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large - distance qcd remains an area , where the concepts of perturbation theory can not be directly applied . to assess this region and make reliable predictions for hadronic processes , the pure perturbative treatment has to be amended by nonperturbative input . in a series of recent papers @xcite , three of us have outlined an approach , based on qcd sum rules with nonlocal condensates @xcite , capable of providing a pion distribution amplitude ( da ) compatible at @xmath2 with the cleo data @xcite on the pion - photon transition . the key feature of this pion da is that its endpoint regions @xmath3 ( @xmath4 being the parton s longitudinal momentum fraction ) are strongly suppressed . this suppression is controlled by the nonlocality of the scalar quark condensate , parameterized by the average quark virtuality @xmath5 in the vacuum , with theoretical estimates in the range @xmath6 @xcite and a preferable value of @xmath7 gev@xmath8 extracted in @xcite from the cleo data . in addition , one can improve the quality of perturbatively calculable observables , notably the factorized hard contribution of the pion s electromagnetic form factor , by trading the traditional power - series perturbative expansion for a non - power - series ( in an analytic ) coupling expansion that avoids _ eo ipso _ the landau singularity rendering all expressions infrared ( ir ) finite @xcite . suffice it here to say that this is achieved through the inclusion into the running coupling of a power - behaved term of nonperturbative origin that removes the landau ghost leaving the ultraviolet behavior of the effective coupling unchanged . crucial for making this analytic approach possible , is the generalization of the analytic running - coupling concept , proposed by shirkov and solovtsov @xcite , to the level of observables depending on more than one scheme scales @xcite , as is , for example , the case for the pion form factor in fixed - order perturbation theory beyond lo @xcite , or performing a sudakov resummation @xcite . along these lines of thoughts , we describe in this contribution our recent works on the pion da , summarizing the main results , and present predictions for the pion s electromagnetic form factor carried out under the imposition of analyticity of the running coupling and its powers using two different procedures . it turns out that if the powers of the coupling have their own analytic ( dispersive ) images , the factorizable hard part of the form factor so calculated bears a minimal dependence on the scheme and scale - setting choice . including also the soft contribution via local duality , this helps improving the quality of the prediction beyond the level of the current experimental - data accuracy . in the context of factorization of hard exclusive processes @xcite , the pion da is a universal , gauge - invariant quantity defined at the twist-2 level by @xmath9 where @xmath10 , @xmath11 mev is the pion decay constant , and @xmath12 $ ] preserves gauge invariance . @xmath13 encapsulates the nonperturbative qcd pion structure in terms of the distribution of the longitudinal momentum fractions between its two valence partons : quark ( @xmath4 ) and antiquark ( @xmath14 ) . together with the da of its first resonance , @xmath15 , it can be related to the nonlocal condensates by means of a sum rule , based on the correlator of two axial currents ( see @xcite ) . due to the finiteness of the vacuum correlation length @xmath16 , the end - point regions @xmath17 are strongly suppressed and by virtue of this fact we can @xcite determine quite accurately the first ten moments @xmath18 of the pion da and _ independently _ also the inverse moment @xmath19 . given that @xmath20 rapidly with increasing @xmath21 , the eigenfunctions decomposition @xmath22 \label{eq : phi024mu0}\end{aligned}\ ] ] can be practically truncated at @xmath23 because all higher coefficients are negligible @xcite . the `` bunch '' of the pion das shown in fig . 1(a ) , parameterized by @xmath24 and @xmath23 , turns out to match all moment constraints for @xmath25 and @xmath26 extracted from the cleo data . the optimum sample out of this `` bunch''bms model@xcite , has at @xmath27 @xmath28 and @xmath29 and is shown in fig . let us close this section with a forward - looking statement : the bms `` bunch '' pion das , though doubly peaked , have their endpoints ( @xmath30 ) strongly suppressed not only relative to @xmath31 but even compared to @xmath32 , substantially reducing the importance of sudakov effects . : bms model @xcite solid line ; cz model @xcite dashed line ; asymptotic da dotted line . ( b ) bms `` bunch '' @xcite in comparison with @xmath33 , @xmath34 ( dashed line ) @xcite , @xmath35 ( dash - dotted line ) @xcite . [ fig : pi - das],title="fig:",scaledwidth=47.0% ] : bms model @xcite solid line ; cz model @xcite dashed line ; asymptotic da dotted line . ( b ) bms `` bunch '' @xcite in comparison with @xmath33 , @xmath34 ( dashed line ) @xcite , @xmath35 ( dash - dotted line ) @xcite . [ fig : pi - das],title="fig:",scaledwidth=47.0% ] it was shown in @xcite at lo and later extended @xcite to nlo of qcd perturbation theory @xcite that the light - cone qcd sum - rule method allows to perform all calculations in the @xmath36 form factor for sufficiently large @xmath37 and analytically continue the results to the limit @xmath38 , hence avoiding problems arising when a photon becomes real . recently @xcite , we have revised and refined this sort of data processing accounting for a correct erbl @xcite evolution of the pion da , including thresholds effects in the running coupling , estimating more accurately the contribution of the twist-4 contribution , and improving the error estimates in determining the @xmath2- and @xmath39-error contours . avoiding here technical details , we gather the results of our analysis in fig . the predictions shown correspond to the following pion da models with associated @xmath40 deviations and designations for the form - factor predictions displayed in the right panel : @xmath41 ( , @xmath42 , upper dashed line ) @xcite ; bms-``nonlocal qcd srs bunch '' ( shaded rectangle ) , @xmath43 ( , @xmath2left panel , shaded strip right panel ) @xcite ; three instanton - based models , viz . , @xcite ( , @xmath44 , dotted line ) , @xcite ( , @xmath39 , dash - dotted line ) , and @xcite ( @xmath45 , @xmath44only left panel ) ; and the asymptotic pion da @xmath46 ( , @xmath44 , lower dashed line ) . a recent transverse lattice result @xcite ( , @xmath39 ) is also shown left panel only . cleo - data analysis in terms of error contours in the ( @xmath24,@xmath23 ) plane . the line assignments are : broken line@xmath2 ; solid line@xmath39 ; dash - dotted line@xmath44 . various pion das are shown , evaluated at @xmath47 gev@xmath8 after nlo erbl evolution .. the slanted shaded rectangle represents the nonlocal qcd sum - rule constraints on ( @xmath48 ) @xcite for @xmath49 gev@xmath50 . ( b ) light - cone sum - rule predictions for @xmath51 in comparison with the cello ( diamonds , @xcite ) and the cleo ( triangles , @xcite ) data evaluated with @xmath52 gev@xmath8 @xcite . [ fig : pi - photonff],title="fig:",scaledwidth=47.0% ] cleo - data analysis in terms of error contours in the ( @xmath24,@xmath23 ) plane . the line assignments are : broken line@xmath2 ; solid line@xmath39 ; dash - dotted line@xmath44 . various pion das are shown , evaluated at @xmath47 gev@xmath8 after nlo erbl evolution .. the slanted shaded rectangle represents the nonlocal qcd sum - rule constraints on ( @xmath48 ) @xcite for @xmath49 gev@xmath50 . ( b ) light - cone sum - rule predictions for @xmath51 in comparison with the cello ( diamonds , @xcite ) and the cleo ( triangles , @xcite ) data evaluated with @xmath52 gev@xmath8 @xcite . [ fig : pi - photonff],title="fig:",scaledwidth=47.0% ] to summarize , the main results obtained in @xcite are : ( i ) both das , @xmath32 @xcite and @xmath31 @xcite are disfavored by the cleo data at @xmath44 and @xmath42 , respectively . in contrast , @xmath53 lies within the @xmath2-error ellipse . model das from instanton - based approaches @xcite are close to but still outside the @xmath39 region . ( ii ) the extracted coefficients @xmath54 and @xmath55 are rather sensitive to the strong radiative corrections and the size of the twist-4 contribution . ( iii ) the value of the vacuum nonlocality extracted from the cleo data is @xmath56 gev@xmath50 . turning to the form - factor predictions , one observes from fig . 2 ( right ) that the bms `` bunch '' of pion das is in good agreement with the cleo data @xcite but also with the cello data @xcite , while the behavior of rival das reflects the situation shown in the left panel : the prediction from the cz model overshoots the data considerably , while that from @xmath32and das close to it are underestimating both sets of experimental data . the crucial new elements of the calculation below are : ( i ) use of the bms pion da , ( ii ) application of two - loop analytic perturbation theory ( apt ) , and ( iii ) a more accurate way , based on local duality , to join the soft part with the hard form - factor contribution . the pion s electromagnetic form factor can be generically written as @xcite @xmath57 where @xmath58 is the factorized part within pqcd and @xmath59 is the `` soft '' part containing subleading power - behaved ( e.g. , twist-4 ) contributions originating from nonperturbative effects . the leading - twist factorizable contribution can be expressed as a convolution in the form @xmath60 , where @xmath61 is the factorization scale between the long- and short - distance dynamics , @xmath62 stands for the renormalization scale . the hard - scattering amplitude , @xmath63 , describing short - distance interactions at the parton level , has been evaluated to nlo accuracy ( @xcite and references cited therein ) using the terminology introduced in @xcite to which we refer for details . then , one obtains @xmath64 , where the lo and nlo terms read , respectively , @xmath65 ^ 2 \ , , \label{eq : q2pfflo}\end{aligned}\ ] ] @xmath66 . \label{eq : q2pffnlo}\ ] ] here @xmath67 marks the maximal number of gegenbauer harmonics taken into account and the calligraphic designation denotes quantities with their @xmath68-dependence pulled out . note that because we take into account the nlo evolution of the pion da , the displayed terms contain diagonal ( d ) as well as ( the nlo term ) non - diagonal ( nd ) components the effects of the lo da evolution are crucial @xcite , while the nlo ones are relatively of less importance . hence , we set here : @xmath69 studying @xmath70 beyond the lo requires an optimal renormalization scheme and scale setting in order to minimize the influence of higher - order loop corrections ( see @xcite for a fully fledged discussion ) . to join the hard with the soft contribution ( the latter being calculated with the aid of local duality ( ld ) , we have to correct the low-@xmath71 behavior of the factorizable part to fulfill the ward identity at @xmath72 , i.e. , @xmath73 with @xmath74 obtained with the bms pion da using standard pqcd within the @xmath75 scheme and adopting @xmath76 ( dashed line ) . the solid line corresponds to the @xmath77 scale setting introduced in @xcite . the experimental data are taken from ( diamonds ) and @xcite ( triangles ) . ( b ) prediction for @xmath78 calculated with the `` maximally analytic '' procedure and with the bms `` bunch '' of pion das . [ fig : softff ] ] the next step is to apply for the calculation of @xmath79 apt . this is done by employing two different analytization procedures : ( i ) a _ maximally analytic _ prescription @xcite , meaning that analyticity has been imposed not only on the coupling , but also on its powers , which , therefore , have their own dispersive images . this amounts to @xmath80_{\rm maxan } \ = \bar{\alpha}_{\rm s}^{(2)}(\mu_{\rm r}^{2})\ , { \cal f}_{\pi}^{\rm lo}(q^2 ) + \frac{1}{\pi}\ , { \cal a}_{2}^{(2)}(\mu_{\rm r}^{2})\ , { \cal f}_{\pi}^{\rm nlo}(q^2;\mu_{\rm r}^{2})\ , , \label{eq : pffmaxan}\ ] ] where @xmath81 is the two - loop analytic coupling and @xmath82 the analytic version of its second power in two - loop order @xcite . ( ii ) another procedure , we call @xcite _ naive analytic _ , replaces in @xmath83 the strong coupling and its powers by the analytic coupling @xmath84 and its powers @xmath85 ^ 2 $ ] , entailing the requirement @xcite @xmath80_{\rm naivan } \ = \bar{\alpha}_{\rm s}^{(2)}(\mu_{\rm r}^{2})\ , { \cal f}_{\pi}^{\rm lo}(q^2 ) + \frac{1}{\pi}\ , \left[\bar{\alpha}_{\rm s}^{(2)}(\mu_{\rm r}^{2})\right]^2\ , { \cal f}_{\pi}^{\rm nlo}(q^2;\mu_{\rm r}^{2})\ , . \label{eq : pffnaivan}\ ] ] the results for @xmath86 vs. the experimental data are displayed in fig . 3(b ) and fig . 4 . using apt and the bms da in conjunction with the `` naive analytic '' ( a ) and `` maximally analytic '' ( b ) procedures : @xmath87 scheme and @xmath76 ( dashed line ) ; blm ( dotted line ) ; @xmath88 ( solid line ) ; @xmath89-scheme ( dash - dotted line ) . the single solid line in panel ( b ) shows the prediction for the soft form - factor part ; below this , the corresponding hard contributions are displayed . [ fig : pidatasum],scaledwidth=86.0% ] the bms pion das @xcite successfully pass the comparison with the cleo data @xcite at the @xmath2 level , as highlighted in fig . 2 ( conforming also with the cello data @xcite ) . employing 2-loop apt naive and maximal we have calculated the hard part of the electromagnetic pion form factor within various renormalization schemes and using different scale settings . joining the hard part with the soft one on the basis of local duality , we have derived predictions that reproduce the available data rather well , especially using the `` maximally analytic '' procedure ( fig . 3(b ) ) . moreover , we found that this procedure minimizes the influence of scheme and scale - setting ambiguities on the form - factor predictions . 0 a.p . bakulev , s.v . mikhailov and n.g . stefanis , phys . * b508 * ( 2001 ) 279 . bakulev , s.v . mikhailov and n.g . stefanis , phys . d * 67 * ( 2003 ) 074012 ; phys . * b578 * ( 2004 ) 91 ; hep - ph/0310267 ; hep - ph/0312141 . mikhailov and a.v . radyushkin , jetp lett . * 43 * ( 1986 ) 712 ; sov . j. nucl . * 49 * ( 1989 ) 494 ; phys . d * 45 * ( 1992 ) 1754 ; a.p . bakulev and a.v . radyushkin , phys . * b271 * ( 1991 ) 223 ; s.v . mikhailov , phys . * 56 * ( 1993 ) 650 . j. gronberg _ et al . _ , phys . d * 57 * ( 1998 ) 33 . bakulev and s.v . mikhailov , phys . d * 65 * ( 2002 ) 114511 . shirkov and i.l . solovtsov , phys . * 79 * ( 1997 ) 1209 . shirkov , theor . * 127 * ( 2001 ) 409 ; eur . j. * c22 * ( 2001 ) 331 ; d.v . shirkov and i.l . solovtsov , phys . nucl . * 32s1 * ( 2001 ) 48 . a.i . karanikas and n.g . stefanis , phys . * b504 * ( 2001 ) 225 ; n.g . stefanis , lect . notes phys . * 616 * ( 2003 ) 153 . bakulev , k. passek - kumeriki , w. schroers and n.g . stefanis , phys . * d 70 * ( 2004 ) 033014 . stefanis , w. schroers and h.c . kim , phys . * b449 * ( 1999 ) 299 ; eur . j. * c18 * ( 2000 ) 137 . efremov and a.v . radyushkin , phys . * b94 * ( 1980 ) 245 ; theor . * 42 * ( 1980 ) 97 ; g.p . lepage and s.j . brodsky , phys . d * 22 * ( 1980 ) 2157 . chernyak and a.r . zhitnitsky , phys . rep . * 112 * ( 1984 ) 173 . m. praszalowicz and a. rostworowski , phys . d * 64 * ( 2001 ) 074003 . dorokhov , jetp lett . * 77 * ( 2003 ) 63 . a. khodjamirian , eur . j. * c6 * ( 1999 ) 477 . a. schmedding and o.i . yakovlev , phys . d * 62 * ( 2000 ) 116002 . f. del aguila and m.k . chase , nucl . b193 * ( 1981 ) 517 ; e.p . kadantseva , s.v . mikhailov , and a.v . radyushkin , sov . j. nucl . * 44 * ( 1986 ) 326 . petrov _ et al . _ , phys . d * 59 * ( 1999 ) 114018 . anikin , a.e . dorokhov and l. tomio , phys . * 31 * ( 2000 ) 509 . s. dalley and b. van de sande , phys . d * 67 * ( 2003 ) 114507 . behrend _ et al . _ [ cello collaboration ] , z. phys . * c49 * 401 ( 1991 ) 401 . b. meli , b. nii and k. passek , phys . d * 60 * ( 1999 ) 074004 . j. volmer _ et al . _ , phys . * 86 * ( 2001 ) 1713 . brown _ et al . _ , d * 8 * ( 1973 ) 92 ; c.j . bebek _ et al . _ , d * 13 * ( 1976 ) 25 .

a pion distribution amplitude , derived from nonlocal qcd sum rules , has been employed to calculate @xmath0 using light - cone sum rules , and @xmath1 in nlo qcd perturbation theory . predictions are presented for both observables and found to be in good agreement with the corresponding data . calculating the hard pion form factor by analytic perturbation theory to two - loop order , it is shown that the renormalization - scheme and scale - setting dependencies are diminished .

