some results about the translation length of isometries , see e.g. bridson and haefliger @xcite or fujiwara @xcite . we start by observing that the translation length of an isometry @xmath37 may be estimated in terms of the distance it moves the basepoint @xmath27 , together with the gromov product of @xmath393 and @xmath925 . [ prop : hyperbolic ] there exists a constant @xmath926 , which depends only on @xmath2 , such that the following holds . for any isometry @xmath37 of a @xmath2-hyperbolic space @xmath1 , if @xmath37 satisfies the inequality @xmath927 then the translation length of @xmath37 is @xmath928 this is well known , but we provide a proof in the appendix for the convenience of the reader . in order to complete the proof of theorem [ theorem : translation ] , we shall now estimate the probability that the translation length is small for a sample path of length @xmath21 . to apply the estimate for translation length we need a lower bound for @xmath929 , which is given by positive drift , and an upper bound for the gromov product @xmath930 , which we now obtain . let @xmath931 ; we shall introduce the notation @xmath932 , and we may think of @xmath933 as an approximate midpoint of the sample path from @xmath27 to @xmath725 , and of @xmath934 as an approximate midpoint of the inverse sample path from @xmath27 to @xmath935 . note that for each @xmath502 , the @xmath0-valued processes @xmath936 and @xmath937 are independent . because of this independence , and the fact that the hitting measures are non - atomic , it is easy to prove the following upper bound on the gromov product @xmath938 . [ lemma : gp upper ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 . if @xmath939 is any function such that @xmath940 as @xmath288 , then we have @xmath941 for all @xmath21 , where @xmath942 . by the definition of shadows , @xmath943 which tends to @xmath184 as @xmath288 . we will now use the fact that if the gromov products @xmath944 and @xmath945 are large , and the gromov product @xmath946 is small , then the two gromov products @xmath930 and @xmath947 are equal , up to bounded additive error depending only on @xmath2 . this follows from the following lemma , which is a standard exercise in coarse geometry . we omit the proof , but the appropriate approximate tree is illustrated in figure [ pic : gromov product ] , with the points labeled according to our application . [ lemma : gp approx ] for any four points @xmath948 and @xmath949 in a gromov hyperbolic space @xmath1 , if there is a number @xmath107 such that @xmath950 , @xmath951 and @xmath952 then @xmath953 . = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( 0,0 ) node [ point , label = below:@xmath27 ] ( 5 , 0 ) node [ point , label = below:@xmath933 ] ; ( 4,0 ) ( 4 , 2 ) node [ point , label = right:@xmath725 ] ; ( 1,0 ) ( 1 , -4 ) node [ point , label = right:@xmath956 ; ( 1,-3 ) ( -1 , -3 ) node [ point , label = below:@xmath935 ] ; we now observe that with high probability , the gromov product @xmath944 is large . recall that by linear progress there exists @xmath16 such that @xmath958 [ lemma : gp lower ] let @xmath0 be a countable group of isometries of a separable gromov hyperbolic space @xmath1 , and let @xmath6 be a non - elementary probability distribution on @xmath0 , and @xmath22 as in eq . . then for any @xmath959 we have @xmath960 for all @xmath21 , where @xmath942 . note that by definition of shadows , we have the equality @xmath961 the result now follows by positive drift . the same argument applied to @xmath935 , which has approximate midpoint @xmath962 , shows that @xmath963 for any @xmath959 . now using lemma [ lemma : gp approx ] , together with the lower bounds on the gromov products of @xmath964 and @xmath965 from lemma [ lemma : gp lower ] , and the upper bound on the gromov product @xmath966 from lemma [ lemma : gp upper ] implies @xmath967 for any function @xmath939 such that @xmath940 as @xmath288 . applying this to the estimate for translation length , shows that the probability that @xmath968 tends to @xmath184 as @xmath288 , as required . if @xmath6 has bounded support in @xmath1 , then this happens exponentially fast , by @xcite . in this section we prove theorem [ theorem : poisson ] , i.e. we show that if the action of @xmath0 on @xmath1 is acylindrical and @xmath6 has finite entropy and finite logarithmic moment , then in fact the gromov boundary with the hitting measure is the poisson boundary . we shall assume from now on that @xmath0 is a non - elementary , countable group of isometries of a separable gromov hyperbolic space @xmath1 , and @xmath6 a probability measure on @xmath0 whose support generates @xmath0 as a semigroup . recall that the entropy of @xmath6 is @xmath969 and @xmath6 is said to have _ finite entropy _ if @xmath970 . the measure @xmath6 is said to have _ finite logarithmic moment _ if @xmath971 let us recall the definition of acylindrical action , which is due to sela @xcite for trees , and bowditch @xcite for general metric spaces . we say a group @xmath0 acts _ acylindrically _ on a gromov hyperbolic space @xmath1 , if for every @xmath40 there are numbers @xmath972 and @xmath973 such that for any pair of points @xmath43 and @xmath44 in @xmath1 , with @xmath45 , there are at most @xmath42 group elements @xmath37 in @xmath0 such that @xmath46 and @xmath47 . for a discussion and several examples of acylindrical actions on hyperbolic spaces , see @xcite . the proof will use kaimanovich s strip criterion from @xcite . briefly , the criterion uses the existence of `` strips '' , that is subsets of @xmath0 which are associated to each pair of boundary points in a @xmath0-equivariant way . in order to apply the criterion , however , one also needs to control the number of elements in the strips ; in fact , we will show that for each strip the number of elements whose images in @xmath1 lie in a ball of radius @xmath103 can grow at most linearly in @xmath103 . in a proper space , one may often choose the strips to consist of all geodesics connecting the endpoints of the sample path , but in our case , this usually gives infinitely many points in a ball of finite radius . instead , we observe that by recurrence , the sample path returns close to a geodesic connecting its endpoints for a positive density of times @xmath974 . using this it can be shown that there are infinitely many pairs of locations @xmath725 and @xmath975 , where the sample path has gone a definite distance along the geodesic in bounded time . in fact , we may choose a suitable group element @xmath976 , and look at all group elements @xmath37 whose orbit points @xmath393 are close to a geodesic @xmath29 , such that both @xmath393 and @xmath977 are close to @xmath29 . we shall call the collection of such group elements _ bounded geometry _ elements , and we will choose our strips to consist of these elements . we will use acylindricality to show that this set is locally finite , and in fact the intersection of its image in @xmath1 with @xmath978 grows at most linearly with @xmath103 . let us now make this precise . let @xmath979 be a group element , @xmath980 two constants , and @xmath981 two boundary points . we say that a group element @xmath37 has _ @xmath982-bounded geometry _ with respect to the pair of boundary points @xmath384 , @xmath983 if the three following conditions hold : 1 . @xmath984 ; 2 . @xmath384 belongs to the interior of the closure ( in @xmath66 ) of @xmath985 ; 3 . @xmath986 belongs to the interior of the closure of @xmath987 . this is illustrated in figure [ pic : bounded geometry ] below . we shall write @xmath988 for the set of bounded geometry elements determined by @xmath384 and @xmath986 ( or @xmath989 if we want to explicitly keep track of the constants ) . this definition is @xmath0-equivariant , i.e. @xmath990 for any @xmath208 . we will refer to the image of a bounded geometry element in @xmath1 under the orbit map as a _ bounded geometry point_. = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( 0,0 ) circle ( 5 cm ) ; ( 60:5 ) node [ point , label = right:@xmath986 ] ; ( -60:5 ) node [ point , label = right:@xmath384 ] ; ( 85:5 ) .. controls ( 1 , 0.5 ) .. ( 25:5 ) node [ right ] @xmath993 ; ( -140:5 ) .. controls ( 1.5 , 0.75 ) .. ( -40:5 ) node [ right ] @xmath994 ; ( 1 , -1 ) node [ point , label = right:@xmath389 ; ( 2 , 2 ) node [ point , label = right:@xmath977 ] ; ( -2 , 2 ) node @xmath1 ; ( -5 , 0 ) node [ left ] @xmath48 ; we say a set of group elements @xmath996 is _ locally finite _ if the set @xmath997 is finite for all @xmath162 and all @xmath998 , and that @xmath996 has _ linear growth _ if there is a constant @xmath374 such that for all @xmath999 @xmath1000 we now show that the set of bounded geometry elements has linear growth . [ p : linear ] there exists @xmath1001 such that for any @xmath1002 , there exists @xmath367 , such that for any @xmath399 , there exists a constant @xmath374 such that we have the estimate @xmath1003 for any @xmath1004 , any @xmath1005 and any group element @xmath979 . in order to prove the proposition , let us start by proving that the number of bounded geometry elements in a ball @xmath1006 is bounded in terms of @xmath23 . [ prop : local bound ] there exists @xmath1001 such that , for any @xmath1002 , there exists @xmath367 such that for any @xmath399 and any group element @xmath979 , we have the estimate @xmath1007 for any pair of boundary points @xmath1008 and @xmath1009 , and any point @xmath1010 . = [ circle , draw , fill = black , inner sep=0pt , minimum width=2.5pt ] ( -1 , 3.5 ) ( 24 , 3.5 ) node [ below ] @xmath29 ; ( -1 , 1.5 ) ( 24 , 1.5 ) node [ below ] @xmath1011 ; ( 4,2 ) node [ point , label = right:@xmath43 ] ; ( 4 , 2 ) circle ( 3.5 cm ) ; ( 4 , 2 ) + ( -60:3.5 ) node [ right ] @xmath1012 ; ( 5,4 ) node [ point , label = right:@xmath389 ; ( 3,1 ) node [ point , label = right:@xmath1013 ; ( 14.5,3 ) node [ point , label = right:@xmath977 ] ; ( 17.5,0.5 ) node [ point , label = right:@xmath1014 ; ( 17.5,3.5 ) node [ point , label = above:@xmath53 ] ; ( 17.5,1.5 ) node [ point , label = above:@xmath1015 ; recall from section [ section : qg ] that we have chosen @xmath142 and @xmath24 to be two numbers such that every pair of points in the gromov boundary @xmath48 are connected by a continuous @xmath897-quasigeodesic . we shall choose @xmath22 to be a morse constant for the @xmath134-quasigeodesics , i.e. for any pair of points @xmath43 and @xmath44 in a @xmath134-quasigeodesic @xmath29 , the segment of @xmath29 between @xmath43 and @xmath44 is contained in an @xmath22-neighbourhood of any geodesic connecting @xmath43 and @xmath44 . let @xmath37 be a bounded geometry element , and write @xmath661 for @xmath1016 and @xmath663 for @xmath1017 . as @xmath37 has bounded geometry , each @xmath417-quasigeodesic from @xmath384 to @xmath986 passes within distance @xmath1018 of both @xmath1019 and @xmath1020 . therefore we may choose @xmath1001 to be sufficiently large such that for any @xmath1002 the distance from any bounded geometry point @xmath393 to a @xmath134-quasigeodesic connecting @xmath384 and @xmath986 is at most @xmath1021 . furthermore , we will choose @xmath1001 to be larger than the quasigeodesic constants @xmath142 and @xmath24 , and also larger than the morse constant @xmath22 . let @xmath393 and @xmath1022 be two bounded geometry elments with respect to the same boundary points @xmath1023 , and the same element @xmath976 . we may write @xmath1024 , for some group element @xmath1025 . this is illustrated in figure [ pic : geodesic ] above . the isometry @xmath105 moves the point @xmath1026 distance at most @xmath1027 . now let @xmath29 be a @xmath417-quasigeodesic joining @xmath384 to @xmath986 . by construction , both @xmath29 and @xmath1028 have endpoints in @xmath1029 and @xmath1030 , hence they both pass within distance @xmath1031 from both @xmath1032 and @xmath1033 . let us now consider @xmath1034 ; we now show that the isometry @xmath105 also moves the point @xmath44 a bounded distance , which yields the claim by definition of an acylindrical action . let @xmath53 be a closest point on @xmath29 to @xmath44 , and @xmath1035 be a closest point on @xmath1011 to @xmath44 , so by construction @xmath1036 as @xmath1037 , this implies that @xmath1038 therefore , since @xmath1039 , and both @xmath1040 and @xmath1035 lie on the quasigeodesic @xmath1028 , we have @xmath1041 and so @xmath1042 therefore , @xmath105 moves each of @xmath43 and @xmath44 distance at most @xmath1043 , and so by acylindricality there are at most @xmath1044 possible choices for @xmath105 , as long as @xmath1045 , as required . let @xmath29 be a @xmath417-quasigeodesic connecting @xmath384 and @xmath986 . we shall choose the number @xmath1001 to be the same as the number @xmath1001 from lemma [ prop : local bound ] . then @xmath23 is sufficiently large such that every element of @xmath988 has an image in @xmath1 which lies within distance at most @xmath1031 of @xmath29 , and any pair of points @xmath1046 and @xmath1047 on the quasigeodesic are distance at most @xmath1031 apart . therefore @xmath1048 is covered by balls of the form @xmath1049 for @xmath1050 , and the claim follows from applying lemma [ prop : local bound ] to each of these balls . given a bi - infinite sample path @xmath1051 , we shall define the forward and backward limit points to be @xmath1052 as the forward and backward random walks converge to the gromov boundary , these limit points are defined for @xmath450-almost all @xmath690 , and the joint distribution of the pair @xmath1053 is @xmath1054 . for any bi - infinite sequence @xmath1055 , we define @xmath1056 to be the set @xmath1057 of bounded geometry elements determined by the limit point @xmath1058 of the forward random walk and the limit point @xmath1059 of the backward random walk . finally , we show that we can choose @xmath23 , @xmath41 , and @xmath976 such that the set of bounded geometry elements is non - empty and locally finite for @xmath1054-almost all @xmath1060 . [ prop : non - empty ] there are constants @xmath980 and a group element @xmath979 such that the set @xmath1061 of bounded geometry elements has linear growth and is non - empty ( in fact , infinite ) for @xmath1054-almost all pairs @xmath1062 . by proposition [ prop : positive ] , we can choose @xmath23 large enough so that for any group element @xmath976 , the closure of the shadow @xmath1063 has positive @xmath14-measure , and the closure of the shadow @xmath1064 has positive @xmath1065-measure . thus , the probability that the group identity element @xmath184 lies in @xmath1056 is positive , because @xmath1066 consider the probability that the location of the random walk @xmath33 lies in @xmath1067 , i.e. @xmath1068 therefore the events @xmath1069 occur with the same positive probability @xmath53 , though they are not independent . by ergodicity of the shift map , the proportion of locations @xmath1070 satisfying @xmath1071 converges to @xmath53 as @xmath42 tends to infinity . as @xmath703 converges to the boundary @xmath450-almost surely , @xmath1056 contains infinitely many elements @xmath450-almost surely . we remind the reader of kaimanovich s strip criterion from @xcite*theorem 6.4 . we shall write @xmath1072 for all group elements whose image in @xmath1 under the orbit map is distance at most @xmath103 from the basepoint @xmath27 , i.e. @xmath1073 let @xmath6 be a probability measure with finite entropy on @xmath0 , and let @xmath1074 and @xmath1075 be @xmath6- and @xmath1076-boundaries , respectively . if there exists a measurable @xmath0-equivariant map @xmath52 assigning to almost every pair of points @xmath1077 a non - empty `` strip '' @xmath1078 , such that for all @xmath37 @xmath1079 for @xmath1080-almost every @xmath1081 , then @xmath1074 and @xmath1082 are the poisson boundaries of the random walks @xmath441 and @xmath1083 , respectively . in order to prove theorem [ theorem : poisson ] , we define the strip @xmath1084 as the set @xmath1085 of bounded geometry elements . by right multiplication by @xmath1086 , the set @xmath1087 has the same cardinality as @xmath1088 furthermore , @xmath1089 and so @xmath1090 proposition [ prop : non - empty ] shows that there are suitable choices of @xmath980 and @xmath976 such that the sets of bounded geometry elements are non - empty almost surely and have linear growth , so there is a number @xmath23 such that @xmath1091 therefore , it suffices to show that almost surely @xmath1092 as @xmath288 , and this follows from the fact that @xmath6 has finite logarithmic moment , as we now briefly explain . finite logarithmic moment implies that @xmath1093 almost surely , and so for any @xmath372 , we have @xmath1094 for all @xmath21 sufficiently large . by the triangle inequality @xmath1095 and so @xmath1096 as this holds for all @xmath372 , this implies that @xmath1097 as @xmath288 , as required . finally , the statement that the map @xmath52 is measurable means that for any @xmath208 , the set @xmath1098 is a borel set ; this holds , since by definition , @xmath1099 if and only if @xmath1023 belongs to the product of the closures of two shadows , which is closed , hence borel . this completes the proof of theorem [ theorem : poisson ] . in this section we provide a proof of proposition [ prop : hyperbolic ] , which estimates the translation length in terms of the distance an isometry moves the basepoint , and the gromov product . there exists a constant @xmath926 , which depends only on @xmath2 , such that the following holds . for any isometry @xmath37 of a @xmath2-hyperbolic space @xmath1 , if @xmath37 satisfies the inequality @xmath1100 then the translation length of @xmath37 is @xmath1101 we start by showing that if @xmath29 is a geodesic segment from @xmath27 to @xmath1102 , then @xmath1103 is contained in a bounded neighbourhood of @xmath29 , for all @xmath1104 . we shall write @xmath279 for @xmath1103 . note that , since the action is isometric , we have for each @xmath280 @xmath1105 let @xmath1107 be a nearest point on @xmath29 to @xmath279 , and let @xmath279 be an element of @xmath1108 furthest from @xmath29 . consider the quadrilateral formed by @xmath1109 and @xmath1110 , as illustrated below in figure [ pic : bounded neighbourhood ] . let @xmath145 be a nearest point to @xmath279 on a geodesic segment from @xmath1117 to @xmath1118 . by the estimate for the gromov product in terms of distance to a geodesic , @xmath1119 ) + o(\delta),\ ] ] and as @xmath37 is an isometry , this implies that @xmath1120 by thin triangles , the point @xmath145 lies within distance @xmath1121 of at least one of the other three sides of the quadrilateral . suppose @xmath145 lies within @xmath1121 of a geodesic from @xmath1117 to @xmath1110 . since @xmath279 is the furthest point from the geodesic @xmath29 , we have @xmath1122 , and since @xmath145 lies in a @xmath1123-neighbourhood of the geodesic from @xmath1117 to @xmath1110 , we have @xmath1124 on the other hand , if we now assume and apply the reverse triangle inequality , we get @xmath1125 hence @xmath1126 which contradicts if @xmath821 is large enough ( depending only on @xmath2 ) . the same argument applies if @xmath145 lies within @xmath1121 of a geodesic from @xmath1118 to @xmath1127 . therefore , @xmath145 lies within @xmath1128 of @xmath29 , and so @xmath1129 , as required . consider a triple of consecutive points , @xmath1131 and @xmath1132 . if their corresponding nearest point projections @xmath1133 and @xmath1134 to @xmath29 do not lie in the same order , then using proposition [ prop : npp2 ] repeatedly one gets , if @xmath1134 lies in between @xmath1107 and @xmath1127 , the equality @xmath1135 which , using and , implies @xmath1136 which contradicts if @xmath821 is large enough . the case where @xmath1127 lies between @xmath1107 and @xmath1134 is completely analogous , therefore the @xmath1107 are monotonically ordered on @xmath29 , and so by @xmath1137 which implies , by proposition [ prop : npp2 ] , a similar bound for @xmath1138 , and so in fact @xmath1139 is quasi - geodesic , with @xmath1140 . the upper bound on @xmath36 follows from the triangle inequality ; indeed , for each @xmath147 one has @xmath1141 and the desired bound follows by taking as @xmath44 the midpoint of the geodesic segment between @xmath27 and @xmath393 , completing the proof of proposition [ prop : hyperbolic ] .

let @xmath0 be a countable group which acts by isometries on a separable , but not necessarily proper , gromov hyperbolic space @xmath1 . we say the action of @xmath0 is weakly hyperbolic if @xmath0 contains two independent hyperbolic isometries . we show that a random walk on such @xmath0 converges to the gromov boundary almost surely . we apply the convergence result to show linear progress and linear growth of translation length , without any assumptions on the moments of the random walk . if the action is acylindrical , and the random walk has finite entropy and finite logarithmic moment , we show that the gromov boundary with the hitting measure is the poisson boundary .

in 1999 m. p. rekalo and myself investigated the possible presence of @xmath0 exchange @xcite in the precise data on the structure function @xmath2 , obtained at the jefferson laboratory ( jlab ) in @xmath3 elastic scattering , up to a value of momentum transfer squared , @xmath4 gev@xmath5 @xcite . a prescription for the differential cross section was derived from general properties of the hadron electromagnetic interaction , as the c - invariance and the crossing symmetry . the discrepancy from the two experiments @xcite , which had been performed in different kinematical conditions , could not be explained in terms of a @xmath0 contribution , but the possibility of @xmath0-corrections was not excluded , starting from @xmath6 gev@xmath5 and the necessity of dedicated experiments was pointed out @xcite . the relative contribution of @xmath0 exchange , through its interference with the main ( i.e. , one - photon ) mechanism is expected to be of the order of the fine structure constant , @xmath7 . but , more than 30 years ago , it was observed @xcite that the relative role of two - photon exchange can essentially increase in the region of high momentum transfer , due to the steep decreasing of the form factors ( ffs ) . this effect can be observed in particular in @xmath8-elastic scattering where it would appear already at momentum transfer squared of the order of 1 gev@xmath5 . the main consequence of the presence of @xmath0 exchange is that the traditional description of the electron - hadron interaction in terms of electromagnetic currents ( and electromagnetic ffs ) can become incorrect . in one - photon exchange , two real amplitudes ( functions of one variable , @xmath9 ) fully describe elastic @xmath10 scattering . if the @xmath0 exchange is present , one has to deal with three complex amplitudes , which are functions of two kinematical variables , @xmath9 and the polarization of the virtual photon , @xmath1 . the presence of @xmath0 exchange leads to very complicated analysis of polarization effects . it destroys the linearity in the variable @xmath1 of the differential cross section for elastic @xmath11 scattering @xcite , and the relatively simple dependence of the ratio @xmath12 ( the components of the final nucleon polarization in the scattering of longitudinally polarized electrons by an unpolarized nucleon target ) on the ratio of the electric and magnetic ffs , @xmath13 , which holds for the one - photon mechanism @xcite . it can be shown that the situation is not so involved , and that even in case of two - photon exchange , one can still use the formalism of ffs , if one takes into account the c - invariance of the electromagnetic interaction of hadrons @xcite . however , in this case only either a specific combination of three t - odd ( or five t - even ) polarization observables or measurements with positron and electron beams in the same kinematical conditions allow a model independent determination of ffs . the exact calculation of the @xmath0-contribution to the amplitude of the @xmath14-process requires the knowledge of the matrix element for the double virtual compton scattering , @xmath15 , in a large kinematical region of colliding energy and virtuality of both photons , and can not be done in a model independent form . therefore we follow another approach : general properties of the hadron electromagnetic interaction , as the c - invariance and the crossing symmetry , give rigorous prescriptions for different observables for the elastic scattering of electrons and positrons by nucleons , in particular for the differential cross section and for the proton polarization , induced by polarized electrons . these concrete prescriptions help in identifying a possible manifestation of the two - photon exchange mechanism and to avoid unjustified assumptions . for example , symmetry properties appear in the spin structure of the amplitudes , with respect to the change @xmath16 with @xmath17 . crossing symmetry allows to connect the matrix elements for the cross - channels : @xmath18 , in @xmath19channel , and @xmath20 , in @xmath21channel . the c - invariance of the electromagnetic hadron interaction and the corresponding selection rules can be applied to the annihilation channel and this allows to find specific properties for one and two photon exchanges . to illustrate this , let us consider firstly the one - photon mechanism for @xmath22 . the conservation of the total angular momentum , @xmath23 , allows one value , @xmath24 , and the quantum numbers of the photon : @xmath25 , @xmath26 . the selection rules with respect to c and p - invariances allow two states for @xmath27 ( and @xmath28 ) : @xmath29 where @xmath30 is the total spin and @xmath31 is the orbital angular momentum . as a result , the @xmath32-dependence of the cross section for @xmath22 , in the one - photon exchange mechanism is : @xmath33 where @xmath34 and @xmath35 are definite quadratic contributions of @xmath36 and @xmath37 , @xmath38 at @xmath39 . let us consider now the @xmath40-dependence of the @xmath41-interference contribution to the differential cross section of @xmath22 . the spin and parity of the @xmath0-states is not fixed , in general , but only a positive value of c - parity , @xmath42 , is allowed . an infinite number of states with different quantum numbers can contribute , and their relative role is determined by the dynamics of the process @xmath43 , with both virtual photons . but the @xmath40-dependence of the contribution to the differential cross section for the @xmath41-interference has a c - odd nature : @xmath44 , \label{eq : sig3}\ ] ] where @xmath45 , @xmath46 are real coefficients , which are functions of @xmath21 , only . the following relation between kinematical variables in the crossing channels holds : @xmath47 . this odd @xmath40 ( or @xmath48)-dependence is essentially different from the even @xmath40-dependence of the cross section for the one - photon approximation . it is , therefore , incorrect to approximate the interference contribution to the differential cross section ( [ eq : sig3 ] ) by a linear function in @xmath49 , because it is in contradiction with the c - invariance of hadronic electromagnetic interaction . it follows that , in presence of @xmath0 exchange , the reduced elastic @xmath10 cross section can be rewritten in the following general form : @xmath50 where @xmath51 is a real function describing the effects of the @xmath41 interference . in order to estimate the upper limit for a possible @xmath0 contribution to the differential cross section and the corresponding changing to @xmath52 , we analyzed four sets of data @xcite , applying eq . ( [ eq : sred2 ] ) with the following parametrization for @xmath51 : @xmath53 } ^2/0.71)^2(1 + q^2\mbox{[gev]}^2/m^2)^2 } , \label{eq : sfit}\ ] ] -1.3 true cm -.7 true cm figure 1 . from top to bottom : @xmath54 , @xmath55 and the two - photon contribution , @xmath56 . @xmath57 is the proton magnetic moment , @xmath58/0.71)^{-2}$ ] . published data are shown as open symbols , the present results which include the @xmath0 contribution as solid symbols . where @xmath56 is a fitting parameter , @xmath59 is the mass of a tensor or vector meson with positive c - parity . for @xmath60 gev , one can predict that the relative role of the @xmath0 contribution should increase with @xmath9 . it is important to stress that eq . ( [ eq : sfit ] ) is a simple expression which contains the necessary symmetry properties of the @xmath41 interference , through a specific ( and non linear ) @xmath1 dependence . therefore , in presence of @xmath0 , the dependence of the reduced cross section on @xmath1 can be parametrized as a function of three parameters , @xmath61 , @xmath62 and @xmath56 , according to eqs . ( [ eq : sred2 ] ) and ( [ eq : sfit ] ) . in fig . 1 , from top to bottom , the electric and magnetic ffs , as well as the two photon parameter @xmath56 , are shown as a function of @xmath9 ( solid symbols ) . the previously published data , derived from the traditional rosenbluth fits , are also shown ( open symbols ) . including a third fitting parameter , @xmath56 , increases the errors on the extracted ffs . the resulting parameter @xmath56 is compatible with zero . from the present analysis it appears that the available data on @xmath10 elastic scattering does not show any evidence of deviation from the linearity of the rosenbluth fit , and hence of the presence of the @xmath0 contribution , when parametrized according to eq . ( [ eq : sfit ] ) . besides the deviation from the linearity of the rosenbluth fit , other possible methods to test the presence of @xmath0-exchange in elastic electron hadron scattering can be listed : comparison of the cross section for scattering of unpolarized electrons and positrons ( by protons or deuterons ) in the same kinematical conditions ; specific polarization phenomena such as the appearance of t - odd polarization observables ; violation of definite relations between t - even polarization observables and structure functions @xcite . the experimental evidence of the presence of the @xmath0-exchange and its quantitative estimation is very important . if this effect appears in elastic @xmath10 scattering already in the range of momentum transfer investigated at jlab , the findings based on the one - photon assumption , will have to be reanalyzed at the light of a new and complicated formalism . in this case , most of the advantages related to the electromagnetic probe would be lost , as it was indicated already long ago @xcite . the results quoted here would not have been obtained wihout a fruitful collaboration and enlightning discussions with professor m. p. rekalo . 9 m. p. rekalo , e. tomasi - gustafsson and d. prout , phys . rev . * c60 * , 042202 ( 1999 ) . l. c. alexa _ et al . _ , lett . * 82 * , 1375 ( 1999 ) . d. abbott _ et al . _ , lett . * 82 * , 1379 ( 1999 ) . j. gunion and l. stodolsky , phys . lett . * 30 * , 345 ( 1973);v . franco , phys . d * 8 * , 826 ( 1973 ) ; v. n. boitsov , l.a . kondratyuk and v.b . kopeliovich , sov . * 16 * , 237 ( 1973 ) ; f. m. lev , sov . * 21 * , 45 ( 1973 ) . m. n. rosenbluth , phys . rev . * 79 * , 615 ( 1950 ) . a. akhiezer and m. p. rekalo , dokl . nauk ussr , * 180 * , 1081 ( 1968 ) ; sov . j. part . nucl . * 4 * , 277 ( 1974 ) ; m. p. rekalo and e. tomasi - gustafsson , lecture notes , arxiv : nucl - th/0202025 . m. p. rekalo and e. tomasi - gustafsson , eur . j. a * 22 * , 331 ( 2004 ) ; nucl . phys . a * 740 * , 271 ( 2004 ) ; nucl . a * 742 * , 322 ( 2004 ) . r. c. walker _ et al . _ , phys . d * 49 * , 5671 ( 1994 ) . l. andivahis _ et al . _ , phys . d * 50 * , 5491 ( 1994 ) . m. e. christy _ et al . _ [ e94110 collaboration ] , phys . rev . c * 70 * , 015206 ( 2004 ) . i. a. qattan _ et al . _ , arxiv : nucl - ex/0410010 .

the presence of @xmath0-exchange in electron proton elastic scattering is discussed . from c - invariance and crossing symmetry , @xmath0 contribution induces a specific dependence of the reduced cross section on the variable @xmath1 . no evidence of such dependence exists in the available experimental data . _ this talk is dedicated to the memory of professor michail p. rekalo _

since it is unlikely that there is a characterization of all those graphs which contain a hamilton cycle it is natural to ask for sufficient conditions which ensure hamiltonicity . one of the most general of these is chvtal s theorem @xcite that characterizes all those degree sequences which ensure the existence of a hamilton cycle in a graph : suppose that the degrees of the graph are @xmath11 . if @xmath12 and @xmath13 or @xmath14 for all @xmath15 then @xmath1 is hamiltonian . this condition on the degree sequence is best possible in the sense that for any degree sequence violating this condition there is a corresponding graph with no hamilton cycle . more precisely , if @xmath16 is a graphical degree sequence ( i.e. there exists a graph with this degree sequence ) then there exists a non - hamiltonian graph @xmath1 whose degree sequence @xmath17 is such that @xmath18 for all @xmath19 . a special case of chvtal s theorem is dirac s theorem , which states that every graph with @xmath12 vertices and minimum degree at least @xmath20 has a hamilton cycle . an analogue of dirac s theorem for digraphs was proved by ghouila - houri @xcite . ( the digraphs we consider do not have loops and we allow at most one edge in each direction between any pair of vertices . ) nash - williams @xcite raised the question of a digraph analogue of chvtal s theorem quite soon after the latter was proved . for a digraph @xmath1 it is natural to consider both its outdegree sequence @xmath21 and its indegree sequence @xmath22 . throughout this paper we take the convention that @xmath3 and @xmath23 without mentioning this explicitly . note that the terms @xmath24 and @xmath25 do not necessarily correspond to the degree of the same vertex of @xmath1 . [ nw ] suppose that @xmath1 is a strongly connected digraph on @xmath12 vertices such that for all @xmath9 * @xmath26 or @xmath27 , * @xmath28 or @xmath29 then @xmath1 contains a hamilton cycle . no progress has been made on this conjecture so far ( see also @xcite ) . it is even an open problem whether the conditions imply the existence of a cycle through any pair of given vertices ( see @xcite ) . as discussed in section [ extremal ] , one can not omit the condition that @xmath1 is strongly connected . at first sight one might also try to replace the degree condition in chvtal s theorem by * @xmath26 or @xmath30 , * @xmath31 or @xmath32 . however , bermond and thomassen @xcite observed that the latter conditions do not guarantee hamiltonicity . indeed , consider the digraph obtained from the complete digraph @xmath33 on @xmath34 vertices by adding two new vertices @xmath35 and @xmath36 which both send an edge to every vertex in @xmath33 and receive an edge from one fixed vertex @xmath37 . the following example shows that the degree condition in conjecture [ nw ] would be best possible in the sense that for all @xmath38 and all @xmath39 there is a non - hamiltonian strongly connected digraph @xmath1 on @xmath2 vertices which satisfies the degree condition except that @xmath40 are replaced by @xmath41 in the @xmath42th pair of conditions . to see this , take an independent set @xmath43 of size @xmath39 and a complete digraph @xmath33 of order @xmath44 . pick a set @xmath45 of @xmath42 vertices of @xmath33 and add all possible edges ( in both directions ) between @xmath43 and @xmath45 . the digraph @xmath1 thus obtained is strongly connected , not hamiltonian and @xmath46 is both the out- and indegree sequence of @xmath1 . a more detailed discussion of extremal examples is given in section [ extremal ] . in this paper we prove the following approximate version of conjecture [ nw ] for large digraphs . [ approxnw ] for every @xmath47 there exists an integer @xmath48 such that the following holds . suppose @xmath1 is a digraph on @xmath49 vertices such that for all @xmath9 * @xmath5 or @xmath6 , * @xmath7 or @xmath50 then @xmath1 contains a hamilton cycle . instead of proving theorem [ approxnw ] directly , we will prove the existence of a hamilton cycle in a digraph satisfying a certain expansion property ( theorem [ expanderthm ] ) . we defer the precise statement to section [ sec4 ] . the following weakening of conjecture [ nw ] was posed earlier by nash - williams @xcite . it would yield a digraph analogue of psa s theorem which states that a graph @xmath1 on @xmath38 vertices has a hamilton cycle if its degree sequence @xmath51 satisfies @xmath13 for all @xmath52 and if additionally @xmath53 when @xmath2 is odd @xcite . note that this is much stronger than dirac s theorem but is a special case of chvtal s theorem . [ nw2 ] let @xmath1 be a digraph on @xmath12 vertices such that @xmath54 for all @xmath55 and such that additionally @xmath56 when @xmath2 is odd . then @xmath1 contains a hamilton cycle . the previous example shows that the degree condition would be best possible in the same sense as described there . the assumption of strong connectivity is not necessary in conjecture [ nw2 ] , as it follows from the degree conditions . the following approximate version of conjecture [ nw2 ] is an immediate consequence of theorem [ approxnw ] . [ posa ] for every @xmath47 there exists an integer @xmath48 such that every digraph @xmath1 on @xmath49 vertices with @xmath57 for all @xmath9 contains a hamilton cycle . in section [ orient ] we give a construction which shows that for oriented graphs there is no analogue of psa s theorem . ( an oriented graph is a digraph with no @xmath58-cycles . ) it will turn out that the conditions of theorem [ approxnw ] even guarantee the digraph @xmath1 to be _ pancyclic _ , i.e. @xmath1 contains a cycle of length @xmath59 for all @xmath60 . [ pancyclic ] for every @xmath47 there exists an integer @xmath48 such that the following holds . suppose @xmath1 is a digraph on @xmath49 vertices such that for all @xmath9 * @xmath61 or @xmath6 , * @xmath62 then @xmath1 is pancyclic . thomassen @xcite proved an ore - type condition which implies that every digraph with minimum in- and outdegree @xmath63 is pancyclic . ( the complete bipartite digraph whose vertex class sizes are as equal as possible shows that the latter bound is best possible . ) alon and gutin @xcite observed that one can use ghouila - houri s theorem to show that every digraph @xmath1 with minimum in- and outdegree @xmath63 is even vertex - pancyclic . here a digraph @xmath1 is called _ vertex - pancyclic _ if every vertex of @xmath1 lies on a cycle of length @xmath59 for all @xmath60 . in proposition [ vertexpan ] we show that one can not replace pancyclicity by vertex - pancyclicity in corollary [ pancyclic ] . minimum degree conditions for ( vertex- ) pancyclicity of oriented graphs are discussed in @xcite . our result on hamilton cycles in expanding digraphs ( theorem [ expanderthm ] ) is used as a tool in @xcite to prove an approximate version of sumner s universal tournament conjecture . theorem [ expanderthm ] also has an application to a conjecture of thomassen on tournaments . a _ tournament _ is an orientation of a complete graph . we say that a tournament is _ regular _ if every vertex has equal in- and outdegree . thus regular tournaments contain an odd number @xmath2 of vertices and each vertex has in- and outdegree @xmath64 . it is easy to see that every regular tournament contains a hamilton cycle . thomassen @xcite conjectured that even if we remove a number of edges from a regular tournament @xmath1 , the remaining oriented graph still contains a hamilton cycle . [ thomconj ] if @xmath1 is a regular tournament on @xmath2 vertices and @xmath65 is any set of less than @xmath64 edges of @xmath1 , then @xmath66 contains a hamilton cycle . in section [ torn ] we prove conjecture [ thomconj ] for sufficiently large regular tournaments . note that conjecture [ thomconj ] is a weakening of the following conjecture of kelly ( see e.g. @xcite ) . [ kelly ] every regular tournament on @xmath2 vertices can be decomposed into @xmath64 edge - disjoint hamilton cycles . in @xcite we showed that every sufficiently large regular tournament can be ` almost ' decomposed into edge - disjoint hamilton cycles , thus giving an approximate solution to kelly s conjecture . this paper is organized as follows . we first give a more detailed discussion of extremal examples for conjecture [ nw ] . after introducing some basic notation , in section [ sec2 ] we then deduce corollary [ pancyclic ] from theorem [ approxnw ] and show that one can not replace pancyclicity by vertex - pancyclicity . our proof of theorem [ approxnw ] uses the regularity lemma for digraphs which , along with other tools , is introduced in section [ sec3 ] . the proof of theorem [ approxnw ] is included in section [ sec4 ] . it relies on a result ( lemma [ cyclelemma ] ) from joint work @xcite of the first two authors with keevash on an analogue of dirac s theorem for oriented graphs . a related result was proved earlier in @xcite . it is a natural question to ask whether the ` error terms ' in theorem [ approxnw ] and corollary [ posa ] can be eliminated using an ` extremal case ' or ` stability ' analysis . however , this seems quite difficult as there are many different types of digraphs which come close to violating the conditions in conjectures [ nw ] and [ nw2 ] ( this is different e.g. to the situation in @xcite ) . as a step in this direction , very recently it was shown in @xcite that the degrees in the first parts of the conditions in theorem [ approxnw ] can be capped at @xmath20 , i.e. the conditions can be replaced by * @xmath67 or @xmath6 , * @xmath68 or @xmath50 the proof of this result is considerably more difficult than that of theorem [ approxnw ] . a ( parallel ) algorithmic version of chvtal s theorem for undirected graphs was recently considered in @xcite and for directed graphs in @xcite . the example given in the introduction does not quite imply that conjecture [ nw ] would be best possible , as for some @xmath42 it violates both ( i ) and ( ii ) for @xmath69 . here is a slightly more complicated example which only violates one of the conditions for @xmath69 ( unless @xmath2 is odd and @xmath70 ) . suppose @xmath71 and @xmath72 . let @xmath33 and @xmath73 be complete digraphs on @xmath74 and @xmath75 vertices respectively . let @xmath1 be the digraph on @xmath2 vertices obtained from the disjoint union of @xmath33 and @xmath73 as follows . add all possible edges from @xmath73 to @xmath33 ( but no edges from @xmath33 to @xmath73 ) and add new vertices @xmath76 and @xmath35 to the digraph such that there are all possible edges from @xmath73 to @xmath76 and @xmath35 and all possible edges from @xmath76 and @xmath35 to @xmath33 . finally , add a vertex @xmath36 that sends and receives edges from all other vertices of @xmath1 ( see figure 1 ) . [ ] [ ] @xmath73 [ ] [ ] @xmath33 [ ] [ ] @xmath36 [ ] [ ] @xmath35 [ ] [ ] @xmath76 , title="fig : " ] thus @xmath1 is strongly connected , not hamiltonian and has outdegree sequence @xmath77 and indegree sequence @xmath78 suppose that either @xmath2 is even or , if @xmath2 is odd , we have that @xmath79 . one can check that @xmath1 then satisfies the conditions in conjecture [ nw ] except that @xmath80 and @xmath81 . ( when checking the conditions , it is convenient to note that our assumptions on @xmath42 and @xmath2 imply @xmath82 . hence there are at least @xmath83 vertices of outdegree @xmath84 and so ( ii ) holds for all @xmath85 . ) if @xmath2 is odd and @xmath70 then conditions ( i ) and ( ii ) both fail for @xmath69 . we do not know whether a similar construction as above also exists for this case . it would also be interesting to find an analogous construction as above for conjecture [ nw2 ] . here is also an example which shows that the assumption of strong connectivity in conjecture [ nw ] can not be omitted . let @xmath86 be even . let @xmath33 and @xmath73 be two disjoint copies of a complete digraph on @xmath20 vertices . obtain a digraph @xmath1 from @xmath33 and @xmath73 by adding all possible edges from @xmath33 to @xmath73 ( but none from @xmath73 to @xmath33 ) . it is easy to see that @xmath1 is neither hamiltonian , nor strongly connected , but satisfies the condition on the degree sequences given in conjecture [ nw ] . as it stands , the additional connectivity assumption means that conjecture [ nw ] does not seem to be a precise digraph analogue of chvtal s theorem : in such an analogue , we would ask for a complete characterization of _ all _ digraph degree sequences which force hamiltonicity . however , it turns out that it makes sense to replace the strong connectivity assumption with an additional degree condition ( condition ( iii ) below ) . if true , the following conjecture would provide the desired characterization . [ nw3 ] suppose that @xmath1 is a digraph on @xmath12 vertices such that for all @xmath9 * @xmath26 or @xmath27 , * @xmath28 or @xmath87 , and such that ( iii ) @xmath88 or @xmath89 if @xmath2 is even . then @xmath1 contains a hamilton cycle . conjecture [ nw3 ] would actually follow from conjecture [ nw ] . to see this , it of course suffices to check that the conditions in conjecture [ nw3 ] imply strong connectivity . this in turn is easy to verify , as the degree conditions imply that for any vertex set @xmath90 with @xmath91 we have @xmath92 and @xmath93 . ( we need ( iii ) to obtain this assertion precisely for those @xmath90 with @xmath94 . ) it remains to check that conjecture [ nw3 ] would indeed characterize all digraph degree sequences which force a hamilton cycle . unless @xmath2 is odd and @xmath70 , the construction at the beginning of the section already gives non - hamiltonian graphs which satisfy all the degree conditions ( including ( iii ) ) except ( i ) for @xmath69 . to cover the case when @xmath2 is odd and @xmath70 , let @xmath1 be the digraph obtained from two disjoint cliques @xmath33 and @xmath73 of orders @xmath83 and @xmath95 by adding all edges from @xmath33 to @xmath73 . if @xmath96 then @xmath1 satisfies ( ii ) ( because @xmath97 ) but not ( i ) . for all other @xmath98 , both conditions are satisfied . finally , the example immediately preceding conjecture [ nw3 ] gives a graph on an even number @xmath2 of vertices which satisfies ( i ) and ( ii ) for all @xmath9 but does not satisfy ( iii ) . nash - williams observed that conjecture [ nw ] would imply chvtal s theorem . ( indeed , given an undirected graph @xmath1 satisfying the degree condition in chvtal s theorem , obtain a digraph by replacing each undirected edge with a pair of directed edges , one in each direction . this satisfies the degree condition in conjecture [ nw ] . it is also strongly connected , as it is easy to see that @xmath1 must be connected . ) a disadvantage of conjecture [ nw3 ] is that it would not imply chvtal s theorem in the same way : consider a graph @xmath1 which is obtained from @xmath99 by removing a perfect matching and adding a spanning cycle in one of the two vertex classes . the degree sequence of this @xmath1 satisfies the conditions of chvtal s theorem . however , the digraph obtained by doubling the edges of @xmath1 does not satisfy ( iii ) in conjecture [ nw3 ] . we begin this section with some notation . given two vertices @xmath100 and @xmath101 of a digraph @xmath1 , we write @xmath102 for the edge directed from @xmath100 to @xmath101 . the order @xmath103 of @xmath1 is the number of its vertices . we denote by @xmath104 and @xmath105 the out- and the inneighbourhood of @xmath100 and by @xmath106 and @xmath107 its out- and indegree . we will write @xmath108 for example , if this is unambiguous . given @xmath109 , we write @xmath110 for the union of @xmath104 for all @xmath111 and define @xmath112 analogously . the _ minimum semi - degree _ @xmath113 of @xmath1 is the minimum of its minimum outdegree @xmath114 and its minimum indegree @xmath115 . 55*proof of corollary [ pancyclic ] . * our first aim is to prove the existence of a vertex @xmath116 such that @xmath117 . such a vertex exists if there is an index @xmath118 with @xmath119 . indeed , at least @xmath120 vertices of @xmath1 have outdegree at least @xmath121 and at least @xmath122 vertices have indegree at least @xmath123 . thus there will be a vertex @xmath100 with @xmath124 and @xmath125 . to prove the existence of such an index @xmath118 , suppose first that there is an @xmath98 with @xmath126 and such that @xmath127 but @xmath128 . then @xmath129 and so @xmath130 as required . the same argument works if there is an @xmath98 with @xmath126 and such that @xmath131 but @xmath132 . suppose next that @xmath133 . then @xmath134 and so @xmath135 . thus we can take @xmath136 . again , the same argument works if @xmath137 . thus we may assume that @xmath138 . but in this case we can take @xmath139 . now let @xmath100 be a vertex with @xmath117 , set @xmath140 and @xmath141 . let @xmath142 and @xmath143 denote the out- and the indegree sequences of @xmath144 . given some @xmath145 and @xmath146 , if @xmath147 then at least @xmath148 vertices in @xmath1 have outdegree at least @xmath149 . thus at least @xmath150 vertices in @xmath144 have outdegree at least @xmath151 and so @xmath152 . thus for all @xmath9 the degree sequences of @xmath144 satisfy * @xmath153 or @xmath154 , * @xmath155 or @xmath156 and so * @xmath157 or @xmath158 , * @xmath159 or @xmath160 hence we can apply theorem [ approxnw ] with @xmath161 replaced by @xmath162 to obtain a hamilton cycle @xmath163 in @xmath144 . we now apply the same trick as in @xcite to obtain a cycle ( through @xmath100 ) in @xmath1 of the desired length , @xmath59 say ( where @xmath164 ) : since @xmath165 there exists an @xmath98 such that @xmath166 and @xmath167 ( where we take the indices modulo @xmath168 ) . but then @xmath169 is the required cycle of length @xmath59 . note that the proof of corollary [ pancyclic ] shows that if conjecture [ nw ] holds and @xmath1 is a strongly @xmath58-connected digraph with * @xmath170 or @xmath171 , * @xmath172 or @xmath173 for all @xmath85 then @xmath1 is pancyclic . the next result implies that we can not replace pancyclicity with vertex - pancyclicity in corollary [ pancyclic ] . [ vertexpan ] given any @xmath174 there are @xmath175 and @xmath176 such that for every @xmath49 there exists a digraph @xmath1 on @xmath2 vertices with @xmath177 for all @xmath85 , but such that some vertex of @xmath1 does not lie on a cycle of length less than @xmath42 . let @xmath178 and suppose that @xmath2 is sufficiently large . let @xmath1 be the digraph obtained from the disjoint union of @xmath179 independent sets @xmath180 with @xmath181 and a complete digraph @xmath33 on @xmath182 vertices as follows . add a new vertex @xmath100 which sends an edge to all vertices in @xmath183 and receives an edge from all vertices in @xmath33 . add all possible edges from @xmath184 to @xmath185 ( but no edges from @xmath185 to @xmath184 ) for each @xmath186 . finally , add all possible edges going from vertices in @xmath33 to other vertices and add all edges from @xmath187 to @xmath33 . then @xmath188 and @xmath189 for all @xmath85 with room to spare . however , if @xmath190 is a cycle containing @xmath100 then the inneighbour of @xmath100 on @xmath190 must lie in @xmath33 . but the shortest path from @xmath100 to @xmath33 has length @xmath74 and so @xmath191 , as required . in section [ sec1 ] we mentioned ghouila - houri s theorem which gives a bound on the minimum semi - degree of a digraph @xmath1 guaranteeing a hamilton cycle . a natural question raised by thomassen @xcite is that of determining the minimum semi - degree which ensures a hamilton cycle in an oriented graph . hggkvist @xcite conjectured that every oriented graph @xmath1 of order @xmath192 with @xmath193 contains a hamilton cycle . the bound on the minimum semi - degree would be best possible . the first two authors together with keevash @xcite confirmed this conjecture for sufficiently large oriented graphs . psa s theorem implies the existence of a hamilton cycle in a graph @xmath1 even if @xmath1 contains a significant number of vertices of degree much less than @xmath20 , i.e. of degree much less than the minimum degree required to force a hamilton cycle . in particular , psa s theorem is much stronger than dirac s theorem . in the same sense , conjecture [ nw2 ] would be much stronger than ghouila - houri s theorem . the following proposition implies that we can not strengthen hggkvist s conjecture in this way : there are non - hamiltonian oriented graphs which contain just a bounded number of vertices whose semi - degree is ( only slightly ) smaller than @xmath194 . to state this proposition we need to introduce the notion of dominating sequences : given sequences @xmath195 and @xmath196 of numbers we say that @xmath196 _ dominates _ @xmath195 if @xmath197 for all @xmath198 . [ orientedprop ] for every @xmath199 , there is an integer @xmath200 and infinitely many oriented graphs @xmath1 whose in- and outdegree sequences both dominate @xmath201 but such that @xmath1 does not contain a hamilton cycle . define @xmath202 where @xmath203 is chosen such that @xmath204 . let @xmath2 be sufficiently large and such that @xmath205 divides @xmath2 and define vertex sets @xmath206 and @xmath207 of sizes @xmath208 and @xmath209 respectively . let @xmath1 be the oriented graph obtained from the disjoint union of @xmath206 and @xmath207 by defining the following edges : @xmath1 contains all possible edges from @xmath65 to @xmath210 , @xmath210 to @xmath190 , @xmath190 to @xmath211 , @xmath65 to @xmath190 , @xmath210 to @xmath211 and @xmath211 to @xmath65 . @xmath207 sends out all possible edges to @xmath65 and @xmath210 and receives all possible edges from @xmath190 and @xmath211 . @xmath210 and @xmath190 both induce tournaments that are as regular as possible ( see figure 2 ) . [ ] [ ] @xmath65 [ ] [ ] @xmath210 [ ] [ ] @xmath190 [ ] [ ] @xmath211 [ ] [ ] @xmath207 [ ] [ ] @xmath209 [ ] [ ] @xmath212 [ ] [ ] @xmath213 [ ] [ ] @xmath214 [ ] [ ] @xmath209 [ ] [ ] @xmath215 [ ] [ ] @xmath216 [ ] [ ] @xmath217 [ ] [ ] @xmath218 in proposition [ orientedprop],title="fig : " ] so certainly @xmath219 for all @xmath220 . furthermore , currently , @xmath221 , @xmath222 , @xmath223 and @xmath224 for all @xmath225 and all @xmath226 . partition @xmath65 into @xmath215 and @xmath216 where @xmath227 and thus @xmath228 . write @xmath229 and @xmath230 . let @xmath215 induce a tournament that is as regular as possible . in particular , every vertex in @xmath215 sends out at least @xmath231 edges to other vertices in @xmath215 . we define the edges between @xmath215 and @xmath216 as follows : add the edges @xmath232 to @xmath1 for all @xmath233 and @xmath234 . note that we can partition both @xmath235 and @xmath236 into @xmath59 sets of size @xmath237 . for each @xmath238 add all possible edges from @xmath239 to @xmath240 and from @xmath241 to @xmath242 . if @xmath243 and @xmath244 are such that the edge @xmath245 has not been included into @xmath1 so far then add the edge @xmath246 to @xmath1 . thus , @xmath247 for all @xmath243 and @xmath248 for all @xmath244 . partitioning @xmath211 into @xmath217 and @xmath218 ( where @xmath249 ) and defining edges inside @xmath211 in a similar fashion to those inside @xmath65 , we can ensure that @xmath250 for all @xmath251 and @xmath252 for all @xmath253 . so indeed @xmath1 has the desired degree sequences . @xmath207 is an independent set , so if @xmath1 contains a hamilton cycle @xmath254 then the inneighbour of each vertex in @xmath207 on @xmath254 must lie in @xmath255 while its outneighbour lies in @xmath256 . so @xmath254 contains at least @xmath257 disjoint edges going from @xmath256 to @xmath255 . however , all such edges in @xmath1 have at least one endvertex in @xmath258 . so there are at most @xmath259 such disjoint edges in @xmath1 . thus @xmath1 does not contain a hamilton cycle ( in fact , @xmath1 does not contain a @xmath260-factor ) . in the proof of theorem [ approxnw ] we will use the directed version of szemerdi s regularity lemma . before we can state it we need some more definitions . the _ density _ of an undirected bipartite graph @xmath261 with vertex classes @xmath65 and @xmath210 is defined to be @xmath262 we will write @xmath263 if this is unambiguous . given any @xmath264 we say that @xmath1 is _ @xmath265-regular _ if for all @xmath266 and @xmath267 with @xmath268 . given disjoint vertex sets @xmath65 and @xmath210 in a digraph @xmath1 , we write @xmath269 for the oriented bipartite subgraph of @xmath1 whose vertex classes are @xmath65 and @xmath210 and whose edges are all the edges from @xmath65 to @xmath210 in @xmath1 . we say @xmath269 is _ @xmath265-regular and has density @xmath270 _ if the underlying bipartite graph of @xmath269 is @xmath265-regular and has density @xmath270 . ( note that the ordering of the pair @xmath271 is important here . ) the diregularity lemma is a variant of the regularity lemma for digraphs due to alon and shapira @xcite . its proof is similar to the undirected version . we will use the degree form of the diregularity lemma which is derived from the standard version in the same manner as the undirected degree form ( see e.g. the survey @xcite for a sketch of the undirected version ) . [ dilemma ] for every @xmath272 and every integer @xmath273 there are integers @xmath274 and @xmath275 such that if @xmath1 is a digraph on @xmath276 vertices and @xmath277 $ ] is any real number , then there is a partition of the vertex set of @xmath1 into @xmath278 and a spanning subdigraph @xmath144 of @xmath1 such that the following holds : * @xmath279 , * @xmath280 , * @xmath281 , * @xmath282 for all vertices @xmath283 , * @xmath284 for all vertices @xmath283 , * for all @xmath285 the digraph @xmath286 $ ] is empty , * for all @xmath287 with @xmath288 the pair @xmath289 is @xmath265-regular and has density either @xmath290 or density at least @xmath270 . we call @xmath291 _ clusters _ , @xmath292 the _ exceptional set _ and the vertices in @xmath292 _ exceptional vertices_. we refer to @xmath144 as the _ pure digraph_. the last condition of the lemma says that all pairs of clusters are @xmath293-regular in both directions ( but possibly with different densities ) . the _ reduced digraph @xmath294 of @xmath1 with parameters @xmath265 , @xmath270 and @xmath273 _ is the digraph whose vertices are @xmath295 and in which @xmath296 is an edge precisely when @xmath289 is @xmath265-regular and has density at least @xmath270 . given @xmath297 , we call a digraph @xmath1 a _ @xmath298-outexpander _ if @xmath299 for all @xmath300 with @xmath301 . the main tool in the proof of theorem [ approxnw ] is the following result from @xcite . [ cyclelemma ] let @xmath302 be positive integers and let @xmath303 be positive constants such that @xmath304 . let @xmath1 be an oriented graph on @xmath276 vertices such that @xmath305 . let @xmath294 be the reduced digraph of @xmath1 with parameters @xmath265 , @xmath270 and @xmath273 . suppose that there exists a spanning oriented subgraph @xmath306 of @xmath294 with @xmath307 which is a @xmath298-outexpander . then @xmath1 contains a hamilton cycle . here we write @xmath308 to mean that we can choose the constants @xmath309 from right to left . more precisely , there are increasing functions @xmath310 and @xmath311 such that , given @xmath312 , whenever we choose some @xmath313 and @xmath314 , all calculations needed in the proof of lemma [ cyclelemma ] are valid . our next aim is to show that any digraph @xmath1 as in theorem [ approxnw ] is an outexpander . in fact , we will show that even the ` robust outneighbourhood ' of any set @xmath300 of reasonable size is significantly larger than @xmath90 . more precisely , let @xmath315 . given any digraph @xmath1 and @xmath300 , the _ @xmath316-robust outneighbourhood @xmath317 of @xmath90 _ is the set of all those vertices @xmath100 of @xmath1 which have at least @xmath318 inneighbours in @xmath90 . @xmath1 is called a _ robust @xmath298-outexpander _ if @xmath319 for all @xmath300 with @xmath320 . [ robustg ] let @xmath275 be a positive integer and @xmath321 be positive constants such that @xmath322 . let @xmath1 be a digraph on @xmath276 vertices with * @xmath323 or @xmath324 , * @xmath325 or @xmath326 for all @xmath15 . then @xmath327 and @xmath1 is a robust @xmath328-outexpander . clearly , if @xmath329 then @xmath330 . if @xmath331 then ( i ) implies that @xmath332 . thus @xmath1 has at least @xmath333 vertices of indegree @xmath84 and so @xmath330 . it follows similarly that @xmath334 . consider any non - empty set @xmath300 with @xmath335 and @xmath336 . let us first deal with the case when @xmath337 . then @xmath90 contains a set @xmath45 of @xmath338 vertices , each having outdegree at least @xmath339 . let @xmath340 be the set of all those vertices of @xmath1 that have at least @xmath341 inneighbours in @xmath45 . then @xmath342 and so @xmath343 . so suppose next that @xmath344 . since @xmath334 we may assume that @xmath345 ( otherwise @xmath346 and we are done ) . thus @xmath347 by ( i ) and ( ii ) . ( here we use that @xmath336 . ) so @xmath1 contains at least @xmath348 vertices @xmath100 of indegree at least @xmath349 . if @xmath350 then @xmath351 contains such a vertex @xmath100 . but then @xmath100 has at least @xmath341 neighbours in @xmath90 , i.e. @xmath352 , a contradiction . if @xmath353 then considering the outneighbourhood of a subset of @xmath90 of size @xmath354 shows that @xmath355 . the next result implies that the property of a digraph @xmath1 being a robust outexpander is ` inherited ' by the reduced digraph of @xmath1 . for this ( and for lemma [ orientexp ] ) we need that @xmath1 is a robust outexpander , rather than just an outexpander . [ robustr ] let @xmath302 be positive integers and let @xmath303 be positive constants such that @xmath356 and such that @xmath357 . let @xmath1 be a digraph on @xmath276 vertices with @xmath358 and such that @xmath1 is a robust @xmath298-outexpander . let @xmath294 be the reduced digraph of @xmath1 with parameters @xmath265 , @xmath270 and @xmath273 . then @xmath359 and @xmath294 is a robust @xmath360-outexpander . let @xmath144 denote the pure digraph , @xmath361 , let @xmath362 be the clusters of @xmath1 ( i.e. the vertices of @xmath294 ) and @xmath292 the exceptional set . . then @xmath364 consider any @xmath365 with @xmath366 . let @xmath367 be the union of all the clusters belonging to @xmath90 . then @xmath368 . since @xmath369 for every @xmath370 this implies that @xmath371 however , in @xmath144 every vertex @xmath372 receives edges from vertices in at least @xmath373 clusters @xmath374 . thus by the final property of the partition in lemma [ dilemma ] the cluster @xmath375 containing @xmath100 is an outneighbour of each such @xmath184 ( in @xmath294 ) . hence @xmath376 . this in turn implies that @xmath377 as required . the strategy of the proof of theorem [ approxnw ] is as follows . by lemma [ robustg ] our given digraph @xmath1 is a robust outexpander and by lemma [ robustr ] this also holds for the reduced digraph @xmath294 of @xmath1 . the next result gives us a spanning oriented subgraph @xmath306 of @xmath294 which is still an outexpander . the somewhat technical property concerning the subdigraph @xmath378 in lemma [ orientexp ] will be used to guarantee an oriented subgraph @xmath379 of @xmath1 which has linear minimum semidegree and such that @xmath306 is a reduced digraph of @xmath379 . ( @xmath379 will be obtained from the spanning subgraph of the pure digraph @xmath144 which corresponds to @xmath306 by modifying the neighbourhoods of a small number of vertices . ) finally , we will apply lemma [ cyclelemma ] with @xmath306 playing the role of both @xmath294 and @xmath306 and @xmath379 playing the role of @xmath1 to find a hamilton cycle in @xmath379 and thus in @xmath1 . [ orientexp ] given positive constants @xmath380 , there exists a positive integer @xmath275 such that the following holds . let @xmath294 be a digraph on @xmath276 vertices which is a robust @xmath298-outexpander . let @xmath254 be a spanning subdigraph of @xmath294 with @xmath381 . then @xmath294 has a spanning oriented subgraph @xmath306 which is a robust @xmath382-outexpander and such that @xmath383 . consider a random spanning oriented subgraph @xmath306 of @xmath294 obtained by deleting one of the edges @xmath384 ( each with probability 1/2 ) for every pair @xmath385 for which @xmath386 , independently from all other such pairs . given a vertex @xmath100 of @xmath294 , we write @xmath387 for the set of all those vertices of @xmath294 which are both out- and inneighbours of @xmath100 and define @xmath388 similarly . let @xmath389 . clearly , @xmath390 if @xmath391 . so suppose that @xmath392 . let @xmath393 . then @xmath394 and so a standard chernoff estimate ( see e.g. ( * ? ? ? * cor . a.14 ) ) implies that @xmath395 where @xmath396 is an absolute constant ( i.e. it does not depend on @xmath316 , @xmath397 or @xmath161 ) . similarly it follows that @xmath398 . consider any set @xmath399 . let @xmath400 and define @xmath401 similarly . we say that @xmath90 is _ good _ if all but at most @xmath402 vertices in @xmath403 are contained in @xmath401 . our next aim is to show that @xmath404 to prove ( [ eq : good ] ) , write @xmath405 for the set of all those vertices @xmath406 for which @xmath407 . note that every vertex in @xmath408 will automatically lie in @xmath401 . we say that a vertex @xmath409 _ fails _ if @xmath410 . the expected size of @xmath411 is at least @xmath412 . so as before , a chernoff estimate gives @xmath413 let @xmath340 be the number of all those vertices @xmath409 which fail . then @xmath414 . note that the failure of distinct vertices is independent ( which is the reason we are only considering vertices in the external neighbourhood of @xmath90 ) . so we can apply the following chernoff estimate ( see e.g. ( * ? ? ? * theorem a.12 ) ) : if @xmath415 we have @xmath416 setting @xmath417 this gives @xmath418 ( the last inequality follows since @xmath419 if @xmath2 is sufficiently large . ) this completes the proof of ( [ eq : good ] ) . since @xmath420 ( if @xmath2 is sufficiently large ) this implies that there is an outcome for @xmath306 such that @xmath383 and such that every set @xmath365 is good . we will now show that the latter property implies that such an @xmath306 is a robust @xmath382-outexpander . so consider any set @xmath421 with @xmath422 . let @xmath423 and @xmath424 . so @xmath425 . since @xmath90 is good and @xmath426 all but at most @xmath402 vertices in @xmath427 are contained in @xmath428 . now consider any partition of @xmath90 into @xmath429 and @xmath430 such that every vertex @xmath431 satisfies @xmath432 for @xmath433 . ( the existence of such a partition follows by considering a random partition . ) then @xmath434 . but since @xmath430 is good this implies that all but at most @xmath402 vertices in @xmath435 are contained in @xmath436 . similarly , since @xmath429 is good , all but at most @xmath402 vertices in @xmath437 are contained in @xmath438 . altogether this shows that @xmath439 as required . as indicated in section [ sec1 ] , instead of proving theorem [ approxnw ] directly , we will prove the following stronger result . it immediately implies theorem [ approxnw ] since by lemma [ robustg ] any digraph @xmath1 as in theorem [ approxnw ] is a robust outexpander and satisfies @xmath327 . [ expanderthm ] let @xmath275 be a positive integer and @xmath440 be positive constants such that @xmath441 . let @xmath1 be a digraph on @xmath276 vertices with @xmath327 which is a robust @xmath298-outexpander . then @xmath1 contains a hamilton cycle . pick a positive integer @xmath273 and additional constants @xmath442 such that @xmath443 . apply the regularity lemma ( lemma [ dilemma ] ) with parameters @xmath293 , @xmath270 and @xmath273 to @xmath1 to obtain clusters @xmath362 , an exceptional set @xmath292 and a pure digraph @xmath144 . then @xmath444 by lemma [ dilemma ] . let @xmath294 be the reduced digraph of @xmath1 with parameters @xmath293 , @xmath270 and @xmath273 . lemma [ robustr ] implies that @xmath445 and that @xmath294 is a robust @xmath360-outexpander . let @xmath254 be the spanning subdigraph of @xmath294 in which @xmath446 is an edge if @xmath447 and the density @xmath448 of the oriented subgraph @xmath289 of @xmath144 is at least @xmath449 . we will now give a lower bound on @xmath450 . so consider any cluster @xmath184 and let @xmath451 . writing @xmath452 for the number of all edges from @xmath184 to @xmath453 in @xmath144 , we have @xmath454 it is easy to see that this implies that there are at least @xmath455 outneighbours @xmath375 of @xmath184 in @xmath294 such that @xmath456 . but each such @xmath375 is an outneighbour of @xmath184 in @xmath254 and so @xmath457 . it follows similarly that @xmath458 . we now apply lemma [ orientexp ] to find a spanning oriented subgraph @xmath306 of @xmath294 which is a ( robust ) @xmath459-outexpander and such that @xmath460 . let @xmath389 . our next aim is to modify the pure digraph @xmath144 into a spanning oriented subgraph of @xmath1 having minimum semi - degree at least @xmath461 . let @xmath379 be the spanning subgraph of @xmath144 which corresponds to @xmath306 . so @xmath379 is obtained from @xmath144 by deleting all those edges @xmath102 that join some cluster @xmath184 to some cluster @xmath375 with @xmath462 . note that @xmath463 is an oriented graph . however , some vertices of @xmath463 may have small degrees . we will show that there are only a few such vertices and we will add them to @xmath292 in order to achieve that the out- and indegrees of all the vertices outside @xmath292 are large . so consider any cluster @xmath184 . for any cluster @xmath464 at most @xmath465 vertices in @xmath184 have less than @xmath466 outneighbours in @xmath375 ( in the digraph @xmath144 ) . call all these vertices of @xmath184 _ useless for @xmath375_. thus on average any vertex of @xmath184 is useless for at most @xmath467 clusters @xmath464 . this implies that at most @xmath468 vertices in @xmath184 are useless for more than @xmath469 clusters @xmath464 . let @xmath470 be a set of size @xmath468 which consists of all these vertices and some extra vertices from @xmath184 if necessary . similarly , we can choose a set @xmath471 of size @xmath468 such that for every vertex @xmath472 there are at most @xmath473 clusters @xmath474 such that @xmath100 has less than @xmath475 inneighbours in @xmath375 . for each @xmath285 remove all the vertices in @xmath476 and add them to @xmath292 . we still denote the subclusters obtained in this way by @xmath362 and the exceptional set by @xmath292 . thus we now have that @xmath477 . moreover , @xmath478 we now modify @xmath379 by altering the neighbours of the exceptional vertices : for every @xmath479 we select a set of @xmath480 outneighbours of @xmath100 in @xmath1 and a set of @xmath480 inneighbours such that these two sets are disjoint and add the edges between @xmath100 and the selected neighbours to @xmath379 . we still denote the oriented graph thus obtained from @xmath379 by @xmath379 . then @xmath481 . since the partition @xmath482 of @xmath483 is as described in the regularity lemma ( lemma [ dilemma ] ) with parameters @xmath484 , @xmath485 and @xmath273 ( where @xmath379 plays the role of @xmath144 and @xmath1 ) we can say that @xmath306 is a reduced digraph of @xmath379 with these parameters . thus we may apply lemma [ cyclelemma ] with @xmath306 playing the role of both @xmath294 and @xmath306 and @xmath379 playing the role of @xmath1 to find a hamilton cycle in @xmath379 and thus in @xmath1 . in this section we prove conjecture [ thomconj ] for sufficiently large regular tournaments . the following observation of keevash and sudakov @xcite will be useful for this . [ kands ] let @xmath486 and let @xmath1 be an oriented graph on @xmath2 vertices such that @xmath487 . then for any ( not necessarily disjoint ) @xmath488 of size at least @xmath489 there are at least @xmath490 directed edges from @xmath90 to @xmath491 . [ thomconjlarge ] there exists an integer @xmath275 such that the following holds . given any regular tournament @xmath1 on @xmath49 vertices and a set @xmath65 of less than @xmath64 edges of @xmath1 , then @xmath66 contains a hamilton cycle . let @xmath492 . it is not difficult to show that @xmath1 is a robust @xmath493-outexpander . indeed , if @xmath109 and @xmath494 then @xmath495 . if @xmath496 then it is easy to see that @xmath497 . so consider the case when @xmath498 . suppose @xmath499 . then by proposition [ kands ] there are at least @xmath490 directed edges from @xmath90 to @xmath500 . by definition each vertex @xmath501 has less than @xmath502 inneighbours in @xmath90 , a contradiction . so @xmath503 as desired . if @xmath507 then there exists precisely one vertex @xmath508 such that either @xmath509 or @xmath510 . without loss of generality we may assume that @xmath509 . note that @xmath511 and let @xmath512 . let @xmath144 be the digraph obtained from @xmath66 by removing @xmath100 and @xmath101 from @xmath66 and adding a new vertex @xmath513 so that @xmath514 and @xmath515 . so @xmath516 and @xmath144 is a robust @xmath517-outexpander . thus by theorem [ expanderthm ] @xmath144 contains a hamilton cycle which corresponds to one in @xmath1 . 10 n. alon and g. gutin , properly colored hamilton cycles in edge colored complete graphs , _ random structures and algorithms _ * 11 * ( 1997 ) , 179186 . n. alon and a. shapira , testing subgraphs in directed graphs , _ journal of computer and system sciences _ * 69 * ( 2004 ) , 354382 . n. alon and j. spencer , _ the probabilistic method _ ( 2nd edition ) , wiley - interscience 2000 . j. bang - jensen and g. gutin , _ digraphs : theory , algorithms and applications _ , springer 2000 . bermond and c. thomassen , cycles in digraphs a survey , _ j. graph theory _ * 5 * ( 1981 ) , 143 . a. bondy , basic graph theory : paths and circuits , in _ handbook of combinatorics _ , 1 , elsevier , amsterdam ( 1995 ) , 3110 . d. christofides , p. keevash , d. khn and d. osthus , a semi - exact degree condition for hamilton cycles in digraphs , submitted . d. christofides , p. keevash , d. khn and d. osthus , finding hamilton cycles in robustly expanding digraphs , submitted . v. chvtal , on hamilton s ideals , _ j. combin . theory b _ * 12 * ( 1972 ) , 163168 . a. ghouila - houri , une condition suffisante dexistence dun circuit hamiltonien , _ c.r . paris _ * 25 * ( 1960 ) , 495497 . r. hggkvist , hamilton cycles in oriented graphs , _ combin . comput . _ * 2 * ( 1993 ) , 2532 . p. keevash , d. khn and d. osthus , an exact minimum degree condition for hamilton cycles in oriented graphs , _ j. london math . soc . _ * 79 * ( 2009 ) , 144166 . p. keevash and b. sudakov , triangle packings and 1-factors in oriented graphs , _ j. combin . theory b _ * 99 * ( 2009 ) , 709727 . l. kelly , d. khn and d. osthus , a dirac - type result on hamilton cycles in oriented graphs , _ combin . prob . _ * 17 * ( 2008 ) , 689709 . l. kelly , d. khn and d. osthus , cycles of given length in oriented graphs , _ j. combin . theory b _ , to appear . d. khn , r. mycroft and d. osthus , an approximate version of sumner s universal tournament conjecture , submitted . d. khn and d. osthus , embedding large subgraphs into dense graphs , in _ surveys in combinatorics _ ( s. huczynska , j.d . mitchell and c.m . roney - dougal eds . ) , _ london math . . lecture notes _ * 365 * , 137167 , cambridge university press , 2009 . d. khn , d. osthus and a. treglown , hamilton decompositions of regular tournaments , submitted . j.w . moon , _ topics on tournaments _ , holt , rinehart and winston , new york , 1968 . nash - williams , problem 47 , _ proceedings of colloq . tihany 1966 _ , academic press 1968 , p. 366 . nash - williams , hamilton circuits in graphs and digraphs , _ the many facets of graph theory _ , springer verlag lecture notes 110 , springer verlag 1968 , 237243 . nash - williams , hamiltonian circuits , _ studies in math . _ * 12 * ( 1975 ) , 301360 . l. psa , a theorem concerning hamiltonian lines , _ magyar tud . akad . mat . fiz . oszt . kozl . _ * 7 * ( 1962 ) , 225226 . g.n . srkzy , a fast parallel algorithm for finding hamiltonian cycles in dense graphs , _ discrete math . _ * 309 * ( 2009 ) , 16111622 . c. thomassen , an ore - type condition implying a digraph to be pancyclic , _ discrete math . _ * 19 * ( 1977 ) , 8592 . c. thomassen , long cycles in digraphs with constraints on the degrees , in _ surveys in combinatorics _ ( b. bollobs ed . ) , _ london math . soc . lecture notes _ * 38 * , 211228 , cambridge university press , 1979 . c. thomassen , edge - disjoint hamiltonian paths and cycles in tournaments , _ proc . london math . _ * 45 * ( 1982 ) , 151168 .