much of the predictive power of quantum chromodynamics ( qcd ) is provided by universality of the non - perturbative functions , in particular , the parton distribution functions ( pdfs ) , in factorization theorems for hard processes . with the aid of factorization and perturbative calculation of short - distance dynamics , the universality allows us to extract a set of pdfs from some reactions and then use them to predict observables in other reactions . knowledge of pdfs is critical for testing qcd dynamics in asymptotic region at existing facilities , as well as , for making predictions for future facilities , like the large hadron collider ( lhc ) . it is also essential for exploring non - perturbative qcd dynamics when the extracted pdfs are compared with what calculated in lattice qcd or in effective field theory approaches . with only one identified hadron , structure functions of inclusive lepton - hadron deeply inelastic scattering ( dis ) are clean observables for extracting pdfs . after more than 30 years of continuous effort , and many generations of machines and detectors , measurements of proton structure functions have become the benchmark tests of qcd dynamics . with the hera at desy , we are able to explore the kinematic region with the bjorken @xmath1 as low as @xmath2 while staying the dis regime . the continuous growth of structure functions as @xmath1 decreases raises an urgent question : when such growth will hit the unitarity limit and slow down ? the knowledge of structure functions and low @xmath0 physics is extremely important for testing qcd dynamics and our ability to explore new physics beyond the standard model . our working group had a total of 46 talks divided into 9 sessions including one joint session with electroweak and beyond the standard model working group . in this writeup , we summarize the recent achievements , progresses , and open questions that were presented at our working group meetings . we organize this summary into seven parts : 1 . structure function measurements at low @xmath0 2 . structure functions and pdfs at high @xmath0 3 . progress in the determination of pdfs 4 . toward qcd precision tests 5 . low-@xmath0 physics : parton evolution and saturation 6 . nuclear structure functions and nuclear pdfs 7 . new approaches to pdfs measurements of the proton structure function , @xmath3 , in neutral current ( nc ) deep inelastic scattering ( dis ) at hera are vital for testing the predictions of perturbative qcd and in the determination of the parton distribution functions of the proton . recent results from the two general - purpose detectors , h1 and zeus , cover five orders of magnitude in the photon virtually , @xmath4 , and in the bjorken scaling variable , @xmath0 @xcite . the possibility afforded by hera of studying the structure functions down to values of @xmath0 as low as @xmath5 is important , as it gives access to partons which have undergone a large number of qcd branching processes . the density of these partons , both gluons and the so - called `` sea '' quarks has be found to increase dramatically as @xmath0 decreases , which may indicate the need to taken into account non - linear effects in qcd evolution , such as saturation . the double - differential cross section for inclusive nc dis is given by : @xmath6 where @xmath7 , in which @xmath8 is the inelasticity , @xmath9 is the total squared center - of - mass energy and @xmath3 , @xmath10 and @xmath11 are the structure functions of the proton . the quantity , @xmath12 is also defined in this equation , which is known as the reduced cross section . although the precision measurements of @xmath3 exist over such a large kinematic range , the same can not be said for the longitudinal structure function , @xmath10 , which has not been directly measured at hera . @xmath10 is directly sensitive to scaling violations and hence to the gluon content of the proton and is therefore crucial to our understanding of proton structure . the final term in equation contains @xmath13 , or the parity violating structure function . this structure function is , however , only important at high @xmath4 and will not be considered further in this section , where the interest is primarily in low @xmath4 structure function measurements . gev@xmath14 from the h1 , zeus and nmc collaborations . ] the rise of @xmath3 with decreasing @xmath0 persists down to very low values of @xmath4 @xcite , although it is known that as @xmath15 , @xmath16 constant @xmath4 , as it must in order to satisfy the conservation of the electromagnetic current . it is also known that around @xmath17 gev@xmath18 , perturbative qcd begins to breakdown and phenomenological models must be invoked to explain the behavior of the @xmath3 data . from an experimental point of view , accessing very low values of @xmath4 is technically challenging , but has been achieved by the hera experiments using a number of different techniques . the h1 collaboration presented recent measurements of the reduced cross section at low @xmath4 using two of these techniques , namely , via the identification of qed compton events @xcite and using a small sample of data in which the interaction vertex was intentionally shifted by @xmath19 cm toward the outgoing proton beam direction @xcite , effectively extending the acceptance of the h1 detector to values of @xmath4 down as low as @xmath20 gev@xmath18 . using this so - called `` shifted vertex '' data sample , they have also specifically identified events in which an energetic photon was emitted by the incoming lepton prior to its interaction with the proton ; these initial - state radiative ( isr ) events give access not only to even lower values of @xmath4 , but also to higher values of @xmath0 , giving a wide coverage in @xmath0 at low @xmath4 . , obtained from fits of the form @xmath21 to low @xmath0 data ] these measurements are shown in figure [ fig : lowq2 ] , in which it can be seen that @xmath3 , and hence the reduced cross section , rises with decreasing @xmath0 , even at low values of @xmath4 . the only exception to this behavior is at the very lowest values of @xmath0 , at which the contribution to the reduced cross section from the longitudinal structure function , @xmath10 , becomes significant , causing @xmath12 to decrease . as can be seen in equation , the contribution to the reduced cross section from @xmath10 is suppressed for all but the highest values of @xmath22 ( low @xmath0 ) . this behavior can be exploited to perform an extraction of @xmath10 , albeit in a model - dependent way . the resulting @xmath10 points were also shown at this workshop @xcite and are already able to discriminate between different pdf parameterizations . the low @xmath4 @xmath3 data can be fitted in order to quantify the change of the low @xmath0 slope of @xmath3 with @xmath4 . figure [ fig : lam ] shows the result of just such a fit , performed for @xmath23 by the h1 collaboration . the expected change in behavior around @xmath17 gev@xmath14 , is clearly observed . structure functions at high @xmath0 region has brought many attentions at this workshop . recently , it has been realized that it is very important to understand this region in order to achieve precise electroweak measurements and to extract new physics signals from hera , tevatron and lhc at high @xmath4 region . a large uncertainty on the pdfs at very high @xmath0 and low @xmath4 region can make a big impact on the high @xmath4 region even at intermediate @xmath0 due to the effect of dglap evolution . most precise data on high @xmath0 come from the traditional fixed target experiments ( slac / bcdms / nmc ) . but their high @xmath0 data corresponds to low @xmath4 region where we face many challenges in understanding all non - perturbative qcd and nuclear effects . one clean way is to probe the structure functions at high @xmath0 and @xmath4 directly . both h1 and zeus @xcite showed measurements of the cross sections for neutral and charged - current scattering as a function of @xmath4 using polarized beams . the measured cross sections are well described by the standard model . but more data is required to constrain parton distributions functions at high @xmath0 . the zeus @xcite presented a very promising method to probe the pdfs up to @xmath24 using the jet information ( @xmath25 and @xmath26 ) to calculate the value of @xmath0 . events with no jets reconstructed within their fiducial volume is assumed to come from very high @xmath27 to 1 . the measured cross sections using early dataset show good agreements with the predictions using cteq6d pdfs , shown in figure [ fig : zeus_highx ] . however , their highest @xmath0 data tend to be higher than the predictions . thus , it would be interesting to see their results using a full dataset . the ratio of the zeus differential cross sections data and predictions using cteq6d pdfs . ] the nutev @xcite presented their final differential cross sections using neutrino - iron scattering . the extracted @xmath3 and @xmath13 from their differential cross sections are 20% higher than the ccfr measurements , and 10 - 15% higher than the bcdms @xmath3 , as shown in figure [ fig : f2_nutev ] . they explained that two third of the difference between the nutev and ccfr measurements is due to an improved calibration of the magnetic field , and a better modelling in monte carlo . this result implies that that nuclear effect in neutrino scattering is different from that in the charged lepton scattering at high @xmath0 . thus , we need to resolve this difference before the nutev data can be used in a global pdfs analysis to constrain the pdfs at high @xmath0 . it would be interesting to see their qcd fit results . a possible difference in the nuclear effect can be resolved by the chorus data on the lead target , and future minerva / minos results . [ left ] and @xmath13[right ] data are compared with ccfr data and the nlo prediction with mrst2001e.,title="fig : " ] [ left ] and @xmath13[right ] data are compared with ccfr data and the nlo prediction with mrst2001e.,title="fig : " ] the ratio of @xmath28 and @xmath29 quarks at high @xmath0 primary comes from the measurements of @xmath3(deuterium ) and @xmath3(proton ) . because of a large uncertainty of nuclear binding effect on deuterium target , this ratio is poorly known . figure [ fig : du][left ] shows the nmc @xmath30/@xmath31 with and without nuclear correction @xcite . with a nuclear correction , the nmc data favors 0.2 for @xmath32 as @xmath33 , which is of theoretical interest for nuclear physics community . however , the size of nuclear binding correction is still controversial . s.kuhn @xcite presented dedicated jlab efforts to study @xmath32 at high @xmath0 . their programs are to study the effect of nuclear binding on neutron structure , and to measure the structure functions of a free neutron by detecting a slow spectator proton . information on @xmath32 can be also extracted from @xmath34 production data at the tevatron . the cdf collaboration @xcite measured the forward - backward charge asymmetry of electrons from @xmath34 boson decays . in order to get a better @xmath32 sensitivity on higher @xmath0 region , they have looked at a higher electron @xmath35 region . fig [ fig : du][right ] shows comparisons with the nlo resbos predictions using cteq6 m and mrst 2001 pdfs . at high @xmath36 , the cdf data tends to favor higher @xmath32 value at high @xmath0 . thus , it would be interesting to compare with the pdfs which was extracted , assuming a large nuclear correction on the deuterium target . they expect to have a big improvement on this measurement by reconstructing @xmath34 rapidity directly . the ratio of @xmath3 data on the deuterium and hydrogen targets with and without the nuclear corrections . [ right ] the cdf lepton charge asymmetry is compared with predictions with cteq6 m and mrst02 pdfs using a nlo resbos calculation.,title="fig : " ] the ratio of @xmath3 data on the deuterium and hydrogen targets with and without the nuclear corrections . [ right ] the cdf lepton charge asymmetry is compared with predictions with cteq6 m and mrst02 pdfs using a nlo resbos calculation.,title="fig : " ] at the workshop , one of the hot subjects was the phenomenon of a parton - hadron duality which states that the average behavior of the nucleon resonances follows the dis scaling limit curve . jlab has very precise data at high @xmath0 and low @xmath4 region ( where a resonance production occurs ) . besides many theoretical issues discussed by s. liuti @xcite , c. keppel and i. niculescu @xcite demonstrated that the duality holds for @xmath3 proton , emc effect , and even spin structure functions in the region of @xmath37 gev@xmath14 . issues are whether non - perturbative power corrections between dis and resonance region is same , and dglap evolution & factorization works in the resonance region too . s. liuti s studies suggest that the size of the higher twist effect may not be same . however , a. bodek @xcite showed that all dis @xmath3 data and jlab s resonance data are well described by his bodek - yang leading - order model , as shown in figure [ fig : bodek ] . this implies that there is not much difference in the power corrections between two regions . this model uses a new scaling variable @xmath38 and @xmath4 dependent @xmath39 factors to all pdfs to describe both pqcd and non - perturbative qcd regions very smoothly . proton [ left ] , and @xmath10 proton[right].,title="fig : " ] proton [ left ] , and @xmath10 proton[right].,title="fig : " ] a community of neutrino oscillation physics have started to pay attention on this non - perturbative qcd region . for precise measurements of mass splitting and mixing angles , neutrino oscillation experiments ( minos , no@xmath40a , and t2k ) need to have a good understanding of neutrino cross section at low energy . this point was well presented by h. gallagher @xcite . certainly this would be a place where dis , nuclear physics , and neutrino physics communities need to make a coherent effort . at the end of the workshop , pdf uncertainties at the tevatron and impact on various measurements are presented by f. chlebana , and a. harel @xcite . chelbana discussed many ideas to constrain the pdfs using the tevatron data . figure [ fig : jet ] shows the latest status of the tevatron jet data with the nlo predictions where there is no observed discrepancy at high @xmath35 region . these measurements are currently dominated by the jet energy scale uncertainties . in future , it would be important to separate pdf effects from any new physics signal . f. chlebana he also pointed out that it is crucial to measure the size of heavy flavor quarks densities for higgs and top physics . algorithm [ left ] and d0 data using cone algorithm[right].,title="fig : " ] algorithm [ left ] and d0 data using cone algorithm[right].,title="fig : " ] the structure functions discussed in the previous section can be expressed in terms of the parton distribution functions ( pdfs ) of the proton . the structure function , @xmath41 is proportional to the sum of the quark and antiquark pdfs . at low @xmath0 , @xmath3 is therefore sensitive to the sea quark distributions and hence is indirectly sensitive to the gluon density . the longitudinal structure function , @xmath10 , is directly sensitive to the gluon density . the parity - violating structure function , @xmath42 , is proportional to the difference between the quark and antiquark pdfs , making it sensitive to the valence quark distributions . the pdfs of the proton may be extracted by fitting , among other things , the hera structure function data . these fits have been performed by a number of different groups , as well as the experimental collaborations themselves . it is crucial that the best possible understanding of the proton pdfs is achieved , given their central role in predictions for other processes , for example , at the lhc . the traditional method of determining the pdfs of the proton relies on assuming an @xmath0-dependence for each of the different pdfs at some starting scale @xmath43 and then using the dokshitzer - gribov - lipatov - altarelli - parisi ( dglap ) evolution equations @xcite to model the @xmath4 dependence of the pdfs . this approach is used by both the cteq collaboration @xcite and martin et al . ( mrst ) @xcite ; both groups presented progress reports at this workshop . this approach has also been adopted by the zeus collaboration @xcite , who also presented results of their latest fit at this workshop . a number of issues have been addressed by both the cteq and mrst groups recently . in particular , the compatibility of datasets and the stability of the fit results have been studied . both groups have performed studies of fit stability @xcite , by studying the impact of restricting the fits to data at higher @xmath0 . the studies were performed by looking at the next - to - leading order ( nlo ) @xmath44 production cross section predictions from fits with different lower @xmath0 limits . the results of both studies are shown in figure [ fig : cteq ] . the cteq group conclude from their studies that the fits are stable . the mrst group conclude that the uncertainties increase significantly as the lower @xmath0 limit is tightened and that next - to - next - to - leading - order ( nnlo ) is inherently more stable and these fits should become the standard in the future . production cross section at the lhc from the cteq and mrst collaborations . the cross sections are plotted as a function of the lower @xmath0 limit applied to the data used to extract the pdfs . the cteq collaboration have considered two different scenarios : one in which the gluon is forced to be positive - definite and the other in which the gluon is left free . both are indicated by crosses . the mrst predictions are indicated by the dots and here the gluon is left free . ] the mrst group also presented the results of other studies , in particular the inclusion of electroweak corrections and qed effects @xcite . the latter , in particular , have a negligible effect on the pdfs themselves , as expected . however , the inclusion of qed effects does lead to a small isospin violation , which significantly improves the predictions for prompt photon production at hera . the zeus collaboration also presented their latest determination of the proton pdfs using only zeus data @xcite . in comparison to their previous fits , zeus jet cross section data has been included , which is directly sensitive to the gluon density of the proton and benefit from small experimental and theoretical uncertainties . both nc dis jet data and direct - enriched dijet photoproduction data , in which the photon behaves as a point - like object , have been included , significantly improving the uncertainty on the gluon pdf in the range @xmath45 . if a simultaneous fit of both the pdfs and @xmath46 is performed , the result for @xmath46 is very precise and in agreement with the world average . several presentations were also made at this workshop , in which alternative approaches to pdf determination were explained . one such presentation was made by the nnpdf collaboration @xcite , who are developing a neural network approach to pdf fitting . this approach avoids some of the shortcomings of the standard method , such as avoiding any potential bias from the choice of functional form for the pdfs . it should also lead to a better estimation of the pdf uncertainties . so far the structure functions have been successfully determined using this approach , but work is still in progress to successfully determine the pdfs using this method . the estimation and reduction of pdf uncertainties was a key theme at this workshop . one presentation made at this workshop looked at the possibility of averaging the @xmath3 data from the h1 and zeus experiments , prior to including it in any global pdf fit @xcite . this has advantages when it comes to the handling of the systematic uncertainties ; it also provides a model - independent method of checking the consistency of the data from the two experiments . it has been found that several contributions to the systematic uncertainties from each experiment are reduced ; the experiments are effectively constraining each other . this is an interesting approach which , it is hoped , will be pursued further by the two collaborations . another presentation made at this workshop looked at the impact of future hera data on the pdf uncertainties @xcite . this study was performed using the zeus pdf fit as a basis . a number of different scenarios were considered , including simply the expected increase in the amount of luminosity , as well as the inclusion of new cross section measurements which have been optimized to constrain the pdfs as tightly as possible . these improvements would lead to significant improvements in the valence quark distributions , as well as in the high @xmath0 sea quark and gluon distributions . other scenarios which were also considered are the inclusion of precision measurements of @xmath10 from hera data ( low - energy proton running ) and the possibility of @xmath47 running to constrain the sea quark asymmetries . new hera data on unpolarized dis structure functions , combined with the present world data , allow to reduce the experimental error on the strong coupling constant , @xmath48 , the fundamental constant of qcd and strong interaction , to the level of @xmath49% . on the theoretial side , the next - to - leading order ( nlo ) analyses have limitations due to scale variations being present which allow no better than @xmath50% accuracy in the determination of @xmath46 @xcite . in order to match the experimental accuracy , it was stressed @xcite that analyses of dis structure functions need to be carried out at the nnlo level . with the recent computation of the 3loop anomalous dimensions @xcite , a complete nnlo study of dis structure functions is now possible . a full nnlo analysis of unpolarized dis structure functions aiming to obtain a high accuracy determination of @xmath46 was presented at the workshop @xcite . it was pointed out that a combination of standard nnlo qcd analysis and fits based on factorization scheme - invariant evolution of dis structure functions will provide a valuable tool in high - precision analyses aiming at @xmath49% accuracy in the determination of @xmath46 @xcite . the factorization scheme - invariant evolution of dis structure functions can be implemented to different pair of structure functions , such as @xmath3 and @xmath10 or @xmath3 and its @xmath51 derivative . in this approach , @xmath46 is determined by performing an one dimensional fit between the evolution of dis structure functions and the data . work is still ongoing . a full nnlo accuracy evolution for @xmath3 and @xmath52 have been completely implemented for massless flavors . inclusion of heavy flavors and the fit to the data , and the one parameter fit to determine @xmath53 are on the way . one interesting result is that comparing the behavior of slopes of @xmath52 to the slopes extracted experimentally points toward a positive gluon density in small-@xmath0 region @xcite , while nlo global analysis of pdfs points to a negative gluon density in low-@xmath0 and low @xmath4 region @xcite . an effort to develop a next generation of event generators was reported at the workshop @xcite . the effort was aiming to set up a systematic scheme for developing event generators that are consistent to qcd factorization of differential cross sections up to nlo accuracy . it was argued that in order to achieve this accuracy , one has to use unintegrated pdfs to replace the parton shower in the lo event generators . the basic rules have been established for a dis event generator , but , there are still works to be done @xcite . at hadron colliders , it is the pdfs that determine partonic flux of hard collisions . full discovery potentials of the lhc and precision tests of qcd are sensitive to pdfs at large @xmath0 . in the form of qcd factorization , extraction of pdfs depends on short - distance dynamics and perturbatively calculated coefficient functions . the coefficient functions often have high powers of logarithms like @xmath54 and @xmath55 , which become large as @xmath0 near 1 and 0 . resummation of these large logarithms is necessary for observables dominated by those kinematic regions . a presentation made at this workshop looked at the effect of large-@xmath0 resummation on the extraction of pdfs @xcite . large-@xmath0 resummation was performed for coefficient functions of dis structure functions in massless approximation as well as in an approach that includes heavy quark - mass effects . after performing fits to the fixed target dis data from nutev , bcdms and nmc collaborations , using nlo and nll - resummed coefficient functions , it was found that the resummation has a visible impact on the extraction of quark distributions at large @xmath0 , and was stressed that large-@xmath0 partonic resummation is needed whenever a high precision is required for cross sections evaluated near partonic threshold @xcite . a precise knowledge of pdfs , in particular , gluon distribution at @xmath56 are vital for understanding almost all standard production processes at the lhc . when we move away from zero rapidity , much smaller @xmath0 partons , as small as @xmath2 , are required for some observables . although perturbative qcd has been very successful in interpreting data on scaling violation of pdfs in terms of dglap evolution in @xmath4 @xcite , the extrapolation of pdfs to smaller @xmath0 has not been very consistent with the bfkl evolution in @xmath0 ( or in energy ) @xcite . although the strong rise of proton structure function @xmath3 with energy , observed at hera , can be well described by a simple ( 3 parameters ) lo bfkl fit @xcite , a much too small effective @xmath57 is needed while the world average is @xmath58 for hera kinematics . a phenomenological study of confronting nlo bfkl with new hera data on @xmath3 structure function was presented at the workshop @xcite . a big discrepancy between theory and data , especially at low @xmath4 , was clearly evident @xcite . a further study is needed although more progresses have been made recently @xcite . one of the challenging problems in qcd is to understand the behavior of hadronic cross sections in high energy limit . experimental data on the total cross section show a slow but distinct rise with collision energy @xmath59 . this rise could be parametrized by a power of @xmath9 , @xmath60 , which is consistent with an exchange of soft pomerons @xcite . on the other hand , after resumming leading powers of @xmath61 contributions , perturbative qcd calculation , in the form of bfkl evolution in energy ( or in @xmath0 ) , predicts a much stronger rise with a much large power of @xmath9 @xcite . as @xmath62 ( or @xmath63 ) , the power - like rise is not compatible with the unitarity of the s - matrix in the high energy limit , or in contradiction with the froissart bound @xcite , which allows at most a logarithmic increase with collision energy . bfkl equation is a linear evolution equation and predicts a large number of low-@xmath0 partons due to the strength of soft gluon radiation in qcd . on the other hand , the large number of soft partons generated by parton radiation are likely to interact and recombine . parton recombination introduces non - linear terms into the bfkl equation , slows down the small-@xmath0 evolution , and removes the apparent violation of the unitarity . when parton recombination is strong enough to balance parton radiation , pdfs saturate as @xmath63 @xcite . the state of saturated partons is sometime referred as the color glass condensate ( cgc ) @xcite . a lot of work , both theoretical and experimental , have been done and many progresses have been made in recent years to understand this saturation phenomenon , especially , in a nuclear environment because of an @xmath64 length enhancement in parton density at a given impact parameter . two sessions at this workshop were devoted to the presentations related to this novel phenomenon . a simple modification to the bfkl equation is balitsky - kovchekov ( bk ) equation @xcite , which adds a quadratic term to the bfkl equation . the bk equation is a non - linear integro - differential equation for unintegrated pdfs . its non - linearity leads to many interesting features that could be seen in high energy reactions . it was shown @xcite that the bk equation is in the equivalence class of the fisher kolmogorov petrovsky piscounov ( fkpp ) non - linear partial differential equation , which has so - called traveling wave solutions . the similarity leads to an interesting point of view that high energy qcd is equivalent to a reaction diffusion system @xcite . a detailed numerical studies of the mean field approximation to the bk equation was presented at the workshop @xcite . it was demonstrated that the numerical solutions of the bk equation does show features of traveling wave solutions . it was also confirmed that the influence of the initial condition disappears for large @xmath65 , so that a universal propagation speed is approached , which should help establish statistical interpretations of the phenomena observed in qcd scattering at high energy . a study of discrete version of the bk equation was presented at this workshop @xcite . by noting that the number of gluons in the hadron wave functions is discrete , and their formation in the chain of small @xmath0 evolution occurs in the discrete intervals of @xmath55 , a discrete version of bk equation was formulated @xcite . it was found that numerical solutions of the discrete bk equation behave chaotically in the phenomenologically interesting kinematic region . it was concluded @xcite that the evolution of the scattering amplitude at high energies in the saturation region might be chaotic , while the scattering amplitude in the normal perturbative region is not affected by the discretization . although the model used in the numerical calculations neglected the diffusion in transverse momentum , stochasticity of gluon emission and the dynamical fluctuations beyond the mean field approximation , it was hoped that at least some of the features of discrete quantum evolution at small @xmath0 will survive a more realistic treatment . the chaotic features of small @xmath0 evolution open a new intriguing prospective on the studies of hadron and nuclear interactions at high energies . recent developments of cgc theory were reported at the workshop @xcite . the cgc theory is an effective theory of the strong interactions at very high energies @xcite . the basic equation of cgc theory is the jimwlk equation @xcite , which governs the evolution of a weight function of a color medium ( or a hadron target ) with rapidity . the evolution kernel is often referred as the jimwlk hamiltonian . the weight function is needed for calculating physical scattering amplitude when it is averaged over the medium s color charge . in the large @xmath66 limit and in the dipole scattering picture , the jimwlk equation reduces to the closed and relatively simple bk equation . in a language of feynman diagrams , the jimwlk equation includes both bfkl ladder diagrams and the fan diagrams of triple ladder interactions , and it naturally describes the physics of scattering on a dense medium ( or a target ) with multiple scattering corrections . it naturally interprets the geometric scaling observed in the data @xcite . however , the jimwlk equation does not include pomeron ( or ladder ) splittings or all the pomeron loops . modifications and improvements to the jimwlk equation were proposed @xcite . in addition , a similar evolution equation was derived for a dilute target @xcite , while the jimwlk equation is suitable for scatterings on a dense target . a striking result is that the evolution kernels of these two equations are apparently dual to each other @xcite . the selfduality of the kernel is somewhat similar ( although different in detail ) to the duality symmetry , @xmath67 , @xmath68 in the hamiltonian of a harmonic oscillator @xcite . relativistic heavy ion collider ( rhic ) is a unique place to test the theory of cgc because of the high density of partons involved . in order to probe small @xmath0 partons and the phenomena of cgc at rhic , one has to go to extremely forward and backward region in rapidity because of the relatively low colliding energy . three major experimental collaborations , brahms , phenix , and star , at rhic carried out the effort and presented their early results at the workshop @xcite . rapidity dependence of high-@xmath69 particle suppression was measured in d - au collisions at @xmath70 gev by brahms collaboration and presented at the workshop @xcite . the data collected from d - au collisions at rhic is compared to p - p in figure [ fig : brahms ] using the nuclear modification factor defined as @xmath71 where @xmath72 is the number of binary collisions estimated to be @xmath73 for minimum biased d+au collisions . . statistical errors are shown with error bars . systematic errors are shown with shaded boxes with widths set by the bin sizes . the shaded band around unity indicates the estimated error on the normalization to @xmath74 . dashed lines at @xmath75 gev / c show the normalized charged particle density ratio @xmath76 nuclear modification factors for charged hadrons at pseudo - rapidities @xmath77 and 3.2 were shown as a function of hadron transverse momentum @xmath69 . at the central region , or zero rapidity , data confirm the cronin type enhancement in large @xmath69 region . however , as the pseudo - rapidity increases , the enhancement vanishes , and the modification factor is less than the unity for entire measured @xmath69 region . the forward region is , the measurement probes target partons with smaller @xmath0 . the observed rapidity dependence of the suppression , which increases with rapidity , fits naturally into the picture of cgc @xcite , and can be also interpreted by the recombination model of hadronization @xcite . in addition , the suppression is consistent with the perturbative qcd calculation based on resummation of coherent multiple scattering @xcite . phenix collaboration reported its measurement of charged hadron production in the same d - au collisions at rhic @xcite . it covers pseudo - rapidities from -2.0 to -1.4 and 1.4 to 2.2 with the forward coverage overlaps with some of brahms measurements . phenix also observes a suppression in hadron yields in d - au collision relative to binary collision . the data was presented in terms of a different nuclear modification factor , @xmath78 , which is defined as the ratio of the particle yield in central collisions to the particle yield in peripheral collisions , each normalized by the averaged number of binary collisions @xmath72 @xmath79 as shown in figure [ fig : phenix ] , phenix data are consistent with brahms data , and are in qualitative agreement with theoretical expectation . quantitatively , the ratio @xmath78 for the most central over the most peripheral collisions is more suppressed than theoretical calculations . as a function of @xmath80 at forward rapidities shown as the average of the two methods . note that the brahms results are for negative hadrons at @xmath81 and their centrality ranges ( @xmath82 and @xmath83 ) are somewhat different from ours . ] measurements of the inclusive yields of @xmath84 mesons in p - p and d - au collisions at rhic were presented by star collaboration @xcite . with a forward @xmath84 detector installed at the solenoidal tracker at rhic ( star ) , it can detect high energy @xmath84 mesons with pseudo - rapidity as large as @xmath85 @xcite . the inclusive yield in p - p collisions at @xmath86 gev are consistent with nlo pqcd calculations . the nuclear modification factor , @xmath87 in figure [ fig : star ] , shows a strong suppression at the large pseudo - rapidity . it was argued that the d - au yield is consistent with a model calculation treating the au nucleus as a cgc for forward particle production @xcite . comparisons with other production models will be interesting to perform . additional measurements with different final - state particles and at different centralities will help elucidate the cause of the observed strong suppression , which covers a broad range of nuclear gluon momentum fraction with a peak value @xmath88 @xcite . yield for p+p [ left ] and d+au collisions normalized by p+p [ right ] . the pion energy ( @xmath89 ) is correlated with the transverse momentum ( @xmath69 ) , as the fpd was at fixed values of pseudo - rapidity ( @xmath36 ) . the inner error bars are statistical , while the outer combine these with the @xmath89- ( @xmath69- ) dependent systematic errors , and are often smaller than the points . the curves ( left ) are nlo pqcd calculations evaluated at fixed @xmath36 , using different fragmentation functions . the x s and stars ( right ) are brahms data for @xmath90 production at smaller @xmath36.,title="fig : " ] yield for p+p [ left ] and d+au collisions normalized by p+p [ right ] . the pion energy ( @xmath89 ) is correlated with the transverse momentum ( @xmath69 ) , as the fpd was at fixed values of pseudo - rapidity ( @xmath36 ) . the inner error bars are statistical , while the outer combine these with the @xmath89- ( @xmath69- ) dependent systematic errors , and are often smaller than the points . the curves ( left ) are nlo pqcd calculations evaluated at fixed @xmath36 , using different fragmentation functions . the x s and stars ( right ) are brahms data for @xmath90 production at smaller @xmath36.,title="fig : " ] two theory talks were presented at the workshop to specifically address the strong suppression observed in the forward region of d - au collisions at rhic @xcite . two completely different pictures were presented on how a leading hadron was produced in the d - au collisions at rhic energies . in one approach @xcite , single hadron production was assumed to be proportional to gluon production . under this approximation , the nuclear modification factor @xmath87 is the same for both hadron and gluon production . the gluon production in d - au collisions was calculated in the framework of cgc physics @xcite . in the other approach @xcite , a single hadron was produced via recombination of partons available during the collisions . it was argued that p - p , p(d)-au , and au - au collisions produce different shapes of parton spectra . recombination of partons with different spectra naturally leads to different hadron distribution and a nontrivial nuclear modification factor @xcite . a striking fact is that both of these approaches provided calculations that are consistent with the observed data . it was observed about two decades ago that dis structure functions of nuclei differ from simple sum of those in the free nucleon @xcite . as a result , pdfs of a nucleus of atomic weight @xmath91 also differ from those in the free proton , @xmath92 . in order to understand the overwhelming data from the rhic and make predictions for the heavy ion programs at the future facilities , like the lhc and electron ion collider ( eic ) , we need precise information of nuclear pdfs ( npdfs ) , in particular , at small @xmath93 . a brief overview of the global dglap analyses of npdfs was presented at the workshop @xcite . the npdfs are defined in terms of the same operators that define the free nucleon pdfs with the free nucleon state replaced by a nuclear state . therefore , npdfs and free nucleon pdfs should share the same dglap evolution equations , and only difference between npdfs of different nuclei and free nucleon pdfs is the input distributions to dglap equations at a scale @xmath94 . once a set of the nonperturbative input distributions are chosen , dglap evolution equations predict npdfs at a larger momentum scale @xmath4 . there are typically two approaches to choose the input distributions : calculated by the nuclear models and determined by fit to the data @xcite . the second approach shares the same procedures as that used in the determination of pdfs , and often referred as the global analyses of npdfs . there are three groups who have been carrying out the global analyses and reanalyses of npdfs : eskola _ et al . _ ( usually called as _ eks98 _ ) @xcite , hirai _ et al . _ @xcite , and de florian and sassot @xcite . the first two groups use lo dglap evolution while the third uses nlo evolution . all analyses , only dis and drell - yan data on nuclear targets were used in the fits . because of the large error in nuclear data and the lack of direct information on gluon initiated processes , all fits have reasonable constrains and consistencies on quark distributions , but , not on gluon distributions @xcite . a hard probe often refers to a scattering process with a large momentum exchange @xmath95 whose invariant mass @xmath96 , and it can probe a distance scale much smaller than size of a nucleon at rest , @xmath97 fm . however , when an active parton s momentum fraction @xmath98 , a hard probe might interact with more than one partons of the nucleon coherently @xcite . when @xmath99 , the hard probe can cover a whole lorentz contracted nucleus and interact with partons from different nucleons . although such coherent multi - parton interactions are power suppressed by hard scales of the scattering , they are enhanced by the nuclear size and could be one of the important sources of nuclear dependence observed in high energy nuclear collisions . a presentation made at this workshop looked at the impact of coherent multiple scattering in dis on nuclear targets and leading particle production in p(d)-au collisions @xcite . an all power resummation of nuclear enhanced power corrections to dis structure functions on nuclear targets was achieved . the calculated results for the bjorken @xmath0- , @xmath4- and @xmath91-dependence of nuclear shadowing in @xmath100 and the nuclear modifications to @xmath101 are consistent with the existing data @xcite . predictions were made for the dynamical shadowing from final state interactions in @xmath102 reactions for sea and valence quarks in the structure functions @xmath100 and @xmath103 , respectively . in addition , calculations for the centrality and rapidity dependent nuclear suppression of single and double inclusive hadron production at moderate transverse momenta in @xmath104 collisions were presented and consistent with the rhic data @xcite . in the gribov - glauber picture , nuclear shadowing and antishadowing observed in nuclear structure functions are due to the destructive and constructive interference of amplitudes arising from the multiple - scattering of quarks in the nucleus , respectively . a calculation of shadowing and antishadowing of nuclear structure functions in the gribov - glauber picture were presented at the workshop @xcite . the coherence of multi - step nuclear processes leads to shadowing and antishadowing of the electromagnetic nuclear structure functions in agreement with the data . but , the same picture leads to substantially different antishadowing for charged and neutral current reactions , thus affecting the extraction of the weak - mixing angle @xmath105 @xcite . this is due to the fact that reggeon couplings depend on the quantum numbers of the struck quark implies non - universality of nuclear antishadowing for charged and neutral currents @xcite . moments of pdfs are matrix elements of local gauge invariant operators which in principle can be calculated by using lattice qcd . a brief review of recent lattice effort in determining the pdfs was presented at the workshop @xcite . lattice qcd calculations of three representative observables , the transverse quark distribution , momentum fraction , and axial charge , were presented @xcite . it was emphasized that lattice calculations of nucleon structure are beginning to realize their promise to elucidate qcd and make contact with the experimental programs . it was concluded that recent calculations are painting a qualitative three dimensional picture of nucleon structure revealing a significant @xmath0 dependence of the transverse size of the nucleon . quantitative calculations of moments of pdfs are progressing , in particular , the calculation of @xmath106 may soon reach a few percent accuracy . an analytical approach to understand the three dimensional picture of nucleon structure was presented at the workshop @xcite . a concept of the quantum phase - space ( wigner ) distributions for the quarks and gluons in the nucleon was introduced . the quark wigner functions were related to the transverse - momentum dependent pdfs and generalized pdfs with emphasis on the physical role of the skewness parameter . any knowledge on the generalized pdfs can be immediately translated into the correlated coordinate and momentum distributions of partons . in particular , the generalized pdfs can be used to visualize the phase - space motion of the quarks , and hence allow studying the contribution of the quark orbital angular momentum to the spin of the nucleon . it was concluded that measurements of generalized pdfs and/or direct lattice qcd calculations of them will provide a fantastic window to the quark and gluon dynamics in the proton @xcite . another presentation made at the workshop looked at quark asymmetries in nucleons @xcite . instead of fitting the data , a physical model for the non - perturbative @xmath0-shape of pdfs was developed . the model was based on gaussian fluctuations in momenta , and quantum fluctuations of the proton into meson - baryon pairs . it was found that the model gives a good description of the proton structure function and a natural explanation of observed quark asymmetries , such as the difference between the anti - up and anti - down sea quark distributions and between the up and down valence distributions @xcite . within this model , there is an asymmetry in the momentum distributions of strange and anti - strange quarks in the nucleon , and the asymmetry is large enough to reduce the nutev anomaly to a level which does not give a significant indication of physics beyond the standard model . effective field theory was used to investigate the nuclear modification to the pdfs and a recent result was presented at the workshop @xcite . it was found that the universality of the shape distortion in npdfs ( the factorization of the bjorken @xmath0 and atomic weight @xmath91 dependence ) is model independent and emerges naturally in effective field theory . for a simple parameterization of nonperturbative functions in the approach , fits to the data confirm the factorization @xcite . we would like to thank all the members of our working group for the excellent presentations and for lively discussions they provoked . we would also like to thank the conveners of the electroweak and beyond the standard model working group for their assistance in the joint session on high @xmath4 structure function measurements . we would also like to thank all the session chairs for agreeing to be involved . last , but not least , we would like to thank the organizers of dis 2005 for interesting and well - organized meeting . dokshitzer , soviet phys . jetp * 46 * , 641 ( 1977 ) ; v. n. gribov and l. n. lipatov , soviet j. nucl * 15 * , 438,675 , ( 1972 ) ; l. n. lipatov , soviet j. nucl . phys . _ 20 _ , 95 , ( 1975 ) ; g. altarelli and g. parisi , nucl . phys * b126 * , 298 ( 1977 ) . l. n. lipatov , _ sov . * 23 * , 338 ( 1976 ) ; e. a. kuraev , l. n. lipatov and v. s. fadin , _ sov . phys . jetp _ * 45 * , 199204 ( 1977 ) ; i. i. balitsky and l. n. lipatov , _ sov . j. nucl . _ * 28 * , 822829 ( 1978 ) . i. balitsky , _ nucl . * b463 * , 99 ( 1996 ) ; _ phys . _ * 81 * , 2024 ( 1998 ) ; _ phys _ * b518 * , 235 ( 2001 ) ; y.v . kovchegov , _ phys . rev . _ * d60 * , 034008 ( 1999 ) ; _ phys . _ * d61 * , 074018 ( 2000 ) .

we report a summary of the structure function working group which covers a wide range of the recent results from hera , tevatron , rhic , and jlab experiments , and many theoretical issues from low @xmath0 to high @xmath0 . address = h.h . wills physics laboratory , university of bristol , bristol , bs8 1tl , uk address = department of physics and astronomy iowa state university , iowa 50011 , usa address = enrico fermi institute , university of chicago , chicago , illinois 60637 , usa

the classical schur - horn theorem @xcite characterizes diagonals of self - adjoint ( hermitian ) matrices with given eigenvalues . it can be stated as follows , where @xmath2 is @xmath3 dimensional hilbert space over @xmath4 or @xmath5 , i.e. , @xmath6 or @xmath7 . [ horn ] let @xmath8 and @xmath9 be real sequences in nonincreasing order . there exists a self - adjoint operator @xmath10 with eigenvalues @xmath11 and diagonal @xmath12 if and only of @xmath13 the necessity of is due to schur @xcite and the sufficiency of is due to horn @xcite . it should be noted that can be stated in the equivalent convexity condition @xmath14 this characterization has attracted a significant interest and has been generalized in many remarkable ways . some major milestones are the kostant convexity theorem @xcite and the convexity of moment mappings in symplectic geometry @xcite . moreover , the problem of extending theorem [ horn ] to an infinite dimensional dimensional hilbert space @xmath15 has attracted a great deal of interest . neumann @xcite gave an infinite dimensional version of the schur - horn theorem phrased in terms of @xmath16-closure of the convexity condition . neumann s result can be considered an initial , albeit somewhat crude , solution of this problem . the first fully satisfactory progress was achieved by kadison . in his influential work @xcite kadison discovered a characterization of diagonals of orthogonal projections acting on @xmath15 . the work by gohberg and markus @xcite and arveson and kadison @xcite extended the schur - horn theorem [ horn ] to positive trace class operators . this has been further extended to compact positive operators by kaftal and weiss @xcite . these results are stated in terms of majorization inequalities as in . other notable progress includes the work of arveson @xcite on diagonals of normal operators with finite spectrum . moreover , antezana , massey , ruiz , and stojanoff @xcite refined the results of neumann @xcite , and argerami and massey @xcite studied extensions to ii@xmath17 factors . for a detailed survey of recent progress on infinite schur - horn majorization theorems and their connections to operator ideals we refer to the paper of kaftal and weiss @xcite . the authors @xcite have recently shown a variant of the schur - horn theorem for a class of locally invertible self - adjoint operators on @xmath15 . this result was used to characterize sequences of norms of a frame with prescribed lower and upper frame bounds . the second author @xcite has extended kadison s result @xcite to characterize the set of diagonals of operators with three points in the spectrum . in this work we shall continue this line of research by giving a characterization of diagonals of self - adjoint operators with finite spectrum . unlike @xcite we shall only consider a characterization which neglects multiplicities of eigenvalues . the investigation of the corresponding problem for operators with prescribed multiplicities is postponed to a future paper . our main result can be thought as an analogue of the work by arveson @xcite who identified some necessary conditions which must be satisfied by diagonals of normal operators with finite spectrum . unlike @xcite our main result deals only with self - adjoint operators . on the other hand , theorem [ npt ] gives a complete characterization of diagonals of self - adjoint operators with finite spectrum . [ npt ] let @xmath18 be an increasing sequence of real numbers such that @xmath19 and @xmath20 , @xmath21 . let @xmath22 be a sequence in @xmath23 $ ] with @xmath24 . for each @xmath25 , define @xmath26 there exists a self - adjoint operator @xmath27 with diagonal @xmath22 and @xmath28 if and only if either : 1 . @xmath29 or @xmath30 , or 2 . @xmath31 and @xmath32 , @xmath33and thus @xmath34 for all @xmath25@xmath35 , and there exist @xmath36 and @xmath37 such that : @xmath38 and for all @xmath39 , @xmath40 we remark that the assumption that @xmath24 is not a true limitation of theorem [ npt ] . indeed , the summable case @xmath41 , or its symmetric variant @xmath42 , leads to a finite rank schur - horn theorem which is discussed below . this case requires a different set of conditions which are closely related to the classical schur - horn majorization . finally , the assumption @xmath19 is made only for simplicity ; the general case follows immediately by a translation argument . the schur - horn theorem and its extensions @xcite are usually stated with eigenvalues in nonincreasing order . this is because positive diagonal entries can be easily arranged into a nonincreasing sequence indexed by @xmath43 , or a finite subset . in particular , we have the following result for finite rank positive operators which can be deduced from results in @xcite , see also ( * ? ? ? * theorems 3.2 and 3.3 ) . the main innovation in the formulation of theorem [ horn - ninc ] is that it does not require a sequence @xmath44 to be globally nonincreasing . this allows the possibility that @xmath44 has infinitely many positive terms and some zero terms . at the same time it also gives us flexibility in arranging small diagonal terms . [ horn - ninc ] let @xmath8 be a positive nonincreasing sequence . let @xmath45 be a nonnegative sequence such that : 1 . @xmath46 for @xmath47 , 2 . the subsequence @xmath48 is nonincreasing . there exists a positive rank @xmath3 operator @xmath27 on a hilbert space @xmath49 with ( positive ) eigenvalues @xmath8 and diagonal @xmath45 if and only if @xmath50 observe that is simply an equivalent way of writing the majorization condition @xmath51 if we insist on arranging diagonal entries into a nondecreasing sequence , then we should instead use @xmath52 as part of the index set . this seemingly trivial observation leads to the following reformulation of theorem [ horn - ninc ] with eigenvalues in nondecreasing order which , as we shall see , has non - trivial consequences . [ horn - ndec ] let @xmath8 be a positive nondecreasing sequence . let @xmath53 be a nonnegative sequence such that : 1 . @xmath54 for @xmath55 , 2 . the subsequence @xmath48 is nondecreasing . there exists a positive rank @xmath3 operator @xmath27 on a hilbert space @xmath49 with ( positive ) eigenvalues @xmath8 and diagonal @xmath53 if and only if @xmath56 we will make an extensive use of kadison s theorem @xcite which characterizes diagonals of orthogonal projections . theorem [ kadison ] serves as a prototype for our theorem [ npt ] . the common feature of both of these results is a trace condition . the main distinction between them is a lack of majorization inequalities in kadison s theorem which are present in theorem [ npt ] . [ kadison ] let @xmath22 be a sequence in @xmath57 $ ] and @xmath58 . define @xmath59 there exists an orthogonal projection on @xmath60 with diagonal @xmath22 if and only if either : 1 . @xmath61 or @xmath62 , or 2 . @xmath63 and @xmath64 , and @xmath65 [ rpart ] note that if there exists a partition of @xmath66 such that @xmath67 then for all @xmath68 we have @xmath63 and @xmath64 and @xmath69 thus , in the presence of a partition satisfying , @xmath70 is a necessary and sufficient condition for a sequence to the be the diagonal of a projection . we will find use for these more general partitions in the sequel . finally , the following `` moving toward @xmath71-@xmath72 '' lemma plays a key role in our arguments . lemma [ movelemma ] is simply a concatenation of ( * ? ? ? * lemmas 4.3 and 4.4 ) stated in a convenient form . [ movelemma ] let @xmath22 be a sequence in @xmath23 $ ] . let @xmath73 be two disjoint finite subsets such that @xmath74 . let @xmath75 and @xmath76 ( i ) there exists a sequence @xmath77 in @xmath23 $ ] satisfying @xmath78 ( ii ) for any self - adjoint operator @xmath79 on @xmath49 with diagonal @xmath77 , there exists an operator @xmath27 on @xmath49 unitarily equivalent to @xmath79 with diagonal @xmath22 . in this section we will show the necessity in theorem [ npt ] . in order to do this it is convenient to formalize the concept of interior majorization with the following definition . [ mar ] let @xmath18 be an increasing sequence such that @xmath19 and @xmath20 , @xmath21 . let @xmath80 be a sequence in @xmath23 $ ] . let @xmath81 and @xmath82 be as in . we say that @xmath44 satisfies _ interior majorization _ by @xmath83 if the following 3 conditions hold : 1 . @xmath31 and @xmath32 , and thus @xmath63 and @xmath84 for all @xmath25 , 2 . there exist @xmath36 and @xmath85 such that @xmath86 3 . for all @xmath87 , @xmath88 despite its initial appearance , the interior majorization conditions and are equivalent with and in theorem [ npt ] . indeed , by remark [ rpart ] , is equivalent to the statement that for all @xmath25 there exists @xmath89 such that @xmath90 fix @xmath91 , where @xmath92 . then , can be rewritten as @xmath93 using , we can remove the presence of @xmath94 in to obtain @xmath95 this is precisely , and the above process is reversible . [ int - nec ] let @xmath27 be a self - adjoint operator on @xmath15 with spectrum @xmath96 where @xmath18 is an increasing sequence such that @xmath19 and @xmath20 , @xmath21 . let @xmath97 be a diagonal of @xmath27 with respect to some orthonormal basis @xmath98 of @xmath49 . assume that for some @xmath99 , @xmath63 and @xmath64 . then , @xmath80 satisfies interior majorization by @xmath18 . by the spectral decomposition , we can write @xmath100 where @xmath101 s are mutually orthogonal projections satisfying @xmath102 . let @xmath103 be the diagonal of @xmath101 . hence , we have @xmath104 for convenience let @xmath105 and @xmath106 . by our assumption @xmath107 summing @xmath108 over @xmath109 yields @xmath110 using we have that @xmath111 . summing this over @xmath112 yields @xmath113 combining and and applying theorem [ kadison ] yields @xmath114 moreover , since @xmath115 by we have @xmath116 by , , theorem [ kadison ] and remark [ rpart ] applied to the projection @xmath117 we have @xmath118 thus , @xmath119 for convenience we let @xmath120 . in particular , by letting @xmath121 the above shows with @xmath122 , where @xmath123 it remains to show the interior majorization inequality . fix @xmath87 , and let @xmath124 and @xmath125 . by the fact that @xmath122 , we have @xmath126 thus , the required majorization is equivalent to @xmath127 by , we have for @xmath128 , @xmath129 thus , can be rewritten as @xmath130 since @xmath131 is an increasing sequence , the left hand side of is @xmath132 . on the other hand , the right hand side of is @xmath133 as it is dominated by @xmath134 in the last step we used . this shows , which implies , thus proving . this completes the proof of theorem [ int - nec ] . the goal of this section is to show the sufficiency in theorem [ npt ] . the sufficiency of condition ( i ) , that is @xmath135 , is a consequence of a result established by the second author , see ( * ? ? ? * corollary 4.5 ) . [ 3intsuff ] let @xmath18 be an increasing sequence such that @xmath19 and @xmath20 , @xmath21 . assume @xmath22 is a sequence in @xmath23 $ ] such that for some ( and hence all ) @xmath136 we have @xmath137 then there is a self - adjoint operator @xmath27 with @xmath138 and diagonal @xmath22 . next , we must demonstrate the sufficiency of condition ( ii ) of theorem [ npt ] . to achieve this we shall introduce an alternative variant of interior majorization which allows us to apply theorems [ horn - ninc ] and [ horn - ndec ] in the crucial case when @xmath12 can be indexed in nondecreasing order by @xmath139 . [ rim ] suppose that @xmath21 and @xmath140 is an increasing sequence in @xmath4 such that @xmath19 and @xmath20 . let @xmath141 be a nondecreasing sequence which takes values in @xmath142 , each at least once . let @xmath143 be a nondecreasing sequence in @xmath23 $ ] such that @xmath144 . we say that @xmath44 satisfies _ riemann interior majorization _ by @xmath18 if there exists such a sequence @xmath141 as above , so that the following two hold : @xmath145 to distinguish between two distinct types of interior majorization we shall frequently refer to the concept introduced in definition [ mar ] as lebesgue interior majorization . this is done purposefully as an analogy between riemann and lebesgue integrals . theorem [ eqmajs ] shows the equivalence of the concepts of riemann and lebesgue interior majorization for nondecreasing sequences . [ eqmajs ] let @xmath18 be an increasing sequence in @xmath4 with @xmath152 and @xmath20 . let @xmath143 be a nondecreasing sequence in @xmath23 $ ] . then , the sequence @xmath12 satisfies interior majorization by @xmath18 if and only if @xmath12 satisfies riemann interior majorization by @xmath18 . we need to set some notation first . without loss of generality we may assume that @xmath153 @xmath154 @xmath155 . for @xmath87 , we set @xmath156 with this notation for @xmath87 , we have @xmath157 and @xmath158 finally , given @xmath159 , we set @xmath160 assume that @xmath12 satisfies either riemann or lebesgue interior majorization by @xmath140 . in the first case we let @xmath161 and we fix @xmath162 such that @xmath163 in the second case we let @xmath164 and @xmath165 be as in and . in either case we have @xmath166 first , we will show the equivalence of with . for @xmath167 we have @xmath168 by , for @xmath169 @xmath170 combining and for @xmath171 yields @xmath172 since the last series converges , by letting @xmath173 , is equivalent with with @xmath174 . next , we will show the equivalence of with . assume that lebesgue interior majorization holds . in the current notation , takes the following form @xmath175 we must demonstrate that @xmath176 for all @xmath177 . since @xmath178 for @xmath179 , we have @xmath176 for @xmath180 . moreover , since @xmath181 for all @xmath182 and @xmath183 as @xmath184 , this implies @xmath176 for all @xmath185 . we will prove by induction on @xmath186 that @xmath187 for @xmath188 . the base case @xmath189 was shown above . assume the inductive hypothesis is true for @xmath190 , where @xmath191 . we will show that that @xmath176 for all @xmath192 . there are two cases to consider . * case 1 . * assume that @xmath193 . first we will show that @xmath194 . if @xmath195 , then the inductive hypothesis implies that @xmath194 , so we may assume @xmath196 . using and then @xmath197 this shows that @xmath194 . * assume @xmath199 . using and then @xmath200 by and , @xmath201 for all @xmath202 combining this with @xmath203 implies that @xmath176 for all @xmath204 . this completes the inductive step and shows . conversely , assume that @xmath44 satisfies riemann interior majorization . we must show that holds for each @xmath205 . suppose that @xmath206 . using @xmath194 and the fact that @xmath207 for @xmath208 , we have @xmath209 next suppose that @xmath210 . using @xmath194 and the fact @xmath211 for @xmath212 , we have @xmath213 this proves that @xmath44 satisfies interior majorization as in definition [ mar ] . the next result gives the crucial sufficiency of riemann interior majorization for the existence of a self - adjoint operator with finite spectrum and prescribed diagonal . for nondecreasing sequences ordered by @xmath139 , the necessity of riemann interior majorization follows by combining theorems [ int - nec ] and [ eqmajs ] . let @xmath18 be an increasing sequence in @xmath4 with @xmath152 and @xmath20 . let @xmath143 be a nondecreasing sequence in @xmath23 $ ] which satisfies riemann interior majorization by @xmath18 . then , there is a self - adjoint operator @xmath27 with @xmath214 and diagonal @xmath143 . let @xmath141 be the sequence as in definition [ rim ] . by possibly shifting both sequences @xmath12 and @xmath11 we may assume without loss of generality that @xmath215 is given by with @xmath216 , that is @xmath148 @xmath154 @xmath155 and @xmath150 @xmath154 @xmath217 for @xmath218 . the special case when there exists @xmath219 such that @xmath220 follows directly from theorem [ horn - ndec ] applied to the sequences @xmath221 and @xmath222 with @xmath223 . moreover , by the symmetry considerations , as explained later in case 3 , one can also deal with the reciprocal case when @xmath224 thus , without loss of generality we can assume that neither nor holds , and since @xmath12 is nondecreasing we have @xmath225 for all @xmath226 . for convenience we note that by for any @xmath227 we have @xmath228 fix an integer @xmath229 $ ] such that @xmath230 obviously , @xmath231 . the proof of theorem [ srim ] splits into three cases . * @xmath232 . there are finite subsets @xmath233 $ ] and @xmath234 such that @xmath235 we apply lemma [ movelemma ] ( i ) to the sequence @xmath236 on the interval @xmath23 $ ] with @xmath237 , to obtain a sequence @xmath238 . observe that @xmath239 for @xmath55 , and @xmath240 is nondecreasing . by and , for @xmath241 we have @xmath242 with equality when @xmath243 . by theorem [ horn - ndec ] there is a positive rank @xmath244 operator @xmath245 with eigenvalues @xmath71 and @xmath246 , and diagonal @xmath247 . define @xmath248 and @xmath249 for @xmath250 . by , and , for @xmath251 we have @xmath252 with equality for @xmath253 . by theorem [ horn - ninc ] there is a positive rank @xmath254 operator @xmath255 with positive eigenvalues @xmath256 and diagonal @xmath257 . then the operator @xmath258 has eigenvalues @xmath146 and @xmath259 , and diagonal @xmath260 thus , @xmath261 has the desired spectrum and diagonal @xmath262 . lemma [ movelemma ] ( ii ) implies there is an operator @xmath27 , unitarily equivalent to @xmath263 with diagonal @xmath236 . this completes the proof of case 1 . * case 2 . * @xmath264 . the proof of case 2 breaks into two subcases . in subcase 1 we assume that there is a ( finite or infinite ) set @xmath265 $ ] such that @xmath266 in subcase 2 we assume that there exists a * finite * set @xmath265 $ ] such that @xmath267 observe that @xmath268 which implies that @xmath269 from we see that if subcase 2 fails , then we must have @xmath270 and we are in subcase 1 . first , assume we are in subcase 1 . if @xmath271 is finite , then @xmath272 and the sequence @xmath273 , consisting of @xmath274 zeros and @xmath275 , satisfy majorization property of the schur - horn theorem [ horn ] ( after reversing indexing ) . if @xmath271 is infinite , then the assumption that @xmath12 is nondecreasing guarantees that the assumptions of theorem [ horn - ndec ] are also met . the fact that @xmath276 s for @xmath55 are indexed by @xmath271 does not cause any problem here since one can temporarily reindex @xmath277 into @xmath278 . therefore , either theorem [ horn ] or theorem [ horn - ndec ] implies that there is a positive rank @xmath279 operator @xmath245 with diagonal @xmath272 and spectrum @xmath280 . we shall establish that a similar conclusion holds in subcase 2 albeit with appropriately modified diagonal terms . next , we assume we are in subcase 2 . set @xmath281 observe that @xmath282 the strict inequality above is a consequence of our assumption that fails . hence , there is a finite set @xmath283 such that @xmath284 we apply lemma [ movelemma ] ( i ) to the sequence @xmath236 on the interval @xmath23 $ ] with @xmath285 to obtain sequence @xmath238 . in particular , we have @xmath286 combining the fact that @xmath287 for @xmath288 with yields @xmath289 with equality when @xmath290 . since @xmath291 for all @xmath109 this shows that the sequence @xmath292 and the sequence @xmath273 , consisting of @xmath274 zeros and @xmath275 , satisfy majorization property of the schur - horn theorem [ horn ] ( with reverse ordering ) . thus , there exists an operator @xmath293 with diagonal @xmath294 and @xmath295 . this was also shown in subcase 1 albeit with @xmath296 and @xmath297 . one can think of a trivial application of lemma [ movelemma ] ( i ) with @xmath298 in subcase 1 . thus , both subcases yield the same conclusion . to finish the proof we set @xmath299)\setminus i_0 $ ] . by and we have @xmath300 by theorem [ kadison ] there is a projection @xmath301 such that @xmath302 has diagonal @xmath303 . since @xmath304 for @xmath305 we see that @xmath306 . consequently , @xmath307 has the desired spectrum and the diagonal @xmath308 . by lemma [ movelemma ] ( ii ) there is an operator @xmath27 which is unitarily equivalent to @xmath79 with diagonal @xmath143 . this completes the proof of case 2 . * @xmath309 . define the sequences @xmath310 and @xmath311 for all @xmath226 . one can verify that @xmath312 satisfies riemann interior majorization by @xmath313 . with this modification the sequence @xmath312 satisfies the requirements of case 2 . hence , there exists an operator @xmath314 with the spectrum @xmath315 and diagonal @xmath316 . therefore , the operator @xmath317 has the desired properties . this completes the proof of case 3 and theorem [ srim ] . by combining theorems [ eqmajs ] and [ srim ] we can show the sufficiency of the lebesgue interior majorization . in essence , we still need to deal with sequences which satisfy lebesgue interior majorization , but do not conform to more restrictive riemann interior majorization . theorem [ inthorn ] shows the sufficiency of theorem [ npt ] . [ inthorn ] let @xmath18 be an increasing sequence in @xmath4 with @xmath152 and @xmath20 . let @xmath22 be a sequence in @xmath23 $ ] which satisfies interior majorization by @xmath18 and @xmath318 . then , there is a self - adjoint operator @xmath27 with spectrum @xmath319 and diagonal @xmath22 . set @xmath320 and @xmath321 for @xmath322 . let @xmath323 be the identity operator on a space of dimension @xmath324 and let @xmath325 be the zero operator on a space of dimension @xmath326 . since @xmath31 and @xmath32 , the only possible limit points of @xmath327 are @xmath71 and @xmath146 . the argument breaks into four cases depending on the number of limit points . * case 1 : * assume both @xmath71 and @xmath146 are limit points of the sequence @xmath327 . this implies that there is a bijection @xmath328 such that @xmath329 is in nondecreasing order . since @xmath327 still satisfies interior majorization , by theorem [ eqmajs ] the sequence @xmath330 satisfies riemann interior majorization . by theorem [ srim ] there is a self - adjoint operator @xmath314 with diagonal @xmath327 and @xmath331 . the operator @xmath332 is as desired . this completes the proof of case 1 . * case 2 : * assume @xmath71 is the only limit point of @xmath327 . since @xmath333 we must have @xmath334 . there is a bijection @xmath335 such that @xmath329 is in nondecreasing order . the sequence @xmath336 satisfies interior majorization by @xmath140 , and theorem [ eqmajs ] implies that it also satisfies riemann interior majorization by @xmath140 . by theorem [ srim ] there is a self - adjoint operator @xmath245 with diagonal @xmath330 and @xmath337 . the operator @xmath338 has the same spectrum and diagonal @xmath12 . this completes the proof of case 2 . * case 4 : * assume @xmath327 has no limit points . this implies that @xmath339 is finite and since @xmath318 we also have @xmath340 . there is a bijection @xmath341 so that @xmath329 is nondecreasing . theorem [ eqmajs ] implies that @xmath342 satisfies riemann interior majorization by @xmath140 . theorem [ srim ] implies that there is a self - adjoint operator @xmath27 with diagonal @xmath342 and @xmath214 . this completes the proof of case 4 and the theorem . consider the sequence @xmath343 by kadison s theorem [ kadison ] there does not exist a projection with diagonal @xmath44 . however , in @xcite it was shown that the set of possible @xmath344 point spectra of operators with the diagonal @xmath44 @xmath345 consists of exactly @xmath346 points @xmath347 . with the help of _ mathematica _ and the characterization from theorem [ npt ] we can find the corresponding set of possible @xmath348 point spectra of operators . the following figure shows the set @xmath349 r. kadison , _ non - commutative conditional expectations and their applications _ , operator algebras , quantization , and noncommutative geometry , 143179 , contemp . math . , * 365 * , amer . soc . , providence , ri , 2004 . v. kaftal , g. weiss , _ a survey on the interplay between arithmetic mean ideals , traces , lattices of operator ideals , and an infinite schur - horn majorization theorem _ , hot topics in operator theory , 101135 , theta ser . , 9 , theta , bucharest , 2008 .

given a finite set @xmath0 we characterize the diagonals of self - adjoint operators with spectrum @xmath1 . our result extends the schur - horn theorem from a finite dimensional setting to an infinite dimensional hilbert space analogous to kadison s theorem for orthogonal projections @xcite and the second author s result for operators with three point spectrum @xcite .