we show that for each @xmath0 every digraph @xmath1 of sufficiently large order @xmath2 is hamiltonian if its out- and indegree sequences @xmath3 and @xmath4 satisfy ( i ) @xmath5 or @xmath6 and ( ii ) @xmath7 or @xmath8 for all @xmath9 . this gives an approximate solution to a problem of nash - williams @xcite concerning a digraph analogue of chvtal s theorem . in fact , we prove the stronger result that such digraphs @xmath1 are pancyclic . # 1&#2 = by -4 1= 1 < @xmath10 to [ firstthm]theorem [ firstthm]proposition [ firstthm]lemma [ firstthm]corollary [ firstthm]problem [ firstthm]definition [ firstthm]conjecture [ firstthm]claim

the periodic anderson model ( pam ) is one of the basic models describing strongly correlated systems whose characteristics fit to be interpreted in a two - band picture @xcite . the model is largely applied in the study of heavy - fermion systems @xcite , intermediate - valence compounds @xcite , or even high critical temperature superconductors @xcite . in contrast however with other models used in the understanding of phenomena created by strong correlation effects , where at least in one dimension exact solutions exist ( for example the hubbard model @xcite ) , the physics described by pam is almost exclusively interpreted based on approximations . this is due to the fact that only few results are exactly known about the behavior of pam . indeed , what we know about the physical behavior of pam in rigorous terms can be summarized as follows : the first results , related to the ground - state of decorated hyper - cubic lattices in the limit of infinite interaction strength has been provided by brandt and giesekus @xcite followed by a non - magnetic ground - state restricted to 1d and on - site repulsion @xmath1 case obtained by strack @xcite . this solution has been extended to @xmath2 as well , at @xmath1 @xcite . again for infinite on - site hubbard repulsion has been demonstrated that at quarter filling the ground - state is unique with a defined total spin @xcite . it has been also underlined that the model becomes solvable in the case of constant and infinite range hopping @xcite . we further know , that for only on - site hybridization and without direct hopping in the correlated band : the symmetric half - filled case is spin - singlet @xcite and in 1d also pseudo - spin singlet @xcite , at half filling anti - ferromagnetic correlations are present @xcite , and in 1d,2d long - range order of ferro , anti - ferro and pairing type is absent @xcite . recently have been published the first exact ground - states at finite @xmath3 in 1d @xcite and 2d @xcite , respectively . concerning 2d , the reported ground - states @xcite require next to nearest - neighbor ( nnn ) one - particle terms as well in the hamiltonian ( @xmath4 ) , and from the obtained solutions , especially the physical properties of the itinerant one has been described in detail . the deduction of exact ground - states in @xmath0 dimensions is of large interest in general , and for strongly correlated systems in special . in the case of pam , at finite and nonzero value of the interaction , the only one procedure doable working at the moment in this respect , is based on plaquette operators introduced in ref.@xcite , and a decomposition of the on - site hubbard interaction as described in ref . @xcite . during this procedure , @xmath4 is such transformed to contain in a positive semidefinite form products of plaquette operators . since in 2d the product of plaquette operators generates nnn one - particle terms as well , the general impression created by the method suggests that the applicability of the procedure is intimately connected to nnn contributions in 2d , and the deduced exact ground - states are fingerprints of this fact . in this paper , presenting for the first time exact ground - states for pam in 2d at finite value of the interaction , in restricted regions of the parameter space and without nnn type of extension terms in @xmath4 , we demostrate that the plaquette operator procedure and the ground - states deduced with it are not necessarily connected to the presence of nnn extensions in @xmath4 . in order to clarify these aspects ( i ) the plaquette operator technique is analysed in detail in general terms and 2d , ( ii ) a completely new type of plaquette operator is introduced which allows the deduction of the presented results , ( iii ) the obtained new localized exact ground - state is analyzed and described in detail , and ( iv ) implications of the results relating the metal - insulator transition in frame of pam are presented . the deduced new ground - state is a completely localized state . in order to characterize this phase , after obtaining the exact ground - state , all relevant ground - state expectation values and correlation functions have been exactly calculated and analyzed . the obtained state is paramagnetic , and based on a coherent control which it has on the occupation number of all lattice sites , it introduces long - range density - density correlations into the system , producing a localized state . concerning implications to physical systems , we mention the intense activity in the field related to the understanding of the metal - insulator transition ( mit ) in frame of pam . the subject has an almost 30 years of history @xcite , and gained renewed interest in the last period based on the observed mit similarities between the hubbard model and pam , used for example in the understanding of the iso - structural electronically driven mit transitions ( like the @xmath5 transition in @xmath6 compounds ) @xcite . since the exactly deduced ground - state energy values presented in this paper are not containing exponential factors characteristique to kondo type of behavior , the results reported here underline that at least in some regions of the parameter space , a localization - delocalization transition in frame of pam is not necessarily connected to kondo physics . the remaining part of the paper is structurated as follows : section ii . presents the hamiltonian we use , section iii . describes the plaquette operator technique , section iv . presents the transformation of the starting hamiltonian into a new expression containing plaquette operators , section v. describes the detected new exact ground - states , section vi . concludes the paper , and appendices a.- d. containing the mathematical details of the starting points of the paper close the presentation . we start with a generic pam hamiltonian written for 2d square lattice as @xmath7 where , the contributing terms in order , are representing the kinetic energy of @xmath8 electrons ( @xmath9 ) , the kinetic energy of @xmath10 electrons ( @xmath11 ) , the on - site @xmath10 electron energy ( @xmath12 ) , the hybridization ( @xmath13 ) , and the on - site hubbard interaction written for @xmath10 electrons @xmath14 , the last contribution representing the interaction term , and @xmath15 being considered during this paper . the presence of @xmath11 in @xmath4 is motivated by the overwhelming evidence , that the heavy - fermion materials contain in their real band structure around the fermi level ( @xmath16 ) very narrow , hybridized bands , which exist already at temperatures far above the thermodynamically determined kondo temperature , being relatively @xmath17 independent and holding an accentuate @xmath10 character @xcite . the interaction term during this paper is exactly transformed in the form @xmath18 where , the positive semidefinite operator @xmath19 defined by eq.([e9 ] ) requires for its lowest zero eigenvalue at least one @xmath10 electron on every lattice site @xcite . as will be clarified in section v. , the representation presented in eq.([e9 ] ) is a key feature from the point of view of the interaction term in the process of the deduction of exact ground - states in the frame presented here . the hybridization @xmath13 is considered to be build up from a local @xmath20 , and a nonlocal @xmath21 contribution , @xmath22 . thus , the local one - particle terms of the hamiltonian are @xmath12 and @xmath20 whose expressions become @xmath23 the non - local one - particle contributions remain to be presented . in order to make the notations clear , instead of the site - numbering notation ( i ) we use here the @xmath24 vectorial notation for the lattice sites . the kinetic energy contribution @xmath25 thus is given by @xmath26 and the non - local hybridization becomes @xmath27 the notation of the non - local hybridization matrix elements by the superscripts @xmath28 and @xmath29 is given by mathematical convenience , and through the paper @xmath30 will be considered . we note that at the level of one - particle contributions , @xmath4 , and as a consequence @xmath31 and @xmath21 , contain at the start all contributions entering in an elementary plaquette ( unit cell for the system ) @xcite . in these circumstances , for both eqs.([e11],[e12 ] ) , it is important that @xmath32 to be rigorously defined . this is because ( i ) we would like to represent different contributions correctly in term of plaquette operators , and ( ii ) we must avoid multiple counting of different terms entering in the expression of @xmath4 . for this reason , we mention that for a given lattice site , taking into account nearest neighbor ( nn ) and nnn contributions as well ( these will be present in an elementary plaquette ) , 8 hopping possibilities exist . from these , only 4 are taken into account explicitly by @xmath33 , and , the remaining 4 contributions are introduced into the hamiltonian by the @xmath34 operation . in these conditions , the defined 4 different @xmath32 contributions entering in @xmath33 are @xmath35 where @xmath36 is the lattice constant , and @xmath37 , @xmath38 , are the versors of the @xmath39 , @xmath40 axis , respectively . the hopping and hybridization matrix elements generated by the contributions from eq.([e14 ] ) are represented for clarity in fig . 1 . also for clarity , the explicit expressions of @xmath31 and @xmath21 from eqs.([e11],[e12 ] ) are presented in appendix . a. in the case of concrete materials , the nnn one - particle contributions are small , and as a consequence are often neglected . furthermore , it is important to know in rigorous terms if nnn contributions are introducing small corrections into the results or are able to provide qualitatively new effects . in the case of pam , which is in a relatively early stage of its exact description , this issue must be also clarified . because of this reason , during this paper , even if we start with nnn terms in @xmath4 for technical reasons , we try to obtain exact ground - state solutions for pam in the absence of nnn contributions as well . this task is also enhanced by the aim to extend the potential possibilities of the plaquette operator procedure we use . starting from these motivations , we are reporting in this paper for the first time exact ground - states for pam in 2d at finite and nonzero @xmath3 , in the absence of nnn extension terms in @xmath4 . in order to be able to obtain exact ground - states in @xmath41 dimensions for the pam hamiltonian presented in eq.([e8 ] ) , we use a plaquette operator procedure which will be described in details in the following section . concerning the method itself , in our knowledge , it is used now for the second time ( see also @xcite ) , and other methods in obtaining exact ground - states for pam in @xmath41 dimensions are not known at the moment . in principle , the procedure can be applied for other model hamiltonians as well containing itinerant degrees of freedom . the technique needs the transformation of @xmath4 in a positive semidefinite expression based on plaquette operators . in 2d , the plaquette operators ( as block operator units used for description ) generate local , nn and nnn terms as well . this suggests that also the studied 2d hamiltonian must contain such type of contributions . if this would be the case , the application possibility of the procedure in 2d would be strongly limited to hamiltonians that contain nnn extension terms as well . we demonstrate in this paper that this impression is not correct , and the procedure can be extended and applied even in the absence of nnn contributions in @xmath4 . we further mention that for hamiltonians containing main long range terms ( next to nnn or higher range contributions ) , the block unit used for description must be enlarged . let us consider a 2d finite @xmath42 square lattice , with lattice constant @xmath36 . in order to identify the lattice sites , we are numbering them by @xmath43 starting from the left - down corner , taking into account first the lowest row , and inside a row counting from left to right ( we mention that for a vectorial position notation we are going to use @xmath24 instead of i , when this is necessary ) . for example , in the simple case of @xmath44 , we obtain the lattice site numbering presented in fig . 2 . as can be seen in this figure , we are denoting by @xmath45 the elementary plaquettes . using this notation , we start from the lowest elementary plaquette row , counting from left to right inside a row , and then going upward with the notation . taking periodic boundary conditions into account in both directions , the number of plaquettes becomes equal to the number of lattice sites @xmath46 . in this case , it is advantageous to denote every plaquette @xmath45 by its down - left corner @xmath47 , as @xmath48 . concerning the notation of a plaquette through its down - left corner , for clarity we mention that for example , in fig . 2 . , the plaquette @xmath49 defined by the lattice sites @xmath50 becomes @xmath51 , or , the plaquette @xmath52 , defined by the lattice sites @xmath53 becomes @xmath54 , etc . let us now consider for pedagogical reasons , some @xmath55 fermionic operators creating particles on lattice sites within the system . in general , the @xmath56 operators can be labelled also by a supplementary @xmath57 index which contains all relevant quantum numbers as well ( in the case of pam , @xmath58 , where @xmath59 denotes the spin , and @xmath60 the type of particle ) . in this section , being interested in the presentation of the method , we are neglecting the @xmath57 index for simplicity . if the reader understands how the procedure works , the presented relations can be easily generalized for @xmath61 as well . using the @xmath56 operators , plaquette operators can be constructed by a linear combination of @xmath56 acting on the corners of an elementary plaquette . we are denoting the coefficients of this linear combination by @xmath62 , where @xmath63 denotes the plaquette , @xmath43 labels a given corner of the plaquette @xmath63 analyzed , and @xmath64 denotes the type of operator considered ( this becomes the @xmath57 index when @xmath65 is used instead of @xmath56 ) , respectively . for example , in case of the plaquettes @xmath66 and @xmath67 from fig . 3a . we obtain @xmath68 working with plaquettes , we must observe that all one - particle contributions of a given hamiltonian can be obtained starting from plaquette operators . for example , let us consider the hopping matrix element connecting the nearest - neighbor lattice sites @xmath69 from fig . 3a , namely @xmath70 . this hamiltonian contribution can be obtained for example , from the expression @xmath71 . indeed , we have @xmath72 where , the operator @xmath73 concentrates all the other terms obtained from the left side of eq.([e2 ] ) . the operator @xmath73 will not be neglected in our considerations . it contains 30 terms that can be easily calculated from eq.([e1 ] ) ( see also appendix b. ) . the important aspect here , which must be keeped in mind , is that @xmath73 do not contains contributions entering in @xmath74 . otherwise , the concrete expression of @xmath73 is not important at the moment . the relation from eq.([e2 ] ) is obtained since , @xmath75 gives rise to @xmath76 , and @xmath77 creates the term @xmath78 , respectively . because the bond @xmath79 is not present in other elementary plaquettes , even if we take into consideration all the plaquettes from the whole lattice in a sum of the form @xmath80 , the hopping matrix element @xmath81 becomes unambiguously expressed as @xmath82 the obtained eq.([e3 ] ) shows that the hamiltonian parameters ( at least the one - particle once in the present case ) , can be expressed in term of plaquette operator parameters if we succeed to express the corresponding hamiltonian terms into a sum of the form @xmath80 . similarly , the next - nearest - neighbor hopping amplitude for the @xmath83 hopping from the plaquette @xmath66 of fig . 3a . , contained in the hamiltonian term @xmath84 becomes @xmath85 in eq.([e3a ] ) only the plaquette operator product @xmath86 contributes , because , the nnn hopping described by @xmath87 is contained only in the plaquette @xmath66 . these examples illustrate that plaquette operators can be extremely useful in the study of different model hamiltonians @xmath4 , since as seen from eq.([e2 ] ) , different emerging contributions in @xmath4 can be represented in diagonal , or positive semidefinite form via the operators @xmath88 . as can be observed from eqs.([e3],[e3a ] ) , a such type of representation in term of plaquette operators , from the point of view of @xmath4 parameters , simply means a parametrization in term of plaquette operator coefficients @xmath62 . for this to be possible , the plaquette operator products summed up over lattice sites of the form @xmath80 ( i ) must generate terms present in @xmath4 , or ( ii ) must generate terms that are constants of motion ( for example total number of particles , or lattice sites ) , or ( iii ) must generate terms that can be cancelled out if the ( i ) and ( ii ) conditions can not be applied . we will return back to this problem after presenting the new plaquette operators defined in this paper ( see after eq.([e7 ] ) ) , and the following section exemplifies in detail a such type of transformation . when the one - particle @xmath4 parameters are not local ( for example @xmath89 for all vertical nearest - neighbor hoppings ) , which means @xmath90 the parameters @xmath62 of different plaquette operators are not independent . in the case of translational invariant hamiltonians , we can chose for example translational invariant plaquette operator parameters as illustrated by fig . 3b . denoting the sites inside a given plaquette starting from the down - left corner and counting anti - clockwise , the corners of the plaquette @xmath66 ( and @xmath67 ) in fig.3b . , will be denoted by @xmath91 ( and @xmath92 ) , respectively . given by the considered translational invariance of plaquette operators , the plaquette operator parameters of the plaquettes @xmath66 and @xmath67 with @xmath93 equal indices will have the same value @xmath94 , @xmath95 . this property is extended as well to all plaquettes . in the examples contained in fig . 3b . , the plaquette operators @xmath96 , @xmath97 become in this case @xmath98 from eqs.([e3],[e4],[e5 ] ) , the unique nn hopping matrix element in @xmath99 direction of @xmath64 particles , based on eq.([e3 ] ) becomes @xmath100 and , from eqs.([e3a],[e5 ] ) , the unique nnn hopping of the same particles along the main diagonal of every elementary plaquette will be described by @xmath101 similarly , all one - particle hamiltonian matrix elements can be expressed in term of plaquette operator parameters . when the so obtained equations ( as eqs.([e6],[e6a ] ) ) allow solutions for the plaquette operator parameters ( this is possible usually in a restricted parameter space region @xmath102 determined by the values of @xmath4 parameters ) , the one - particle part of the hamiltonian can be expressed via @xmath103 ( see eq.([e2 ] ) ) . based on these relations and using for example properties related to positive semidefinite operators , the hamiltonian of the system can be diagonalized exactly , at least for the ground - state , inside @xmath102 . after testing this method in 1d @xcite ( using bonds instead of plaquettes ) , a such type of procedure has been recently used by us @xcite in order to provide the first exact ground - state wave - functions for the periodic anderson model in 2d in restricted regions of the parameter space . this has been done by choosing @xmath104 for @xmath105 electrons with @xmath106 @xmath107 in pam , and defining based on this choice , the @xmath108 spin - dependent plaquette operators containing spin - independent @xmath109 parameters with @xmath60 , @xmath110 as follows ( the example is taken for the plaquette @xmath66 of fig . 3b . ) @xmath111 the obtained ground - state solutions based on eq.([e6b ] ) were connected to @xmath112 filling @xcite , and are highly non - trivial states . one of them is a completely localized state , and the second one is itinerant , with the momentum distribution function for the half - filled upper diagonalized band as shown in fig . 4 , presenting a clear evidence of ( exactly deduced ) non - fermi liquid behavior in normal phase and @xmath0 spatial dimensions . this shows that the procedure detects ground - states which are far to be trivial . however , the inconvenience of the plaquette operator from eq.([e6b ] ) is that via @xmath113 it creates nnn terms , these must be present in @xmath4 as well , so the deduced ground - states , and the procedure itself , seem to be related to the presence of nnn extensions in the hamiltonian . we present below how this inconvenience can be removed . for this reason , we must observe , that the choice of the operators @xmath56 in eq.([e5 ] ) and the form of the plaquette operator itself is not fixed a @xmath114 . this means that the possibility presented in eq.([e6b ] ) for the plaquette operators is not unique , even if we are interested in the study of a fixed model ( as pam in the present case ) . as a consequence , we can chose other possible forms for the plaquette operators , and using them , we can deduce other ground - states in other regions of the @xmath115 phase diagram of the model . to exemplify this statement , in the present paper we define for the decomposition of the studied pam hamiltonian in translational invariant case , a completely new type of plaquette operators @xmath116 and each of these has different plaquette operator parameters @xmath118 and @xmath119 , @xmath110 , @xmath60 , @xmath120 . furthermore , both plaquette operators @xmath116 and @xmath117 are containing both spin components with different numerical prefactors , i.e. @xmath118 , @xmath119 are considered independent , and @xmath121 dependent . exemplifying the new form for the case of the plaquette @xmath66 of fig . 3b . , where @xmath24 denotes the vectorial position of the site @xmath122 , the new @xmath116 operator is defined as @xmath123 similar expression is used for the @xmath117 operator as well for the same plaquette @xmath66 , in which , the plaquette operator parameters are considered @xmath119 , instead of @xmath124 . note the plaquette independent values of the @xmath118 and @xmath119 parameters , which , as explained in this section , is given by the translational invariance of the considered system . comparing eq.([e6b ] ) and eq.([e7 ] ) , we realize that the @xmath108 plaquette operators for both @xmath125 values have 8 independent @xmath109 parameters , while in the present case , for both @xmath126 , @xmath127 operators , the number of independent plaquette operator parameters is 32 . this enlargement of the number of parameters give us the possibility to demonstrate that the described procedure is able to detect also ground - states whose presence do not require the nnn terms in @xmath4 of the system , even if the @xmath128 products are providing such type of terms at the start . the key feature for this to work is the presence of two plaquette operators @xmath88 and @xmath129 containing different spin - dependent coefficients . indeed , in this case , by @xmath130 we can cancel out not only the @xmath131 terms created by @xmath132 which are not present in @xmath4 ( these would represent for example hopping terms containing spin - flip ) , but also the nnn terms generated by @xmath80 . because of this reason becomes to be possible to obtain the expression of a hamiltonian not containing nnn contributions in term of plaquette operator products which create such type of elements . the concrete transformation of the hamiltonian is presented in the following section . comparing the expression of the hamiltonian presented in the previous section together with the explicitations contained in appendices a. and b. , eqs . ( [ e10],[a1],[a2],[b1 ] ) , we realize that the following relation holds @xmath133 \sum_{\bf i}^{n_{\lambda } } \hat d^{\dagger}_{{\bf i},\sigma } \hat d_{{\bf i } , \sigma } + \sum_{\sigma } [ \sum_{n=1}^4(|a_{n , f,\sigma}|^2 + |b_{n , f,\sigma}|^2 ) ] \sum_{\bf i}^{n_{\lambda } } \hat f^{\dagger}_{{\bf i},\sigma } \hat f_{{\bf i } , \sigma } \ : , \label{e15}\end{aligned}\ ] ] if , the hopping and hybridization matrix elements are related to the parameters of the plaquette operators @xmath116 and @xmath117 via @xmath134 where the non - linear system of equations from eq.([e16 ] ) is presented explicitly in the appendix . c. , and @xmath135 . these equations arise as eqs.([e6],[e6a ] ) in section i. the system of equations eq.([e16 ] ) must be considered as containing known @xmath4 parameters ( @xmath136 ) , and unknown plaquette operator numerical prefactors ( @xmath137 ) . in fact , a simple ( but lengthy ) algebraic calculation shows that eq.([e15 ] ) exactly holds if the relations between the parameters of @xmath4 and the numerical prefactors of the plaquette operators , presented explicitly in appendix c , are satisfied . the number of equations contained in eq.([e16 ] ) is @xmath138 , and the 32 unknown complex plaquette operator parameters provides 64 unknown variables ( the real and imaginary parts ) . these are entering in eq.([e16 ] ) in a nonlinear , but complex - algebraic manner . since the number of equations is higher that the number of unknown variables , solutions will be allowed only if some inter - dependences ( fixed by eq.([e16 ] ) ) will be present between the @xmath4 parameters . these relations contribute in the definition of @xmath139 ( see also the observations below eq . ( [ e24 ] ) ) . we underline that since the structure of plaquette operators used in this paper ( see eq.([e7 ] ) ) is completely different from the structure of the plaquette operators from eq.([e6b ] ) ( in that case , instead of eq.([e16 ] ) , we have 17 equations presented in eq.(9 ) of ref.@xcite , containing 16 unknown variables ) , the problem set up here , from mathematical point of view , is completely different from that analysed in our previous work . we also note that as can be seen from appendix b , @xmath140 introduces @xmath141 like terms as well , which are missing from the hamiltonian . because of this reason we need a second plaquette operator product @xmath142 , whose role is exactly to cancel out these supplementary terms not present in @xmath4 , eq.([e8 ] ) . furthermore , the presence of @xmath143 allows also to cancel out the nnn terms created by @xmath144 . via eq.([e15 ] ) , this give as the possibility to express @xmath4 through @xmath145 even in the absence of nnn terms in the hamiltonian , and to obtain ground - state wave - functions in this case as well . using now eqs.([e9],[e10 ] ) , we have @xmath146 where , the positive semidefinite operator @xmath147 has been defined in sec.ii . adding eq.([e17 ] ) to eq.([e15 ] ) and using for the plaquette operators the anti - commutation property presented in eq.([b2 ] ) , we find @xmath148 where , the introduced constants are defined by @xmath149 , with @xmath60 . imposing the relations @xmath150 the expression of @xmath4 from eq.([e19 ] ) becomes @xmath151 since we are working at fixed number of particles @xmath152 , from eq.([e21 ] ) we obtain @xmath153 where @xmath154 for @xmath15 is a positive semidefinite operator defined by @xmath155 and the number @xmath156 is given by @xmath157 the transformation of eq.([e8 ] ) into eq.([e22 ] ) is possible only if the system of equations eqs.([e16],[e20 ] ) allows solutions for the plaquette operator parameters . the presence of these solutions will be possible only on restricted domains @xmath102 of the parameter space of the problem given by the inter - dependences between the @xmath4 parameters mentioned below eq.([e16 ] ) . as a consequence , the solutions that will be presented below are valid only in this @xmath102 region . in this section we are presenting first the derivation of the exact ground - states , then we discuss the possible solutions for the plaquette operator parameters , and finally , we analyse in extreme details the solution obtained for zero nnn contributions . starting from eq.([e22 ] ) , taking into account that @xmath154 is a positive semidefinite operator , we realize that the ground - state of @xmath158 is the wave function @xmath159 , for which @xmath160 . to find @xmath159 , we have to keep in mind eq.([e23 ] ) which defines @xmath154 . given by @xmath161 we observe that the plaquette operator part of eq.([e23 ] ) applied to @xmath162 gives zero . furthermore , since @xmath147 requires for its zero ( and minimum ) eigenvalue at least one @xmath10-electron on every lattice site , we add to the ground - state the contribution @xmath163 , where @xmath164 are arbitrary coefficients . as a consequence , the ground - state with the property @xmath160 becomes @xmath165 |0 \rangle \ : , \label{e26}\end{aligned}\ ] ] where , @xmath166 is the bare vacuum with no fermions present . the product in eq.([e26 ] ) must be taken over all lattice sites . because of this reason , the product of the creation operators in eq.([e26 ] ) introduces @xmath167 particles within the system , so the deduced ground - state wave - function corresponds to @xmath112 filling . all degeneration possibilities of the ground - state are contained in eq.([e26 ] ) , since the wave function with the property @xmath168 at @xmath112 filling always can be written in the presented @xmath159 form . we underline however that pam contains two hybridized bands , and @xmath112 filling for a two - band system means in fact half filled upper hybridized band ( the lower band being completely filled up ) . the wave - vector @xmath159 represents the ground - state of the starting hamiltonian , only if eq.([e8 ] ) can be transformed in eq.([e22 ] ) . this is possible only if we are situated inside the region @xmath102 of the parameter space , i.e. the system of equations eq.([e16 ] ) detailed in appendix c. allows solutions for the plaquette operator parameters , in conditions in which also the constrains from eqs.([e13],[e20 ] ) hold . in the remaining part of the paper we will concentrate on these possible solutions . we underline , that @xmath159 presented in eq.([e26 ] ) describes rigorously only the @xmath15 case , since the presence of the @xmath169 operator into the ground - state is required only by the non - zero @xmath3 value . as a consequence , the ground - state at @xmath170 can not be expressed in the form presented in eq.([e26 ] ) . we emphasize that the differences between eq.([e26 ] ) and the ground - states deduced previously @xcite are present because instead of @xmath171 obtained in the old case with @xmath172 defined by eq.([e6b ] ) , we now have in the ground - state wave function @xmath173 . before going further , we mention that the physical properties of the ground - state wave - function written mathematically in eq.([e26 ] ) strongly depend on the nature of the concrete solutions provided by eq.([e16 ] ) . the solutions for the plaquette operator parameters which lead to the ground - state @xmath159 must be obtained solving together eqs.([e13],[e20],[c1 ] ) . these taken together represent 74 nonlinear complex - algebraic coupled equations , so a relatively difficult mathematical task . a study of the next - nearest neighbor contributions entering in eq.([c1 ] ) shows that the solutions exist only if the following inter - dependences are present between the plaquette operator parameters @xmath174 where , @xmath175 are complex , finite , nonzero , otherwise arbitrary parameters defined by the ratios presented in eq.([e27 ] ) . using eq.([e27 ] ) , the studied system of equations can be completely transcribed for the @xmath119 unknown variables with @xmath110 ; @xmath176 ; @xmath125 ( the @xmath118 parameters being given through @xmath119 via eq.([e27 ] ) ) . since the so obtained equations for the @xmath119 variables are representing the starting point of the description of physical properties provided by the deduced ground - states , they are presented in eqs.([d1],[d2 ] ) of appendix d. starting from this moment , we must solve the system of equations presented in appendix d. we have found for the system of equations eqs.([d1],[d2 ] ) several mathematical solutions , which will be briefly presented below . \a ) taking @xmath177 , we find the first class of solutions . the interesting aspect of this case is that the 20 equations contained in eq.([d1 ] ) are automatically satisfied , and we must concentrate only on equations presented in eq.([d2 ] ) . this last system provides a solution for @xmath178 , @xmath135 , which however do not has new aspects in comparison to the solutions we find in ref.(@xcite ) . \b ) as can be seen , in order to obtain new solutions , we must take @xmath179 into account . the first attempt that can be made , is to consider @xmath180 , @xmath181 , and @xmath182 . this solution presents the interesting property that reduces the system to 1d case . this means that the solution emerges only for @xmath183 and @xmath184 . new aspects related to pam in comparison with those reported in refs.(@xcite ) are not present . this case merits however attention in the future , since it allows to study at the level of exact ground - states ( taken in the form of eq.([e26 ] ) ) the modification of the 1d properties to 2d characteristics by taking into account small and smooth deviations from the @xmath185 condition . \c ) the third solution that we have found was deduced in @xmath180 , @xmath181 , and @xmath186 case . this solution will be presented here in details , since presents a 2d ground - state that emerges for zero next - nearest - neighbor @xmath4 contributions . a such type of exact solution for pam is completely new , because it can not be obtained by the decomposition used previously @xcite . \d ) we have studied also the general @xmath187 , @xmath188 , case as well , obtaining only localized solution which require the presence of next - nearest neighbor @xmath4 terms as well . herewith , we analyze in detail the solution c ) described above requiring @xmath186 . this emerges at @xmath189 so it describes a ground - state wave function for pam not containing in its hamiltonian nnn extension terms . a such type of exact ground - state in 2d at finite nonzero value of the interaction is presented for the first time in this paper . solving for the plaquette operator parameters the system of equations eqs.([d1],[d2 ] ) we have found @xmath190 the conditions imposed for the parameters entering in eq.([e29 ] ) are @xmath191 , @xmath186 , @xmath192 , @xmath193 real , @xmath194 real . together with eq.([e29 ] ) , the nonzero @xmath4 parameters become @xmath195 the @xmath196 , @xmath197 factors present in this relation are given by @xmath198 we further mention , that the obtained solution , for @xmath199 and @xmath200 reduces to the isotropic case , where @xmath201 , @xmath60 , and @xmath202 . the ground - state wave function from eq.([e26 ] ) in the case of the solution from eq.([e29 ] ) reduces to the simple form @xmath203 the result presented in eq.([e32 ] ) is obtained because @xmath204 , in the studied case , is unable to introduce three particles on the same lattice site . since @xmath205 introduces @xmath206 electrons in the system , being unable to put three electrons on a given site , an uniform particle distribution will be obtained with two electrons per site , which generates the product @xmath207 in @xmath208 in eq.([e32 ] ) . the added term contained in @xmath169 ( see eq.([e26 ] ) ) introduces one more @xmath10 electron on each site , and as a consequence , do not modify the created uniform particle distribution within the system , and eq.([e32 ] ) arise . this wave - function has a well defined norm @xmath209 and as mentioned above , coherently maintains three particles on every lattice site @xmath210 \label{e34}\end{aligned}\ ] ] denoting by @xmath211 the ground - state expectation values , we obtain long - range density - density correlations within the system @xmath212 furthermore , it can be observed that @xmath208 prohibits in the same time the hopping and non - local hybridization between all site pairs @xmath213 since , for @xmath214 , we have @xmath215 , for all @xmath216 . as a consequence , the ground - state @xmath208 clearly represents a completely localized state . the remaining non - zero ground - state expectation values of different @xmath4 terms are given by @xmath217 because of @xmath15 , and as seen from eqs.([e30],[e31 ] ) , @xmath218 , the non - zero on - site hybridization , coupling the two bands , decreases the energy of the system ( @xmath219 ) . the ground - state energy becomes @xmath220 . in average , the number of @xmath10-electrons per site becomes @xmath221 where , @xmath222 and we have @xmath223 in the isotropic case , and @xmath224 , @xmath225 in general , for the considered solution . since in concrete physical situations @xmath226 , the number of @xmath10 electrons per site is close to one , but not exactly one in the ground - state . excepting the small number of sites with double @xmath10-electron occupancy , the local @xmath10-moments are not compensated . in fact , defining @xmath227 , with @xmath60 , we have @xmath228 , and @xmath229 , where as presented before , @xmath164 are arbitrary . concentrating on the magnetic properties of the ground - state , the total spin of the system can be standardly expressed via @xmath230 , @xmath231 , @xmath232 , and @xmath233 . calculating the ground - state expectation values , we find @xmath234 taking now two extremum @xmath235 distributions , a ) for @xmath236 , we find @xmath237 , and @xmath238 . this situation corresponds to maximum total spin in the system , with average total spin absolute value per site @xmath239 , which is of order @xmath240 for large @xmath46 . b ) dividing however the square lattice into two equal sub - lattices with @xmath241 in one sub - lattice , and @xmath242 in the other one , we obtain @xmath243 , i.e. in the thermodynamic limit , the total spin in absolute value per site is zero in this case . as can be observed , the degeneration of the ground - state physically is given by the fact that all possible total spin values are contributing in its construction . as a consequence , the ground - state behaves paramagnetically . we must further observe , that not all different @xmath235 sets provide linearly independent ground - state wave - function contributions . for example , choosing for all @xmath24 the values @xmath244 , or @xmath245 , or @xmath246 , we recover the same ground - state with maximum value of @xmath247 . as a consequence , in order to find orthogonal wave - functions that belong to the ground - state , the @xmath164 coefficients can not be chosen completely random and independent . also the normalization to unity of @xmath208 represents a constraint to the value of these coefficients . neglecting the trivial multiplicity obtained from the spatial orientation of the total spin @xmath248 , the degree of the degeneration of the ground - state is @xmath249 . the spin - spin correlation functions can be also calculated , and for @xmath250 we obtain in the case of a fixed @xmath251 set @xmath252 since , as shown before , the @xmath164 coefficients are not completely independent , the spin - spin correlations given by eq.([e40 ] ) are quasi - random . resembling behavior is experimentally seen in heavy - fermion cases @xcite . the phase diagram region where the solution occurs is presented in fig . 5 . for the isotropic case . the general aspect of this region remains the same in the anisotropic case as well . it represents a surface in the parameter space which extends from the low @xmath3 region up to the high @xmath3 region as well . this region is completely different from that obtained in ref.(@xcite ) which emerges for nonzero values of next - nearest - neighbor hopping and non - local hybridizations . the non - local nearest - neighbor hybridization matrix element in the isotropic case is related to hopping matrix elements by @xmath253 . in the anisotropic case this relation becomes @xmath254 , @xmath225 . modifying the values of hopping or / and hybridization matrix elements , we can leave @xmath139 , destroying the ground - state character of @xmath208 . this process can be tuned by pressure which strongly influences the @xmath255 parameters ( see for example ref.(@xcite ) ) . since the reduction of eq.([e26 ] ) into the completely localised @xmath208 from eq.([e32 ] ) it is itself based on a delicate balance between @xmath4 parameters ( contained in eqs.([e30 ] ) ) , the loss of the localization character of particles in principle can be easily achieved . this localization - delocalization transition represents in fact a mit transition provided by pam . since the exactly deduced ground - state energy do not contains the exponential term characteristic for a kondo type behavior ( see for example the discussion presented in ref.(@xcite ) ) , a such type of mit transition can not be connected to kondo physics . instead , the mit transition connected to the destruction of the localized @xmath208 ground - state is related to the break - down of the long - range density - density correlations . we are presenting for the first time exact ground - states for the periodic anderson model ( pam ) at finite @xmath3 in @xmath0 dimensions in the case in which the hamiltonian does not contain next to nearest neighbor ( nnn ) extension terms . for this reason , and based on this frame ( i ) the used plaquette operator procedure is presented in detail and it is underlined that its applicability is not connected to the presence of nnn extension terms in the hamiltonian . we underline that this is the only one procedure known at the moment , which is able to provide exact ground - states for pam in @xmath0 dimensions and finite @xmath3 interaction values . ( ii ) a new plaquette operator has been introduced for the study of the pam hamiltonian . the plaquette operator contains contributions coming from all spin components , possesses spin dependent numerical prefactors , and allows the detection of ground - states even in the absence of nnn extension terms in the hamiltonian in restricted regions of the parameter space . ( iii ) the physical properties of the deduced ground - state have been analysed in detail . all relevant ground - state expectation values and correlation functions have been deduced for this reason . ( iv ) the implications of the deduced results relating the metal - insulator transition in the frame of pam have been analysed . it has been pointed out that the lost of the localization character in the studied case is connected to the break - down of the long - range density - density correlations rather than kondo physics . the obtained new exact ground - state emerges at @xmath112 filling , not requires the presence of nnn extension terms in the hamiltonian , it is paramagnetic , and presents quasi - random spin - spin correlations . it represents a fully quantum - mechanical state ( in the sense that it is far to be quasi - classical ) , and it is build up through superposition effects . the ground - state wave - function coherently controls the occupation number on all lattice sites , introducing in this manner long - range density - density correlations within the system , and prohibiting in the same time the hopping and non - local hybridizations . the local @xmath10 moments are not compensated and the @xmath10 electron occupation number per site in average is close to , but not exactly one . concerning the question of the physical relevance , we would like to mention that in general terms , even a solution detected in a restricted parameter - space region which behaves completely repulsively in the renormalization group language could have significant physical implications @xcite . in the present case , besides presenting open roots toward the deduction possibilities of exact ground - states in @xmath0 dimensions for strongly correlated systems , the presented results provide exact theoretical data which can be used in the process of understanding and description of the metal - insulator transition in the frame of pam . the research has been supported in 2002 by the contract otka - t-037212 of hungarian founds for scientific research . the author kindly acknowledge extremely valuable discussions on the subject with dieter vollhardt . he also would like to thank for the kind hospitality of the department of theoretical physics iii . , university augsburg in autumn 2001 , 4 months of working period relating this field spent there , and supported by alexander von humboldt foundation . the kinetic energy term for @xmath8 electrons has the explicit form @xmath256 \ : . \label{a1}\end{aligned}\ ] ] the kinetic energy term for @xmath10 electrons can be simply obtained from eq . ( [ a1 ] ) interchanging the @xmath8 index with the @xmath10 index , and @xmath257 by @xmath258 . the explicit expression of the non - local hybridization becomes @xmath259 \ : . \label{a2}\end{aligned}\ ] ] the expression @xmath260 summed up over the whole lattice considered with periodic boundary conditions in both directions is presented below in condensed form ( @xmath261 ; @xmath262 ) . @xmath263 + [ \hat g^{\dagger}_{{\bf i},\sigma } \hat g'_{{\bf i } + { \bf y } , \sigma ' } ( a^{*}_{1,g,\sigma } a_{4,g',\sigma ' } + a^{*}_{2,g,\sigma } a_{3,g',\sigma ' } ) + h.c . ] + \nonumber\\ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % & & [ \hat g^{\dagger}_{{\bf i},\sigma } \hat g'_{{\bf i } + ( { \bf x}+{\bf y } ) , \sigma ' } ( a^{*}_{1,g,\sigma } a_{3,g',\sigma ' } ) + h.c . ] + [ \hat g^{\dagger}_{{\bf i},\sigma } \hat g'_{{\bf i } + ( { \bf y}-{\bf x } ) , \sigma ' } ( a^{*}_{2,g,\sigma } a_{4,g',\sigma ' } ) + h.c . ] + \nonumber\\ % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % & & [ \hat g^{\dagger}_{{\bf i},\sigma } \hat g'_{{\bf i } , \sigma ' } ( \sum_{n=1}^4 a^{*}_{n , g,\sigma } a_{n , g',\sigma ' } ) + h.c . ] [ 1 - \frac{1}{2 } \delta_{g , g ' } \delta_{\sigma,\sigma ' } ] \ } \ : . % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % \label{b1}\end{aligned}\ ] ] furthermore , the following property is satisfied @xmath264 for the @xmath117 plaquette operators the eqs.([b1],[b2 ] ) hold as well by changing the coefficients @xmath118 to @xmath119 , where @xmath135 . the explicit expression of the system of equations eq.([e16 ] ) containing 70 equalities is presented below in condensed form . the used abreviations are @xmath262 , @xmath265 , and @xmath266 represents @xmath267 or @xmath268 . after using eq.([e27 ] ) , the remaining equations for the plaquette operator parameters are presented in detail in this appendix . these equations can be divided in two parts : a homogeneous part ( eq.([d1 ] ) ) , and a non - homogeneous one ( eq.([d2 ] ) ) , see below . these two system of equations are presented here in detail in condensed form . written explicitely , eq.([d1 ] ) ( eq.([d2 ] ) ) contains 20 ( 41 ) different equations , respectively . the non - homogeneous part of the equations is presented below . the abbreviations used here are @xmath273 @xmath267 or @xmath268 , and the presence of @xmath121 means that two equations are simultaneously present with @xmath274 and @xmath275 . for a single @xmath276 index we have @xmath60 . @xmath277

the derivation procedure of exact ground - states for the periodic anderson model ( pam ) in restricted regions of the parameter space and @xmath0 dimensions using plaquette operators is presented in detail . using this procedure , we are reporting for the first time exact ground - states for pam in 2d and finite value of the interaction , whose presence do not require the next to nearest neighbor extension terms in the hamiltonian . in order to do this , a completely new type of plaquette operator is introduced for pam , based on which a new localized phase is deduced whose physical properties are analyzed in detail . the obtained results provide exact theoretical data which can be used for the understanding of system properties leading to metal - insulator transitions , strongly debated in recent publications in the frame of pam . in the described case , the lost of the localization character is connected to the break - down of the long - range density - density correlations rather than kondo physics .

the modification of the @xmath0 meson properties ( masses and widths ) in hot , dense matter has consequences for scenarios of @xmath4 suppression , e.g. , by the processes of the type @xmath5 which couple hidden charm to open charm states and thus lead to a dissociation of charm in the medium @xcite , see refs . @xcite for early controversial estimates of the cross sections of such processes . the reverse process @xcite of charmonium regeneration by open charm recombination should play a dominant role for @xmath4 production at lhc @xcite where charm is abundant in the medium . since either dropping masses @xcite or increasing widths @xcite of the @xmath0 mesons in a hot and dense medium can lead to a lowering of the reaction threshold and thus to a strong increase of the rate for the processes ( [ flip ] ) , both effects may contribute to an explanation of the anomalous @xmath4 suppression found in the na50 experiment @xcite and subsequently confirmed by na60 @xcite and phenix @xcite . for a recent review , see @xcite . in contrast to results from a relativistic mean - field model of @xmath0 mesons in nuclear matter which predicts a strong downwards shift of @xmath1 meson masses due to the renormalization with a scalar mean field @xcite , the consideration of the quark substructure of these mesons leads to qualitatively different behavior . as we will show in this work on the basis of a chiral quark model of the nambu - jona - lasinio ( njl ) type and its polyakov - loop extension ( pnjl ) , the pauli blocking effect in the bethe - salpeter equation for the @xmath0-mesons largely compensates the dropping masses of their quark constituents . as a result , @xmath0-meson masses do not drop but stay almost constant or rather increase with increasing density ( and temperature ) of the matter . their decay width , however , increases rapidly and reaches values which allow for a subthreshold quark rearrangement dissociation of @xmath4 . therefore , scenarios for @xmath4 suppression built on the quark rearrangement reaction ( [ flip ] ) have to be reconsidered . as has been demonstrated in @xcite , a sufficient width of @xmath0 mesonic correlations in the quark plasma is essential for understanding charm thermalisation and diffusion in rhic experiments , see @xcite for a review . it is interesting to note that a recent self - consistent coupled channel approach for @xmath0 mesons in hot , dense nuclear matter @xcite supports the picture of a spectral broadening with a negligible mass shift up to temperatures @xmath6 mev and densities @xmath7 with @xmath8 @xmath9 being the nucleon density of nuclear matter at saturation . therefore , it seems likely that a quark hadron duality similar to that discussed for low - mass dilepton production @xcite can be observed also in the @xmath0 meson channel at the deconfinement transition . the spectral broadening of @xmath0 mesons rather than their mass shift has been suggested for an explanation of anomalous @xmath4 suppression in @xcite . this question , however , awaits a thorough investigation . in the present note , we investigate the extension of previous exploratory calculations of @xmath0 mesonic correlations in quark matter , based on the njl model @xcite to the domain of finite baryon densities which will become accessible in the cbm experiment at fair . of particular interest will be the question whether the suggested mass splitting of @xmath0 meson states @xcite will be observable or rather washed out by spectral broadening . furthermore , a strong isospin dependence of the @xmath0 meson broadening could result in observable signatures , possibly relevant for quark - gluon plasma diagnostics . chiral dynamics has been applied successfully not only in the light quark sector but also especially for the investigation of heavy - light pseudoscalar meson properties . this has been most impressively demonstrated within the dyson - schwinger equation approach in ref . @xcite which reproduced a complete set of heavy meson observables . a particular feature of these systems is the heavy quark symmetry which allows to separate the physics of the heavy and the light quark components from each other and absorb the chiral dynamics of the latter into the universal isgur - wise function @xcite . for more details , see the reviews on heavy quark effective theory @xcite . it is interesting to note that already the chiral dynamics encoded in the rather schematic njl model reproduces features like heavy - quark symmetry and isgur - wise function @xcite when extended to the heavy quark sector . these properties remain unaffected when confining properties are mimicked in the heavy - quark extended njl model by an infrared cutoff procedure @xcite or by a confining interquark potential in a relativistic potential model of heavy mesons @xcite . @xmath0 meson properties as reported in our study come from a simultaneous solution of two types of nonperturbative equations : the gap equations for the quark masses and the bethe - salpeter equations for the meson masses . the results for the dependence of pseudoscalar meson masses @xmath10 on the current quark masses @xmath11 of the heavier quark in the meson nicely reproduce the transition from a gell - mann oakes renner like behaviour for pion and kaon ( @xmath12 ) to the additive quark model like behaviour ( @xmath13 ) for @xmath0 and @xmath14 mesons . these results are then readily generalized from the vacuum to finite temperature and chemical potential within the matsubara formalism . we employ a four - flavor model with njl - type interaction kernel as a straightforward generalization of recent work on the su@xmath15(3 ) scalar and pseudoscalar meson spectrum @xcite developed on the basis of ref . @xcite and its generalization by coupling to the polyakov loop @xcite , @xmath16 \nonumber\\ & - & \mathcal{u}\left(\phi[a],\bar\phi[a];t\right ) . \end{aligned}\ ] ] here @xmath17 denotes the quark field with four flavors , @xmath18 , @xmath19 , and three colors , @xmath20 ; @xmath21 are the flavor su@xmath15(4 ) gell - mann matrices ( @xmath22 ) , @xmath23 is a coupling constant . the global symmetry of the lagrangian ( [ lagr ] ) is explicitly broken by the current quark masses @xmath24 . the covariant derivative is defined as @xmath25 , with @xmath26 ( polyakov gauge ) ; in euclidean notation @xmath27 . the strong coupling constant @xmath28 is absorbed in the definition of @xmath29 , where @xmath30 is the ( su@xmath31(3 ) ) gauge field and @xmath32 are the ( color ) gell - mann matrices . the polyakov loop field @xmath33 appearing in the potential term of ( [ lagr ] ) is related to the gauge field through the gauge covariant average of the polyakov line @xcite @xmath34 where @xmath35\ , . \label{eq : loop}\ ] ] concerning the effective potential for the ( complex ) @xmath33 field , we adopt the form and parametrization proposed in ref . @xcite . this effective chiral field theory has the same chiral symmetry of qcd , which is also shared by the quark interaction terms . the ( p)njl model is a primer for describing the dynamical breakdown of this symmetry in the vacuum and its partial restoration at high temperatures and chemical potentials . at the same time it provides a field - theoretic description of pseudoscalar meson properties which is in accordance with the low energy theorems ( such as the goldstone theorem ) of qcd . in the vacuum the pnjl model with the lagrangian ( [ lagr ] ) goes over to the njl one and the pseudoscalar meson properties are described in the standard way by analyzing the polarization operators @xmath36 \\ % { \cal p}_{ij } = s_i ( p ) ( i \gamma_5 ) s_j ( p+p)(i \gamma_5 ) \right]~,\end{aligned}\ ] ] where @xmath37 is the trace over dirac matrices , @xmath38 is the quark green function with the dynamical quark mass @xmath39 . the polarization operators can be presented in terms of two integrals which for mesons at rest in the medium are given by @xmath40 i_2^{ij}(p_0 ) \bigr\ } , \end{aligned}\ ] ] where @xmath41 @xmath42 where @xmath43 is the quark energy . as the lagrangian ( [ lagr ] ) defines a nonrenormalizable field theory , we introduce the 3 - momentum cutoff with the parameter @xmath44 to regularize the integrals . when @xmath45 it is necessary to take into account the imaginary part of the second integral . it may be found , with help of the @xmath46 prescription @xmath47 , that @xmath48 where @xmath49 is the momentum and @xmath50 the corresponding energy . the quark mass @xmath39 we find from the gap equation @xmath51 the meson mass spectrum we obtain from the condition @xmath52 the pseudoscalar meson - quark - antiquark coupling constants are defined as @xmath53 { \big| _ { p_0=m_p}}. \end{aligned}\ ] ] note that when @xmath54 , then eq . ( [ mass ] ) has to be solved in their complex form in order to determine the mass of the resonance @xmath10 and the respective decay width @xmath55 . thus , we assume that this equation can be written as a system of two coupled equations @xmath56 = \nonumber \\ & & \frac{(8g_s)^{-1}-(i_1^i+i_1^j)}{|i_2(p_0=m_p+i\epsilon)|^2 } \mbox{re } i_2(p_0=m_p+i\epsilon),\\ & & -\left [ m_p \gamma_p \right ] = \nonumber \\ & & \frac{(8g_s)^{-1}-(i_1^i+i_1^j)}{|i_2(p_0=m_p+i\epsilon)|^2 } \mbox{im } i_2(p_0=m_p+i\epsilon ) , \label{bse2}\end{aligned}\ ] ] which have solutions of the form @xmath57 as shown in @xcite , the model ( [ lagr ] ) succesfully describes meson properties in the vacuum at @xmath58 . we use here the parametrization given in table iii of ref . @xcite for the case of the njl model . the value of the coupling constant is @xmath59 , the three - momentum cutoff is at @xmath60 mev , see also @xcite for an online tool fo the three - flavor njl model . the parametrization of the current - quark masses for the heavy flavors @xmath61 is performed with the above formulae for the masses of the corresponding heavy - light pseudoscalar mesons @xmath62 . the results are summarized in table [ tab : par ] . the dependence of the heavy - light meson mass on the current mass of the heavier quark is shown by the solid line in fig . [ fig : mass ] and provides satisfactory agreement with the particle data group listings @xcite . .results for pseudoscalar meson properties in the light , strange , charm and bottom sectors for the corresponding current quark masses and dynamically generated quark masses of the model in the vacuum at @xmath58 . [ tab : par ] [ cols="<,^,^,^,^,^ " , ] in dependence on the mass @xmath63 of the heavy flavor in the bound state with a light antiquark within the present model ( red solid line ) is compared to the masses of heavy - light pseudoscalar mesons ( @xmath64 , @xmath65 , @xmath0 , @xmath14 ) and the limits for the corresponding current quark masses ( symbols with error bars ) according to the particle data group @xcite . [ fig : mass ] , scaledwidth=45.0% ] the generalization of the model ( [ lagr ] ) to the case of nonzero temperature and density ( details in @xcite ) is done within the imaginary time formalism by introducing the matsubara frequencies @xmath66 , @xmath67 , so that @xmath68 with the chemical potential @xmath69 and the temperature @xmath70 . instead of the integration over @xmath71 , we have now to perform a sum over matsubara frequencies . in the result we obtain the integrals @xmath72 @xmath73 , \end{aligned}\ ] ] where @xmath74 are the generalized fermion distribution functions @xcite for quarks of flavor @xmath75 with positive ( negative ) energies in the presence of the polyakov loop values @xmath33 and @xmath76 @xmath77 which go over to the ordinary fermi functions in the case of the njl model , where @xmath78 @xmath79 note that we put the chemical potential for charm quarks to zero in the calculations discussed below . a small finite value might be necessary to ensure exactly vanishing net charm at high baryon density when , as we demonstrate below , the symmetry between masses of @xmath0 mesons with charm and those with anticharm is broken by medium effects . we have performed selfconsistent solutions of the gap equations for the dynamically generated light quark masses at finite temperatures and chemical potentials for the pnjl model and its njl model limit within the standard setting as summarized above . in fig . [ quarks ] we display the results for the restoration of the approximate chiral symmetry in the @xmath80 quark sector along trajectories in the qcd phase diagram with a constant ratio @xmath81 , where @xmath82 . as a function of the quark chemical potential along trajectories in the quark matter phase diagram for @xmath83 ( black solid line ) , @xmath84 ( green dotted line ) , @xmath85 ( blue dash - dotted line ) and @xmath86 ( red dashed line ) . results of the njl model ( thin lines ) are compared to those of the pnjl model . they coincide for @xmath83 . [ quarks ] , scaledwidth=45.0% ] the ( pseudo)critical temperatures for the njl model and for the pnjl model are shown in the phase diagram in fig . [ phd ] together with the trajectories along which we investigate the @xmath0 meson properties . along which we investigate the @xmath0 meson properties : @xmath87 . [ phd ] ] when lowering the ratio @xmath88 , the phase transition turns from crossover to first order . the chiral restoration is a result of the phase space occupation ( pauli blocking ) which effectively reduces the interaction strength in the gap equation . navely , one would expect that heavy - light mesons such as @xmath89 mesons , should suffer a mass reduction when embedded in a hot and dense medium , as a result of the strong reduction of the light quark constituent mass ( the charm quark mass is approximately unaffected ) . the solution of the in - medium @xmath2 meson bethe - salpeter equation ( bse ) , however , shows a different result , displayed in figs . [ dmesons - tmu],[dmesons - t3mu ] for the njl case . this can be understood from inspecting the kernel of the bse : the mass shift of the light quarks , which affects the energy denominators and would lead to a lowering of the meson masses , when evaluated in free space , gets compensated or even overcompensated by the pauli blocking factor in the numerator . it is clear that @xmath0 mesons containing a light quark , as the dominant quark species in a dense medium , feel a stronger pauli blockig than those containing light antiquarks . therefore , the @xmath2 mass is shifted upwards while the @xmath1 mass stays approximately constant . meson masses ( upper panel ) and widths ( lower panel ) along the trajectory @xmath86 in the qcd phase diagram for the njl model case . while the @xmath0 meson bound state masses are almost constant or moderately rise with increasing density the continuum threshold ( @xmath90 ) gets lowered dramatically due to the chiral symmetry restoration transition . at about twice nuclear density , the decay channel into their quark - antiquark constituents opens and the bound states become resonant scattering states in the quark - antiquark continuum ( mott effect ) . [ dmesons - tmu ] , scaledwidth=45.0% ] along the trajectory @xmath84 . [ dmesons - t3mu ] , scaledwidth=45.0% ] there is an important consequence of this fact that the heavy - light continuum threshold is lowered while the bound state masses are not : at a critical density the bound states merge the continuum ( mott effect ) and become unstable against decay into their quark constituents , as signalled by the nonvanishing decay widths shown in the lower panels of figs . [ dmesons - tmu ] , [ dmesons - t3mu ] . the stronger pauli blocking for the @xmath2 mesons leads not only to a lower mott density , but also to a larger decay width as compared to the @xmath1 . from these figures one can also read off the critical densities where the phenomenon of @xmath0 meson mott effect could be expected for the njl model . in figs . [ dmu_njl],[dmu_pnjl ] we present results as a function of the quark chemical potential in isospin symmetric matter for the njl model and its extension with coupling to the polyakov loop . the results are rather similar , except for the smaller width of the transition region in the pnjl case and the different position of the critical point in the phase diagram . the higher temperature of the critical endpoint entails that along the trajectory for @xmath91 the medium undergoes a first order transition and the @xmath0 meson properties change discontinuously . meson masses and widths along the trajectories @xmath86 , @xmath85 and @xmath84 in the qcd phase diagram for the njl model . [ dmu_njl ] ] for the pnjl model . [ dmu_pnjl ] ] the modification of @xmath0 meson properties in hot and dense nuclear matter as reported in the present work is essentially different from the one suggested in ref . @xcite , where a lowering of the @xmath0 meson mass had been conjectured with consequences for charmonium dissociation in heavy - ion collision experiments . in this reference the pauli blocking effect was neglected . it is interesting to note that the pauli blocking effect occurs not only on the quark level but also on the hadronic level of description , when coupled channel equations for @xmath0 mesons in nuclear matter are solved selfconsistently as , e.g. , in ref . @xcite . also in this approach the @xmath0 meson mass remains almost unshifted while a considerable spectral broadening is obtained under similar conditions of density and temperature as considered in the present work . in the isospin - symmetric quark matter case holds @xmath92 and @xmath93 . the @xmath0 mesons containing light quarks ( @xmath94 , @xmath95 ) suffer a positive mass shift due to the effective reduction of the coupling by pauli blocking , since the phase space is occupied by light quarks abundant in the medium . the @xmath0 mesons composed of light antiquarks ( @xmath96 , @xmath97 ) suffer no pauli shift since there are no antiquarks in the medium . their mass at the mott transition density is approximately the same as in vacuum . it is interesting to compare the decay widths of the @xmath0 mesons into their quark constituents . due to the repulsive pauli shift , the @xmath2 and @xmath98 are clearly above the threshold already at the first order chiral phase transition and have a non - negligible decay width , whereas the @xmath1 and @xmath99 mesons are still very good resonances with negligible decay width at the transition . the effect of coupling the chiral quark model to the polyakov loop in the pnjl model is an effective suppression of the quark distribution functions in the bse ( [ bse1]),([bse2 ] ) as long as @xmath100 which leads to a narrowing @xcite of the chiral transition region where medium effects on the d meson properties occur in the present model . in this region the dissociation of d mesons ( mott effect ) occurs and is signalled by an increase in the spectral width of these states . summarizing the results of this model , we find no support for dropping @xmath0 meson masses in the vicinity of the chiral / deconfinement phase transition in hot and dense matter but rather a strong spectral broadening . therefore , scenarios of @xmath4 suppression in dense matter via dissociation processes like ( [ flip ] ) , which are built on increasing widths @xcite for the @xmath0 mesons should be conceptually preferable over those built on dropping masses . the work of yu.l.k was supported by dfg under grant no . bl 324/3 - 1 and by the russian fund for basic research ( rfbr ) under grant no . 09 - 01 - 00770a . d.b . acknowledges partial support from the polish ministry of science and higher education ( mnisw ) under grant no . nn 202 2318 37 and by rfbr under grant no . 11 - 02 - 01538-a . the work of p.c . was supported by fct under project no . cern / fp/116356/2010 . d. kharzeev , h. satz , phys . lett . * b340 * , 167 - 170 ( 1994 ) . k. martins , d. blaschke , e. quack , phys . rev . * c 51 * , 2723 ( 1995 ) . s. g. matinyan , b. muller , phys . rev . * c58 * , 2994 - 2997 ( 1998 ) . c. -y . wong , e. s. swanson , t. barnes , phys . rev . * c62 * , 045201 ( 2000 ) . c. m. ko , b. zhang , x. n. wang , x. f. zhang , phys . * b444 * , 237 ( 1998 ) . p. braun - munzinger , k. redlich , nucl . * a661 * , 546 ( 1999 ) . r. l. thews , m. schroedter , j. rafelski , phys . rev . * c63 * , 054905 ( 2001 ) . a. andronic , p. braun - munzinger , k. redlich , j. stachel , phys . * b652 * , 259 ( 2007 ) . h. satz , nucl . * a783 * , 249 ( 2007 ) . a. sibirtsev , k. tsushima , k. saito and a. w. thomas , phys . b * 484 * , 23 ( 2000 ) . c. fuchs , b. v. martemyanov , a. faessler and m. i. krivoruchenko , phys . c * 73 * , 035204 ( 2006 ) . g. r. g. burau , d. b. blaschke and y. l. kalinovsky , phys . b * 506 * , 297 ( 2001 ) . d. blaschke , g. burau , yu . l. kalinovsky and v. l. yudichev , prog . suppl . * 149 * , 182 ( 2003 ) . m. gonin _ et al . _ [ na50 collaboration ] , nucl . a * 610 * , 404c ( 1996 ) . m. c. abreu _ et al . _ [ na50 collaboration ] , phys . b * 410 * , 337 ( 1997 ) . b. alessandro _ et al . _ [ na50 collaboration ] , eur . j. c * 39 * , 335 ( 2005 ) . r. arnaldi _ et al . _ [ na60 collaboration ] , nucl . a * 774 * , 711 ( 2006 ) . r. arnaldi _ et al . _ [ na60 collaboration ] , phys . lett . * 99 * , 132302 ( 2007 ) . a. adare _ et al . _ [ phenix collaboration ] , phys . lett . * 98 * , 172301 ( 2007 ) . r. rapp , d. blaschke and p. crochet , prog . part . phys . * 65 * , 209 ( 2010 ) . h. van hees and r. rapp , phys . c * 71 * , 034907 ( 2005 ) . h. van hees , m. mannarelli , v. greco and r. rapp , phys . rev . lett . * 100 * , 192301 ( 2008 ) . r. rapp and h. van hees , arxiv:0903.1096 [ hep - 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ph/0602238 ] . f. sandin , _ 3fcs parametrization tool _ , http://3fcs.pendicular.net

we study @xmath0-meson resonances in hot , dense quark matter within the njl model and its polyakov - loop extension . we show that the mass splitting between @xmath1 and @xmath2 mesons is moderate , not in excess of @xmath3 mev . when the decay channel into quasifree quarks opens ( mott effect ) at densities above twice saturation density , the decay width reaches rapidly the value of 200 mev which entails a spectral broadening sufficient to open @xmath4 dissociation processes . contrary to results from hadronic mean - field theories , the chiral quark model does not support the scenario of a dropping @xmath0-meson masses so that scenarios for @xmath4 dissociation by quark rearrangement built on the lowering of the threshold for this process in a hot and dense medium have to be reconsidered and should account for the spectral broadening .