the phase transition between the superconducting and normal states in many two - dimensional ( 2d ) systems is of kosterlitz - thouless ( kt ) type @xcite . the thermally excited vortices in a large enough sample , interacting via a logarithmic potential , are bound in neutral pairs below the kt transition temperature @xmath3 @xcite , and as the temperature @xmath4 is increased across @xmath3 from below these pairs start to unbind . the kt transition has been observed in experiments on superconducting films @xcite , 2d josephson junction arrays @xcite , and cuprate superconductors @xcite . in these experiments , the current - voltage ( @xmath1-@xmath2 ) characteristics have been commonly measured to detect the transition . for small enough currents it has a power law form @xmath5 ( or equivalently , @xmath6 with the electric field @xmath7 and the current density @xmath8 ) , where the @xmath1-@xmath2 exponent @xmath9 is known to have a universal value 3 precisely at the transition @xcite ; for @xmath10 one has @xmath11 , whereas @xmath12 for @xmath13 @xcite . the dynamic critical exponent @xmath14 which relates the relaxation time @xmath15 to the vortex correlation length @xmath16 via @xmath17 , is connected with @xmath9 through the relation @xmath18 consequently , @xmath14 has the value 2 at @xmath3 . indeed , the values @xmath19 and @xmath20 at the kt transition have been confirmed in many numerical simulations , e.g. , the lattice coulomb gas with monte carlo dynamics @xcite , the langevin - type molecular dynamics of coulomb gas particles @xcite , and the 2d _ xy _ model both with the resistively shunted junction dynamics ( rsjd ) and the relaxational dynamics ( rd ) @xcite . that is why @xmath21 obtained by pierson _ _ in ref . at the resistive transition in many 2d systems is very intriguing . we in ref . have re - analyzed the experimental @xmath1-@xmath2 characteristics for an ultra thin ybco sample in ref . and compared with the 2d rsjd model . this led to the suggestion that a novel finite - size type scaling effect of the @xmath1-@xmath2 characteristics ( which can possibly be caused not only by the actual finite size of the sample but also by a finite perpendicular penetration depth or a residual weak magnetic field @xcite ) is responsible for the large value of @xmath21 in ref . , but at the same time it was concluded that this exponent has nothing to do with the true dynamic critical exponent which at @xmath3 has value two @xcite . since similar conclusions were also reached in ref . , we believe that there now seems to emerge some consensus @xcite , although the physical mechanism causing this finite - size scaling behavior is still unclear . in the present paper we search for a possible reason of this finite - size scaling behavior . we use the rsjd @xcite , the rd ( often referred to as time - dependent ginzburg - landau dynamics ) @xcite , and the monte carlo dynamics ( mcd ) simulations @xcite to study the @xmath1-@xmath2 characteristics of three different models of 2d superconductors : the villain model @xcite , the @xmath0 model in its original form , and the @xmath0 model modified with @xmath22-type of potential ( see ref . for details ) . these three models differ from each other by the density of the thermally created vortices . we come to the conclusion that the qualitative features of the scaling suggested in ref . are ubiquitous and independent of both the vortex density and the type of dynamics . we also demonstrate that the @xmath1-@xmath2 curves obtained with different types of dynamics , up to constant scale factors , coincide very well in a broad range of the external current density . this opens a possibility to use mcd simulation , which from a simulation point of view is more efficient than rsjd or rd , to examine long - time dynamic properties of the models . the layout of the paper is as follows : in sec . [ sec : models ] we recapitulate the hamiltonians for the generalized 2d _ xy _ model with the @xmath22-type of potential ( the usual _ xy _ model is recovered when @xmath23 ) and the villain model , and describe details of the dynamics used ( rsjd , rd , and mcd ) . the results from the usual _ xy _ model with @xmath23 subject to different dynamics ( rsjd , rd , and mcd ) are presented in sec . [ sec : xy ] , while the results from rsjd simulations applied to the different types of models , i.e. , the usual _ xy _ model with @xmath23 , the _ xy _ model with @xmath24 and the villain model , are described and analyzed in sec . [ sec : rsj ] . we summarize and make final remarks in sec . [ sec : conc ] . the 2d @xmath0 model defined on a square lattice , where each lattice point @xmath25 is associated with the phase @xmath26 of the superconducting order parameter , is often used for studies of the kt transition . the phase variables in this model interact via the hamiltonian with nearest neighbor coupling , which in the absence of frustration is given by @xmath27 where @xmath28 denotes sum over nearest neighbor pairs , @xmath29 is the angular difference between nearest neighbors , and the interaction potential @xmath30 is written as @xmath31 with the josephson coupling strength @xmath32 . the dominant characteristic physical features close to the kt transition are associated with vortex pair fluctuations . one interesting aspect is then how the density of the thermally excited vortex fluctuation effects the critical properties . to study this we generalize the interaction potential by using a parameter @xmath22 @xcite : @xmath33 , \end{aligned}\ ] ] where @xmath34 corresponds to the potential of the usual @xmath0 model [ see eq . ( [ eq : uxy ] ) ] . the practical point with such generalization is that the vortex density increases with increasing @xmath22 @xcite . the variation of the parameter @xmath22 can also change the nature of the transition : for @xmath22 exceeding some maximum value ( @xmath35 ) the type of the phase transition changes from kt to the first order @xcite . in the present paper we choose @xmath24 which is well inside the kt transition region , yet is large enough to ensure substantially more vortex fluctuation over a temperature region around the phase transition in comparison with the usual @xmath23 @xmath0 model given by eq . ( [ eq : uxy ] ) . while the @xmath0 model with the @xmath22-type potential with @xmath36 has more vortices than the usual @xmath23 @xmath0 model , we also study the villain model @xcite which has less vortex - antivortex pairs @xcite . the interaction potential @xmath30 in the villain model is given by @xmath37 .\ ] ] to simulate the dynamic behaviors of these models we use several types of dynamics : rsjd , rd , and mcd . all these dynamics should result in the same equilibrium static behaviors if we apply them to the models with the same interaction potential . however , dynamic properties of the systems can be different . of course , different types of dynamics have their own advantages and disadvantages . the rsjd is constructed from the elementary josephson relations for single josephson junction that forms the array units , plus kirchhoff s current conservation condition at each lattice site @xcite . therefore , this type of dynamics has a firm physical realization . on the other hand , rsjd is quite slow which leads to the limitation in the time scale one can probe in simulations . although the rd @xcite is much easier to implement than rsjd , it does not converge much faster than rsjd and it does not have a similar direct physical realization as rsjd . however , a superconductor has been argued to have a rd type of dynamics rather than a rsjd @xcite . the mcd simulations @xcite are much faster than rsjd or rd , which allows one to investigate dynamic behaviors in much longer time scale ( one can also study dynamic behaviors at much lower temperatures with mcd ) . however , since there is no direct physical realization of the mcd in practice the applicability of this dynamics to a specific physical system must then be explicitly demonstrated . in the following discussions on the details of the different dynamics used , we focus on the original 2d @xmath0 model with the interaction potential in eq . ( [ eq : uxy ] ) since the extensions to a modified 2d @xmath0 model ( [ eq : up ] ) and villain model ( [ eq : uvillain ] ) are straightforward . in this section we briefly review the dynamical equations of motion for rsjd , rd , and mcd , in the presence of the fluctuating twist boundary condition ( ftbc ) @xcite . we perform simulations of unfrustrated square @xmath38 lattices with @xmath39 , 8 , and 10 at various temperatures to measure the voltage across the lattice as a function of the external current . although the system sizes are relatively small , which is inevitable because of the low temperatures and the small external currents used here , the ftbc has been shown to be very efficient in reducing the artifact due to small system sizes @xcite , and reliable results can be established @xcite . in the ftbc , the twist variable @xmath40 is introduced and the phase difference @xmath41 on the bond @xmath42 is changed into @xmath43 , with the unit vector @xmath44 from site @xmath25 to site @xmath45 , while the periodicity on @xmath46 is imposed : @xmath47 . the hamiltonian of 2d @xmath38 @xmath0 model under ftbc without external current has been introduced in ref . , and is written as [ compare with eq . ( [ eq : hxy ] ) ] @xmath48 which later in ref . has been extended to the system in the presence of an external current and written as @xmath49 where @xmath8 the current density in the @xmath50 direction . we introduce first the rsjd equations of motion for phase variables and twist variables , which are generated from the local ( global ) current conservation for the phase ( twist ) variables ( see ref . for details and discussions ) . the net current @xmath51 from site @xmath25 to site @xmath45 is the sum of the supercurrent @xmath52 , the normal resistive current @xmath53 , and the thermal noise current @xmath54 : @xmath55 . the supercurrent is given by the josephson current - phase relation , @xmath56 , where @xmath57 is the critical current of the single junction . the normal resistive current is given by @xmath58 , where @xmath59 is the potential difference across the junction , and @xmath60 is the shunt resistance . finally the thermal noise current @xmath61 in the shunt at temperature @xmath4 satisfies @xmath62 and @xmath63 , where @xmath64 is thermal average , and @xmath65 and @xmath66 are dirac and kronecker delta , respectively . using the current conservation law at each site of the lattice together with the josephson relation @xmath67 one can derive the rsjd equations of motion for phase variables : @xmath68 where the primed summation is over the four nearest neighbors of @xmath45 , @xmath69 is the lattice green function for 2d square lattice , and @xmath70 is the dimensionless thermal noise current defined by @xmath71 . the time , the current , the distance , the energy , and the temperature are normalized in units of @xmath72 , @xmath73 , the lattice spacing @xmath9 , the josephson coupling strength @xmath32 , and @xmath74 , respectively . in order to get a closed set of equations we further specify the dynamics of the twist variable @xmath75 from the condition of the global current conservation that the summation of the all currents through the system in each direction should vanish @xcite : @xmath76 where @xmath77 denotes the summation over all nearest neighbor links in the @xmath50 direction , and we apply the external dc current with the current density @xmath8 in the @xmath50 direction . here , the thermal noise terms @xmath78 and @xmath79 obey the conditions @xmath80 , and @xmath81 . in the rd , the equations of motion for the phase variables are written as @xcite @xmath82 where @xmath83 is a dimensionless constant ( we set @xmath84 from now one ) , @xmath85 is in eq . ( [ eq : hftbcj ] ) , @xmath86 is in units of @xmath87 , and the thermal noise @xmath88 at site @xmath25 satisfies @xmath89 and @xmath90 . the equation of motion for the twist variables in the absence of an external current is of the form ( see ref . for more details ) @xmath91 which is the same as eqs . ( [ eq : delta_x ] ) and ( [ eq : delta_y ] ) for rsjd . accordingly , to some extent the rd may be viewed as a simplified version of the rsjd where the global current conservation is kept but the local current conservation is relaxed . consequently , the equations for the phase variables are different for rsjd and rd [ eqs . ( [ eq : rsjphase ] ) and ( [ eq : rdphase ] ) , respectively ] while the same equations ( [ eq : delta_x ] ) and ( [ eq : delta_y ] ) apply to the twist variables for both dynamics . these coupled equations of motion are discretized in time with the time step @xmath92 and @xmath93 for rsjd and rd , respectively , and numerically integrated using the second order runge - kutta - helfand - greenside algorithm @xcite . the voltage drop @xmath2 across the system in the @xmath50 direction is written as @xmath94 ( see ref . ) in units of @xmath95 for rsjd and in units of @xmath96 for rd , respectively . we measure the electric field @xmath97 to obtain @xmath1-@xmath2 characteristics , where @xmath98 denotes the time average performed over @xmath99 time steps for large currents for both rsjd and rd , and @xmath100 and @xmath101 steps for small currents for rsjd and rd , respectively . the technique to simulate 2d _ xy _ model with mcd is based on the hamiltonian ( [ eq : hftbcj ] ) and the standard metropolis algorithm @xcite . the one mc step , which we identify as a time unit , is composed as follows @xcite : 1 . [ step : pickphase ] pick one lattice site and try to rotate the phase angle at the site by an amount randomly chosen in @xmath102 $ ] ( we call @xmath103 the trial angle range ) . the twist variable @xmath104 is kept constant during the update of the phase variables . [ step : dephase ] compute the energy difference @xmath105 before and after the above try ; if @xmath106 or if @xmath107 is greater than a random number chosen on the interval @xmath108 , accept the trial move . [ step : repeatphase ] repeat steps [ step : pickphase ] and [ step : dephase ] for all the lattice sites to update the phase variables . [ step : picktwist ] update the fluctuating twist variables @xmath109 in the similar way that @xmath110 is tried to rotate within the angle range @xmath111 with @xmath46 and @xmath112 kept unchanged . ( for convenience , we use @xmath113 ) . [ step : detwist ] compute the energy difference @xmath105 before and after the trial step [ step : picktwist ] for @xmath114 . accept the step [ step : picktwist ] , if @xmath106 otherwise accept it with probability @xmath115 like in step [ step : dephase ] . [ step : repeattwist ] repeat steps [ step : picktwist ] and [ step : detwist ] to update @xmath112 . in the mcd simulation the trial angle @xmath116 has been chosen since it is sufficiently small in order to obtain the correct @xmath1-@xmath2 characteristics while it is big enough to make mcd much faster than the other dynamic methods @xcite . the time - averaged electric field is obtained after equilibration from the averages over @xmath101 ( at large currents ) to @xmath117 ( at small currents ) mc steps . in this section we use three different types of dynamics , the rsjd , the rd , and the mcd to study dynamic behavior of the 2d _ xy _ model with @xmath23 under the ftbc . the dynamic behavior of the system can be obtained from the complex conductivity , the flux noise spectrum , as well as the @xmath1-@xmath2 characteristics which is commonly measured in experiments . we will focus on the @xmath1-@xmath2 characteristics in the present paper . as pointed out in sec . [ sec : models ] the rd to some extent may be considered as a simplified version of the rsjd . thus from this point of view it is perhaps not surprising that these two models ( as we will see ) contain similar features of the vortex dynamics . in ref . from the study of a simple dynamic model of isolated magnetic particles in a uniform field it has been shown that the actual dynamics of the model and mcd are in a good agreement when the acceptance ratio of the metropolis step is low enough . this implies that the mcd should give the same @xmath1-@xmath2 characteristics as the rsjd after an appropriately chosen normalization of time , when the trial angle @xmath118 is sufficiently small ( it was shown in ref . that @xmath119 is sufficiently small ) . we will confirm this further in the present simulations . in fig . [ fig : iv ] we compare @xmath1-@xmath2 characteristics in the form of the electric field @xmath7 versus the current density @xmath8 obtained from rsjd , rd , and mcd simulations in the temperature range @xmath120 ( the temperature interval is 0.05 if @xmath121 and 0.10 otherwise ) in log scales for the system size @xmath122 . in order to make @xmath1-@xmath2 data of rsjd and rd simulations coincide , @xmath7 obtained with rd ( @xmath123 ) is multiplied by a temperature - independent factor represented by the horizontal line in the inset of fig . [ fig : iv ] . since the measured time - averaged electric field is inversely proportional to the time scale for a given dynamics , one can from the ratio @xmath124 infer the correspondence between times of rd and rsjd . > from this comparison , we find for @xmath121 that one unit of time in rd approximately corresponds to 0.526 time unit in rsjd , independent of the temperature . note that the @xmath1-@xmath2 curves corresponding to the temperatures exceeding 1.00 are almost a straight lines . therefore the collapse of data between the different dynamics is trivial for @xmath125 . also the @xmath1-@xmath2 characteristics obtained from mcd can be made to collapse on top of the corresponding curves for the rsjd and rd , as shown in fig . [ fig : iv ] . however , in this case the time scale factor , describing how many rsjd time steps one mcd step corresponds to , depends on the temperature , as shown in the inset of fig . [ fig : iv ] , where this factor is shown to be a linear function of temperature for system size @xmath39 , 8 , and 10 up to @xmath126 . this is in accordance with the model studied in ref . , where the same linear behavior in terms of the temperature has been found . thus for the @xmath1-@xmath2 curves we have a precise relation between rsjd and mcd : for example , at @xmath127 we get @xmath128 mcs = @xmath129rsjd time unit . this opens a practical possibility to study some aspects of dynamic behavior of the @xmath0 model by using mcd simulation , which is usually much more efficient than rsjd and rd . we have shown in this section that the rsjd , the rd , and the mcd applied to the usual @xmath0 model gives basically the identical @xmath1-@xmath2 characteristics up to some constant factors . from this observation , one can conclude that the dynamic critical behaviors of the @xmath0 model inferred from the @xmath1-@xmath2 characteristics should be identical for all these dynamic models . we in next section use the rsjd to study the villain model and @xmath0 models with @xmath24 and presume from the observation in the present section that the conclusion drawn in sec . [ sec : rsj ] for the rsjd case should be also valid in the other dynamics ( rd and mcd ) . to study the critical behavior of the system in the vicinity of the transition one can use scaling relations . fisher , fisher , and huse ( ffh ) in ref . proposed that the nonlinear @xmath1-@xmath2 characteristics in a @xmath130-dimensional superconductor scales as @xmath131 where @xmath16 and @xmath14 are the correlation length and the dynamic critical exponent , respectively , and @xmath132 is the scaling function above ( @xmath133 ) and below ( @xmath134 ) the transition . _ in ref . have applied a variant of this ffh scaling approach to the @xmath1-@xmath2 data for thin ( @xmath135 ) superconductors and superfluids and suggested a phase transition with @xmath21 . in ref . another scaling relation has been introduced and it has been shown that a certain finite - size effect which is not included in the ffh scaling may have caused the large @xmath14 in spite of the fact that the finite size effect precludes the possibility of a real phase transition . this finite - size scaling around and below the kt transition is given by the form ( see ref . for the details ) @xmath136 where @xmath137 is a finite - size induced resistance without external current , @xmath138 , @xmath139 for large @xmath50 , and @xmath140 is a function of at most @xmath4 and @xmath141 such that a finite limit function @xmath142 exists in the large-@xmath141 limit . for small values of the variable @xmath143 the @xmath4-dependence of the scaling is absorbed in a function @xmath144 for each fixed size @xmath141 giving rise to the scaling form @xmath145 in fig . [ fig : scaling ] we demonstrate the existence of the finite - size scaling given by eq . ( [ eq : scale ] ) for @xmath122 within the temperature intervals @xmath146 . the data are obtained for mcd and scaled by the appropriate factor so as to correspond to rsjd and rd ( compare fig . [ fig : iv ] ) . because of the correspondence between the three types of dynamics ( see sec . [ sec : xy ] ) , this also means that the existence of the finite - size scaling given by eq . ( [ eq : scale ] ) is insensitive to the choice of dynamics . the ffh scaling given by eq . ( [ eq : p1 ] ) is correct only in the thermodynamic limit @xmath147 . however , from a practical point of view there is a connection between pierson method , which is based on ffh scaling , and the finite - size scaling introduced by eq . ( [ eq : scale ] ) . if we assume that @xmath16 in eq . ( [ eq : p1 ] ) is proportional to @xmath148 , where @xmath149 is @xmath4-independent constant , the connection between the two different scaling approach becomes : @xmath150 with @xmath151 being a constant which may depend on @xmath141 . in fig . [ fig : g_l ] there are presented three different functions @xmath144 corresponding to @xmath39 , 8 , and 10 . these functions are determined from the condition that curves corresponding to the different temperatures should collapse when plotted as @xmath152 vs @xmath153 . since it is well established that the 2d @xmath0 model on the square lattice has the kt transition at @xmath154 ( ref . ) , fig . [ fig : g_l ] shows that @xmath144 over a limited region in the vicinity of the kt transition is very well represented by the @xmath148 with @xmath155 for all investigated system sizes . next we demonstrate how the finite - size scaling given by eq . ( [ eq : scale ] ) works for the @xmath1-@xmath2 data obtained by simulations of the modified @xmath0 model . these simulations are done with rsjd . [ villain](a ) verifies that this scaling indeed exists for the @xmath0 model modified with the villain type of potential introduced by eq . ( [ eq : uvillain ] ) . the inset of fig . [ villain](a ) shows the scaling function @xmath144 determined by finding the best data collapse for small values of @xmath153 . one can see that in the vicinity of @xmath3 , which is approximately equals to @xmath156 for this model , @xmath144 can be fitted by @xmath157 with @xmath155 . the data collapse in fig . [ villain](b ) shows that @xmath152 is only a function of the scaling variable @xmath158 ( when @xmath158 is small enough ) for the @xmath0 model modified with @xmath22-type of potential ( eq . ( [ eq : up ] ) ) , where @xmath24 . the inset shows the function @xmath144 together with @xmath157 . since @xmath159 for this model one can again see that the scaling function @xmath144 in the vicinity of kt transition is proportional to @xmath148 with the same exponent @xmath155 as for the original @xmath0 and villain models . the crucial difference between the villain model , the usual @xmath0 model and the @xmath24 @xmath0 model in the present context is the vortex density . the kt - transitions for these models occur at the coulomb gas temperatures @xmath160 , 0.2 , and 0.1 , respectively ( @xmath161 ( see ref . ) ) . lower @xmath162 means higher vortex density . thus the finite - size scaling property given by eq . ( [ eq : scale ] ) appears to be independent of vortex density . we have simulated 2d _ xy _ model with three types of dynamics : rsjd , rd , and mcd . the main conclusion of the paper is that the qualitative features of the finite - size scaling given by eq . ( [ eq : scale ] ) are independent of both the vortex density and the type of dynamics . therefore the finite - size scaling behavior given by eq . ( [ eq : scale ] ) of the finite - size induced tails of the @xmath1-@xmath2 characteristics appears to be a robust feature . from the comparisons of the current - voltage characteristics obtained for each type of dynamics we found that , up to some scale factor , @xmath1-@xmath2 curves at a given temperature are identical over a broad range of external currents . this makes it possible to use mcd simulations , which are more computer efficient than rsjd and rd simulations , to obtain @xmath1-@xmath2 curves corresponding to both rsjd and rd dynamics . the phase transition for the 2d @xmath0-type models are of kt type with @xmath19 . this raises the intriguing question of the origin of the large @xmath21 obtained by pierson _ et al . _ @xcite . in ref . it was argued that the pierson scaling in relation to the finite - size scaling given by eq . ( [ eq : scale ] ) corresponds to the proportionality @xmath163 where @xmath164 is the exponent which corresponds to the `` @xmath14 '' obtained by the pierson scaling . the reason for the existence of this scaling like behavior is still unclear . in the present paper we have shown that , within the class of 2d @xmath0-type models studied , a value @xmath165 is obtained independently of the type of dynamics , as well as , of the vortex density . the origin of this seemingly robust behavior calls for further investigations . for a general review see , e.g. , p. minnhagen , rev . * 59 * , 1001 ( 1987 ) ; for connections to high-@xmath166 superconductors see , e.g. , p. minnhagen , in _ models and phenomenology for conventional and high - temperature superconductors _ , proceedings of the international school of physics , `` enrico fermi '' course cxxxvi ( ios press , amsterdam , 1998 ) , p. 451 . see , for instance , i.g . gorlova , and yu.i . latyshev , physica c * 193 * , 47 ( 1992 ) ; n .- c . yeh and c.c . tsuei , phys . b * 39 * , 9708 ( 1989 ) ; s. martin , a.t . fiory , r.m . fleming , g.p . espinosa and a.s.cooper , phys . lett . * 62 * , 677 ( 1989 ) ; q.y . ying and h.s . kwok , phys . b * 42 * , 2242 ( 1990 ) . s.w . pierson , m. friesen , s.m . ammirata , j.c . hunnicutt , and l.a . gorham , phys . b * 60 * , 1309 ( 1999 ) ; s.m . ammirata , m. friesen , s.w . pierson , l.a . gorham , j.c . hunnicutt , m.l . trawick , and c.d . keener , physica c * 313 * , 225 ( 1999 ) ; s.w . pierson and m. friesen , physica b * 284 * , 610 ( 2000 ) .

two - dimensional ( 2d ) @xmath0 model subject to three different types of dynamics , namely monte carlo , resistivity shunted junction ( rsj ) , and relaxational dynamics , is numerically simulated . from the comparisons of the current - voltage ( @xmath1-@xmath2 ) characteristics , it is found that up to some constants @xmath1-@xmath2 curves at a given temperature are identical to each other in a broad range of external currents . simulations of the villain model and the modified 2d @xmath0 model allowing stronger thermal vortex fluctuations are also performed with rsj type of dynamics . the finite - size scaling suggested in medvedyeva _ et al . _ [ phys . rev . b ( in press ) ] is confirmed for all dynamic models used , implying that this finite - size scaling behaviors in the vicinity of the kosterlitz - thouless transition are quite robust . 2

1 . comparison of charged - higgs signal and principal backgrounds in the @xmath112 channel at @xmath21 tev , including branching fractions and acceptance cuts but excluding @xmath3-tag factors , with @xmath69 gev : ( a ) cross sections versus @xmath9 for @xmath113 gev ; ( b ) cross sections versus @xmath8 for @xmath92 . 2 . comparison of charged - higgs signals and summed backgrounds in the distribution versus reconstructed charged - higgs mass @xmath88 , with two counts per event . the cases @xmath114 gev are shown for ( a ) @xmath115 and ( b ) @xmath116 .

we discuss the viability of @xmath0 charged - higgs signals at the proposed lhc @xmath1 supercollider , in the decay channel @xmath2 . here one top quark decays hadronically and one semileptonically , with all three @xmath3-quarks giving flavor - tagged jets . the principal backgrounds come from @xmath4 and @xmath5 continuum production , with possible mis - tagging of @xmath6 and @xmath7 . we conclude that significant signals can be separated from these backgrounds , for limited but interesting ranges of the parameters @xmath8 and @xmath9 , with the lhc energy and luminosity . 6.5 in 8.5 in .25 in # 1#20=1=to0@xmath100=1=to0#2#1 - 01 @=11 tempcntc citex[#1]#2@fileswauxout tempcnta@tempcntb@neciteaciteforciteb:=#2citeo#1 citeotempcnta > tempcntbciteacitea , tempcnta = tempcntbtempcnta @=12 = cmssbx10 scaled 2 to * heavy charged higgs signals + at the lhc * + v. barger@xmath11 , r.j.n . phillips@xmath12 , and d.p . roy@xmath13 + _ @xmath11physics department , university of wisconsin , madison , wi 53706 , usa + @xmath12rutherford appleton laboratory , chilton , didcot , oxon ox11 0qx , uk + @xmath13tata institute of fundamental research , bombay 400005 , india _ the search for higgs bosons is in the forefront of present research effort in particle physics@xcite . while there is a single higgs boson in the standard model ( sm ) , the minimal supersymmetric extension ( mssm ) has five of them three neutral @xmath14 and two charged @xmath15 . phenomenological interest here has concentrated largely on the neutral sector@xcite . as regards @xmath16 , it is recognized that top decay would provide viable signals at hadron colliders if @xmath17 . on the other hand , the region @xmath18 is favored by constraints from @xmath19 data@xcite , if there are no light charginos@xcite ; this region has been considered problematical , since the principal signal @xmath20 would suffer from large qcd backgrounds at a hadron collider@xcite . however , the possibility of efficient @xmath3-tagging could transform this situation by discriminating against the background , as in the case of neutral higgs signals in the intermediate mass region@xcite . the present letter is devoted to a quantitative exploration of this possibility ; our results apply to two - higgs - doublet models in general , though we shall refer to particular features of the mssm from time to time . some preliminary results from a similar study by gunion@xcite have recently appeared ; these are complementary to the present work , since his methods of calculation and analysis differ somewhat from ours . we show below that viable signals may indeed be expected , over a limited but interesting range of @xmath16 mass and coupling parameter space , in the proposed large hadron collider ( lhc ) @xcite with @xmath1 collisions at cm energy @xmath21 tev . in two - higgs - doublet models , where it is usually assumed that up - type and down - type quarks get masses from different vevs , the main @xmath16 interactions with quarks are given by @xmath22 b + \rm h.c . \;,\ ] ] neglecting terms suppressed by small quark masses or small km matrix elements @xmath23 , where @xmath24 is the usual ratio of vevs . the principal hadroproduction and decay mechanisms for a heavy charged higgs boson are therefore @xmath25 plus the corresponding charge - conjugate channel . ( in the mssm , an alternative decay mode to the same final state , @xmath26 , is suppressed in the mass range @xmath18 of present interest@xcite ) . as a tag for top production , we shall assume that one of the @xmath27-bosons decays leptonically @xmath28 ( with @xmath29 ) . to enhance the event rate and facilitate event reconstruction , we assume that the other @xmath27-boson decays hadronically @xmath30 , with invariant mass @xmath31 . thus we consider the signal @xmath32 where all five quarks give separate jets and the lepton is isolated . we also assume that all three @xmath3-jets are tagged by a vertex detector ; tagging via semileptonic @xmath3-decays is less desirable , since the additional missing neutrinos blur the kinematics , but on the other hand it distinguishes @xmath3 from @xmath33 and removes some ambiguity in the event reconstruction . this final state implies a spectator @xmath3-quark in one of the beams ; however , we expect that this spectator will be produced at small angle and will not appear in the acceptance region described below . our approach differs here from gunion@xcite who calculates the subprocess @xmath34 where the spectator is explicit . the principal background sub - processes are qcd production @xmath35 and fake backgrounds from @xmath36 where the @xmath37 jet or one of the @xmath38 jets is mistakenly tagged ; @xmath39 decays are understood . there is an electroweak contribution to eq.([qcdprod ] ) from @xmath16 exchange in the @xmath40-channel , but this is much smaller than the signal ( suppressed by additional propagators ) and we henceforth neglect it . there is also a possible background from intermediate - mass neutral higgs boson production and decay : @xmath41 where one of the final @xmath3-quarks does not give a separate jet within acceptance cuts . in the mssm , this neutral boson could be @xmath42 or @xmath43 or @xmath44 ; with our present heavy @xmath16 scenario , we would then have @xmath43 and @xmath44 equally heavy @xmath45 with their @xmath46 contributions suppressed by competing channels @xmath47 and @xmath48 ) while @xmath42 couplings are approximately those of the sm . however , the total @xmath49 production @xcite is then an order of magnitude smaller than @xmath5 production via eq.([qcdprod ] ) , so we henceforth neglect the channel of eq.([i - m h^0 ] ) . it is already known@xcite that these backgrounds are potentially much larger than the signal . however , we shall show that the background of eq.([qcdprod ] ) can be reduced to the same order as the signal ( in favorable cases ) by a choice of kinematic cuts , while the fake background eq.([fake ] ) is also reduced to a comparable level by the additional @xmath3-tagging requirement . we here choose the following acceptance cuts on the 3 tagged plus 2 untagged jets ( collectively labelled @xmath50 ) , the lepton @xmath51 and missing transverse momentum @xmath52 : @xmath53 where @xmath54 and @xmath55 denote transverse momentum and pseudorapidity . we also require minimum separations @xmath56^{1/2}$ ] between the jets and lepton , @xmath57 to simulate some effects of jet - finding and lepton isolation criteria . we take account of possible invisible neutrino energy in @xmath58 decays by monte carlo modelling , and thereafter regard all partons as jets if they pass the above cuts . we simulate calorimeter resolution by a gaussian smearing of @xmath54 , with @xmath59 for jets and @xmath60 for leptons ( taking the same resolution for @xmath61 and @xmath62 for simplicity ) . the @xmath52 is evaluated from the vector sum of lepton and jet momenta , after resolution smearing . we require the invariant mass of the two untagged jets to be consistent with @xmath63 : @xmath64 we assume branching fractions @xmath65 , and tagging efficiencies @xmath66 for individual @xmath3-jets , @xmath7-jets and gluon ( or light quark ) jets respectively . we calculate production rates using the mrsd@xmath67 parton distributions@xcite at scale @xmath68 for both the signal and the backgrounds , assuming @xmath69 gev throughout . since the @xmath3-quark distribution is inferred via qcd evolution from descriptions of deep inelastic scattering data , there is room for controversy here ; however , both the signal and the true " background of eq.([qcdprod ] ) depend on the same input @xmath3-distribution . the net signal and background cross sections , with these cuts and branching / tagging factors , are illustrated in fig . 1 for @xmath1 collisions at @xmath21 tev . figure 1 , which does not include tag - factors , shows that the charged - higgs signal has an appreciable size for some ranges of the parameters @xmath70 and @xmath9 . the @xmath9 dependence is given by a factor @xmath71 , with a minimum at @xmath72 . the neighbourhood of this minimum is unpromising for @xmath16 detection , but many @xmath73@xmath74 models suggest that @xmath9 lies near @xmath75 or alternatively is very large@xcite . tagging reduces the major @xmath76 and @xmath77 backgrounds by a factor @xmath78 relative to the signal , making them roughly comparable for favourable @xmath9 . to improve the signal / background ratio further and to estimate the mass @xmath70 , we propose the following strategy for event reconstructions . 1 . reconstruct the missing neutrino momentum , by equating @xmath79 and fixing the longitudinal component @xmath80 by the invariant mass constraint @xmath81 . the latter gives two solutions in general ; if they are complex we discard the imaginary parts and the solutions coalesce . we note that the sign @xmath82 of this @xmath27 ( and hence by inference the other @xmath27 too ) is determined by the sign of the lepton charge . 2 . there are now 6 ways in which two of the @xmath3-jets can be paired with the two @xmath27 s to form top candidates ( unless some of the @xmath3-jets are also lepton - tagged and thus have known signs ) . together with the two - fold ambiguity from ( a ) , this gives 12 candidate reconstructions , in each of which there are two top mass values @xmath83 . we select the assignment with best fit to the top mass ( that will be known ) , determined by minimizing @xmath84 subject to the requirements @xmath85 gev and @xmath86 gev . if these requirements can not be met , we reject the event as unreconstructable . 3 . in the selected best - fit assignment above , there are 2 ways in which the remaining @xmath3-jet can be paired with one of the top candidates , so we have 2 candidate values for the reconstructed charged - higgs mass @xmath87 . unless the charge of the @xmath3-jet can be identified , there is no way to choose between them ( unless the @xmath3-jet is also lepton - tagged ) , so we retain both values ; thus even the signal events contain an irreducible combinatorial background . however , the correct pairings will give a peak in the @xmath88 distribution while the incorrect pairings and background events will be more broadly distributed . this strategy is more ambitious than that of ref.@xcite , where a @xmath3-jet is combined only with a reconstructed @xmath89 hadronic system . figure 2 compares the signal and background contributions to the @xmath90 distributions , for @xmath91 gev with either @xmath92 or @xmath93 ; there are two possible values and hence two counts per event in this graph . for the most favourable of the cases illustrated , namely @xmath94 gev with @xmath93 , the signal integrated over the range @xmath95 gev is 5 counts over a total background of 4 counts for each fb@xmath96 of luminosity . with @xmath97 of luminosity ( one years running at design luminosity @xmath98 ) this signal would be very significant . as @xmath99 increases , both the signal and background fall at comparable rates ; for @xmath100 gev , the signal in a 60 gev bin is @xmath101 over a background of @xmath102 counts / fb@xmath96 that would still be very significant with @xmath97 luminosity . if we take @xmath103 instead , the background remains essentially the same while all the signals drop by a factor 2.8(11 ) ; hence the regions @xmath104 and @xmath105 are very promising while the region @xmath106 is problematical . thus far we have assumed @xmath69 gev ; for @xmath107 gev instead , the @xmath108 signals shown here increase by about @xmath109 ( except near threshold @xmath110 ) while the net background falls by about @xmath111 . lastly we remark that the assumed cuts above are rather stringent , reducing the higgs signal by factors of order 1030 depending on @xmath8 , and the tagging efficiencies may prove to be better than we have assumed here@xcite ; in these respects our event rates may be viewed as conservative . we conclude that the outlook is promising . with our assumed tagging efficiencies and cuts , significant @xmath20 charged - higgs signals would be detectable for a limited but interesting range of the parameters @xmath8 and @xmath9 . we thank alan stange and rahul sinha for helpful discussions . 99 for a review see j.f . gunion , h. haber , g.l . kane and s. dawson , the higgs hunter s guide " ( addison - wesley , reading , ma , 1990 ) . h. baer et al . , phys . rev . * d46 * ( 1992 ) 1067 . j.f . gunion , r. bork , h.e . haber , and a. seiden , phys . rev . * d46 * ( 1992 ) 2040 ; j.f . gunion , l.h . orr , ibid . * d46 * ( 1992 ) 2052 ; j.f . gunion , h.e . haber , c. kao , ibid . * d46 * ( 1993 ) 2907 . z. kunszt and f. zwirner , nucl . phys . * b46 * ( 1992 ) 4914 . v. barger , m.s . berger , a.l . stange , and r.j.n . phillips , phys . rev . * d45 * ( 1992 ) 4128 ; v. barger , k. cheung , r.j.n . phillips , and a.l . stange , ibid . * d46 * ( 1992 ) 4914 . v. barger , j.l . hewett , and r.j.n . phillips , phys . rev . * d41 * ( 1990 ) 3421 . a.c . bawa , c.s . kim , and a.d . martin , z. phys . * c47 * ( 1990 ) 75 . r.m . godbole and d.p . roy , phys . rev . * d43 * ( 1991 ) 3640 ; m. drees and d.p . roy , phys . lett . * b269 * ( 1991 ) 155 ; d.p . roy , ibid . * b277 * ( 1992 ) 183 ; ibid . * b283 * ( 1992 ) 403 . r.m . barnett , j.f . gunion , r.cruz , and b. hubbard , phys . rev . * d47 * ( 1993 ) 1048 . j.l . hewett , phys . rev . lett . * 70 * ( 1993 ) 1045 ; v. barger , m.s . berger , r.j.n . phillips , ibid . * 70 * ( 1993 ) 1368 ; cleo collaboration , report to washington aps meeting , april 1993 . s. bertolini et al . , nucl . phys . * b353 * ( 1991 ) 591 ; r. barbieri and g.f . giudice , phys . lett . * b309 * ( 1993 ) 86 . j.f . gunion , g.l . kane , and j. wudka , nucl . phys . * b299 * ( 1988 ) 231 . t. garavaglia , w. kwong , and d.d . wu , phys . rev . * d48 * ( 1993 ) 1899 . j. dai , j.f . gunion , r. vega , phys . rev . lett . * 71 * ( 1993 ) 2699 . j.f . gunion , work in progress , quoted by j.f . gunion and s. geer , report of ssc higgs working group , ucd-93 - 32 . for latest parameters , see lhc news no.4 and cern / ac/93 - 03 . a.d . martin , r.g . roberts , and w.j . stirling , phys . lett . * b309 * ( 1993 ) 492 . e.g. s. dimopoulos , l.j . hall and s. raby , phys . rev . * d45 * ( 1992 ) 4192 ; v. barger , m.s . berger , and p. ohmann , phys . rev . * d47 * ( 1993 ) 1093 .