similarly to ordinary calculus , we can find in the literature distinct definitions for fractional derivatives and for fractional integrals , which are generalizations of the integer - order derivatives and multiple integrals , respectively . the most common ones and consequently more studied are the riemann liouville , caputo and grnwald letnikov definitions . we deal in this paper with the hadamard fractional derivative , introduced in @xcite . recently , it has call the attention of researchers and numerous results have appeared , with an extensive study of properties of such kind of operators @xcite . for recent results we suggest @xcite . due the complexity of solving equations involving fractional operators , in most cases is impossible to determine the exact solution and so numerical methods are used to determine an approximated solution of the problem . this is an emerging field , and we can find already in the literature several methods to deal with these problems , at least for the most common fractional derivative types . for the hadamard fractional derivative , we mention the recent paper @xcite , where the fractional operator is replaced by a finite sum involving only integer - order derivatives of the function . replacing the fractional derivative by this sum , we rewrite the initial problem in terms of integer - order derivatives and thus we are able to apply classical known methods . in @xcite another approximation formula is obtained , using also integer - order derivatives only . the disadvantage is that in order to have a good approximation , we need to use higher - order derivatives , which may not be adequate for fractional problems . in this paper we follow a different path , by discretizing the fractional derivative , and then convert continuous problems into discrete ones . to start , let us recall the definition of the hadamard fractional derivative . let @xmath0 be two reals with @xmath1 , and @xmath2\to\mathbb{r}$ ] be a function . for @xmath3 , the left hadamard fractional derivative of order @xmath4 is defined by @xmath5 while the right hadamard fractional derivative of order @xmath4 by @xmath6 where @xmath7 denotes the gamma function . when @xmath8 is an absolutely continuous function , there exists an equivalent definition ( cf . @xcite ) @xmath9 and @xmath10 more properties can be found in references at the end . the paper is organized in the following way . in section [ sec:2 ] we present the main result of the paper . starting with the definition , and with an appropriate grid on time , we present a new discrete version for the left hadamard fractional derivative . to show the efficiency of the method , in section [ sec : example ] we compare the exact expression of a fractional derivative with some numerical experiments , for different values of @xmath4 and different step sizes @xmath11 . in section [ sec : app ] we appply the technique to solve a fractional differential equation and a fractional calculus of variation problem . the discretization method is described in the following way . given a function @xmath2\to\mathbb{r}$ ] , fix a positive integer @xmath11 , and define the time step @xmath12 given @xmath13 , denote the time and space grid by @xmath14{\left(\frac{b}{a}\right)^n } \quad \mbox{and } \quad x_n = x(t_n).\ ] ] let @xmath2\to\mathbb{r}$ ] be a function of class @xmath15 and @xmath16 . denote @xmath17 then , for all @xmath18 , @xmath19 where @xmath20 and @xmath21 @xmath22.\\ \end{array}\ ] ] thus , we get the desired approximation formula . now , let us determine an upper bound for the error when we use formula . the error is given by @xmath23 let @xmath24}\left|x^{(i)}(\tau)\right|\ , , \quad i=1,2.\ ] ] then , using taylor s theorem , we get that , for all @xmath25 and for all @xmath26 $ ] , @xmath27 . \end{array}\ ] ] therefore , the error is bounded by @xmath28 \,d\tau\\ & = \displaystyle \frac{m_1+\frac32 m_2b}{\gamma(1-\alpha)}\sum_{k=1}^n ( t_k - t_{k-1 } ) \int_{t_{k-1}}^{t_k } \left(\ln\frac{t_n}{\tau}\right)^{-\alpha}\frac{1}{\tau}\ , d\tau . \end{array}\ ] ] having into consideration that , for all @xmath25 and for all @xmath29 , @xmath30 we have that @xmath31 also , since @xmath32 then @xmath33 in conclusion , we obtain the upper bound formula for our approximation : @xmath34 which converges to zero as @xmath35 . in opposite to the classical case , where the concept of derivative is local , a fractional derivative contains memory , and thus to compute the approximation obtained in eq . at a point @xmath36 , we need to know the values of @xmath37 from the beginning until de end - point , i.e. , from @xmath38 to @xmath39 for the right hadamard fractional derivative , we have in a similar way the following approximation formula : @xmath40 let @xmath41 , for @xmath42 $ ] . then ( see @xcite ) @xmath43 in figure [ intexp ] we show the accuracy of the procedure , for different values of @xmath44 and for different values of @xmath45 . the error of the numerical experiments is measured using the norm @xmath46 we can see that , for a greater value of @xmath11 , the error decreases as expected . [ ex1 ] consider a fractional differential equation with dependence on the left hadamard fractional derivative : @xmath47\\ x(a)=x_a . \end{array}\right.\ ] ] the procedure to solve numerically the system is described next . fix @xmath16 and for @xmath18 , define @xmath48 replacing the fractional operator by the approximation given in eq . , we obtain a classical difference equation with @xmath11 unknown points @xmath49 , @xmath50 for example , consider the system @xmath51\\ x(1)=0 . \end{array}\right.\ ] ] the obvious solution is @xmath52 . applying the discussed method , we obtain @xmath53 with @xmath54 in figure [ example1 ] we show the numerical results , for different values of @xmath44 and for different values of @xmath55 . [ ex2 ] for our next application , we show how to solve fractional variational problems with the lagrangian depending on the hadamard fractional derivative . consider the functional @xmath56 on the set on functions that satisfy the boundary conditions @xmath57 where @xmath58 are two fixed reals . the procedure how to find a numerical approximation is explained next . first , divide the interval @xmath59 $ ] into @xmath11 subintervals @xmath60 $ ] for @xmath18 , where @xmath61 denoting @xmath62 , applying the trapezoidal rule and taking into consideration eq . , the variational integral is approximated as @xmath63 observe that we used here the approximation @xmath64 we can regard this sum as a function of @xmath65 unknown variables @xmath66 , and then to find the optimal solution one needs to solve the system @xmath67 and with this we track the desired values @xmath68 . observe that , in opposite to the classical case , @xmath69 depends on the points @xmath70 . for example , we want the global minimizer for @xmath71 with the restrictions @xmath72 the optimal solution is @xmath52 since the functional takes only non - negative values and vanishes when evaluated at @xmath73 . in figure [ example2 ] we show the solution of the problem , for different values of @xmath44 and for different values of @xmath55 . for all numerical experiments presented above we used matlab to obtain the results . in examples [ ex1 ] and [ ex2 ] , when we take a small number of mesh points ( @xmath74 ) we get a not so good solution . however , increasing the value of @xmath11 , the error decreases and the numerical solution approaches the analytic solution , converging to it . from the numerical results we also notice that , for the same values of @xmath11 , as @xmath75 increases the error also increases , which makes sense taking into account formula . we fix all parameters except @xmath75 , it is easy to check that the maximum for error value increases as we increase the value of @xmath75 . .number of mesh points , @xmath11 , with corresponding error , @xmath76 from formula . [ cols="^,^,^,^,^,^,^,^",options="header " , ] this work was supported by portuguese funds through the cidma - center for research and development in mathematics and applications , and the portuguese foundation for science and technology ( fct - fundao para a cincia e a tecnologia ) , within project uid / mat/04106/2013 . a. a. kilbas and a. a. titioura , nonlinear differential equations with marchaud - hadamard - type fractional derivative in the weighted space of summable functions , math . model . * 12 * ( 2007 ) , no . 3 , 343 - 356 . s. pooseh , r. almeida and d. f. m. torres , expansion formulas in terms of integer - order derivatives for the hadamard fractional integral and derivative , numer . funct . * 33 * ( 2012 ) , no . 3 , 301319 . m. d. qassim , k. m. furati and n .- e . tatar , on a differential equation involving hilfer - hadamard fractional derivative , abstract and applied analysis , vol . 2012 , article i d 391062 , 17 pages , 2012 . doi:10.1155/2012/391062 d. qian , z. gong and c. li , a generalized gronwall inequality and its application to fractional differential equations with hadamard derivatives , 3rd conference on nonlinear science and complexity ( nsc10 ) , cankaya university , ankara , turkey , 2831 july , 2010 .

we present a new discretization for the hadamard fractional derivative , that simplifies the computations . we then apply the method to solve a fractional differential equation and a fractional variational problem with dependence on the hadamard fractional derivative . * mathematics subject classification 2010 : * 26a33 , 49m25 , 49m25 . * keywords : * fractional calculus , discretization methods .

it is widely accepted that the hard ( 2kev ) emission from active galactic nuclei ( agns ) is primarily produced via unsaturated inverse compton scattering of uv soft- photons from the accretion disk by a corona of hot , likely thermal , electrons ( e.g. , haardt & maraschi 1991 ; zdziarski , poutanen , & johnson 2000 ; kawaguchi , shimura , & mineshige 2001 ) . the emitted photon spectrum in the energy range is best described by a power - law of the form @xmath4 , where @xmath5 , the photon index , takes a fairly constant value of @xmath62 , and is predicted to be only weakly sensitive to large changes in the electron temperature and the optical depth in the corona ( e.g. , haardt & maraschi 1991 ) . throughout this work , we refer to @xmath5 only as the photon index in the hard ( 2kev ) band . nandra & pounds ( 1994 ) found a @xmath7=1.95@xmath80.15 for a sample of bright seyfert1 galaxies , confirming the mean predicted @xmath5 and its small dispersion . brandt , mathur , & elvis ( 1997 ) have studied the spectra of a sample of nearby agns , including low moderate luminosity narrow - line seyfert 1 ( nls1 ) galaxies , defined as type 1 agns having h@xmath2 full width at half - maximum intensity ( fwhm)2000 ^ -1 km s@xmath9 ( osterbrock & pogge 1985 ) . the addition of nls1s , which were previously overlooked due to observational biases , broadened the range of @xmath5 values , and allowed brandt et al . ( 1997 ) to find that @xmath5 is anticorrelated with fwhm(h@xmath2 ) . this extended the analogous relation between fwhm(h@xmath2 ) and the effective spectral slope in the band to harder ( e.g. , boller , brandt , & fink 1996 ; laor et al . the pseudo - thermal soft- excess radiation may be responsible for cooling the corona , thus steepening the spectrum ( e.g. , pounds , done , & osborne 1995 ) , and the pronounced soft excesses of nls1s ( e.g. , puchnarewicz et al . 1995 ) may readily explain their steep spectra and the @xmath5-fwhm(h@xmath2 ) anticorrelation . the remarkable dependence of the spectral shape of the emission , originating30@xmath10 from the central engine , on the width of a broad - emission line region ( belr ) line , emitted at @xmath1110@xmath12 , was interpreted as a fundamental dependence of @xmath5 on the accretion rate , since , as explained below , fwhm(h@xmath2 ) is considered an accretion - rate indicator in type 1 agns ( e.g. , boroson & green 1992 ; brandt & boller 1998 ) . a high accretion rate would increase the disk temperature hence producing more soft- radiation ( the `` soft excess '' ) and , at the same time , increase the compton cooling of the corona and steepen the hard- power law . recent studies of nearby ( @xmath130.5 ) type 1 radio - quiet quasars ( rqqs ) have consistently shown that @xmath14 , and some of those studies confirmed the brandt et al . ( 1997 ) @xmath5-fwhm(h@xmath2 ) relationship ( e.g. , leighly 1999 ; reeves & turner 2000 ; porquet et al . 2004 ; piconcelli et al . 2005 , hereafter p05 ; brocksopp et al . photon indices of @xmath15 are also observed for 0.5@xmath136 rqqs ( e.g. , reeves & turner 2000 ; page et al . 2005 ; shemmer et al . 2005 ) , however , the spread in their @xmath5 values seems smaller than that for nearby sources ( see fig.3 of shemmer et al . 2005 ) . on the other hand , grupe et al . ( 2006 ) report @xmath16 for a sample of @xmath17 rqqs , and suggest that the relatively steep spectra of such sources may be attributed to high accretion rates , in analogy with nls1s . lcccccccc & 13 48 44.08 & @xmath1803 53 25.0 & 2.370 & 2.47 & 2005 jul 05 & 10.4 / 90 & 10.4 / 144 & 9.4 / 520 + & 22 20 06.77 & @xmath1828 03 23.9 & 2.414 & 1.28 & 2005 oct 31 & 25.1 / 732 & 25.1 / 790 & 21.2 / 2968 in this _ letter _ we test the hypothesis that the accretion rate largely determines the hard- spectral slope in agns , and argue that previous studies have not been able to break the degeneracy between the dependence of @xmath5 on fwhm(h@xmath2 ) and the accretion rate , since highly luminous agns were left out of the analyses ( e.g. , porquet et al . this is performed by investigating and h@xmath2 spectral - region data for a well - defined sample of 30 moderate high - luminosity rqqs , selected for having high - quality and spectroscopy . five of the sources are luminous rqqs at @xmath0 , allowing , for the first time in this context , expansion of the agn parameter space by @xmath3 orders of magnitude in luminosity and black - hole mass ( @xmath19 ) . two of the @xmath0 rqqs , and , were selected for observations from the shemmer et al . ( 2004 ; hereafter s04 ) sample of luminous and high accretion - rate quasars based on their expected high fluxes in that sample . in [ observations ] we present the new observations , their reduction , and the spectral fitting of these two sources ; their and optical properties are presented in [ individual ] . in [ indicator ] we discuss our results , and in particular , the dependence of @xmath5 on the accretion rate for agns . throughout this work we consider only radio - quiet agns to avoid any contribution from jet - related emission to the spectra . luminosity distances are computed using the standard cosmological model with parameters @xmath20 , @xmath21 , and @xmath22 ^ -1 km s@xmath9mpc@xmath9 . table[obs_log ] gives a log of the ( jansen et al . 2001 ) imaging spectroscopic observations for and ; the data were processed using standard science analysis system v6.5.0 tasks . the event files of the observation of were filtered to remove @xmath635ks of flaring - activity periods ; the net exposure times in table[obs_log ] reflect the filtered data ( time filtering was not required for the observation ) . the spectra of the quasars were extracted from the images of all three european photon imaging camera ( epic ) detectors using apertures with radii of 30 . background regions were at least as large as the source regions . the spectra of ( ) were grouped with a minimum of 10 ( 50 ) counts per bin . joint spectral fitting of the data from all three epic detectors for each quasar was performed with xspec v11.3.2 ( arnaud 1996 ) . we employed galactic - absorbed power - law models at observed - frame energies @xmath230.6kev , corresponding to @xmath232kev in the rest - frame of each quasar , where the underlying power - law hard- spectrum is less prone to contamination due to any potential soft excess emission and ionized absorption . the best - fit @xmath5 values and power - law normalizations from these fits are given in table[properties ] , and the data , their joint , best - fit spectra , and residuals appear in fig.[spectra ] . we searched for intrinsic absorption in each quasar by jointly fitting the spectra with a galactic - absorbed power law model including an intrinsic ( redshifted ) neutral - absorption component with solar abundances in the same energy range as before . none of the quasars shows significant intrinsic absorption ; upper limits on intrinsic @xmath24 values appear in table[properties ] . fig.[spectra ] includes a confidence - contour plot from this fitting for each quasar . by applying @xmath25-tests between the models including intrinsic absorption and those that exclude it , we found that neither data set requires an intrinsic absorption component . we also note that our spectra show no indication of compton - reflection components or lines ; such features are expected to be relatively weak , below our detection threshold , in the luminous sources under study in this work ( e.g. , the equivalent width is expected to be50ev ; page et al . 2004a ) . finally , we searched for rapid ( @xmath61hr timescale in the rest frame ) variations in the data of the two quasars by applying kolmogorov - smirnov ( ks ) tests to the lists of photon arrival times from the event files , but no significant variations were detected . lccccccccc & @xmath26 & @xmath27 & 37 ( 52 ) & @xmath28 & 46.9 & 5100 & 10.0 & 0.3 & @xmath181.69 + & @xmath29 & @xmath30 & 50 ( 50 ) & @xmath31 & 47.2 & 5200 & 10.3 & 0.3 & @xmath181.69 basic and optical properties of and are given in table[properties ] . the luminosity at a rest - frame wavelength of 5100 is given in column(6 ) , and fwhm(h@xmath2 ) is given in column(7 ) ; these data were obtained from s04 . column(8 ) gives @xmath19 , determined as @xmath32^{0.7}\left[\frac{{\rm fwhm}({\rm h}\beta)}{10 ^ 3\,{\rm km\,s^{-1}}}\right]^2,\ ] ] and based on the recent reverberation - mapping results of peterson et al . ( 2004 ) and the kaspi et al . ( 2005 ) belr size - luminosity relation ( see also kaspi et al . we note that this relation relies on a sample of agns having luminosities up to @xmath1110@xmath33ergss@xmath9 , and extrapolating it to higher luminosities is somewhat uncertain ; a reverberation - mapping effort is underway to test the validity of such extrapolations ( s. kaspi et al . , in preparation ) . however , as @xmath19 is expected to scale with luminosity , for a given h@xmath2 width , and since in this work we perform only nonparametric statistical ranking tests , our results are not significantly sensitive to the precise coefficient values in this relation . using equation ( [ eq : mbh ] ) , the accretion - rate ratios ( col . [ 9 ] ) , @xmath34/@xmath35 ( where @xmath34 is the bolometric luminosity ; hereafter @xmath36 ) , are given by @xmath37^{0.3}\left[\frac{{\rm fwhm}({\rm h}\beta)}{10 ^ 3\,{\rm km\,s^{-1}}}\right]^{-2},\ ] ] where we have employed equation ( 21 ) of marconi et al . ( 2004 ) to obtain @xmath38 , which is the luminosity - dependent bolometric correction to @xmath39 . the bolometric correction is for the @xmath1 ( @xmath0 ) sources in this work . the optical spectral slopes in column(10 ) are defined as @xmath40@xmath41 , where @xmath42 and @xmath43 are the flux densities at 2kev and 2500 , respectively . the @xmath40 values were derived using the photon indices and fluxes in columns ( 2 ) and ( 3 ) , respectively , and the optical luminosities in column(6 ) , assuming a uv continuum of the form @xmath44 ( vanden berk et al . the @xmath40 values for our sources are consistent with the expected values , given their optical luminosities ( e.g. , steffen et al . 2006 ) . in fig.[ge ] we plot @xmath5 vs. @xmath45 , fwhm(h@xmath2 ) , @xmath19 , and @xmath36 for and . fig.[ge ] also includes 28 unabsorbed palomar green ( pg ) rqqs ( schmidt & green 1983 ) which have high - quality and optical data ; 25 of the quasars are at @xmath1 and three , pg1247@xmath46267 , pg1630@xmath46377 , and pg1634@xmath46706 , are at . photon indices in the 212kev rest - frame band for 27 of the pg quasars were obtained from table3 of p05 ; the photon index of pg1247@xmath46267 was obtained from page et al . ( 2004b ) . optical data for the pg quasars were obtained from neugebauer et al . ( 1987 ) , boroson & green ( 1992 ) , nishihara et al . ( 1997 ) , and mcintosh et al . ( 1999 ) , and processed with equations ( [ eq : mbh ] ) and ( [ eq : lledd ] ) . the fact that the and optical data are not contemporaneous may provide a source for scatter in fig.[ge ] . however , since most of the pg quasars do not exhibit significant variations in @xmath5 ( e.g. , george et al . 2000 ) and h@xmath2 width ( kaspi et al . 2000 ) , and their optical - continuum flux variations are typically at a level of 50% over timescales of several years ( kaspi et al . 2000 ) , the data in fig.[ge ] are not expected to be strongly affected by variability ( see also equations [ [ eq : mbh ] ] and [ [ eq : lledd ] ] ) . we computed spearman rank - correlation coefficients between @xmath5 and @xmath39 , fwhm(h@xmath2 ) , @xmath19 , and @xmath36 for the 25 low - luminosity , @xmath1 sources . we found that , with the exception of @xmath39 , the investigated properties are significantly correlated with @xmath5 ( with 99.9% confidence ) , in agreement with porquet et al . ( 2004 ) and p05 . the @xmath5-@xmath19 and @xmath5-@xmath36 correlations are largely a consequence of the @xmath5-fwhm(h@xmath2 ) correlation ( brandt et al . 1997 ) , since both @xmath19 and @xmath36 depend strongly on fwhm(h@xmath2 ) , and the sample spans a relatively narrow luminosity range [ @xmath4710@xmath48@xmath49ergss@xmath9 ; see eqs . ( [ eq : mbh ] ) and ( [ eq : lledd ] ) , and fig.[ge ] _ a _ ] . the absence of a @xmath5-@xmath50 correlation has been observed for larger agn samples spanning broader luminosity ranges ( e.g. , shemmer et al . 2005 ; vignali et al . 2005 ) . we repeated the above correlations , adding the five luminous quasars at @xmath0 to the analysis . we found that @xmath5 remains uncorrelated with @xmath39 , and that the @xmath5-@xmath19 correlation disappeared . this is a consequence of extending the luminosity and @xmath19 ranges by @xmath3 orders of magnitude , while our measured @xmath5 values are typical of high - redshift quasars ( e.g. , shemmer et al . the @xmath5-fwhm(h@xmath2 ) correlation remained significant with 99.9% confidence , but the ( anti-)correlation coefficient dropped from 0.61 to 0.44 . the @xmath5-@xmath36 correlation remained significant , with 99.9% confidence , and the correlation coefficient , 0.60 , has not changed . this supports the hypothesis that the photon index depends primarily on @xmath36 . to test the hypothesis that @xmath5 depends primarily on @xmath36 , we considered agns from fig.[ge]_b _ with 3500@xmath51fwhm(h@xmath2)@xmath516000 ^ -1 km s@xmath9 [ which is the fwhm(h@xmath2 ) interval of the five luminous @xmath0 sources as well as that of the majority of the s04 quasars ] , and checked the significance of the deviation between the @xmath5 values of the five luminous sources and the @xmath5 values of the eight @xmath1 pg quasars in that fwhm(h@xmath2 ) interval . a mann - whitney ( mw ) nonparametric rank test shows that the @xmath5 distributions of the five high - luminosity quasars and the eight moderate - luminosity quasars are significantly different with 99.9% confidence . we also performed a ks test on the two @xmath5 distributions and found that the @xmath5 values of the two groups of quasars can not be drawn from a single distribution with 99% confidence . on the other hand , mw and ks tests showed that , in the 0.2@xmath360.5 range ( representing the range of @xmath36 values of the five luminous quasars as well as that of the majority of the s04 sources ) , the @xmath5 values of the two groups of quasars are not significantly different ( as can be clearly seen from fig.[ge]_d _ ) . compton - reflection contributions to the spectra of all 30 quasars are expected to be negligible given their moderate high luminosities ( e.g. , page et al . 2004a , but see also jimnez - bailn et al . however , page et al . ( 2004b ) detected compton reflection in pg1247@xmath46267 . we reanalyzed the spectrum of this source , and found that the hard- excess has no significant effect on @xmath5 , whether fitted with or without a compton - reflection component ( consistent with page et al . we also searched for systematic differences between the analyses of and and the p05 sources by reanalyzing the spectra of pg1630@xmath46377 , pg1634@xmath46706 , and the eight @xmath1 pg quasars with 3500@xmath51fwhm(h@xmath2)@xmath516000 ^ -1 km s@xmath9 . we reproduced the p05 photon indices in the rest - frame band with no systematic deviations from their values . additionally , fitting the data of and from rest - frame ( observed - frame 0.63.5kev ) did not alter our results significantly . we note that for five @xmath1 pg quasars with 3500@xmath51fwhm(h@xmath2)@xmath516000 ^ -1 km s@xmath9 the @xmath5 values from p05 are consistent with those from porquet et al . ( 2004 ) , and are somewhat lower than those from brocksopp et al . ( 2006 ) , due perhaps to analyses differences . our results suggest that @xmath5 depends primarily on @xmath36 , as pointed out by e.g. , brandt & boller ( 1998 ) and laor ( 2000 ) . the key difference between this work and previous studies ( e.g. , porquet et al . 2004 ; wang , watarai , & mineshige 2004 ; bian 2005 ) is that we expanded the quasar parameter space by adding highly luminous sources to the correlations ( see fig.[ge]_a _ ) . such luminous quasars are found only at , and their h@xmath2 lines , required for @xmath19 and @xmath36 determinations ( see e.g. , s04 ; baskin & laor 2005 ) , are shifted into the observed near - infrared band . and h@xmath2 spectral - region data of additional high - luminosity quasars are crucial to break the degeneracy between the @xmath5-fwhm(h@xmath2 ) and correlations and conclude that the power - law photon index depends on the accretion rate . ultimately , a large enough inventory of and h@xmath2 spectral - region data for luminous , high - redshift quasars will allow testing of the hypothesis that such sources are analogous to nls1s ( e.g. , grupe et al . 2006 ) . this work is based on observations obtained with , an esa science mission with instruments and contributions directly funded by esa member states and the usa ( nasa ) . we thank an anonymous referee for constructive comments , and george chartas and aaron steffen for fruitful discussions . we gratefully acknowledge the financial support of nasa grant ( o.s , w.n.b ) , nasa ltsa grant ( o.s , w.n.b ) , and the zeff fellowship at the technion ( s.k . ) . this work is supported by the israel science foundation grant 232/03 .

we present new observations of two luminous and high accretion - rate radio - quiet active galactic nuclei ( agns ) at @xmath0 . together with archival and rest - frame optical spectra of three sources with similar properties as well as 25 moderate - luminosity radio - quiet agns at @xmath1 , we investigate , for the first time , the dependence of the hard ( 2kev ) power - law photon index on the broad h@xmath2 emission - line width and on the accretion rate across @xmath3 orders of magnitude in agn luminosity . provided the accretion rates of the five luminous sources can be estimated by extrapolating the well - known broad - line region size - luminosity relation to high luminosities , we find that the photon indices of these sources , while consistent with those expected from their accretion rates , are significantly higher than expected from the widths of their h@xmath2 lines . we argue that , within the limits of our sample , the hard- photon index depends primarily on the accretion rate .

the analysis of the octet baryon masses in the framework of chiral perturbation theory already has a long history , see e.g. @xcite . in this paper , we present the results of a first calculation including _ all _ terms of order @xmath4 , where @xmath5 is a generic symbol for any one of the light quark masses @xmath6 . we work in the isospin limit @xmath7 and neglect the electromagnetic corrections . previous investigations only considered mostly the so called computable corrections of order @xmath8 or included some of the finite terms at this order . this , however , contradicts the spirit of chiral perturbation theory ( chpt ) in that all terms at a given order have to be retained , see e.g. @xcite . in general , the quark mass expansion of the octet baryon masses takes the form @xmath9 modulo logs . here , @xmath1 is the mass in the chiral limit of vanishing quark masses and the coefficients @xmath10 are state dependent . furthermore , they include contributions proprotional to some low energy constants which appear beyond leading order in the effective lagrangian . in this letter , we evaluate these coefficients for the octet baryons @xmath11 and @xmath12 . in addition , we also calculate the pion nucleon @xmath0term which is intimately related to the quark mass expansion of the baryon masses @xcite . for some comprehensive reviews , see e.g. @xcite . to perform the calculations , we make use of the effective meson baryon lagrangian . our notation is identical to the one used in @xcite and we discuss here only the new terms . denoting by @xmath13 the pseudoscalar goldstone fields ( @xmath14 ) and by @xmath15 the baryon octet , the effective lagrangian takes the form @xmath16 where the chiral dimension @xmath17 counts the number of derivatives and/or meson mass insertions . the baryons are treated in the extreme non relativistic limit @xcite , i.e. they are characterized by a four velocity @xmath18 . in this approach , there is a one to one correspondence between the expansion in small momenta and quark masses and the expansion in goldstone boson loops , i.e. a consistent power counting scheme emerges . the form of the lowest order meson baryon lagrangian is standard , see e.g. @xcite , and the meson lagrangian is given in @xcite . the dimension two meson baryon lagrangian can be written as ( we only enumerate the terms which contribute ) @xmath19 with the @xmath20 monomials in the fields of chiral dimension two . typical forms are @xmath21)$ ] , @xmath22)$ ] or @xmath23 , with @xmath24 , @xmath25 and @xmath26 collects the pseudoscalars . the form of @xmath27 is @xcite , @xmath28 ) + b_0 \ , { \rm tr}(\bar b b ) \ , { \rm tr}(\chi_+ ) \ , \ , , \label{leff2st}\ ] ] i.e. it contains three low energy constants and @xmath29 is proportional to the quark mass matrix @xmath30 since @xmath31 . here , @xmath32 and @xmath33 is the pseudoscalar decay constant . all low energy constants in @xmath34 are finite . a subset of the @xmath35 has been estimated in @xcite by analyzing kaon nucleon scattering data . the splitting of the dimension two meson baryon lagrangian in eq.([leff2 ] ) is motivated by the fact that while the first three terms appear in tree and loop graphs , the latter ten only come in via loops . there are seven terms contributing at dimension four , @xmath36 with typical forms of the @xmath37 are @xmath38 or @xmath39 ) $ ] . at this stage , we take @xmath40 . for @xmath41 , there is an additional term in @xmath42 . the explicit expressions for the various terms in eqs.([leff2],[leff4 ] ) can be found in @xcite . we point out that there are 20 a priori unkown constants . in addition , there are the @xmath43 and @xmath44 coupling constants ( subject to the constraint @xmath45 ) from the lowest order lagrangian @xmath46 . what we have to calculate are all one loop graphs with insertions from @xmath47 and tree graphs from @xmath48 . we stress that we do not include the spin3/2 decuplet in the effective field theory @xcite , but rather use these fields to estimate the pertinent low energy constants ( resonance saturation principle ) . we therefore strictly count in small quark masses and external momenta with no recourse to large @xmath49 arguments . the form of the terms @xmath50 and @xmath51 for the baryon masses and @xmath52 is standard , we use here the same notation as ref.@xcite . the @xmath53 contribution to any octet baryon mass @xmath54 takes the form @xmath55 with @xmath56 and @xmath57 the scale of dimensional regularization . the explicit form of the state dependent prefactors @xmath58 can be found in ref.@xcite . notice the appearance of mixed terms @xmath59 which were not considered in most existing investigations . the fourth order contribution to the baryon masses contains divergences proportional to ( using dimensional regularization ) @xmath60 \biggr\rbrace \ , \ , , \label{l}\ ] ] with @xmath61 . these are renormalized by an appropriate choice of the low energy constants @xmath62 , @xmath63 in fact , all seven @xmath62 are divergent . the appearance of these divergences is in marked contrast to the @xmath64 calculation which is completely finite ( in the heavy fermion approach ) . in what follows , we set @xmath65 gev . similarly , the fourth order contribution to the pion nucleon @xmath0term can be written as @xmath66 \ , \ , , \label{sigma4}\ ] ] with the @xmath67 given in @xcite . here , the renormalization is somewhat more tricky . it can most efficiently performed in a basis of a linearly independent subset of the operators @xmath68 , @xmath69 , as detailed in ref.@xcite . a good check on the rather lengthy expressions for the nucleon mass and @xmath52 is given by the feynman hellmann theorem , @xmath70 , with @xmath71 the average light quark mass , @xmath72 . we remark here that in contrast to the order @xmath64 calculation , the shift to the cheng dashen point , @xmath73 , is no longer finite , i.e. there appear @xmath74dependent divergences . we therefore do not consider this @xmath0term shift in what follows . we will also not discuss in detail the two kaon nucleon @xmath0terms , @xmath75 , for similar reasons in this letter . clearly , we are not able to fix all the low energy constants appearing in @xmath48 from data , even if we would resort to large @xmath49 arguments . we will therefore use the principle of resonance saturation to estimate these constants . this works very accurately in the meson sector @xcite . in the baryon case , one has to account for excitations of meson ( @xmath76 ) and baryon ( @xmath77 ) resonances . one writes down the effective lagrangian with these resonances chirally coupled to the goldstones and the baryon octet , calculates the pertinent feynman diagrams to the process under consideration and , finally , lets the resonance masses go to infinity ( with fixed ratios of coupling constants to masses ) . this generates higher order terms in the effective meson baryon lagrangian with coefficients expressed in terms of a few known resonance parameters . symbolically , we can write @xmath78 \to { \cal l}_{\rm eff } [ \ , u , b \ , ] \ , \ , .\ ] ] here , there are two relevant contributions . one comes from the excitation of the spin-3/2 decuplet states and the second from t channel scalar and vector meson excitations , cf . fig.[fig1 ] . it is important to stress that for the resonance contribution to the baryon masses , one has to involve goldstone boson loops . this is different from the normal situation like e.g. in form factors or scattering processes . 1.2 in = 1.5 in consider first the decuplet contribution . we treat these field relativistically and only at the last stage let the mass become very large . the pertinent interaction lagrangian between the spin3/2 fields ( denoted by @xmath79 ) , the baryon octet and the goldstones reads ( suppressing su(3 ) indices ) @xmath80 with @xmath81 the standard @xmath82 isospin transition operator , @xmath83 mev the ( average ) pseudeoscalar decay constant and @xmath84 determined from the decays @xmath85 . the dirac matrix operator @xmath86 is given by @xmath87 for the off shell parameter @xmath88 , we use @xmath89 from the determination of the @xmath79 contribution to the @xmath90 scattering volume @xmath91 @xcite . the tadpole graphs shown in fig . 1b with an intermediate vector meson only contribute at order @xmath92 . this is evident in the conventional vector field formulation . in the tensor field formulation , the vertex @xmath93 seems to lead to a contribution at order @xmath53 . however , in a tadpole graph one needs the index contraction @xmath94 and thus has no term at order @xmath53 . matters are different for the scalars . denoting by @xmath95 and @xmath96 the scalar octet and singlet with @xmath97 gev , respectively , the lowest order coupling to the baryon octet reads @xmath98 ) + d_{s_1 } \,s_1 \ , { \rm tr}(\bar b b ) \label{sb}\ ] ] where the coupling constants @xmath99 , @xmath100 and @xmath101 are of the order one . in fact , there are no empirical data to really pin them down . from the decay pattern of the low lying baryons one can , however , estimate these numbers to be small . we will generously vary them between zero and one . for the couplings of the scalars to the goldstones , we use the notation of @xcite and the parameters determined therein . putting pieces together , all low energy constants are expressed via resonance parameters and the baryon masses take the form @xmath102 with @xmath103 the contribution of @xmath104 and @xmath105 gev is the average decuplet mass . notice that it is convenient to lump the @xmath106 and the @xmath4 corrections togther as it was done in eq.([massres ] ) ( in the @xmath107 and @xmath108 ) . we have kept explict the baryon mass in the chiral limit . of course , in the fourth order terms it could be substituted by the corresponding physical values . at second order , however , we would get a state dependent shift , see e.g. @xcite , and we thus prefer to work with @xmath1 . alternative representations for the baryon masses are given in @xcite . let us briefly explain the origin of the various terms in eq.([massres ] ) . the @xmath57 contributions are tadpoles with @xmath109 insertions from @xmath34 . similarly , the @xmath110 and @xmath111 terms stem from tadpole graphs with insertions proportional to the low energy constants @xmath112 and @xmath35 , respectively . note , however , that in the resonance exchange approximation not all of the ten @xmath35 are contributing . the @xmath113 terms subsume the contributions from @xmath42 , these are proportional to the low energy constants @xmath62 . finally , the terms of the type @xmath108 are the scalar meson contributions to the mass . they amount to a constant , state dependent shift . these consist of terms of the types @xmath114 , @xmath115 and @xmath116 , see @xcite . to be specific , we give the coefficients @xmath117 and @xmath57 for the nucleon , @xmath118 m_\eta^2 \ln \frac{m_\eta^2}{\lambda^2 } \nonumber \\ & + & \biggl [ \frac{9}{8}(d + f)^2 \bigl ( \frac{m_k^2}{4 } + m_\pi^2 \bigr ) \biggr ] m_\pi^2 \ln \frac{m_\pi^2}{\lambda^2 } \nonumber \\ & + & \biggl [ -\frac{3}{4}(d - f)^2 m_k^2 + \frac{1}{96 } ( 223d^2 - 318df + 351f^2 ) m_\pi^2 \biggr ] m_k^2 \ln \frac{m_k^2}{\lambda^2 } \biggr\rbrace \ , \ , , \nonumber \\ & d_n^\delta & = \frac{{\cal c}^2}{2f_p^2 } \biggl\lbrace -\frac{415}{288 } m_\pi^4 + \frac{83}{72 } m_\pi^2 m_k^2 - \frac{31}{72 } m_k^4 \biggr\rbrace \ , \ , , \nonumber \\ & d_n^\epsilon & = \frac{{\cal c}^2}{8 \pi^2 f_p^2 } \biggl\lbrace - \frac{1}{2 } m_\pi^4 \ln \frac{m_\pi^2}{\lambda^2 } - \frac{1}{8 } m_k^4 \ln \frac{m_k^2}{\lambda^2 } \biggr\rbrace \ , \ , . \label{dn}\end{aligned}\ ] ] the corresponding coefficients for the @xmath119 , @xmath120 and @xmath12 and also the @xmath108 can be found in @xcite . the tree contribution from @xmath27 is subsumed in the @xmath107 . the numerical values of the @xmath121 , @xmath122 , @xmath123 and @xmath108 are given in table 1 ( using @xmath124 , @xmath125 and @xmath126 from @xcite ) . .numerical values of the state dependent coefficients in eq.([massres ] ) . the @xmath107 are dimensionless . the @xmath108 are for @xmath127 , @xmath128 from @xcite and @xmath129 gev . [ cols="^,^,^,^,^,^",options="header " , ] note , however , that this is only a subset of the coefficients considered in this work . the full list will be given in @xcite . we remark that the procedure used in @xcite involves the summation of arbitrary high orders via a lippmann schwinger equation and is thus afflicted with some uncertainty not controled within chpt . the only free parameter in the formula for the baryon masses , eq.([massres ] ) , is the mass of the baryons in the chiral limit , @xmath1 , since all low energy constants are fixed in terms of resonance parameters . in particular , in contrast to ref.@xcite , the parameters @xmath130 , @xmath131 and @xmath132 are no longer free . also , at quadratic order in the quark masses the ambiguity between @xmath1 and @xmath130 is resolved , it is not necessary to involve any one of the @xmath0terms in the fitting procedure @xcite . in fact , one can not find one single value of @xmath1 to fit all four octet masses , @xmath133 gev , @xmath134 gev , @xmath135 gev and @xmath136 gev exactly . we therefore fit these masses individually and average the corresponding values for @xmath1 . the contribution from scalar meson exchange only enters the uncertainties of the numbers given . this is justified since the numerical values for the couplings @xmath100 and @xmath99 are supposedly small . a more thorough discussion on this point can be found in @xcite . we have @xmath137 , @xmath138 , @xmath139 , @xmath140 , with the following average @xmath141 this number is compatible with the one found in the analysis of the pion nucleon @xmath0term , where approximately 130 mev to the nucleon mass were attributed to the strange matrix element @xmath142 ( with a sizeable uncertainty ) @xcite . the spread of the various values is a good measure of the uncertainties related to this complete @xmath53 calculation . let us take a closer look at the quark mass expansion of the nucleon mass , in the notation of eq.([massform ] ) , @xmath143 this looks similar for the other octet baryons . we conclude that the quark mass corrections of order @xmath5 , @xmath144 and @xmath8 are all of the same size . can be significantly smaller . however , the nucleon mass is much more sensitive to the scalar contribution than the other octet masses . ] in ref.@xcite , it was argued that only the leading non analytic corrections ( lnac ) @xmath51 are large and that further terms like the ones @xmath145 are modestly small , of the order of 100 mev . this would amount to an expansion in @xmath146 with a large leading term . this expectation is not borne out by our results , the next corrections are as large as the lnacs . these findings agree with the meson cloud model calculation of gasser @xcite . a last remark about the baryon masses concerns the deviation from the gell - mann - okubo relation , @xmath147 which empirically is about 6.5 mev . we find @xmath148 mev , which is larger in magnitude than the value found in @xcite . we remark that in our case the decuplet contribution is contained in the @xmath8 contributions and not in the @xmath144 as in @xcite . therefore , in our case , @xmath149 is dominated by the @xmath8 piece . the sizeable uncertainty in the chiral limit masses , eq.([mkrig ] ) , does not allow for a very accurate statement about this very small quantity . it is also very sensitive to the scalar couplings . to get a better handle on this issue , one either has to be able to fix all pertinent low energy constants at order @xmath8 from data or improve upon the resonance saturation estimate used here by including e.g. the mass splitting within the decuplet and the su(3 ) breaking of the decuplet - octet - meson couplings . a better understanding of this topic is , of course , at the heart of the determination of the quark mass ratio @xmath150 from the baryon masses ( once the electromagnetic corrections have been included ) . nucleon @xmath0term is completely fixed . using for @xmath1 its average , we find ( no scalar resonance contribution ) @xmath151 with an uncertainty of about @xmath152 mev due to the spread in @xmath1 . this number compares favourably with the one found in ref.@xcite , @xmath153 mev . we stress that this result reflects a very non trivial consistency for the complete calculation to quadratic order in the quark masses using the resonance saturation principle in the scalar sector . the additional contribution from the scalar meson exchange is accounted for in the @xmath152 mev uncertainty . to be specific , we have @xmath154 . it is furthermore instructive to disentangle the various contributions to @xmath155 of order @xmath156 , @xmath64 and @xmath53 , respectively , @xmath157 which shows a moderate convergence , i.e. the terms of increasing order become successively smaller . still , the @xmath53 contribution is important . also , at this order no @xmath158 rescattering effects are included . we notice that using the values for @xmath112 and @xmath35 as determined in ref.@xcite leads to a much increased fourth order contribution . in this paper , we have used heavy baryon chiral perturbation theory to calculate the octet baryon masses to quadratic order in the quark masses , including 20 local operators with unknown coeffcients . these low energy constants were fixed by resonance exchange . the dominant contribution comes indeed from the excitation of the spin3/2 decuplet fields . tadpole graphs with scalar meson exchange only lead to small corrections . this left us with one free parameter , the baryon mass in the chiral limit , which could be determined within 18% accuracy , and is compatible with the value inferred for the strange matrix element @xmath159 in ref.@xcite . furthermore , the pion nucleon @xmath0term comes out surprisingly close to its empirical value , @xmath160 mev , with an uncertainty of about @xmath152 mev . this first exploratory @xmath53 study of the three flavor scalar sector of baryon chpt points towards a significant improvement compared to previous investigations which were mostly confined to so called `` computable '' corrections and/or fitted a few of the pertinent low energy constants . however , the calculation is not yet accurate enough to determine the quark mass ratio @xmath76 reliably from the octet masses . furthermore , we did not address the kaon nucleon @xmath0terms , the corresponding shifts to the pertinent cheng - dashen points together with the strangeness content of the proton here . we will come back to these topics in ref.@xcite . we are grateful to daniel wyler for a very useful remark . s. mallik , `` massive states in chiral perturbation theory '' , saha institute preprint sinp - tnp/94 - 10 , 1994 [ hep - ph/9410344 ] m.k . banerjee and j. milana , `` baryon mass splittings in chiral perturbation theory '' , preprint umpp 95 - 058 , 1994 [ hep - ph/9410398 ]

we analyze the octet baryon masses and the pion nucleon @xmath0term in the framework of heavy baryon chiral perturbation theory . in contrast to previous investigations , we include _ all _ terms up - to - and - including quadratic order in the light quark masses . the pertinent low energy constants are fixed from resonance exchange . this leaves as the only free parameter the baryon mass in the chiral limit , @xmath1 . we find @xmath2 mev together with @xmath3 mev . we discuss various implications of these results . tk 95 21 ,

numerical studies of hamiltonian su(@xmath0 ) lattice gauge theory in ( 3 + 1 ) dimensions have shown that the gauge fields exhibit chaotic behavior in the classical limit @xcite . the numerical value of the largest positive lyapunov exponent @xmath1 has been obtained for su(2 ) and su(3 ) with the result @xcite @xmath2 where @xmath3 is the average energy per plaquette , @xmath4 for su(2 ) , and @xmath5 for su(3 ) . for the su(2 ) gauge theory the complete spectrum of lyapunov exponents was obtained on small lattices @xcite . these calculations , which follow the evolution of a classical gauge field configuration in minkowski space , also showed that the energy density distribution on the lattice rapidly approaches a thermal distribution @xcite . this finding confirms the expectation of a finite growth rate of the coarse - grained entropy density of the gauge field , which follows from the observation that the sum over all positive lyapunov exponents at fixed energy density grows like the volume @xcite . hence , at any given level of coarse - graining , the classical gauge field `` self - thermalizes '' on a time scale of the order of the inverse lyapunov exponent . in order to determine the value of the maximal lyapunov exponent @xmath1 , the evolution of the gauge field configurations must be followed over periods @xmath6 . the lyapunov exponent is therefore effectively obtained for gauge fields that are members of a thermal ensemble , and we can identify the average energy per plaquette @xmath3 in ( 1 ) with that of a thermalized lattice . at high temperature the gauge field is a collection of weakly coupled harmonic oscillators , hence the average energy per independent degree of freedom of the classical gauge field is equal to the temperature @xmath7 , yielding @xmath8 for su(@xmath0 ) . the factor @xmath9 accounts for the restrictions imposed by gauss law . we can therefore rewrite the result ( [ 1.1 ] ) as @xmath10 as already noted in @xcite these values for @xmath1 coincide , apart from a factor 2 , with those of the damping rate of a thermal plasmon at rest , obtained by braaten and pisarski @xcite in the framework of thermal perturbation theory : @xmath11 the goal of the present article is to establish this connection and to explain the origin of the factor @xmath12 . we approach this goal in several steps . first we review the numerical `` measurement '' of the lyapunov exponent in classical lattice gauge theory . we point out that the exponential growth rate of a small perturbation in the magnetic energy density used in those calculations is equal to twice of the growth rate of fluctuations in the elementary field variable , in the continuum limit the vector potential . this explains the factor 2 between @xmath1 and @xmath13 . in the next step we demonstrate that in classical calculations the linear perturbation propagation corresponding to the equations of motion of a chaotic dynamical system has in general a fourier spectrum of imaginary frequencies . the lyapunov exponent is equal to the magnitude of those imaginary frequencies . then we argue that the chaotic dynamics of the classical system acts like a thermal ensemble averaging the perturbation propagation equation over stochastic frequencies . the square of these frequencies can either be positive or negative . in this case the damping rate and the plasma frequency of the classical elementary field fluctuations are related to the mean value and the width of the probability distribution of frequency squares . the final result of these considerations is that the lyapunov exponent as defined in [ 1 ] measures twice the damping rate of classical gauge field fluctuations on the lattice . it is left to show that the quantum field theoretical calculation of the thermal damping rate at rest in hot perturbation theory in the leading @xmath14 order survives in the classical @xmath15 limit . we begin with the discussion of this point in order to establish connection with thermal quantum field theory . we begin by briefly reviewing the derivation of the plasmon damping rate . nonabelian gauge field fluctuations in a thermal background have been studied extensively in the framework of perturbation theory @xcite . the gauge field develops massive collective modes ( plasmons ) with frequency @xmath16 due to interaction with `` hard '' thermal gauge bosons , i.e. excitations with energy of order @xmath7 . the energy of a plasmon at rest is @xmath17 in su(@xmath0 ) gauge theory . for our purpose it is important that the dispersion relation @xmath18 can be obtained in the framework of semiclassical transport theory , where classical field fluctuations @xmath19 are coupled to the quantized thermal excitations of the gauge field @xcite . the gauge invariant description of the collective modes requires the introduction of effective @xmath20-point vertices @xcite , which can be systematically derived from the effective action @xcite : @xmath21 where @xmath22 is a null four - vector , and the integral is over all directions of the spatial unit vector @xmath23 . @xmath24 stands for the gauge - covariant derivative . we have denoted the collective gauge potential @xmath19 and field strength @xmath25 by lower - case letters to indicate that these describe fluctuations around a thermal background . note that @xmath26 is a classical construction , with the sole exception that the plasmon rest mass @xmath27 depends on the energy distribution @xmath28 of quantized thermal excitations of the gauge field : @xmath29 at leading order in @xmath30 , ( [ 1.11 ] ) is evaluated for hard thermal quanta with @xmath31 . braaten and pisarski@xcite showed that the collective plasmon modes are unstable due to the effective interaction ( [ 1.10 ] ) . the plasmon damping rate @xmath32 is defined as imaginary part of the plasmon pole in the feynman propagator corresponding to decaying plane wave solutions . the rate of instability for a plasmon at rest can be expressed as the imaginary part of the polarization function of the gauge field at the plasmon pole @xcite : @xmath33 where the transverse polarization function @xmath34 only depends on soft modes described by ( [ 1.10 ] ) . the plasmon rest mass exactly cancels from the expression ( [ 1.13 ] ) and the result ( [ 1.3 ] ) is a pure number multiplied by @xmath35 , which is a classical inverse length scale . in fact , the calculation explicitly makes use of the classical limit of the bose distribution , @xmath36 , in the evaluation of the loop integral ( see eq . ( 23 ) of ref . @xcite ) . since the effective action ( [ 1.10 ] ) can be derived from classical considerations @xcite , assuming a given spectrum of thermal excitations , it also applies to the collective excitations of the _ classical _ gauge field on a lattice . the sole modification is that the spectrum of thermal fluctuations is now given by the limit of the bose distribution . denoting the lattice spacing by @xmath37 we find @xmath38 in the weak - coupling , large volume limit . the plasmon mass ( [ 1.12 ] ) is a purely classical quantity of dimension ( length)@xmath39 not containing @xmath40 , but it diverges in the continuum limit @xmath41 . this is not surprising , since the lattice spacing serves as a cut - off that is required to regularize the ultraviolet divergences of the classical thermal gauge theory . the exponential growth rate of small classical field fluctuations is not affected by this divergence because it does not depend on the value of @xmath27 , as mentioned above . the result ( [ 1.3 ] ) for the plasmon damping rate @xmath13 remains valid if the correct plasmon mass @xmath27 in the effective action ( [ 1.10 ] ) is replaced by the value ( [ 1.12 ] ) for the classical gauge field defined on a lattice . more intuitively , the independence of @xmath13 from the value of @xmath27 can be understood as follows . the cross section for scattering of a thermal gluon on a slow plasmon is : @xmath42 where @xmath43 is the inverse debye color screening length . the scattering rate @xmath44 is obtained by multiplying with the gluon density in the initial state and with the bose factor in the final state , yielding : @xmath45 where we have made use of ( [ 1.10 ] ) . from this result , which has the same structure as the expression ( [ 1.3 ] ) for @xmath13 , it is obvious that the plasmon mass @xmath27 as well as @xmath40 cancel from the scattering rate . the lyapunov exponents measure the growth rate of infinitesimal perturbations around an exact solution of the classical lattice yang - mills equations . since the maximal lyapunov exponent @xmath1 was shown to be independent of the lattice spacing , we assume that we can work in the continuum limit whenever adequate . if @xmath46 is an exact solution of the yang - mills equations , the linearized equation for a small perturbation @xmath47 around @xmath48 is @xmath49 = 0 . \label{1.4}\ ] ] here @xmath50 $ ] is the gauge covariant derivative where the bracket denotes the lie algebra commutator , and @xmath51 is the field strength tensor associated with the background field @xmath48 . the numerical approach to the determination of @xmath1 proceeds by solving ( [ 1.4 ] ) for an arbitrary initial condition @xmath52 and measuring the growth rate of the norm of @xmath47 . to be precise , the maximal lyapunov exponent was determined in @xcite from the logarithmic growth rate of the `` distance '' between neighboring field configurations , defined on the lattice as @xmath53 = { 1\over 2n_p } \sum_p\big\vert { \tr}\ ; u_p - { \tr } \ ; u'_p \big\vert , \label{1.5}\ ] ] where @xmath54 are the group valued link variables , @xmath55 denotes the elementary plaquette operator , and @xmath56 is the total number of spatial plaquettes . in the continuum limit , the distance measure ( [ 1.5 ] ) takes the form @xmath57 \propto \int d^3x \left\vert { \tr}\ ; b'(x)^2 - { \tr}\ ; b(x)^2\right\vert , \label{1.6}\ ] ] where @xmath58 are the magnetic fields associated with the gauge potential @xmath59 . in going from ( [ 1.5 ] ) to ( [ 1.6 ] ) we have suppressed the constant factor @xmath60 , since we are interested only in the growth rate of @xmath61 . for an infinitesimal perturbation @xmath19 that is a solution of the linearized equation ( [ 1.4 ] ) , we obtain : @xmath62 \equiv{\cal d}[a_{\mu}+a_{\mu},a_{\mu } ] \propto \nonumber \\ \int d^3x \left\vert { \rm tr } \left ( { \partial ( { \rm tr}\ ; b^2 ) \over\partial a_{\mu } } a_{\mu}\right ) + { 1\over 2 } { \rm tr } \left ( { \partial^2 ( { \rm tr}\ ; b^2)\over \partial a_{\mu}\partial a_{\nu } } a_{\mu } a_{\nu}\right ) \right\vert . \label{1.7}\end{aligned}\ ] ] the maximal lyapunov exponent is then defined as @xmath63 = \lim_{t_0\to\infty } { 1\over t_0 } \ln \frac{{\cal d}[a_{\mu}(t_0)\vert a_{\mu}]}{{\cal d}[a_{\mu}(0)\vert a_{\mu}]}. \label{1.7a}\ ] ] in practice , every randomly chosen initial configuration @xmath64 with a fixed average energy density has been found to yield the same value for the maximal lyapunov exponent @xmath1 . the numerical calculations show that the maximal lyapunov exponent depends only weakly on the lattice size and extrapolates smoothly to the limit of spatially homogeneous gauge potentials on a @xmath65 lattice . we take this as an indication that @xmath1 is associated with long wavelength perturbations @xmath47 in an appropriately chosen gauge . we now propose to make use of the fact , noted in the introduction , that the background gauge field @xmath46 rapidly approaches thermal configurations , by replacing the _ long - time _ average of the growth rate of @xmath66 by the _ canonical _ average over background gauge fields @xmath48 , where the temperature @xmath7 is chosen such that the thermal energy density equals the average energy density of the time - dependent background field @xmath46 . the replacement of the temporal average by the canonical average relies on two conditions : the autocorrelation function of the background field @xmath46 must decay on a time scale that is short compared with the time @xmath67 required for the calculation of the lyapunov exponent , and the time evolution of the background field must be ergodic on the time scale @xmath67 . the ergodicity of the background gauge field is assured by its dynamical chaoticity on time scales long compared to the inverse of the positive lyapunov exponents , hence the second condition is fulfilled @xcite . on the other hand , if the first condition were violated , the lyapunov exponent would depend on the starting configuration @xmath46 . in numerical studies [ 1 - 4 ] we have found that this is not the case . a direct study of the autocorrelation function performed by us has shown that the first condition is also satisfied . these conditions are in accordance with the @xmath68 hierarchy assumed in hot perturbative gauge theory . the maximal lyapunov exponent is then obtained from the relation @xmath69\rangle_t , \label{1.15a}\ ] ] where the distance measure ( [ 1.7 ] ) in a thermal background is @xmath70\rangle_t & \propto & \int d^3x \left\vert { \rm tr } \left ( \left\langle { \partial({\rm tr}\ ; b^2)\over \partial a_{\mu}}\right\rangle_t a_{\mu}^{(t)}\right ) \right . \nonumber \\ \nonumber \\ & + & \left . { 1\over 2 } { \rm tr } \left ( \left\langle { \partial^2 ( { \rm tr}\ ; b^2 ) \over \partial a_{\mu } \partial a_{\nu}}\right\rangle_t a_{\mu}^{(t ) } a_{\nu}^{(t)}\right ) \right\vert\ , . \label{1.8}\end{aligned}\ ] ] the first term in ( [ 1.8 ] ) vanishes , because the thermal average of any quantity transforming under the adjoint representation is zero . in the second term , the thermal average projects on to the singlet part of @xmath71 , yielding @xmath72\rangle_t \propto \int d^3x \left\vert \left\langle { \partial^2({\rm tr}\ ; b^2)\over\partial a_{\mu}\partial a_{\nu}}\right\rangle_t { \rm tr}\ ; \bigl(a_{\mu}^{(t)}a_{\nu}^{(t)}\bigr ) \right\vert . \label{1.9}\ ] ] since the averaged value of @xmath73 is quadratic in the field fluctuations @xmath74 the lyapunov exponent defined through the magnetic energy distance measure is twice as large as the one defined by the dominant exponential growth rate of the fluctuations of the elementary field @xmath63 = 2 \,\ , \lim_{t_0\to\infty } { 1\over t_0 } \ln \frac{||a_{\mu}(t_0)||}{||a_{\mu}(0)||}.\ ] ] solving the classical equations of motion one deals with a problem essentially different from perturbative field theory : instead of investigating transition amplitudes between scattering states we follow the evolution of a given initial configuration from a time @xmath75 forwards . the appropriate method to analyze this evolution is not the fourier transformation as in quantum field theory , but the laplace transformation . its inverse transformation is then calculated along a path which has all poles of the spectral function on its same side ; the path s position is shifted accordingly , compared with the fourier transformation . the classical solution of the equations of motion for field perturbations therefore explores in forward time direction all poles of a free oscillator ( or wave ) equation . in case of chaotic hamiltonian dynamics the solutions are both exponentially growing and damped giving rise to poles of the laplace transform with positive as well as negative real parts . making the formal connection between laplace and fourier transformation through a complex rotation of the frequency variable , @xmath76 , the inverse laplace transformation path runs * above * all poles in the complex @xmath77-plane . as a consequence in either case ( oscillatory or chaotic ) the integration path for the inverse laplace transformation includes * all * poles for positive time and * none * for negative time while the fourier transformation includes upper half plane poles for the advanced ( negative time ) and lower half plane poles for the retarded ( positive time ) propagator . = 6.0 cm the position of the poles obtained in a classical time forward calculation may have in general both positive and negative imaginary parts . therefore a better quantity for comparison is the spectral function which also considers poles in the whole complex @xmath77 plane . summarizing this argument the position of all poles of a spectral function can be obtained from the linearized classical equations of motion for field perturbations ( in the leading order of an @xmath40 expansion ) , but the retarded and advanced propagators used to solve scattering problems in perturbative field theory discard the unsuitable ones due to their very definition . a positive lyapunov exponent in hamiltonian ( energy conserving ) dynamical systems , on the other hand , always occurs together with its negative counterpart liouville s theorem ensures it . therefore studying positive exponential rates gives an information about the position of the poles of damped retarded and advanced propagators as well . the growth or damping rate , or the oscillation frequency of small amplitude fluctuations in a classical dynamical system is studied by linearizing the classical equations of motion . this procedure leads to a new differential operator whose spectrum gives the poles of the classical spectral function . odd parity under time reflection , real valuedness and normalization conditions then determine the relative weights of the pole terms . the differential operator belonging to the linear perturbation propagation equation ( 10 ) is identical with the second variation of the classical action , @xmath78 $ ] , taken at the background field configuration @xmath79 which is a solution of the classical equation of motion @xmath80=0 $ ] . here the prime means variation with respect to @xmath79 . considering the generating functional of connected green functions , the two point function is just the inverse of this differential operator , @xmath81 = \langle aa ' \rangle - \langle a \rangle \langle a ' \rangle = ( s''[a])^{-1},\ ] ] in the gaussian approximation to the small amplitude fluctuations . so the linear perturbation propagation in classical equations of motion gives information about the saddle point approximated generating functional . now aiming at the description of long wavelength plasmon damping we may neglect spatial derivatives and write the general form of the classical , linearized perturbation propagation equation ( 10 ) schematically as @xmath82 a(t ) = 0.\ ] ] the spectrum of this operator contains two poles on the real axis @xmath83 if @xmath84 is a positive constant . this case , familiar from zero - temperature perturbative field theory , describes small oscillations determining the real poles of the spectral function and the familiar retarded and advanced propagators . in classically chaotic , highly excited systems , however , it happens that @xmath84 is negative . this causes exponentially growing fluctuations a typical source of chaotic behavior . in order to gain a qualitative understanding about the ( classical ) spectral function of chaotic systems we consider @xmath84 as a gauss - distributed stochastic variable @xcite . it can have both negative and positive values , and its time variation is replaced by the ensemble variation due to the ergodic property of classically chaotic dynamical systems discussed in the previous section . in this limit the probability distribution of the frequency squares , @xmath85 , is determined by its two lowest moments , @xmath86 parametrized by two real parameters @xmath87 and @xmath88 . this parametrization reflects the fact that while @xmath89 can either be positive or negative , the width of its distribution is always positive . the stochastic average of the differential operator for the fluctuations has to be carried out on the quadratic level , because with the gaussian distribution we assumed white noise property of the stochastic quantity . we get @xmath90 this result exhibits the symmetric four pole structure typical for a spectral function describing classical plasma oscillations @xmath91 yielding the lorentz shape @xmath92 the relative signs of the pole terms follow from the definition of the spectral function as the difference between the advanced and retarded propagators and from its odd time parity @xmath93 the normalization factor @xmath94 ensures that @xmath95 so in each mode exactly one boson is counted by the spectral function @xmath96 @xcite . this particular , four pole spectral function describes a general solution of the stochastically averaged perturbation propagation equation which behaves like @xmath97 after some initial oscillations the exponential growth dominates the long time behavior of @xmath98 . it is exactly this , which has been seen in numerical calculations . the conclusion of this argument is that the lyapunov exponent of elementary field fluctuations averaged ergodically is equal to the classical gluon damping rate as expressed by the imaginary part of the pole positions in the spectral function @xmath99 . we note that in a recent publication @xcite a similar gaussian model for the chaotic instability in general hamiltonian flows has been investigated . our result presented above recovers the more general one of ref @xcite for vanishing expectation value of the noisy oscillator frequency square ( @xmath100 after substituting a characteristic timescale @xmath101 in the general formula ( 19 ) of ref . finally we argue again that the leading order gluon damping rate ( 3 ) obtained in hot perturbative qcd ( pqcd ) is classical , i.e. it retains its value in the classical limit @xmath102 this fact has been argued before in section ii . here we briefly reconstruct the argument and resolve some technical issues . this concludes our reasoning about the equality of the lyapunov exponent of chaotic classical lattice gauge theory and the gluon damping rate at rest in a hot plasma . the gluon damping rate in hot pqcd is obtained from the definition ( 6 ) dividing the imaginary part of the self energy by the thermal gluon mass @xmath103 the general one - loop form of the self energy contains an integral over hard momenta , a factor of @xmath104 and the phase space distribution of thermal gluons @xmath105 where the complicated algebraic expression @xmath106 depends only on scaled momentum variables . using now the long wavelength approximation the phase space distribution of thermal gluons is replaced with its classical counterpart , @xmath107 leading to @xmath108 scaling the integration variable with the debye mass which is of quantum origin containing the planck constant we see that the imaginary part of the 1-loop gluon self energy in a hot plasma is proportional to @xmath27 . it follows that the gluon damping rate obtained using `` classical '' thermal gluons does not depend on the debye mass and planck s constant , @xmath109 showing that the result ( 3 ) is essentially classical . finally , it is still to show whether non - pole contributions to the self energy in the field theoretical calculation do not interfere with the above arguments . the one - loop spectral function used there as an input contains a pole term picking up the zeroes @xmath110 of the inverse propagator corresponding to collective plasma modes to the lowest order and a cut term describing the effect of scattering on thermally excited spacelike modes : @xmath111 the cut coefficient @xmath112 is related to the real and imaginary parts of the self energy @xmath113 : @xmath114 the respective self energies for the transverse and longitudinal excitations to leading order in hot perturbative qcd are@xcite @xmath115 and @xmath116 with @xmath117 and @xmath118 . using these forms one obtains the following cut parts of the retarded fourier transform of the spectral function , @xmath119 for small @xmath120 @xmath121 and @xmath122 since the integrand is bounded , the cut contribution can not grow exponentially with time and hence does not contribute to the maximal lyapunov exponent ( [ 1.15a ] ) . in fact , the cut contribution vanishes in the long wavelength limit @xmath123 . this leaves us with the pole part , which remains finite in this limit . this concludes our argument establishing a connection between the classical lyapunov exponent and the gluon damping rate in hot perturbative qcd . we note that some elements of the argument are heuristic , in particular , the replacement of the long - time average of the growth rate of fluctuations around a specific field configuration by the thermal average . this reasoning assumes that the growth rate , or equivalently the plasmon damping rate , depends only on coarse - grained properties of the gauge field . we believe that this is so , because the one - loop calculation of the damping rate @xmath13 only involves soft loop momenta @xcite and hence does not depend on details of the short - distance fluctuations of the gauge field . because of the general nature of our argument , we conjecture that the complete spectrum of lyapunov exponents obtained in @xcite reflects the spectrum of damping rates @xmath32 of excitations in a thermal bath . if this were true , it would confirm our assumption that @xmath124 . since , at present , it is not known whether @xmath32 is a quantity with a classical limit for @xmath125 , the identification with the lyapunov spectrum remains a conjecture . we finally note that if the correspondence between ergodic and canonical averages holds up for other physical quantities , transport coefficients of nonabelian gauge fields at the classical scale @xmath126 , such as magnetic screening @xcite or color diffusion @xcite , could possibly also be calculated by real - time evolution of classical gauge fields on a lattice . * acknowledgements : * we thank u. heinz , s. g. matinyan , h. b. nielsen , g. k. savvidy , and m. thoma for illuminating discussions and the referee for pointing out a mistake in our original manuscript . this work was supported in part by the u.s . department of energy ( grant de - fg05 - 90er40592 ) and in part by the collaboration agreement between the norwegian research council ( nfr ) and the hungarian academy of science ( mta ) ( grant 422.92/001 ) . one of us ( b.m . ) thanks the physics department of tokyo metropolitan university , especially h. minakata , for their hospitality and support during his visit there . t.s.b . acknowledges the hospitality and the support of physics department of bergen university and of l. p. csernai during his visit there . the sign conventions in ref . 5 are such that a positive imaginary part of @xmath127 implies a negative imaginary part of the pole energy . in writing ( 6 ) we have already continued @xmath128 to minkowski space . note that the infrared limit of the yang - mills equation has been shown not to be completely ergodic , with some very small regions of stable trajectories remaining in phase space . see : p. dahlquist and g. russberg , _ phys . 65 * _ , 2837 ( 1990);r . marcinek , e. pollak , and j. zakrzewski , _ phys . lett . * b327 * _ , 67 ( 1994 ) . it is unknown , if such a phenomenon persists in the case of spatially dependent yang - mills fields .