nowadays physics at small scales is established to the characteristic distances for the nucleons substructure of about @xmath0 m. there are many interesting theories which explore physics at even smaller distances , e.g. , theories of extra dimensions and string theories . yet another type of models constitute so - called composite models @xcite . the composite models which contain colored preons predict a possibility of leptoquarks and leptogluons . leptoquarks are also present in many theories of grand unification @xcite , r - parity - violating supersymmetric models @xcite and in the extended technicolor models @xcite . many theories containing the particles that carry lepton number and color simultaneously predict color - singlet leptohadrons , in particular the bound states of leptoquark - antiquark pairs @xcite . however , their experimental signatures are poorly addressed in the literature . in this paper we consider the possible phenomenology of a particle that interacts with a lepton and a meson and can be referred to as leptomeson ( lm ) . since lms are colorless they can be lighter ( and more accessible for collider probes ) than the leptoquarks with their heavy color dressing . at low energies the new heavy particles may reveal themselves in the tiny effects on the lepton intrinsic properties ( magnetic moment , charge radius , etc . ) . currently one of the tools most sensitive to the new physics is the precise measurement of the muon anomalous magnetic moment ( amm ) @xmath1 , which reveals a discrepancy with its theoretical prediction within the standard model ( sm ) @xcite . in the theories of compositeness the new contributions to the lepton amm can arise from the relativistic bound state description of leptons @xcite . among the possible new heavy composites , purely leptogluons may contribute to the lepton amm starting only from the two - loop level . however , one - loop contributions can be generated by leptogluons plus color - octet @xmath2 bosons @xcite , the colorless excited leptons @xcite , and leptoquarks @xcite . leptomesons ( leptodiquarks ) may effectively provide a new one - loop ( two - loop ) contribution to the muon amm . a not too large size for this contribution can be achieved due to large lm masses of @xmath3 gev and/or small lm couplings , e.g. , of @xmath4 . many kinds of one - loop contributions to @xmath5 were classified @xcite by the types of particles that propagate in the loop , which typically include light sm fermions ( leptons and light quarks ) and heavy vectors , scalars and fermions ( @xmath6 , @xmath2 , @xmath7 , @xmath8 , and new heavy particles ) . in addition , there is a possibility of very light [ with the masses of @xmath9 gev ] superweakly coupled new particles @xcite propagating in the loop , which is hard to test in collider experiments . however , the discussed leptomeson contribution to @xmath5 may involve light vector and/or scalar particles ( vector and scalar mesons ) , and can be tested at colliders . in this paper we investigate the effects of lms on the muon @xmath10 . to our knowledge , these effects have not been considered yet in the literature . in the next section we give analytical expressions for the characteristic lm contributions to @xmath5 . in section [ sec : results ] we present and discuss the numerical results , and we conclude in section [ sec : conclusion ] . the present discrepancy between the experimental data and the sm calculation for the muon amm is @xcite @xmath12 which is 3.3 @xmath13 . recent progress in calculating the hadronic contribution increases this discrepancy up to 4.0 @xmath13 @xcite . a further essential increase is possible after more precise measurements of @xmath5 in the near - future experiments at fermilab @xcite and j - parc @xcite . we consider the possibilities of explaining this anomaly using lm contributions . for each lepton flavor one can expect several neutral lms @xmath14 ( @xmath15 ) , which interact with the mesons of scalar ( vector ) type @xmath16 ( @xmath17 ) and decay into the lepton - meson pairs : @xmath18 , @xmath19 , @xmath20 ( @xmath21 , @xmath22 , @xmath23 ) . concerning the charged lms , we presume the singly charged @xmath24 and the doubly charged @xmath25 states , which can decay into \{@xmath26 , @xmath27 } ( \{@xmath28 , @xmath29 } ) and @xmath30 ( @xmath31 ) , respectively . due to the large number of mesons and possible variety of lms we restrict our consideration by typical scalar and vector meson - lm - lepton interactions , which may give significant corrections to the lepton amm @xmath32 . the lowest - dimension lagrangian for the effective interactions of neutral , singly charged and doubly charged lms with the charged leptons and the mesons can be written as @xmath33 where @xmath34 is the charged lepton , and @xmath35 and @xmath36 are the new dimensionless couplings with suppressed flavor indices . for simplicity we require real couplings ( to avoid constraints from the electric dipole moment of the electron ) and flavor conserving interactions in eq . . for the leptomesons : neutral @xmath37 ( left ) and singly charged @xmath38 ( right ) . , title="fig:",scaledwidth=30.0% ] for the leptomesons : neutral @xmath37 ( left ) and singly charged @xmath38 ( right ) . , title="fig:",scaledwidth=30.0% ] for the leptomesons : neutral @xmath39 ( left ) and singly charged @xmath40 ( right ) . , title="fig:",scaledwidth=30.0% ] for the leptomesons : neutral @xmath39 ( left ) and singly charged @xmath40 ( right ) . , title="fig:",scaledwidth=30.0% ] the leading contributions to the lepton amm from the one - loop processes with neutral and singly charged scalar and vector lms are represented in figs . [ fig : diagrams1 ] and [ fig : diagrams2 ] . using the generic analytic formulas for the one - loop contributions to the lepton amm @xcite in the limit of large lm masses with respect to the lepton mass @xmath41 and the meson masses , these contributions can be written as @xmath42 where @xmath43 ( @xmath44 ) is the scalar ( vector ) meson mass , @xmath45 ( @xmath46 ) is the mass of the charged ( neutral ) lm , and the loop functions are @xmath47 \nonumber\\ & \approx & \frac{1}{3 } + \frac{11}{6}x + x\,\text{ln}\,x , \label{eq : tilda_s } \\ f_\text{ffs}(x ) & = & \frac{1}{6(1-x)^4 } [ 1 - 6x + 3x^2 + 2x^3 - 6x^2\,\text{ln}\,x ] \nonumber\\ & \approx & \frac{1}{6 } - \frac{x}{3 } , \label{eq : tilda_s0 } \\ f_\text{vvf}(x ) & = & \frac{1}{6(1-x)^4 } [ 4 - 49x + 78x^2 - 43x^3 + 10x^4 - 18x\,\text{ln}\,x ] \nonumber\\ & \approx & \frac{2}{3 } - \frac{11}{2}x - 3x\,\text{ln}\,x , \\ f_\text{ffv}(x ) & = & \frac{1}{6(1-x)^4 } [ 5 - 14x + 39x^2 - 38x^3 + 8x^4 + 18x^2\,\text{ln}\,x ] \nonumber\\ & \approx & \frac{5}{6 } + x , \label{eq : ffv}\ ] ] where in the approximate expressions we neglected the terms of @xmath48 . clearly , these functions are positive for small @xmath49 , and the contributions in eqs . and are negative , while in eqs . and they are positive . the relations to the loop functions given in eq . ( a.1 ) of ref . @xcite are as follows @xmath50 the contributions of the scalar lms in eqs . and are suppressed by the second power of the meson - to - lm mass ratio . the case of vector lms is special since their contributions in eqs . and do not have this suppression . hence smaller values of the couplings are required for the vector lm interactions to not to exceed the discrepancy in eq . . notice that in the case of one - loop contributions to @xmath32 of a scalar @xmath16 ( @xmath51 ) and a pseudoscalar @xmath52 ( @xmath53 ) the lepton mass @xmath41 receives large loop corrections unless the chirally symmetric limit @xmath54 is satisfied @xcite . for the doubly charged leptomesons : @xmath55 ( upper ) and @xmath56 ( lower ) . , title="fig:",scaledwidth=30.0% ] for the doubly charged leptomesons : @xmath55 ( upper ) and @xmath56 ( lower ) . , title="fig:",scaledwidth=30.0% ] + for the doubly charged leptomesons : @xmath55 ( upper ) and @xmath56 ( lower ) . , title="fig:",scaledwidth=30.0% ] for the doubly charged leptomesons : @xmath55 ( upper ) and @xmath56 ( lower ) . , title="fig:",scaledwidth=30.0% ] the leading contributions to the lepton amm from the one - loop processes with doubly charged scalar and vector lms are shown in fig . [ fig : diagrams3 ] , and can be written as @xmath57 where @xmath58 is the mass of the doubly charged lm , and the loop functions are @xmath59 \nonumber\\ & = & \frac{5}{2}x + x\,\text{ln}\,x + \mathcal{o}(x^2),\nonumber\\ f_\text{vf}(x ) & = & -f_\text{vvf}(x ) + 2f_\text{ffv}(x ) \label{eq : fvf}\\ & = & \frac{1 + 2x}{2(1-x)^4 } [ 2 + 3x - 6x^2 + x^3 + 6x\,\text{ln}\,x ] \nonumber\\ & = & 1 + \frac{15}{2}x + 3x\,\text{ln}\,x + \mathcal{o}(x^2),\nonumber\end{aligned}\ ] ] where the factors of 2 in eqs . and come from the fact that the electromagnetic interaction of the doubly charged lm is twice as strong as that of a singly charged particle . both scalar and vector doubly charged lm contributions are negative . we demonstrate in section [ sec : results ] that a single _ mumeson _ ( muonic leptomeson)which is either a charged scalar @xmath60 or neutral vector @xmath61 , and dominantly interacts with a specific meson can provide the observed value of @xmath5 for the mass of a few hundred gev with the couplings of @xmath62 and @xmath63 , respectively . the case with either several mumesons or one mumeson that has a significant interaction with several mesons , which essentially contribute to @xmath5 , is more involved and potentially has richer phenomenology . we discuss this case in several examples . the new particles in various theories can be generically constrained using the parameters @xmath16 and @xmath64 @xcite . however , large contributions to the @xmath64 parameter are excluded in the case of approximate mass degeneracy of the components of the new weak multiplets , while a significant change of the @xmath16 parameter is avoided in case of vector - like couplings of new fermions to the gauge bosons ( this simultaneously ensures the cancellation of axial - vector gauge anomalies ) @xcite . the existence of the three generations of the sm leptons supposes analogous generations of the leptohadrons . this allows one to accommodate the assumption of minimal flavor violation @xcite , which may help to avoid the constraints from the nonobservation of the flavor - violating processes ( such as @xmath65 ) and to protect the muon mass from large corrections induced by vector fermions @xcite . the nonobservation of new fermions in @xmath66 collisions at the lep collider at center - of - mass energies @xmath67 gev yields a generic lower bound on the lm mass of @xmath68 gev . moreover @xmath2-pole precision measurements at lep strongly constrain the vector lm mixings . allowed regions for the charged mumeson(s ) interacting with the scalar meson(s ) . + _ left _ : mumeson mass @xmath45 vs coupling @xmath69 for a single meson - mumeson - lepton interaction . the light grey region is allowed by the current @xmath70 data within the 1@xmath13 range . the dark grey regions represent the generic lep mass bound of @xmath71 gev and the perturbativity bound of @xmath72 . _ right _ : mass @xmath45 vs ratio @xmath73 in case of two mumeson contributions to @xmath5 . the light grey region is allowed by the current @xmath5 data ( within 1@xmath13 ) and the perturbativity bound of @xmath74 . the dark grey region is disfavored by the lep data . , title="fig:",scaledwidth=45.0% ] allowed regions for the charged mumeson(s ) interacting with the scalar meson(s ) . + _ left _ : mumeson mass @xmath45 vs coupling @xmath69 for a single meson - mumeson - lepton interaction . the light grey region is allowed by the current @xmath70 data within the 1@xmath13 range . the dark grey regions represent the generic lep mass bound of @xmath71 gev and the perturbativity bound of @xmath72 . _ right _ : mass @xmath45 vs ratio @xmath73 in case of two mumeson contributions to @xmath5 . the light grey region is allowed by the current @xmath5 data ( within 1@xmath13 ) and the perturbativity bound of @xmath74 . the dark grey region is disfavored by the lep data . , title="fig:",scaledwidth=45.0% ] one - loop leptomeson @xmath38 ( left ) and @xmath75 ( right ) contributions to the effective four - lepton interactions @xmath76 . , title="fig:",scaledwidth=30.0% ] one - loop leptomeson @xmath38 ( left ) and @xmath75 ( right ) contributions to the effective four - lepton interactions @xmath76 . , title="fig:",scaledwidth=30.0% ] for scalar mumesons the value of @xmath5 is insensitive to the meson masses for @xmath77 . the proper value of @xmath5 can be provided for a single charged mumeson @xmath60 with a mass @xmath45 below 280 gev as shown in fig . [ fig : g_m ] ( left ) . in this figure we assume a significant effect on @xmath5 of only one coupling @xmath78 , where @xmath79 is either @xmath80 or @xmath81 . it is clear that small values of the coupling @xmath69 below 1.2 are ruled out . constraints from four - lepton contact interactions are absent if the meson @xmath82 is self - conjugate since the box diagram shown in fig . [ fig : box_v ] ( left ) is canceled by a second diagram with crossed fermion lines in the final state . in the case of two significant scalar mumeson contributions to @xmath5 , the lightest mumeson mass @xmath45 can be as large as 400 gev , which is shown in fig . [ fig : g_m ] ( right ) for the charged mumesons . in this figure the ratio @xmath73 is defined as @xmath83 where @xmath84 ( @xmath85 ; @xmath86 ) and @xmath87 ( @xmath88 ) are the mumeson masses and couplings , respectively . in particular , in the case of two scalar mumesons @xmath89 and @xmath90 that interact with a meson @xmath82 with the same coupling this ratio is reduced to @xmath91 , while in the case of one scalar mumeson , which significantly interacts with two mesons @xmath92 and @xmath93 with different couplings @xmath94 , the ratio is reduced to @xmath95 . the lower bound on @xmath96 mass can be significantly increased through the searches for their pair production at the lhc . however in this case the final state is composed of a dimuon and neutral meson - antimeson pair ( the decay of which may give photons , leptons , @xmath97 , etc . ) instead of the final state of dilepton plus either gluonic or quark jets , which was considered in the searches for leptoquarks @xcite and leptogluons @xcite . another alternatives include searches for single @xmath96 productions , and @xmath98 production in meson - meson fusion via @xmath8-channel exchange of @xmath96 . allowed regions in the vector meson - mumeson mass plane for the neutral mumeson(s ) with flavor - universal lm interactions . the dark grey area in the bottom is covered by the lep searches for new particles . the top dark grey areas with solid and dot - dashed boundaries are disfavored by lep constraints on @xmath76 contact interactions to explain @xmath5 ( within the 1@xmath13 range ) for the left - chiral and right - chiral case , respectively . _ left _ : case of a single mumeson @xmath61 with the coupling @xmath99 , where the light grey regions with dotted , solid , and dashed boundaries are allowed by the current @xmath5 data ( within 1@xmath13 ) for @xmath100 , 0.03 and 0.05 , respectively . _ right _ : case of two mumeson - meson - lepton couplings @xmath101 , where the light grey regions with dotted and solid boundaries are allowed by @xmath5 data ( within 1@xmath13 ) and correspond to @xmath102 and @xmath103 , respectively . , title="fig:",scaledwidth=45.0% ] allowed regions in the vector meson - mumeson mass plane for the neutral mumeson(s ) with flavor - universal lm interactions . the dark grey area in the bottom is covered by the lep searches for new particles . the top dark grey areas with solid and dot - dashed boundaries are disfavored by lep constraints on @xmath76 contact interactions to explain @xmath5 ( within the 1@xmath13 range ) for the left - chiral and right - chiral case , respectively . _ left _ : case of a single mumeson @xmath61 with the coupling @xmath99 , where the light grey regions with dotted , solid , and dashed boundaries are allowed by the current @xmath5 data ( within 1@xmath13 ) for @xmath100 , 0.03 and 0.05 , respectively . _ right _ : case of two mumeson - meson - lepton couplings @xmath101 , where the light grey regions with dotted and solid boundaries are allowed by @xmath5 data ( within 1@xmath13 ) and correspond to @xmath102 and @xmath103 , respectively . , title="fig:",scaledwidth=45.0% ] for vector mumesons @xmath104 the value of @xmath5 is sensitive to the meson masses and almost insensitive to the mumeson masses . for one neutral mumeson interaction the allowed range of the mumeson mass values is @xmath105 gev , which corresponds to the allowed area between the lower and upper dark grey bands in fig . [ fig : g_m_vec ] ( left ) . the three light grey areas in this figure correspond to the ranges allowed by the current @xmath5 data ( within @xmath106 ) for the chosen values of the coupling of @xmath100 , 0.03 , and 0.05 , where @xmath107 ( @xmath79 is either @xmath81 or @xmath80 ) . clearly only small values of @xmath108 are allowed . the neutral leptomesons @xmath75 generate four - lepton contact interactions @xmath76 through the box diagram shown in fig . [ fig : box_v ] ( right ) . these interactions can be calculated and compared to the effective expression containing the contact interaction scale @xmath109 as @xcite @xmath110 where we restored the flavor index of the couplings @xmath111 and assumed a common mass scale @xmath46 of the lms @xmath39 , the parameter @xmath112 for @xmath113 ( @xmath114 ) , and the loop function can be written as @xmath115>0.\end{aligned}\ ] ] for the flavor - universal couplings @xmath116 the limit in eq . can be rewritten using eq . as @xmath117 where @xmath118 is the minimal allowed value of @xmath119 , e.g. , within the @xmath106 range : @xmath120 from eq . . in the considered case of constructive interference between the sm process and the contact interactions , the constraints from the lep measurements of @xmath121 processes correspond to the lower limits of @xmath122 tev for @xmath123 and @xmath124 tev for @xmath125 ( for @xmath126 , which gives stronger limits ) @xcite , and exclude parts of the parameter space , which are painted dark grey in the top of fig . [ fig : g_m_vec ] ( left ) . in the case of two neutral vector - mumeson contributions to @xmath5 , which have the common lm and meson mass scales @xmath46 and @xmath44 , the left - hand side of eq . gets an additional factor of @xmath127 where @xmath128 is the ratio of the two couplings . the minimal value of 1/2 of this factor is achieved for the equal couplings ( up to the sign ) and corresponds to the weakest constraint from the contact interactions . figure [ fig : g_m_vec ] ( right ) illustrates the case of either two vector mumesons with approximately equal masses @xmath129 interacting with the same charged meson or one mumeson with the mass @xmath130 , which significantly interacts with the two mesons with close masses @xmath131 . in this figure we assumed equal meson - mumeson - lepton couplings @xmath132 , where @xmath133 with fixed @xmath79 . the allowed range for the values of the mumeson mass of @xmath134 gev corresponds to the space between the lower and upper dark grey bands . for flavor - nonuniversal couplings @xmath135 the limit in eq . can be rewritten as @xmath136 which is also valid for the two neutral vector - mumeson contributions to @xmath5 with the common lm and meson mass scales . figure [ fig : g_m_vec_nonfu ] shows that in this flavor - nonuniversal case the limits from the contact interaction @xmath137 can be suppressed by a small value of the coupling @xmath138 . concerning the lhc searches for @xmath139 , in the case of a sm gauge singlet , drell - yan production of singlet pairs may be not possible . then cascade decays from heavier charged particles can be considered @xcite . notice that in the case of long - lived neutral lms , which can escape the detector , the generic lep lower bound for the new particle masses of 100 gev may be weakened for their masses . mev ( which we do not consider in this paper ) were discussed in refs . @xcite . ] same as fig . [ fig : g_m_vec ] but for the flavor - nonuniversal lm interactions . the dark grey regions with solid and dashed ( dot - dashed and dotted ) boundaries are disfavored by lep constraints on the @xmath137 contact interaction to explain @xmath5 within the 1@xmath13 range for the left - chiral and right - chiral case , respectively , taking @xmath140 ( @xmath141 ) . , title="fig:",scaledwidth=45.0% ] same as fig . [ fig : g_m_vec ] but for the flavor - nonuniversal lm interactions . the dark grey regions with solid and dashed ( dot - dashed and dotted ) boundaries are disfavored by lep constraints on the @xmath137 contact interaction to explain @xmath5 within the 1@xmath13 range for the left - chiral and right - chiral case , respectively , taking @xmath140 ( @xmath141 ) . , title="fig:",scaledwidth=45.0% ] we have found the regions for lm model parameters which are allowed by the muon @xmath10 data and the lep data . we considered various minimal models and have shown that the scenarios with one ( two ) scalar meson - lm - lepton interactions limit the value of the lightest charged lm mass from above by 280 ( 400 ) gev . however the case of vector meson - lm - lepton interactions is , in general , less predictive . this is very useful to investigate further collider restrictions on lm model parameters . in particular , the charged scalar lm interactions can potentially be either strongly bounded or even ruled out through the searches for their drell - yan production at the lhc . future collider experiments such as the ilc and the fcc have great prospects to probe a significant part of the leptomeson ( leptodiquark ) model parameter space , which is still allowed . the author thanks the listeners of the seminar of the division of field theory and elementary particles of the university of silesia for useful comments . this work was supported in part by the polish national science centre , grant number dec-2012/07/b / st2/03867 , and german research foundation dfg under contract no . collaborative research center crc-1044 . the author used jaxodraw @xcite to draw the feynman diagrams . j. c. pati and a. salam , `` lepton number as the fourth color , '' phys . rev . d * 10 * ( 1974 ) 275 [ phys . d * 11 * ( 1975 ) 703 ] . h. terazawa , k. akama and y. chikashige , `` unified model of the nambu - jona - lasinio type for all elementary particle forces , '' phys . d * 15 * ( 1977 ) 480 . y. neeman , `` irreducible gauge theory of a consolidated weinberg - salam model , '' phys . b * 81 * ( 1979 ) 190 . m. a. shupe , `` a composite model of leptons and quarks , '' phys . b * 86 * ( 1979 ) 87 . h. harari , `` a schematic model of quarks and leptons , '' phys . b * 86 * ( 1979 ) 83 . e. j. squires , `` qdd : a model of quarks and leptons , '' phys . lett . b * 94 * ( 1980 ) 54 . h. harari and n. seiberg , `` a dynamical theory for the rishon model , '' phys . b * 98 * ( 1981 ) 269 . h. fritzsch and g. mandelbaum , `` weak interactions as manifestations of the substructure of leptons and quarks , '' phys . lett . b * 102 * ( 1981 ) 319 . r. barbieri , r. n. mohapatra and a. masiero , `` compositeness and a left - right symmetric electroweak model without broken gauge interactions , '' phys . b * 105 * ( 1981 ) 369 [ erratum - ibid . b * 107 * ( 1981 ) 455 ] . s. f. king and s. r. sharpe , `` is the @xmath2 degenerate with an exotic quarkonium ? 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many models on the market allow for particles carrying both lepton number and color , e.g. , leptoquarks and leptogluons . some of the models with this feature can also accommodate color - singlet . we have found that the long - standing discrepancy between the experimental result and the standard model prediction for the muon anomalous magnetic moment can be explained by the effect of leptomesons within the wide allowed range of their masses and couplings . these new particles are testable at the current run of the lhc .

immediately after the discovery of quantum mechanics , it was realized that it contains an interesting feature in quantum correlations between two particles . it was first discussed in seminal paper of einstein , podolsky and rosen ( epr ) for the coordinate and momentum of a pair of massive particles @xcite and after a time reformulated for spin - entangled systems @xcite . assuming a pair of maximally entangled spin-1/2 particles we can perfectly predict the results of the complementary measurements on one particle from an appropriate measurements of the other one . an ability of the precise prediction of complementary variables arises from quantum nature of the correlations between the particles . from a fundamental point of view it was proved that for such the particles the measurements of correlated quantities should yield a different result in the quantum mechanical case to those expected in local realism . a condition derived in a form of bell inequality has to be satisfied within the local realism @xcite . the predictions of quantum mechanics were satisfactorily experimentally proved using pairs of photons entangled in the polarization @xcite . from a practical point of view , such the entangled particles distributed at a distance can be used to securely distribute classical information . an experiment demonstrating the interesting attribute of the correlations between quantum systems can be build up assuming a generally mixed state @xmath7 of qubit @xmath4 and meter qubit @xmath6 . this experiment is schematically depicted in fig . 1 . performing two projective ( ideal ) measurements @xmath5 on the qubit @xmath6 , the prediction of the results of the complementary measurements @xmath3 on the qubit @xmath4 can be improved . the complementarity of the measurements means that @xmath10 . for example , having maximally entangled state @xmath11 , the results of arbitrary measurement @xmath12 on the qubit @xmath4 can be precisely predicted performing the same measurement @xmath12 on the qubit @xmath6 . however , without this measurement we have vanishing knowledge since the state of @xmath4 is maximally random . the total ability to predict the result of measurement was quantified by a concept of _ knowledge _ defined in @xcite . generally , there are such states for which the both complementary knowledges @xmath13 and @xmath14 obtained from the measurements @xmath5 are larger than these @xmath15 and @xmath16 without the measurements . then corresponding knowledge excesses @xmath17 and @xmath18 can be introduced , respectively for the complementary measurements . a duality between the knowledge excesses for any mixed state @xmath7 can be derived , analogically as in ref . @xcite . for @xmath19 , the knowledge excesses satisfy the inequality @xmath20 . thus performing a single measurement on @xmath6 , the sum of the squares of knowledge excesses can not be larger than unity . in this paper , we analyze a duality between knowledge excesses beyond the condition @xmath19 and derive the following restriction of the complementary knowledge excesses @xmath21 and @xmath22 , namely @xmath23^{2},\end{aligned}\ ] ] where @xmath9 is maximal violation of bell inequalities @xcite after optimal local filtering operations on a single copy of the state @xmath7 @xcite . thus @xmath9 restricts the ability to enhance both the knowledge excesses by the different measurements on @xmath6 . to overcome the unit value of sum of squares of the knowledge excesses we need a state violating the bell inequality . for any state having vanishing the a priori knowledge in any basis ( for example , bell - diagonal state ) we achieve the equality only by choosing appropriate measurements @xmath3 and @xmath5 . such the state , exhibiting the maximal accessible @xmath9 under local filtering on a single copy and unitary operations , can be probabilistically but uniquely obtained from an arbitrary two - qubit state using local single - copy filtrations @xcite . thus the inequality ( [ main1 ] ) with @xmath9 can be always saturated assuming an appropriate local filtration before the suitable measurements @xmath3 and @xmath5 . we analyze simple and experimentally feasible examples covering such the cases in which noise prevents an maximal extraction of knowledge . in this section we define the complementary knowledge excesses @xmath1 and @xmath22 and discuss a duality between them which arises from a single measurement on @xmath6 . assume two - component projective measurement @xmath24 giving results either @xmath25 or @xmath26 , respectively . we expand the state @xmath7 as @xmath27 , @xmath28 and the meter operators @xmath29 depend on a choice of the measurement @xmath30 . to predict a result of @xmath30 on @xmath4 we can unambiguously discriminate the mixed states @xmath31 by a projective two - component measurement @xmath32 ( @xmath33 , @xmath34 ) on the qubit @xmath6 . after projection @xmath35 , the local state of the qubit @xmath4 collapses to @xmath36,\end{aligned}\ ] ] where @xmath37 is the probability of the projection . thus after the meter measurement we obtain two sub - ensembles of the states @xmath38 and @xmath39 weighted with probabilities @xmath40 and @xmath41 . we denote @xmath42 and @xmath43 . adopting maximum likelihood estimation strategy , we guess for each event that the measurement @xmath30 gave the most likely results either @xmath25 or @xmath26 . our strategy is maximize the likelihood function @xmath44 . the knowledge in a binary decision problem is the fractional excess of right guesses over wrong guesses in many experiments repeated under identical conditions @xcite . if we have @xmath45 of right guesses and @xmath46 of wrong guesses than our knowledge is @xmath47 . in our task a priori knowledge without the measurement @xmath48 is @xmath49 . a sub - ensemble knowledge after the particular projection @xmath35 is @xmath50 . then after the meter measurement , an amount of the knowledge @xmath51 is the @xmath52-weighted sum of sub - ensembles knowledges @xcite @xmath53 where @xmath54 . the knowledge excess @xmath1 is that amount of knowledge which exceeds the apriori knowledge @xmath55 , explicitly @xmath56 where @xmath57 . the knowledge excess quantifies only a part of knowledge which is gained from the measurement on @xmath6 . if we are not able to extract any extra knowledge from the measurement on @xmath6 then the knowledge excess is vanishing . using the expansion ( [ state ] ) we have @xmath58 the largest @xmath1 over all @xmath48 is the distinguishability excess @xmath59 @xmath60 where @xmath61 , and thus @xmath62 . the analogical quantities @xmath63 and @xmath64 can be defined for the complementary measurement @xmath65 . now we shortly prove a relation between the knowledge excesses for @xmath19 . the derivation is inspired by a similar one in ref . the sub - ensemble knowledge about prediction of the complementary projection along the state @xmath66 , after a particular projection @xmath35 , is @xmath67\leq 2\sqrt{w_{i}w_{i}^{\bot}}|c_{i}|$ ] . since @xmath68 we have @xmath69 and using schwarz inequality @xmath70 , we obtain @xmath71 then using ( [ know1 ] ) and ( [ help ] ) we can straightforwardly derive that for @xmath72 the knowledge excesses satisfy @xmath73 using the same measurement on @xmath6 , the sum of squares of the knowledge excesses can never overcome unity . it is a duality between the knowledge excesses from a single meter measurement . accomplishing generally different @xmath74 , @xmath48 , the sum of squares of the knowledge excesses can be larger than unity if the state violates the bell inequalities . let us discuss in this case a limitation of the knowledge excesses for any mixed state of the two - qubit system @xmath75 where @xmath26 stands for the identity operator , @xmath76 are vectors in @xmath77 , @xmath78 are the standard pauli operators and @xmath79 , @xmath80 are the eigenstates of @xmath81 . the coefficients @xmath82 form a real correlation matrix @xmath83 and vectors @xmath84 and @xmath85 determine the local states @xmath86 , @xmath87 . we assume a subset of states @xmath88 with the diagonal correlation tensor @xmath89 , @xmath90 and with vectors @xmath91 and @xmath92 . any mixed state @xmath7 can be uniquely converted to some @xmath88 using appropriate local unitary operations . further we have the following ordering of the diagonal elements @xmath93 or @xmath94 . since the prove for @xmath94 is only analogical we will shortly discuss afterward . according to the strongest correlation @xmath95 , one measurement on @xmath4 is naturally chosen as @xmath96 and from the expansion ( [ state ] ) we obtain using ( [ dist ] ) we have @xmath97 and either @xmath98 or @xmath99 . and according to the second strongest correlation @xmath100 , the complementary measurement is chosen as @xmath101 , @xmath102 and then we analogically have @xmath103 and either @xmath104 or @xmath105 . we can also express a violation of bell inequalities in a simple way for such states with a diagonal @xmath106 . adopting criterion in ref . ( @xcite ) , a state @xmath7 violates a bell inequality if its maximal bell factor is @xmath107 since the factor @xmath108 is invariant under local unitary transformations @xmath109 of the state . similarly , we can derive analogical result for the case when @xmath94 , either we have @xmath110 or @xmath99 and for the complementary measurement , either @xmath111 or @xmath112 and maximum of the bell factor is @xmath113 . then we can generally obtain that @xmath114 for arbitrary state @xmath7 with diagonal @xmath83 , where @xmath115 corresponds to the two largest correlations . the equality is obtained for the states having vanishing priori knowledge in any basis . now we generalize the discussion for any state @xmath7 assuming arbitrary measurements @xmath116 . any mixed two - qubit state can be uniquely prepared from the set of states @xmath88 by appropriate local unitary transformations @xmath117 on the qubits @xmath4 and @xmath6 . further , the transformation of the measurements @xmath118 and @xmath119 to the arbitrary but still complementary ones effectively corresponds the extra local unitary transformation @xmath120 of the state qubit @xmath4 . since the distinguishabilities @xmath59 , @xmath121 are generally invariant under any local unitary transformation on the meter @xmath6 , it is sufficient to implement a joint unitary transformation @xmath122 only on the qubit @xmath4 . further we can preserve the measurements @xmath118 and @xmath119 in a general prove . for any unitary transformations @xmath123 there are unique rotations @xmath124 such that @xmath125 . if a state @xmath7 with diagonal @xmath106 is subjected to the @xmath126 transformation its correlation matrix transforms as follows @xcite @xmath127 thus a joint unitary transformation @xmath128 can be represented as a transformation of the correlation tensor @xmath129 , where @xmath130 is the matrix of rotation in @xmath77 space @xmath131 with the elements @xmath132 where @xmath133 , @xmath134 , @xmath135 . we explicitly calculate @xmath59 and @xmath136 for the state after the previous transformation . first we assume that @xmath93 . for the first measurement from the complementary measurements we obtain either @xmath137 or @xmath138 for the second complementary measurement is either @xmath139 or @xmath140 and using the transformation @xmath129 we can derive that @xmath141 consequently , assuming @xmath142 we can prove the following inequality @xmath143 for @xmath94 , by repeating our calculation we have for the first measurement the result ( [ disc1 ] ) the second complementary measurement is either @xmath144 or @xmath145 using ( [ elem ] ) we derive @xmath146 and subsequently , @xmath147 finally , since @xmath148 and @xmath149 we prove that @xmath150 is generally satisfied . thus the maximum of bell factor represents an important bound on the squares of the excess of knowledge which can be extracted from the meter measurements . for class of the states with vanishing priori knowledge for any measurements @xmath3 , i.e. @xmath151 , this inequality can be saturated only by an appropriate choice of the measurements @xmath116 . for a mixture of bell states ( [ mixbell ] ) we can find that @xmath152 , @xmath153 and @xmath154 which saturates the relation ( [ ineq1 ] ) . it was shown that local filtering operations on single copy of the state can increase the degree of violation of bell inequalities @xcite . there is a unique local ( stochastically reversible ) filtering operation @xmath155 and @xmath156 ( @xmath157 and @xmath158 ) on single copy of the state @xmath159 which transforms with some non - zero probability any two - qubit mixed state into a state which is diagonal in bell basis @xmath160 having the largest @xmath161 @xcite . a two - qubit mixed state can be uniquely bring to this bell diagonal form with the maximal violation either with finite probability or asymptotically . for the bell diagonal states , the bell violation can not be increased by any local filtering on a single copy . naturally , after the local filtering the inequality ( [ ineq1 ] ) is still satisfied also for the remaining state @xmath162 . the bell diagonal states have both the local states maximally disordered , both the apriori knowledges vanish and we can always saturate the inequality ( [ ineq1 ] ) with the upper bound given by @xmath163 only by an appropriate choice of the measurements @xmath116 . thus assuming that @xmath9 in the inequality ( [ ineq1 ] ) is then maximum under local filtering and unitary operations on a single copy of the state , the inequality is satisfied for any mixed state and can be for any mixed state saturated if we use an appropriate local filtering . we analyze two interesting and experimentally feasible examples of the bell - diagonal states . in both cases , we can use as a source state the maximally entangled state @xmath164 produced by the spdc process . evidently , the source state maximally violates bell inequalities and has @xmath165 . in the first example , we simultaneously perform a depolarization by ( i ) random flip of linear polarizations @xmath166 with the probability @xmath167 and simultaneously , by ( ii ) phase - shift @xmath168 between linear polarizations @xmath169 . as a result of the depolarizing procedure we prepare a mixture of bell states @xmath170 for this case , the distinguishability excess is @xmath171 and vanishes only if the random polarization flip has larger probability , whereas the complementary distinguishability excess is @xmath172 and decreases only with an increasing probability of the random polarization phase - shift . the maximal value of bell factor is @xmath173 and the inequality ( [ ineq1 ] ) is saturated . thus we can independently control both the complementary distinguishability excesses . for @xmath174 ( @xmath175 ) and changing @xmath176 ( @xmath177 ) we are able to extract the maximal unit distinguishability excess @xmath178 ( @xmath179 ) irrespective to the complementary one @xmath180 ( @xmath178 ) . for @xmath181 ( @xmath182 ) and controlling @xmath176 ( @xmath177 ) , the depolarization prevents us to extract any knowledge excess since @xmath183 ( @xmath111 ) however we are still able to obtain the complementary knowledge excess @xmath184 ( @xmath184 ) . in the second example , we show as both @xmath178 and @xmath185 can be gradually enhanced sharing separable state , entangled state which satisfies the bell inequalities and consequently , sharing entangled state violating bell inequalities . we assume a loss of both the knowledge excesses from the state @xmath186 by an extraction of the photon in the meter beam with probability @xmath187 and its substitution by another photon with a completely random polarization . thus we detect the results produced by a mixture of the bell states which is known as werner state @xcite @xmath188 in this case , a loss of the distinguishability excesses with decreasing @xmath189 are identical @xmath190 . we know that the werner s state is entangled only for @xmath191 and violates bell inequalities , having maximum of bell factor @xmath192 , only if @xmath193 . also in this case , the inequality ( [ ineq1 ] ) can be saturated . since we have an entangled state non - violating bell inequalities for @xmath194 and non - entangled state for @xmath195 it means that we can observe both non - vanishing distinguishability excesses @xmath196 and @xmath197 even in a case of classical correlated states . * acknowledgments * we would like to thank j. fiur ' aek , l. mita jr . for the fruitful discussions . the work was supported by the projects 202/03/d239 of gacr and ln00a015 and cez : j14/98 of the ministry of education of czech republic . theoretically , for a given state @xmath198 maximal value of the bell factor @xmath199 can be theoretically calculated using the following formula @xmath200 , where @xmath201 is real - valued function of the density matrix @xmath198 @xcite . to define @xmath201 , one needs a @xmath202 matrix @xmath83 with elements @xmath203 . then , @xmath201 is the sum of the two largest eigenvalues of the hermitian matrix @xmath204 . if @xmath205 then bell inequalitie are violated and all experimental outcomes which state @xmath198 generate can not be explained only by local realism and quantum theory must be used @xcite .

a constraint on two complementary knowledge excesses by maximal violation of bell inequalities for a single copy of any mixed state of two qubits @xmath0 is analyzed . the complementary knowledge excesses @xmath1 and @xmath2 quantify an enhancement of ability to predict results of the complementary projective measurements @xmath3 on the qubit @xmath4 from the projective measurements @xmath5 performed on the qubit @xmath6 . for any state @xmath7 and for arbitrary @xmath3 and @xmath5 , the knowledge excesses satisfy the following inequality @xmath8 , where @xmath9 is maximum of violation of bell inequalities under single - copy local operations ( local filtering and unitary transformations ) . particularly , for the bell - diagonal states only an appropriate choice of the measurements @xmath3 and @xmath5 are sufficient to saturate the inequality .

the ising model in a transverse field is widely studied for being the simplest system of interacting spins with quantum dynamics . the most striking feature is that the competition between thermal and quantum fluctuations reduce the critical temperature up to a point when a quantum phase transition occurs at @xmath4 , at a quantum critical point(qcp)@xcite . we will not discuss here the extensive literature on results for several versions of the model , but we will concentrate instead in the quantum ising spin glass in a transverse field . this is represented by a hamiltonian in which only one component of the spins , say the z - component , interact among themselves with a random interaction while a uniform , constant field @xmath5 is applied in the transverse x - direction . the experimental realizations of this model are the @xmath6 compounds@xcite . in the calculation of the quantum mechanical partition function special tools are needed to deal with the non - commuting operators forming the hamiltonian . the method more currently used in the study of short - range @xcite and infinite range @xcite spin glasses in a transverse field is the trotter - suzuki formula @xcite , that maps a system of quantum spins in d - dimensions to a classical system of spins in ( d + 1)-dimensions , and it is suited to perform numerical studies . another way of dealing with the non - commutativity of quantum mechanical spin operators is to use feynman s path integral formulations @xcite and to introduce time - ordering by means of an imaginary time @xmath7 , where @xmath8 is the inverse temperature . the work by bray and moore@xcite established the basis for recent developments in the theory of the quantum heisenberg spin glass@xcite . a still different functional integral formulation consists in using grassmann variables to write a field theory with an effective action where the spin operators in the hamiltonian are expressed as bilinear combinations of fermions@xcite . the advantage of the fermionic formulation is that it has a natural application to problems in condensed matter theory , where the fermion operators represent electrons that also participate in other physical processes , like superconductivity@xcite and the kondo effect@xcite . in the present paper we use two fermionic models within a grassmannian field theory@xcite to analyze the long range ising spin glass in a transverse field.the novelty resides in the method , as all previous results rely on the trotter - suzuki approximation@xcite . a criticism to the fermionic formulation may be that the spin eigenstates at each site do not belong to one irreducible representation @xmath9 , but they are labeled instead by the fermionic occupation numbers @xmath10 or 1 , giving two more states with @xmath11 . we call this the `` four states '' ( 4s ) model , and despite the presence of these two unwanted states the 4s - ising spin glass model describes a spin glass transition with the same characteristics as the sherrington - kirkpatrick ( sk ) model @xcite in a replica symmetric theory . a way to get rid of the unwanted states was introduced before by wiethege and sherrington@xcite for non - random interactions and it consists in fixing the occupation number @xmath12 by means of an integral constraint at every site . we refer to this as the `` two states '' ( 2s)-ising model . in sect . 2 we analyse the 4s - ising and 2s - ising spin glass models in a transverse field , within the static approximation in a replica symmetric theory . the static ansatz neglects time fluctuations and may be considered an approximation similar to mean field theory . numerical monte carlo solutions of bray and moore s equations indicate that the static approximation reproduces the correct results at finite temperatures@xcite . when @xmath13 the static approximation reproduces the exact results obtained by other methods , in particular for the 2s - ising spin glass model we recover sk equations @xcite . the results in both models are very similar ; they both exhibit a critical spin glass temperature @xmath2 that decreases when the strength @xmath0 of the transverse field increases , until it reaches a quantum critical point(qcp ) at @xmath14 , @xmath15 . the value of @xmath3 is the same for both models and the 4s - ising and 2s - ising models are identical close to the qcp . we obtained for both models that the replica symmetric solution is unstable @xcite in the whole spin glass phase , in agreement with previous results with the trotter - suzuki method@xcite . we left sect . 3 for discussions . the ising spin glass in a transverse field is represented by the hamiltonian @xmath16 where the sum is over the n sites of a lattice and @xmath17 is a random coupling among all pairs of spins , with gaussian probability distribution : @xmath18 the spin operators are represented by auxiliary fermions fields : @xmath19\nonumber\\ s^{x}_{i}=\frac{1}{2}[a_{i \uparrow}^{\dagger}a_{i \downarrow } + a_{i \downarrow}^{\dagger}a_{i \uparrow } ] \label{2.3}\end{aligned}\ ] ] where the @xmath20 are creation ( destruction ) operators with fermion anticommutatiom rules and @xmath21 or@xmath22 indicates the spin projections . the number operators @xmath23 or 1 , then @xmath1 in eq.([2.3 ] ) has two eigenvalues @xmath24 corresponding to @xmath25 , and two vanishing eigenvalues when @xmath26 . we shall use the lagrangian path integral formulation in terms of anticommuting grassmann fields described in previous publications @xcite , so we avoid giving repetitious details . we consider two models : the unrestrained , four states model that has been used previously @xcite , and also the two states model of wiethege and sherrington where the number operators satisfy the restraint @xmath27 , what gives @xmath28 , at every site @xcite the partition function in the 4s - model is given by @xmath29 while in the restrained model it takes the form : @xmath30 \label{2.5}\end{aligned}\ ] ] where @xmath31 is the inverse temperature . by using the integral representation for the kronecker @xmath32-function : @xmath33 } \label{2.6}\end{aligned}\ ] ] we can express @xmath34 and @xmath35 in the compact functional integral form @xmath36 where : @xmath37-\nonumber \\ & & h(\varphi_{j \sigma}^{\ast}(\tau),\varphi_{j \sigma}(\tau)\ } \label{2.8}\end{aligned}\ ] ] and @xmath38 for the 4s - model while @xmath39 for the 2s - model . going to fourier representation we introduce the spinors : @xmath40 and the pauli matrices : @xmath41 to write the spin glass part of the action @xmath42 where @xmath43 with matsubara s frequencias @xmath44 and @xmath45 . in the static approximation , we retain just the term @xmath46 in the sum over the frequency @xmath47 . the transverse part of the action is given by : @xmath48 where the inverse propagator is @xmath49 and the total action can be rebuild as @xmath50 where @xmath51 is the static component of eq.([2.11 ] ) . we are now able to follow the standard procedures to get the configurational averaged free energy per site by using the replica formalism : @xmath52 where the configurational averaged , replicated , partition function @xmath53 becomes , after averanging over @xmath17 : @xmath54 with the replica index @xmath55 , and @xmath56 \label{2.18}\end{aligned}\ ] ] we indicate by @xmath57 the static component @xmath58 of eq.([2.12 ] ) . we assume a replica symmetric solution of the saddle point equations : @xmath59 where q is the spin glass order parameter and @xmath60 is related to the static susceptibility by @xmath61 . the sums over @xmath62 in the spin part of the action produce again quadratic terms that can be linearized by introducting new auxiliary fields , with the result @xmath63 where @xmath64 and @xmath65 with @xmath66 @xmath67 the gaussian integral over grassmann variables is straigthforward@xcite , giving the result : @xmath68 \label{2.24}\ ] ] @xmath69^{2 } + ( \beta \gamma)^{2 } \label{2.25}\ ] ] the sum over frequencias can be also easily performed@xcite and we obtain @xmath70 from eq.([2.17 ] ) , eq.([2.20 ] ) and eq.([2.26 ] ) we obtain at the saddle point : @xmath71}\prod_{j } \ { \displaystyle \int_{- \infty}^{\infty}dz \displaystyle \prod_{\alpha } \displaystyle \int_{-\infty}^{\infty}d \xi_{\alpha } \frac{1}{2 \pi } \nonumber \\ & & \displaystyle \int_{0}^{2 \pi}dx_{j \alpha}[e^{- \mu_{j \alpha}}+ e^{\mu_{j \alpha}}+2 \cosh{\sqrt{\delta_{\alpha } } } ] \ } \label{2.27}\end{aligned}\ ] ] for the four states ( 4s ) model there is no restraint and @xmath72 , then the integrals over @xmath73 equal unity in eq.([2.27 ] ) . for the restrained two states ( 2s ) model we have @xmath74 from eq.([2.16 ] ) , then the integrals over the exponential terms identically vanish in eq.([2.27 ] ) . we then obtain for the model with @xmath75 states , @xmath76 or @xmath77 : @xmath78- \int_{-\infty}^{\infty}dz \log [ 2 k_{p}(q,\bar{\chi},z ) ] \label{2.28}\ ] ] where @xmath79 the saddle point equations for the order parameters are : @xmath80 @xmath81 where @xmath82 is given in eq.([2.23 ] ) . we obtain for the de almeida - thouless eigenvalue @xcite and entropy in both models : @xmath83 -[\displaystyle \int_{-\infty}^{\infty}d \xi\frac{\lambda } { \sqrt{\delta}}\sinh \sqrt{\delta}]^{2 } \ } \label{2.32}\end{aligned}\ ] ] @xmath84 the landau expansion of the free energy in powers of q gives : @xmath85 where the coefficients are : @xmath86[d_{p}-2]\nonumber\\ c_{p}=- \frac{4}{3}[d_{p}-1]\{2(d_{p}-1)^{2}+ 3 ( d_{p}-2)\ } \label{2.35}\end{aligned}\ ] ] and @xmath87 @xmath88 in eq.([2.37 ] ) we need also : @xmath89 as @xmath90 , the spin glass phase is characterized by @xmath91 , giving a maximum instead of a minimum @xcite of the free energy . the critical temperature is obtained by solving simultaneously : @xmath92 the numerical results for the critical temperature @xmath2 and the entropy @xmath93 are shown in fig . 1 for the 4s - model and fig . 2 for the 2s - model . for large values of the transverse field @xmath0 the 2s - model and 4s - model are undistinguishable . the analytic soluction of eq.([2.39 ] ) when @xmath94 gives the critical value @xmath95 for both models . when @xmath96 , the equations ( 30)-(33 ) for the 2s - model ( p=0 ) reproduce the sherrington - kirkpatrick @xcite results , while for the 4s - model ( p=1 ) , we recover our previous results @xcite . finally , we comment on the de almeida - thouless instability . the exact soluction for @xmath97 in eq.([2.32 ] ) in both limits , @xmath13 and @xmath98 , shows that @xmath99 at the transition point both for the 2s - model and the 4s - model , while numerical results confirm that @xmath100 for both models on the critical line @xmath2 . this is a correct result and a.t . acknowledges a flaw in a previous publication @xcite . we performed a new study of two quantum ising spin glass models in a transverse field by means of a path integral formalism where the spin operators are represented by bilinear combinations of fermionic fields . all previous results in this problem were obtained with the trotter - suzuki approximation@xcite . in the unrestricted four - states ( 4s)-model the fermionic representation gives for the diagonal @xmath1-operator two eigenvalues @xmath101 and two vanishing eigenvalues , while in the state ( 2s)-model the vanishing eigenvalues are suppressed by means of an integral constraint . the results in both models were obtained with the static approximation and the phase diagram coincides with previous results with the trotter - suzuki method @xcite . regarding the de almeida - thouless instability@xcite , we obtained that the replica symmetric solution is unstable in the whole spin glass phase . in future work we will apply the fermionic representation of the transverse ising spin glass to problems in condensed mather theory and also the replica symmetry breaking in the ordered state will be investigated . we acknowledge partial financial support from the conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq),financiadora de estudos e projetos ( finep ) and fundao de amparo pesquisa do estado de rio grande do sul ( fapergs ) . fig1 . critical temperature @xmath2 and entropy on the critical line @xmath102 for the 4s - model same as fig1 . for the 2s - model \a ) h. rieger and a. p. young , phys . lett . * 72 * 4141(1994 ) . + b ) muyu guo , r.n . bhatt and david a. huse , phys . lett . * 72 * 4137(1994 ) . a ) k. d. usadel and b. schmitz , solid st . comm . * 64 * 975(1987 ) . + b ) d. thirumalai , qiang li and t. r. kirkpatrick , j phys . a : math . gen . * 22 * 3339(1989 ) . + c ) g. bttner and k. d. usadel , phys . rev . b * 41 * 428(1990 ) . + d ) yadin y goldschmidt and pik - 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we analyze the long range ising spin glass in a transverse field @xmath0 by using grassmann variables in a field theory where the spin operators are represented by bilinear combinations of fermionic fields . we compare the results of two fermionic models . in the four state ( 4s)-model the diagonal @xmath1 operator has two vanishing eigenvalues , that are suppressed by a restraint in the two states ( 2s)-model . within a replica symmetric theory and in the static approximation we obtain similar results for both models . they both exhibit a critical temperature @xmath2 that decreases when @xmath0 increases , until it reaches a quantum critical point ( qcp ) at the same value of @xmath3 and they are both unstable under replica symmetry breaking in the whole spin glass phase .