we explain why the maximal positive lyapunov exponent of classical su(@xmath0 ) gauge theory coincides with ( twice ) the damping rate of a plasmon at rest in the leading order of thermal gauge theory . = cmbx10 scaled2 = cmbx10 scaled1 = cmcsc10

the study of topological states of matter has undergone a vertiginous growth since the theoretical prediction @xcite and the experimental observation @xcite of the quantum spin hall ( qsh ) effect . unlike the quantum hall ( qh ) effect , which is generated by an external magnetic field , or the quantum anomalous hall ( qah ) effect , which requires time - reversal symmetry ( trs ) to be spontaneously broken without applying an external magnetic field , the qsh state is characterized by trs and is generally driven by the intrinsic spin - orbit ( iso ) coupling.@xcite nevertheless , it has been recently shown that in absence of spin - flip terms the qsh effect survives even if the trs is broken . this state has been dubbed a _ weak qsh _ state@xcite ( or _ trs broken qsh _ state@xcite ) , where the weakness refers to the absence of protection by trs . indeed , the gap and the topological chern and spin chern numbers associated with the topological phase remain robust if the trs is broken by an exchange term @xcite or by an additional magnetic field,@xcite and there is a quantum phase transition to a topologically distinct or to a trivial phase only when the gap is closed.@xcite an interesting open question is what kind of competition could originate in systems where there is an externally applied magnetic field in addition to intrinsic magnetic moments , which on their own would lead to the qah effect . recently , the qah effect has been studied thoroughly for several theoretical models of two - dimensional topological insulators , including hgte quantum wells@xcite and thin films of bi@xmath0se@xmath1@xcite doped with transition metal elements such as mn , fe , or cr . in addition , graphene has been proposed as a candidate for the observation of this effect.@xcite the influence of a magnetic field on mn - doped hgte quantum wells has been partially investigated in ref . , with the aim of polarizing the mn magnetic moment to be eventually able to generate the qah effect upon shutting down the magnetic field . however , here we concentrate on aspects not considered so far . since for a certain range of parameters the zeeman coupling can have a similar effect on the landau - level ( ll ) spectrum as the iso interaction@xcite and can also lead to the ( trs broken ) qsh effect , it is interesting to explore the interplay between the usual zeeman term , which is linear in the applied magnetic field , and the non - linear effect arising from the exchange coupling with the magnetic moment of the mn atoms . in this paper , we show that within a model with a spin - conserving hamiltonian , the trs broken qsh phase occurs , and that a _ reentrant _ behavior is present for a certain range of parameters . a reentrant integer qh effect has been experimentally observed a few years ago in gaas quantum wells for filling factors between @xmath2 and @xmath3 , in the first ll.@xcite the phenomenon was later understood to occur due to a sequence of first - order quantum phase transitions between electron - solid ( wigner crystal or bubble phases ) and electron - liquid phases,@xcite and it was grounded on the strong electron - electron interactions that dominate the physics at non - integer filling factors . due to the self - similarity of the hall conductance curve,@xcite which displays a fractal behavior , a similar phenomenon was predicted to occur also for a second - generation of composite fermions.@xcite in that case , a series of reentrant plateaus would turn out to be quantized at the nearby fractional laughlin values.@xcite a second possibility is to observe reentrant integer qh effects solely due to the ll structure of the system . for instance , in si / sige heterostructures , reentrant behavior can be driven at the single - particle level by varying the in - plane magnetic field while keeping the perpendicular field fixed , as to modify the ratio between the cyclotron energy and the zeeman splitting.@xcite in these systems , the crossings of the lls for spin up and spin down are responsible for the reentrant behavior of the quantized hall conductivity . for hgte@xcite and inas / gasb@xcite quantum wells , the reentrance of the hall conductivity has been used as a practical method to prove the existence of such a ll crossing and consequently the inverted order of the bands . in this paper , we show that by applying a magnetic field perpendicular to a mn - doped hgte quantum well , the charge and spin hall conductivities may reenter concomitantly , i.e. , there can be a reentrance of the same topological phase , characterized by both its charge and spin topological invariants . this effect is caused by the nonmonotonic behavior of the ll energies due to the nonlinear dependence of the zeeman term on the externally applied magnetic field . a rich panoply of ll crossings , combined with the nonmonotonicity of the ll energies , provides us with regimes of parameters where this reentrant behavior could be experimentally accessed . quantum wells are ideal candidates for the observation of these effects , because they have a strong iso coupling and a large zeeman @xmath6-factor . we use the effective four - band model of ref . to compute the ll spectrum@xcite together with the relevant chern numbers in order to identify the qsh state and other qh - like states . band structure calculations are performed for different values of the quantum well thickness and of the mn doping fraction to set realistic parameters for this model . we then study the ll spectra including the charge and spin hall conductivities and determine the conditions for the observation of reentrant topological phases . the outline of this paper is as follows . in sect . [ sect_model ] , we define the effective model that we use to derive our results . in sect . [ sect_results ] , we compute the ll spectrum , explain the mechanisms that lead to the reentrant effects , and explore the parameter regimes for which they can be observed . we conclude by discussing in sect . [ sect_discussion ] the possibilities to resolve the reentrant effects in experiments . hgte and related materials have a zincblende lattice structure , so that the physics of the low - energy electronic bands is well described by the eight - band kane model.@xcite by using perturbation theory ( see the appendix for details ) , higher - energy bands are projected out in order to reduce this model to an effective four - band model.@xcite in this reduced model , the bands under consideration are referred to as @xmath7 , in this order . here , @xmath8 and @xmath9 refer to electron- and hole - like bands , respectively , and @xmath10 and @xmath11 distinguish the two members of each of the two kramers pairs @xmath12 and @xmath13 , hereafter referred to as spin components . the symmetry properties under the parity and time - reversal transformations dictate the quadratic - order hamiltonian @xmath14 , with@xcite @xmath15 where @xmath16 here , @xmath17 denotes the pauli matrices and @xmath18 , @xmath19 , @xmath20 , @xmath21 , and @xmath22 are parameters that depend on the material composition and on the thickness of the quantum well . in particular , the variations of these parameters induce the topological phase transition from a regime where the electronic bands are ordered normally to a regime where the order is inverted and where the qsh effect is present.@xcite the system is subjected to a perpendicular magnetic field @xmath23 , which we will express in terms of the dimensionless variable @xmath24 , which denotes the magnetic flux per unit cell measured in units of the flux quantum @xmath25 . with these definitions , @xmath24 relates to the magnetic field @xmath26 and to the magnetic length @xmath27 as @xmath28 . for hgte , with lattice constant @xmath29 , the flux value @xmath30 corresponds to a magnetic field of @xmath31 . in the remainder of this text , we set @xmath32 for convenience , and we set @xmath33 as the unit of length , so that @xmath18 , @xmath19 , @xmath20 , and @xmath22 all have the dimension of energy . the materials under consideration show a large zeeman effect , with land @xmath6 factors of the order of @xmath34.@xcite we therefore consider the zeeman term in the hamiltonian , with different @xmath6 factors for electrons and holes , @xmath35 where @xmath36 is a rescaled zeeman parameter , proportional to the bohr magneton @xmath37 and to the @xmath6 factor @xmath38 for electrons ( holes).@xcite in quantum wells of hgte doped with mn ( with molar fraction @xmath39 , i.e. , we consider hg@xmath4mn@xmath40te ) , the presence of mn has a significant effect on the magnetic properties of the material . it has been found that for low mn concentrations ( @xmath41 ) , the material behaves paramagnetically , so that its response to the magnetic field is nonlinear : in addition to the zeeman effect ( linear in the magnetic field strength ) , there is also a nonlinear contribution from the exchange interaction between mn ions and band states . the exchange interaction term is given by@xcite @xmath42 where @xmath43 and @xmath44 are the exchange energies for the electron and hole bands , respectively , @xmath45 is the brillouin function,@xcite @xmath46 is an exchange parameter , with @xmath47 , and @xmath48 is an effective temperature , where @xmath49.@xcite since the electron wave function @xmath12 is a linear combination of the wave functions of the @xmath50 and @xmath51 bands , the exchange energy @xmath43 is a linear combination of the exchange energies @xmath52 and @xmath53 associated with these bands , respectively . the only contribution to the hole wave function @xmath13 comes from the @xmath51 bands , so that @xmath44 is proportional to @xmath53.@xcite for a more detailed explanation , we refer the reader to the appendix . the energy splitting due to the exchange interactions can be considered as an effective zeeman splitting , by virtue of the similarity between hamiltonians and . here , one writes the zeeman energy as @xmath54 , where @xmath55 is the effective , field - dependent @xmath6 factor for the electron ( hole ) band . in the low - field limit ( @xmath56 ) , the effective @xmath6 factor is approximately constant , @xmath57 , derived by using a linear approximation to @xmath58 . in the high - field limit @xmath59 , the exchange interaction energy is almost constant ( @xmath60 ) as a function of the field , because @xmath61 for @xmath62 , and as a consequence , it depends also very weakly on the temperature . in order to derive the ll spectrum , we model the effect of the magnetic field @xmath23 by the peierls substitution : in the hamiltonian , the momentum @xmath63 is replaced by @xmath64 , where @xmath65 is the gauge potential , such that @xmath66 . the freedom of the gauge choice allows us to choose the symmetric gauge , @xmath67 . subsequently , we replace @xmath68 and @xmath69 by the ladder operators @xmath70 and @xmath71 , respectively . these operators raise and lower the ll index by @xmath72 , and their prefactors are chosen such that @xmath73=1$].@xcite in the model presented here , we neglect the coupling between the two spin bands which would arise in the presence of bulk - inversion asymmetry and rashba spin - orbit coupling . by virtue of this decoupling , the two spin bands can be treated separately . thus , the eigenvalues and eigenvectors of the hamiltonian are given by the solutions to the equation @xmath74 , with the appropriate values for @xmath75 . here , the eigenvalues @xmath76 give the energies of the lls , where @xmath77 is the ll index , @xmath78 refers to the spin components , and @xmath79 distinguishes between the two solutions that exist for each spin component.@xcite for the hamiltonian that includes the ( effective ) zeeman effect , the resulting ll energies are given by@xcite @xmath80(2\pi\phi ) \pm\sqrt{[m+(-\sigma d-2nb+\tfrac{1}{2}\sigma\tilde{g}^\mathrm{eff}_-)(2\pi\phi)]^2 + 2na^2(2\pi\phi ) } \quad(n\geq1),\nonumber\\ e_{+,0}&= m -(d+b-\tilde{g}^\mathrm{eff}_\mathrm{e})(2\pi\phi),\qquad\qquad e_{-,0 } = -m -(d - b+\tilde{g}^\mathrm{eff}_\mathrm{h})(2\pi\phi),\label{eqn_ll_ladder_zeeman}\end{aligned}\ ] ] where @xmath81 are the sum and the difference of the effective ( field - dependent ) @xmath6 factors given by eq . , including the effect of the mn doping . the ll spectra presented here are computed using eq . , where the relevant parameters have been derived numerically from band structure calculations based on the eight - band kane model,@xcite as explained in more detail in the appendix . these parameters have been computed for several values of the quantum well thickness @xmath82 and mn fraction @xmath39 . in particular , the dependence of @xmath18 on @xmath39 has dramatic consequences : increasing @xmath39 leads to an increase of @xmath18 , such that it drives the system from the inverted regime ( @xmath83 ) to the topologically trivial regime ( @xmath84).@xcite the charge ( spin ) hall conductivity in a specific bulk gap is defined as the sum of the charge ( spin ) chern numbers over all occupied lls below it . here , by virtue of the decoupling of the two spin components in the hamiltonian , the charge and spin chern numbers of each ll are equal to the sum and difference of the chern numbers @xmath85 associated with each of the two components . these chern numbers are well - defined due to the spin - conserving nature of the hamiltonian . thus , the charge and spin hall conductivity expressed in units of their respective quanta , @xmath86 and @xmath87 , are computed as @xmath88 where the summation is over the occupied lls . these values are robust , even in the absence of trs.@xcite in this model , each ll contributes a chern number of @xmath72 , so that the analysis is simplified to merely counting the lls . the presented values have been verified by analysis of the edge states in a ribbon geometry ; see e.g. refs . for the details of this alternative approach . the absence of coupling between the two spin states has an important consequence for the qsh phase . since the qsh state may be viewed as a combination of two independent qh effects for spin up and spin down , it persists even in the absence of time - reversal symmetry.@xcite additional symmetry - breaking terms , for instance due to bulk - inversion asymmetry and rashba spin - orbit coupling , would cause an opening of a small gap between the edge states , which allows for some backscattering in the presence of impurities.@xcite in fig . [ fig_ll](a ) , we have displayed the ll spectrum for an undoped ( @xmath89 ) quantum well with @xmath90 . this system is in the inverted regime , so that the spectrum displays a ( weak ) qsh gap , ( with @xmath91 ) , for magnetic fields up to @xmath92 , where the lls cross , at @xmath93 $ ] . in addition to the ( weak ) qsh gap , we observe several spin - filtered ( e.g. , @xmath94 ) , spin - imbalanced ( e.g. , @xmath95 ) and ordinary ( e.g. , @xmath96 ) qh gaps , and a trivial gap ( @xmath97 ) . the ( weak ) qsh gap is the only gap which exhibits a helical edge state structure ; all other nontrivial gaps are chiral . within this formalism , no other inverted gaps form besides the one at @xmath98 , because the involved lls do not cross anywhere else than at @xmath99 . in contrast , a tight - binding description of a honeycomb lattice in a perpendicular magnetic field does allow for other gaps with helical edge structures at higher flux and fermi energy values.@xcite the ll spectrum of fig . [ fig_ll](a ) shows two mechanisms that lead to reentrant behavior of the hall conductivity and spin hall conductivity . the first mechanism is illustrated for a fermi energy of @xmath100 ( lower dashed line ) , which lies just below the energy value at which the two lowest landau levels ( llls ) , i.e. , the lls with energies @xmath101 and @xmath102 , cross . holding the fermi energy fixed and increasing the magnetic field , we successively traverse the weak qsh gap with @xmath91 , the spin - filtered qh gap with @xmath103 , and the trivial gap , where @xmath97 . thus , the charge hall conductivity is @xmath104 for low and high magnetic fields , and @xmath105 for intermediate values , which characterizes a reentrance of a charge - insulating state ( see fig . [ fig_ll](c ) ) . at a fermi energy slightly above the crossing ( e.g. , @xmath106 , see fig . [ fig_ll](b ) ) , a similar sequence is observed , but with a different intermediate state ( @xmath107 ) . in both cases , the spin hall conductivity takes the values @xmath108 , @xmath72 , and @xmath104 , and does therefore not show reentrant behavior . clearly the reentrance of the hall conductivity is caused by the structure of the spectrum around the crossing of the llls . to observe the reentrance of the charge hall conductivity , it is essential that the derivatives @xmath109 of the two llls at the crossing differ in sign , which can happen only in the inverted regime . thus , experimental observation of this type of reentrance provides a proof that the hgte quantum well can indeed be described as an inverted dirac system.@xcite one may verify that if the signs of the derivatives would be equal , then the charge hall conductivity does not reenter . instead , we would observe reentrant spin hall conductivity . we note that crossings of the latter type are ubiquitous for higher lls ( @xmath110 ) , but they are difficult to observe due to the vicinity of other lls . however , later we will show that , under some circumstances , the crossings of the llls may also be of this type . in fig . [ fig_ll](d ) , we show the effect of doping ( @xmath111 ) on the ll spectrum . two effects are visible . first , the size of the ( weak ) qsh gap has decreased , consistent with the increase of @xmath18 . secondly , the energies @xmath112 of the two llls are no longer linear in the magnetic field . in fact , one of these llls shows a nonmonotonic dependence on the field . as can be observed in fig . [ fig_ll](d ) , this nonmonotonic lll attains its maximum for a flux value less than @xmath113 . thus , if the fermi energy is located between the energy of the crossing and that of the maximum ( e.g. , if @xmath114 , see the inset of fig . [ fig_ll](d ) and fig . [ fig_ll](e ) ) , the spin - filtered qh gap reenters , and the intermediate state is the ( weak ) qsh gap . thus , the system goes from a chiral , to a helical , and back to the ( same ) chiral phase again . this simultaneous reentrance of the charge and spin hall conductivity should be contrasted with the reentrant behavior around the lll crossing , where only one of them reenters , but not both . we remark that such a sequence is possible only if the intermediate phase is the ( weak ) qsh phase , and consequently only if the ll involved is one of the llls , since the higher lls are all monotonic . therefore , this behavior can not be observed in the undoped system , where the lll energies are linear . as can be observed in the inset of fig . [ fig_ll](d ) , the maximum of this lll has an energy close to that of the ll crossing . the sequence of charge and spin hall conductivities is therefore affected by both mechanisms . we shall call this phenomenon _ compound _ reentrant behavior . above the energy of the crossing , the aforementioned sequence ( spin - filtered qh , weak qsh , spin - filtered qh ) is followed by the trivial gap , so that we get an additional reentrance of the zero charge hall conductivity . just below the crossing energy ( e.g. , @xmath115 , see fig . [ fig_ll](f ) ) , the sequence of gaps is spin - filtered qh @xmath116 , weak qsh @xmath117 , spin - filtered qh @xmath118 , and trivial @xmath119 . in this situation , the two spin - filtered qh phases are _ different _ gaps , unlike the sequence above the crossing . these examples show that the rich compound reentrant behavior will appear if the crossing and the maximum of the llls are close to each other . in order to be able to observe the reentrant effects in experiments , we study the qualitative structure of the ll spectrum as a function of the well width @xmath82 , the doping fraction @xmath39 , and the temperature @xmath120 . more specifically , for a fixed choice of parameters , we analyze whether one of the lll is nonmonotonic , and whether the llls cross . furthermore , if the nonmonotonicity and crossing appear at the same time , we determine the position of the maximum / minimum and the crossing relative to each other . for simplicity , we restrict ourselves to the structure of the two llls . the bottom row of fig . [ fig_diag_reentrant_types ] displays five qualitatively different lll spectra , which distinguish the regimes as given by fig . [ fig_diag_reentrant_types](a)(c ) . these regimes are characterized as follows . for regimes ( i)(iii ) , the band gap has inverted order ( i.e. , @xmath83 ) , and therefore shows the qsh phase at zero magnetic field . in regime ( i ) , the llls are monotonic , so that the only mechanism that leads to reentrant effects is the crossing . in regimes ( ii ) and ( iii ) one lll is nonmonotonic , so that we have compound reentrant behavior . these two regimes are distinguished by the flux value of the maximum , which is smaller ( ii ) or greater ( iii ) than the flux value of the crossing . in the case ( iii ) , both llls are increasing at the crossing , so that we observe reentrant spin hall conductivity , as argued before . in regimes ( iv ) and ( v ) , the band gap is normally ordered ( i.e. , @xmath84 ) , so that we find a trivial phase at zero magnetic field . in this situation , the llls do not cross , and the only mechanism that can lead to reentrant behavior is the presence of a nonmonotonic lll , as is the case ( iv ) . for regime ( v ) , both lll are monotonic and do not cross , thus preventing any type of reentrant behavior . let us finally comment on the ability to resolve these reentrant effects , based on the range of the fermi energies for which they are present . in order to estimate the observability , we compare this energy range to the broadening of the lls , that will cause the change of conductivity across a ll to be smooth rather than step - like . here , we consider a gaussian broadening with width @xmath121,@xcite that incorporates both ll broadening due to disorder and the smooth variation of the fermionic filling function at finite temperatures . the broadening due to disorder has a field - dependent width @xmath122 , where @xmath123@xmath124 and @xmath125.@xcite the thermal broadening is approximated by a gaussian with width @xmath126 . in fig . [ fig_ll](b , c , e , f ) , we illustrate the smooth transitions of the conductivities due to the combined effects of both types of broadening . we consider the compound reentrant effects displayed in fig . [ fig_diag_reentrant_types](ii ) and ( iii ) to be well resolvable if the difference between the energies at the crossing and at the maximum exceeds twice the broadening width @xmath127 . indeed , in that case , the difference between the actual conductivity values and the quantized ones in absence of broadening effects is @xmath128 . however , the value of @xmath127 is not a hard limit : variations of the quantized hall conductivity , even if they are far from the quantized values , may already be considered as a signature for a reentrant effect , for example as demonstrated in fig . [ fig_ll](e ) . in the diagrams of fig . [ fig_diag_reentrant_types](a)(c ) , the different shadings indicate this distance compared to @xmath121 . in the brightest regions , the distance between the lls is larger than @xmath127 , sufficient for the reentrant effect to be observed . we find that for @xmath129 , situation ( ii ) is difficult to observe due to the small energy difference between the lls , whereas the observation of ( iii ) is easier close to the critical doping , above which the system goes to the trivial regime ( iv ) . for the observation of the compound reentrant effects , thicker wells are favorable because the energy range where the effects appear is larger . the simple reentrant effect due to nonmonotonicity , as in situation ( iv ) , is generally present in a large energy regime and therefore its observation is less affected by the ll broadening . transport experiments with hgte quantum wells have so far concentrated on the charge hall conductivity of the system . for example , the reentrance of the charge hall conductivity has been utilized to identify the possible regime of the qsh phase.@xcite observation of the simultaneous reentrant behavior of the charge and spin hall conductivity would also require the availability of a spin - sensitive detector , e.g. , a contact consisting of a tunneling barrier and a ferromagnet.@xcite however , this technique has the drawback that it only works at low fields , within the hysteresis range of the ferromagnet . another detection mechanism could be a local magneto - optical kerr effect ( moke ) experiment,@xcite although this measurement would be difficult due to the small bandgap of the semiconductor . the inverse spin hall effect may provide a way to measure the spin polarization of the edge states.@xcite nevertheless , the measurement of the charge hall conductivity at multiple fermi energies together with knowledge of the structure of the spectrum may provide indirect evidence for the existence of these reentrant effects . in conclusion , we have demonstrated that the nonmonotonic behavior of the lls in the presence of mn doping leads to reentrant topological phases , and that the vicinity of ll crossings leads to rich compound reentrant behavior . five different qualitative forms of the structure of the llls were shown to occur in the parameter space characterized by mn doping , well thickness , and temperature . furthermore , we have investigated the effects of ll broadening to estimate the ability to resolve the reentrant effects in experiments . we thank v. jurii and e. m. hankiewicz for useful discussions . this work was supported by the netherlands organisation for scientific research ( nwo ) ( w. b. and c. m. s. ) , the german research foundation dfg [ spp 1285 halbleiter spintronik , dfg - jst joint research program ( l. w. m. ) , and grant no . as327/2 ( e. g. n. ) , the alexander von humboldt foundation ( c. x. l. ) , and the eu erc - ag program ( l. w. m. ) . in this appendix , we illustrate the used numerical method and relate it to the perturbation theory which allows us to determine the parameters of a four - band effective model . in the kane model , the band structure of the material consists of eight bands.@xcite however , the two bands @xmath130 are separated by approximately @xmath131 from the other six bands and will be neglected here . the resulting six - band modified kane hamiltonian is written in the basis @xmath132 , @xmath133 , @xmath134 , @xmath135 , @xmath136 and @xmath137 , which we denote as @xmath138 , @xmath139 , @xmath140 , @xmath141 , @xmath142 and @xmath143 for short in the following . the hamiltonian can then be written as @xmath144 where @xmath145 is the six - band kane hamiltonian,@xcite @xmath146 is the linear zeeman term and @xmath147 is due to the exchange interaction between the mn ions and the band states in a magnetic field @xmath26 in the @xmath148 direction . the zeeman term reads @xmath149 for the ( decoupled ) conduction ( @xmath138 and @xmath139 ) and valence ( @xmath140 , @xmath141 , @xmath142 , and @xmath143 ) band parts of the hamiltonian . here @xmath150 is the angular momentum operator , @xmath151 is the bare zeeman @xmath6-factor of hgte , and @xmath152 is a phenomenological parameter.@xcite the exchange term , induced by the @xmath153-@xmath82 coupling between the mn @xmath82 level electrons and conduction or valence band electrons , has a similar form as the zeeman term , and reads @xmath154 where @xmath155 and @xmath156 are the coupling constants between the mn spin @xmath157 and the conduction band ( @xmath52 ) or the valence band ( @xmath53 ) , respectively , and @xmath39 is the mole fraction of mn@xmath158 ions . the polarization of the mn spin @xmath157 is assumed to be in the @xmath148 direction . we regard the mn spin as a classical spin and use the mean field value @xmath159 instead of @xmath157 , which yields @xmath160 where @xmath161 is the brillioun function as given by eq . , @xmath162 , @xmath47 , and @xmath49 for mn.@xcite the argument of @xmath161 in this equation is equal to @xmath163 , cf . eqs . and . now , we consider the above model in a periodic superlattice grown in the @xmath148-direction with well width @xmath82 and barrier width @xmath164 . in the limit of large @xmath164 , it becomes equivalent to a single quantum well . due to the periodic boundary condition along the @xmath148-direction , according to bloch s theorem , we can write the wave function as @xmath165 where @xmath166 and @xmath167 . the in - plane wave vector @xmath168 is a good quantum number for the system , and @xmath169 is the superlattice wave number in the @xmath148 direction , taken to be zero , because the quantum wells are effectively decoupled for large barrier thickness @xmath164 . @xmath170 is a multi - component periodic wave function @xmath171 of the @xmath172-band , which is expanded in terms of a plane - wave basis as @xmath173 where @xmath174 denotes the component @xmath175 of the wave function , and the expansion coefficients @xmath176 are functions of @xmath168 . the eigenequation for these states is given by @xmath177 where @xmath178 depends on @xmath168 . with the expansion , we find @xmath179 a truncation method is applied and a finite number of basis vectors ( @xmath180 ) is used to solve this eigenvalue problem to obtain the coefficients @xmath176 . given the fact that we are only interested in the low - energy physics , taking @xmath181 yields a solution that is sufficiently accurate . ( color online ) comparison of the energy dispersion calculated from the full hamiltonian ( blue solid curves ) and from the effective model ( red dashed curves ) for @xmath90 and @xmath182 . the two results exhibit a good overlap in the low - energy regime , which demonstrates the reliability of the effective model.,width=325 ] next , we relate the perturbation theory to the previous numerical method . the hamiltonian is divided into @xmath183 where @xmath184 is treated as the zero - order hamiltonian and @xmath185 as the perturbation . the wave function at the @xmath121 point ( @xmath186 ) can be obtained from the numerical calculation , which is denoted as @xmath187 with @xmath188 . only the subbands @xmath189 , which are denoted as @xmath190 for short , are concerned in this calculation . using symmetry arguments , we obtain @xmath191,@xmath192,@xmath193 . under two - dimensional spatial reflection , @xmath194 have even parity and @xmath195 have odd parity . furthermore , in order to take into account the contribution of the other subbands in second - order perturbation theory , additional states @xmath196 , @xmath197 , @xmath198 , and @xmath199 ( the second electron , light hole , and second and third heavy hole bands , respectively ) are also solved numerically and can be written in a similar way.@xcite dd|ddddddddd & & & & & & & & & & + 5.5 & 0.00 & -16.9 & 8.8 & 0.60 & -1.15 & -0.73 & 15.8 & 1.22 & 0.62 & 0.37 + 5.5 & 0.01 & -5.8 & 20.0 & 0.62 & -1.05 & -0.63 & 14.4 & 1.29 & 0.64 & 0.35 + 5.5 & 0.02 & 5.6 & 31.5 & 0.64 & -0.96 & -0.55 & 13.2 & 1.36 & 0.66 & 0.33 + 5.5 & 0.03 & 17.4 & 43.3 & 0.66 & -0.89 & -0.48 & 12.2 & 1.42 & 0.68 & 0.31 + 6.5 & 0.00 & -24.4 & -4.9 & 0.58 & -1.45 & -1.04 & 20.0 & 1.22 & 0.58 & 0.41 + 6.5 & 0.01 & -13.9 & 5.7 & 0.60 & -1.30 & -0.88 & 18.0 & 1.28 & 0.61 & 0.38 + 6.5 & 0.02 & -3.0 & 16.6 & 0.62 & -1.17 & -0.75 & 16.3 & 1.35 & 0.63 & 0.36 + 6.5 & 0.03 & 8.4 & 28.0 & 0.65 & -1.06 & -0.65 & 14.9 & 1.42 & 0.66 & 0.34 + 7.5 & 0.00 & -29.9 & -14.6 & 0.55 & -1.87 & -1.45 & 24.3 & 1.21 & 0.55 & 0.44 + 7.5 & 0.01 & -19.9 & -4.6 & 0.58 & -1.62 & -1.20 & 21.8 & 1.28 & 0.58 & 0.41 + 7.5 & 0.02 & -9.5 & 5.8 & 0.61 & -1.42 & -1.00 & 19.5 & 1.34 & 0.61 & 0.39 + 7.5 & 0.03 & 1.4 & 16.8 & 0.63 & -1.26 & -0.85 & 17.6 & 1.41 & 0.64 & 0.36 with the obtained zero - order wave function , we apply the second - order perturbation formalism@xcite @xmath200 to obtain the effective model given by eqs . . here @xmath201 are the states chosen from @xmath202 , @xmath203 , @xmath204 , and @xmath205 while @xmath206 is one of the intermediate states @xmath196 , @xmath197 , @xmath198 , and @xmath199 . with this approach , we relate the parameters of the effective model to the parameters of the six - band modified kane model . we find that for the effective mass parameters @xmath22 and @xmath20 and for the effective @xmath6 factor @xmath207 , we need to take into account the second - order perturbation , while for the other parameters the first - order term is accurate enough for our purpose . as the derivation is straightforward and the expressions for the parameters are quite lengthy , we do not write them explicitly here . as an example , the exchange parameters @xmath43 and @xmath44 are given by @xmath208 where @xmath209 and @xmath210 , and @xmath163 is the argument of @xmath161 in eqs . and . in table [ tbl_parameters ] , we show the numerical values of these parameters for several different well thicknesses and different mn doping . in fig . [ fig_dispcom ] , the energy dispersion calculated from the effective model using the parameters in table [ tbl_parameters ] is shown to fit well with that calculated from the full kane model at small @xmath211 . this result justifies the use of the effective model to discuss the low - energy physics , in particular in the energy range where the reentrant behavior occurs . in this paper , we have restricted ourselves to wells with a thickness @xmath212 , because above this value , the @xmath213 band lies between the @xmath214 and @xmath215 bands , and in that case the four - band model is no longer accurate , especially in the energy regime of the valence band . nevertheless , the mechanisms for appearance of the reentrant effects may still be present for thicker wells . 40ifxundefined [ 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/physrevlett.95.146802 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.106802 [ * * , ( ) ] link:\doibase 10.1126/science.1133734 [ * * , ( ) ] link:\doibase 10.1126/science.1148047 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.82.3045 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/97/23003 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.066602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.096601 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.125327 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.146802 [ * * , ( ) ] link:\doibase 10.1126/science.1187485 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.165101 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.134408 [ * * , ( ) ] link:\doibase 10.1103/physrevb.82.161414 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.155447 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.165453 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.085427 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.83.1193 [ * * , ( ) ] @noop ( ) , link:\doibase 10.1103/physrevlett.88.076801 [ * * , ( ) ] link:\doibase 10.1103/physrevb.68.241302 [ * * , ( ) ] link:\doibase 10.1103/physrevb.69.115327 [ * * , ( ) ] link:\doibase 10.1038/422391a [ * * , ( ) ] link:\doibase 10.1209/epl / i2004 - 10215 - 5 [ * * , ( ) ] link:\doibase 10.1103/physrevb.69.155324 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.93.216802 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.86.866 [ * * , ( ) ] link:\doibase 10.1143/jpsj.77.031007 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.85.2364 [ * * , ( ) ] link:\doibase 10.1007/s11467 - 011 - 0204 - 1 [ ( ) ] , link:\doibase 10.1016/0022 - 3697(57)90013 - 6 [ * * , ( ) ] link:\doibase 10.1103/physrevb.72.035321 [ * * , ( ) ] link:\doibase 10.1063/1.341700 [ * * , ( ) ] link:\doibase 10.1209/epl / i2003 - 10094 - 2 [ * * , ( ) ] link:\doibase 10.1063/1.1397275 [ * * , ( ) ] link:\doibase 10.1038/nphys1914 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.166803 [ * * , ( ) ] link:\doibase 10.1038/nphys543 [ * * , ( ) ] link:\doibase 10.1126/science.1105514 [ * * , ( ) ] link:\doibase 10.1038/nphys1655 [ * * , ( ) ] @noop ( ) , link:\doibase 10.1088/1367 - 2630/12/6/065012 [ * * , ( ) ]

quantum wells of hgte doped with mn display the quantum anomalous hall effect due to the magnetic moments of the mn ions . in the presence of a magnetic field , these magnetic moments induce an effective nonlinear zeeman effect , causing a nonmonotonic bending of the landau levels . as a consequence , the quantized ( spin ) hall conductivity exhibits a reentrant behavior as one increases the magnetic field . here , we will discuss the appearance of different types of reentrant behavior as a function of mn concentration , well thickness , and temperature , based on the qualitative form of the landau - level spectrum in an effective four - band model .

networks are useful constructs to schematize the organization of interactions in social and biological systems . networks are particularly valuable for characterizing _ interdependent _ interactions , where the interaction between components a and b influences the interaction between components b and c , and so on . for most such integrated systems , it is a flow of some entity passengers traveling among airports , money transferred among banks , gossip exchanged among friends , signals transmitted in the brain that connects a system s components and generates their interdependence . network structures constrain these flows . therefore , understanding the behavior of integrated systems at the macro - level is not possible without comprehending the network structure with respect to the flow , the _ dynamics on _ the network . one major drawback of networks is that , for visualization purposes , they can only depict small systems . real - world networks are often so large that they must be represented by coarse - grained descriptions . deriving appropriate coarse - grain descriptions is the basic objective of community detection @xcite . but before we decompose the nodes and links into modules that represent the network , we must first decide what we mean by appropriate . " that is , we must decide which aspects of the system should be highlighted in our coarse - graining . if we are concerned with the process that _ generated _ the network in the first place , we should use methods based on some underlying stochastic model of network formation . to study the formation process , we can , for example , use modularity @xcite , mixture models at two @xcite or more @xcite levels , bayesian inference @xcite , or our cluster - based compression approach @xcite to resolve community structure in undirected and unweighted networks . if instead we want to infer system behavior from network structure , we should focus on how the structure of the extant network constrains the dynamics that can occur on that network . to capture how local interactions induce a system - wide flow that connects the system , we need to simplify and highlight the underlying network structure with respect to how the links drive this flow across the network . for example , both markov processes on networks and spectral methods can capture this notion @xcite . in this paper , we present a detailed description of the flow - based and information - theoretic method known as the map equation @xcite . for a given network partition , the map equation specifies the theoretical limit of how concisely we can describe the trajectory of a random walker on the network . with the random walker as a proxy for real flow , minimizing the map equation over all possible network partitions reveals important aspects of network structure with respect to the dynamics on the network . to illustrate and further explain how the map equation operates , we compare its action with the topological method modularity maximization @xcite . because the two methods can yield different results for some network structures , it is illuminating to understand when and why they differ . there is a duality between the problem of compressing a data set , and the problem of detecting and extracting significant patterns or structures within those data . this general duality is explored in the branch of statistics known as mdl , or minimum description length statistics @xcite . we can apply these principles to the problem at hand : finding the structures within a network that are significant with respect to how information or resources flow through that network . to exploit the inference - compression duality for dynamics on networks , we envision a communication process in which a sender wants to communicate to a receiver about movement on a network . that is , we represent the data that we are interested in the trace of the flow on the network with a compressed message . this takes us to the heart of information theory , and we can employ shannon s source coding theorems to find the limits on how far we can compress the data @xcite . for some applications , we may have data on the actual trajectories of goods , funds , information , or services as they travel through the network , and we could work with the trajectories directly . more often , however , we will only have a characterization of the network structure along which these objects can move , in which case we can do no better than to approximate likely trajectories as random walks guided by the directed and weighted links of the network . this is the approach that we take with the map equation . in order to effectively and concisely describe where on the network a random walker is , an effective encoding of position will necessarily exploit the regularities in patterns of movement on that network . if we can find an optimal code for describing places traced by a path on a network , we have also solved the dual problem of finding the important structural features of that network . therefore , we look for a way to assign codewords to nodes that is efficient with respect to the dynamics on the network . a straightforward method of assigning codewords to nodes is to use a huffman code @xcite . huffman codes are optimally efficient for symbol - by - symbol encoding and save space by assigning short codewords to common events or objects , and long codewords to rare ones , just as morse code uses short codes for common letters and longer codes for rare ones . figure [ fig1](b ) shows a prefix - free huffman coding for a sample network . it corresponds to a lookup table for coding and decoding nodes on the network , a _ codebook _ that connects nodes with codewords . in this codebook , each huffman codeword specifies a particular node , and the codeword lengths are derived from the ergodic node visit frequencies of a random walk ( the average node visit frequencies of an infinite - length random walk ) . because the code is prefix - free , that is , no codeword is a prefix of any other codeword , codewords can be sent concatenated without punctuation and still be unambiguously decoded by the receiver . with the huffman code pictured in fig . [ fig1](b ) , we are able to describe the nodes traced by the specific 71-step walk in 314 bits . if we instead had chosen a uniform code , in which all codewords are of equal length , each codeword would be @xmath0 bits long ( logarithm taken in base 2 ) , and @xmath1 bits would have been required to describe the walk . this huffman code is optimal for sending a one - time transmission describing the location of a random walker at one particular instant in time . moreover , it is optimal for describing a list of locations of the random walker at arbitrary ( and sufficiently distant ) times . however , if we wish to list the locations visited by our random walker in a sequence of successive steps , we can do better . sequences of successive steps are of critical importance to us ; after all , this is flow . many real - world networks are structured into a set of regions such that once the random walker enters a region , it tends to stay there for a long time , and movements between regions are relatively rare . as we design a code to enumerate a succession of locations visited , we can take advantage of this regional structure . we can take a region with a long persistence time and give it its own separate codebook . so long as we are content to reuse codewords in other regional codebooks , the codewords used to name the locations in any single region will be shorter than those in the global huffman code example above , because there are fewer locations to be specified . we call these regions `` modules '' and their codebooks `` module codebooks . '' however , with multiple module codebooks , each of which re - uses a similar set of codewords , the sender must also specify which module codebook should be used . that is , every time a path enters a new module , both sender and receiver must simultaneously switch to the correct module codebook or the message will be nonsense . this is implemented by using one extra codebook , an index codebook , with codewords that specify which of the module codebooks is to be used . the coding procedure is then as follows . the index codebook specifies a module codebook , and the module codebook specifies a succession of nodes within that module . when the random walker leaves the module , we need to return to the index codebook . to indicate this , instead of sending another node name from the module codebook , we send the `` xit command '' from the module codebook . the codeword lengths in the index codebook are derived from the relative rates at which a random walker enters each module , while the codeword lengths for each module codebook are derived from the relative rates at which a random walker visits each node in the module or exits the module . here emerges the duality between coding a data stream and finding regularities in the structure that generates that stream . using multiple codebooks , we transform the problem of minimizing the description length of places traced by a path into the problem of how we should best partition the network with respect to flow . how many modules should we use , and which nodes should be assigned to which modules to minimize the map equation ? figure [ fig1](c ) illustrates a two - level description that capitalizes on structures with long persistence time and encodes the walk in panel ( a ) more efficiently than the one - level description in panel ( b ) . we have implemented a dynamic visualization and made it available for anyone to explore the inference - compression duality and the mechanics of the map equation ( http://www.tp.umu.se/~rosvall/livemod/mapequation/ ) . figure [ codebooks ] visualizes the use of one or multiple codebooks for the network in fig . the sparklines show how the description length associated with between - module movements increases with the number of modules and more frequent use of the index codebook . contrarily , the description length associated with within - module movements decreases with the number of modules and with the use of smaller module codebooks . the sum of the two , the full description length , takes a minimum at four modules . we use stacked boxes to illustrate the rates at which a random walker visits nodes and enters and exits modules . the codewords to the right of the boxes are derived from the within - module relative rates and within - index relative rates , respectively . both relative rates and codewords change from the one - codebook solution with all nodes in one module , to the optimal solution , with an index codebook and four module codebooks with nodes assigned to four modules ( see online dynamic visualization @xcite ) . we have described the huffman coding process in detail in order to make it clear how the coding structure works . but of course the aim of community detection is not to encode a particular path through a network . in community detection , we simply want to find the modular structure of the network with respect to flow and our approach is to exploit the inference - compression duality to do so . in fact , we do not even need to devise an optimal code for a given partition to estimate how efficient that optimal code would be . this is the whole point of the map equation . it tells us how efficient the optimal code would be for any given partition , without actually devising that code . that is , it tells us the theoretical limit of how concisely we can specify a network path using a given partition structure . to find an optimal partition of the network , it is sufficient to calculate this theoretical limit for different partitions of the network and pick the one that gives the shortest description length . for a module partition @xmath2 of @xmath3 nodes @xmath4 into @xmath5 modules @xmath6 , we define this lower bound on code length to be @xmath7 . to calculate @xmath8 for an arbitrary partition , we first invoke shannon s source coding theorem @xcite , which implies that when you use @xmath3 codewords to describe the @xmath3 states of a random variable @xmath9 that occur with frequencies @xmath10 , the average length of a codeword can be no less than the entropy of the random variable @xmath9 itself : @xmath11 ( we measure code lengths in bits and take the logarithm in base 2 ) . this provides us with a lower bound on the average length of codewords for each codebook . to calculate the average length of the code describing a step of the random walk , we need only to weight the average length of codewords from the index codebook and the module codebooks by their rates of use . this is the map equation : @xmath12 here @xmath13 is the frequency - weighted average length of codewords in the index codebook and @xmath14 is frequency - weighted average length of codewords in module codebook @xmath15 . further , the entropy terms are weighted by the rate at which the codebooks are used . with @xmath16 for the probability to exit module @xmath15 , the index codebook is used at a rate @xmath17 , the probability that the random walker switches modules on any given step . with @xmath18 for the probability to visit node @xmath19 , module codebook @xmath15 is used at a rate @xmath20 , the fraction of time the random walk spends in module @xmath15 plus the probability that it exits the module and the exit message is used . now it is straightforward to express the entropies in @xmath21 and @xmath22 . for the index codebook , the entropy is @xmath23 and for module codebook @xmath15 the entropy is @xmath24 by combining eqs . [ map_master ] and [ map_module ] and simplifying , we can write the map equation as : @xmath25 in this expanded form of the map equation , we note that the term @xmath26 is independent of partitioning , and elsewhere in the expression @xmath22 appears only when summed over all nodes in a module . consequently , when we optimize the network partition , it is sufficient to keep track of changes in @xmath27 , the rate at which a random walker enters and exits each module , and @xmath28 , the fraction of time a random walker spends in each module . they can easily be derived for any partition of the network , and updating them is a straightforward and fast operation . any numerical search algorithm developed to find a network partition that optimizes an objective function can be modified to minimize the map equation . [ cols= " < , < , < , < " , ] for undirected networks , the node visit frequency of node @xmath19 simply corresponds to the relative weight @xmath29 of the links connected to the node . the relative weight is the total weight of the links connected to the node divided by twice the total weight of all links in the network , which corresponds to the total weight of all link - ends . with @xmath29 for the relative weight of node @xmath19 , @xmath30 for the relative weight of module @xmath15 , @xmath31 for the relative weight of links exiting module @xmath15 , and @xmath32 for the total relative weight of links between modules , the map equation takes the form @xmath33 for directed weighted networks , we use the power iteration method to calculate the steady state visit frequency for each node . to guarantee a unique steady state distribution for directed networks , we introduce a small teleportation probability @xmath34 in the random walk that links every node to every other node with positive probability and thereby converts the random walker into a _ random surfer_. the movement of the random surfer can now be described by an irreducible and aperiodic markov chain that has a unique steady state by the perron - frobineous theorem . as in google s pagerank algorithm @xcite , we use @xmath35 . the results are relatively robust to this choice , but as @xmath36 , the stationary frequencies may poorly reflect the important nodes in the network as the random walker can get trapped in small clusters that do not point back into the bulk of the network @xcite . the surfer moves as follows : at each time step , with probability @xmath37 , the random surfer follows one of the outgoing links from the node @xmath19 that it currently occupies to the neighbor node @xmath38 with probability proportional to the weights of the outgoing links @xmath39 from @xmath19 to @xmath38 . it is therefore convenient to set @xmath40 . with the remaining probability @xmath34 , or with probability @xmath41 if the node does not have any outlinks , the random surfer `` teleports '' with uniform probability to a random node anywhere in the system . but rather than averaging over a single long random walk to generate the ergodic node visit frequencies , we apply the power iteration method to the probability distribution of the random surfer over the nodes of the network . we start with a probability distribution of @xmath42 for the random surfer to be at each node @xmath19 and update the probability distribution iteratively . at each iteration , we distribute a fraction @xmath37 of the probability flow of the random surfer at each node @xmath19 to the neighbors @xmath38 proportional to the weights of the links @xmath39 and distribute the remaining probability flow uniformly to all nodes in the network . we iterate until the sum of the absolute differences between successive estimates of @xmath22 is less than @xmath43 and the probability distribution has converged . given the ergodic node visit frequencies @xmath22 for @xmath44 and an initial partitioning of the network , it is easy to calculate the ergodic module visit frequencies @xmath28 for module @xmath15 . the exit probability for module @xmath15 , with teleportation taken into account , is then @xmath45 where @xmath46 is the number of nodes in module @xmath15 . this equation follows since every node teleports a fraction @xmath47 and guides a fraction @xmath48 of its weight @xmath22 to nodes outside of its module @xmath15 . if the nodes represent objects that are inherently different it can be desirable to nonuniformly teleport to nodes in the network . for example , in journal - to - journal citation networks , journals should receive teleporting random surfers proportional to the number of articles they contain , and , in air traffic networks , airports should receive teleporting random surfers proportional to the number of flights they handle . this nonuniform teleportation nicely corrects for the disproportionate amount of random surfers that small journals or small airports receive if all nodes are teleported to with equal probability . in practice , nonuniform teleportation can be achieved by assigning to each node @xmath19 a normalized teleportation weight @xmath49 such that @xmath50 . with teleportation flow distributed nonuniformly , the numeric values of the ergodic node visit probabilities @xmath22 will change slightly and the exit probability for module @xmath15 becomes @xmath51 this equation follows since every node now teleports a fraction @xmath52 of its weight @xmath22 to nodes outside of its module @xmath15 . conceptually , detecting communities by mapping flow is a very different approach from inferring module assignments for underlying network models . whereas the former approach focuses on the interdependence of links and the dynamics on the network once it has been formed , the latter one focuses on pairwise interactions and the formation process itself . because the map equation and modularity take these two disjoint approaches , it is interesting to see how they differ in practice . to highlight one important difference , we compare how the map equation and the generalized modularity , which makes use of information about the weight and direction of links @xcite , operate on networks with and without flow . for weighted and directed networks , the modularity for a given partitioning of the network into @xmath5 modules is the sum of the total weight of all links in each module minus the expected weight @xmath53 here @xmath54 is the total weight of links starting and ending in module @xmath15 , @xmath55 and @xmath56 the total in- and out - weight of links in module @xmath15 , and @xmath57 the total weight of all links in the network . to estimate the community structure in a network , eq . [ modularity ] is maximized over all possible assignments of nodes into any number @xmath5 of modules . figure [ compare ] shows two different networks , each partitioned in two different ways . both networks are generated from the same underlying network model in the modularity sense : 20 directed links connect 16 nodes in four modules , with equal total in- and out - weight at each module . the only difference is that we switch the direction of two links in each module . because the weights @xmath57 , @xmath54 , @xmath55 , and @xmath56 are all the same for the four - module partition of the two different networks in fig . [ compare](a ) and ( c ) , the modularity takes the same value . that is , from the perspective of modularity , the two different networks and corresponding partitions are identical . however , from a flow - based perspective , the two networks are completely different . the directed links shown in the network in panel ( a ) and panel ( b ) induce a structured pattern of flow with long persistence times in , and limited flow between , the four modules highlighted in panel ( a ) . the map equation picks up on these structural regularities , and thus the description length is shorter for the four - module network partition in panel ( a ) than for the unpartitioned network in panel ( b ) . by contrast , for the network shown in panels ( c ) and ( d ) , there is no pattern of extended flow at all . every node is either a source or a sink , and no movement along the links on the network can exceed more than one step in length . as a result , random teleportation will dominate and any partition into multiple modules will lead to a high flow between the modules . for networks with links that do not induce a pattern of flow , the map equation will always be minimized by one single module . the map equation captures small modules with long persistence times , and modularity captures small modules with more than the expected number of link - ends , incoming or outgoing . this example , and the example with directed and weighted networks in ref . @xcite , reveal the effective difference between them . though modularity can be interpreted as a one - step measure of movement on a network @xcite , this example demonstrates that one - step walks can not capture flow . any greedy ( fast but inaccurate ) or monte carlo - based ( accurate but slow ) approach can be used to minimize the map equation . to provide a good balance between the two extremes , we have developed a fast stochastic and recursive search algorithm , implemented it in c++ , and made it available online both for directed and undirected weighted networks @xcite . as a reference , the new algorithm is as fast as the previous high - speed algorithms ( the greedy search presented in the supporting appendix of ref . @xcite ) , which were based on the method introduced in ref . @xcite and refined in ref . @xcite . at the same time , it is also more accurate than our previous high - accuracy algorithm ( a simulated annealing approach ) presented in the same supporting appendix . the core of the algorithm follows closely the method presented in ref . @xcite : neighboring nodes are joined into modules , which subsequently are joined into supermodules and so on . first , each node is assigned to its own module . then , in random sequential order , each node is moved to the neighboring module that results in the largest decrease of the map equation . if no move results in a decrease of the map equation , the node stays in its original module . this procedure is repeated , each time in a new random sequential order , until no move generates a decrease of the map equation . now the network is rebuilt , with the modules of the last level forming the nodes at this level . and exactly as at the previous level , the nodes are joined into modules . this hierarchical rebuilding of the network is repeated until the map equation can not be reduced further . except for the random sequence order , this is the algorithm described in ref . @xcite . with this algorithm , a fairly good clustering of the network can be found in a very short time . let us call this the core algorithm and see how it can be improved . the nodes assigned to the same module are forced to move jointly when the network is rebuilt . as a result , what was an optimal move early in the algorithm might have the opposite effect later in the algorithm . because two or more modules that merge together and form one single module when the network is rebuilt can never be separated again in this algorithm , the accuracy can be improved by breaking the modules of the final state of the core algorithm in either of the two following ways : * _ submodule movements . _ first , each cluster is treated as a network on its own and the main algorithm is applied to this network . this procedure generates one or more submodules for each module . then all submodules are moved back to their respective modules of the previous step . at this stage , with the same partition as in the previous step but with each submodule being freely movable between the modules , the main algorithm is re - applied . * _ single - node movements . _ first , each node is re - assigned to be the sole member of its own module , in order to allow for single - node movements . then all nodes are moved back to their respective modules of the previous step . at this stage , with the same partition as in the previous step but with each single node being freely movable between the modules , the main algorithm is re - applied . in practice , we repeat the two extensions to the core algorithm in sequence and as long as the clustering is improved . moreover , we apply the submodule movements recursively . that is , to find the submodules to be moved , the algorithm first splits the submodules into subsubmodules , subsubsubmodules , and so on until no further splits are possible . finally , because the algorithm is stochastic and fast , we can restart the algorithm from scratch every time the clustering can not be improved further and the algorithm stops . the implementation is straightforward and , by repeating the search more than once , 100 times or more if possible , the final partition is less likely to correspond to a local minimum . for each iteration , we record the clustering if the description length is shorter than the previously shortest description length . in practice , for networks with on the order of 10,000 nodes and 1,000,000 directed and weighted links , each iteration takes about 5 seconds on a modern pc . in this paper and associated interactive visualization @xcite , we have detailed the mechanics of the map equation for community detection in networks @xcite . our aim has been to differentiate flow - based methods such as spectral methods and the map equation , which focus on system behavior once the network has been formed , from methods based on underlying stochastic models such as mixture models and modularity methods , which focus on the network formation process . by comparing how the map equation and modularity operate on networks with and without flow , we conclude that the two approaches are not only conceptually different , they also highlight different aspects of network structure . depending on the sorts of questions that one is asking , one approach may be preferable to the other . for example , to analyze how networks are formed and to simplify networks for which links do not represent flows but rather pairwise relationships , modularity @xcite or other topological methods @xcite may be preferred . but if instead one is interested in the dynamics on the network , in how local interactions induce a system - wide flow , in the interdependence across the network , and in how network structure relates to system behavior , then flow - based approaches such as the map equation are preferable .

many real - world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent . as the tools for doing these simplifications proliferate within the network literature , researchers would benefit from some guidelines about which of the so - called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking . here we show that different methods highlight different aspects of a network s structure and that the the sort of information that we seek to extract about the system must guide us in our decision . for example , many community detection algorithms , including the popular modularity maximization approach , infer module assignments from an underlying model of the network formation process . however , we are not always as interested in how a system s network structure was formed , as we are in how a network s extant structure influences the system s behavior . to see how structure influences current behavior , we will recognize that links in a network induce movement across the network and result in system - wide interdependence . in doing so , we explicitly acknowledge that most networks carry flow . to highlight and simplify the network structure with respect to this flow , we use the map equation . we present an intuitive derivation of this flow - based and information - theoretic method and provide an interactive on - line application that anyone can use to explore the mechanics of the map equation . the differences between the map equation and the modularity maximization approach are not merely conceptual . because the map equation attends to patterns of flow on the network and the modularity maximization approach does not , the two methods can yield dramatically different results for some network structures . to illustrate this and build our understanding of each method , we partition several sample networks . we also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation .

in the main text , we have mostly discussed and presented results for an initial filling of @xmath28 with @xmath18 . here we present additional material for @xmath105 ( and , for comparison , the @xmath28 data from the main text ) to elucidate the dependence of the quantum distillation on @xmath29 . such data are presented for @xmath106 in figs . [ fig : u8]-[fig : u100 ] , respectively . we find that for all @xmath70 and all values of @xmath53 considered here , quantum distillation takes place . we distinguish between the quantum distillation in a _ weak _ sense , namely for some sites @xmath107 . such behavior is seen in the case of @xmath60 ( fig . [ fig : u8 ] ) . the more pronounced effect of the formation of approximate fock states , _ i.e. _ , @xmath108 , is observed for large @xmath29 and large @xmath53 ( see figs . [ fig : u20][fig : u100 ] ) . [ [ expansion - from - a - box - trap - dependence - on - the - particle - number - at - a - fixed - initial - filling - n_mathrminit ] ] expansion from a box trap : dependence on the particle number at a fixed initial filling @xmath29 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ we have also studied the dependence on the number of particles @xmath3 at a fixed @xmath29 . the respective results are presented in fig . [ fig : ndep ] for @xmath67 and @xmath28 . qualitatively , the spatial extension and stability of the approximate fock states grows with @xmath3 . the figure further shows [ see fig . [ fig : ndep](d ) ] that by plotting the data vs @xmath109 , @xmath33 , calculated for different @xmath3 , all results collapse onto essentially the same curve . this illustrates that the time for the formation of the quasi - fock state scales linearly with the number of particles ( or the number of doublons in the initial state , respectively ) . most importantly , the _ qualitative _ behavior of the quantum distillation process is independent of @xmath3 at a fixed @xmath29 the quantum distillation effect can be nicely illustrated by plotting the radius @xmath68 of the double occupancies : @xmath111 where @xmath112 is the total number of double occupancies . this radius is shown in fig.[fig : rd ] for several values of @xmath53 and an initial density of @xmath57 . in this quantity , the effect is visible for @xmath113 . [ [ expansion - from - the - harmonic - trap - reduction - of - the - entanglement - entropy ] ] expansion from the harmonic trap : reduction of the entanglement entropy ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ here we provide results for the reduction of the entanglement entropy for the expansion from a harmonic trap and the parameters of fig . 4 of the main text ( @xmath99 , @xmath101 , @xmath100 ) , yet for two values of the interaction @xmath19 and @xmath67 after the quench . we have also performed runs with different discarded weights @xmath114 to check the quality of our results . the data presented in fig . [ fig : entropy_app](b ) suggest that the reduction of the entanglement entropy is ( for @xmath115 and @xmath67 ) @xmath116 note that we have not yet reached the minimum in the time - dependent reduction of @xmath117 . at long times , the simulation is slowed down due to the relatively large entanglement of blocks with @xmath118 . the figure unveils that the more accurate the simulation is ( _ i.e. _ , the smaller the discarded weight ) , the better the efficiency of the quantum distillation in reducing @xmath119 is captured . the last figure , fig . [ fig : trap_n ] , illustrates the dependence of the quantum distillation on first , @xmath53 [ panel ( a ) ] and second , the effective density @xmath120 [ panel ( b ) ] . we use the parameters of fig . 4 from the main text , except for varying @xmath53 in fig . [ fig : trap_n](a ) and varying @xmath3 in fig . [ fig : trap_n](b ) . we plot the average double occupancy and we observe that similar to the case of a box trap that was discussed in the main text , quantum distillation works as long as some doublons ( signified by sites with @xmath121 at @xmath43 ) are present in the initial state . also , as expected , the final @xmath53 needs to be larger than @xmath122 to induce the formation of a metastable state with doublons grouping together in the leftmost sites . on the number of particles at a fixed filling of @xmath28 ( @xmath67 ) : ( a ) @xmath18 , @xmath131 ( @xmath132 , @xmath133 ) ; ( b ) @xmath134 , @xmath135 ( @xmath136 , @xmath137 ) ; ( c ) @xmath134 , @xmath135 ( @xmath138 , @xmath137 ) ; ( d ) @xmath139 vs. time @xmath109 ( _ i.e. _ , rescaled on the number of particles ) . the same qualitative behavior is observed , independently of @xmath3 . the time for building up the quasi - fock state is simply proportional to the total number of particles . slight differences between the results for different @xmath3 at short times are due to differences in the initial state ( _ i.e. _ , boundary and @xmath3-dependent oscillations in the density profile ) . , scaledwidth=48.0% ] for @xmath141 ( bottom to top ) and the parameters of fig . 4 of the main text ( @xmath99 , @xmath101 , @xmath98=2 t ) : ( a ) @xmath19 ; ( b ) @xmath67 . the runs were performed with a time step of @xmath127 and different discarded weights @xmath114 ( see the figure s legend).,title="fig:",scaledwidth=45.0% ] for @xmath141 ( bottom to top ) and the parameters of fig . 4 of the main text ( @xmath99 , @xmath101 , @xmath98=2 t ) : ( a ) @xmath19 ; ( b ) @xmath67 . the runs were performed with a time step of @xmath127 and different discarded weights @xmath114 ( see the figure s legend).,title="fig:",scaledwidth=45.0% ]

correlations between particles can lead to subtle and sometimes counterintuitive phenomena . we analyze one such case , occurring during the sudden expansion of fermions in a lattice when the initial state has a strong admixture of double occupancies . we promote the notion of quantum distillation : during the expansion , and in the case of strongly repulsive interactions , doublons group together , forming a nearly ideal band insulator , which is metastable with a low entropy . we propose that this effect could be used for cooling purposes in experiments with two - component fermi gases . one of the most exciting features about ultracold atom gases is the possibility of experimentally studying strongly correlated systems with time - dependent interactions and in out - of - equilibrium situations ( for a review , see @xcite ) . for instance , a time - dependent tuning of parameters has been used to illustrate the collapse and revival of coherence properties of bosons in an optical lattice suggested to be well described by the bose - hubbard model @xcite . also , the question of the relaxation of strongly correlated systems initially prepared in a high - energy state that typically is not an eigenstate has been studied @xcite , as well as the intimately related question of thermalization @xcite . a third context is the expansion of originally trapped particles into an empty optical lattice or a waveguide . while expansions after turning off all trapping potentials and lattices are commonly used to measure the momentum distribution of the originally trapped system @xcite , recently , several studies have investigated the expansion in one dimensional ( 1d ) geometries with and without the presence of an optical lattice along the 1d tubes . in 1d cases , in contrast to the usual time - of - flight experiments , interactions play a fundamental role . the findings include phenomena such as the fermionization of the momentum distribution function of hard - core bosons and anyons @xcite , the asymptotic transformation of generic lieb - liniger wave - functions to a tonks - girardeau structure @xcite , the emergence of quasi - coherence in bosonic @xcite and fermionic systems @xcite , or disorder - induced effects @xcite . such expansions have already experimentally been realized in 1d ultracold atomic gases @xcite . while strongly correlated phases of bosons in optical lattices have been observed and studied in many experiments , reaching the quantum degenerate regime of fermions is more difficult @xcite . only recently , evidence for the observation of the mott - insulator ( mi ) transition of a two - component fermi gas in optical lattices has been presented @xcite . unfortunately , the temperatures in those experiments are still too high to observe the expected low - temperature insulating antiferromagnetic phase , and to make contact with the regime of interest to high-@xmath0 superconductivity @xcite . hence , intense efforts are currently under way to develop efficient cooling schemes for fermions in optical lattices @xcite . in this rapid communication , we study the expansion of a two - component fermi gas in an optical lattice from an initial state with a strong admixture of doubly occupied sites . our main result is the observation that double occupancies ( doublons ) are dynamically separated from the rest of the system and that they group together into a metastable state , which is very close to a fock state . the condition for this to happen is that interaction energies need to be much larger than the kinetic energy . our results are in qualitative agreement with recent studies that have argued , invoking a similar reasoning , that relaxation times can be rather long , and one may thus encounter metastable states ( see , _ e.g. _ , ref . @xcite ) . experimentally , this phenomenon has been observed as repulsively bound pairs in the case of bosons @xcite . we will also discuss the potential of our findings as a cooling technique where one dynamically generates a low - entropy region in a system with arbitrarily large values of the on - site repulsion . _ model and methods _ we consider the one - dimensional ( 1d ) hubbard model : @xmath1 standard definitions are employed in eq . ( [ eq : ham ] ) @xcite . open boundary conditions are imposed , @xmath2 is the number of sites and @xmath3 the number of particles ( with a filling factor @xmath4 ) . the nonequilibrium dynamics is studied using the adaptive time - dependent density matrix renormalization group method ( tdmrg ) @xcite and time - dependent exact diagonalization ( ed ) techniques @xcite . we prepare initial states with particles confined into a finite region of an optical lattice , with a filling factor of @xmath5 in that region and zero otherwise . to this end @xcite , we apply onsite energies @xmath6 to only a portion of the system : @xmath7 ( @xmath8 for @xmath9 and @xmath10 elsewhere ) . hence , at time @xmath11 , we have @xmath12 , while we turn off @xmath13 at @xmath14 . in our tdmrg runs , we use either a third - order trotter - suzuki time - evolution scheme or a krylov - space based method @xcite , with time steps of @xmath15 and @xmath16 ( we set @xmath17 ) . during the time - evolution , we keep up to 1600 states . ( @xmath18 , @xmath19 ) : ( a ) density @xmath20 and ( c ) local charge fluctuations @xmath21 for times @xmath22 . insets : ( b ) : @xmath20 vs time for @xmath23 . ( d ) average double occupancy @xmath24 for @xmath25 ( @xmath26).,scaledwidth=38.0% ] _ results _ figure [ fig : ni](a ) shows snapshots of density profiles @xmath27 at several times for @xmath19 and @xmath28 ( for results for other @xmath29 , see @xcite ) . since @xmath30 , single - particle hopping can only propagate a particle out of a doubly occupied site into the empty sites to the right of the initially occupied region if , at the same time , the remaining particle is moved to the left into a site that was previously singly occupied , in order to preserve the total energy . for that reason , the density in the first sites _ increases _ as the remaining of the block slowly `` melts '' . from fig . [ fig : ni](b ) , one can see that the density in the first sites gets very close to @xmath31 , which promotes the picture that dynamically , doublons are separated from the rest and group together in a region of the lattice . we introduce the term quantum distillation for this process . figure [ fig : ni](c ) shows the local charge fluctuations @xmath32 . consistent with the picture described before , these fluctuations are the largest in the interface region ( sites 7 - 11 in the figure ) , while they die out in the first sites as time increases . to gain a better understanding of how this happens with time , we compute the average double occupancy @xmath33 ( @xmath34 ; @xmath35 ) on the @xmath36 first sites , counting from the left . figure [ fig : ni](d ) depicts this quantity for different values of @xmath36 . for @xmath37 and , _ e.g. _ , @xmath38 , the average double occupancy increases towards @xmath39 , which is quite stable over a long time window for @xmath40 [ fig . [ fig : ni](d ) ] . for @xmath41 , _ i.e. _ , the block with @xmath42 at @xmath43 , this quantity remains almost constant : this is basically equivalent to saying that only a small portion of the interaction energy @xmath44 is converted into kinetic energy during the times simulated . ( @xmath28 ) : vs ( a ) time @xmath45 and ( b ) vs @xmath46 for @xmath47 ( tdmrg ) . insets in ( b ) : @xmath48 for @xmath49 ( @xmath50 ) ; @xmath51 for @xmath52 ( @xmath19 , thin dashed line ) ( ed ) . , scaledwidth=38.0% ] _ time scales _ we next address the question on which time scales ( i ) the fock state is formed and ( ii ) it decays . the latter is equivalent to asking on what time scale a band insulator ( bi ) in the large @xmath53 limit would decay . while from energy considerations it is clear that a piece of doubly occupied sites , _ i.e. _ , a bi , delocalizes on a time - scale of @xmath54 @xcite , we are primarily interested in the case of @xmath5 . to address this situation , we first perform ed calculations for a short initial state with @xmath5 , specifically @xmath55 on two and three sites , respectively [ see the insets of fig . [ fig : double](b ) ] . in this case , one can clearly see that doublons form on the first sites on a @xmath53-independent time - scale , governed by the bare hopping matrix element @xmath56 . doublons then get delocalized over the neighboring empty sites on a time - scale @xmath54 ( times a prefactor that scales with @xmath3 ) , consistent with the decay - rate of a bi . our tdmrg results presented in figs . [ fig : double](a ) and ( b ) unveil that the same picture holds for @xmath57 with @xmath18 . figure [ fig : double](a ) shows the average double occupancy on the first three sites for several values of @xmath53 . quantum distillation is best realized for @xmath58 ( as opposed to @xmath59 ) , since , at @xmath60 , the average double occupancy increases with time ( @xmath61 ) , but it reaches a maximum value that is well below one . nevertheless , as the large-@xmath53 results show , the formation of a quasi - fock state happens over a @xmath53-independent time - scale . figure [ fig : double](b ) displays the same data versus @xmath62 , and the small-@xmath53 curves fall on top of each other at times @xmath63 , while the large-@xmath53 data seem to approach the same curve , too . the results of fig . [ fig : double](b ) show , in the example of @xmath64 , that the fock state will start to delocalize after a time @xmath65 in the @xmath19 case ( @xmath66 for @xmath67 ) . while we have presented results for @xmath18 , the physics does not qualitatively change upon increasing @xmath3 at a fixed @xmath29 @xcite . essentially , the time scale for building up the fock state scales linearly with @xmath3 @xcite . note that in typical experiments with 1d optical lattices , on the order of 100 atoms are confined in one 1d structure @xcite . it should be readily possible to experimentally observe the quantum distillation by measuring , _ e.g. _ , the radius @xmath68 of the double occupancies [ @xmath69 , see @xcite ] . we predict that , for @xmath30 and @xmath70 , this quantity should decrease as a function of time . as @xmath71 depends exponentially on the lattice depth @xmath72 , one can enter into the regime @xmath30 by tuning @xmath72 @xcite . _ entanglement entropy _ further quantitative information can be gained by invoking the entanglement entropy @xmath73 ( ref . @xcite ) , where @xmath74 is the reduced density matrix of a subsystem of length @xmath36 @xcite . the subsystems of primary interest here are those where double occupancy increases as time evolves , as illustrated in fig . [ fig : ni](d ) ( @xmath36 counts the sites starting from the left end ) . figure [ fig : entropy ] depicts one of our most important results , constituting a defining property of the quantum distillation process , namely , the spontaneous reduction of the entanglement entropy in the metastable region where doublons group together . we plot @xmath75 for several @xmath76 at @xmath19 , which ought to be contrasted against the corresponding @xmath77 data , shown in the inset . figure [ fig : entropy ] shows that for the leftmost sites , @xmath78 is nonzero and remains approximately constant for some @xmath36-dependent time window ( _ e.g. _ , @xmath79 for @xmath64 ) . at later times , the behavior strongly depends on the value of @xmath71 . for @xmath80 , @xmath81 increases as the density decreases , while for @xmath82 , @xmath83 decreases , on the time scales simulated , by a factor of 15 as the metastable state with @xmath31 is generated . we relate this decrease of @xmath84 to the quantum distillation , which creates regions with low entanglement . this is complementary to experiments aiming at preparing maximally entangled states @xcite . ( @xmath28 , @xmath19 ) . inset : @xmath84 for @xmath85 at @xmath77 . , scaledwidth=38.0% ] _ harmonic trap _ while there are experimental efforts directed at engineering box - like traps ( see , _ e.g. _ , ref . @xcite ) , which is the case our results presented so far directly apply to , in most typical experiments with optical lattices , the atoms experience a harmonic confinement @xcite . we now consider this case to show that quantum distillation works even in the more difficult case of strongly inhomogeneous initial states , using @xmath86 , where @xmath87 is the curvature of the harmonic potential . the confinement gives rise to a shell structure in the fermionic density @xcite : metallic and mi shells alternate , depending on the characteristic density @xmath88 @xcite . in order to achieve optimal distillation and thus the formation of a low - entropy region , one does not need to follow the approach often discussed in the literature ( see , _ e.g. _ , ref . @xcite ) , in which @xmath71 is increased adiabatically . instead , we propose that one can start with a trapped system with a low value of @xmath89 . in such cases , double occupancy is energetically favorable against trapping energy and hence ideal for our distillation scheme , and moreover , the presence of mi shells is suppressed . at time @xmath43 , one can then quench @xmath71 to a large value @xcite and turn off the trapping potential , but keeping the optical lattice on . in many experiments , the trapping potential is provided by the same lasers that produce the lattice , which , however , is not necessary . in order for our scheme to work , one needs a trapping potential that can be controlled independently of the lattice @xcite . the tdmrg results for the time - evolution of the quench described above are displayed in fig . [ fig : trap ] . we have taken the final values of @xmath71 after the quench to be @xmath19 [ fig . [ fig : trap](a ) ] and @xmath90 [ fig . [ fig : trap](b ) ] . both figures show the average double occupancy @xmath91 for several values of @xmath36 . in fig . [ fig : trap](a ) , one can see that @xmath91 first decreases and then increases to form the quasi - fock state . it is also illustrative to compare the initial density profile to one at later times [ inset of fig . [ fig : trap](b ) ] : while fast particles escape , a block of particles remains in the initial region of the system . a very different behavior of @xmath92 can be seen in fig . [ fig : trap](b ) . there , since the final value of @xmath71 is small , the system simply melts and no signature of quantum distillation is seen . during the expansion , the entanglement growth @xcite induced by the quench from @xmath93 to @xmath94 competes with the reduction of the entanglement entropy @xmath84 of blocks with an increasing double occupancy due to the quantum distillation . it is thus an amazing result that , at sufficiently long times and despite the large final value of @xmath53 chosen in the quench , the quantum distillation wins , as is shown in fig . [ fig : trap](c ) . there , we display @xmath84 for @xmath95 , and indeed , these @xmath84 first increase due to the quench , but eventually drop below their initial value ( dotted lines ) at the maximum times simulated . for the time - scales considered in fig . [ fig : trap](c ) , we find that @xmath96 . the same behavior emerges for other values of @xmath97 , @xmath98 , and @xmath53 @xcite . during the expansion from a harmonic trap ( @xmath99 , @xmath100 , @xmath101 , @xmath102 ) with ( a ) a quench to @xmath19 and ( b ) no quench . dotted line in ( a ) : @xmath103 . inset in ( b ) : density profile at @xmath104 . ( c ) entanglement entropy @xmath84 for @xmath95 ( @xmath19 ) ; horizontal lines are @xmath78 . , scaledwidth=38.0% ] so far , our results suggest that quantum distillation is robust against a variation of initial conditions and quenches . we envision it as a way to experimentally achieve very low temperatures in fermionic systems by creating a very low entropy band insulating state with arbitrarily large values of @xmath71 once such a state is created , it can be used as an initial state to achieve an antiferromagnetic mi with one particle per site . the idea would be to load such a bi in a trap for which that state is close to the ground state . this step is important in ensuring a low - entropy initial state that is in equilibrium @xcite . one can then adiabatically reduce the strength of that trapping potential and of the ratio @xmath71 . that way one could produce a final low temperature mott insulating state starting from a bi . in conclusion , we demonstrated that low - entropy states of two - component fermi gases can be dynamically created in optical lattices utilizing the expansion of particles into the empty lattice in the limit of strong interactions . after having the low entropy state in an appropriate trap , an adiabatic lowering of trapping potential and interaction strength can lead to a final low temperature mi . we stress that the quantum distillation we discussed here for fermions will also work in the bosonic case . we thank l. hackermller , g. refael , a. rosch , u. schneider , and d. s. weiss for fruitful discussions . m.r . was supported by startup funds from georgetown university and by the us office of naval research . a.m. acknowledges partial support by the dfg through sfb / trr21 . e.d . was supported in part by the nsf grant dmr-0706020 and the division of materials science and engineering , u.s . doe , under contract with ut - battelle , llc . m.r . and a.m. are grateful to the aspen center for physics for its hospitality .

measurements of the charged track mutliplicity distribution in @xmath0 meson decay are used to constrain unmeasured or poorly measured branching fractions in monte carlo simulations so that generated event samples more closely represent actual data . the cleo monte carlo parameterization of @xmath0 meson decays has been tuned to agree with our measurements and our model is used by other experimental groups @xcite . charged particle multiplicity in heavy meson decay has been studied by several groups @xcite@xcite@xcite . in this paper we present a measurement of the charged particle multiplicity in inclusive @xmath6 decays that is an improvement over our previous result @xcite . we also present improved measurements of the charged particle multiplicities in semileptonic and nonleptonic decays . for clarity , we use the term `` observed multiplicity '' to denote the number of well reconstructed charged particle tracks in a given event . we use the term `` decay multiplicity '' to denote the number of @xmath37 , @xmath38 , @xmath39 , @xmath40 and @xmath41 that come from the decay of the primary @xmath0 mesons and also from the subsequent decays of any secondary or tertiary particles other than neutrons , @xmath42 , or @xmath43 . the decay multiplicity excludes any tracks produced through interactions with the detector or surrounding material . not all charged decay products will result in reconstructed tracks , and not all observed tracks come from the primary decay , so the observed multiplicity may be less than , equal to , or greater than the decay multiplicity for a given event . the cleo detector is located at the cornell electron storage ring , a high luminosity @xmath44 collider operated at or near the @xmath1 resonance . the results presented here are derived from a sample of 1.4 fb@xmath45 , corresponding to @xmath2 @xmath0 meson pairs , collected with the cleo ii detector @xcite . charged particle tracks are measured by cylindrical wire drift chambers inside a 1.5 t superconducting solenoid . a csi crystal calorimeter is also inside the magnet , and energy deposition information from both the calorimeter and the drift chamber is used for particle identification . muon counters are layered in the steel yoke surrounding the coil . to obtain a clean sample of candidate @xmath6 events , we select hadronic events by requiring that an event have three or more reconstructed tracks , energy deposition in the calorimeter greater than 15% of the center of mass energy and an event vertex consistent with the interaction region . for additional background suppression , the total reconstructed event energy , including charged and neutral particles , is required to be between 4 gev and 12 gev , and the total reconstructed vector momentum of the event is required to have a magnitude less than 3 gev/@xmath46 . this hadronic event sample contains events from both @xmath6 and continuum processes such as @xmath47 and @xmath48 production . we remove the continuum contribution by rescaling and subtracting the observed multiplicity distribution of a separate 0.7 fb@xmath45 data sample collected 65 mev/@xmath49 below the @xmath1 resonance . to be counted in our observed multiplicity , drift chamber tracks are required to be well reconstructed and consistent with having originated from the event vertex . tracks must not be within 25.8 degrees of the @xmath44 beam axis . once the event selection , continuum subtraction , and track selection are completed , we count the selected tracks in each event to obtain the _ observed _ charged track multiplicity distribution in fig . there are events with fewer than three selected tracks because not all reconstructed tracks pass the track selection criteria . to obtain the true decay multiplicity distribution from the observed multiplicity distribution , we must account for detector effects . if the number of events with observed multiplicity @xmath50 is @xmath51 and the number of events with decay multiplicity @xmath52 is @xmath53 , these quantities are related by eqn . [ e : migration ] . @xmath54 where @xmath55 is the probability that an event with decay multiplicity @xmath52 will be reconstructed with observed multiplicity @xmath50 . we have assumed that charge is conserved , so the index @xmath52 can only take even values . in principle , there can be events with decay multiplicity of zero , where two neutral @xmath0 mesons decay to all neutral final states . however , we do not include zero decay multiplicity events in our analysis both because of the very low branching ratio for such events and also because our event selection criteria make detection of such events extremely unlikely . the upper bound in eqn . [ e : migration ] is , in principle , the maximum decay multiplicity in a @xmath6 event , which is not known . we vary the maximum decay multiplicity in our analysis as described below . the fact that the sum of @xmath53 and the sum of @xmath51 are both equal to the total number of events is used to constrain the values of the @xmath53 , as expressed in eqn . [ e : constraint ] . @xmath56 where the upper bound @xmath57 is the maximum value of our observed multipicity which is 20 . the coefficients @xmath55 in eqn . [ e : migration ] are obtained from monte carlo simulation and depend primarily on the detector s track finding efficiency and also on the probability of producing extra charged particles that pass the track selection cuts and that are counted . while these coefficients depend on accurate simulation of detector response and processes such as photon conversion and decays in flight , they do not depend significantly on the exact tuning of the branching fractions or the decay multiplicity distribution in the simulation . the parameters @xmath53 in eqn . [ e : migration ] are determined by a @xmath58 fit . this fit unfolds the detector effects to give the decay multiplicity distribution of events that pass the event selection criteria . these selection criteria are biased against very low decay multiplicity events , particularly because of the requirement of three or more reconstructed tracks . we remove the event selection bias using monte carlo simulation to determine the probability for events of a given decay multiplicity to pass the event selection cuts . after unfolding detector and reconstruction effects , we obtain the decay multiplicity distribution in fig . the error bars represent the statistical uncertainty in both the @xmath58 fit and the event bias correction , but do not include systematic errors . the dashed lines represent the high multiplicity and low multiplicity statistical fluctuations in the fit . the large error bar on the @xmath59 point is due to the event bias correction . from the distribution in fig . [ fig2 ] , we obtain a mean of @xmath60 charged particles for inclusive @xmath6 decay , where the error is statistical only . the points with @xmath61 show no evidence for events with such high multiplicities . we have tested our fitting procedure by varying the maximum decay multiplicity included in the fit between 20 and 28 . the fit is stable and the unfolded mean decay multiplicity does not change significantly when decay multiplicities of 22 and higher are included or excluded from the fit . the most important systematic effect in this analysis is the accuracy of modeling the detector s track finding efficiency . our studies indicate that the overall efficiency for finding a single track is known to within @xmath62 for tracks with momentum greater than @xmath63 mev/@xmath46 and with decreasing accuracy as track momentum decreases . the uncertainty in single track finding efficiency gives a @xmath64 uncertainty in the measured mean decay multiplicity . removal of the event selection bias shifts the measured mean decay multiplicity by @xmath65 , which we also take as the systematic uncertainty for event selection bias . because the track finding efficiency depends on the momentum of the track , we account for the uncertainty due to this dependence . this analysis uses all tracks without regard to momentum . when we add a track selection cut requiring a reconstructed momentum of at least @xmath66 mev/@xmath46 , we observe a shift in the decay multiplicity of @xmath67 . we assign an additional @xmath68 uncertainty due to this momentum dependence . charged pions produced by the decays of @xmath69 will have lower track finding efficiency than average charged pions , so our result depends on the rate of @xmath69 production in our simulation . based on our studies of inclusive @xmath69 production in @xmath0 meson decay , we assign a systematic uncertainty of @xmath70 from this source . interactions of neutral and charged decay products with the detector material produce additional charged tracks , some of which satisfy the track selection criteria . we study the effect of these extra tracks by varying the rates of photon conversion and hadronic interactions in our monte carlo sample . misidentification of these extra tracks contributes an additional @xmath71 systematic uncertainty . extra tracks also come from the decay of particles in the detector volume , adding another @xmath72 uncertainty . additional uncertainty comes from contamination by non-@xmath6 events , most notably from beam - gas interactions . we study the effect of non-@xmath6 events by varying the size of the continuum subtraction by @xmath73 , which is the uncertainty in our measured luminosity , and we find a @xmath74 uncertainty in our mean decay multiplicity . we have looked for other potential systematic effects by varying our track selection cuts but do not observe any significant change in mean decay multiplicity . when all systematic errors are added in quadrature , we obtain a total systematic uncertainty of @xmath75 , which gives a final result of @xmath3 for the mean inclusive charged particle decay multiplicity in the decay of a @xmath6 pair . systematic errors are summarized in table [ tab1 ] . the results presented here are a significant improvement over the previous cleo result @xcite . the earlier analysis found a mean multiplicity of @xmath76 . in addition to using a much larger sample of @xmath6 events , the present analysis uses a substantially different method . the former analysis treated the coefficients @xmath55 in eqn . [ e : migration ] as a matrix and used matrix inversion to solve for the @xmath53 . such matrix inversion is very sensitive to singularities and can produce unstable results . in the current analysis , we do not attempt to invert the @xmath55 matrix , but instead we perform a more stable @xmath58 fit for the parameters @xmath53 . .systematic errors for the inclusive decay multiplicity measurement for @xmath6 pairs . [ cols="<,^ " , ] we gratefully acknowledge the effort of the cesr staff in providing us with excellent luminosity and running conditions . j.r . patterson and i.p.j . shipsey thank the nyi program of the nsf , m. selen thanks the pff program of the nsf , m. selen and h. yamamoto thank the oji program of doe , j.r . patterson , k. honscheid , m. selen and v. sharma thank the a.p . sloan foundation , m. selen and v. sharma thank the research corporation , f. blanc and s. von dombrowski thank the swiss national science foundation , and h. schwarthoff and e. von toerne thank the alexander von humboldt stiftung for support . this work was supported by the national science foundation , the u.s . department of energy , and the natural sciences and engineering research council of canada .

we have used the cleo ii detector to study the multiplicity of charged particles in the decays of @xmath0 mesons produced at the @xmath1 resonance . using a sample of @xmath2 @xmath0 meson pairs , we find the mean inclusive charged particle multiplicity to be @xmath3 for the decay of the pair . this corresponds to a mean multiplicity of @xmath4 for a single @xmath0 meson . using the same data sample , we have also extracted the mean multiplicities in semileptonic and nonleptonic decays . we measure a mean of @xmath5 charged particles per @xmath6 decay when both mesons decay semileptonically . when neither @xmath0 meson decays semileptonically , we measure a mean charged particle multiplicity of @xmath7 per @xmath6 pair . 6.5 in 9.0 in -0.50 in 0.00 in 0.00 in g. brandenburg,@xmath8 a. ershov,@xmath8 y. s. gao,@xmath8 d. y .- j . kim,@xmath8 r. wilson,@xmath8 t. e. browder,@xmath9 y. li,@xmath9 j. l. rodriguez,@xmath9 h. yamamoto,@xmath9 t. bergfeld,@xmath10 b. i. eisenstein,@xmath10 j. ernst,@xmath10 g. e. gladding,@xmath10 g. d. gollin,@xmath10 r. m. hans,@xmath10 e. johnson,@xmath10 i. karliner,@xmath10 m. a. marsh,@xmath10 m. palmer,@xmath10 c. plager,@xmath10 c. sedlack,@xmath10 m. selen,@xmath10 j. j. thaler,@xmath10 j. williams,@xmath10 k. w. edwards,@xmath11 r. janicek,@xmath12 p. m. patel,@xmath12 a. j. sadoff,@xmath13 r. ammar,@xmath14 p. baringer,@xmath14 a. bean,@xmath14 d. besson,@xmath14 r. davis,@xmath14 s. kotov,@xmath14 i. kravchenko,@xmath14 n. kwak,@xmath14 x. zhao,@xmath14 s. anderson,@xmath15 v. v. frolov,@xmath15 y. kubota,@xmath15 s. j. lee,@xmath15 r. mahapatra,@xmath15 j. j. oneill,@xmath15 r. poling,@xmath15 t. riehle,@xmath15 a. smith,@xmath15 s. ahmed,@xmath16 m. s. alam,@xmath16 s. b. athar,@xmath16 l. jian,@xmath16 l. ling,@xmath16 a. h. mahmood,@xmath17 m. saleem,@xmath16 s. timm,@xmath16 f. wappler,@xmath16 a. anastassov,@xmath18 j. e. duboscq,@xmath18 k. k. gan,@xmath18 c. gwon,@xmath18 t. hart,@xmath18 k. honscheid,@xmath18 h. kagan,@xmath18 r. kass,@xmath18 j. lorenc,@xmath18 h. schwarthoff,@xmath18 m. b. spencer,@xmath18 e. von toerne,@xmath18 m. m. zoeller,@xmath18 s. j. richichi,@xmath19 h. severini,@xmath19 p. skubic,@xmath19 a. undrus,@xmath19 m. bishai,@xmath20 s. chen,@xmath20 j. fast,@xmath20 j. w. hinson,@xmath20 j. lee,@xmath20 n. menon,@xmath20 d. h. miller,@xmath20 e. i. shibata,@xmath20 i. p. j. shipsey,@xmath20 y. kwon,@xmath21 a.l . lyon,@xmath22 e. h. thorndike,@xmath22 c. p. jessop,@xmath23 k. lingel,@xmath23 h. marsiske,@xmath23 m. l. perl,@xmath23 v. savinov,@xmath23 d. ugolini,@xmath23 x. zhou,@xmath23 t. e. coan,@xmath24 v. fadeyev,@xmath24 i. korolkov,@xmath24 y. maravin,@xmath24 i. narsky,@xmath24 r. stroynowski,@xmath24 j. ye,@xmath24 t. wlodek,@xmath24 m. artuso,@xmath25 r. ayad,@xmath25 e. dambasuren,@xmath25 s. kopp,@xmath25 g. majumder,@xmath25 g. c. moneti,@xmath25 r. mountain,@xmath25 s. schuh,@xmath25 t. skwarnicki,@xmath25 s. stone,@xmath25 a. titov,@xmath25 g. viehhauser,@xmath25 j.c . wang,@xmath25 a. wolf,@xmath25 j. wu,@xmath25 s. e. csorna,@xmath26 k. w. mclean,@xmath26 s. marka,@xmath26 z. xu,@xmath26 r. godang,@xmath27 k. kinoshita,@xmath28 i. c. lai,@xmath27 p. pomianowski,@xmath27 s. schrenk,@xmath27 g. bonvicini,@xmath29 d. cinabro,@xmath29 r. greene,@xmath29 l. p. perera,@xmath29 g. j. zhou,@xmath29 s. chan,@xmath30 g. eigen,@xmath30 e. lipeles,@xmath30 m. schmidtler,@xmath30 a. shapiro,@xmath30 w. m. sun,@xmath30 j. urheim,@xmath30 a. j. weinstein,@xmath30 f. wrthwein,@xmath30 d. e. jaffe,@xmath31 g. masek,@xmath31 h. p. paar,@xmath31 e. m. potter,@xmath31 s. prell,@xmath31 v. sharma,@xmath31 d. m. asner,@xmath32 a. eppich,@xmath32 j. gronberg,@xmath32 t. s. hill,@xmath32 d. j. lange,@xmath32 r. j. morrison,@xmath32 t. k. nelson,@xmath32 j. d. richman,@xmath32 d. roberts,@xmath32 r. a. briere,@xmath33 b. h. behrens,@xmath34 w. t. ford,@xmath34 a. gritsan,@xmath34 h. krieg,@xmath34 j. roy,@xmath34 j. g. smith,@xmath34 j. p. alexander,@xmath35 r. baker,@xmath35 c. bebek,@xmath35 b. e. berger,@xmath35 k. berkelman,@xmath35 f. blanc,@xmath35 v. boisvert,@xmath35 d. g. cassel,@xmath35 m. dickson,@xmath35 s. von dombrowski,@xmath35 p. s. drell,@xmath35 k. m. ecklund,@xmath35 r. ehrlich,@xmath35 a. d. foland,@xmath35 p. gaidarev,@xmath35 r. s. galik,@xmath35 l. gibbons,@xmath35 b. gittelman,@xmath35 s. w. gray,@xmath35 d. l. hartill,@xmath35 b. k. heltsley,@xmath35 p. i. hopman,@xmath35 c. d. jones,@xmath35 d. l. kreinick,@xmath35 t. lee,@xmath35 y. liu,@xmath35 t. o. meyer,@xmath35 n. b. mistry,@xmath35 c. r. ng,@xmath35 e. nordberg,@xmath35 j. r. patterson,@xmath35 d. peterson,@xmath35 d. riley,@xmath35 j. g. thayer,@xmath35 p. g. thies,@xmath35 b. valant - spaight,@xmath35 a. warburton,@xmath35 p. avery,@xmath36 m. lohner,@xmath36 c. prescott,@xmath36 a. i. rubiera,@xmath36 j. yelton,@xmath36 and j. zheng@xmath36 @xmath8harvard university , cambridge , massachusetts 02138 + @xmath9university of hawaii at manoa , honolulu , hawaii 96822 + @xmath10university of illinois , urbana - champaign , illinois 61801 + @xmath11carleton university , ottawa , ontario , canada k1s 5b6 + and the institute of particle physics , canada + @xmath12mcgill university , montral , qubec , canada h3a 2t8 + and the institute of particle physics , canada + @xmath13ithaca college , ithaca , new york 14850 + @xmath14university of kansas , lawrence , kansas 66045 + @xmath15university of minnesota , minneapolis , minnesota 55455 + @xmath16state university of new york at albany , albany , new york 12222 + @xmath18ohio state university , columbus , ohio 43210 + @xmath19university of oklahoma , norman , oklahoma 73019 + @xmath20purdue university , west lafayette , indiana 47907 + @xmath22university of rochester , rochester , new york 14627 + @xmath23stanford linear accelerator center , stanford university , stanford , california 94309 + @xmath24southern methodist university , dallas , texas 75275 + @xmath25syracuse university , syracuse , new york 13244 + @xmath26vanderbilt university , nashville , tennessee 37235 + @xmath27virginia polytechnic institute and state university , blacksburg , virginia 24061 + @xmath29wayne state university , detroit , michigan 48202 + @xmath30california institute of technology , pasadena , california 91125 + @xmath31university of california , san diego , la jolla , california 92093 + @xmath32university of california , santa barbara , california 93106 + @xmath33carnegie mellon university , pittsburgh , pennsylvania 15213 + @xmath34university of colorado , boulder , colorado 80309 - 0390 + @xmath35cornell university , ithaca , new york 14853 + @xmath36university of florida , gainesville , florida 32611

recently , extremely high superconducting upper - critical fields were observed in ultra thin single - crystals of transition - metal dichalcogenides for magnetic fields applied along a planar direction @xcite . in effect , in single - layered nbse@xmath12 and in electric field - induced superconducting mos@xmath12 , upper critical fields exceeding the weak coupling pauli limiting field , respectively by factors of 6 and 4 , were reported @xcite . single atomic layers of both compounds are characterized by lack of inversion symmetry and a strong spin orbit coupling , which couples ( or locks ) the spins of the carriers along the orbital moments pointing perpendicularly to the conducting planes . valley degeneracy and time reversal symmetry imply that the orbital moments and coupled spins , must have opposite polarization between the valleys at the _ k _ and the _ -k _ points . while multi - layers are spin degenerate , in single atomic layers the aforementioned coupling induces a zeeman - like spin - splitting of the valence and conduction bands as observed at the _ k_-point of their hexagonal brillouin zone @xcite . ising - like superconducting pairing is proposed to occur between carriers ( of opposite spin- and valley - polarization ) at the _ k _ and _ -k _ valleys . this state would be particularly robust against an external in - plane magnetic field due to the spin - valley locking leading to the extremely large values of @xmath7 seen experimentally @xcite . extremely high upper critical fields were also recently reported for a new family of pd based quasi - one - dimensional transition - metal chalcogenides containing nb and ta @xcite . although these compounds display relatively low @xmath9s , i.e. ranging from @xmath13 k to @xmath14 k for nb@xmath12pd@xmath15s@xmath16 @xcite and ta@xmath0pd@xmath0te@xmath17 @xcite respectively , they tend to display upper critical fields that greatly exceed the weakly coupling pauli limiting field value @xmath18 . for example , for nb@xmath12pd@xmath15s@xmath16 one obtains @xmath19 37 t for fields along its @xmath20axis @xcite . these compounds are characterized by structural disorder due , for instance , to the off stoichiometry of the pd atoms . disorder was proposed to suppress the paramagnetic pair - breaking effect due to strong spin - orbit scattering thus producing their very high @xmath7s @xcite . a very large spin - orbit coupling parameter was extracted from a fit to the werthamer - helfand - hohenberg formalism of the superconducting to metallic phase - boundary of nb@xmath0pd@xmath21se@xmath2 @xcite . in nb@xmath12pd@xmath1s@xmath16 the isovalent substitution of pd for the heavier element pt was found to enhance the ratio of @xmath7 to @xmath9 , while this ratio is slightly suppressed for ni doping @xcite . this represents additional evidence indicating that spin - orbit coupling on the pd sites plays a relevant role for its superconducting properties contributing to its large upper critical fields . according to density functional theory calculations @xcite the fermi surface of these compounds tend to be complex being composed of corrugated quasi - one - dimensional sheets and quasi - two - dimensional surfaces as in nb@xmath12pd@xmath21s@xmath16 @xcite , in addition to a complex three dimensional network as in nb@xmath0pd@xmath21se@xmath2 @xcite . in both compounds the superconducting anisotropy @xmath22 , defined as the ratio of @xmath23s for fields along the needle or the _ b_-axis with respect to @xmath7s for fields applied along the other crystallographic orientations , is temperature dependent as seen in the fe pnictide superconductors and is interpreted as evidence for multi - band superconductivity . this conclusion is supported by ref . , which claims that ta@xmath12pds@xmath16 should be considered as a two - band strong - coupled superconductor in the dirty limit argues that the nearly linear dependence of @xmath7 on temperature @xmath8 displayed by most of these pd based compounds would correspond to experimental evidence for such scenario . based on the previous paragraphs one would conclude that disorder , multi - band superconductivity and spin - orbit coupling are the basic ingredients leading to the very large @xmath7s observed in the pd based ta and nb chalcogenides . however , scanning tunnelling spectroscopy measurements on ta@xmath3pd@xmath0te@xmath4 , with a middle point transition at @xmath24 k @xcite , reveals evidence for either an anisotropic superconducting @xmath25wave gap , i.e. having gap minima , or the possibility of gap nodes ( a @xmath26wave component ) in a multi - band system @xcite . in addition , its extreme anisotropy leads to elongated vortices with a core anisotropy of @xmath27 @xcite . furthermore , low - temperature thermal conductivity measurements in this compound reveal a significant residual electronic term at zero magnetic field which , similarly to the cuprates , increases rapidly as the field increases hence corresponding to evidence for nodes in its gap function @xcite . this , coupled to the existence of a superconducting dome in its temperature as a function of pressure phase - diagram would , according to ref . , correspond to evidence for an unconventional superconducting state in ta@xmath3pd@xmath0te@xmath4 . here , we report the phase diagram of nb@xmath0pd@xmath28se@xmath29 single - crystals displaying a middle point superconducting transition at @xmath30 k which is nearly a factor two higher than the previously reported value . this suggests that this series also displays a superconducting dome as a function of the pd content . more importantly , for fields along their needle axis these crystals display an anomalous superconducting to metallic phase - boundary with @xmath23 displaying a @xmath31 dependence on temperature over the entire @xmath8 range . this is the functional dependence observed in single - crystals composed of single- or few atomic - layers of transition - metal dichalcogenides for fields applied along a planar direction @xcite . such a dependence leads to a nearly divergent superconducting anisotropy @xmath22 upon approaching @xmath9 which is consistent with two - dimensional superconductivity . this contrasts with the more conventional phase - diagram extracted from samples displaying lower @xmath9s , indicating a remarkable increase in electronic and hence in superconducting anisotropy and possibly a dependence of the superconducting gap symmetry on pd stoichiometry . this anomalous @xmath32dependence might be attributable to the locking of the spins along a direction perpendicular to the needle axis due to strong spin - orbit ( so ) coupling , although our band - structure calculations indicate that so - coupling is relatively weak for this compound . furthermore , we observe considerably smaller values of @xmath7 for ta@xmath3pd@xmath0te@xmath4 , which is also monoclinic and composed of considerably heavier elements , albeit with its @xmath7s displaying an anomalous linear dependence on temperature over the entire @xmath8 range . for both compounds the absence of saturation in the low temperature values of @xmath7 , particularly when the magnetic field exceeds the pauli limiting value , is at odds with conventional singlet superconductivity . our observations suggest that ta@xmath3pd@xmath0te@xmath4 probably is an orbital limited superconductor over the entire @xmath8 range . as for nb@xmath0pd@xmath21se@xmath2 , we argue , based on band structure calculations , that ising pairing scenario is unlikely to explain our observations . instead , its non - stoichiometric composition is likely to induce a very weak coupling between superconducting planes leading to rather small superconducting coherence lengths and possibly to the concomitantly high upper critical fields observed by us . as of temperature for several randomly oriented ta@xmath33pd@xmath34te@xmath4 single crystals . @xmath35 measured under zero - field cooled conditions is depicted by blue makers , while @xmath35 measured under field - cooled conditions is indicated by red markers . the pronounced deviation among both curves observed below @xmath36 k indicates a bulk superconducting transition . ( b ) electronic contribution to the heat capacity @xmath37 , normalized by the temperature @xmath8 , as obtained after subtracting a @xmath38 term ( i.e. phonon contribution ) . a pronounced anomaly is observed at @xmath9.,width=7 ] ta@xmath0pd@xmath0te@xmath39 single - crystals were grown _ via _ a self - flux method where high purity ( @xmath40 % ) elements were mixed in the ratio ta : pd : te of 2:3:15 and then sealed in an evacuated quartz ampoule . the ampoule was heated up to @xmath41 c at a rate of @xmath42 c / h and held at this temperature for 24 h. the ampoule was subsequently cooled to @xmath43 c at a rate of @xmath44 c / h from which temperature it was finally cooled down to room temperature @xcite . this procedure produces brittle single - crystals with typical dimensions of @xmath45 @xmath46m@xmath47 . single - crystals produced by this procedure displayed a middle - point superconducting transition @xmath48 k. we also followed the original synthesis method of ref . with a starting mixture of elements in the ratio ta : pd : te of 2:1:11 which was heated up to @xmath49 c and held at this temperature for 12 days . this procedure yielded single - crystals showing a range of @xmath9s , i.e. from 3.9 k to 4.7 k. nb@xmath0pd@xmath1se@xmath2 single - crystals were grown using the previously reported method @xcite showing a distribution of @xmath9s ranging from @xmath50 k which depends on the pd content @xcite . the stoichiometric composition was determined by energy dispersive @xmath51-ray spectroscopy ( eds ) . for the nb@xmath0pd@xmath1se@xmath2 single - crystals displaying a higher @xmath9 of @xmath52 k , eds indicates that the pd content is closer to the stoichiometric value of 1 , or @xmath53 , although it also reveals a considerable excess of se , i.e. nearly 20% in some of the crystals . as for ta@xmath3pd@xmath0te@xmath4 eds reveals samples with compositions very close to the stoichiometric values , but also samples with a pd deficiency of @xmath54 % in addition to samples apparently belonging to the ta@xmath0pd@xmath0te@xmath17 phase @xcite . electrical transport measurements were performed by using a combination of superconducting and resistive magnets , coupled to @xmath55he and dilution refrigerators . figure 1 ( a ) displays the magnetic susceptibility @xmath56 as a function of the temperature @xmath8 for several randomly oriented ta@xmath0pd@xmath0te@xmath39 single - crystals . blue markers depict @xmath57 measured under zero - field cooled conditions , while red markers correspond to @xmath57 measured under field cooled conditions . both curves were acquired under a field @xmath58 oe . notice the pronounced diamagnetic signal observed below @xmath59 k indicating bulk superconductivity @xcite . bulk superconductivity is further supported by the size of the anomaly seen in the electronic contribution to the heat capacity @xmath60 at the superconducting transition , which is shown in fig . 1 ( b ) . this curve was collected from several crystals . @xmath60 was obtained after subtracting a @xmath38 phonon contribution term . nevertheless , the size of the anomaly @xmath61 mj / molk@xmath62 relative to the electronic contribution to the heat capacity in the metallic state , i.e. @xmath63 mj / molk@xmath62 , is considerably smaller than the bcs value @xmath64 . this indicates that the superconducting volume fraction is @xmath65 while also reflecting the distribution in @xmath9s among the different crystals . figure 2 ( a ) shows the resistivity @xmath66 of a ta@xmath3pd@xmath0te@xmath4 single - crystal for currents flowing along the @xmath20axis . in contrast to both nb@xmath12pd@xmath1s@xmath16 and nb@xmath0pd@xmath1se@xmath2 , see refs . , this compound displays i ) rather small residual resistivities , in the order of just a few @xmath67 cm , and ii ) fermi liquid like behavior as indicated by the red line which corresponds to a fit to @xmath68 . notice also that the onset of the resistive transition , i.e. the temperature where the resistivity reaches 90 % of its value in the metallic state just above the transition , is @xmath69 k which is slightly higher than the value previously reported @xcite . figure 2(b ) displays @xmath66 as a function of the applied field @xmath70 applied along the @xmath71-axis for several temperatures . here , we follow the nomenclature used by ref . to identify the crystallographic axes of the single - crystal(s ) , with the @xmath72 and the @xmath71 axes being perpendicular to the @xmath20axis ( and to each other ) but not aligned along the crystallographic @xmath73 and @xmath74 axes . figure 2(b ) displays @xmath66 as a function of the applied field @xmath70 applied along the @xmath72-axis for several temperatures . colored lines depict data collected from a second single - crystal having a similar @xmath9 but measured at temperatures well below 1 k. data from both samples collected over an extended range in temperatures are included in the phase - diagram shown below . by comparing all three panels , one can see that i ) this compound is mildly anisotropic , ii ) that its superconducting transition does not broaden considerably under field and iii ) that the transition field does not saturate at a given value as @xmath8 is lowered , as one would expect for a conventional superconductor . subsequently , we compare the upper critical fields observed in ta@xmath3pd@xmath0te@xmath4 with those of nb@xmath0pd@xmath1se@xmath75 , since both compounds belong to the same monoclinic @xmath76 space group and display comparatively similar @xmath9s . the former compound contains heavier elements , therefore one would expect a greater influence of the spin - orbit coupling on its superconducting phase diagram . in reality , nb@xmath0pd@xmath1se@xmath75 single - crystals with @xmath77 and @xmath9s approaching 2 k , already display considerably higher @xmath7s than those of ta@xmath3pd@xmath0te@xmath4 crystals with @xmath78 k. figure 3 ( a ) displays the resistivity @xmath66 , for electrical currents along the _ b_-axis , as a function of @xmath8 . similarly to our previous report this sample also displays an anomaly in the resistivity centered around 100 k of unknown origin , but the onset of its superconducting transition starts at a considerably higher temperature , i.e. @xmath79 k. figures 3 ( a ) and ( b ) depict @xmath66 as a function of the magnetic field applied along the @xmath80 and the @xmath81axis respectively , and for several temperatures . notice that in contrast to ta@xmath3pd@xmath0te@xmath4 the values of its upper critical fields tend to saturate as @xmath8 is lowered , as expected for a singlet - paired superconductor . figure 3 ( c ) displays @xmath66 as a function of @xmath70 applied along its needle or @xmath20axis . in this panel we have included data from two samples , one measured under @xmath82 k ( solid lines ) which displays a sharper transition , and a second one characterized by a broader transition under @xmath83 k ( solid markers ) . both samples display similar @xmath9s and hence similar values of @xmath7 at intermediary temperatures , e.g. at 0.65 k. the residual resistivities were similar for both crystals hence to allow a comparison between both samples we renormalized the resistivity of one of the samples . by comparing all three panels one concludes i ) that the @xmath7s are anisotropic , increasing considerably for fields along the @xmath84-axis and ii ) that the transition also broadens considerably when tilting the field from the @xmath73 to the @xmath20 axis suggesting a prominent role for superconducting fluctuations . broader transitions , which are common to the two - dimensional high @xmath9 cuprates @xcite , suggest that the effective electronic dimensionality is reduced by the application of fields along the @xmath81 and more particularly along the @xmath20axis as if the field renormalized the transfer integrals along crystallographic directions perpendicular to these axes . in figure 4 we plot the superconducting phase diagrams and concomitant superconducting anisotropies for both compounds . as seen in fig . 4 ( a ) , the upper critical fields for ta@xmath3pd@xmath0te@xmath4 follow a simple linear dependence on temperature over the _ entire _ @xmath8 range . here , for all three field orientations we used the 90 % criteria , or the onset of the resistive transition in order to avoid any possible influence from phases related to vortex matter . nevertheless , for fields along the needle - axis we also included points corresponding to the middle point of the resistive transition ( brown markers ) , or the 50 % criterium , to indicate that such anomalous linear dependence can not be attributed to superconducting fluctuations . for @xmath85axis , one observes non - saturation of @xmath86 when @xmath70 approaches or surpasses the value of the pauli limiting field @xmath87 which in the weak coupling regime , or for @xmath88 , leads to @xmath89 t which is smaller than the measured value . here , @xmath90 reflects the strength of the electron - phonon or of the electron - boson coupling . this observation contrasts with what is expected for singlet pairing ; for dirty and nearly isotropic superconductors ( single crystal _ x_-ray diffraction indeed indicates a sizeable amount of site disorder ) the influence of a reduced mean free path is usually described through the maki - de gennes relation @xcite : @xmath91 where @xmath92 is the digamma function and @xmath93 the diffusion constant , with @xmath22 being the electronic coefficient in the heat - capacity and @xmath94 the residual resistivity . in contrast to what is seen here , such expression leads to a saturation of the upper - critical fields at temperatures well below @xmath9 see , for example , ref . . in fe pnictide superconductors one observes a similar linear dependence of @xmath7 on @xmath8 for fields along certain orientations , but not for _ all _ orientations , which was claimed to result from the orbital limiting effect @xcite . a linear dependence of @xmath7 on @xmath8 was also observed in few layered transition metal dichalcogenides for fields applied perpendicularly to the conducting planes @xcite and described in terms of a phenomenological two - dimensional ginzburg - landau model : @xmath95 where @xmath96 correspond to the quantum of flux and @xmath97 to an average in - plane ginzburg - landau coherence length at @xmath98 k. nevertheless , it is seemingly unphysical for a bulk single - crystal to display two - dimensional superconductivity for every orientation of an external magnetic field . the contrast in phase - diagrams between ta@xmath3pd@xmath0te@xmath4 and nb@xmath34pd@xmath21s@xmath99 , with both compounds being characterized by disorder and as seen below multi - band superconductivity , is at odds with the scenario proposed by ref . to explain this linear dependence on @xmath8 . instead , it might indicate that ta@xmath3pd@xmath0te@xmath4 is orbital - limited over the entire temperature range , or equivalently that the pauli limiting field has been renormalized to higher values due to correlations , i.e. @xmath100 as seen in strongly coupled superconductors . this is supported by a ginzburg - landau analysis of @xmath101 as a function of @xmath102 in the vicinity of @xmath9 ( similar plots and related discussion can be found in refs . ) which yield unreasonably high values for the pauli limiting field . finally , the mild dependence on temperature of the superconducting anisotropy @xmath103 as seen in fig . 4 ( b ) suggests multi - band superconductivity as previously argued @xcite for nb@xmath104pd@xmath21s@xmath105 and nb@xmath34pd@xmath21s@xmath99 . at the lowest temperatures a @xmath106 is comparable to @xmath107 previously obtained for nb@xmath34pd@xmath21s@xmath99 . for nb@xmath34pd@xmath21s@xmath99 single - crystals , with a middle point @xmath30 k , the phase - boundary between the superconducting and the metallic states for fields applied perpendicularly to the needle - axis behave quite similarly to those displaying a @xmath108 k , namely a linear dependence on @xmath8 followed by saturation of the @xmath7s at the lowest temperatures . roughly , the @xmath7s between both sets of samples scale linearly with @xmath9 . notice how for both sets of samples , and for fields nearly along the @xmath71-axis , @xmath109 for @xmath110 already surpasses the pauli limiting value . orange lines are fits to the ginzburg - landau expression : @xmath111 where @xmath112 is the reduced temperature and @xmath113 is the coherence length along the @xmath72 , or @xmath84 , or @xmath71 axis . as seen , @xmath114s for fields applied perpendicularly to the needle - axis , can be well - described by conventional behavior , albeit leading to @xmath115 for fields applied along the @xmath74axis . the observed saturation in @xmath7s seen at the lowest @xmath8s clearly indicates that nb@xmath34pd@xmath21s@xmath99 is a pauli limited superconductor . in contrast , for fields along the @xmath20axis we observe a fast increase of @xmath7 as @xmath8 is lowered with respect to @xmath9 . we observed this behavior in 3 crystals with @xmath9s approaching 3.5 k ; for both samples whose raw data are displayed in fig . 3 ( d ) and which were used to build the extended phase - boundary ( black markers ) shown in fig . 4(c ) , and a third sample ( diamonds ) measured in the vicinity of @xmath9 . notice that for this orientation @xmath86 does _ not _ saturate but increases as the @xmath8 is lowered , contrasting markedly with what is observed for fields along the @xmath72-axis . magenta line is a fit to the phenomenological two - dimensional ginzburg - landau expression used to describe @xmath116 for fields along the planes of ultra - thin crystals of transition metal dichalcogenides @xcite : @xmath117 where @xmath118 corresponds to the effective thickness of the superconducting layers . as discussed in ref . this expression can be derived from eq . ( 1 ) in the vicinity of @xmath9 by modifying it to describe not the pair - breaking effect produced by impurities but the pauli pair breaking effect . remarkably , this expression fits the data over the entire @xmath8 range , indicating that for fields along the @xmath20axis nb@xmath34pd@xmath21s@xmath99 behaves is if it was a two - dimensional superconductor , or as if the spins were locked at a direction perpendicular to the @xmath84-axis through a strong spin - orbit coupling @xcite . further evidence for two - dimensional superconductivity is provided by the superconducting anisotropy @xmath22 which seemingly diverges " as one approaches @xmath9 , while displaying at the lowest @xmath8s the same value @xmath107 previously observed in the lower @xmath9 samples . for @xmath119 t , one obtains @xmath120 @xmath62 . if one assumed that the effective thickness of the superconducting slabs was comparable to the lattice constant along the @xmath74axis @xcite or 21.0370 , one would obtain that @xmath97 would extend over @xmath121 lattice constants along the @xmath73axis . the fermi surface of all of the pd - chalcogenide based superconductors is quite similar , that is composed of quasi - one - dimensional and two - dimensional sheets along with three - dimensional fermi surface networks , see fig . 5 for the fermi surface of ta@xmath3pd@xmath0te@xmath4 resulting from the density functional theory ( dft ) calculations discussed below , and also refs . . therefore , it would be quite difficult to explain the marked differences between the phase - diagrams of nb@xmath0pd@xmath1se@xmath2 and ta@xmath3pd@xmath0te@xmath4 based solely on their similar crystallography and electronic structures at the fermi level . however , and as previously mentioned , in transition metal dichalcogenides the strong spin - orbit interaction is known to split the valence and the conduction bands aligning or locking " the spins of the charge carriers along a direction perpendicular to the conducting planes with each spin - orbit split sub - band developing a spin texture , namely spins pointing in opposite directions at the @xmath122 and @xmath123 valleys . this moderately large , and valley - dependent , zeeman - like spin splitting in the vicinity of the @xmath122 points , would protect singlet cooper pairing among carriers originally located at the @xmath122 and @xmath123 valleys : each carrier would have a locked , out of the plane spin polarization of opposite polarity . this inter - valley ising like pairing is claimed , by the authors of refs . , to enhance their upper critical fields well beyond the bcs pauli limiting field . perhaps a similar scenario might be applicable to nb@xmath0pd@xmath1se@xmath2 and also to the nb@xmath12pd@xmath1(se , s)@xmath16 series , namely ising pairing among carriers that have their spins locked along a direction perpendicular to the @xmath20axis , probably along the @xmath124 or @xmath72-axis ( since both systems display the lowest @xmath7s along this orientation ) , due to strong spin - orbit coupling . similarly to the transition metal dichalcogenide single - layers , this would explain the extremely large values of @xmath86 observed in the pd based superconductors , which far exceed their pauli limiting field . such scenario might explain why nb@xmath0pd@xmath1se@xmath2 would have a relatively higher value of @xmath86 in the limit of zero temperature , i.e. @xmath125 t versus @xmath126 t for nb@xmath12pd@xmath1s@xmath16 , given their relative @xmath9s of @xmath52 k and @xmath127 k , respectively . nb@xmath0pd@xmath1se@xmath2 has a slightly higher content of nb relative to the chalcogen element with se being heavier than s thus leading to a potentially stronger spin - orbit coupling . in this scenario , if ta@xmath3pd@xmath0te@xmath4 was not an orbital limited superconductor characterized by higher values of @xmath113 relative to nb@xmath0pd@xmath1se@xmath2 , it should have displayed even higher values of @xmath86 . in this respect , and in order to understand the role of spin - orbit coupling , we evaluated the effect of so - coupling on the band structure of both compounds through dft calculations using the wien2k implementation . we find that the bands of ta@xmath3pd@xmath0te@xmath4 are in general more dispersive and , not surprisingly , exhibit stronger so - splitting than those of nb@xmath0pd@xmath1se@xmath2 , as can be seen in fig . 6 . there is a relatively flat band in ta@xmath3pd@xmath0te@xmath4 that crosses the fermi level ; it has primarily pd - character hybridized with te and might therefore be expected to exhibit reasonably strong spin - orbit effects due to the two heavy elements , see fig . 6 . however , our calculations with and without the so term show that there is nearly no effect at all on this band , though so - splitting can be seen in other ( ta / te - derived ) bands and cause small changes in the fermi surfaces . in nb@xmath0pd@xmath1se@xmath2 , on the other hand , the effects of the spin - orbit are much smaller and are not even visible to the eye at the fermi energy , despite the heavy pd - character of most bands crossing @xmath128 . therefore , the ising - like pairing scenario driven by so - coupling does not seem to be appropriate for the superconducting state of the pd chalcogenides . in addition , by analyzing fig . 6 one realizes that one can not find , for either compound , two nearly degenerate minima ( or valleys " ) in the conduction band(s ) ( or two nearly degenerate maxima in the valence band(s ) ) which should , in the ising pairing scenario , be located at high symmetry points and provide the charge carriers for pairing . from the perspective of the proximity to magnetism both compounds behave differently . first , the nb@xmath0pd@xmath1se@xmath2 is already near a magnetic instability . in our calculations , a spin - polarized state with small moments on some of the nb ions is lower in energy than a spin - unpolarized state , though an ordered state is not observed in experiment . this kind of discrepancy is often a sign of disordered local moments ( paramagnetism ) . in ta@xmath3pd@xmath0te@xmath4 , the energetic ground state is fully non - magnetic . in addition to the standard spin - orbit coupling term , we can also gauge the response to an external magnetic field by adding a term of the form @xmath129 to the potential , where @xmath130 is the angular momentum and @xmath131 the spin . this field is applied only within spheres drawn around each constituent ion ( consistent with our muffin - tin methodology ) and the resulting orbital moments are calculated within the same sphere . similarly , a spin moment is calculated within the sphere , but because polarization of the charge does not require an evaluation of orbital occupancy , we can also gauge the spin response of the interstitial charge , _ i.e. _ that charge which can not be assigned to any given site . we apply this field along three directions : along the _ b_-axis ( _ b_-axis " ) , perpendicular to the _ b_-axis along the nb - se chain direction ( chain " ) and perpendicular to the _ b_-axis along the interchain direction ( interchain " ) . in nb@xmath0pd@xmath1se@xmath2 , there is a clear spin response and an anisotropic orbital response to the field . regardless of field direction , the nb(1 ) and nb(2 ) atoms carry a local spin moment of @xmath132 @xmath133 and the nb(3 ) atom that sits nearest the pd(2 ) sites carries almost no moment at all . the se ions are spin- and orbital - unpolarized . the pd sites show an intriguing anisotropy in orbital moment polarization . both pd sites have a _ c2/m _ square planar symmetry , and each site shows a much stronger tendency toward orbital polarization when the field points perpendicular the plane of the square . for pd(1 ) the easy field direction is the _ b_-axis " direction where an orbital moment of @xmath134 @xmath133 forms in response to the applied field , compared to @xmath135 @xmath133 in the other two directions . the pd(2 ) sites are oriented perpendicularly such that the easy field direction is approximately along the chain " direction . both pd sites carry a spin moment of less than 0.1 @xmath133 with no noticeable anisotropy as a function of field direction . the increase in orbital polarization as a function of the field is likely to pin the spin moment to the lattice and hence contribute to the high upper critical fields . in contrast , the overall response of ta@xmath3pd@xmath0te@xmath4 to the applied field is very different from that of nb@xmath0pd@xmath1se@xmath2 : none of ions carry a significant spin moment and the orbital moment on pd is almost entirely anisotropic . the interstitial spin moment is very close to that of nb@xmath0pd@xmath1se@xmath2 at @xmath136 @xmath133/f.u . and the orbital moments on the ta ions are approximately @xmath134 @xmath133 ( identical to the nb ) . the pd ions have a spin moment of @xmath137 @xmath133 and an orbital moment of @xmath138 @xmath133 , almost identical to the values found for pd in nb@xmath0pd@xmath1se@xmath2 . however , the orbital response of the pd ions in ta@xmath3pd@xmath0te@xmath4 is insensitive to the field direction . it should be noted that there is a slight tendency for the interstitial charge to spin - polarize more strongly when the field is along the @xmath139-axis , with the moment growing to @xmath140 @xmath133/f.u . the smaller spin - orbit coupling , particularly near the fermi energy , combined with the larger spin response to the application of a magnetic field , suggests that nb@xmath0pd@xmath1se@xmath2 might have a smaller critical field ratio than ta@xmath3pd@xmath0te@xmath4 , and yet we see precisely the opposite in our experiments . hence , we are led to conclude that the strength of the spin - orbit coupling and proximity to magnetism is of little relevance for the superconducting phase - diagram of ta@xmath3pd@xmath0te@xmath4 further supporting the orbital - limiting scenario over the entire temperature range . it is tempting to tie both the variation in the superconducting temperature as a function of pd content and the anisotropy and strength of @xmath7 to the strongly anisotropic ( 2d ) orbital response of the pd ions . to gauge the effect of changing pd content on the system , we doubled the unit cell and removed one of the pd(2 ) atoms from the compound . this results in nb@xmath0pd@xmath141se@xmath2 , a lower pd content than our model compound nb@xmath0pdse@xmath2 and also lower than the observed stoichiometry of nb@xmath0pd@xmath142se@xmath29 . nonetheless , we believe that the trends seen in this calculation should mirror the effects of variation of the pd(2 ) content in real materials . interestingly , very little changes overall with the decreased pd content . the previously unpolarized nb(3 ) ion gains a small amount of spin polarization , while all other nb , pd(1 ) , and se spin and orbital moments are indistinguishable from the original ( high pd content ) calculation . the remaining pd(2 ) ion , however , loses nearly all of its spin polarization , but retains most of the orbital polarization . two - dimensional superconductivity in nb@xmath0pd@xmath142se@xmath29 could result from weakly - coupled superconducting planes which are themselves formed by chains composed of square - planar and trigonal - prismatic se polyhedra approximately centered around the pd and the nb atoms , see fig . 3 in ref . these planes are connected just by square coordinated pd(2 ) atoms , whose molar fraction is non - stoichiometric . in addition , one would expect that the excess se @xmath143 became interstitial se randomly placed in between adjacent planes acting perhaps as pair - breaking impurities . unconventional superconductors are particularly susceptible to non - magnetic impurities @xcite . hence , given the experimental evidence , we speculate that the cooper pair coherence length is rather anisotropic but becomes comparable to the inter - planar distance at low @xmath8s . nevertheless , we do not have an explanation for the lack of saturation in the @xmath86s of nb@xmath0pd@xmath1se@xmath2 which is observed in the _ same _ compound @xcite when it displays lower @xmath9s . it is tempting to attribute this lack of saturation to a possible lack of local inversion symmetry , which is indeed observed on the pd sites of ta@xmath3pd@xmath0te@xmath4 which displays a @xmath144 instead of the @xmath76 symmetry . lack of inversion symmetry , which leads to an asymmetric spin - orbit coupling , was claimed to minimize the role of the pauli limiting effect in spin - singlet superconductors @xcite and even to an admix of spin - singlet and spin - triplet superconducting pairing channels @xcite . however , the nb compound preserves inversion symmetry while the contrasting behavior as a function of the pd stoichiometry suggests instead a possible change in the superconducting pairing symmetry probably triggered by the rapid increase in superconducting anisotropy . the correlation of @xmath9 with the decrease in superconducting and/or electronic anisotropy is difficult to reconcile with conventional @xmath25wave pairing . insofar , contradictory evidence for unconventional pairing has only been reported for the ta@xmath3pd@xmath0te@xmath4 compound @xcite , but a similar crystalline space group and an akin electronic structure suggests that unconventional superconductivity is a possibility for all of the pd and chalcogenide based superconductors . unconventional superconductivity is further supported by the anomalous temperature dependence of the upper critical fields reported in this manuscript . we acknowledge t. besara and t. siegrist for the _ x_-ray diffraction measurements on the ta@xmath3pd@xmath0te@xmath4 single - 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both nb@xmath0pd@xmath1se@xmath2 and ta@xmath3pd@xmath0te@xmath4 crystallize in a monoclinic point group while exhibiting superconducting transition temperatures as high as @xmath5 and @xmath6 k , respectively . disorder was claimed to lead to the extremely large upper critical fields ( @xmath7 ) observed in related compounds . despite the presence of disorder and heavier elements , @xmath7s in ta@xmath3pd@xmath0te@xmath4 are found to be considerably smaller than those of nb@xmath0pd@xmath1se@xmath2 while displaying an anomalous , non - saturating linear dependence on temperature @xmath8 for fields along all three crystallographic axes . in contrast , crystals of the latter compound displaying the highest @xmath9s display @xmath10 , which in monolayers of transition metal dichalcogenides is claimed to be evidence for an ising paired superconducting state resulting from strong spin - orbit coupling . this anomalous @xmath8-dependence indicates that the superconducting state of nb@xmath0pd@xmath1se@xmath2 is quasi - two - dimensional in nature . this is further supported by a nearly divergent anisotropy in upper - critical fields , i.e. @xmath11 , upon approaching @xmath9 . hence , in nb@xmath0pd@xmath1se@xmath2 the increase of @xmath9 correlates with a marked reduction in electronic dimensionality as observed , for example , in intercalated fese . for the nb compound , density functional theory ( dft ) calculations indicate that an increase in the external field produces an anisotropic orbital response , with especially strong polarization at the pd sites when the field is perpendicular to their square planar environment . the field also produces an anisotropic spin moment at both pd sites . therefore , dft suggests the field - induced pinning of the spin to the lattice as a possible mechanism for decoupling the superconducting planes . overall , our observations represent further evidence for unconventional superconductivity in the pd chalcogenides .