the @xmath9 expansion of matrix - valued field theories is probably the most important non - perturbative and non - numerical theoretical tool presently available in the study of such models as non - abelian gauge theories and two - dimensional quantum gravity . a resolution of the above - mentioned models in the large-@xmath0 limit would be the starting point for many analytical developments . in particular when a lattice formulation is involved one must consider different possibilities in the search for the continuum limit ; for the case of asymptotically free theories one must explore the limit of vanishing coupling @xmath10 ( trivial fixed point ) while keeping a physical mass scale fixed , while in the case of quantum gravity one must search for a nontrivial fixed point @xmath11 and reach the limit with a specific power - law dependence on @xmath0 of @xmath12 , which is known as `` double scaling limit '' @xcite . therefore it is useful to achieve a full knowledge of the coupling dependence of such models , from extreme weak coupling to strong coupling , in order to explore those regions that may turn out to be physically most interesting . as a matter of fact , notwithstanding many recent efforts toward an understanding of the possible properties of large-@xmath0 solutions to nontrivial quantum field theories , our present analytical knowledge is limited to a small number of few - matrix systems . this number is even smaller if we restrict our attention to the case of unitary matrix fields , which is especially relevant to the problem of lattice qcd . to the best of our knowledge , the only solved examples are gross - witten s single - link problem @xcite and its generalizations , the external field problem @xcite and @xmath13 chiral chains @xcite . we stress that extending the number of solved few - matrix systems is not at all a pointless exercise . indeed apart from purely theoretical informations that might be achieved , not only does every few - matrix system have a reinterpretation , via the double scaling limit , as some different kind of matter coupled to 2-dimensional quantum gravity , but also every few - matrix system involving unitary matrices can be reinterpreted as the generating functional for a class of integrals over unitary groups , and these integrals in turn are the essential missing ingredient in the context of a complete algorithmization of the strong coupling expansion of many interesting models @xcite . following ref . @xcite , we may introduce the notion of a `` superskeleton '' , that is a graph whose vertices are joined by at most one link ( simple graph ) . as has been shown , knowledge of all the group integrals involved in the strong coupling expansion of a lattice model with nearest - neighbor interactions defined on such a graph provides sufficient information for the algorithmic reconstruction of the strong coupling series for a model enjoying the same global symmetry and defined on an arbitrary lattice . these were basic motivations for us to begin the study of the class of lattice chiral models which we termed `` simplicial chiral models '' @xcite . in particular we focused on principal chiral models , with a global @xmath14 symmetry , defined on a @xmath1 dimensional simplex formed by connecting @xmath5 vertices by @xmath15 links , and explored specifically the large-@xmath0 limit of such models , whose relevance we have just been discussing . our fundamental result is the reduction of the above problem to that of solving a single inhomogeneous integral equation for the eigenvalue distribution of a single hermitian semi - positive definite matrix . although we could not find a closed form solution to this equation for arbitrary @xmath5 , we are able to solve it in several interesting special cases and we set up a systematic numerical approach to the solutions which led us to a conjecture about the location of the critical surface as a function of @xmath5 . we have also studied in detail the related topic of chiral chains , their strong coupling expansion and critical behavior . as a result of these analyses , we are confident that the critical surface is defined by @xmath16 for all @xmath5 . moreover , by treating @xmath5 as a continuous parameter , there are two distinct regions . for small @xmath5 , @xmath17 , the models exhibit the third order gross - witten transition . indeed for @xmath18 they coincide exactly with the chiral chains studied earlier by brower , rossi and tan @xcite . in this region , the criticality is related to that of @xmath19 spin models on random surfaces , as discussed by gaudin - kostov @xcite . for @xmath20 , however , there is a first order transition ending at the `` upper critical '' dimensions @xmath21 , which we scrutinize in some detail . this paper is organized as follows : * in section [ matrix ] we set up our formalism for simplicial chiral model and derive the large-@xmath0 effective action and a representation for the internal energy . we begin in section [ spin ] by illustrating the formalism for the simpler case of vector @xmath22 spin models on a simplicial lattice , deriving the closed form large-@xmath0 solution for arbitrary values of @xmath5 and studying its properties . sec [ largenlimit ] gives the large-@xmath0 effective action for the simplicial chiral model and section [ saddlepoint ] , the saddle - point equation ( large-@xmath0 schwinger - dyson equation ) for the eigenvalue distribution , discussing its features and converting it into a standard inhomogeneous fredholm equation of the second kind . * in section [ sec5 ] we analyze the solvable examples . we begin with integer values of @xmath23 ( which correspond to @xmath24 chiral chains ) , followed in section [ sec6 ] by a detailed discussion of the @xmath2 case , including features of the weak and strong coupling expansions and the asymptotic expansion around the critical point @xmath25 . we present in section [ sec8 ] the exact result for the limit @xmath3 and develop in section [ sec8a ] a treatment based on the @xmath26 expansion . this large-@xmath5 analysis , which works best outside the critical region : @xmath27 , has provided us with the first numerical confirmation for the conjecture that @xmath4 , and is discussed in greater details in appendix d. * in section [ sec7 ] we solve the models at criticality for arbitrary @xmath28 and sketch the peculiar features of the critical behavior when @xmath29 . also in section [ sec9a ] we present numerical methods based on the gaussian integration techniques for @xmath29 on the critical surface @xmath16 . * some technical extensions are included in other appendices . appendix [ appa ] is devoted to a discussion of the double - scaling limit of critical chiral chains @xmath30 and appendix [ appb ] extends the discussion of chiral chains to @xmath31 by an analysis of strong coupling expansion . whereas appendix [ appc ] is devoted to details of the weak and strong coupling expansions and series analysis for the @xmath2 simplicial chiral model . a @xmath1 dimensional simplex is formed by connecting in a fully symmetric way @xmath5 vertices by @xmath32 links . let us assign a @xmath33 matrix to each vertex . the partition function for principal chiral models on a simplicial lattice is obtained by integrating over unitary @xmath34 matrices with a normalized invariant haar measure : @xmath35\right\}\;. \label{zdm}\ ] ] thus the @xmath5-matrix simplicial model has an underlying permutation symmetry instead of the cyclic symmetry of the @xmath5-matrix chiral chains . for @xmath5 = 1 , 2 and 3 these two symmetries and the associated models are equivalent . we shall explore the @xmath14 symmetry of the system and , in particular , study its critical behavior in the large-@xmath0 limit . for this purpose , it is sufficient to study the bulk thermodynamic " properties , _ e.g. _ , the free energy density , internal energy , and specific heat , which are respectively given by @xmath36 we focus in this paper on computing the free energy and determine the critical point @xmath37 for all values of the parameter @xmath5 in the large @xmath0-limit . we find that there is a sequence of critical theories with @xmath16 , which exhibit a third order gross - witten singularities for @xmath38 and a first order transition for @xmath39 . special attention will be give to the marginal dimension at @xmath2 . scaling exponents and finite size ( or double scaling ) properties will be presented in some special cases , but a thorough investigation for all @xmath5 is beyond the scope of this paper . because of the the permutation symmetry of the vertices , the simplicial chiral models can be reformulated in terms of a lagrange multiplier field which decouples the original degrees of freedom . the resulting effective theory is very reminiscent of the mean field approximation to standard lattice models , but in contrast with mean field this reformulation is exact . we therefore replace the direct interaction between unitary matrices with the coupling to an auxiliary field , which in this case is a complex @xmath34 matrix , @xmath40 , introduced in the following representation of the identity , @xmath41\right\ } \over \int d a \exp \left\{-n\beta\,{\rm tr } \,\left [ a a^\dagger \right]\right\}}\;. \label{unv}\ ] ] again by exchanging the order of integrations and representing the partition function in the form @xmath42 , we may obtain @xmath43 + \left[a\sum_i u_i^\dagger\right ] + \left[a^\dagger \sum_i u_i\right ] - d\right]\right\}\;. \label{tzm}\ ] ] by performing the single - link external field integral , we may introduce the auxiliary function , @xmath44\right\ } \ ; , \label{fnv}\ ] ] and re - express @xmath45 , up to an irrelevant multiplicative factor , in the form @xmath46\ ; , \label{tzm2}\ ] ] where @xmath47 replaces @xmath48 . a first crucial point in our analysis is the observation that the integrand in eq . ( [ tzm2 ] ) is a function of the eigenvalues @xmath49 of the hermitian semi - positive definite matrix @xmath50 . moreover morris @xcite has shown that , when integrating over complex matrices , a proper parameterization may offer in specific cases , like ours , the possibility of performing the `` angular '' integrations exactly and reducing the problem to that of integrating over the @xmath0 variables @xmath49 . referring to morris paper for a proof of the angular integration , we apply it to our eq . ( [ tzm2 ] ) thus obtaining , again up to irrelevant numerical factors , @xmath51\;. \label{tzm3}\ ] ] the second crucial observation concerns the function @xmath52 . it is known exactly for all @xmath33 groups @xcite , while integral representations exist for @xmath53 groups @xcite , and it takes on a relatively simple form in the large-@xmath0 limit . before proceeding further , we shall first provide an even simpler illustrative example whose large-@xmath0 solution can be obtained fairly straightforwardly . consider instead of our simplicial chiral models , an example of an @xmath22 symmetric nonlinear model defined on a simplex . the same basic methods used for the chiral models are easily illustrated in this much simpler context . the partition function is obtained by integrating over the @xmath54 independent components of @xmath5 vectors , @xmath55\;. \label{zds}\ ] ] the effective field is a single unconstrained @xmath0-component vector , @xmath56 , which again can be introduced as lagrange multiplier field via an identity , @xmath57 upon substituting this identity into eq . ( [ zds ] ) and inverting the order of the integrations , we may then represent the partition function by @xmath42 , where @xmath58\;. \label{ztds}\ ] ] it is now possible to perform the decoupled constrained integrations . to this end we may define the auxiliary function @xmath59 where @xmath60 . this function is known explicitly for all values of @xmath0 and admits a large @xmath0 limit @xcite : @xmath61\;. \label{fns2}\end{aligned}\ ] ] as a consequence for large @xmath0 we have the following representation of @xmath45 , @xmath62\right\}\ ; . \label{ztds2}\ ] ] the large-@xmath0 value of the integral in eq . ( [ ztds2 ] ) may be obtained by a saddle - point estimate . after some simple manipulations , the saddle - point equation may be reduced to @xmath63 the solution of this equation is @xmath64\;. \label{sps}\ ] ] by taking the logarithmic derivative of the partition function with respect to @xmath65 , we obtain an expression for the internal energy ( per unit link ) @xmath66 of simplicial spin models in the large-@xmath0 limit @xmath67\;. \label{energys}\ ] ] we may check many special cases of this result , and in particular we may notice that the r.h.s . of eq . ( [ energys ] ) is zero when @xmath68 , while when @xmath69 @xmath70\ ; , \label{enes2}\ ] ] consistent with the single - link model result . finally let us notice that in the large-@xmath5 limit , as a trivial consequence of the structure of the model , the solution we found coincides with the mean field solution , which is exact in this limit . it is worth observing that , while eq . ( [ energys ] ) is formally correct for all values of @xmath65 , in order to recover the standard strong and weak coupling expansions of the solution we must separately consider the two different regimes @xmath71 and @xmath72 , where for all @xmath5 we obtain @xmath4 . returning to the simplicial chiral models , we are again interested in the large-@xmath0 limit . for the free energy function @xmath52 resulting from a one - link integral over a @xmath33 matrix , the limiting form can be extracted by solving the schwinger - dyson equations and written in a simple closed form @xcite , @xmath73 we must distinguish two different phases , a weak coupling regime where @xmath74 and @xmath75 and a strong coupling regime where @xmath76 is dynamically determined by the condition , @xmath77 it is important for future developments to observe that eq . ( [ cond ] ) also leads to the condition @xmath78 it is completely legitimate to apply the above results to a saddle - point evaluation of the large-@xmath0 limit of the integral appearing in eq . ( [ tzm3 ] ) . to this end we may define an effective action @xmath79 and derive a saddle - point equation @xmath80 very simple manipulations , including the use of eq . ( [ cond2 ] ) , lead to a reformulation of eq . ( [ sdeq ] ) , which can be turned into the relationship @xmath81 this equation , supplemented with the condition @xmath82 and with the constraint @xmath74 ( weak coupling ) or eq . ( [ cond ] ) ( strong coupling ) is the fundamental saddle - point equation of principal chiral models on a simplicial lattice . it is the starting point of most of the developments presented in the following sections . we recall that , once eq . ( [ sdeq2 ] ) is solved , knowledge of the saddle - point value of the eigenvalues @xmath83 allows the large-@xmath0 evaluation of @xmath45 via the relationship , @xmath84 \ ; , \label{zdsp}\ ] ] and we can also extract the internal energy per unit link by taking a logarithmic derivative of @xmath85 with respect to @xmath65 which leaves us with the relationship , @xmath86 in order to study eq . ( [ sdeq2 ] ) we shall start by applying well - established techniques , and in particular by introducing an eigenvalue density function . it is however convenient first to introduce a new variable @xmath87 whose formal definition is @xmath88 subject to the condition @xmath89 . we may assume that the eigenvalue variable @xmath49 lies in a single interval @xmath90 $ ] , @xmath91 . in terms of the new variable @xmath87 , one has @xmath92 $ ] where @xmath93 , @xmath94 and @xmath95 for weak coupling , @xmath74 , and we expect in general @xmath96 . for strong coupling , one expects @xmath97 so that @xmath98 . we shall be interested in the weak - strong transition as one varies @xmath5 and @xmath65 . in a third - order transition , typical of large-@xmath0 transition previously studied , @xmath99 . in a first - order transition , which we will encounter for @xmath29 , @xmath100 when approached from the strong coupling regime . denoting the large-@xmath0 eigenvalue density by @xmath101 ; it vanishes outside the interval @xmath102 $ ] . we may now turn eq . ( [ sdeq ] ) into the following integral equation , @xmath103 the function @xmath101 , and therefore also the extremes @xmath104 and @xmath105 of the integration region , are thus determined dynamically . in particular the normalization condition , @xmath106 must be satisfied . in addition to the positivity requirement , @xmath107 over the interval @xmath102 $ ] , the desired solution to eq . ( [ s4e3 ] ) must also satisfy either the weak coupling inequality , eq . ( [ wcond ] ) or the strong coupling constraint , eq . ( [ cond ] ) . in the large-@xmath0 limit , ( [ wcond ] ) becomes @xmath108 whereas eq . ( [ cond ] ) becomes @xmath109 the determination of the transition point , @xmath37 , and of the critical behavior around this value is one of the interesting physical problems concerning this model . ( [ s4e3 ] ) has a somewhat unconventional form when compared to other integral equations , because of the special structure of its kernel . we may however perform a few manipulations in order to obtain a more familiar relationship . our starting point is the introduction of an analytic function of @xmath110 , by the definition @xmath111 by construction , the analyticity domain of @xmath112 is the complex @xmath110 plane with the exception of a cut on the positive real axis in the interval @xmath102 $ ] . the discontinuity on the cut may be parameterized by writing @xmath113 when @xmath114 $ ] and it is easy to recognize that @xmath115 it follows that @xmath116 is itself an analytic function of @xmath110 , with a cut on the negative real axis in the interval @xmath117 $ ] . let us now notice that the normalization condition implies @xmath118 as a consequence in the same limit we obtain @xmath119 and in turn @xmath120\goto { 1\over i\pi}\left ( { z\over 4\beta } - { d\over 2 } + { d-4\over 2z}\right ) \,+\ , o\left ( { 1\over z^2}\right)\;. \label{s4e11}\ ] ] this equation can in principle be used in order to determine relationships between the constants @xmath104 and @xmath105 in place of the normalization condition . we must now distinguish between weak and strong coupling regimes . in both cases , by exploiting analyticity properties of the function @xmath112 and defining appropriate auxiliary functions , it is relatively easy to reduce eq . ( [ s4e3 ] ) to the following forms @xmath121 \;\;\;\;\;\;\;\;\;\;{\rm for}\;\;\;\ ; \beta > \beta_c\ ; , \label{s4e12a}\ ] ] @xmath122 \;\;\;\;\;\;\;\;\;\;{\rm for}\;\;\;\ ; \beta < \beta_c\;. \label{s4e12b}\ ] ] the values of @xmath104 and @xmath105 as functions of @xmath65 are determined by enforcing the asymptotic condition ( [ s4e11 ] ) . ( [ s4e12a]-[s4e12b ] ) are inhomogeneous fredholm equations of the second kind . it is therefore in principle possible to apply standard methods of ( approximate ) resolution by expressing the kernels in terms of appropriate orthonormal sets of eigenfunctions . the weak and strong coupling constraints , eq . ( [ wcond ] ) and eq . ( [ cond ] ) , can be expressed in terms of the analytic function @xmath112 as @xmath123 and @xmath124 respectively . alternatively , writing @xmath125 and analytically continue this expression outside of the interval @xmath102 $ ] , eq . ( [ wcond ] ) and eq . ( [ cond ] ) , can also be expressed as @xmath126 and @xmath127 respectively . note that eq . ( [ s4e12b ] ) is parameterized so that the strong coupling constraint ( [ s4e16strong ] ) is automatically satisfied . the transition point @xmath37 can be determined approaching from the weak coupling regime by enforcing the equality @xmath128 . we shall return to a general discussion of this criticality in section iv . a final comment concerns the explicit evaluation of @xmath45 . instead of directly substituting @xmath101 in the expression of the partition function , it is convenient to apply eq . ( [ enem ] ) in the form @xmath129 and perform an integration with respect to @xmath65 to recover the free energy . here we present the solutions at @xmath130 as a function of @xmath5 . they can be broken into three classes for @xmath131 , @xmath132 and @xmath20 respectively . for @xmath131 , they are equivalent to chiral chain models with @xmath24 studied earlier@xcite all of which exhibit the third order gross - witten transition at @xmath37 . for @xmath133 there is a first order transition , which ends exactly at @xmath132 , consequently the end point at @xmath132 is of special interest . as we mentioned in the introduction , when @xmath5 is integer and less than 4 simplicial chiral models are only reformulation of trivial or already solved models . it is however quite instructive to consider even these examples in our new language . let us begin with the only apparently trivial case @xmath134 . obviously @xmath135 , however @xmath136 is nontrivial and we need to know its value in order to compute @xmath85 . as a matter of fact eq . ( [ tzm2 ] ) already implies that , up to a constant @xmath137 we would like , as a consistency check , to derive this result from the saddle point equation . a straightforward manipulation of eq . ( [ s4e3 ] ) leads to @xmath138 this equation is solved by @xmath139 with the only constraint @xmath140 . however by keeping in mind that only the combination @xmath141 is physically meaningful because of eq . ( [ s4e1 ] ) , we recognize that @xmath142 and the physical solution is unique and leads by a trivial integration , to eq . ( [ s5e1 ] ) . because of our definitions @xmath143 , @xmath144 . the eigenvalue distribution @xmath101 however is the generating function for the moments of the linear combination of a complex and a unitary matrix , and these moments can be highly nontrivial , even if the complex matrix itself has a gaussian probability distribution , as a consequence of the averaging over unitary matrices . as a matter of fact we were not able to solve explicitly the saddle - point equation associated to the @xmath68 models , even though the solution probably has reasonably simple mathematical properties . let us now turn to the @xmath69 case . as a straightforward application of eqs . ( [ s4e12a]-[s4e12b ] ) , we immediately find both the weak and strong coupling solutions , @xmath145 @xmath146 for the strong coupling region , @xmath147 . it is easy to recognize that the @xmath69 model corresponds to the gross - witten single - link problem , which in turn is equivalent to large-@xmath0 qcd@xmath148 with wilson action on the lattice @xcite . the properties of this model are well known , and in particular it is known that @xmath149 , consistent with eqs . ( [ s5e5a]-[s5e5b ] ) . another consistency check is easily made by applying eq . ( [ s4e14 ] ) and verifying that the known expressions for @xmath150 are reproduced . finally let us comment about the @xmath151 case . this model in its original formulation is completely equivalent to the three - link chiral chain studied in refs . we therefore know that it must possess a third - order phase transition at the critical value @xmath152 . however , as we already observed , our reformulation leads to exploring quite different classes of correlation functions and there is no obvious relationship between old and new results apart from bulk thermodynamical properties . again we have no analytical solution for the @xmath151 model equation , whose known properties stand as a benchmark for future attempts . turning to @xmath2 leads us to a new situation , where we are no longer guided by known results , since the 3-dimensional simplex ( tetrahedron ) is distinct from the solved four - link chain . actually it would be instructive and convenient to embed both models in a more general case interpolating between them and including many more interesting situations . we are studying the most general four - site system with bilinear interactions of four unitary matrices , which turns out to be reducible to an interacting two - complex matrix system . a separate paper will be devoted to a discussion of this system . here we only discuss the solutions of the saddle point equation obtained from eq . ( [ s4e3 ] ) in the @xmath2 case , @xmath154 in order to solve this equation , let us separately consider the weak and the strong coupling regimes , while changing variables for convenience to @xmath155 and defining the distribution @xmath156 by @xmath157 . the special structure of eq . ( [ s6e7 ] ) makes it convenient to follow a special procedure not directly related to eqs . ( [ s4e12a]-[s4e12b ] ) derived for the general case . in the weak coupling phase , we define the functions @xmath158 and @xmath159 subject to the normalization constraint @xmath160 the functions @xmath161 and @xmath162 are real analytic , with a cut along the interval @xmath163 $ ] on the real axis . on this interval the relationship , @xmath164 holds , while analyticity and eq . ( [ s6e7 ] ) imply @xmath165 however eqs . ( [ s6e8]-[s6e9 ] ) imply that @xmath166 in order to determine @xmath167 and @xmath168 we may use eq . ( [ s6e10 ] ) and the observation that in the complex @xmath169 plane when @xmath170 @xmath171 as a consequence one obtains that @xmath172 around @xmath163 $ ] , that is @xmath173 in strong coupling we adopt a similar strategy by defining @xmath174 and @xmath175 with the constraint @xmath176 and the boundary condition @xmath177 we then find @xmath178 and @xmath179 the boundary condition ( [ s6e20 ] ) leads to the relationship @xmath180 where @xmath181 all the integrals appearing in eqs . ( [ s6e13 ] ) , ( [ s6e16 ] ) , ( [ s6e22 ] ) and ( [ s6e23 ] ) are elliptic integrals . it is therefore possible to re - express both the weak and the strong coupling results in terms of known functions . in particular it is convenient to re - express everything in terms of `` natural '' rescaled variables , by defining @xmath182 @xmath183 and setting @xmath184 . it is not too difficult to eliminate completely the parameters @xmath104 and @xmath105 in favor of @xmath185 by making use of eqs . ( [ s6e16 ] ) and ( [ s6e23 ] ) respectively . as a consequence we obtain the weak coupling expression @xmath186\ ; , \label{s6e26}\ ] ] and the strong coupling counterpart @xmath187 ^ 2 } \left [ k^2{\sqrt{1-\zeta^2}\over \sqrt{k^2-\zeta^2}}k(k ) -\sqrt{k^2-\zeta^2}\sqrt{1-\zeta^2 } \pi(\zeta^2,k)\right]\ ; , \label{s6e27}\ ] ] where @xmath188 are the elliptic integrals of the first , second and third kind respectively , and the domain of @xmath189 is the interval @xmath190 $ ] , @xmath191 . obviously , in order for the problem to be completely solved , one must try expressing @xmath185 as a function of @xmath65 . this is achieved in principle by enforcing the normalization condition , which takes the form @xmath192 by symbolically writing @xmath193 in agreement with eqs . ( [ s6e26 ] ) and ( [ s6e27 ] ) , it is actually possible to express all results as functions of @xmath185 by the relationship @xmath194 in practice this form of our results is sufficient for both numerical evaluation and asymptotic expansions , not to mention the possibility of exploring the region around the criticality . criticality is characterized by the limit @xmath195 , where simple mathematical properties of elliptic integrals allow us to show that both weak and strong coupling results lead to @xmath25 and @xmath196 in order to obtain the usual weak and strong coupling expansion of physical quantities , like the internal energy , as power series in @xmath197 and @xmath65 respectively , one must consider in turn the @xmath198 limit and the expansion in powers of @xmath185 . obviously the different structure of @xmath199 in the two phases will lead to different expressions . in particular we have the asymptotic behaviors @xmath200 @xmath201 and it is conceptually straightforward to obtain power series expansions in the powers of @xmath185 for such quantities as the internal energy and to convert them into standard weak and strong coupling series . a few details will be discussed in appendix [ appc ] . the expansion around the critical point @xmath25 , @xmath202 , is slightly subtler because the expansion of elliptic integrals around @xmath203 is asymptotic . however by exploiting a few known or previously derived results , we have managed to obtain the following relationships , holding in weak coupling near the criticality : @xmath204 \left [ \zeta \ln { 1+\zeta\over 1-\zeta}- k'^2\left ( \ln { 4\over k ' } -{1\over 2}\right){\zeta^2\over 1-\zeta^2 } + o(k'^2)\right]\ ; , \label{s6e32}\ ] ] where @xmath205 . from eq . ( [ s6e29 ] ) we then obtain @xmath206\ ; , \label{s6e33}\ ] ] and as a consequence @xmath207\ ; , \label{s6e34}\ ] ] therefore @xmath208 apart from logarithms . by properly applying eq . ( [ s4e14 ] ) we may also extract the result @xmath209 and by simple manipulations , from the specific heat relationship @xmath210 we may obtain near the criticality @xmath211 a similar analysis can be performed in strong coupling near criticality , @xmath212 \left [ \zeta \ln { 1+\zeta\over 1-\zeta}+ k'^2\left ( \ln { 4\over k ' } + { 1\over 2}\right){\zeta^2\over 1-\zeta^2 } + o(k'^2)\right]\ ; , \label{s6e38}\ ] ] where again @xmath205 , and @xmath213\;. \label{s6e39}\ ] ] we then find @xmath214 the strong and weak coupling expressions of @xmath215 near the criticality show that the critical behavior around @xmath216 corresponds to a limiting case of a third order phase transition with critical exponent of the specific heat @xmath217 near the boundary with weak second order critical behavior . notice that in terms of double scaling limit @xmath218 would correspond to a central charge @xmath219 . for the interested readers we mention that in the derivation of eqs . ( [ s6e32 ] ) and ( [ s6e38 ] ) we made use of the following formula ( which appeared with some misprints in ref . @xcite ) @xmath220 for the asymptotic expansion of the elliptic integral of the third kind @xmath221 in the region @xmath222 . while at present we are not aware of any general method to get an analytic solution of the saddle - point equation ( [ s4e3 ] ) for arbitrary @xmath5 , the @xmath3 limit provides another interesting instance in which the equation is solvable . it is easy to show that for larger and larger values of @xmath5 the distribution @xmath101 becomes narrower and narrower , with a width decreasing like @xmath224 and a peak value @xmath225 which can easily be determined by replacing in eq . ( [ s4e3 ] ) @xmath226 and obtaining the large-@xmath0 , large-@xmath5 equation @xmath227 a consistent solution is obtained by assuming the limit to be taken at a fixed value of @xmath228 , in which case @xmath229 with the obvious restriction @xmath230 ( weak coupling phase ) . when @xmath231 one must recognize that the saddle - point condition , when correctly applied to the original expression for the effective action eq . ( [ seff ] ) in the large-@xmath5 limit , unambiguously leads to the prediction @xmath232 , @xmath233 ( strong coupling phase ) . the most interesting features of this result are : * the large-@xmath5 prediction for the location of the critical point , @xmath234 , amazingly enough , seems to be satisfied for all values of @xmath5 . * the complete equivalence with the mean field solution of infinite volume principal chiral models on a @xmath235-dimensional hypercubic lattice such that @xmath236 @xcite , where we may observe that this last relationship enforces the constraint that corresponding models have the same coordination number . it is easy to compute the large-@xmath5 expression for the internal energy in the weak coupling , @xmath237 at the criticality . the weak coupling value of @xmath238 is @xmath239 , while the strong coupling value is @xmath240 . therefore the large-@xmath0 , large-@xmath5 prediction for the nature of the criticality is that of a first - order phase transition . it is however important to notice that the large-@xmath5 prediction for the specific heat in the weak - coupling phase , @xmath241\ ; , \label{s8e5b}\ ] ] shows a divergence at the phase transition , with no indication for the existence of a metastable phase . it is interesting to compare the specific heat behavior for @xmath242 . in fig . [ dc ] we plotted @xmath243 versus @xmath244 . the large-@xmath5 result may also be the starting point for a systematic @xmath26 expansion of eq . ( [ s4e3 ] ) , and for a numerical approximation scheme which turns out to be quite efficient at least in the weak coupling domain away from criticality . the essential ingredient for both these developments is the observation that , substituting the definition of @xmath112 , eq . ( [ s4e6 ] ) , into eq . ( [ s4e3 ] ) we obtain the functional equation , @xmath246 subject to the following constraints : ( a ) eq . ( [ s8e6 ] ) is satisfied in the interval @xmath102 $ ] , with @xmath104 and @xmath105 dynamically determined ; ( b ) @xmath112 is real analytic outside the interval @xmath102 $ ] ; ( c ) the asymptotic behavior of @xmath112 when @xmath247 is @xmath248 . let us now define @xmath249 where @xmath250 is a constant whose value lies in the interval @xmath102 $ ] and will be dynamically generated . substituting the definition ( [ s8e8 ] ) into eq . ( [ s8e6 ] ) , we obtain @xmath251 postponing the discussion of the numerical approximation scheme , let us illustrate here the procedure for a systematic @xmath26 expansion of eq . ( [ s8e9 ] ) . we introduce the following ansatz for the function @xmath252 , @xmath253 and assume the functions @xmath254 , @xmath255 to be real analytic in the interval between the roots of the polynomial @xmath256 including the point @xmath257 . moreover we require @xmath258 and the boundary condition @xmath259 these requirements fix the constants @xmath250 , @xmath260 and @xmath261 dynamically , which in turn determine @xmath104 and @xmath105 . finally we assume all functions and constants to be expandable in @xmath26 , with non vanishing leading order . as an illustration let us find the solution to first nontrivial order . all leading order quantities will be independent of @xmath5 and labeled by a subscript 0 . after an expansion of eq . ( [ s8e10 ] ) in powers of @xmath26 we obtain @xmath262 by imposing the asymptotic boundary conditions we have the stricter condition ( forced by analyticity of @xmath263 , @xmath264 ) , @xmath265 as a consequence we may also predict the @xmath266 asymptotic behavior , @xmath267 now by substituting the above results into eq . ( [ s8e9 ] ) we obtain , after expansion in powers of @xmath26 , @xmath268 where we always assume the large-@xmath5 limit to be taken while keeping @xmath228 finite . substituting the condition @xmath269 into eq . ( [ s8e14 ] ) we may solve it in the form , @xmath270 @xmath271\ ; , \label{s8e15b}\ ] ] while the implementation of eq . ( [ s8e12 ] ) fixes @xmath272 and in conclusion we may also write @xmath273 let us notice that the leading order is completely determined in terms of the parameter @xmath274 , which in turn is fixed through eq . ( [ s8e15a ] ) to take the value @xmath275 hence we recognize that @xmath250 is nothing but a generalization of the mean field parameter which ( roughly speaking ) describes the center of the eigenvalue distribution , while the width of the distribution itself is @xmath276 as one may easily see by studying the roots of the polynomial under the square root sign . the eigenvalue distribution itself may be recovered ( order by order in @xmath26 ) by the relationship @xmath277 and it is not too difficult to check that the large-@xmath5 limit of eq . ( [ s8e19 ] ) may be taken and the result is @xmath278 as expected . we have worked out higher orders of the @xmath26 expansion . the ansatz ( [ s8e10 ] ) can also be used as a starting point for numerical approximations , and this will be described in appendix d. for instance , with the precision of about @xmath279 , one can quickly verify numerically the conjecture that @xmath4 . we would like to be able to understand in more detail the behavior of the critical properties as a function of d. eq . ( [ s4e3 ] ) , to the best of our knowledge , does not lend itself to an exact treatment for arbitrary values of @xmath65 and @xmath5 . however , on the critical surface for weak - strong transition , we find that it is possible to turn eqs . ( [ s4e12a])-([s4e12b ] ) into a homogeneous ( eigenvalue ) equation . for @xmath280 , this eigenvalue problem can be solved analytically . for @xmath20 , the problem can be solved numerically with great precision . it is also worth pointing out that at criticality for @xmath281 , eq . ( [ s4e3 ] ) can be solved by applying a method of gaudin and kostov for the study of @xmath19 spins on random surfaces . there is in fact an exact mapping of our weak - coupling critical saddle point equation to that of ref . @xcite , with @xmath282 . however , for @xmath20 , the gaudin - kostov s solution become pathological . in contrast , for the simplicial models , we find that a consistent solution exists for @xmath283 . the determination of @xmath37 can be achieved by considering eq . ( [ s4e12a ] ) in the limit when the equality in the weak coupling constraint , eq . ( [ s4e16 ] ) , is reached . on the weak coupling side of criticality the condition @xmath128 implies @xmath284 \;=\;0\;. \label{s4e17}\ ] ] there are two possible solutions , ( i ) @xmath285 , and ( ii ) @xmath286 , with @xmath287 if eq . ( [ s4e17 ] ) is solved by @xmath285 at @xmath288 , eqs . ( [ s4e12a ] ) and ( [ s4e12b ] ) both reduce to @xmath289\;. \label{s4e18}\ ] ] this indeed applies for @xmath290 , and we shall find an explicit solution of eq . ( [ s4e18 ] ) , which agrees with a result previously found by gaudin and kostov @xcite . in particular , we find that @xmath4 for @xmath291 . when @xmath29 , eq . ( [ s4e17 ] ) must be solved with @xmath292 . substituting eq . ( [ s4e19 ] ) into eqs . ( [ s4e12a])-([s4e12b ] ) , one arrives at a homogeneous equation @xmath293 where @xmath294 . ( [ s4e20 ] ) can be solved numerically with great accuracy . surprisingly , the relationship @xmath4 was found to be satisfied within machine precision . the essential feature of the solution for @xmath29 is the strong coupling relationship @xmath295 , while in weak coupling necessarily @xmath74 . since @xmath296 is the same on both sides of the transition , at criticality one finds from eq . ( [ s4e14 ] ) @xmath297 and as a consequence a first order phase transition is observed . when @xmath290 , eq . ( [ s4e3 ] ) at criticality can be solved by assuming that @xmath285 , as suggested by our analytic results discussed in section iii . let us therefore focus on eq . ( [ s4e18 ] ) . this equation on the first sight suggests that @xmath101 would vanish at @xmath298 as @xmath299 . however , it is easy to verify that , upon substituting this behavior into the right hand side of the equation , this square - root behavior is in fact inconsistent . based on our earlier exact analytic solutions , we assume that @xmath296 vanishes at @xmath298 faster the @xmath299 ; it follows that the square - bracket in eq . ( [ s4e18 ] ) must also vanish at @xmath298 . as a consequence , we have @xmath300 and we again arrive at a homogeneous equation @xmath301 note that this homogeneous equation connects smoothly with that appropriate for @xmath29 , eq . ( [ s4e20 ] ) , with @xmath285 . it is convenient to change variable from @xmath110 to @xmath302 , @xmath303 . in solving for @xmath302 in terms of @xmath110 , we shall choose the branch @xmath304 so that eq . ( [ s4e23 ] ) becomes @xmath305 where @xmath306 . although @xmath307 is originally defined only for the interval @xmath308 , the right - hand side of eq . ( [ s7e26 ] ) provides a natural extension to the region @xmath309 . with this extension , one finds that , over the positive axis , @xmath310 , @xmath311 it is then straightforward to verify that this extension can also be made for the right - hand side of eq . ( [ s7e26 ] ) so that it becomes @xmath312 let us next treat eq . ( [ s7e28 ] ) as an eigenvalue problem , and consider the ansatz where @xmath313 $ ] . using the technique of contour - integration , it is easy to verify that this indeed is an eigenvector with eigenvalue @xmath314 since @xmath315 , it follows that @xmath316 . with @xmath290 , one has @xmath317 real and @xmath318 , which allows a solution where @xmath296 is positive definite ! using the normalization condition for @xmath319 together with the criticality condition @xmath320 , we can fix the normalization constant @xmath321 and the end point @xmath322 one then obtains @xmath323 where @xmath324 and @xmath325 . one can show that eq . ( [ s7e6 ] ) reproduces the known critical solution when @xmath69 and 4 . when substituted into the critical equation at @xmath151 eq . ( [ s7e6 ] ) is numerically found to be a satisfactory solution . to determine the critical value @xmath37 , we can re - express eq . ( [ s4e22 ] ) in terms of @xmath307 as @xmath326 again , by an contour integration , one arrives at the remarkable result @xmath327 the solution discussed above does not apply to the case @xmath29 because the analytic continuation of eq . ( [ s7e6 ] ) for the critical density would no longer be positive - definite in the interval @xmath329 $ ] ; we must choose the alternative , @xmath286 . we have previously seen , with @xmath286 , how the criticality condition for @xmath37 , eq ( [ s4e19 ] ) , and the homogeneous integral equation for @xmath319 , eq . ( [ s4e20 ] ) , can be obtained , approaching from the weak coupling regime , by enforcing the condition @xmath128 with @xmath74 . it is instructive to see how these equations can be similarly derived from the strong coupling regime . starting with eq . ( [ s4e12b ] ) , one is working within the strong coupling regime where the constraint , eq . ( [ cond ] ) , is automatically satisfied with @xmath98 . it can be shown that the criticality condition , eq . ( [ s4e19 ] ) , corresponds to a situation where a zero of @xmath101 enters at @xmath330 . that is , as one increases @xmath65 beyond @xmath37 , the positivity of @xmath101 would be violated , thus terminating the validity of the strong coupling solution . as pointed earlier , with eq . ( [ s4e19 ] ) , the strong coupling equation , eq . ( [ s4e12b ] ) , again leads to eq . ( [ s4e20 ] ) . as a consequence , for @xmath29 , when one approaches @xmath37 from the strong coupling regime , one finds that @xmath331 and a first - order phase transition occurs . the solution to eq . ( [ s4e20 ] ) , on the critical point , subject to eq . ( [ s4e4 ] ) and eq . ( [ s4e5 ] ) , can be found numerically . in order to have a better behaved kernel when @xmath5 is close to @xmath332 we define a new function @xmath333 , @xmath334 with @xmath335 , the integral equation eq . ( [ s4e20 ] ) , and the constraints , eqs . ( [ s4e4])-([s4e5 ] ) , and the equation which determines @xmath37 , eq . ( [ s4e19 ] ) , become @xmath336 @xmath337 @xmath338 @xmath339 the solution of the integral equation eq . ( [ inteq ] ) can be done numerically by discretizing the kernel . after the discretization , the problem is reduced to an eigenvalue problem of a real non symmetric matrix . there are several ways to discretize the kernel . any rule of numerical integration is a discretization rule for the kernel . it is known that for integral equations the best discretization rules are the gauss quadrature rules @xcite . there are several gauss quadrature rules . we used the simplest possible : the gauss - chebyshev rule . all these rules require to map the integration interval to @xmath340 $ ] . thus we perform the following change of variables @xmath341 or @xmath342 under this change of variables the integral equation becomes @xmath343 where @xmath344 . the constraints take the following form @xmath345 @xmath346 and the equation for @xmath37 is @xmath347 the solution to eq . ( [ finteq ] ) can be found up to an overall constant @xmath215 , ( assuming that @xmath348 is a nondegenerate eigenvalue ) . this constant and the upper bound , @xmath105 , of the support of @xmath319 can be computed using the constrains eq . ( [ fcon1 ] ) , eq . ( [ fcon2 ] ) . let s denote an eigenfunction of ( [ finteq ] ) by @xmath349 . then @xmath350 , which is the function that is positive in [ -1,1 ] and satisfies the constraints , ( [ fcon1 ] ) and ( [ fcon2 ] ) , is related to @xmath349 by @xmath351 . if we now define @xmath352 @xmath353 then @xmath354 @xmath355 from the above formulas it is obvious that one can fix @xmath356 , solve eq . ( [ finteq ] ) and then find the eigenvalue @xmath348 which has a positive definite eigenfunction . because the problem is well defined one expects that there exists only one such function . this expectation is confirmed by the numerical results . it turns out that the eigenfunction with the largest eigenvalue is the one which is positive definite in @xmath340 $ ] . the @xmath357th eigenfunction has @xmath358 zeros in @xmath340 $ ] . thus for a given @xmath356 one computes @xmath359 , @xmath360 , @xmath361 , @xmath362 , @xmath363 . using the above numerical method we have computed @xmath364 with great precision for @xmath5 in the interval ( 4.4,250 ) . combining the numerical results with the analytical for @xmath365 , @xmath366 and @xmath367 are plotted in fig . [ acbc ] as functions of @xmath26 for @xmath368 . the functions @xmath369 , @xmath370 are continuous functions of @xmath5 at @xmath2 . several interesting features now emerge from an analysis of this data . on the one hand we can fit the functions @xmath369 , @xmath370 with great accuracy as power series of @xmath371 around @xmath2 and they agree with the corresponding weak coupling expressions up to very high orders . on the other hand , if one does a careful extrapolation of @xmath372 to @xmath2 a new feature is seen ( figure . [ logacbc ] ) . the upper limit , @xmath370 , extrapolates linearly in @xmath371 to @xmath373 , consistent with analyticity in the @xmath371 series expansion . the data alone determines the intercept @xmath374 to be @xmath373 to an accuracy of @xmath375 . when one examines the lower limit @xmath366 , it also approaches zero as @xmath5 approaches @xmath332 . however it does not go to zero as a simple power . the more we improved our data near @xmath2 the higher the effective power became . it appears that @xmath366 may have an essential singularity at @xmath2 vanishing faster than any power . since the discontinuity of the internal energy on the first order line is given by @xmath376 , this is pertinent to the critical properties at the end of the first order transition . the log - log plots of @xmath369 and @xmath377 in fig . [ logacbc ] clearly support these observations . in fig . [ acbc ] we have also been helped by an expansion of the functions @xmath378 and @xmath379 in powers of @xmath26 . the coefficients of this expansion have been determined by best fits on the numerical results and found to be consistent with integer numbers within the precision of our determination . this result is also consistent with the results of the @xmath26 expansion which we shall discuss in the next section . in particular we found @xmath380 the last integer term in both equations is uncertain . furthermore from the numerical computation of the @xmath37 for @xmath29 , we can see the @xmath381 result @xmath4 still holds above the critical point . we performed the numerical calculations in double precision and we see no deviation at all from the @xmath26 law . the deviation of @xmath382 from zero is determined to be less than @xmath383 for @xmath5 in the interval ( 4.4,250 ) . the 1/n expansion of matrix models has recently been used as a discrete representation for summing over random surfaces and , through the double - scaling " limit , for studying low - dimensional string theories . even more importantly , the large-@xmath0 expansion has provided us with a scheme for addressing non - perturbative issues in non - abelian gauge theories . for instance , many qualitative features of qcd , _ e.g. _ , confinement , the ozi rule , etc , can best be understood in a large-@xmath0 setting . however , quantitative progress in these directions has been slow due partly to the technical difficulties associated with the large number of independent loop " variables in this limit . nevertheless , it has been possible to gain useful insights into various interesting situations by utilizing as guides solvable models involving a small number of matrices , _ e.g. _ , models involving two hermitian matrices . much less is known for models involving unitary matrices . our current work not only adds to the list of solvable models in this category but also introduces new techniques for addressing matrix model studies in the large-@xmath0 limit . in this paper , we have studied the large-@xmath0 structure of simplicial chiral models defined on a @xmath1 dimensional simplex as one varies @xmath5 and the coupling @xmath65 . by exploring the global @xmath384 symmetry and by introducing an auxiliary complex matrix field , we are able to reduce the problem to that of solving for the eigenvalue distribution of a single hermitian semi - positive - definite matrix in the large-@xmath0 limit . in addition to providing exact large-@xmath0 solutions for several specific values of @xmath5 , we are able to identify and solve the strong - weak criticality problem for all values of @xmath5 , @xmath385 . for @xmath290 , analytic solutions for @xmath319 can be found . interestingly we find that the criticality occurs precisely at @xmath4 , as suggested by our previous studies @xcite . we find that the transition is third order . for @xmath386 , the criticality can also be studied by solving a homogeneous integral equation . however , we are only able to carry this out numerically . within numerical accuracy , we have shown that the criticality again takes place at @xmath4 , but with a first order transition . since we are able to reduce a @xmath1 dimensional simplicial chiral model to a model involving a single complex matrix , with @xmath5 entering as a parameter in the effective action , the large-@xmath0 limit can thus be solved by finding a density function for the eigenvalue distribution . unlike usual matrix models where all eigenvalues lie in a single connected band in the large-@xmath0 limit , this model effectively involves two bands , a right - band " where @xmath387 and a left - band " where @xmath388 . however , unlike other two - band problems @xcite , the distribution over these two bands are correlated . this new feature presents a challenge which can not be handled by a conventional large-@xmath0 treatment . our key result is the reduction of the above problem to that of solving a single inhomogeneous integral equation for the eigenvalue distribution of a single hermitian semi - positive definite matrix . although we could not find a closed form solution to this equation for arbitrary @xmath5 , we are able to solve it in several interesting special cases and we set up a systematic numerical approach to the solutions . we have found that the critical surface is defined by @xmath16 for all @xmath5 . for small @xmath5 , @xmath389 , the models exhibit the third order gross witten transition . indeed for @xmath390 they coincide exactly with the chiral chains studied earlier by brower , rossi and tan @xcite . in this region , the criticality is related to that of @xmath19 spin models on random surfaces , as discussed by gaudin - kostov @xcite . for @xmath20 , however , there is a first order transition ending at the `` upper critical '' dimensions @xmath21 . it therefore appears that , from the perspective of the double - scaling limit , the most interesting situation corresponds to @xmath391 . we have found that the point @xmath2 , having a logarithmic singularity , corresponds to @xmath392 . in the language of the double - scaling limit , this corresponds to having a vanishing string susceptibility `` , @xmath393 , where @xmath394 , which formally correspond to that resulted from a @xmath219 cft theory . this calls for further studies around @xmath2 which can provide further insight into possible different mechanisms for generating @xmath219 physics . one way is to vary @xmath5 near @xmath2 . another approach is to stay at @xmath2 , and embellish the model by relaxing the ' ' permutation symmetry " of the original @xmath2 simplicial chiral model . this will be presented in a subsequent publication . we are deeply indebted with prof . g. cicuta for bringing ref . @xcite to our attention and for useful conversations . this work was supported in part by the u. s. department of energy , under grant de - fg02 - 91er400688 , task a. when @xmath398 , @xmath399 can be reduced to the partition function of the gross - witten single - link problem @xcite @xmath400\ ; , \label{ae2b}\ ] ] thus sharing the same thermodynamic properties . the free energy density at @xmath401 , @xmath402 , is piecewise analytic with a third order transition at @xmath403 between the strong coupling and weak coupling domains . the large-@xmath0 limit of the specific heat is @xmath404 the behavior of @xmath405 around @xmath37 can be characterized by a specific heat critical exponent @xmath406 . it is worth noting that an analysis of the double scaling limit , @xmath407 and @xmath408 , allows the determination of the correlation length critical exponent , @xmath409 @xcite , and that @xmath410 and @xmath411 satisfy a hyperscaling relationship associated to a two - dimensional critical phenomenon , @xmath412 . this fact is related to the equivalence of the double scaling limit with the continuum limit of a two - dimensional gravity model with central charge @xmath413 . it has been shown that in the context of single - matrix models the parameter @xmath0 plays a role quite analogous to the volume in ordinary systems , and double scaling limit turns out to be very similar to finite size scaling in a two - dimensional critical phenomenon @xcite . as a manifestation of this fact , it has been observed that in the gross - witten single - link problem , ( i ) the asymptotic approach of the complex @xmath414 zeroes closest to @xmath37 , @xmath415 , toward the real axis occurs at a rate determined by the correlation length exponent @xcite , @xmath416 and , ( ii ) for sufficient large @xmath0 the position of the peak of the specific heat , @xmath417 , behaves as @xcite @xmath418 we recall that in ordinary critical behaviors finite size scaling leads to relations of the type ( [ ae6]-[ae7 ] ) with @xmath0 replaced by the size of the system . we recall that the chiral chain with @xmath421 is equivalent to the simplicial model with @xmath151 . in this case @xmath152 and the phase transition is still third order . in the weak coupling region , @xmath422 , the @xmath401 specific heat is given by @xmath423 therefore close to @xmath37 , @xmath424 similarly in the strong coupling region , @xmath425 , and close to @xmath37 , @xmath426 then the strong and weak coupling expressions of @xmath427 show that the critical point @xmath152 is third order and @xmath428 . for @xmath429 the study of the critical behavior around @xmath430 is slightly subtler . in the weak coupling domain the @xmath401 internal energy can be expressed as @xmath431 where @xmath432 is implicitly determined by the equation @xmath433\;=\;1\;. \label{ae12}\ ] ] ( this equation comes from the normalization of the eigenvalue distribution @xmath434 introduced in ref . @xcite ) . since @xmath435 at @xmath430 , in order to study the critical behavior close to @xmath37 we expand eq . ( [ ae12 ] ) around @xmath435 , obtaining the following relation @xmath436 and therefore @xmath437 apart from logarithms . furthermore we have @xmath438 we then obtain for the specific heat @xmath439 when @xmath440 . a similar analysis can be performed in strong coupling , where the internal energy can be written as @xmath441 with @xmath189 implicitly defined by the equation @xmath442 since @xmath257 at @xmath37 , we expand eq . ( [ ae17 ] ) around @xmath257 obtaining @xmath443 consequently , @xmath444 apart from logarithms and @xmath445 we then obtain when @xmath446 , @xmath447 a comparison of eqs . ( [ ae15 ] ) and ( [ ae20 ] ) leads to the conclusion that the phase transition is again third order with a critical exponent @xmath218 . the critical exponent @xmath411 could then be determined by using the 2-d hyperscaling relationship , obtaining @xmath448 . this value of @xmath411 was confirmed by a numerical monte carlo study of the scaling of the specific heat peak position at finite @xmath0 , we indeed observed a behavior like eq . ( [ ae7 ] ) compatible with @xmath448 within a few per cent of uncertainty . in fig . [ lc ] we plot the specific heat versus @xmath65 for @xmath449 . in conclusion we have seen that @xmath449 have a third order phase transition at increasing critical values @xmath450 , with specific heat critical exponents @xmath451 , respectively . notice the behavior of @xmath410 with respect to @xmath452 , which is increasing for @xmath453 reaching the limit of a third order critical behavior , but then in large-@xmath452 limit it returns to @xmath406 . strong coupling series of the free energy density of chiral chain models are best generated by means of the character expansion , which leads to the following result @xmath454 where @xmath455 is the free energy of the single unitary matrix model ( @xmath456 and @xmath457 is given by eq . ( [ ae2b ] ) ) , and @xmath458 @xmath459 denotes the sum over all irreducible representations of @xmath33 , @xmath460 and @xmath461 are the corresponding dimensions and character coefficients . the calculation of the strong coupling series of @xmath462 is much simplified in the large-@xmath0 limit , due to the following relationships @xcite @xmath463 and @xmath464 where @xmath465 is independent of @xmath65 and @xmath357 is the order of the representation @xmath466 . explicit expressions of @xmath460 and @xmath465 are given in ref . notice that the large-@xmath0 strong coupling expansion of @xmath467 is actually a series in @xmath468 , @xmath469 @xmath470 represents also the generating functional for the `` potentials '' @xmath471 introduced in ref . @xcite , in the context of the strong coupling expansion of more general models , indeed the following relationship holds @xmath472 it is important to recall that the large-@xmath0 character coefficients have jumps and singularities at @xmath473 @xcite , and therefore the relevant region for a strong - coupling character expansion is @xmath474 . we have analyzed the strong coupling series of chiral chain models in order to investigate their large-@xmath0 critical behaviors for @xmath475 . given the simple behavior of the large-@xmath0 limit of @xmath455 , we considered only the contributions from @xmath476 , thus working with series in @xmath468 . we generated about 15 terms for each @xmath477 and analyzed , as series in @xmath468 , the specific heat derivative , which diverges at the critical point in a third phase transition . we employed the integral approximant technique @xcite , which at present seems to be one of the most powerful method of resummation . in particular we considered integral approximants obtained from first order linear differential equations . let us begin with the results obtained for the known cases @xmath13 . for @xmath421 already 15 terms in the series ( in @xmath468 ) suffice to get @xmath478 and @xmath479 with a precision of about @xmath480 and @xmath481 respectively . however it is worth noticing that in the analysis of the specific heat derivative we found spurious non - diverging singularities on the positive real axis for @xmath71 . concerning the @xmath429 case , it is known that the integral approximant resummation analysis can not reproduce an @xmath218 singularity type @xcite and therefore it is not really suitable to this case . anyway , we obtained a good determination of @xmath37 , we found @xmath482 up to about @xmath483 , and a rather stable but wrong exponent , @xmath484 , which should somehow simulate the logarithmic corrections found in the appendix [ appa ] , given that they can not be generated by the differential equation solution . again we found spurious non - diverging singularities for @xmath71 . the strong - coupling analysis starts giving new information when @xmath475 . due to the persistent presence of spurious singularities , guided by the @xmath13 analysis , in all cases we considered the first diverging singularity on the positive real axis as an estimate of the true critical point . for @xmath485 we obtained quite stable results : @xmath486 and @xmath487 . we should say that the @xmath429 analysis suggests some caution in accepting this estimate of @xmath410 , it could still be a masked @xmath218 . the analysis of @xmath488 series gave a rather stable estimate of the critical point @xmath489 , but unstable exponents ( although negative and small ) . similar results were found for @xmath490 : @xmath491 for @xmath492 , @xmath493 for @xmath494 , @xmath495 for @xmath496 . notice that , unlike the @xmath497 cases , these values can not be considered as an estimate of the critical point . they are indeed larger than @xmath498 , that is out of the region where a strong coupling analysis can be predictive , and therefore something else must happen before , breaking the validity of the strong coupling expansion . an example of this phenomenon comes from the gross - witten single - link model ( recovered when @xmath398 ) , where the strong coupling expansion of the @xmath401 free energy leads to an analytical function not having singularity at all , @xmath499 , thus @xmath500 can not be determined from a strong coupling analysis . of course we can not consider this analysis satisfactory , but from it we may hint at a possible scenario . as for @xmath501 , for @xmath502 , that is when the estimate of @xmath37 coming from the above strong coupling analysis is smaller than @xmath498 and therefore acceptable , the term @xmath476 in eq . ( [ be1 ] ) should be the one relevant for the critical properties , determining the critical points and giving @xmath503 ( maybe @xmath218 as in the @xmath429 case ) . for @xmath490 the critical point may not be a singular point in strong or weak coupling , but just the point where weak coupling and strong coupling curves meet each other . this would cause a softer phase transition with @xmath406 , as for the gross - witten single - link problem . we expect @xmath504 also for @xmath490 . this scenario would be consistent with the analysis of green and samuel @xcite , who studied the behavior of the link determinant ( i.e. @xmath505 ) to determine the critical points . the values of @xmath37 we found for @xmath502 are consistent with their estimates . we have briefly discussed in section [ sec6 ] the possibility of performing weak and strong coupling expansion in the @xmath2 model starting from an expansion in the powers of @xmath506 . here we want to give more details on the concrete implementation of this program . let us first of all for convenience define a few auxiliary functions of @xmath506 , the labels s and w are to remind the strong and weak coupling expansions , which we shall treat on the same footing . in terms of standard elliptic integrals we define @xmath507 @xmath508\ ; , \label{ce1b}\ ] ] @xmath509 @xmath510 let us introduce the function @xmath511 and notice that , because of the properties of elliptic integrals and of eq . ( [ s6e29 ] ) we may obtain from eqs . ( [ s6e26]-[s6e27 ] ) @xmath512 holding for the proper choice of indices both in the weak and in the strong coupling phase . in order to set up our expansion we therefore need power series representation for the functions @xmath513 , @xmath514 and @xmath515 . the elliptic integrals of the first and second kind have simple known expansions , @xmath516 @xmath517 where @xmath518 the elliptic integral of the third kind , needed in the construction of @xmath515 , admits the following expansion in the powers of the first argument , @xmath519 where @xmath520 satisfy the recursion equation @xmath521 with boundary condition , @xmath522\;. \label{ce10}\end{aligned}\ ] ] in turn we may show that @xmath523 are related to @xmath524 by @xmath525 as a consequence the function @xmath515 is completely determined once @xmath526 are known , by the relationship @xmath527 the recursion eq . ( [ ce9 ] ) may be solved if we expand the functions @xmath526 in a power series of @xmath506 , @xmath528 it is tedious but straightforward to verify that @xmath529 ^ 2 } \label{ce15}\ ] ] satisfy the recursion . one may also for convenience define @xmath530 and finds from eq . ( [ ce12 ] ) that @xmath531 direct substitution of the series expansions thus obtained in eq . ( [ ce4 ] ) allows to construct an expansion of @xmath65 ( or @xmath532 respectively ) in powers of @xmath506 which a simple symbolic manipulation program can easily extend to extremely high orders . in order to make our analysis complete we must manage to extend our discussion to the evaluation of a physical observable . by recalling eq . ( [ s4e14 ] ) , we notice that the expression for the internal energy in the @xmath2 case is @xmath533 which gives rise in weak coupling to @xmath534 while in strong coupling we obtain @xmath535 by considering the explicit form of the functions @xmath536 and @xmath537 which we obtain from eqs . ( [ s6e26]-[s6e27 ] ) , it is straightforward to parameterize @xmath238 by @xmath538n(k^2)^2 } -{1\over 3}-{2\over 3}{d(k^2)-p(k^2)\over n(k^2)^2}\ ; , \label{ce21}\ ] ] where the functions @xmath539 can in turn be reconstructed , by using the results presented in this appendix , in terms of the functions @xmath524 and @xmath540 . the results for the two regimes are @xmath541 ( k^2)^l \label{ce22}\ ] ] and @xmath542 ( k^2)^l \label{ce23}\ ] ] respectively . by expanding eq . ( [ ce21 ] ) in power series of @xmath506 , inverting eq . ( [ ce4 ] ) and substituting @xmath506 as a function of @xmath532 or @xmath65 respectively , we obtain the standard weak and strong coupling series for the internal energy . the ansatz ( [ s8e10 ] ) can also be used as a starting point for numerical approximations based on the very simple consideration that real analytic functions of @xmath189 can be approximated with any assigned precision by polynomials of sufficiently high degree in @xmath189 itself . we may therefore introduce the @xmath357-th truncations of @xmath254 and @xmath255 respectively by the definitions @xmath545 we may now explicitly perform a laurent series expansion around the point @xmath257 in the form , @xmath546\;\equiv\ ; \sum_{i=1}^n p_i\zeta^i + p_0 + \sum_{j=1}^\infty p_{-j}\zeta^{-j}\ ; , \label{s9e2}\ ] ] where the coefficients @xmath547 , @xmath548 and @xmath549 are completely determined in terms of @xmath550 , @xmath260 and @xmath261 . we may now notice that the asymptotic condition on @xmath551 forces us to impose the relationships @xmath552 at this stage @xmath553,@xmath554,@xmath555 are completely determined in terms of the coefficients @xmath550 and the parameter @xmath556 . we may now consider the effect of substituting @xmath551 into eq . ( [ s8e9 ] ) and power - series expanding in @xmath189 ; if we define @xmath557 to be the @xmath185-th derivative of the function @xmath558 evaluated at the point @xmath559 , we may turn eq . ( [ s8e9 ] ) into the following approximate relationship , @xmath560 + ( d-2)\sum_{k=0}^n { ( -\zeta)^k\over k!}\phi_n^{(k)}(-2\tilde{z})\ ; , \label{s9e4}\ ] ] which in turn decomposes into @xmath561 equations in the @xmath561 unknowns @xmath562 , @xmath563 these equations may be solved numerically for arbitrary values of @xmath65 and @xmath5 and offer better and better approximations to the true eigenvalue distribution ( by applying eq . ( [ s8e19 ] ) with increasing values of @xmath357 ) . numerical experiments have shown that when @xmath564 even extremely small values of @xmath357 give quite accurate predictions , while around the criticality , corresponding in this language to the condition @xmath565 which determines @xmath37 , but the accuracy is definitely weaker . this fact did not prevent us from determining the location of criticality , by the use of @xmath566 , in the cases @xmath567 , with a precision of about 1% . in the context of this discussion it is important to observe that ( even approximate ) knowledge of @xmath252 implies an ( approximate ) knowledge of the moments of the eigenvalue distribution , which may be obtained from the laurent series expansion via the relationship @xmath568 which in particular implies that @xmath569 and in turn we may extract the internal energy via the relationship @xmath570