the ligo and virgo gravitational wave interferometric detectors are approaching their design sensitivity @xcite,@xcite and in the near future , coincidences between the three detectors ( ligo - hanford , ligo - livingston and virgo ) will be possible . in order to reconstruct the direction of the astrophysical sources in the sky , it is well known @xcite that a minimum of three detectors is mandatory even if an ambiguity remains between two positions symmetric with respect to the plane defined by the 3 detectors . the source direction is also provided by the coherent searches for bursts @xcite , @xcite , @xcite or coalescing binaries @xcite , @xcite , @xcite where one of the outputs of the detection algorithm is an estimation of the source direction . in this paper , we propose a method for estimating the source position using only the arrival time of the gravitational signal in each detector . the event detection is supposed to have been previously done by dedicated algorithms ( [ 10 - 23 ] for bursts , [ 24 - 29 ] for coalescing binaries ) and is not within the scope of this article . the direction reconstruction is based on a @xmath0 minimization as described in section ii . this technique can be easily extended to any set of detectors . moreover , the method can be applied to several types of sources ( burst , coalescence of binary objects ... ) as soon as an arrival time can be defined for the event . section ii also deals with the simulation procedure which will be used in the following sections to evaluate the reconstruction quality in several configurations . sections iii and iv describe the performances of the ligo - virgo network first neglecting ( iii ) , then including ( iv ) the angular response of the detectors . in section v , we consider the addition of other gravitational wave detectors ( supposing a similar sensitivity ) and investigate their impact on the reconstruction . in real conditions , systematic errors on arrival time are likely to exist and their impact on the reconstruction is tackled in section vi . within a network of _ n _ interferometers , we suppose that each detector @xmath3 measures the arrival time @xmath4 of the gravitational wave . of course , the definition of the arrival time depends on the source type and is a matter of convention , for example : peak value in the case of a supernova signal , end of the coalescence for binary events . in the following , it is assumed that all interferometers use consistent conventions . the error on the arrival time , @xmath5 , depends on the estimator used and on the strength of the signal in the detector @xmath3 , strength ( for a given distance and a given signal type ) which is related to the antenna pattern functions ( see @xcite and references therein ) at time @xmath4 . at that time , the antenna pattern depends on the longitude and the latitude of the detector location , as well as its orientation , the angle between the interferometer arms , the sky coordinates @xmath6 ( right ascension ) and @xmath7 ( declination ) of the source , and the wave polarization angle @xmath8 . the timing uncertainty @xmath5 can be parametrized by @xcite : @xmath9 where @xmath10 is the measured snr in detector @xmath3 , @xmath11 and @xmath12 are constants depending on the detection algorithm and the signal shape . for example , a burst search with a 1-ms gaussian correlator leads to @xmath13 ms and @xmath14 . typically , for an snr equal to 10 , the error on the arrival time is a few tenth of milliseconds and weakly depends on @xmath12 for snr values between 4 and 10 . the @xmath15 measured arrival times @xmath4 and their associated errors @xmath5 are the input for the reconstruction of the source direction in the sky , direction defined by @xmath6 and @xmath7 . in the 3-detector configuration , the angles ( @xmath16 and @xmath17 ) of the source in the detector coordinate system ( see ref.@xcite and @xcite for exact definitions ) are given by @xcite : @xmath18 where d@xmath19 is placed at ( 0,0,0 ) , d@xmath20 at ( @xmath21,0,0 ) and d@xmath22 at ( @xmath23,@xmath24,0 ) . when performing a coherent analysis of the gw detector streams the position of the source in the sky is part of the output parameters , corresponding to the stream combination which maximizes the snr . however , for a burst search , it is known that thousands of possible positions have to be tested to obtain the solution @xcite , @xcite or a least - square function involving the integration of detector streams has to be minimized @xcite . this minimization also implies the test of hundreds of initial conditions in order to reach the right minimum . concerning coalescing binaries , it implies the definition of a five - parameter bank of filters including the chirp mass , the three euler angles and the inclination angle @xcite of the orbital plane or a three - parameter bank of thousands filters for the two source angles and the chirp time @xcite . in all coherent techniques , the extraction of the source direction is an heavy process imbedded in the detection procedure . in this paper , we propose a simpler approach where @xmath6 and @xmath7 are found through a least - square minimization using separately triggered events obtained by a coincidence search . we suppose that the detection is already performed applying suitable algorithms ( matched filter for coalescing binaries , robust filters for bursts ) . the @xmath0 is defined by : @xmath25 where @xmath26 is the arrival time of the gravitational wave at the center of the earth and @xmath27 is the delay between the center of the earth and the i@xmath28 detector which only depends on @xmath6 and @xmath7 . the first advantage of this definition is that it deals with absolute times recorded by each detector rather than time differences where one detector has to be singled out . otherwise , the best choice for the reference detector is not obvious : the detector with the lower error on the arrival time , the detector which gives the larger time delays or the detector leading the best relative errors on timing differences ? this definition leads to uncorrelated errors on fitted measurements . the second advantage is that the network can be extended to any number of detectors and the addition of other detectors is straightforward . obviously , the method requires that the event is seen by all detectors . the least - square minimization provides the estimation of @xmath29 and the covariance matrix of the fitted parameters . when the number of detectors is greater than 3 , the @xmath0 value at the minimum can also be used as a discriminating variable , as the system is overconstrained . a list of coincident events are defined by the three arrival times and their associated errors . no detection procedure is performed in these simulations as stated before . the simulation proceeds in two steps in order to study two coupled effects : antenna - patterns and location with respect to the 3-detector plane . the first step is a simplified approach : the antenna - pattern functions are ignored and the same error @xmath5 is assumed for arrival times . the second step is more realistic : we assume the same sensitivity for each detector and the signal strength is adjusted to have the mean ( over the three detectors ) snr equal to 10 . however , this implies that sometimes , due to the antenna - pattern , the signal is seen in a given detector with an snr lower than 4.5 , which remains an acceptable threshold for a real detection . the same threshold equal to 4.5 will be used later to see the effect of a reasonable detection scheme on the angular reconstruction error . we evaluate the errors @xmath5 for each arrival time @xmath4 using equation [ eq : sigma_t ] with @xmath14 . for a given simulation , the coordinates ( @xmath30 ) of the source are chosen and the true arrival times @xmath31 on each detector are computed taken @xmath32 equal to 0 ( it is obvious that the timing origin can always been chosen such as @xmath33 ) the measured arrival times @xmath34 are drawn according to a gaussian distribution centered on @xmath35 and of width @xmath5 . the simulated values @xmath34 are then used as inputs for the least - square minimization . as the minimization is an iterative procedure , some initial values for the parameters have to be given . @xmath26 is initialized by the average of @xmath34 . for the angles , it appears that the initial values for the angles have no influence on the minimization convergence and a random direction is adequate . in the case of three interferometers , it is well known that there is a twofold ambiguity for the direction in the sky which can lead to the same arrival times in the detectors . these two solutions are symmetric with respect to the 3-detector plane . in order to resolve the ambiguity , a fourth detector is needed . for the evaluation of the reconstruction accuracy , only the solution closest to the source is retained . in this section , we only deal with the ligo - virgo network and the effect of the antenna - pattern functions are not included and it is assumed that all detectors measure the arrival time with the same precision . as previously said , it allows to decouple the effect of the antenna - patterns and of the location with respect to the 3-detector plane . first of all , as an example , in order to evaluate the accuracy of the reconstruction , we choose a given position in the sky ( coordinates of the galactic center @xmath36 ) and we perform the simulation with @xmath37 s at a fixed time . the results are shown on figure [ fig : reco_pos ] . a resolution of about 0.7 degrees can be achieved both on @xmath6 and @xmath7 . the angular error is defined as the angular distance on the sphere between the true direction and the reconstructed one ( it does not depend on the coordinate system and in particular there is no divergence ( only due to the coordinate system ) when @xmath7 is equal to 90 degrees ) . this variable will be used in the following steps as the estimator of the reconstruction quality . the mean angular error is 0.8 degrees . as shown on table [ tab : covar ] , the estimated errors ( given by the covariance matrix ) obtained by the @xmath0 minimization are in perfect agreement with these resolutions . .reconstruction accuracies on @xmath6 and @xmath7 and errors given by the covariance matrix . three digits are given in order to show the adequacy between rms and errors given by the covariance matrix . [ cols="^,^,^ " , ] all angular errors quoted previously suppose that arrival time measurements are only subject to gaussian noise . systematic biases can also be introduced by the analysis and their effect can be evaluated . in order to do so , we modify equation [ eq : t_measured_bias ] introducing a timing bias for only one detector : @xmath38 as in sections [ source ] and [ including ] , we only consider the ligo - virgo network . it appears that the widths of the distribution for reconstructed @xmath6 and @xmath7 are not modified by the bias but the central values are shifted from the true ones . the differences between the reconstructed value and the true one are proportional to the bias and are significantly different from zero when the bias and the statistical error have the same order of magnitude . table [ tab : bias ] shows the effect of the bias for a given direction ( we check that the effect is independant of the source location ) . in this example , the bias has been applied to the livingstone interferometer . the width of statistical errors on arrival time was .1 ms leading to a statistical angular error about 0.8@xmath39 . for the tested configurations , we do not observe significant differences between the three interferometers of the network . we described a method for the reconstruction of the source direction using the timing information ( arrival time and associated error ) delivered by gravitational wave detectors such as ligo and virgo . the reconstruction is performed using a least - square minimization which allows to retrieve the angular position of the source and the arrival time at the center of the earth . the minimization also gives an estimation of errors and correlations on fitted variables . for a given position , the angular error is proportional to the timing resolution and the systematic errors ( if they exist ) introduce a significant bias on reconstructed angles when they reach the level of the statistical one . when the antenna - pattern effect is included and imposing a mean snr value of 10 in the ligo - virgo network , a precision of @xmath40 can be reached for half of the sky . in order to reproduce a realistic case , we apply a threshold on the snr in each detector ( snr@xmath1 4.5 leading to a false alarm rate about @xmath41 hz when performing a threefold coincidence ) . this condition is satisfied for 60 % of the sky and the median angular error in this case is @xmath42 . as a resolution of @xmath2 is obtained for 30 % of the events satisfying the snr condition , it means that about 20 % of the whole sky is seen with an angular error lower than @xmath2 . adding other gravitational waves detectors allows to reduce the blind regions and to lower the mean resolution . in the best considered case ( 6 detectors ) , the resolution is about @xmath43 and 99% of the sky is seen with a resolution lower than @xmath44 . all quoted resolutions ( about one degree ) are similar to those delivered by @xmath45-ray satellites when the first grb counterparts have been identified . so , we can expect it will be also sufficient for the first identification of gravitational wave sources .

this paper deals with the reconstruction of the direction of a gravitational wave source using the detection made by a network of interferometric detectors , mainly the ligo and virgo detectors . we suppose that an event has been seen in coincidence using a filter applied on the three detector data streams . using the arrival time ( and its associated error ) of the gravitational signal in each detector , the direction of the source in the sky is computed using a @xmath0 minimization technique . for reasonably large signals ( snr@xmath14.5 in all detectors ) , the mean angular error between the real location and the reconstructed one is about @xmath2 . we also investigate the effect of the network geometry assuming the same angular response for all interferometric detectors . it appears that the reconstruction quality is not uniform over the sky and is degraded when the source approaches the plane defined by the three detectors . adding at least one other detector to the ligo - virgo network reduces the blind regions and in the case of 6 detectors , a precision less than @xmath2 on the source direction can be reached for 99% of the sky .

it has been well established by now that the milky way is not axisymmetric with both a central bar and spiral structure perturbing its disk . due to our location in the galactic plane both spiral and bar structure is impossible to observe directly . galactic bar parameters such as orientation and pattern speed have been inferred indirectly from both asymmetries around the galactic center ( e.g. , @xcite ) and its effect on the local velocity distribution of old stars , i.e. , the hercules stream @xcite . spiral structure parameters , however , are much more uncertain . current spiral density wave models @xcite strongly disagree on the strength of the spiral structure , the number of arms , and the pattern speed . these models differ in their predictions of the induced velocity streaming at different angular positions in the galaxy . for example , a four - armed density wave with velocity perturbations of @xmath1 km / s will exhibit rapidly varying radial and tangential velocity components with azimuth across distances of a few kpc , and we could expect to detect @xmath2 km / s variations in the mean line - of - sight stellar velocity as a function of the distance from the sun . however , the strength of the spiral arm perturbation remains controversial . based on the ogle number counts , @xcite estimated that the sagittarius - carina arm has a factor of two increase in density compared to the underlying disk . this model is inconsistent with cobe studies which find a much smaller contrast ( @xmath3 ) and show that the perseus and scu - cru arms are more dominant @xcite . hi , co , cepheid , and far - infrared observations suggest that the galactic disk contains a four - armed tightly wound structure . on the other hand , @xcite have shown that the near - infrared observations are consistent with a dominant two - armed structure . @xcite suggest that locally the milky way can be modeled by the superposition of a two- and four - armed structure moving at the same pattern speed . by studying the nearby spiral arms , @xcite find that the sagittarius - carina arm is a superposition of two features , moving at different pattern speeds . the effect of a two- and four - armed structure , moving at different angular velocities , on the velocity dispersion of a galactic disk has been explored numerically by @xcite . estimates for the pattern speed of the milky way spiral structure , or equivalently , the sun s position with respect to resonances associated with spiral structure , span a large range of values . reviewing previous work , @xcite finds a clustering of estimates for the pattern speed of local spiral structure near @xmath4 , though other studies suggest @xmath5 . the model by @xcite places the sun near the corotation resonance @xmath6 ) , and was fit to cepheid kinematics the recent gas dynamical studies @xcite match the properties of the gas in nearby arms with a spiral pattern speed of @xmath7 . @xcite propose that a two - armed stellar structure consistent with the stellar distribution inferred from cobe could cause four - arms in the gas distribution near the sun . the gas dynamical model proposed by @xcite with a similar spiral pattern speed matches hi and co kinematics . the pattern speed of a spiral density wave can be tightly constrained from the location of its resonances . for example , @xcite associated stellar streams in the solar neighborhood with the 4:1 ilr resonance of a two - armed pattern and were then able to tightly constrain the pattern speed of the driving spiral density wave to within 5% . independent constraints on the pattern speed come from recent surveys of nearby open clusters ( e.g. @xcite ) where the older clusters are found to have drifted further from their original density wave location . these authors concluded that the sun is located near the cr . a solar circle near the cr is also favored by @xcite and @xcite . in this paper we investigate how spiral structure parameters can be inferred from velocity and density maps resulting from pencil - beam and large - scale surveys of the galaxy . at present the influence of spiral arms on the observed kinematic properties of the galactic disk is very poorly understood . with the advent of future galactic all - sky ( gaia , segue ) and pencil - beam ( argos , brava ) radial velocity surveys , large amounts of kinematic data will be collected . the types of dynamical constraints made possible with these new data sets is not currently known . we address that issue here with synthetic models for the purpose of exploring how spiral structure might be constraint from these data . we perform 2d test - particle simulations of an initially axisymmetric exponential galactic disk . in order to reproduce the observed kinematics of the galactic disk , we use disk parameters consistent with observations ( table [ table : par ] ) . the reader is referred to @xcite for a more detailed description of our simulation set up . in all of our simulations we start with an initially warm disk , i.e. , the radial velocity dispersion at @xmath8 is @xmath9 where @xmath10 is the velocity of the local standard of rest . the background axisymmetric potential due to the disk and halo has the form @xmath11 , corresponding to a flat rotation curve . we treat the spiral pattern as a small perturbation to the axisymmetric model of the galaxy by viewing it as a quasi - steady density wave in accordance with the lin - shu hypothesis @xcite . the spiral wave gravitational potential perturbation is expanded in fourier components as @xmath12.\ ] ] the parameter @xmath13 is related to the pitch angle of the spiral wave , @xmath14 , as @xmath15 , negative for trailing spirals with rotation counterclockwise , and @xmath16 are plane polar coordinates . the pattern speed is given by @xmath17 and the spiral strength by @xmath18 . for a two - armed structure the @xmath19 term dominates . upon taking the real part of equation [ eq : sp ] the perturbation due to the two - armed spiral density wave becomes lcc solar neighborhood radius & @xmath8 & 1 + circular velocity at @xmath8 & @xmath10 & 1 + radial velocity dispersion & @xmath20 & @xmath21 + @xmath22 scale length & @xmath23 & @xmath24 + disk scale length & @xmath25 & @xmath26 + spiral strength & @xmath27 & @xmath28 + pitch angle & @xmath14 & @xmath29 @xmath30 integrations are performed forward in time . the perturbation is grown from zero to its maximum strength in four rotation periods at @xmath8 . in order to improve statistics , positions and velocities are time averaged for 10 spiral periods . we distribute particles ( stars ) between in inner and outer galactic radii @xmath31 . new particles are added until the final number of outputs is @xmath32 . in addition , the two - fold symmetry of our model galaxy is used to double this number . we present our results by changing the spiral pattern speed , @xmath17 , and keeping the solar radius fixed at @xmath33 . for a two - armed spiral pattern the primary resonances are the 2:1 inner and outer lindblad resonances ( ilr and olr ) . those are achieved when @xmath34 , respectively , where @xmath35 is the epicyclic frequency . similarly , the second order resonances are the 4:1 lindblad resonances ( lrs ) at @xmath36 . we examine a region of parameter space for a range of pattern speeds placing the sn between the 4:1 lrs . interpretation of line - of - sight velocities with galactic longitude and distance from the sun is not straightforward . to help out we first discuss morphology as seen by an outside viewer . in figure [ fig : den_m2 ] we present stellar number density contour plots for simulations of galactic disks with different pattern speeds . the background axisymmetric disk is subtracted to emphasize the spiral structure . the quantity plotted is @xmath37 , where @xmath38 and @xmath39 are the perturbed and axisymmetric stellar number densities . concentric circles represent the 2:1 lrs ( dashed ) , the solar radius ( solid ) , and the cr ( dash - dotted ) . darker colors correspond to lower density . the inner @xmath40 disk is not plotted since we do not model the galactic center . each panel represents a simulation with a distinct pattern speed , @xmath17 , and all other parameters kept the same ( see table [ table : par ] ) . pattern speeds considered range approximately between the 4:1 lrs , @xmath41\omega_0 $ ] in units of 0.1 . the minima of the two - armed spiral potential are graphed in each panel as solid curves . note the crowding of resonances as the pattern speed increases . in general , changing the solar radius in a simulation with the same pattern speed is equivalent to changing the pattern speed and keeping the solar radius fixed . however , this is exactly true only if the stellar density and velocity dispersion varied linearly with radius . this is not the case in real galaxies ; both of these are found to vary exponentially with radius . @xcite estimated the number density and radial velocity dispersion scale lengths in the milky way to be @xmath42 , and @xmath43 , respectively . thus we need to perform different simulation runs when changing the pattern speed . note the disruption of the spirals near the 2:1 lrs ( dashed circles ) . a rapid decrease of spiral strength at the 2:1 olr was also observed by @xcite in a galactic disk model consisting of solving numerically the boltzmann moment equations . it has also been suggested by @xcite that strong ( nonlinear ) spiral structure can not extend beyond the 4:1 ilr since at that location the stellar orbits are not in phase with the imposed spiral . as pointed out by @xcite , however , this limited extent of the spirals found by @xcite is probably related to the restrictive assumptions they make in order to constructing self - consistent spiral structure . another remarkable feature in the plots of figure [ fig : den_m2 ] is the overdensity of stellar orbits just outside the 2:1 lrs ( where they exist ) and near the cr ( dash - dotted circles ) . in the case of the cr , the enhancement is due to the stable lagrange points @xmath44 associated with this resonance . assuming the primary spiral structure in the solar neighborhood is two - armed , the question arises : how can we determine any spiral structure parameters , given our inconvenient position in the galaxy ? one way to do this is by collecting a large number of stars with known velocities , distances , etc , and constructing velocity and density maps by plotting these versus galactic longitude , @xmath45 and heliocentric distance , @xmath46 . in the next section we show what such maps would look like when a two - armed spiral structure perturbs a stellar disk . we examine different pattern speeds and solar positions with respect to a spiral arm . we call these maps `` pencil beam maps '' or pbms . to investigate the global structure of the galaxy we require accurate stellar velocities and distances . in a pencil - beam spectroscopic survey line - of - sight velocities can be measured to great distances . on the other hand , proper motions are hard to measure for stars farther than about two kpc from the sun . thus those can not be used in our investigation . for a complete kinematic study accurate distance estimates are also needed . due to the large distances involved in a such survey trigonometric parallax measurements are not possible . instead , photometric distances can be estimated given accurate photometry . this way of computing distances , however , is hampered by the dust obscuration in the galactic plane aside from several known windows , e.g. , baade s window at @xmath47 . another distance estimator is the use of standard candles such as cepheids , etc . here we do not attempt to model the reddening resulting from dust extinction but present an idealized model as a first attempt to tackle this problem . a future paper will be dedicated to a more detailed modeling . due to this shortcoming our model can be directly applied only to the known low extinction galactic plane windows . like baade s window , the scutum window at @xmath48 has low extinction and we can observe stars at 10 kpc distances towards the inner disk . clump giants of the intermediate - age and older population of the disk and thick disk will be abundant in these fields . this line - of - sight at @xmath48 is tangent to the scu - cru spiral arm , with an av extinction of about 3 mag at the distance of the spiral arm tangent point ( @xmath49 kpc ) . in the scutum window , the hi and h@xmath13 profiles clearly show the presence of spiral arms @xcite . in figure [ fig : pbm20 ] we present pbms of the line - of - sight velocity @xmath50 ( left column ) , the corresponding velocity dispersion @xmath51 ( middle column ) , and the stellar number density @xmath38 ( right column ) . this is a simulation of a galactic disk perturbed by a two - armed spiral density wave moving with @xmath52 . to create the contour plots in this figure we bin the disk in galactic longitude @xmath45 ( x - axis ) , and heliocentric distance @xmath53 ( y - axis ) as seen from an observer at a solar orientation with respect to the concave spiral arm of @xmath54 this is in contrast to figure [ fig : den_m2 ] where we present number density plots of a face - on view . the contours in the first row in figure [ fig : pbm20 ] show the results of an axisymmetric disk , which is indicated by the subscript @xmath55 . in the second row the disk is perturbed by an @xmath19 spiral density wave . the third row in figure [ fig : pbm20 ] plots contours of the difference between the perturbed and axisymmetric disks for the mean velocity and its dispersion : @xmath56 and @xmath57 . the number density , on the other hand , is obtained as in figure [ fig : den_m2 ] : @xmath58 we showed a number density plot in the x - y plane of this particular simulation in figure [ fig : den_m2 ] ( panel with @xmath59 ) . in figure [ fig : pbm20 ] , however , we plot observables from a point of view centered on the sun , as pencil - beam surveys would see the galaxy . the shaded contours are equally spaced with darker color corresponding to lower density . on top of each panel we show the minimum and maximum contour values ( in units of @xmath10 in the case of the velocities and the velocity dispersion ) . as it is commonly accepted , the galactic longitude is zero in the direction of the galactic center , with the galactic anticenter at @xmath60 . the inner @xmath40 disk has been removed everywhere as in the density plots in figure [ fig : den_m2 ] . the white curve in each panel represents the projection of the solar circle in a pbm and the vertical line shows the galactic longitude @xmath61 . note that in the background subtracted pbms ( third row ) the spiral structure is much better pronounced compared to the raw values ( first row ) . knowing the global galactic potential is imperative to extracting information from a pbm . this presents a problem since the milky way potential is not very well known . we could , however , use our model axisymmetric data to subtract from the observational data . how well do we need to know the axisymmetric structure so that spiral features are not wiped out ? from the third row of figure [ fig : pbm20 ] we can estimate that distance uncertainties of @xmath62 do not prevent detection of spiral structure . we also need line - of - sight velocity precision of @xmath1 km / s and the axisymmetric potential must be known to within @xmath63 . what information about the spiral structure can we infer from pbms such as figure [ fig : pbm20 ] ? to answer this question we need to vary the parameters and look at how structure in these pbms changes . we do this in the following sections . we would like to know how to infer the spiral pattern speed from structure in the pbms . in figure [ fig : phi20 ] we plot the variation of pbms with a change in the spiral pattern speed in the range @xmath64 ; the solar orientation with respect to the concave spiral arm is kept fixed at @xmath54 . this range places the sun from just inside the 4:1 ilr to the cr . in contrast to figure [ fig : pbm20 ] here all panels have the axisymmetric background subtracted to reveal the symmetry of the spiral residuals . we are now looking for features that are strong enough to be detected in an actual pencil - beam survey . the @xmath65 values indicated above each panel give the maximum error introduced by spiral structure in the otherwise axisymmetric background disk . these values for the line - of - sight velocity @xmath50 , and its standard deviation @xmath51 , are @xmath0 km / s for @xmath66 km / s ( left and middle columns in figure [ fig : phi20 ] ) , which is well above the resolution of upcoming radial velocity surveys ( @xmath67 km / s ) . strong features showing marked variation with the change in pattern speed are apparent in all three observables : ( 1 ) in the case of @xmath50 high positive and negative velocity groups resulting from the effect of the 2:1 ilr are found at @xmath68 and @xmath69 , respectively , for @xmath70 ( top left panel ) . with the increase of pattern speed , these clumps spiral in a clockwise direction toward the galactic center . ( 2 ) the standard deviation of the line - of - sight velocities , @xmath51 , which can also be described as the `` heating '' ( or the random motions ) of stars , peaks at a particular ring - like shape around the galactic center for all pattern speeds ( middle column ) . these rings are associated with the 2:1 ilr induced by the spiral density wave . it is clear that the radii of these rings are changing with the change of the pattern speed and thus the location of the 2:1 ilr . beyond the cr ( @xmath71 ) this resonance falls inside the inner three kpc or inside the galactic bulge . thus , using the radius of this hot ring to infer the location of the 2:1 ilr ( and thus the pattern speed ) is only valid if @xmath17 values are in the range considered in figure [ fig : phi20 ] . ( 3 ) lastly , the number density pbms in figure [ fig : den_m2 ] ( right column ) are also indicative of the changing pattern speed . the disruption of the spiral arms near the 2:1 lrs and variation in spiral strength due to the encounter the second order resonances and cr , creates a large contrast in these axisymmetric background subtracted pbms . many of these features can be used in addition to the information extracted from the velocities . all of the features described above can be used to identify the location of the 2:1 ilr and thus the patter speed . as we mentioned , however , if the solar circle is placed at or beyond the cr , the 2:1 ilr falls inside the galactic bulge . consequently , the features created by it disappear . fortunately , just as this happens the 2:1 olr enters the galactic disk ( our disks extend to a radius of @xmath72 ) and similarly to the 2:1 ilr case , resonant features are created , this time in the outer parts of the disk . how can we infer the sun s azimuth with respect to the galactocentric line passing through the intersection of the solar circle and a concave spiral arm ? to find out we plot pbms of simulation runs with the same pattern speed and different solar phase angle . figure [ fig : om0.7 ] shows such plots for a fixed @xmath52 and a solar phase angle changing from top to bottom in the range @xmath73 $ ] . similarly to figure [ fig : phi20 ] we now look for strong features in the three observables that can be used to estimate @xmath74 : ( 1 ) inspection of the left column of figure [ fig : om0.7 ] reveals a negative velocity stream which changes position with a change of phase angle . for @xmath75 ( top left ) this feature is centered on @xmath76 at a heliocentric distance of @xmath77 . as the angle is increased this steam moves to larger longitudes roughly preserving its distance from the sun . note that the high positive and negative features discussed in the context of @xmath50 in figure [ fig : phi20 ] , do not vary as the angle is changed since the pattern speed is kept fixed . ( 2 ) the line - of - sight velocity dispersion ( middle column of figure [ fig : phi20 ] ) does not seem to be particularly useful for constraining the solar phase angle . ( 3 ) finally , the structure in the number density pbms shows prominent variation with the change in solar angle . for example pencil - beam observations at @xmath78 would be drastically different depending on the phase angle . in an actual survey we would first try to infer the position of the inner or outer lr as discussed in section [ sec : om ] and thus find the pattern speed . so far we have only discussed simulations involving a two - armed spiral density wave perturbation . in this section we show the effect of a four - armed structure , make comparison with the two - armed case and suggest a way to distinguish between the two . figure [ fig : phi20m4 ] shows pbms of a galactic disk , similarly to figure [ fig : phi20 ] , but perturbed by a four - armed spiral structure . as in figure [ fig : phi20 ] pattern speed changed from top to bottom in the range @xmath79\omega_0 $ ] and the solar orientation with respect to a concave arm is kept fixed at @xmath54 . in this case the first order resonances are the 4:1 ilr / olr which occur at @xmath80 , respectively . as expected , more structure is apparent in all pbms in the case of the four - armed structure . note that the hot rings in @xmath81 present in the case of the two - armed structure ( figure [ fig : phi20 ] ) are also apparent in the four - armed case although not as pronounced . the reason for this is the fact that the 2:1 ilr which causes these is a second order when @xmath82 . these hot rings can be used in both the two- and four - armed cases to estimate @xmath17 but are expected to be much stronger for @xmath19 . inspecting @xmath83 and @xmath84 ( left and right columns ) in figure [ fig : phi20 ] and [ fig : phi20m4 ] it is clear that pencil - beam observations along galactic longitudes @xmath61 or @xmath85 , for example , can unambiguously distinguish between @xmath19 and @xmath82 spiral structure , as the oscillation frequency doubles when @xmath82 . upcoming galactic disk surveys will reveal the age , composition and phase space distribution of stars within the various galactic components . these stellar excavations will provide essential clues for understanding the structure , formation and evolution of our galaxy . to facilitate the interpretation of the huge amounts of data resulting from these surveys , galactic disk models , such as the one presented here , are needed to interpret the observations . we have investigated how the milky was spiral structure parameters , such as pattern speed and solar phase angle , can be estimated in a deep all - sky survey . we performed a series of test - particle simulations of a warm galactic disk approximating the disk kinematics of the milky way . we considered both two- and four - armed spiral structure and suggested a way to distinguish between the two using velocity and number density maps . we found that the axisymmetric potential needs to be known to @xmath86 , line - of - sight velocities to @xmath1 km / s , and distance uncertainties need to be less than @xmath62 . the mean line - of - sight velocity and the velocity dispersion are affected by up to @xmath0 km / s which is well within the detectable limit for forthcoming radial velocity surveys . pattern speed can be constrained by a hot ring at the 2:1 ilr in both two- and four - armed spiral structure . to distinguish between the two , however , we also need information related to the velocities and stellar number density . if the pattern speed is such that the 2:1 ilr is hidden inside the galactic bulge the 2:1 olr would be present in the outer galaxy and thus can equivalently be used to estimate the pattern speed . once the pattern speed is known the solar angle can be estimated from the number density variation with heliocentric distance ; @xmath74 is also reflected in the @xmath50 pbms . future work needs to address the issue of how to obtain the axisymmetric background potential needed to subtract from the observational data as discussed at the end of section [ sec : pbm ] . also , it is important to know what type of tracer stars are needed that would allow the estimation of photometric parallaxes with errors less than @xmath62 , and the distribution of those stars . while here we only considered steady state spiral stricture , other theories of spiral structure , such as transient and swing - amplified spirals , need to be investigated as well . it has also been suggested that the galaxy harbors two sets of spiral structure moving at the same @xcite or different @xcite pattern speeds . we expect in all those cases it will be again resonant features to relate to the pattern speed and solar angle . in the case of non - steady state spirals , however , the structure in the pbms will vary with integration time and interpretation will become more complicated . we refer these cases to a future study . it is also known that the milky way is a barred galaxy . the simulations performed here do not include the influence of the bar . this is not necessarily a shortcoming since most of the features in the pbms we use to infer spiral structure parameters are caused by resonances and , unless a resonance overlap with the central bar exists in the same location , those would not be different when a bar is included in the simulations . future work should also look at this problem . support for this work was in part provided by national science foundation grant asst-0406823 , and the national aeronautics and space administration under grant no . nng04gm12 g issued through the origins of solar systems program .

we investigate the effect of spiral structure on the galactic disk as viewed by pencil beams centered on the sun , relevant to upcoming surveys such as argos , segue , and gaia . we create synthetic galactic maps which we call pencil beam maps ( pbms ) of the following observables : line - of - sight velocities , the corresponding velocity dispersion , and the stellar number density that are functions of distance from the observer . we show that such maps can be used to infer spiral structure parameters , such as pattern speed , solar phase angle , and number of arms . the mean line - of - sight velocity and velocity dispersion are affected by up to @xmath0 km / s which is well within the detectable limit for forthcoming radial velocity surveys . one can measure the pattern speed by searching for imprints of resonances . in the case of a two - armed spiral structure it can be inferred from the radius of a high velocity dispersion ring situated at the 2:1 ilr . this information , however , must be combined with information related to the velocities and stellar number density in order to distinguish from a four - armed structure . if the pattern speed is such that the 2:1 ilr is hidden inside the galactic bulge the 2:1 olr will be present in the outer galaxy and thus can equivalently be used to estimate the pattern speed . once the pattern speed is known the solar angle can be estimated from the line - of - sight velocities and the number density pbms . forthcoming radial velocity surveys are likely to provide powerful constraints of the structure of the milky way disk .

consider the following piecewise smooth ( pws ) system : @xmath3 for @xmath4 $ ] , and where , for @xmath5 , @xmath6 are open , disjoint and connected sets , and @xmath7 . system is subject to initial condition @xmath8 , prescribed in one of the regions @xmath9 s . in , for any @xmath5 , each @xmath10 is smooth in @xmath9 , so that there is a classical solution in each region @xmath9 , but the solution is not properly defined on the boundaries of these regions . we assume that these regions are separated ( locally ) by an implicitely defined smooth manifold @xmath0 of co - dimension @xmath11 . that is , we have @xmath12 and for all @xmath13 : @xmath14 , @xmath15 , @xmath16 , @xmath17 , and @xmath18 , are linearly independent on ( and in a neighborhood of ) @xmath0 . it is useful to think of @xmath19 , where @xmath20 , and @xmath21 , are co - dimension 1 manifolds . finally , without loss of generality we will henceforth use the following labeling of the four regions @xmath9 , @xmath5 : @xmath22 for later use , we will also adopt the notation @xmath23 to denote the set of points @xmath24 or @xmath25 , for which we also have @xmath26 or @xmath27 . e.g. , @xmath28 . finally , we will denote with @xmath29 the projections of the vector fields in the normal directions to the manifolds . for , a classical solution in general can not exist on the boundaries of the given regions , and several concepts of generalized solution have been proposed during the years ( see @xcite for a beautiful exposition on different solutions concepts ) . we will restrict attention to filippov solutions , @xcite , consisting of absolutely continuous functions whose derivative is in the convex hull of the neighboring vector fields almost everywhere . in the case of one single discontinuity manifold of co - dimension 1 , filippov methodology has provided a widely accepted mathematical framework to understand motion on the discontinuity surface . consider a discontinuity manifold @xmath30 , separating two regions @xmath31 ( where @xmath32 ) and @xmath33 ( where @xmath34 ) , with respective vector fields @xmath35 and @xmath36 . assuming that @xmath0 is attractive , a condition that is satisfied when @xmath37 then _ sliding motion _ on @xmath0 takes place with vector field @xmath38 filippov theory provides also _ first order exit conditions _ : if one of @xmath35 or @xmath36 ( but not both ) become tangent to @xmath0 , then @xmath39 or @xmath40 , @xmath0 loses attractivity , and the solution trajectory generically will leave @xmath0 tangentially ( and smoothly ) to enter in @xmath31 with vector field @xmath35 , or in @xmath33 with vector field @xmath36 . furthermore , it has been understood for a long time that the limiting behavior of the iterates obtained with euler method near @xmath0 leads to the selection of the filippov sliding vector field itself ( e.g. , see @xcite , chapter 3 , section 1.1 ) whenever @xmath0 is attractive , and that the euler iterates leave a neighborhood of @xmath0 when @xmath0 loses attractivity . however , when @xmath0 is of the form , an obvious lack of uniqueness ( in general ) arises in the construction of a filippov vector field sliding on @xmath0 . in fact , on @xmath0 , filippov methodology now leads to the requirement that the vector field satisfies the following ( for positive values of @xmath41 ) : @xmath42 which is clearly an underdetermined system of equations . indeed , even when @xmath0 is attractive , in general has a one parameter family of solutions and hence of possible filippov sliding vector fields . @xmath43 * characterization of attractivity of @xmath0 in the present co - dimension 2 case is considerably more elaborate than in the case of co - dimension 1 . we will assume that @xmath0 is either _ attractive by subsliding _ ( see @xcite ) or _ attractive by spiralling _ ( see @xcite ) , in the form recalled below in definition [ attrsigma ] . * in the present case , the general lack of uniqueness is not resolved by considering the limiting behavior of the euler iterates near @xmath0 . in fact , as already noted in @xcite , the limit of the euler iterates selects one specific element of the one - parameter family of filippov solutions . [ attrsigma ] @xmath0 ( or a portion of it ) is _ attractive upon sliding _ if it is reached in finite time by solution trajectories for any given nearby initial condition , and further there is sliding motion towards @xmath0 along at least one of the @xmath44 . when there is sliding motion towards @xmath0 along all of the @xmath23 s then we say that @xmath0 is _ nodally attractive_. @xmath0 ( or portion of it ) is _ attractive by spiralling _ if @xmath0 is reached in finite time by trajectories for any given nearby initial condition , and there is clockwise or counter clockwise motion around @xmath0 ( and no sliding on @xmath23 ) for the functions @xmath45 and @xmath46 . when @xmath0 is attractive , in the literature there have been at least two systematic proposals to select the coefficients @xmath47 s in , leading to sliding vector fields of flilippov type on @xmath0 : the _ bilinear _ and the _ moments _ vector fields . the former has been extensively studied , see @xcite for example , and it consists in choosing the sliding vector field in the form @xmath48+\alpha[(1-\beta)f_3+\beta f_4]\ , \\ ( \alpha,\beta ) : & \ , \begin{bmatrix } w^1_1 & w^1_2 & w^1_3 & w^1_4 \\ w^2_1 & w^2_2 & w^2_3 & w^2_4 \end{bmatrix } \begin{bmatrix } ( 1-\alpha)(1-\beta ) \\ ( 1-\alpha)\beta \\ \alpha(1-\beta ) \\ \alpha\beta \end{bmatrix } = \begin{bmatrix } 0 \\ 0 \end{bmatrix}\ . \end{split}\ ] ] the moments method has been recently introduced in @xcite and consists in solving the linear system in , by appending to it the extra relation @xmath49 the two choices above generally lead to distinct solution trajectories , a chief difference between them being the behavior of the bilinear and moments trajectories at so - called _ generic tangential exit points_. these were introduced in @xcite , and are values of @xmath13 , where one ( and just one ) of the sliding vector fields on @xmath50 or @xmath51 is itself tangent to @xmath0 ( _ exit vector field _ ) . the moments vector field automatically aligns with the exit vector field , whereas the bilinear vector field does not . an important question , and one which we will try to answer in this work is : what should happen to trajectories of in a neighborhood of @xmath0 , when @xmath0 loses attractivity ( according to generic first order exit conditions ) ? if _ natura non facit saltum _ , then is either a description of an innatural system or a wrong model . probably , whenever it is set forth , it is neither innatural nor wrong , though it may be a little of both things , perhaps because the model should be complemented by some missing information ( the 20 years old exposition of seidman in @xcite is still worthwhile reading ) . regardless , the above 18th century motto suggests considering a regularized version of , by replacing it with a smooth differential system . of course , we must assume that we do not have knowledge of where comes from , if from anywhere at all , otherwise we should surely use this knowledge . however , in the absence of further insight , it is not obvious how one should globally regularize the system , and several possibilities for globally regularizing the system have been proposed in the literature . arguably , the most studied regularization techniques are what we may call the `` singular perturbation '' and `` sigmoid blending '' techniques . the paper @xcite was the first seminal work on the singular perturbation approach , followed by the more recent works @xcite , as well as @xcite , @xcite , @xcite and , in the context of gene regulatory networks , @xcite . the first systematic exposition of blending was the beautiful work @xcite . but , in the end , these techniques are all rather similar and amount to a regularization of the bilinear vector field . this can be done locally , just in a neighborhood of @xmath0 , say using a cutoff function ( as we do in section [ br_section ] ) , or more globally , perhaps through use of hyperbolic @xmath52 functions to connect different vector fields . in this work , we are exclusively concerned with local regularizations , hence those that alter the given problem only in a neighborhood of @xmath0 . in this case , the above mentioned regularization proposals share some common traits , the most important ones being that , outside of a neighborhood of @xmath0 , the regularized vector field effectively reduces to the original vector fields , and that the regularization depends on a small parameter ( or several small parameters ) in such a way that as the parameter(s ) go to @xmath53 , the neigborhood collapses onto @xmath0 . the first fact is surely a reasonable property , since , away from @xmath0 , there is a well defined smooth vector field depending on where the trajectory is , be it one of the original @xmath54 or one of the filippov sliding vector fields on the surfaces @xmath55 . the second fact may be a bit more controversial , since the regularized trajectory will typically select a specific sliding motion on @xmath0 ( when @xmath0 is attractive ) , as the neighborhood collapses onto @xmath0 ; however , as it is well understood , and as we will also see , different regularizations do behave differently . as noted by utkin ( @xcite ) , given that in a neighborhood of @xmath0 there are non - unique dynamics , the inherited dynamics on @xmath0 ( sliding motion ) does depend on the choice of regularization . but , this being the case , is there an appropriate way to evaluate different regularizations ? to answer this question , we must first decide how we should evaluate dynamics . the first observation is that when we evaluate the dynamics of we should distinguish between the two cases ( i ) and ( ii ) below . * the pws smooth system is just a `` convenient '' formalism : there is a `` true '' smooth system , defined globally , but it is simpler to replace it by a pws one . for example , this is the case for problems arising in gene regulatory networks ( @xcite , @xcite ) . in these cases , one knows what is the desired behavior in a neighborhood of @xmath0 : it is the behavior of the original problem ! however , one must be careful in replacing the true system by , since it is not clear that the dynamics of the purely pws system reflect the dynamics of the original problem ; this was already noted in works such as @xcite and @xcite relative to a discontinuity surface of co - dimension 1 . but , does nt this discrepancy mean that representing the original problem as a pws one was not an appropriate modeling simplification in the first place ? * the problem arises as piecewise smooth problem , or we do not have sufficient knowledge of an underlying `` true '' problem ( if any ) ; for example , in bang - bang control , or in dry friction models . in our opinion , in these cases when there is no ( knowledge of an ) underlying `` true '' dynamics that one is trying to recover , the choices we make must be consistent with the pws formulation . this is the case in which we are interested . the above consideration ( ii ) leads us to restrict to the model as * the system * we are given and it is this system that we will study . this realization motivates us ( and it has motivated us for several years ) to look at the global properties of the discontinuity surface , and to decide on what is appropriate based on whether or not the surface attracts the dynamics of the pws system for initial conditions off the discontinuity surface itself . in our opinion , it is not easy to justify studying by assuming a specific form of an underlying system from where arises as some form of limiting process . in other words , whereas it is surely legitimate ( and by no means trivial ) to study the limiting behavior of a specific choice of regularized vector field defined in a neighborhood of @xmath0 , this may actually have a restricted scope of applicability compared to . as already noted by utkin ( see @xcite ) , once one has replaced with a smoothed version of it , the stability of sliding motion on @xmath0 is inherited by that of the dynamics of the regularized field . but , is this consistent with the formulation ? for the given pws system , we believe that it is its dynamics near the discontinuity manifold that determines the appropriate behavior of a trajectory . this dynamics is what we will try to capture in this work . moreover , our main interest is in the case when the discontinuity manifold transitions from being attractive to not attractive for the trajectories of the original discontinuous system : in this case , will ( or should ) a trajectory ( even an ideal trajectory , sliding on the manifold ) feel this loss of attractivity , and hence leave a neighborhood of @xmath0 ? therefore , without assuming any form of idealized motion on @xmath0 , we can reformulate our main task as follows : _ `` how should we evaluate the dynamics of in a neighborhood of @xmath0 ? '' _ in principle , one may want to do this by studying the dynamics of a regularized problem , and we have already mentioned some possibilities , such as sigmoid blending and singular perturbation techniques . e.g. , see @xcite . we emphasize once more that these choices ( as noted by alexander and seidman for blending , and by teixeira et alia . for singular parturbation ) remove the ambiguity of how sliding on @xmath0 occurs , but these choices are effectively modeling assumptions , and we should ask if they render a behavior of the dynamics on @xmath0 that is consistent with that of . other possibilities have also been set forth in the literature , see @xcite for further references . * euler broken line approximation . this is simple to do , and it consists in replacing by a euler method approximation with constant stepsize , call it @xmath56 . we have experimented extensively with this technique , see below . [ the eventuality , of probability 0 , that an euler iterate lands exactly on a discontinuity surface can be handled in different ways ; e.g. , by randomly selecting one of the neighboring vector fields , or by retaining the last vector field used . ] * hysteresis ( or delay ) approximation . this approach appears in @xcite for a surface of co - dimension 1 . for the case of @xmath0 of co - dimension 2 , it has been studied first in @xcite and then in @xcite , always in the case of nodally attractive @xmath0 . the idea here is that one has a region @xmath57 around @xmath0 ( called a chatterbox in @xcite ) , and uses the same vector field , say @xmath35 , not only in the region @xmath31 , but until the boundary of @xmath57 in a different region is reached ; at that point , a switch to the appropriate vector field in the new region is performed . the rationale for this approach is that one does not notice immediately that a discontinuity surface is reached , but there is a `` delay '' in appreciating this fact . * replacing with a stochastic de of ito type : @xmath58 . again we note that there is zero probability of landing exactly on the discontinuity surface(s ) . the interesting feature of this approach is that it is bound to sample different vector fields around @xmath0 . the disadvantage is that it makes quantitative predictions possible only in a statistical sense . with the exception of ( c ) , the other choices above effectively replace the original pws with another deterministic dynamical system , possibly discrete ( as in the case of euler method ) . but , unfortunately , these new systems have their own dynamics , and it is unclear whether or not these are consistent with that of . see section [ numerics_section ] for ample illustration of this fact . at the same time , each of the cases above has some distinguished features , that we will attempt to retain . our proposal to evaluate the dynamics of in a neighborhood of @xmath0 is to : _ `` consider the euler iterates with random steps , '' _ with steps uniformly distributed about a reference stepsize . in other words , we want to retain the simplicity of looking at euler iterates , but aim at retaining a certain amount of randomness in the process to avoid getting trapped by the purely euler dynamics . furthermore , in this work we will restrict to well - scaled vector fields @xmath10 , @xmath59 , with none of the @xmath60 s in exceeding @xmath40 in absolute value . the reason for this is to avoid the numerical trajectory going too far away from @xmath0 , and at the same time to attempt retaining the flavor of a hysteretical trajectory . we will complement our experiments made with the above strategy , by also using other approaches . for example , euler method with constant stepsizes , singular perturbation regularizations , and also numerical integration of performed with variable stepsize integrators . we emphasize that our goal is to reach some conclusions insofar as what should happen to a solution trajectory of in a ( small ) neighborhood of @xmath0 . we are not comparing methods , or different recipes of ideal sliding motion , but simply trying to evaluate the dynamics of in the most plausible and honest way we can think of , without superimposing on any extra modeling assumption . finally , one more caveat . our examples are all of systems in @xmath1 and @xmath2 , and the discontinuity surfaces @xmath61 and @xmath25 are planes ; this makes it easier to visualize and understand things . higher dimensional state space , and non - planar discontinuity surfaces , can surely bring new phenomena into play , but we have no reason to suspect that the basic picture that emerges in our study , with the dichotomy between attractivity and lack of attractivity of @xmath0 , will be modified substantially . the remainder of this work is structured as follows . in section [ br_section ] , we consider a prototypical regularization of the bilinear vector field , and give sufficient ( and sharp ) conditions guaranteeing that the regularized solution converges to a sliding solution on @xmath0 according to . in section [ euler_section ] , we give some details of how we implemented the above mentioned proposal , particularly of what practical criteria we adopted to detect `` exiting '' from a neighborhood of @xmath0 . in section [ numerics_section ] , we illustrate through several examples the different things that can happen , and their dependence on the adopted simulation choice . space regularizations are often employed in literature as an analytical mean to model the switching mechanism of a discontinuous system , see @xcite . typically , these regularizations are one parameter families of vector fields with different time scales in a neighborhood of the sliding surface @xmath0 , namely a slow dynamics tangent to @xmath0 and a fast one normal to @xmath0 . in what follows , we consider regularizations for filippov discontinuous systems as in , but other approaches are available in the literature for non - smooth systems that are not of filippov s type , for example control systems with nonlinear control , as in @xcite . when the regularization parameter goes to zero , the regularized solution converges to a solution of filippov s differential inclusion ( ( * ? ? ? * theorem 1 , it follows that , if @xmath0 is a codimension @xmath40 discontinuity surface , for any regularization that satisfies the assumptions of ( * ? ? ? * theorem @xmath40 , @xmath62 ) , the corresponding solution will converge to the unique sliding filippov solution on @xmath0 as the regularization parameter goes to zero . but if @xmath0 has codimension @xmath63 , then the ambiguity of filippov s selection will reflect also in the amibiguity of the limit of regularized solutions ( in other words , the limit will depend on the chosen regularization ) . our goal in this section is to study the limiting behavior of the solutions of a certain regularization , namely the bilinear interpolant , or simply bilinear , regularization below . this regularization ( or a close relative ) has often been employed and studied in the literature ; see @xcite . more specifically , we will study when the solution of the bilinear regularization converges to the solution of a particular selection of the filippov s vector field : the sliding bilinear vector field . when @xmath0 is nodally attractive , this convergence has been shown in ( * ? ? ? * theorem 5.1 ) , and -under the same assumption- in @xcite it is shown that the bilinear regularized vector field converges to the bilinear sliding vector field . in what follows , we relax the hypothesis on attractivity of @xmath0 , and simply assume that @xmath0 is _ attractive in finite time _ , i.e. all trajectories with initial conditions in a neighborhood of @xmath0 will reach @xmath0 in finite time . this can be achieved either upon sliding along @xmath61 and/or @xmath25 ( @xmath0 is attractive upon sliding ) , or spiralling around @xmath0 ( @xmath0 is spirally attractive ) ; see definition [ attrsigma ] . for later reference , and under the stated attractivity assumptions of @xmath0 , we note that the algebraic system has a unique solution @xmath64 in @xmath65 . for simplicity , we assume that @xmath0 is the intersection of the hyperplanes @xmath66 and @xmath67 . then , we consider the @xmath68-neighborhood @xmath69 of @xmath0 : @xmath70 , and two smooth ( at least @xmath71 ) monotone functions ( the functions of course depend on @xmath68 , but for notational simplicity this dependence on @xmath68 is omitted ) @xmath72 $ ] interpolating at @xmath73 , as follows : @xmath74 we call _ bilinear regularization _ the following one parameter family of vector fields @xmath75 + \\ \alpha(x_1)(\left[(1-\beta(x_2))f_3(x)+\beta(x_2)f_4(x)\right]\ . \end{split}\ ] ] to be specific , in what follows , we have taken @xmath76 \\ 0 & x_1<-\epsilon \end{matrix } \right . , \,\ \beta(x_2)=\left \ { \begin{matrix } 1 & x_2>\epsilon \\ \frac 12 + \frac{x_2}{4 \epsilon } ( 3-(\frac{x_2}\epsilon)^2 ) & x_2 \in [ -\epsilon , \epsilon ] , \\ 0 & x_2<-\epsilon \end{matrix } \right .\ ] ] but other choices of a @xmath71 monotone interpolant could be considered . clearly , a choice of two different parameters for @xmath77 and @xmath78 , respectively @xmath79 and @xmath80 , could also be considered ( e.g. , see @xcite ) , and this would be justified if , for example , it is known a priori that the trajectories of un underlying physical system approach the two surfaces @xmath61 and @xmath25 at different rates . naturally , the choice of different parameters might lead to different qualitative behavior of the corresponding solutions , as we will see in section [ numerics_section ] . [ c0-int ] often ( e.g. , see @xcite ) , a simpler @xmath81 regularization is adopted , namely @xmath82 , and similarly for @xmath83 , possibly with different @xmath84 and @xmath85 . proposition [ qinvariant_prop ] is not meaningful in this case , whereas proposition [ as_stable_prop ] holds with essentially the same proof . see also remark [ remdummy ] . our goal is to study the behavior of solutions of @xmath86 for @xmath87 , and to see when , for @xmath88 , they converge to the sliding solution on @xmath0 with vector field @xmath89 as given in . in order to do so , we split @xmath86 into fast and slow motions . for @xmath90 in @xmath69 , we can rewrite with respect to the variables @xmath91 . for the sake of more compact notation , we let @xmath92 and further @xmath93 . in these variables , the full system rewrites as @xmath94 where @xmath95 is the standard @xmath96-th unit vector in @xmath97 , @xmath98 . notice that @xmath99 and @xmath100 depend on @xmath68 and are strictly positive ( inside @xmath69 ) ; from , they are equal to @xmath101 , with @xmath98 . we refer to @xmath77 and @xmath78 as the fast variables and to @xmath102 as the slow variable . we denote the solution of as @xmath103 . now , using , in , because of monotonicity of @xmath77 and @xmath78 , we can rewrite @xmath104 as a function of @xmath77 and @xmath105 as a function of @xmath78 . from , let @xmath106 and rewrite the first of , for @xmath107 $ ] , as : @xmath108 . using vieta s substitution @xmath109 , we get the equation @xmath110 . then @xmath111 and its third roots have modulus one , so that @xmath112 is real . let @xmath113 , then for @xmath77 in [ 0,1 ] , @xmath114 is in @xmath115 $ ] , and for @xmath116 , the corresponding @xmath117 can be rewritten as @xmath118 and satisfies @xmath119 and @xmath120 , this @xmath117 is the function we are looking for . same reasoning applies for @xmath78 . then system rewrites as @xmath121 with @xmath122 . notice that the function @xmath123 is strictly positive for @xmath124 and it is @xmath53 at @xmath125 . following standard approaches for singularly perturbed systems , we set @xmath126 in and obtain the following system @xmath127 notice that solutions of are sliding solutions on @xmath0 with bilinear vector field @xmath89 as in . let @xmath128 denote the solution of @xmath129 ( recall that , under the assumptions of attractivity by sliding or by spiralling , @xmath128 is the unique solution of in @xmath130 ^ 2 $ ] ) . our goal in this section is twofold : to see if solutions of converge to solutions of as @xmath88 and to explore the behavior of solutions of in the neighborhood of generic exit points . from , we introduce the time variable @xmath131 and consider the fast system @xmath132 where the `` prime '' denotes differentiation with respect to @xmath56 , and @xmath102 is considered as a vector of parameters . notice that the solution @xmath128 of is an equilibrium of . we denote with @xmath133 the solution of with initial condition @xmath134 . in case we take different @xmath84 and @xmath85 in , we can write @xmath135 , perform the change of time variable @xmath136 , and obtain ( similarly to ) the fast system @xmath137 the following result by artstein is a stronger version of a classical result in singular perturbation theory ( ( * ? ? ? * theorem 2.1 ) ) . [ classicsingpert ] assume that * the solution @xmath128 of is continuous in @xmath102 ; * @xmath128 is a locally asymptotically stable equilibrium of the fast system ; * the initial condition @xmath138 of is such that the @xmath139-limit set of @xmath140 , is @xmath141 ; * the problem @xmath142 , @xmath143 , has a unique solution , denote it as @xmath144 . then , the solution of with initial condition @xmath138 , is such that as @xmath88 : * @xmath145 converges to @xmath146 , uniformly in time on intervals of the form @xmath147 $ ] ; * @xmath148 converges to @xmath149 uniformly in time on intervals of the form @xmath150 $ ] , @xmath151 . therefore , when conditions ( i)-(iv ) of theorem [ classicsingpert ] are verified , the solution of the regular system converges to the sliding solution with bilinear vector field . [ remdummy ] if we used ( see remark [ c0-int ] ) the @xmath81 regularization @xmath82 , @xmath152 , and let @xmath153 , then we would have obtained the system @xmath154 instead of . now , is precisely the `` dummy '' system of @xcite and @xcite . however , is not orbitally equivalent to . as we will see in proposition [ as_stable_prop ] , under appropriate conditions of attractivity of @xmath0 , @xmath128 is an asymptotically stable equilibrium for both and . however , condition @xmath155 in theorem [ classicsingpert ] , implies that the limiting behavior of the solution of , depends also on the basin of attraction of @xmath128 and this , in general , is not the same for the two systems . as an illustration of this , see example [ example_2 ] in section [ numerics_section ] . system is the system we need to study in order to understand the limiting behavior of the regularized solution in the case of the @xmath71 regularization . moreover , we note that if we had used the @xmath81 regularization with different parameters @xmath84 and @xmath85 , namely @xmath156 and @xmath157 , then we would obtain a system not orbitally equivalent to . in what follows , we first assume that @xmath0 is attractive in finite time ( upon sliding or spirally ) and we want to verify if and when ( i ) , ( ii ) and ( iii ) are satisfied . this in turn will imply that solutions of the regularized problem converge to the sliding solution on @xmath0 with vector field @xmath89 as given in . the following hold , both when @xmath0 is attractive upon sliding or spirally . * system has a unique equilibrium @xmath128 in @xmath65 . * the functions @xmath158 are smooth functions of @xmath102 . ( for @xmath0 attractive upon sliding , this is in ( * ? ? ? * theorem 8) , and the case of @xmath0 spirally attractive is analogous . ) notice that this point ( b ) implies ( iv ) of theorem [ classicsingpert ] . [ unique_eq_remark ] we emphasize that the assumption that @xmath0 is attractive in finite time is sufficient but not necessary for the uniqueness of @xmath128 ; e.g. , see @xcite . as we will see in section [ numerics_section ] , this in turn will impact the behavior of the regularized solution , that might remain close to @xmath0 even when @xmath0 is not attractive . in proposition [ qinvariant_prop ] and proposition [ as_stable_prop ] , we study the dynamics of to verify asymptotic stability of @xmath159 . [ qinvariant_prop ] the phase space of is the square @xmath160 ^ 2 $ ] . moreover , the boundary of @xmath161 , denoted as @xmath162 , is an invariant set for all values of @xmath102 . the vertices of @xmath161 are equilibria . if any of the sliding vector fields @xmath163 is well defined ( i.e. , if there is sliding on any of @xmath23 ) , the corresponding value of @xmath164 in is an equilibrium of for @xmath165 . for @xmath166 , or @xmath167 , it must be @xmath168 or @xmath169 , and hence @xmath170 or @xmath171 . then , for all @xmath102 , the boundary of @xmath161 is invariant under the flow of . it also follows that the vertices of @xmath161 are equilibria of . notice that the corresponding values of @xmath172 and @xmath173 at the equilibria are @xmath174 for @xmath175 , @xmath176 for @xmath177 , @xmath178 for @xmath179 , and @xmath180 for @xmath181 , where the @xmath60 s are defined in . in addition to the vertices of @xmath161 , other equilibria of might belong to @xmath162 , as follows . * if , for @xmath165 , there is sliding on @xmath182 , then there exists @xmath183 such that @xmath184 and @xmath185 is an equilibrium of . the vector field for @xmath186 is @xmath187 . * if , for @xmath165 , there is sliding on @xmath188 then there exists @xmath189 such that @xmath190 and @xmath191 is an equilibrium of . the vector field for @xmath192 is @xmath193 . * if , for @xmath165 , there is sliding on @xmath194 for @xmath165 , then there exists @xmath195 such that @xmath196 and @xmath197 is an equilibrium of . the vector field for @xmath198 is @xmath199 . * if , for @xmath165 , there is sliding on @xmath200 for @xmath165 , then there exists @xmath201 such that @xmath202 and @xmath203 is an equilibrium of . the vector field for @xmath204 is @xmath205 . in proposition [ as_stable_prop ] we give sufficient conditions for @xmath159 to be locally asymptotically stable . [ as_stable_prop ] assume that @xmath0 is attractive in finite time upon sliding along at least two of the @xmath206 , and that there is attractive sliding along these codimension 1 surfaces . then @xmath128 is exponentially asymptotically stable for . we prove the result for a particular configuration of vector fields in a neighborhood of @xmath0 . the proof for all the other configurations is analogous . we consider the case in figure [ cases1ps2p1_fig ] , where @xmath0 is attractive upon sliding along @xmath44 , as characterized by the signs of the @xmath60 s in table [ cases1ps2p1_table ] , where the @xmath60 s are defined in . . [ cols="^,^,^,^,^",options="header " , ] * unregularized integration . the numerical solution obtained with the stiff matlab integrator ode23s , with reltol@xmath207 , stays close to @xmath0 up to @xmath208 and then it leaves @xmath0 to enter one of the @xmath209 s . for lower values of reltol , the integrator takes more than half an hour in the time interval @xmath130 $ ] . the numerical solution obtained with ode15s and reltol = abstol=@xmath210 is very inaccurate as it is evident from the plot in figure [ ode15s_fig ] . lower tolerances do not produce better results . + [ ode15s_fig ] + . unregularized integration with ode15s and reltol = abstol=@xmath210 . ] [ example_4 ] we consider the following vector fields @xmath211 and @xmath0 is the @xmath212 plane . the circle @xmath213 ( see figure [ ambiguous_fig ] ) , divides @xmath0 in two regions . outside @xmath214 , @xmath0 is attractive upon sliding along @xmath215 and @xmath216 . for all points on @xmath214 , the vector field @xmath217 is tangent to @xmath0 and , inside @xmath214 , it points away from @xmath0 , so that @xmath0 is not attractive from inside @xmath214 . we would expect the solution of to leave @xmath0 once it reaches @xmath214 . note that there is a family of filippov sliding vector fields on @xmath0 , namely : @xmath218 , @xmath219 , @xmath220 , with @xmath221 . * random euler . the ambiguity of a filippov sliding vector field is clearly reflected in the numerical solutions computed with random euler . in figure [ ambiguous_fig ] we plot the @xmath222 and @xmath223 components of @xmath224 trajectories computed with @xmath225 , and same initial condition @xmath226 . the dotted circle in the plot is the curve @xmath214 . the two bold darker lines are two sample trajectories . the shaded region is obtained by plotting all @xmath224 trajectories . the plot suggests that the choice of random stepsizes covers the region obtained by choosing one of the possible vector fields in filippov s differential inclusion . + : plot in the plane @xmath212 of @xmath224 approximations obtained with random euler and @xmath225 , and initial condition @xmath226 . ] + for completeness , in figure [ ambiguous_one_traj_fig ] we plot in function of time the first and second component of the average trajectory computed with random euler with @xmath227 . at @xmath228 , the third and fourth components of the average trajectory are on @xmath214 and indeed the plotted solution leaves @xmath0 and starts sliding on @xmath216 at @xmath229 . first and second component of the numerical solution computed with random euler with @xmath227 . ] * regularized integration . here , the computed approximations of the regularized system move away from @xmath0 once they reach @xmath214 . we note that @xmath214 is a curve of saddle - node bifurcation values for for any parameter value @xmath79 and/or @xmath80 . on @xmath214 , @xmath230 is a double root of , while inside @xmath214 there is no solution of . in this work we have been interested in studying the behavior of solutions of piecewise smooth systems in the neighborhood of a co - dimension @xmath11 discontinuity surface @xmath0 , intersection of two co - dimension @xmath40 discontinuity surfaces . it has long been accepted that if solution trajectories can not leave @xmath0 ( @xmath0 is attractive ) , some form of sliding motion on @xmath0 should be taking place . precisely which sliding motion has been the subject of much investigation , but it has not been our concern in this paper . our chief interest in this work has been trying to understand what should happen when @xmath0 loses attractivity ( at generic first order exit points ) . to our knowledge , this type of study had not been carried out before . we took the point of view that the piecewise smooth system was the only information at our disposal , and treated this model with its own mathematical dignity . naturally , if arose as a simplified model for some other known differential system , then this original system should ultimately guide the search for appropriate dynamics near @xmath0 , and it may well be that the dynamics of this `` true '' system are not matched by those of . if this is the case , we should legitimately question the validity of the model in the first place . on the other hand , in the absence of knowledge of an underlying `` true '' system , when is the only datum we have , then we believe that we should try to modify this model so that the dynamics of the modified system match those of . to obtain information on the dynamics of , we proposed use of a simple euler method with random steps , uniformly chosen with respect to a reference , small , stepsize . our study unambiguously show that : ( i ) when @xmath0 is attractive , solution trajectories remain near @xmath0 ( thereby validating an idealized sliding motion on @xmath0 ) ; ( ii ) when @xmath0 loses attractivity , solution trajectories leave a neighborhood of @xmath0 . several other possibilities have also been considered in this work : regularization techniques , plain and simple euler method with fixed stepsize , and direct numerical integration of with sophisticated off - the - shelf solvers for differential equations . none of these options satisfactorily resolved the dynamics of , and often produced misleading behavior . ultimately , this occurred because each of these choices either superimposed its own dynamics on those of ( as euler method and regularization techniques do , further producing different behaviors depending on how the regularization is made ) , or just failed to produce reliable answers in too many cases ( this was the case with directly solving with existing software , where the outcome dramatically dependend on the solver used , or on the tolerances values , or both ) . unfortunately , our conclusions are not fully satisfactory either . our analysis tells us that is the dynamics of around @xmath0 that must be used to tell us what should happen in a neighborhood of @xmath0 , but we know of no general foolproof mean to regularize the system so that the regularized trajectory will be following the dynamics of . perhaps , and again as long as the model is appropriate , the most reliable and practically efficient way to proceed is to accept some form of idealized sliding motion on @xmath0 as long as @xmath0 is attractive , while also demanding that a sliding trajectory leaves @xmath0 when the latter loses its attractivity . the construction of appropriate sliding vector fields fulfilling these requests remains an outstanding and challenging task . , a comparison of filippov sliding vector fields in co - dimension @xmath11 . _ journal of computational and applied mathematics _ , 262 ( 2014 ) , 161 - 179 . corrigendum in _ journal of computational and applied mathematics _ , 272 ( 2014 ) , pp . 273 - 273 . , uniqueness of filippov sliding vector field on the intersection of two surfaces in @xmath1 and implications for stability of periodic orbits . , to appear jnls - d-14 - 00195.1 ( 2015 ) , doi : 10.1007/s00332 - 015 - 9265 - 6 . , the residue of model reduction . the residue of model reduction , in hybrid systems iii . verification and control , 1066 ; r. alur , t.a . henzinger , e.d . sontag , eds . 201 - 207 , springer - verlag , berlin ( 1996 ) .

we consider a piecewise smooth system in the neighborhood of a co - dimension 2 discontinuity manifold @xmath0 ( intersection of two co - dimension 1 manifolds ) . within the class of filippov solutions , if @xmath0 is attractive , one should expect solution trajectories to _ slide _ on @xmath0 . it is well known , however , that the classical filippov convexification methodology does not render a uniquely defined sliding vector field on @xmath0 . the situation is further complicated by the possibility that , regardless of how sliding on @xmath0 is taking place , during sliding motion a trajectory encounters so - called _ generic first order exit points _ , where @xmath0 ceases to be attractive . in this work , we attempt to understand what behavior one should expect of a solution trajectory near @xmath0 when @xmath0 is attractive , what to expect when @xmath0 ceases to be attractive ( at least , at generic exit points ) , and finally we also contrast and compare the behavior of some regularizations proposed in the literature , whereby the original piecewise smooth system is replaced in a neighborhood of @xmath0 by a smooth differential system . through analysis and experiments in @xmath1 and @xmath2 , we will confirm some known facts , and provide some important insight : ( i ) when @xmath0 is attractive , a solution trajectory indeed does remain near @xmath0 , viz . sliding on @xmath0 is an appropriate idealization ( of course , in general , one can not predict which sliding vector field should be selected ) ; ( ii ) when @xmath0 loses attractivity ( at first order exit conditions ) , a typical solution trajectory leaves a neighborhood of @xmath0 ; ( iii ) there is no obvious way to regularize the system so that the regularized trajectory will remain near @xmath0 as long as @xmath0 is attractive , and so that it will be leaving ( a neighborhood of ) @xmath0 when @xmath0 looses attractivity . we reach the above conclusions by considering exclusively the given piecewise smooth system , without superimposing any assumption on what kind of dynamics near @xmath0 ( or sliding motion on @xmath0 ) should have been taking place . the only datum for us is the original piecewise smooth system , and the dynamics inherited by it .

in his surprising proof @xcite , @xcite of the irrationality of @xmath4 , r. apry introduced the sequence @xmath5 which has since been referred to as the apry sequence . it was shown by i. gessel ( * ? ? ? * theorem 1 ) that , for any prime @xmath6 , these numbers satisfy the _ lucas congruences _ @xmath7 where @xmath8 is the expansion of @xmath9 in base @xmath6 . initial work of f. beukers @xcite and d. zagier @xcite , which was extended by g. almkvist , w. zudilin @xcite and s. cooper @xcite , has complemented the apry numbers with a , conjecturally finite , set of sequences , known as apry - like , which share ( or are believed to share ) many of the remarkable properties of the apry numbers , such as connections to modular forms @xcite , @xcite , @xcite or supercongruences @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . after briefly reviewing apry - like sequences in section [ sec : aperylike ] , we prove in sections [ sec : lucas ] and [ sec : lucas2 ] our main result that all of these sequences also satisfy the lucas congruences . for all but two of the sequences , we establish these congruences in section [ sec : lucas ] by extending a general approach provided by r. mcintosh @xcite . the main difficulty , however , lies in establishing these congruences for the sequence @xmath2 . for this sequence , and to a lesser extent for the sequence @xmath1 , we require a much finer analysis , which is given separately in section [ sec : lucas2 ] . in the approaches of gessel and mcintosh , binomial sums , like , are used to derive lucas congruences . other known approaches to proving lucas congruences for a sequence @xmath10 are based on expressing @xmath10 as the constant terms of powers of a laurent polynomial or as the diagonal coefficients of a multivariate algebraic function . however , neither of these approaches is known to apply , for instance , to the sequence @xmath2 . in the first approach , one seeks a laurent polynomial @xmath11 such that @xmath12 is the constant term of @xmath13 . in that case , we write @xmath14 for brevity . if the newton polyhedron of @xmath15 has the origin as its only interior integral point , the results of k. samol and d. van straten @xcite ( see also @xcite ) apply to show that @xmath10 satisfies the _ dwork congruences _ @xmath16 for all primes @xmath6 and all integers @xmath17 , @xmath18 . the case @xmath19 of these congruences is equivalent to the lucas congruences for the sequence @xmath10 . for instance , in the case of the apry numbers , we have ( * ? ? ? * remark 1.4 ) @xmath20^n,\ ] ] from which one may conclude that the apry numbers satisfy the congruences , generalizing . similarly , for the sequence @xmath2 , one may derive from the binomial sum , using g. egorychev s method of coefficients @xcite , that its @xmath9th term is given by @xmath21 , where @xmath22 however , @xmath23 is not a laurent polynomial , and it is unclear if and how one could express the sequence @xmath24 as constant terms of powers of an appropriate laurent polynomial . as a second general approach , e. rowland and r. yassawi @xcite show that lucas congruences hold for a certain class of sequences that can be represented as the diagonal taylor coefficients of @xmath25 , where @xmath26 is an integer and @xmath27 $ ] is a multivariate polynomial . again , while such representations are known for some apry - like sequences , see , for instance , @xcite , no suitable representations are available for the sequences @xmath24 or @xmath1 . it was conjectured by s. chowla , j. cowles and m. cowles @xcite and subsequently proven by i. gessel @xcite that @xmath28 the congruences show that the apry numbers are periodic modulo @xmath3 , and it was recently demonstrated by e. rowland and r. yassawi @xcite that they are not eventually periodic modulo @xmath29 , thus answering a question of gessel . the apry numbers are also periodic modulo @xmath30 ( see ) and their values modulo @xmath31 are characterized by an extension of the lucas congruences @xcite ; see also the recent generalizations @xcite of c. krattenthaler and t. mller , who characterize generalized apry numbers modulo @xmath31 . as an application of the lucas congruences established in sections [ sec : lucas ] and [ sec : lucas2 ] , we address in section [ sec : periodic ] the natural question to which extent results like are true for apry - like numbers in general . in particular , we show in theorem [ thm : az8 ] that the almkvist zudilin numbers are periodic modulo @xmath3 as well . the primes @xmath32 do not divide any apry number @xmath33 , and e. rowland and r. yassawi @xcite pose the question whether there are infinitely many such primes . while this question remains open , we offer numerical and heuristic evidence that a positive proportion of the primes , namely , about @xmath34 , do not divide any apry number . in section [ sec : primes ] , we investigate the analogous question for other apry - like numbers , and prove that cooper s sporadic sequences @xcite behave markedly differently . indeed , for any given prime @xmath6 , a fixed proportion of the last of the first @xmath6 terms of these sequences is divisible by @xmath6 . in the case of sums of powers of binomial coefficients , such a result has been proven by n. calkin @xcite . along with the apry numbers @xmath35 , defined in , r. apry also introduced the sequence @xmath36 which allowed him to ( re)prove the irrationality of @xmath37 . this sequence is the solution of the three - term recursion @xmath38 with the choice of parameters @xmath39 and initial conditions @xmath40 , @xmath41 . because we divide by @xmath42 at each step , it is not to be expected that the recursion should have an integer solution . inspired by f. beukers @xcite , d. zagier @xcite conducted a systematic search for other choices of the parameters @xmath43 for which the solution to , with initial conditions @xmath40 , @xmath41 , is integral . after normalizing , and apart from degenerate cases , he discovered four hypergeometric , four legendrian as well as six sporadic solutions . it is still open whether further solutions exist or even that there should be only finitely many solutions . the six sporadic solutions are reproduced in table [ tbl : sporadic2 ] . note that each binomial sum included in this table certifies that the corresponding sequence indeed consists of integers . similarly , the apry numbers @xmath35 , defined in , are the solution of the three - term recurrence @xmath44 with the choice of parameters @xmath45 and initial conditions @xmath40 , @xmath41 . systematic computer searches for further integer solutions have been performed by g. almkvist and w. zudilin @xcite in the case @xmath46 and , more recently , by s. cooper @xcite , who introduced the additional parameter @xmath47 . as in the case of , apart from degenerate cases , only finitely many sequences have been discovered . in the case @xmath46 , there are again six sporadic sequences , which are recorded in table [ tbl : sporadic3 ] . moreover , by general principles ( see ( * ? ? ? * eq . ( 17 ) ) ) , each of the sequences in table [ tbl : sporadic2 ] times @xmath48 is an integer solution of with @xmath49 . apart from such expected solutions , cooper also found three additional sporadic solutions , including @xmath50 , \label{eq : s18}\ ] ] for @xmath51 , with @xmath52 , as well as @xmath53 and @xmath54 , which are included in table [ tbl : sporadic3 ] . remarkably , these sequences are again connected to modular forms @xcite ( the subscript refers to the level ) and satisfy supercongruences , which are proved in @xcite . indeed , it was the corresponding modular forms and ramanujan - type series for @xmath55 that led cooper to study these sequences , and the binomial expressions for @xmath53 and @xmath1 were found subsequently by zudilin ( sequence @xmath54 was well - known before ) . it is a well - known and beautiful classical result of lucas @xcite that the binomial coefficients satisfy the congruences @xmath56 where @xmath6 is a prime and @xmath57 , respectively @xmath58 , are the @xmath6-adic digits of @xmath9 and @xmath59 . that is , @xmath8 and @xmath60 are the expansions of @xmath9 and @xmath59 in base @xmath6 . correspondingly , a sequence @xmath61 is said to satisfy _ lucas congruences _ , if the congruences @xmath62 hold for all primes @xmath6 . it was shown by i. gessel ( * ? ? ? * theorem 1 ) that the apry numbers @xmath33 , defined in , satisfy lucas congruences . e. deutsch and b. sagan ( * ? ? ? * theorem 5.9 ) show that the lucas congruences in fact hold for the family of generalized apry sequences @xmath63 with @xmath64 and @xmath65 positive integers . this family includes the sequences ( a ) , ( b ) from table [ tbl : sporadic2 ] , and the sequences ( @xmath66 ) , @xmath54 from table [ tbl : sporadic3 ] . the purpose of this section and section [ sec : lucas2 ] is to show that , in fact , all the apry - like sequences in tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] satisfy the lucas congruences . using and extending the general framework provided by r. mcintosh ( * ? ? ? * theorem 6 ) , which we review below , we are able to prove this claim for all of the sequences in the two tables , with the exception of the two sequences @xmath24 and @xmath1 , for which we require a much finer analysis , which is given in section [ sec : lucas2 ] . [ thm : lucas]each of the sequences from tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] satisfies the lucas congruences . the lucas congruences , in general , do not extend to prime powers . however , it is shown in @xcite , and generalized in @xcite , that the lucas congruences modulo @xmath30 for the apry numbers extend to hold modulo @xmath31 . on the other hand , numerical evidence suggests that all the apry - like sequences from tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] in fact satisfy the dwork congruences . while theorem [ thm : lucas ] proves the case @xmath19 of these congruences , it would be desirable to establish the corresponding congruences modulo higher powers of primes . following @xcite , we say that a function @xmath67 has the _ double lucas property _ ( * dlp * ) if @xmath68 , for @xmath69 , and if @xmath70 for every prime @xmath6 . here , as in , @xmath57 and @xmath58 are the @xmath6-adic digits of @xmath9 and @xmath59 , respectively . equation shows that the binomial coefficients @xmath71 are a * dlp * function . more generally , it is shown in ( * ? ? ? * theorem 6 ) that , for positive integers @xmath72 , @xmath73 is a * dlp * function . for instance , choosing the exponents as @xmath74 , we find that the multinomial coefficient @xmath75 is a * dlp * function for any integer @xmath76 . suppose that @xmath77 is a * dlp * function and that @xmath78 and @xmath79 are * lp * functions , that is , the sequences @xmath78 and @xmath80 satisfy the lucas congruences . then , as shown in ( * ? ? ? * theorem 5 ) , @xmath81 is an * lp * function . note that and combined are already sufficient to prove that the generalized apry sequences , defined in , satisfy lucas congruences . in order to apply this machinery more generally , and prove theorem [ thm : lucas ] , our next results extend the repertoire of * dlp * functions . in fact , it turns out that we need a natural extension of the lucas property to the case of three variables . we say that a function @xmath82 has the _ triple lucas property _ ( * tlp * ) if @xmath83 , for @xmath84 , and if @xmath85 for every prime @xmath6 , where @xmath57 , @xmath58 and @xmath86 are the @xmath6-adic digits of @xmath9 , @xmath59 and @xmath87 , respectively . it is straightforward to prove the following analog of for * tlp * functions . [ lem : tlp : sum]if @xmath88 is a * tlp * function , then @xmath89 satisfies the double lucas congruences . in particular , if @xmath90 , for @xmath69 , then @xmath77 is a * dlp * function . let @xmath6 be a prime . it is enough to show that , for any nonnegative integers @xmath91 such that @xmath92 and @xmath93 , @xmath94 since the sum defining @xmath77 is naturally supported on @xmath95 , we may extend it over all @xmath96 . modulo @xmath6 , we have @xmath97 which is what we had to prove . [ lem : tlp1]the function @xmath98 is a * tlp * function . clearly , @xmath83 , for @xmath84 . in order to show that @xmath99 is a * tlp * function , we therefore need to show that , for any prime @xmath6 , @xmath100 provided that @xmath101 and @xmath102 . observe that in the case @xmath103 both sides of the congruence vanish because of the lucas congruences for the binomial coefficients . we may therefore proceed under the assumption that @xmath104 . writing @xmath105 f ( x)$ ] for the coefficient of @xmath106 in the polynomial @xmath107 , we begin with the simple observation that @xmath108 ( 1 + x)^{k + j } .\ ] ] modulo @xmath6 , we have @xmath109 since @xmath110 , extracting the coefficient of @xmath111 from this product results in the congruence @xmath112 note that , under our assumption that @xmath104 , the second term on the right - hand side of this congruence vanishes ( since @xmath113 ) . this , along with , proves . [ cor : dlp1]the function @xmath114 is a * dlp * function . set @xmath115 in lemma [ lem : tlp1 ] . [ lem : dlp2]the function @xmath116 is a * dlp * function . let @xmath6 be a prime . as usual , we write @xmath117 and @xmath118 where @xmath119 and @xmath120 . in light of and , the simple observation @xmath121 demonstrates that the sequence of central binomial coefficients is an * lp * function . we claim that @xmath122 is an * lp * function as well . from the lucas congruences for the central binomials , that is @xmath123 we observe that @xmath124 is divisible by @xmath6 if @xmath125 . hence , we only need to show the congruences @xmath126 under the assumption that @xmath127 . note that @xmath128 ( 1 + x)^{3 k}\\ & \equiv & [ x^{k_0 } ( x^p)^{k ' } ] ( 1 + x)^{3 k_0 } ( 1 + x^p)^{3 k ' } \pmod{p}\\ & = & \binom{3 k_0}{k_0 } \binom{3 k'}{k ' } + \binom{3 k_0}{k_0 + p } \binom{3 k'}{k ' - 1 } + \binom{3 k_0}{k_0 + 2 p } \binom{3 k'}{k ' - 2 } . \end{aligned}\ ] ] in the case @xmath127 , we have @xmath129 , so that the last two terms on the right - hand side vanish . this proves . next , we claim that @xmath130 by congruence , both sides vanish modulo @xmath6 if @xmath131 . on the other hand , if @xmath132 , then the usual argument shows that @xmath133 ( 1 + x)^{n_0 } ( 1 + x^p)^{n ' } = \binom{n_0}{3 k_0 } \binom{n'}{3 k ' } \pmod{p } .\ ] ] in combination with , this proves . finally , the congruences @xmath134 , that is @xmath135 follow from fermat s little theorem and the fact that both sides vanish if @xmath136 or @xmath137 . we are now in a comfortable position to prove theorem [ thm : lucas ] for all but two of the sporadic apry - like sequences . to show that sequences @xmath2 and @xmath1 satisfy lucas congruences as well requires considerable additional effort , and the corresponding proofs are given in section [ sec : lucas2 ] . recall from that the sequence of central binomial coefficients is an * lp * function . further armed with as well as corollary [ cor : dlp1 ] and lemma [ lem : dlp2 ] , the claimed lucas congruences for the sequences @xmath138 , @xmath139 , @xmath140 , @xmath141 , @xmath142 , @xmath143 , @xmath144 , @xmath145 , @xmath54 , @xmath53 follow from . it remains to consider the sequences @xmath146 , @xmath147 , @xmath148 as well as @xmath24 and @xmath1 . sequence @xmath146 can be written as @xmath149 where @xmath150 are the franel numbers ( sequence @xmath151 ) , which we already know to be an * lp * function . as a consequence of fermat s little theorem , the sequence @xmath152 is an * lp * function for any integer @xmath153 . hence , equation applies to show that @xmath154 is an * lp * function . in order to see that sequence @xmath155 satisfies the lucas congruences as well , it suffices to observe that @xmath156 is almost a * dlp * function , that is , it satisfies the congruences @xmath157 but does not vanish for @xmath69 . this is enough to conclude from lemma [ lem : dlp2 ] that @xmath158 is a * dlp * function . since this is the summand of sequence @xmath147 , the desired lucas congruences again follow from . on the other hand , for sequence @xmath148 , we observe that @xmath159 satisfies the congruences @xmath160 by lemma [ lem : tlp : sum ] because the summand is a * tlp * function by lemma [ lem : tlp1 ] . hence , @xmath161 is a * dlp * function . writing sequence @xmath148 as @xmath162 the claimed congruences once more follow from . the proof of the lucas congruences in the previous section does not readily extend to the sequences @xmath24 and @xmath1 from table [ tbl : sporadic3 ] , because , in contrast to the other cases , the known binomial sums for these sequences do not have the property that their summands satisfy the double lucas property . let us first note that the binomial sums for @xmath1 and sequence @xmath2 , given in @xmath163 and table [ tbl : sporadic3 ] , can be simplified at the expense of working with binomial coefficients with negative entries . namely , we have @xmath164 and @xmath165 where , as usual , for any integer @xmath76 and any number @xmath166 , we define @xmath167 for instance , the equivalence between and is a simple consequence of the fact that , for integers @xmath168 and @xmath169 , @xmath170 for the first equality , we used that , for integers @xmath171 , @xmath172 the following result generalizes the lucas congruences for the sequence @xmath173 . [ thm : s18:x : lucas]suppose that @xmath174 is a * dlp * function with the property that @xmath175 . then , the sequence @xmath176 is an * lp * function , that is , @xmath33 satisfy the lucas congruences . let @xmath6 be a prime and let @xmath168 be an integer . write @xmath177 and @xmath118 , where @xmath178 and @xmath179 and @xmath180 are nonnegative integers . we have to show that @xmath181 in the sequel , we denote @xmath182 for @xmath183 , we have @xmath184 and @xmath185 . hence , by the usual argument , we have @xmath186 ( 1 + x)^{2 n_0 - 3 k_0 } ( 1 + x^p)^{2 n ' - 3 k ' } \pmod{p}\\ & \equiv & \binom{2 n_0 - 3 k_0}{n_0 } \binom{2 n ' - 3 k'}{n ' } \pmod{p } . \end{aligned}\ ] ] hence , we find that , when @xmath183 , @xmath187 for @xmath188 , we have @xmath189 . by the same argument as above , we find that @xmath190 and hence @xmath191 modulo @xmath6 . finally , consider the case @xmath192 and @xmath193 . in that case , @xmath194 or , equivalently , @xmath195 . hence , we have , modulo @xmath6 , @xmath186 ( 1 + x)^{2 n_0 - 3 k_0 + p } ( 1 + x^p)^{2 n ' - 3 k ' - 1 } \nonumber\\ & \equiv & \binom{2 n_0 - 3 k_0 + p}{n_0 } \binom{2 n ' - 3 k ' - 1}{n ' } \nonumber\\ & \equiv & \binom{2 n_0 - 3 k_0}{n_0 } \binom{2 n ' - 3 k ' - 1}{n ' } , \label{eq : bin23:3 } \end{aligned}\ ] ] because , for any integers @xmath196 and @xmath197 such that @xmath198 , @xmath199 set @xmath200 . applying to the second binomial factor in , we find that @xmath201 in combination with the assumed symmetry of @xmath174 , we therefore have that , when @xmath192 and @xmath202 , @xmath203 we are now ready to combine all cases . first , suppose that @xmath192 . noting that @xmath204 implies @xmath205 , and using , and , we conclude that , modulo @xmath6 , @xmath206 \sum_{k ' = 0}^{n ' } c ( n ' , k')\\ & = a ( n_0 ) a ( n ' ) , \end{aligned}\ ] ] which is what we wanted to prove . the case @xmath207 is simpler , and we only have to use to again conclude that holds . [ cor : lucas : s18]the sequence @xmath173 satisfies the lucas congruences . recall from the discussion in section [ sec : lucas ] that @xmath208 is a * dlp * function . obviously , @xmath175 . hence , theorem [ thm : s18:x : lucas ] applies to show that @xmath173 , in the form satisfies the lucas congruences . next , we prove that the sequence @xmath2 , which corresponds to the choice @xmath209 in theorem [ thm : lucas : eta ] , satisfies lucas congruences as well . [ thm : lucas : eta]let @xmath210 . then , the sequence @xmath211 is an * lp * function , that is , @xmath33 satisfy the lucas congruences . let @xmath6 be a prime and let @xmath168 be an integer . as in the proof of theorem [ thm : s18:x : lucas ] , we write @xmath117 and @xmath212 , where @xmath178 and @xmath120 and @xmath213 are nonnegative integers . again , we have to show that @xmath214 throughout the proof , let @xmath215 . if @xmath216 , then @xmath217 and @xmath218 . since @xmath219 , we thus have @xmath220 and @xmath221 . therefore , modulo @xmath6 , @xmath222 ( 1 + x)^{4 n_0 - 5 k_0 - d p } ( 1 + x^p)^{4 n ' - 5 k ' + d}\\ & \equiv & \binom{4 n_0 - 5 k_0 - d p}{3 n_0 - d p } \binom{4 n ' - 5 k ' + d}{3 n ' + d}\\ & \equiv & \binom{4 n_0 - 5 k_0}{3 n_0 } \binom{4 n ' - 5 k ' + d}{3 n ' + d } , \end{aligned}\ ] ] where in the last step we used that , modulo @xmath6 , @xmath223 which follows from because @xmath224 . in particular , we have @xmath225 and we observe that , for @xmath226 , @xmath227 to see this , note that the the sum of the @xmath59-th and @xmath228-th term does not depend on the value of @xmath226 . indeed , using , pascal s relation and again , we deduce that @xmath229 + \left [ \binom{4 n - 5 k}{3 n + 1 } - \binom{4 n - 5 k - 1}{3 n + 1 } \right]\\ & = & \binom{4 n - 5 k}{3 n } + \binom{4 n - 5 k - 1}{3 n}\\ & = & \binom{4 n - 5 k}{3 n } + ( - 1)^n \binom{4 n - 5 ( n - k)}{3 n } . \end{aligned}\ ] ] next , suppose that @xmath192 and @xmath230 . in that case , @xmath231 or , equivalently , @xmath232 . hence , we have , modulo @xmath6 , @xmath222 ( 1 + x)^{4 n_0 - 5 k_0 + p } ( 1 + x^p)^{4 n ' - 5 k ' - 1}\\ & \equiv & \binom{4 n_0 - 5 k_0 + p}{3 n_0 - d p } \binom{4 n ' - 5 k ' - 1}{3 n ' + d } . \end{aligned}\ ] ] we rewrite the first binomial factor as follows , applying first and then twice , to find that , with @xmath233 , modulo @xmath6 , @xmath234 here , we proceeded under the assumption that @xmath235 . it is straightforward to check that the final congruence also holds when @xmath236 , because then the binomial coefficients vanish modulo @xmath6 . we conclude that , when @xmath192 and @xmath230 , @xmath237 in particular , we have @xmath238 and we observe that , for integers @xmath239 , @xmath240 because , by , @xmath241 therefore , we can combine and into @xmath242 which holds for all @xmath119 ( recall from the discussion at the beginning of this section that @xmath243 , like sequence @xmath2 , can be represented as in table [ tbl : sporadic3 ] ) . on the other hand , suppose that @xmath244 . set @xmath245 . since @xmath246 , we have @xmath247 . the usual arguments show that , modulo @xmath6 , @xmath222 ( 1 + x)^{4 n_0 - 5 k_0 - f p } ( 1 + x^p)^{4 n ' - 5 k ' + f } \nonumber\\ & \equiv & \binom{4 n_0 - 5 k_0 - f p}{3 n_0 - d p } \binom{4 n ' - 5 k ' + f}{3 n ' + d } \nonumber\\ & \equiv & \binom{4 n_0 - 5 k_0}{3 n_0 - d p } \binom{4 n ' - 5 k ' + f}{3 n ' + d } . \label{eq : eta : bin2 } \end{aligned}\ ] ] we are now in a position to begin piecing everything together . to do so , we consider individually the cases corresponding to the value of @xmath248 . first , suppose @xmath46 or @xmath249 . congruence coupled with implies that @xmath250 to conclude the desired congruence , it therefore only remains to show that @xmath251 this is easily seen in the case @xmath46 , because then each term of this sum vanishes modulo @xmath6 . equivalently , for @xmath46 , vanishes whenever @xmath244 ( because @xmath252 ) . on the other hand , if @xmath249 , we claim that the sum vanishes modulo @xmath6 because the terms corresponding to @xmath253 and @xmath254 cancel each other . to see that , observe first that , for @xmath249 , vanishes whenever @xmath244 and @xmath255 ( because @xmath256 if @xmath257 ) . therefore , for the term corresponding to @xmath253 , @xmath258 while , for the term corresponding to @xmath254 with @xmath259 , @xmath260 where we applied for the second congruence it is now immediate to see that the sum indeed vanishes modulo @xmath6 for @xmath249 . it remains to prove the lucas congruences in the case @xmath261 . using , we have @xmath262 where @xmath263 combining this congruence with the identity @xmath264,\ ] ] which can be deduced along the same lines as , we find that @xmath265 we have , by , modulo @xmath6 , @xmath266 where the last congruence is a consequence of the identity @xmath267 ( which follows from and replacing @xmath59 with @xmath268 ) and the fact that vanishes for @xmath244 if @xmath269 . using this value of @xmath270 in , we find that the desired lucas congruence follows , if we can show that @xmath271 note that , if @xmath216 , then , by and , @xmath272 a similar argument , combined with , shows that the congruence also holds if @xmath273 . we therefore find that is equivalent to @xmath274 the next lemma proves that this congruence indeed holds provided that @xmath275 . [ lem : lucas : eta:0]let @xmath6 be a prime , and @xmath276 . then we have , for all @xmath9 such that @xmath277 , @xmath278 to prove these congruences we employ n. calkin s technique @xcite for proving similar divisibility results for sums of powers of binomials . denoting @xmath279 , we have , by and , @xmath280 clearly , @xmath281 where @xmath282 denotes the pochhammer symbol ( in particular , @xmath283 ) . likewise , @xmath284 since @xmath285 and @xmath286 are not divisible by @xmath6 , we have to show that @xmath287 since the polynomials @xmath288 form an integer basis for the space of all polynomials with integer coefficients and degree at most @xmath59 , there exist integers @xmath289 with @xmath290 so that @xmath291 then the left - hand side of becomes @xmath292 where we used @xmath293 to evaluate @xmath294 the desired congruence therefore follows if we can show that @xmath295 for all @xmath296 . since @xmath297 and @xmath298 , the numerator @xmath299 is always divisible by @xmath6 . the congruences thus follow if @xmath300 for all @xmath87 , or , equivalently , @xmath301 . since @xmath302 we have @xmath301 if and only if @xmath303 clearly , this inequality holds for all @xmath18 if and only if @xmath304 . numerical evidence suggests that the values @xmath210 in theorem [ thm : lucas : eta ] are the only choices for which the sequence satisfies lucas congruences . in light of lemma [ lem : lucas : eta:0 ] , it is natural to ask if there are additional values of @xmath153 and @xmath305 , for which the sequence @xmath306 satisfies lucas congruences . empirically , this does not appear to be the case . in particular , for @xmath307 this sequence does not satisfy lucas congruences for either @xmath308 or @xmath309 . the apry numbers satisfy @xmath310 and so are periodic modulo @xmath30 . as in the case of the congruences , which show that the apry numbers are also periodic modulo @xmath3 , the congruences were first conjectured in @xcite and then proven in @xcite . we say that a sequence @xmath12 is _ eventually periodic _ if there exists an integer @xmath311 such that @xmath312 for all sufficiently large @xmath9 . an initial numerical search suggests that each sporadic apry - like sequence listed in tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] can only be eventually periodic modulo a prime @xmath6 if @xmath313 . as an application of theorem [ thm : lucas ] , we prove this claim next . [ cor : notperiodic]none of the sequences from tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] is eventually periodic modulo @xmath6 for any prime @xmath314 . gessel @xcite shows that , if a sequence @xmath10 satisfies the lucas congruences modulo @xmath6 and is eventually periodic modulo @xmath6 , then @xmath315 modulo @xmath6 for all @xmath316 . for instance , let @xmath10 be the almkvist zudilin sequence ( @xmath317 ) . then , @xmath318 , @xmath319 and @xmath320 . suppose @xmath10 was eventually periodic modulo @xmath6 . then @xmath6 has to divide @xmath321 , which implies that @xmath322 . in table [ tbl : periodic ] we list , for each sequence , the primes dividing both @xmath323 and @xmath324 . the fact , that all these primes are at most @xmath325 , proves our claim . .[tbl : periodic]the primes dividing both @xmath323 and @xmath324 , for each sequence @xmath10 from tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] example [ eg : a:7 ] shows that the first @xmath326 values of the apry numbers modulo @xmath326 are palindromic . our next result , which was noticed by e. rowland , shows that this is true for all primes . [ lem : a : palin]for any prime @xmath6 , and integers @xmath9 such that @xmath327 , the apry numbers @xmath33 satisfy the congruence @xmath328 for @xmath9 such that @xmath329 , we employ and to arrive at @xmath330 as claimed . theorem [ thm : lucas ] and lemma [ lem : a : palin ] , considered together , suggest that @xmath331 of the primes do not divide any apry number . indeed , let us make the empirical assumption that the values @xmath33 modulo @xmath6 , for @xmath332 , are independent and uniformly random . since one of the values @xmath33 is congruent to @xmath333 modulo @xmath6 with probability @xmath334 , it follows that the probability that @xmath6 does not divide any of the @xmath335 first values is @xmath336 by the lucas congruences , shown in theorem [ thm : lucas ] , and lemma [ lem : a : palin ] , @xmath6 does not divide any of the @xmath335 first values if and only if @xmath6 does not divide any apry number . in the limit @xmath337 , the proportion becomes @xmath338 . observe that this empirical prediction matches the numerical data in table [ tbl : primes ] rather well . we therefore arrive at the following conjecture . [ conj : prop : apery]the proportion of primes not dividing any apry number @xmath33 is @xmath339 . proportion of primes ( up to @xmath340 ) not dividing the sequences @xmath155 , @xmath341 , @xmath144 , @xmath148 , @xmath145 , with the dotted line indicating @xmath339 . the apry sequence is plotted in blue . ( we thank arian daneshvar for producing this plot . ) ] while lemma [ lem : a : palin ] does not hold for the other apry - like numbers @xmath12 from tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] , we make the weaker observation that if a prime @xmath342 divides @xmath12 , where @xmath343 , then @xmath6 also divides @xmath344 . we expect that this empirical observation can be proven in the spirit of the proof of lemma [ lem : a : palin ] , but do not pursue this theme further . we only note that it allows us to extend the heuristic leading to conjecture [ conj : prop : apery ] to the apry - like sequences @xmath155 , @xmath341 , @xmath144 , @xmath148 from table [ tbl : sporadic3 ] . in other words , we conjecture that , for each of these sequences , the proportion of primes not dividing any of the terms is again @xmath339 . figure [ fig : primes ] visualizes some numerical evidence for this conjecture . on the other hand , for sequence @xmath24 as well as the sequences from table [ tbl : sporadic2 ] , the proportion of primes not dividing any of their terms appears to be about half of that , that is @xmath345 . to explain this extra factor of @xmath346 , we note that , for the apry - like numbers @xmath347 stienstra and beukers @xcite proved that , modulo @xmath6 , @xmath348 ( and conjectured that the congruence should hold modulo @xmath349 , which was later proved by ahlgren and ono @xcite ) . in particular , congruence makes it explicit that every prime @xmath350 divides a term of this apry - like sequence . note that , by a classical congruence of gauss , the congruences are equivalent , modulo @xmath6 , to the congruences @xmath351 which are valid for any prime @xmath352 . the more general result in @xcite also includes the cases @xmath353 and @xmath354 . similar divisibility results appear to hold for the other apry - like numbers from table [ tbl : sporadic2 ] , and it would be interesting to make these explicit . on the other hand , the extra factor of @xmath346 in case of sequence @xmath24 is explained by the following congruences , which resemble remarkably well . for any prime @xmath355 , we have that , modulo @xmath6 , @xmath356 suppose that @xmath357 , and write @xmath358 . the congruence can be checked directly for @xmath359 and @xmath360 , and so we may assume @xmath342 in the sequel . applying to the definition of sequence @xmath24 in table [ tbl : sporadic3 ] , we have @xmath361 since @xmath362 and @xmath363 , the term @xmath364 is always divisible by @xmath6 , unless @xmath365 ( because , otherwise , one of the @xmath366 factors of @xmath367 is divisible by @xmath6 , while @xmath368 is not ) . note that @xmath369 and @xmath370 are equivalent to @xmath371 and @xmath372 , respectively . however , @xmath373 can not be an integer ( since @xmath374 ) . we thus find that vanishes modulo @xmath6 unless @xmath375 and @xmath376 , in which case is congruent to @xmath377 modulo @xmath6 . combined with the analogous discussion for the corresponding second term in , we conclude that @xmath378 applying this to the sum and combining the signs properly , we arrive at the congruences when @xmath379 . the case @xmath380 is similar and a little bit simpler . in summary , we conjecture that the proportion of primes not dividing any term of the apry - like sequences in tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] is as follows . [ conj : prop : aperyx]@xmath381 * let @xmath12 be one of the sequences of table [ tbl : sporadic2 ] or sequence @xmath24 . then the proportion of primes not dividing any @xmath382 is @xmath383 . * let @xmath12 be one of the sequences @xmath155 , @xmath341 , @xmath384 , @xmath148 , @xmath145 from table [ tbl : sporadic3 ] . then the proportion of primes not dividing any @xmath12 is @xmath339 . in stark contrast , cooper s sporadic sequences @xmath53 , @xmath54 , @xmath1 from table [ tbl : sporadic3 ] are divisible by all primes . indeed , let @xmath12 denote any of these three sequences . then , @xmath385 for all primes @xmath6 . in fact , we can prove much more . for any given prime @xmath6 , the last quarter ( or third ) of the first @xmath6 terms of these sequences are divisible by @xmath6 . in the case of sequence @xmath54 , the sum of fourth powers of binomial coefficients , this is proved by n. calkin @xcite . indeed , among other divisibility results on sums of powers of binomials , calkin shows that , for all integers @xmath386 , the sums @xmath387 are divisible by all primes @xmath6 in the range @xmath388 in particular , in the case @xmath307 , we conclude that @xmath389 is divisible by all primes @xmath6 that satisfy @xmath390 . equivalently , we have @xmath391 whenever @xmath392 . our final result proves the same phenomenon for cooper s sporadic sequences @xmath393 . we note that in each case , empirically , the bounds on @xmath87 can not be improved ( with the expection of the case @xmath394 for @xmath1 ; see remark [ rk : s18:mod3 ] ) . [ thm : cc]for any prime @xmath6 , we have @xmath395 whenever @xmath396 , and @xmath397 whenever @xmath392 . for the sequence @xmath53 , we want to show @xmath398 for @xmath396 . note that for @xmath399 or @xmath400 the summand is already zero . therefore , we assume that @xmath401 . we write the summand as @xmath402 and observe that the denominator is not divisible by @xmath6 if @xmath403 . on the other hand , the factorial @xmath404 in the numerator is divisible by @xmath6 since @xmath405 where we used @xmath406 to verify the final inequality . thus , for @xmath396 , the congruences @xmath407 hold modulo @xmath6 , as claimed . we proceed similarly for @xmath408 , which is given by @xmath409 and , using , write the summand as @xmath410 none of the terms in the denominator is divisible by @xmath6 since @xmath403 . on the other hand , @xmath411 in the numerator is divisible by @xmath6 since @xmath412 where we used @xmath413 for the final inequality . therefore , for @xmath392 , each of the terms in the sum for @xmath408 is a multiple of @xmath6 , and we obtain the desired congruences . [ rk : s18:mod3]employing , we observe that @xmath414 for @xmath51 , which reaffirms corollary [ cor : periodic ] for this sequence . finally , as noted in @xcite , each of the sequences in table [ tbl : sporadic2 ] times @xmath48 is an integer solution of with @xmath49 . observe that @xmath48 is divisible by a prime @xmath6 for all @xmath9 such that @xmath415 . this results in a ( weaker ) analog of theorem [ thm : cc ] for these apry - like sequences , and implies , in particular , that these sequences are again divisible by all prime numbers . in sections [ sec : lucas ] and [ sec : lucas2 ] , we showed that all sporadic solutions of and , given in tables [ tbl : sporadic2 ] and [ tbl : sporadic3 ] , uniformly satisfy lucas congruences . however , for two of these sequences , especially sequence @xmath2 , we had to resort to a rather technical analysis . we therefore wonder if there is a representation of these sequences from which the lucas congruences can be deduced more naturally , based on , for instance the approaches of @xcite and @xcite , or @xcite . more generally , it would be desirable to have a uniform approach to these congruences , possibly directly from the shape of the defining recurrences and associated differential equations . in another direction , it would be interesting to show that , as numerical evidence suggests , _ all _ of the apry - like sequences in fact satisfy the dwork congruences . the congruences show that the apry numbers are periodic modulo @xmath3 , alternating between the values @xmath416 and @xmath325 . as a consequence , the other residue classes @xmath417 modulo @xmath3 are never attained . on the other hand , the observations in section [ sec : primes ] show that certain primes do not divide any apry number . this can be rephrased as saying that the residue class @xmath333 is not attained by the apry numbers modulo these primes . this leads us to the question of which residue classes are not attained by apry - like numbers modulo prime powers @xmath418 . in particular , are there interesting cases which are not explained by sections [ sec : periodic ] and [ sec : primes ] ? the second part of congruence makes it explicit that every prime @xmath419 divides a term of the apry - like sequence . is there a similarly explicit result which demonstrates that the apry numbers are divisible by infinitely many distinct primes ? this paper builds on experimental results obtained together with arian daneshvar , pujan dave and zhefan wang during an illinois geometry lab ( igl ) project during the fall 2014 semester at the university of illinois at urbana - champaign ( uiuc ) . the aim of the igl is to introduce undergraduate students to mathematical research . we wish to thank arian , pujan and zhefan ( at the time undergraduate students in engineering at uiuc ) for their great work . in particular , their experiments predicted corollaries [ cor : notperiodic ] and [ cor : periodic ] , and provided the data for table [ tbl : primes ] , which lead to conjecture [ conj : prop : apery ] . g. p. egorychev . , volume 59 of _ translations of mathematical monographs_. american mathematical society , providence , ri , 1984 . translated from the russian by h. h. mcfadden , translation edited by lev j. leifman . d. zagier . integral solutions of apery - like recurrences . in j. harnad and p. winternitz , editors , _ groups and symmetries . from neolithic to john mckay _ , volume 47 . american mathematical society , 2009 .

in 1982 , gessel showed that the apry numbers associated to the irrationality of @xmath0 satisfy lucas congruences . our main result is to prove corresponding congruences for all sporadic apry - like sequences . in several cases , we are able to employ approaches due to mcintosh , samol van straten and rowland yassawi to establish these congruences . however , for the sequences often labeled @xmath1 and @xmath2 we require a finer analysis . as an application , we investigate modulo which numbers these sequences are periodic . in particular , we show that the almkvist zudilin numbers are periodic modulo @xmath3 , a special property which they share with the apry numbers . we also investigate primes which do not divide any term of a given apry - like sequence .

states in manifolds with non - trivial topologies and entanglement involving such states have close implications to quantum information @xcite and quantum computing @xcite . more specifically , generation of coherent states and their superpositions ( cat states ) @xcite or and ther entanglement ( as epr states @xcite ) in intricate geometrical or topological configurations can give rise to interesting effects . recent developments of topological qubits @xcite , quantum optics in curved space - time @xcite , topological defects @xcite , graphene and topological isulators @xcite have increased the interest in systems with non - trivial topologies . here we concentrate on torus and mbius topologies . different from the known coherent state elaborations for quantum optics @xcite , in non - trivial manifolds , coherent states acquire topological properties , with consequences that affect periodicity and normalization . this is the case of non - trivial manifolds with topologies of torus @xcite and mobius strip @xcite . we remark that there is no general method for a construction of coherent states for a particle on an arbitrary manifold , specially when the space is described in a non - trivial topology @xcite . a general algorithm was introduced by perelomov @xcite for construction of coherent states for homogeneous spaces @xmath0 which are quotients @xmath1 of a lie - group manifold @xmath2 and the stability subgroup @xmath3 . unfortunately , in many interesting cases for physics as a particle on a circle , sphere or torus , the phase space whose points should label the coherent states , more precisely a cotangent bundle @xmath4 , where @xmath5 is the configuration space , is not a homogeneous space . a general theory of coherent states when the configuration space has non - trivial topology is far from complete . in view of the lack of the general method for the construction of coherent states one is forced to study each case of a particle on a concrete manifold separately . it is worth to point out that coherent states for a quantum particle on circle , sphere and torus have been introduced very recently @xcite . in such cases different constructions of the coherent states ( cs ) in the boson case are practically straightforward , and a simple addition by hand of @xmath6 to the angular momentum operator @xmath7 for the fermionic case into the corresponding cs remains obscure and non - natural . the question that naturally arises is if there exists any geometry for the phase space in which the cs construction leads precisely to a fermionic quantization condition . in a recent work @xcite , we demonstrated the positive answer to this question showing that the cs for a quantum particle on the mobius strip geometry is the natural candidate to describe fermions exactly as the cylinder geometry for bosons . in this paper , we investigate the entanglement of coherent states on a torus , mobius strip and torus - mobius , i.e. , in the case where two - particle states , one in the torus and other in mobius , are entangled in the intersection of the two manifolds . taking into account the consistent operators acting in each space and their action on states , we consider the generation of entangled states and non - orthogonal projective measurements , a generalization of von - neumann type measurement , to quantify entanglement . we also consider the topological and geometrical consequences associated to cs in torus and mobius and the relations of the toroidal operators and entangled torus states to @xmath8 . we can start with a revision of the quantum mechanics on a torus . such situation , in principle , can be identified with product of two circles , this implies , following @xcite , that we can define the algebra @xmath10&=&\delta_{ij}u_{j } , \\ \left[j_{i},u_{j}^{\dagger}\right]&= & -\delta_{ij}u_{j}^{\dagger } , \\ \left[j_{i},j_{j}\right]&= & 0 , \\ \left[u_{i},u_{j}\right ] & = & \left[u_{i}^{\dagger},u_{j}^{\dagger}\right]= \left[u_{i},u_{j}^{\dagger}\right]=0,\end{aligned}\ ] ] where @xmath11 and @xmath12 . taking @xmath13 , and @xmath14 , @xmath15 , the eigenvalue equation is written as @xmath16 where @xmath17 are ladder operators that satisfy @xmath18 where @xmath19 and @xmath20 . consequently , starting from a vector @xmath21 , and @xmath22 , @xmath23 , a hilbert space can be generated by a set of vectors @xmath24 . the antiunitary operator of time inversion @xmath25 can be also defined by means of the relations @xmath26 this operator is symmetric and the action on @xmath17 is given by @xmath27 as a consequence @xmath28 can take only four possible values @xmath29 , @xmath30 , @xmath31 and @xmath32 . the relation between @xmath33 and @xmath34 is given by @xmath35 in the coordinate representation of the states on a torus , the following eigenvalue equation is satisfied @xmath36where @xmath37 now , let us define the following operator @xmath38 that mixes operators @xmath39 and @xmath25 , @xmath40the action of this operator on @xmath33 will generate superposition states of the form @xmath41@xmath42as a consequence the even exponents of @xmath38 are eigenvectors of @xmath33 , @xmath43we have then @xmath44 we have been seen in previous references , the action of the creation and annihilation operators corresponding to the generators of the heisenberg - weyl algebra with the basic coherent states of the non - compact oscillator is given as follows @xmath46and @xmath47 , with @xmath48 the corresponding eigenvalue of the coherent state . the importance of such construction is because it describes a unitary representation of the @xmath49 group that is of infinite dimension . this representattion was used in the case of coherent states in noncompact oscilatrs and also in supergravity models and strings.this specific action of the operators @xmath50over the basic states @xmath51 is clearly isomorphous with the action of the@xmath52 operators defined on the torus making the identification@xmath53and also 2@xmath54 this operator @xmath55 make a cross over from a compact to a non compact structure of the of the geometry of the configuration space . this point is extreamely important to explain topologically and geometrically speaking the metal / superconductor - insulator transition . there exists , for instance , a clear possibility to map the sugra ( supergravity ) model @xcite into the torus due the specific action of the operators @xmath55 . a concrete adaptation of the model to metal isulator - transition and sugra - torus correspondence will be the scope of a future work @xcite . in particular , let us first propose an entangled state based on such states @xmath56 we will show that there is an explicit operator capable of realize the building of such state . here we can only analyse the following properties . the change under the action of @xmath57 , where @xmath58 has the same action of the operator @xmath59 , is the following @xmath60then , the state is invariant under @xmath57 . we can also check the action under @xmath61 and @xmath62 , @xmath63@xmath64 such that this state generate two equivalent and entangled states . the coherent states on a torus are defined from the relation @xmath65 , where @xmath66 , @xmath67 and @xmath68 . the relation to the states @xmath33 is given by @xmath69 the coherent states can be explicitly written in terms of @xmath70 , using @xmath71 , by means of @xmath72that can also be written as @xmath73 the normalization conditions of ( [ z ] ) are product of jacobi theta functions , here we consider only the case @xmath29 , that is given by @xmath74now , we define two - mode coherent states by means of the product state @xmath75where @xmath76 . we can also write this state as @xmath77 where the relations the eigenvalue relations are written as @xmath78@xmath79 and the normalization factor @xmath80the action of @xmath38 on a two - mode coherent state for operators @xmath81 and @xmath82 acting in each mode , we will have the following relations for even exponents @xmath83it follows that the two - mode coherent states from the torus are eigenstates of the even exponents of @xmath81 and @xmath82 . the coherent states can be reduced from the torus to mobius strip by means of a constraint between the angle variables @xmath84 we will describe this more appropriatelly in a section below . once we have the a mobius strip parametrization , a deformation can be applied by means of following transformation @xmath85that does not change the topological properties of the mobius strip . in this way a coherent state associated to the mobius strip @xcite is defined by means of @xmath86 , where @xmath87 . @xmath88 acts in the @xmath89 mode of @xmath90 and @xmath91 let us consider the action of the following operators @xmath92 on the states @xmath93 , where @xmath94 is are integer values , . this will be given by @xmath95 note that this state is equivalent to the state @xmath96 , discussed previously . this state is built in a consistent way from the state @xmath97 by means of the action of the operator @xmath98 . in particular , the operators @xmath59 and @xmath58 discussed previously are in fact actions of the operator @xmath99 that in appropriate conditions leave the state invariant , i.e. , in the forms @xmath62 and @xmath61 . such operators are important when we search operators that leave a bipartite entangled state invariant @xcite . now , let us consider the bipartite coherent state for a torus , @xmath100 this can be written as @xmath101 the case where @xmath102 , we can write @xmath103it is interesting to observe that the states @xmath104 and @xmath105 are not equivalents , since @xmath106 and @xmath107 , what implies that the above state is in fact a entangled state . note that the state ( [ eqlera ] ) with ( [ eq1er ] ) can be associated to the state of photons with orbital angular momentum states @xmath108 entangled , experimentally verified @xcite , described by @xmath109 where @xmath110 is the azimuthal angle . in the general case the operator @xmath111 acts in the entanglement of two - particle states with torus topology ( figure [ trajtorus2 ] and [ trajtorus21 ] ) . in this section we arise to the question about if the topology of a manifold is sufficient to give a correct description of the physical states living there . in the reference @xcite , we show the reduction of the toroidal geometry to the mobius strip due the use of suitable projection operators , starting from the torus as the original quantum phase space . a position point in a mobius strip geometry can be parameterized by means of specific points @xmath112 and @xmath113 given by @xmath114 where the coordinates of @xmath112 describe a central cylinder , generated from a invariant fiber in the middle of the strip weight , i.e. , @xmath115 this is the topological invariant of the geometry under study . here @xmath116 is the polar angle , measured from the axis z , and @xmath117 the azimuthal angle . the coordinates of @xmath113 , the boundaries of the mbius band , are of @xmath118 , the cylinder , plus @xmath119 the band has weight @xmath120 . this mbius strip configuration has a deep relation with the torus and it is a result of a reduction that changes both topology and geometry by means of a constraint @xmath121 on the torus geometry ( figure [ figtorus ] ) described by @xmath122 an important point to enphasize is that the angles are no more independent in the case of the mbius strip , leading to a special embedding from the torus . in such situation , the geometry turns to be an unnoriented surface and the topological properties are also altered . the consideration @xmath123 implies that the generated mbius strip lives inside the torus . the parametrization associated to the mbius strip ( figure [ figmobius])is then given by taking @xmath124 and inserting ( 5 ) into ( 4 ) we obtain the parametrization of the band @xmath125 taking the parametrization of the torus we can build the corresponding lagrangean@xmath126 \nonumber \\ & + & \frac{m}{2}\left ( r\overset{.}{\theta } \right ) ^{2}-mr\ \sin\theta \ \overset{.}{z}_{0}\overset{.}{\theta } \nonumber \\ & + & \frac{m}{2}\left ( \overset{.}{z}_{0}\right ) ^{2}\end{aligned}\ ] ] we can note that by inserting the contraint associated to the reduction to a mbius strip , eq . ( [ 5 ] ) , the lagrangean describes the motion on mbius strip @xmath127 . the hamiltonian for the torus is easily computed from ( @xmath128 ) @xmath129 where we have @xmath130 this will lead to the following hamiltonian @xmath131 , \nonumber\end{aligned}\ ] ] by the insertion of eq . ( [ 5 ] ) , the torus reduces to a mobius strip , by legendre transform @xmath127 , we arrive at the above with eq . ( [ 5 ] ) , the hamiltonian for mobius @xmath132 . the quantized version of both hamiltonians then describe quantum particles on a torus and mobius . such description has a geometrical base . the deformation of particles turning around on such surfaces will not alter the topological structure . the inclusion of the constraint in the torus , eq . ( [ 5 ] ) , reduces the geometrically to a mobius strip inside the torus . the deformation of the mobius leads to an intersection , but from the topological point of view the topological properties are preserved @xcite . in order to study the coherent states ( cs ) associated to the mobius strip , we analyse again the cs torus . as we saw previously , in the torus case , the coordinates @xmath133 and @xmath134 are absolutely independent . thus , we assume two cylinder type parametrizations , one for @xmath135 @xmath136 cylinder with angular variable @xmath134 and other one with finite @xmath137 @xmath138 @xmath139 where @xmath140 . the above expression correspond to a geometrical factorization , leading to a physical decomposition for @xmath141 that is useful for our proposal is the following @xmath142 @xmath143 i.e. , the mobius strip has its portion splited from the the toroidal space @xmath144 this implies in geometrical and topological changes . it is interesting to note that , although the topology of the torus is considered as the product of two cilinders , the introduction of the parameter @xmath145 involving the @xmath146 coordinate makes that the torus becames the product of one cilinder with infinity longitude and other with longitude @xmath147 . the link between geometrical description of the torus and its topological equivalence to two cilinders is given by the following equivalent form for @xmath148 @xmath149 with this we can perform the projection from torus phase space to the mobius phase space by means of @xmath150 @xmath151 where @xmath152 . it is important to note that we can proceed other time performing the projection from the mobius geometry to the circle straighforwardly obtaining the cs for the bose case . then the procedure of projections can be sinthetized in the following scheme @xmath153 this in fact connects the aceptions of that the coherent states on mobius topology as fermions and in the circle topology as bosons @xcite . a periodic circular trajectory of @xmath147 on torus ( figure [ trajtorus ] ) has a bosonic behaviour that can be deformed topologically without losing such characteristic . the mobius the periodic trajectory of @xmath154 ( figure [ trajmobius ] ) has a fermionic behaviour and can also be deformed without alter the topological properties . such behaviours in qubits operations acting with deformations that preserve topology . by considering the coherent states @xmath155 on the mbius strip , we see that their normalization is given in terms of jacobi theta functions , as in the case of a torus @xmath156 taking the approximation formula for theta functions , @xmath157 then , the the projections relations for coherent states on a mobius strip can be written as @xmath158 where the variables with tilde correspond to the coherent state @xmath159 . on a period in the mobius strip , the state turns to the same initial state . in fact , this property for coherent states defined on a mobius strip is a characteristic of the topology associated to this surface . taking @xmath160 and @xmath161 using the projection in the form ( [ projection ] ) , we can write in a more simplified form @xmath162 but as @xmath163 , we have that the state with a difference in the phase @xmath134 corresponding to the period of a mobius will be the same @xmath164 , as expected from the topology . consequently , under a period of @xmath154 the state turns to the same . such behaviour is a characteristic of fermion spin variables and the entanglement and in fact can be associated to entanglements for fermionic systems . now let us consider the action of the operators restricted to the mobius strip @xmath165 as a consequence the action on the state @xmath166 will be @xmath167 and the state is invariant under the time inversion operator @xmath25 @xmath168 now let us consider the operators @xmath169acting on @xmath170 . this will give @xmath171 where @xmath172 . let us now consider a two - particle coherent state on the mbius strip ( figure [ trajtwomobius ] and [ trajtwomobius2 ] ) given by @xmath173 we will have the entangled state @xmath174 where @xmath175 in a more general form , we can define @xmath176 that are consistent with the time inversion @xmath25 and the operator @xmath177 , where @xmath178 @xmath179 . the action on the state @xmath170 will give @xmath180 and on the state @xmath181 will give @xmath182 we can consider the set of non - orthogonal measurements in terms of coherent states @xmath183 note that , if we consider the states normalized , this measurement turns a von - neumann type measurement in each reduced space @xcite . by acting on the state ( [ st7 ] ) , we have @xmath184 @xmath185 the effective dimensionality of the hilbert space can be computed in terms of the hilbert - schmidt norm @xmath186 consider the density matrix associated to @xmath187 is given by @xmath188 , given by @xmath189 then the associated entangled measurement can be calculated by means of projective measurements @xmath190 compared to the corresponding measurements to the state @xmath191 , that are obtainded by doing @xmath192 and @xmath193 , @xmath194 the ratios @xmath195 give the measurement of the entanglement associated do the action of @xmath196 . we can consider also coherent states pertaining to both spaces , i.e. , torus and mobius , by means of products in the intersection , as @xmath197 ) . although each particle is confined to its surface ( figure [ mobius_torustr ] ) , the intersection can be used as a point of correlation . in fact , if we think in terms of orbital angular momentum and spin , we can associate the interaction to a spin - orbit coupling entanglement . let us consider an operator acting on the intersection of torus and mobius , @xmath198 such that @xmath199 leaves the mobius state invariant and @xmath200 leaves the torus state invariant . the action of @xmath201 on @xmath202 will be given by @xmath203 as an example we can calculate @xmath204 where @xmath205 since the @xmath206 and @xmath207 , the action of the operator on these states will lead to @xmath208 . this will have effect in the coherent states by means of @xmath209 . a corresponding density matrix associated to such state @xmath210 we can consider the set of non - orthogonal measurements in terms of the coherent states acting on torus and mobius @xmath211 @xmath212 this measurement turns to a von - neumann type measurement if we normalize the states . the effective dimensionality of the hilbert space can be computed in terms of the hilbert - schmidt norms @xmath213 the measurement of entanglement can be calculated by means of projective measurements @xmath214 compared to the corresponding measurements to the uncorrelated state @xmath215 , @xmath216 by means of the respective ratios @xmath217 that give the correlations corresponding to each state on torus and mobius . we have considered the entanglement of quantum particles in non - trivial topologies , considering the cases of a torus , mobius strip and torus - mobius . we have derived the corresponding parametrizations in each case and developed appropriate operators . we have derived the geometrical lagrangeans and hamiltonians of torus and mobius that are associated to each other by means to a constraint that , from the topological point of view , is a topological reductions equivalent to cuts in the deformations . the quantum particle dynamics in the quantized form is a consequence the canonical quantization of the hamiltonians . from each case we derived the corresponding one and two - mode coherent states that are entangled by the action of the proper operators in each case . we also have shown the relation of the toroidal operators that lead to entanglement and @xmath8 , that can lead to possible connections with supergravity models . the mobius is obtained by means of a reduction from the torus by means of a constraint in the angular variables . this implies that the mobius strip can be keept inside the torus . by applying a deformation on the strip , the topological properties keep unaltered and we use it to build the corresponding coherent states associated to the mobius topology starting from the torus . we have shown that the entangled state generated in the torus has a characteristic of the experimentally verified photon entanglement by orbital angular momentum , such that the states on torus behave like bosons as a special case . this is an important fact , since we can associate the entanglement in a torus to a photon entanglement by orbital angular momentum as a special case . on the other hand , the entanglement states associated to mobius strip have the periodicity associated to fermions @xmath154 and can be more appropriate to describe the entanglement in fermionic systems . we also have considered the entanglement between torus and mobius in the intersection of torus and mobius with the action of operators defined in the intersection . such situation can be originated in the case of an entanglement by spin - 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we consider states defined on non - trivial topologies of torus , mobius and torus - mobius . adequate operators leading to the construction of coherent states , two - mode coherent states and entangled states are derived and we show a class of entangled states on torus that are related to the entanglement of photons by orbital angular momentum states .

in this section we show how one can trade the dimensions of subsystem and co - subsystem to obtain new codes from a given subsystem or stabilizer code . the results are obtained by exploiting the symplectic geometry of the space . a remarkable consequence is that nearly any stabilizer code yields a series of subsystem codes . our first result shows that one can decrease the dimension of the subsystem and increase at the same time the dimension of the co - subsystem while keeping or increasing the minimum distance of the subsystem code . [ th : shrinkk ] let @xmath23 be a power of a prime @xmath24 . if there exists an @xmath25 subsystem code with @xmath26 that is pure to @xmath27 , then there exists an @xmath28 subsystem code that is pure to @xmath29 . if a pure @xmath30 subsystem code exists , then there exists a @xmath31 subsystem code . see @xcite replacing @xmath32-bases by @xmath14-bases in the proof of the previous theorem yields the following variation of the previous theorem for @xmath14-linear subsystem codes . [ th : fqshrinkr ] let @xmath23 be a power of a prime @xmath24 . if there exists a pure @xmath14-linear @xmath0_q$ ] subsystem code with @xmath33 , then there exists a pure @xmath14-linear @xmath34_q$ ] subsystem code . see @xcite [ cor : generic ] if there exists an ( @xmath14-linear ) @xmath35_q$ ] stabilizer code that is pure to @xmath27 , then there exists for all @xmath36 in the range @xmath37 an ( @xmath14-linear ) @xmath38_q$ ] subsystem code that is pure to @xmath29 . if a pure ( @xmath14-linear ) @xmath0_q$ ] subsystem code exists , then a pure ( @xmath14-linear ) @xmath39_q$ ] stabilizer code exists . see @xcite using this theorem we can derive many families of subsystem codes derived from families of stabilizer codes as shown in table [ table : families ] [ cols="^,^,^,^",options="header " , ] let @xmath40 and @xmath41 be two classical codes defined over @xmath42 . the direct sum of @xmath43 and @xmath44 is a code @xmath45 defined as follows an @xmath48_q$ ] classical code @xmath43 is a subcode in an @xmath49_q$ ] if every codeword @xmath50 in @xmath43 is also a codeword in @xmath44 , hence @xmath51 . we say that an @xmath52_q$ ] subsystem code @xmath53 is a subcode in an @xmath54_q$ ] subsystem code @xmath55 if every codeword @xmath56 in @xmath53 is also a codeword in @xmath55 and @xmath57 . _ notation . _ let @xmath23 be a power of a prime integer @xmath24 . we denote by @xmath14 the finite field with @xmath23 elements . we use the notation @xmath58 to denote the concatenation of two vectors @xmath59 and @xmath60 in @xmath61 . the symplectic weight of @xmath62 is defined as @xmath63 we define @xmath64 for any nonempty subset @xmath65 of @xmath66 . the trace - symplectic product of two vectors @xmath67 and @xmath68 in @xmath66 is defined as @xmath69 where @xmath70 denotes the dot product and @xmath71 denotes the trace from @xmath14 to the subfield @xmath32 . the trace - symplectic dual of a code @xmath72 is defined as @xmath73 we define the euclidean inner product @xmath74 and the euclidean dual of @xmath75 as @xmath76 we also define the hermitian inner product for vectors @xmath77 in @xmath78 as @xmath79 and the hermitian dual of @xmath80 as @xmath81 [ th : oqecfq ] let @xmath16 be a classical additive subcode of @xmath66 such that @xmath82 and let @xmath83 denote its subcode @xmath84 . if @xmath85 and @xmath86 , then there exists a subsystem code @xmath87 such that we first note that for any additive subcode @xmath98 , we can define an additive code @xmath99 by @xmath100 we have @xmath101 . furthermore , if @xmath102 , then @xmath103 is contained in @xmath104 for all @xmath105 in @xmath14 , whence @xmath106 . by comparing cardinalities we find that equality must hold ; in other words , we have @xmath107 by theorem [ th : oqecfq ] , there are two additive codes @xmath16 and @xmath83 associated with an @xmath25 clifford subsystem code such that @xmath108 and @xmath109 we can derive from the code @xmath16 two new additive codes of length @xmath110 over @xmath14 , namely @xmath111 and @xmath112 . the codes @xmath111 and @xmath113 determine a @xmath114 clifford subsystem code . since @xmath115 we have @xmath116 . furthermore , we have @xmath117 . it follows from theorem [ th : oqecfq ] that we know that existence of the pure subsystem code @xmath6 with parameters @xmath25 implies existence of a pure stabilizer code with parameters @xmath128 for @xmath129 and @xmath130 from ( * ? ? ? * theorem 2 . ) . by * theorem 70 ) , there exist a pure stabilizer code with parameters @xmath131 . this stabilizer code can be seen as @xmath132 subsystem code . by using ( * ? ? ? * theorem 2 . ) , there exists a pure @xmath14-linear subsystem code with parameters @xmath133 that proves the claim . we know that existence of the pure subsystem code @xmath6 implies existence of a pure stabilizer code with parameters @xmath134_q$ ] for @xmath129 and @xmath135 by using ( * ? ? ? * theorem 2 . and theorem 5 . ) . by ( * ? ? ? * theorem 70 ) , there exist a pure stabilizer code with parameters @xmath136_q$ ] . this stabilizer code can be seen as an @xmath137_q$ ] subsystem code . by using ( * ? ? ? * theorem 3 . ) , there exists a pure @xmath14-linear subsystem code with parameters @xmath138_q$ ] that proves the claim . by theorem [ th : oqecfq ] , if an @xmath25 subsystem code @xmath6 exists for @xmath96 and @xmath144 , then there exists an additive code @xmath145 and its subcode @xmath146 such that @xmath147 and @xmath148 . furthermore , @xmath149 . let @xmath150 and @xmath151 be two vectors in @xmath61 . , we can assume that the code @xmath152 is defined as @xmath153 let @xmath154 and @xmath155 be two vectors in @xmath156 . also , let @xmath157 be the code obtained by puncturing the first coordinate of @xmath152 , hence @xmath158 since the minimum distance of @xmath152 is at least 2 , it follows that @xmath159 and the minimum distance of @xmath157 is at least @xmath160 . now , let us construct the dual code of @xmath157 as follows . @xmath161 furthermore , if @xmath162 , then @xmath163 . therefore , @xmath164 is a self - orthogonal code and it has size given by @xmath165 we can also puncture the code @xmath16 to the code @xmath139 at the first coordinate , hence @xmath166 clearly , @xmath167 and if @xmath168 , then the vector @xmath169 , therefore , @xmath170 . this gives us that @xmath171 . furthermore , hence @xmath172 . the dual code @xmath173 can be defined as @xmath174 also , if @xmath175 , then @xmath176 , furthermore , @xmath177 therefore there exists a subsystem code @xmath178 . also , the code @xmath157 is pure and has minimum distance at least @xmath160 . we can proceed and compute the dimension of subsystem @xmath179 and co - subsystem @xmath180 from theorem [ th : oqecfq ] as follows . the minimum distance condition follows since the code @xmath6 has @xmath184 and the code @xmath126 has minimum distance as @xmath6 reduced by one . so , the minimum weight of @xmath185 is at least the minimum weight of @xmath186 @xmath187 if the code @xmath6 is pure , then @xmath188 , therefore , the new code @xmath126 is pure since @xmath189 . we also can reduce dimension of the subsystem code for fixed length @xmath18 and minimum distance @xmath12 , and still obtain a new subsystem code with improved minimum distance as shown in the following results . [ lem : reducingk ] if a ( pure)@xmath14-linear @xmath0_q$ ] subsystem code @xmath6 exists for @xmath135 , then there exists an @xmath14-linear @xmath191_q$ ] subsystem code @xmath192 ( pure to d ) such that @xmath193 . the idea of the proof comes by extending the code @xmath83 by some vectors from @xmath200 ) . let us choose a code @xmath201 of size @xmath202 . we also ensure that the code @xmath201 is self - orthogonal . clearly extending the code @xmath83 to @xmath201 will extend both the codes @xmath16 and @xmath203 to @xmath204 and @xmath205 , respectively . hence @xmath206 and @xmath207 . [ lem : reducen - m ] suppose an @xmath0_q$ ] linear pure subsystem code @xmath6 exists generated by the two codes @xmath216 . then there exist linear @xmath217_q$ ] and @xmath218_q$ ] subsystem codes with @xmath219 , @xmath220 , @xmath221 , and @xmath222 for any integer @xmath223 such that there exists a codeword of weight @xmath223 in @xmath224 . [ sketch ] this lemma [ lem : reducen - m ] can be proved easily by mapping the subsystem code @xmath6 into a stabilizer code . by using ( * ? ? ? * theorem 7 . ) , and the new resulting stabilizer code can be mapped again to a subsystem code with the required parameters . we can also construct new subsystem codes from given two subsystem codes . the following theorem shows that two subsystem codes can be merged together into one subsystem code with possibly improved distance or dimension . [ thm : twocodes_n1k1r1d1n2k2r2d2 ] let @xmath53 and @xmath55 be two pure subsystem codes with parameters @xmath225_2 $ ] and @xmath226_2 $ ] for @xmath227 , respectively . then there exists a subsystem code with parameters @xmath228_2 $ ] , where @xmath229 and @xmath230 . existence of an @xmath231_2 $ ] pure subsystem code @xmath232 for @xmath233 , implies existence of a pure stabilizer code @xmath234 with parameters @xmath235_2 $ ] with @xmath236 , see @xcite . therefore , by ( * ? ? ? * theorem 8 . ) , there exists a stabilizer code with parameters @xmath237_2 $ ] , @xmath238 . but this code gives us a subsystem code with parameters @xmath239_2 $ ] with @xmath240 and @xmath230 that proves the claim . [ lem : twocodes_nk1r1d1nk2r2d2 ] let @xmath53 and @xmath55 be two pure subsystem codes with parameters @xmath52_q$ ] and @xmath54_q$ ] , respectively . if @xmath241 , then there exists an @xmath242_q$ ] pure subsystem code with minimum distance @xmath243 and @xmath244 . existence of a pure subsystem code with parameters @xmath245_q$ ] implies existence of a pure stabilizer code with parameters @xmath246_q$ ] using ( * ? ? ? * theorem 4 . ) . but by using ( * ? ? ? * lemma 74 . ) , there exists a pure stabilizer code with parameters @xmath247_q$ ] with @xmath248 . by ( * theorem 2 . , corollary 6 . ) , there must exist a pure subsystem code with parameters @xmath242_q$ ] where @xmath248 and @xmath244 , which proves the claim . [ lem : twocodes_nk1r1d1nk2r2d2another ] let @xmath53 and @xmath55 be two pure subsystem codes with parameters @xmath52_q$ ] and @xmath54_q$ ] , respectively . if @xmath241 , then there exists an @xmath249_q$ ] pure subsystem code with minimum distance @xmath243 . @xmath265 the code @xmath83 has size of @xmath266 also , we can define the code @xmath16 based on the codes @xmath43 and @xmath44 as @xmath267 the code @xmath16 is of size @xmath268 but the trace - alternating dual of the code @xmath83 is therefore , @xmath275 is a self - orthogonal code with respect to the trace alternating product . furthermore , @xmath276 hence , @xmath277 . therefore , there exists an @xmath14-linear subsystem code @xmath278 with the following parameters . if there exist two pure subsystem quantum codes @xmath53 and @xmath55 with parameters @xmath225_q$ ] and @xmath226_q$ ] , respectively . then there exists a pure subsystem code @xmath281 with parameters @xmath282_q$ ] . this lemma can be proved easily from ( * ? ? ? * theorem 5 . ) and ( * ? ? ? * lemma 73 . ) . the idea is to map a pure subsystem code to a pure stabilizer code , and once again map the pure stabilizer code to a pure subsystem code . if there exist two pure subsystem quantum codes @xmath53 and @xmath55 with parameters @xmath225_q$ ] and @xmath226_q$ ] , respectively . then there exists a pure subsystem code @xmath281 with parameters @xmath283_q$ ] . let us choose the codes @xmath16 and @xmath83 as follows . @xmath284 and @xmath285 respectively . from this construction , and since @xmath263 and @xmath264 are self - orthogonal codes , it follows that @xmath83 is also a self - orthogonal code . furthermore , @xmath286 and @xmath287 , then @xmath288 hence @xmath289 . the code @xmath16 is of size latexmath:[\ ] ] where @xmath363 $ ] and + @xmath364 $ ] . the matrix @xmath365 generates the code @xmath366 . now @xmath83 defines a @xmath357_3 $ ] stabilizer code . therefore , @xmath367 . it follows that @xmath368 . by ( * theorem 4 ) , we have a @xmath369_3 $ ] viz . a @xmath324_3 $ ] subsystem code . we can also have a trivial @xmath324_2 $ ] code . this trivial extension seems to argue against the usefulness of subsystem codes and if they will really lead to improvement in performance . an obvious open question is if there exist nontrivial @xmath324_2 $ ] or @xmath342_2 $ ] subsystem codes .

we demonstrate propagation rules of subsystem code constructions by extending , shortening and combining given subsystem codes . given an @xmath0_q$ ] subsystem code , we drive new subsystem codes with parameters @xmath1_q$ ] , @xmath2_q$ ] , @xmath3_q$ ] . we present the short subsystem codes . the interested readers shall consult our companion papers for upper and lower bounds on subsystem codes parameters , and introduction , trading dimensions , families , and references on subsystem codes @xcite and references therein . * subsystem codes . * let @xmath4 be the hilbert space @xmath5 . let @xmath6 be a quantum code such that @xmath7 , where @xmath8 is the orthogonal complement of @xmath6 . recall definition of the error model acting in qubits @xcite . we can define the subsystem code @xmath6 as follows an @xmath0_q$ ] subsystem code is a decomposition of the subspace @xmath6 into a tensor product of two vector spaces a and b such that @xmath9 , where @xmath10 and @xmath11 . the code @xmath6 is able to detect all errors of weight less than @xmath12 on subsystem @xmath13 . subsystem codes can be constructed from the classical codes over @xmath14 and @xmath15 . the euclidean construction of subsystem code is given as follows @xcite . [ lem : css - euclidean - subsys ] if @xmath16 is a @xmath17-dimensional @xmath14-linear code of length @xmath18 that has a @xmath19-dimensional subcode @xmath20 and @xmath21 , then there exists an @xmath22_q\ ] ] subsystem code .

let @xmath0 be an integer greater than one . for a given @xmath0-tuple of analytic functions ( or formal power series ) @xmath1 and of nonnegative integers @xmath2 , hermite considered the following two rational approximation problems . the first is to find @xmath0 polynomials @xmath3 of degree @xmath4 such that @xmath5 where @xmath6 ; i.e. , the left - hand side has a zero of order at least @xmath7 at @xmath8 . the second is to find @xmath0 polynomials @xmath9 of degree @xmath10 such that @xmath11 each system of polynomials @xmath12 and @xmath13 generically turns out to be unique up to simultaneous multiplication by constants , as an elementary consequence of linear algebra . the above approximation problems come from hermite s study on arithmetic properties of the exponential function and are called collectively the _ hermite - pad approximations _ ; often , the former is referred to as the ` type i ' and the latter as the ` type ii ' or as the _ simultaneous pad approximation_. note that if @xmath14 then both of them reduce to the ( usual ) pad approximations . although these two types of approximations were seemingly unrelated , mahler discovered that they were fundamentally connected to each other . put @xmath15 for @xmath16 . [ thm : original ] it holds that @xmath17 with @xmath18 being a diagonal constant matrix . moreover , if every diagonal part @xmath19 and @xmath20 is chosen to be a monic polynomial then @xmath18 becomes the identity matrix . the aim of this paper is to develop an underlying relationship of hermite s two approximations with the theory of linear differential equations in the complex domain , especially with that of isomonodromic deformations . interestingly enough , mahler s duality plays a crucial role in constructing a certain class of _ schlesinger transformations _ , i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant . in sect . [ sect : hpa ] , we begin by introducing the two types of rational approximations for an @xmath0-tuple of functions , which are slightly modified ( in order to fit the construction of schlesinger transformations ) from the original hermite - pad and simultaneous pad approximations . we then prove a variation of mahler s duality between them ( see theorem [ thm : dual ] ) . applying the approximations for the solution of an @xmath21 fuchsian system of linear differential equations yields its schlesinger transformation ( see theorem [ thm : schl ] ) ; in fact , mahler s duality guarantees the absence of apparent singularities in the new fuchsian system after the schlesinger transformation . in sect . [ sect : det ] , we deduce determinantal representations for the approximants and remainders from the approximation conditions ( see propositions [ prop : detq ] and [ prop : detp ] and also remark [ remark : bt ] ) . it should be noted that any key ingredient here is written in terms of _ block - toeplitz determinants_. in sect . [ sect : cf ] , we present an algorithm for constructing the schlesinger transformation via vector continued fraction expansions , which is a variation of the _ jacobi perron algorithm _ ( i.e. a higher dimensional analogue of the euclidean algorithm ) . the last two sections are devoted to the study of isomonodromic deformations . in sect . [ sect : aid ] , we first review the _ schlesinger system _ of nonlinear differential equations , which governs isomonodromic deformations of a fuchsian system . since a schlesinger transformation preserves the monodromy of the fuchsian system , it gives rise to a discrete symmetry of the corresponding schlesinger system . we clarify , based on the above relationship with rational approximations , the determinantal structure in the general solutions of the schlesinger systems . next we concern a particular family of the schlesinger systems , which possesses a unified description as a polynomial hamiltonian system denoted by @xmath22 ( @xmath23 ) ; it includes various noteworthy examples of isomonodromic deformations such as the sixth painlev equation ( @xmath24 ) and the garnier system in @xmath25 variables ( @xmath26 ) . in sect . [ sect : ihi ] , we demonstrate schlesinger transformations on the previously known hypergeometric solution of @xmath22 ( see @xcite ) ; as a result , we obtain solutions of @xmath22 written in terms of iterated hypergeometric integrals through fubini s theorem and the vandermonde determinant ( see theorem [ thm : hgi ] and its sequel ) . in this section we show how rational approximations are useful for constructing schlesinger transformations of linear differential equations . fix an integer @xmath27 . we shall first introduce two different types of rational approximation problems for an @xmath0-tuple @xmath28\!]\ ] ] of formal power series , where we assume @xmath29 without loss of generality . let @xmath30 be a positive integer . consider for each @xmath31 @xmath32 an @xmath0-tuple of polynomials @xmath33 @xmath34 of degree at most @xmath35 , where @xmath36 is the kronecker delta . suppose the approximation condition @xmath37 is fulfilled for each @xmath31 . this condition amounts to a system of @xmath38 homogeneous linear equations for the @xmath39 unknown coefficients of the polynomials @xmath40 @xmath34 . under a generic condition for the power series @xmath41 , these polynomials are uniquely determined up to simultaneous multiplication by constants ; see sect . [ subsect : dethpa ] . we will be concerned with the row vector @xmath42 by construction , the polynomial @xmath43 has no constant term . moreover , the degree of the _ diagonal _ part @xmath44 @xmath32 turns out to be @xmath30 exactly ; see ( [ eq : deg ] ) in sect . [ subsect : dual ] . we treat another type of approximation problem for the same power series @xmath41 . consider for each @xmath45 @xmath34 an @xmath0-tuple of polynomials @xmath46 @xmath32 of degree at most @xmath47 which satisfies the following approximation conditions : @xmath48 these conditions are interpreted as a system of @xmath49 homogeneous linear equations for the @xmath50 unknown coefficients of the polynomials @xmath51 @xmath32 . hence the column vector @xmath52 is generically unique up to multiplication by constants ; see sect . [ subsect : detspa ] . let @xmath53 . it is immediate from @xmath54 \!]$ ] to verify that @xmath55 there is an interesting connection between the two approximation problems ( [ eq : hpa ] ) and ( [ eq : spa ] ) although they are seemingly unrelated . the following theorem is thought of as a variation of _ mahler s duality _ ; see theorem [ thm : original ] or @xcite . we will give a proof of it because our setup is slightly different from the original ; cf . * theorem 8.1.2 ) and @xcite . [ thm : dual ] it holds that @xmath56=w^{nl } \cdot d\ ] ] with @xmath18 being a diagonal constant matrix . let us first estimate the degree of the @xmath57-entry @xmath58 of the left - hand side . the degree of each polynomial @xmath59 or @xmath60 reads @xmath61 hence we have @xmath62 next we shall estimate the multiplicity of @xmath63 at @xmath8 by means of the approximation conditions . consider @xmath64 [ [ section ] ] @xmath65 therefore , @xmath66 [ [ section-1 ] ] @xmath67 therefore , @xmath68 noticing @xmath54\!]$ ] we verify @xmath69 combining this with ( [ eq : degm ] ) , we can conclude that @xmath70 . + consequently , the diagonal entry @xmath71 coincides with the term of highest degree in @xmath72 and thus @xmath73 we henceforth normalize @xmath74 and @xmath75 so that their diagonal parts @xmath76 and @xmath77 become monic polynomials , i.e. @xmath78 and thereby @xmath79 ( the identity matrix ) . [ cor : r ] the polynomial matrix @xmath80\ ] ] satisfies @xmath81 ; \\ { \rm(ii ) } & \det r(z)=1 , \quad \text { i.e. } r(z ) \in { \rm sl}(l , { \mathbb c}[z ] ) . \end{array}\ ] ] theorem [ thm : dual ] shows ( i ) immediately . then , it holds that @xmath82)$ ] and thus @xmath83 . by definition , @xmath84 takes the form @xmath85 namely , its constant term is an upper triangular matrix whose diagonal entries are all one . therefore , we have @xmath86 . consider an @xmath87 fuchsian system @xmath88 of linear ordinary differential equations with @xmath89 regular singularities @xmath90 on the riemann sphere . let @xmath91 and @xmath92 be upper and lower triangular matrices , respectively . assume for simplicity there is no integer difference among the _ characteristic exponents _ @xmath93 at @xmath94 ( resp . @xmath95 at @xmath96 ) , i.e. the eigenvalues of the residue matrix @xmath91 ( resp . @xmath97 ) . then we have a solution @xmath98 of ( [ eq : fs ] ) normalized as @xmath99 with @xmath100 and @xmath101 being an invertible constant matrix ( the _ connection matrix _ between @xmath94 and @xmath96 ) . here @xmath102 is a matrix function holomorphic at @xmath8 ( @xmath96 ) and @xmath103 is invertible and lower triangular , i.e. @xmath104 many literatures adopt a different normalization such that the residue matrix at @xmath96 becomes diagonal . our present normalization treats the two points @xmath94 and @xmath96 equally and it emerges naturally from the similarity reduction of the uc hierarchy , which is a context of infinite - dimensional integrable systems ; see @xcite . furthermore , as clarified by haraoka @xcite , this normalization is effective to find a ` good ' coordinate of the space of fuchsian systems having a given riemann scheme . an analytic continuation along a loop on @xmath105 based at some point @xmath106 induces a linear transformation of @xmath107 according to its multi - valuedness at the branch points @xmath108 . we thus obtain an @xmath0-dimensional representation of the fundamental group @xmath109 , which is called the _ monodromy _ of @xmath107 . a left multiplication @xmath110 of a rational function matrix @xmath111 is said to be a _ schlesinger transformation _ if the new equation @xmath112 satisfied by @xmath113 becomes the same form as the original ( [ eq : fs ] ) . because @xmath114 is rational , @xmath113 and @xmath107 have the same monodromy though they have different characteristic exponents by integers . it is known that if we specify an admissible discrete change of the characteristic exponents then the corresponding rational function matrix @xmath84 of the schlesinger transformation is algebraically computable from @xmath107 ; see @xcite . in fact , the construction problem of schlesinger transformations is naturally related to rational approximation problems ; see also remark [ remark : history ] . in this paper we focus on a class of schlesinger transformations , which is of particular interest from the viewpoint of hermite s two approximation problems and also of vector continued fractions ( see sect . [ sect : cf ] ) . note that , for a general schlesinger transformation other than the present direction , though it can also be controlled by some rational approximation problems but it becomes much more complicated due to the absence of a duality like mahler s ; e.g. @xmath115 seems not to have a concise determinantal representation . let us define the @xmath0-tuple @xmath116 of power series in @xmath117 as the first column of @xmath102 , where @xmath102 is the power series part of the solution @xmath107 of ( [ eq : fs ] ) near @xmath96 . notice that @xmath118 certainly holds . therefore , all the general arguments in sects . [ subsect : hpa][subsect : dual ] are still valid for this specific case , and we are led to the schlesinger transformation through the two approximation problems for @xmath119 . now we state the result . [ thm : schl ] the polynomial matrix @xmath114 given by ( [ eq : smatrix ] ) realizes the schlesinger transformation @xmath120 shifting the characteristic exponents at @xmath96 by @xmath121 . it follows from the hermite - pad approximation condition ( [ eq : hpa ] ) that @xmath122 . by definition , @xmath84 takes the form @xmath123 hence , if we write as @xmath124 then @xmath125 becomes the same form as @xmath126 . also , we verify from ( [ eq : r&r-1 ] ) that @xmath84 does not change the form of the power series expansion of @xmath107 near @xmath94 . on the other hand , the coefficient @xmath127 of the fuchsian system ( [ eq : fs ] ) is transformed as @xmath128 if we remember both @xmath84 and @xmath115 being polynomials ( see corollary [ cor : r ] ) , then @xmath129 turns out to be a rational function matrix having only simple poles at @xmath108 as well as the original @xmath130 . in this sense mahler s duality guarantees the absence of apparent singularities in the new equation @xmath131 satisfied by @xmath132 . in the rank two case ( @xmath14 ) a similar construction of schlesinger transformations as theorem [ thm : schl ] has been established in @xcite based on ( usual ) pad approximations . [ remark : history ] a series of pioneering works was done by d. chudnovsky and g. chudnovsky on the close connection between rational approximation problems and riemann s monodromy problem , involving ( semi - classical ) orthogonal polynomials ; see @xcite and references therein . the ` pad method ' recently proposed by yamada @xcite is a recipe for lax formalism of painlev equations and , at the same time , for their special solutions , which is based on pad approximations ( or interpolations ) of elementary functions ; interestingly enough , it is applicable also for various discrete analogues of painlev equations beyond the originals ; see @xcite . the essential idea of the above works could be exemplified by the following : let us consider a function @xmath133 with @xmath134 . the remainder @xmath135 of its pad approximation then satisfies a second - order linear differential equation denoted by @xmath136 , which may have an apparent singularity besides the four regular singularities @xmath137 . however , the two functions @xmath138 and @xmath139 share the same multi - valuedness since they are rationally related ; thus , the monodromy of @xmath136 is obviously constant with respect to @xmath140 . this fact leads to special solutions of the sixth painlev equation @xmath141 , i.e. the _ isomonodromic deformation _ ( cf . [ sect : aid ] ) of a second - order linear differential equation with four regular singularities . it is interesting to note that such an idea had been recognized implicitly by laguerre ( before the discovery of painlev equations ) ; see @xcite and also @xcite . the approximation conditions ( [ eq : hpa ] ) and ( [ eq : spa ] ) can be interpreted as certain multi - orthogonality relations among the @xmath0-tuples of polynomials @xmath74 and @xmath75 , respectively ; i.e. , these polynomials can constitute multi - orthogonal polynomial systems . in this paper , although we do not enter into details on such aspects , we present below the _ determinantal representations _ for them , which will crucially work in the last two sections . in this section we derive determinantal representations for the approximation polynomials @xmath142 and @xmath143 . we write the power series as @xmath144\!]\ ] ] henceforth ; note that the superscript @xmath31 of @xmath145 is just an index , not an exponent . introduce the @xmath146 _ rectangular toeplitz matrix _ @xmath147 _ { \begin{subarray}{l}1 \leq m \leq k \\1 \leq n \leq l \end{subarray } } \nonumber \\ & = \kbordermatrix{&1&2 & & l \\ 1&a^i_j & a^i_{j-1 } & \cdots & a^i_{j - l+1 } \\ 2&a^i_{j+1 } & a^i_j & \cdots & a^i_{j - l+2 } \\ & \vdots & \vdots & & \vdots \\ k&a^i_{j+k-1 } & a^i_{j+k-2 } & \cdots & a^i_{j+k - l } } \label{eq : rtm}\end{aligned}\ ] ] for the sequence @xmath148 , where @xmath149 if @xmath150 . it holds that @xmath151 by definition . we can calculate separately for each @xmath31 @xmath32 . therefore , for brevity , we shall express the coefficients of the approximation polynomial @xmath152 as @xmath153 with omitting the index @xmath31 . the condition ( [ eq : hpa ] ) implies vanishing of the coefficients of @xmath154 in the left - hand side . consequently , we have a system @xmath155 of homogeneous linear equations for the @xmath156 unknowns @xmath157 where @xmath158.\ ] ] the solution of ( [ eq : lhpa ] ) is unique up to multiplication by constants if and only if the rank of the @xmath159 matrix @xmath160 equals @xmath161 ( which we will always assume ) . interestingly enough , we have the following determinantal representation of @xmath142 . [ prop : detq ] it holds that @xmath162,\ ] ] where @xmath163 are some normalizing constants . consider @xmath164 which is the remainder of the approximation condition ( [ eq : hpa ] ) . substituting ( [ eq : q ] ) shows that @xmath165.\end{aligned}\ ] ] therefore , if we put @xmath166 , then the coefficients read @xmath167 it is immediate from a property of determinants to verify @xmath168 for any @xmath169 less than @xmath161 ; thus , we have @xmath170 indeed . + we will normalize the polynomials so that its diagonal part @xmath44 becomes monic as well as in sect . [ subsect : dual ] . accordingly , the normalizing constant @xmath163 should be @xmath171 \\ & = ( -1)^{n(i+1 ) } \det \begin{bmatrix } a^0_0(nl , n ) & \cdots & a^{i}_{0}(nl , n ) & a^{i+1}_{-1}(nl , n ) & \cdots & a^{l-1}_{-1}(nl , n ) \end{bmatrix}.\end{aligned}\ ] ] thus , the leading coefficient @xmath172 of the remainder is given by @xmath173;\end{aligned}\ ] ] the constant term of the polynomial @xmath174 @xmath175 is given by @xmath176 \\ & = \frac{(-1)^{n i}}{{\rm nq}^{(i ) } } \det \begin{bmatrix } a^0_0(nl , n ) & \cdots & a^{i-1}_{0}(nl , n ) & a^{i}_{-1}(nl , n ) & \cdots & a^{l-1}_{-1}(nl , n ) \end{bmatrix}.\end{aligned}\ ] ] [ remark : bt ] we here restrict ourselves to the case where @xmath177 . let us introduce the _ block - toeplitz determinant _ @xmath178\ ] ] of size @xmath179 for each @xmath31 @xmath180 ; e.g. @xmath181 . we see in particular that @xmath182 these simple formulae will be used later in sect . [ sect : ihi ] . suppose @xmath183 for simplicity . or , equivalently , we may understand that we have renamed @xmath184 @xmath175 as @xmath185 . for a given power series @xmath186 , we employ the notation @xmath187^{b}_{a } = \sum_{k = a}^b f_k w ^k\ ] ] denoting its section between @xmath188 and @xmath189 if @xmath190 . from now on , we set @xmath191 as well as in remark [ remark : bt ] . first we shall construct the formulae for @xmath192 @xmath34 . [ [ section-2 ] ] the approximation condition ( [ eq : spa ] ) requires that @xmath193^{m+n-1}_{m}=0 \quad ( 1 \leq i \leq l-1)\ ] ] since @xmath194 @xmath175 is a polynomial of degree at most @xmath195 . if we write @xmath196 then we find a system @xmath197 of homogeneous linear equations for the @xmath198 unknowns @xmath199 . the solution of ( [ eq : lspa ] ) is unique up to multiplication by constants if and only if the rank of the @xmath200 matrix in the left - hand side equals @xmath201 . [ [ section-3 ] ] similarly , it follows from ( [ eq : spa ] ) that @xmath202^{m+n-1}_{m}&=0 \quad ( 1 \leq i\leq j-1 ) \\ \left [ f_j p_0^{(j ) } \right]^{m+n-2}_{m}&=0 \quad ( i = j ) \\ \left [ f_i p_0^{(j ) } \right]^{m+n-2}_{m-1}&=0 \quad ( j+1 \leq i \leq l-1).\end{aligned}\ ] ] these amount to the simultaneous linear equation @xmath203 for the @xmath201 unknown coefficients @xmath204 of the polynomial @xmath205 [ prop : detp ] the polynomials @xmath206 admit the following determinantal representations : @xmath207 and @xmath208 for @xmath209 , where @xmath210 are some normalizing constants . next the other @xmath51 ( @xmath211 ) can be written as follows : @xmath212^{m-1}_{0 } ; \\ \bullet \ \text{if $ 1 \leq j \leq l-1 $ } & & p_i^{(j ) } = \left [ f_i p_0^{(j ) } \right]^{m-1}_{0 } & \text{for $ 1 \leq i \leq j-1 $ } , \\ & & p_j^{(j ) } = w \left [ f_j p_0^{(j ) } \right]^{m-1}_{0 } & \text{for $ i = j$ } , \\ & & p_i^{(j ) } = w \left [ f_i p_0^{(j ) } \right]^{m-2}_{0 } & \text{for $ j+1 \leq i \leq l-1$}. \end{array}\ ] ] we will choose the normalization so that the diagonal part @xmath213 becomes monic . accordingly , we obtain @xmath214 and @xmath215 for @xmath216 . here we have employed the notation of the block - toeplitz determinant ( see remark [ remark : bt ] ) in view of ( [ eq : tenchi ] ) . if @xmath217 , @xmath218 is divisible by @xmath117 . by construction ( see corollary [ cor : r ] ) , we observe that @xmath115 takes the form ( cf . ( [ eq : r ] ) ) @xmath219 in this section we present an alternative construction of the same schlesinger transformation ( considered in sect . [ sect : hpa ] ) through an algorithm for expanding a vector - valued function into a vector continued fraction . let @xmath220 be the @xmath0-tuple of formal power series ( [ eq : fi ] ) . for simplicity , we assume tentatively that @xmath221 for all @xmath222 . we abbreviate the constant term @xmath223 of @xmath224 as @xmath225 . first we apply a left multiplication of a permutation matrix to @xmath119 : @xmath226 next we eliminate the constant term of @xmath185 by a subtraction of constant multiple of @xmath227 for each @xmath228 and by a multiplication by @xmath117 for @xmath229 : @xmath230 eventually we obtain a new @xmath0-tuple @xmath231 of power series from the original @xmath119 . the above procedure is summarized as a left multiplication @xmath232 of an invertible matrix @xmath233 with @xmath234 . this is an analogue of the euclidean algorithm and can be repeated generically . let @xmath235={}^{\rm t}(f_0[k],\ldots , f_{l-1}[k])$ ] denote the corresponding vector of power series at the @xmath169th step and let @xmath236 $ ] denote their constant terms . hence , we have @xmath237=t[k]{\boldsymbol f}[k ] \quad \text{and } \quad { \boldsymbol f}[0]={\boldsymbol f},\ ] ] where @xmath238=\frac{1}{w } \begin{bmatrix } 0 & & & & & w \\ 1&-\frac{a^0[k]}{a^1[k ] } & & & & \\ & 1&-\frac{a^1[k]}{a^2[k ] } & & & \\ & & \ddots & \ddots & \\ & & & 1&-\frac{a^{l-3}[k]}{a^{l-2}[k ] } & \\ & & & & 1&-\frac{a^{l-2}[k]}{a^{l-1}[k ] } \end{bmatrix}.\ ] ] on the other hand , solving ( [ eq : f ] ) for @xmath239 yields @xmath240 where @xmath241 . let us introduce the inhomogeneous coordinates @xmath242 by @xmath243 . therefore , we have @xmath244 where @xmath245 . let @xmath242 be an @xmath246-vector such that @xmath247 . then the vector @xmath248 is called the _ reciprocal of @xmath249_. note that @xmath250 . under this notation , the correspondence ( [ eq : phi ] ) can be translated into @xmath251 or equivalently into @xmath252 where @xmath253 namely , the vector @xmath254 is determined as the constant term of @xmath255 and the matrix @xmath256 is then specified by @xmath254 . let @xmath257 $ ] denote the inhomogeneous coordinates of the vector @xmath235 \in { \mathbb c}^{l}[\![w]\!]$ ] at the @xmath169th step . taking the reciprocal repeatedly in this way , we obtain formally the _ vector continued fraction _ @xmath258 + \cfrac { w b[0 ] } { { \boldsymbol a}[1 ] + \cfrac { w b[1 ] } { { \boldsymbol a}[2 ] + \cfrac{w b[2 ] } { { \boldsymbol a}[3 ] + \ddots } } } } , \ ] ] which is regarded as an @xmath246-dimensional generalization of the _ stieltjes - type _ continued fraction . refer to ( * ? ? ? * appendix a ) for a classification of continued fractions . our algorithm differs from the other known examples such as the jacobi perron algorithm ; cf . @xcite . note also that some dynamical system , like the toda lattice , has been studied based on the connection among the jacobi perron algorithm , rational approximations and bi - orthogonal polynomials ; see @xcite . the following theorem can be verified straightforwardly through the above algorithm , as well as the case of a stieltjes - type continued fraction ( i.e. , @xmath14 case ) . the @xmath169th convergents ( rational functions ) @xmath259 } , \\ { \boldsymbol \pi}_k&= \cfrac{1}{{\boldsymbol a}[0 ] + \cfrac { w b[0 ] } { { \boldsymbol a}[1 ] + \cfrac { w b[1 ] } { { \boldsymbol a}[2 ] + \ddots + \cfrac{w b[k-2 ] } { { \boldsymbol a}[k-1 ] } } } } \in { \mathbb c}^{l-1}(w ) \quad ( k \geq 2)\end{aligned}\ ] ] of the vector continued fraction ( [ eq : stiel ] ) provide approximants of the vector @xmath260\!]$ ] of power series in the sense that @xmath261 . in calculating @xmath262 , it is convenient to apply the projective transformations ( [ eq : tn ] ) successively as follows : @xmath263^{-1}t[1]^{-1 } \cdots t[k-1]^{-1 } \begin{bmatrix}1 \\0 \\ \vdots \\ 0 \end{bmatrix } \quad \text{and } \quad { \boldsymbol \pi}_k=\frac{1}{\varpi_{0}}\begin{bmatrix}\varpi_1 \\ \varpi_2 \\ \vdots \\ \varpi_{l-1 } \end{bmatrix}.\ ] ] let @xmath98 be a solution ( [ eq : fss ] ) of the fuchsian system ( [ eq : fs ] ) having the local behaviors @xmath264 where the power series parts are normalized by @xmath265 and @xmath101 is the connection matrix . let @xmath116 denote the first column of @xmath102 . it is clear from the construction of the matrix @xmath266 $ ] that @xmath267 near @xmath8 ( @xmath96 ) . on the other hand , it holds that @xmath268 near @xmath94 . after we repeat the same procedure @xmath0 times , the power series part thus recovers its original form : @xmath269\cdots t[1]t[0]\psi = z^{l-1 } \left(\begin{bmatrix } 1 & \cdots & * \\ & \ddots & \vdots \\ & & 1 \end{bmatrix } + o(z ) \right).\ ] ] in conclusion , the matrix @xmath270\cdots t[1]t[0]$ ] turns out to be a polynomial in @xmath271 and to be the multiplier of the schlesinger transformation shifting the characteristic exponents at @xmath96 by @xmath272 ; cf . theorem [ thm : schl ] . in fact , the same approximation problem considered in sect . [ sect : hpa ] appears in the following manner . concerning the polynomial matrix @xmath273\cdots t[1]t[0 ] $ ] , we observe from the form of @xmath274 $ ] that the diagonal entries are all monic linear functions , the strictly upper triangular part is linear and divisible by @xmath117 , and the strictly lower triangular part is a constant . moreover , in view of ( [ eq : fn ] ) we have @xmath275\cdots t[1]t[0 ] { \boldsymbol f}=o(w^l),\ ] ] which coincides with the approximation condition ( [ eq : hpa ] ) , and thus @xmath275\cdots t[1]t[0 ] = \begin{bmatrix } \widetilde{\boldsymbol q}^{(0)}(w ) \\ \vdots \\ \widetilde{\boldsymbol q}^{(l-1)}(w ) \end{bmatrix}\ ] ] under @xmath276 . in this section we first review some basic results on the _ schlesinger system _ , which governs isomonodromic deformations of a fuchsian system of linear ordinary differential equations . as explained in sect . [ subsect : st ] , a schlesinger transformation preserves the monodromy of the fuchsian system under consideration and , thereby , leads to a discrete symmetry of the associated schlesinger system . combining this fact with the result in sect . [ sect : det ] reveals a determinantal nature of isomonodromic deformations . next we treat a particular case of the schlesinger systems unifying various painlev - type differential equations and show its relationship with certain hypergeometric functions , which will be needed later . let us consider again the @xmath21 fuchsian system ( [ eq : fs ] ) : @xmath277 where @xmath278 and @xmath279 . we start with a well - known result on isomonodromic deformations of ( [ eq : fs2 ] ) . the monodromy of a fundamental solution @xmath107 , i.e. @xmath280 , does not depend on @xmath281 if and only if @xmath282 are rational functions in @xmath271 . we henceforth impose on our fuchsian system ( [ eq : fs2 ] ) the following assumptions : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ \(i ) all the residue matrices @xmath283 are _ semi - simple _ , i.e. diagonalizable ; + ( ii ) there is no integer difference other than zero among the eigenvalues of each @xmath283 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let us choose a normalization as before such that @xmath91 and @xmath92 are upper and lower triangular matrices , respectively . then we can take a fundamental solution @xmath98 of the form @xmath264 with @xmath284 and @xmath285 , where each @xmath286 may depend on @xmath287 . moreover , we have @xmath288 near each of the other regular singularities @xmath289 @xmath290 , where @xmath291 and @xmath292 are certain constant matrices satisfying @xmath293 . therefore , the monodromy matrices of @xmath107 attached to loops around @xmath289 @xmath290 and @xmath294 read @xmath295 suppose now that every monodromy matrix of @xmath107 is constant with respect to @xmath296 and , additionally , so is the connection matrix @xmath101 . then the rational functions @xmath297 can be explicitly written as @xmath298 where @xmath299 denotes the lower triangular part of @xmath283 ; see appendix for details . note in particular that the diagonal part @xmath300 of @xmath283 is expressible in terms of @xmath301 as @xmath302 the compatibility condition @xmath303=0\ ] ] of ( [ eq : fs2 ] ) and ( [ eq : defe ] ) is equivalent to a set of nonlinear differential equations for the matrices @xmath283 with respect to @xmath304 , which is called the _ schlesinger system _ @xcite . if @xmath305 , then the schlesinger system reduces to the sixth painlev equation @xmath141 . next we shall investigate how the solution @xmath107 and the coefficient @xmath306 of the fuchsian system ( [ eq : fs2 ] ) are connected with each other . concerning the power series expansion @xmath307 at the point of infinity ( @xmath308 ) , the coefficients @xmath309 turn out to be polynomials in the off - diagonal entries of the lower triangular matrix @xmath310 through frobenius method . conversely , substituting this solution @xmath107 in ( [ eq : fs2 ] ) , we find that @xmath311 { \rm diag \ } ( w^{\varepsilon_{\infty , j } } ) \cdot c\end{aligned}\ ] ] and @xmath312y.\end{aligned}\ ] ] recall @xmath279 here . equating the coefficients of @xmath313 in these power series yields @xmath314 thus @xmath97 becomes a polynomial in the entries of the leading coefficient @xmath315 of the solution , and also @xmath316 moreover , if one needs similar expressions for all other residue matrices @xmath283 besides @xmath97 , it is convenient to use the _ deformation equation _ ( [ eq : defe ] ) ; one can verify in fact @xmath317 in summary , each residue matrix @xmath283 of @xmath318 is expressible as a polynomial in the entries of the coefficients of @xmath107 ( and their derivatives with respect to @xmath304 ) , and vice versa . a schlesinger transformation keeps the monodromy of the fuchsian system invariant but shifts its characteristic exponents by integers ; recall sect . [ subsect : st ] . consequently , it gives rise to a discrete symmetry of the schlesinger system via the above correspondence between the solutions and coefficients of the fuchsian system . on the other hand , any ingredient of schlesinger transformations or of the associated rational approximations is described in terms of block - toeplitz determinants ; recall sect . [ sect : det ] . this fact thus provides a natural explanation for the determinantal structure appearing in solutions of isomonodromic deformations , e.g. painlev equations . refer to @xcite for a detailed investigation of the determinantal structure in jimbo miwa ueno s _ @xmath319-functions _ @xcite for a general framework admitting irregular singularities . we turn now to a particular case of the schlesinger systems , which will be the main subject in the rest of this paper . consider an @xmath21 fuchsian system of the form ( [ eq : fs2 ] ) whose _ spectral type _ is given by the partitions of @xmath0 : @xmath320 which indicate how the characteristic exponents overlap at each of the @xmath89 singularities . fix the characteristic exponents as listed in the following table ( riemann scheme ) : @xmath321 assume the sum of all the characteristic exponents equals zero ( fuchs relation ) , i.e. @xmath322 let @xmath91 and @xmath92 be upper and lower triangular matrices , respectively . then such a fuchsian system , denoted by @xmath323 , can be parametrized as follows : @xmath324 under the relations @xmath325 the last two of which come from the triangularity of @xmath97 . also , we can and will normalize the characteristic exponents at @xmath94 by @xmath326 without loss of generality . as shown in @xcite , the schlesinger system governing isomonodromic deformations of @xmath323 reduces to the multi - time hamiltonian system @xmath22 : @xmath327 here we let @xmath328 and define the hamiltonian function @xmath329 by @xmath330 with @xmath331 , @xmath332 and @xmath333 . therefore , @xmath329 is a polynomial in the unknowns ( _ canonical variables _ ) @xmath334 the number of the constant parameters @xmath335 contained in @xmath22 is essentially @xmath336 in view of ( [ eq : norm1 ] ) and ( [ eq : norm2 ] ) . for example , the case where @xmath14 and any @xmath337 coincides with the garnier system in @xmath25-variables and , thereby , the first nontrivial case @xmath24 does with the hamiltonian form of @xmath141 . we have _ a priori _ known from their spectral type that the fuchsian systems equipped with the riemann scheme ( [ eq : rs ] ) constitute a @xmath338-dimensional family . the coordinates of such a family are called _ accessory parameters _ , which are realized by the @xmath338 canonical variables ( [ eq : canvar ] ) in this instance . although the phase space of @xmath22 is a quite - complicated algebraic variety in general , there exists a family of solutions parametrized by a point in the projective space @xmath340 when the constants @xmath341 take certain special values . in fact , these solutions are written in terms of the _ hypergeometric function _ @xmath342 = \sum_{m_i \geq0 } \frac{(\alpha_1)_{|{\boldsymbol m}| } \cdots ( \alpha_{l-1})_{|{\boldsymbol m}| } ( \beta_1)_{m_1 } \cdots ( \beta_n)_{m_n } } { ( \gamma_1)_{| { \boldsymbol m}| } \cdots ( \gamma_{l-1})_{| { \boldsymbol m}| } } \frac{{x_1}^{m_1 } \cdots { x_n}^{m_n}}{{m_1 } ! \cdots { m_n}!},\ ] ] where @xmath343 and @xmath344 . if @xmath305 , then ( [ eq : hg ] ) is exactly gau s hypergeometric function . to state the result precisely , we introduce the integral representation @xmath345\ ] ] of @xmath339 , where the multi - valued function @xmath346 in @xmath347 is given by @xmath348 and the cycle @xmath349 is chosen to be an @xmath246-simplex @xmath350 also , we introduce supplementarily the integrals @xmath351 we are now ready to state the hypergeometric solution of @xmath22 ; see ( * ? ? ? * theorem 3.2 ) . [ thm : hgsol ] if @xmath352 then the hamiltonian system @xmath22 possesses a solution @xmath353 under the correspondence @xmath354 of constant parameters . the vector - valued function @xmath355 satisfies a certain linear pfaffian system @xmath356 of rank @xmath357 , whose fundamental solution is prepared by collecting admissible cycles along with the foregoing @xmath246-simplex ( [ eq : simplex ] ) . note that the linear space of these cycles , i.e. _ twisted de rham homology group _ , is generated by the chambers framed by the real section of branch locus of @xmath346 ; see @xcite . of course , theorem [ thm : hgsol ] is valid for any solution @xmath358 of @xmath356 . the fuchsian system @xmath323 is specialized as @xmath352 and @xmath359 along the above hypergeometric solution of @xmath22 ; it thus becomes reducible . in fact , via the gauge transformation @xmath360 we have a solution of the form @xmath361 where @xmath362 @xmath363 are holomorphic functions at @xmath8 defined by the integrals @xmath364 note that an @xmath365 matrix @xmath366 can be described by thomae s hypergeometric function @xmath367 . for details we refer to @xcite , in which a curious coincidence between @xmath368 and the _ lax pair _ of @xmath22 , i.e. the pair of the original fuchsian system ( [ eq : fs2 ] ) and its deformation equation ( [ eq : defe ] ) , is also discussed . our aim here is to generalize theorem [ thm : hgsol ] by application of schlesinger transformations starting from this hypergeometric solution at @xmath352 . notice that the schlesinger transformation established in theorem [ thm : schl ] shifts the constant parameters ( [ eq : const ] ) as @xmath369 while all the others are unchanged ; cf . the riemann scheme ( [ eq : rs ] ) . hence , by virtue of the algebraic relation between the solution of @xmath323 and the canonical variables of @xmath22 ( recall sect . [ subsect : ss ] and also ( [ eq : resmat ] ) and ( [ eq : canvar ] ) ) , we know _ in principle _ how to derive a solution @xmath370 of @xmath22 at @xmath371 for any positive integer @xmath30 even though the resulting expression in this way will be terribly complicated . in the next section we explore this problem to achieve much simpler formulae for these special solutions . this section is concerned with the schlesinger transform of the hypergeometric solution of @xmath22 . we present its explicit formula by using the block - toeplitz determinant whose entries are given by the hypergeometric functions . key ingredients of the argument are the determinantal representations for the approximation polynomials ; see sect . [ sect : det ] . moreover , we prove through fubini s theorem and the vandermonde determinant that these block - toeplitz determinants can be written in the form of iterated hypergeometric integrals . our result will be summarized in theorem [ thm : hgi ] , which is regarded as a generalization of theorem [ thm : hgsol ] , i.e. the previously known hypergeometric solution of @xmath22 . let @xmath1 be the functions defined by @xmath372 if the cycle @xmath349 is chosen such that @xmath373 , then @xmath374 is holomorphic at @xmath8 . for instance , it is enough to choose a bounded cycle as @xmath349 . accordingly , we have a power series expansion @xmath375 with the coefficients @xmath376 for @xmath377 . observe that each @xmath378 can be regarded as a moment @xmath379 of the ` measure ' @xmath380 upon the following notations : @xmath381 namely @xmath374 @xmath363 is written as the _ stieltjes transform _ @xmath382 of a function @xmath383 . in parallel , we introduce the functions @xmath384 also . we thus see that @xmath385 where @xmath386 the following linear relations ( contiguity relations ) hold : @xmath387 where @xmath388 denotes the down - shift operator with respect to @xmath389 defined by @xmath390 by definitions ( [ eq : hjk ] ) and ( [ eq : hj ] ) it is immediate to verify these formulae . + if @xmath349 is chosen to be the @xmath246-simplex ( [ eq : simplex ] ) : @xmath391 both ( [ eq : hjk ] ) and ( [ eq : hj ] ) are written by the hypergeometric function @xmath339 . let us introduce the function @xmath392 = \prod_{l=1}^{l-1 } \frac{\gamma(\alpha_l ) \gamma(\gamma_l-\alpha_l)}{\gamma(\gamma_l ) } \times f_{l , n } \left [ \begin{array}{c } { \boldsymbol \alpha } , { \boldsymbol \beta } \\ { \boldsymbol \gamma } \end{array } ; { \boldsymbol x } \right].\ ] ] then it holds by definition that @xmath393 \quad ( 0 \leq k \leq l-1).\ ] ] for example we have @xmath394 , \\ h\left [ \begin{array}{c } \alpha_{1}+j , \ldots , \alpha_{l-1}+j , { \boldsymbol \beta } \\ \gamma_1+j+1,\gamma_{2}+j , \ldots , \gamma_{l-1}+j \end{array } ; { \boldsymbol x } \right ] , \\ h\left [ \begin{array}{c } \alpha_{1}+j+1 , \alpha_{2}+j,\ldots , \alpha_{l-1}+j , { \boldsymbol \beta } \\ \gamma_1+j+1,\gamma_{2}+j+1 , \gamma_{3}+j , \ldots , \gamma_{l-1}+j \end{array } ; { \boldsymbol x } \right]\end{aligned}\ ] ] and so forth . it is convenient to prepare the notation of the block - toeplitz determinant ( cf . remark [ remark : bt ] ) for any @xmath0-tuple of nonnegative integers : @xmath395 let @xmath396 . we set @xmath397 \nonumber \\ & = ( -1)^{\sum_{i < j } n_in_j } \det \begin{bmatrix } a^0 _ { | { \boldsymbol n } | } ( n_0,| { \boldsymbol n } | ) \\ \vdots \\ a^{k-1}_{| { \boldsymbol n } |}(n_{k-1},| { \boldsymbol n } | ) \\ a^{k}_{| { \boldsymbol n } |-1}(n_k,| { \boldsymbol n } | ) \\ \vdots \\ a^{l-1}_{| { \boldsymbol n } |-1}(n_{l-1},| { \boldsymbol n } | ) \end{bmatrix } \label{eq : bt2}\end{aligned}\ ] ] for each @xmath169 @xmath398 . if @xmath399 , we fix @xmath400 . as well as ( [ eq : rtm ] ) , the symbol @xmath401 denotes the @xmath402 rectangular toeplitz matrix for the sequence @xmath403 whose top left corner is @xmath404 , and @xmath405 if @xmath150 . the second equality in ( [ eq : bt2 ] ) can be verified easily from ( [ eq : tenchi ] ) . henceforth we suppose @xmath406 unless expressly stated otherwise . we will often abbreviate @xmath407 for @xmath408 as @xmath409 . note that this convention is consistent with the description in remark [ remark : bt ] . we also prepare the canonical basis @xmath410 of @xmath411 , i.e. @xmath412 as seen in ( [ eq : special ] ) , the coefficient @xmath413 of the fuchsian system @xmath323 attached to the hypergeometric solution of @xmath22 with @xmath352 ( see theorem [ thm : hgsol ] ) is expressed as @xmath414 @xmath415 , where @xmath416 cf . ( [ eq : resmat ] ) . our next task is applying the schlesinger transformation to this fuchsian system . to derive the action of the schlesinger transformation , we need basically to deal with ( [ eq : atoa ] ) : @xmath417 namely , since both @xmath111 and @xmath115 are polynomials in @xmath271 , each residue matrix @xmath418 can be calculated by @xmath419 however , thanks to ( [ eq : aid ] ) , it is rather easy to calculate the diagonal parts even in the _ general _ case . first we will demonstrate it . the multiplier @xmath111 of the schlesinger transformation ( see theorem [ thm : schl ] ) can be written , a little more specifically than ( [ eq : r ] ) , as @xmath420 recall ( [ eq : smatrix ] ) . multiplying @xmath421 by @xmath84 from the left yields @xmath422 with @xmath423 as shown in theorem [ thm : schl ] , where both @xmath301 and @xmath424 are diagonal matrices independent of @xmath100 . we mention , without fear of repetition , that the hermite - pad approximation condition ( [ eq : hpa ] ) assures @xmath425 with @xmath116 denoting the first column of @xmath126 . we thus find the formula @xmath426 by virtue of remark [ remark : bt ] . combining this with ( [ eq : aid ] ) under @xmath328 : @xmath427 we arrive at the formulae @xmath428 for @xmath429 , where @xmath430 denotes the hirota differential with respect to @xmath431 . next we turn to the _ particular _ case , i.e. the fuchsian system @xmath323 upon the substitution ( [ eq : special ] ) corresponding to the hypergeometric solution ( see theorem [ thm : hgsol ] ) . write the residue matrices as @xmath432 @xmath415 . in order to reconstruct the canonical variables @xmath370 of the schlesinger transform of @xmath22 , it is only necessary to know the quantities @xmath433 because we have already known from ( [ eq : dhat ] ) the diagonal entries @xmath434 for @xmath429 ; cf . ( [ eq : canvar ] ) . calculations of ( [ eq : hatc ] ) are a little complicated ; therefore , we will separately carry out the cases @xmath435 and @xmath436 in sects . [ subsect : cal2 ] and [ subsect : cal3 ] , respectively . the result can be found in sect . [ subsect : result ] . consider the row vector @xmath437 in view of ( [ eq : hatai ] ) and @xmath278 , where @xmath438 ; recall ( [ eq : bchg ] ) . [ [ section-4 ] ] it follows from the determinantal representation of the polynomial @xmath439 and @xmath440^{m-1}_{0}$ ] for @xmath441 ( see proposition [ prop : detp ] and its sequel ) that @xmath442 summation over @xmath443 of this formula entails @xmath444 via the contiguity relation ( [ eq : cont1 ] ) . hence the @xmath445th component of @xmath437 reads @xmath446 here we have used ( [ eq : np0 ] ) . [ [ section-5 ] ] similarly it holds for @xmath447 that @xmath448 taking a sum and using ( [ eq : cont1 ] ) thus yield @xmath449 here we have used ( [ eq : npk ] ) . [ [ section-6 ] ] the ratio of ( [ eq : p01 ] ) and ( [ eq : pk1 ] ) leads to the formula @xmath450 consider for @xmath429 the row vector @xmath451 in view of ( [ eq : hatai ] ) and @xmath452 , which is nothing but the top row of the matrix @xmath453 due to @xmath454 ; recall ( [ eq : bchg ] ) . [ [ section-7 ] ] we are interested in the following determinant @xmath455 subtracting the @xmath45th column multiplied by @xmath456 from the @xmath457th column for @xmath458 sequentially and using the contiguity relation ( [ eq : cont2 ] ) , we thus obtain @xmath459 [ [ section-8 ] ] in the same way as above , we find @xmath460 [ [ section-9 ] ] hence we verify from the ratio of ( [ eq : p0x ] ) and ( [ eq : pkx ] ) that @xmath461 summarizing the above we are led to the following result . [ thm : hgi ] let @xmath30 be a positive integer . if @xmath371 then the hamiltonian system @xmath22 possesses a solution @xmath370 given by @xmath462 under the correspondence @xmath463 of constant parameters , where @xmath464 . we have in fact an alternative expression @xmath465 of the above solution with no use of the hirota differentials . this formula can be verified by calculating @xmath466 in the same manner as sects . [ subsect : cal2 ] and [ subsect : cal3 ] ; further details might be left to the reader . as seen in theorem [ thm : hgi ] , we have constructed a particular solution of @xmath22 expressed in terms of the block - toeplitz determinant @xmath407 whose entries are given by the hypergeometric functions . finally we shall rewrite @xmath407 as an iterated hypergeometric integral : if we remember that @xmath467 is defined as a moment ( see sect . [ subsect : prel ] ) , then we observe that @xmath468 _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_0 \end{subarray } } , \ldots , \left [ { s_{k-1,j}}^{n_{k-1}+i - j } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_{k-1 } \end{subarray } } , \left [ { s_{k , j}}^{n_{k}+i - j-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_{k } \end{subarray } } , \ldots , \left [ { s_{l-1,j}}^{n_{l-1}+i - j-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_{l-1 } \end{subarray } } \right ] \\ & \nonumber \qquad \times \prod_{a=0}^{l-1 } \prod_{j=1}^{n_a } { \rm d}\mu_a(s_{a , j } ) \\ \nonumber & = \int \det \left [ \left [ { s_{0,j}}^{i-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_0 \end{subarray } } , \left [ { s_{1,j}}^{i-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_1 \end{subarray } } , \ldots , \left [ { s_{l-1,j}}^{i-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_{l-1 } \end{subarray } } \right ] \prod_{a=0}^{k-1 } \prod_{j=1}^{n_a } { s_{a , j } } \\ & \qquad \times \prod_{a=0}^{l-1 } \prod_{j=1}^{n_a } { s_{a , j}}^{n_a - j } { \rm d}\mu_a(s_{a , j } ) \label{eq : intdelta}\end{aligned}\ ] ] through fubini s theorem . the vandermonde determinant shows that @xmath469 _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_0 \end{subarray } } , \left [ { s_{1,j}}^{i-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_1 \end{subarray } } , \ldots , \left [ { s_{l-1,j}}^{i-1 } \right ] _ { \begin{subarray}{l } 1 \leq i \leq |{\boldsymbol n}|\\ 1 \leq j \leq n_{l-1 } \end{subarray } } \right ] = \prod_{(a , b)>(c , d ) } ( s_{a , b}-s_{c , d}),\ ] ] where @xmath470 is defined to mean @xmath471 also it holds that @xmath472 where @xmath473 denotes the symmetric group on @xmath474 . therefore , because the value of integral ( [ eq : intdelta ] ) is invariant under a permutation of the variables @xmath475 for each @xmath476 , we conclude that @xmath477 usually , we often normalize the fuchsian system ( [ eq : fs2 ] ) so that @xmath479 ( the residue matrix at @xmath96 ) is diagonal when considering its isomonodromic deformation ; see e.g. @xcite . but , in this paper , we adopt a different normalization treating the two points @xmath94 and @xmath96 equally ; i.e. we choose @xmath91 ( the residue matrix at @xmath94 ) and @xmath97 to be upper and lower triangular , respectively . note that the latter normalization is more versatile in a general setting than the former . in this appendix , for a supplement to sect . [ subsect : ss ] , we demonstrate how to determine the coefficient @xmath297 of the deformation equation ( [ eq : defe ] ) : @xmath480 of ( [ eq : fs2 ] ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath482 if @xmath483 then @xmath484 is holomorphic at @xmath289 ; + @xmath482 if @xmath485 then @xmath486 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ similarly , we observe that @xmath487 and @xmath488 here we have used the assumption that the connection matrix @xmath101 does not depend on @xmath489 . consequently , @xmath490 is a rational function matrix in @xmath271 that has only a simple pole at @xmath289 with residue @xmath491 and thus @xmath492 where @xmath493 is a constant matrix . substituting @xmath96 ( @xmath8 ) in ( [ eq : solatinfty ] ) shows that @xmath493 is a lower triangular matrix . therefore , substituting @xmath94 in ( [ eq : solatzero ] ) , we conclude that @xmath494 . the authors are deeply grateful to shuhei kamioka for giving them an exposition on various multi - dimensional continued fractions . they appreciate satoshi tsujimoto his kind information about literature on rational approximations . also , they have benefited from invaluable discussions with yasuhiko yamada . this work was supported in part by a grant - in - aid from the japan society for the promotion of science ( grant number : 25800082 and 25870234 ) . haraoka , y. : regular coordinates and reduction of deformation equations for fuchsian systems . in : balser , w. , filipuk , g. , ysik , g. , michalik , s. ( eds . ) , formal and analytic solutions of differential and difference equations , pp . 3958 . polish acad . inst . math . , warsaw ( 2012 ) noumi , m. , tsujimoto , s. , yamada , y. : pad interpolation for elliptic painlev equation . in : iohara , k. , morier - genoud , s. , rmy , b. ( eds . ) , symmetries , integrable systems and representations , pp . springer , london ( 2013 )

we develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations . we show that a certain duality in hermite s two approximation problems for functions leads to the schlesinger transformations , i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant . since approximants and remainders are described by block - toeplitz determinants , one can clearly understand the determinantal structure in isomonodromic deformations . we demonstrate our method in a certain family of hamiltonian systems of isomonodromy type including the sixth painlev equation and garnier systems ; particularly , we present their solutions written in terms of iterated hypergeometric integrals . an algorithm for constructing the schlesinger transformations is also discussed through vector continued fractions .