the large-@xmath0 saddle - point equations for the principal chiral models defined on a @xmath1 dimensional simplex are derived from the external field problem for unitary integrals . the saddle point equation are studied analytically and numerically in many relevant instances , including @xmath2 and @xmath3 , with special attention to the critical domain , which is found to correspond to @xmath4 for all @xmath5 . related models ( chiral chains ) are discussed and large-@xmath0 solutions are analyzed . # 1currentlabel@secnum>0 .[#1 ] critical behavior of simplicial chiral models @xmath6 physics department , boston university , boston , ma 02215 , usa . + @xmath7 dipartimento di fisica delluniversit and i.n.f.n . , i-56126 pisa , italy . + @xmath8 department of physics , brown university , providence ri 02912 , usa

redshifted absorption systems lying along the sight - lines to distant quasars are important probes of the early to present day universe . in particular , damped lyman-@xmath0 absorption systems ( dlas ) , where @xmath4 , are useful since they account for at least 80% of @xmath5 in the universe @xcite . since the lyman-@xmath0 transition occurs in the ultra - violet band , direct ground based observations of neutral hydrogen are restricted to redshifts of @xmath6 . however , observations of the spin - flip transitions at @xmath7 cm can probe from @xmath8 , thereby providing a useful complement to the high redshift optical data . provided the 21-cm and lyman-@xmath0 absorption arise in the same cloud complexes , the column density @xmath9 [ ] of the absorbing gas in a homogeneous cloud is related to the velocity integrated optical depth of the 21-cm line via @xmath10 here @xmath11 [ k ] is the spin temperature of the gas and so in principle , armed with the neutral hydrogen column density from an observation of the lyman-@xmath0 line , this quantity may be derived from the optical depth of the 21-cm absorption . however , the observed optical depth of the 21-cm line also depends upon on how effectively the background radio continuum is covered by the absorber via , @xmath12 , where @xmath13 is the depth of the line relative to the flux density and @xmath14 is the covering factor of the flux by the absorber . in the optically thin regime ( @xmath15 ) , which applies to all but one of the known 21-cm absorbing dlas , equation [ enew ] reduces to @xmath16 thus giving a direct measure of the spin temperature of the gas for a known column density ( from the lyman-@xmath0 line ) and covering factor . however , in the absence of any direct measurement of the size of the radio absorbing region , this latter value is often assumed or at best estimated from the size of the background emission region . from the literature , 16 of the 35 dlas searched have been found to exhibit 21-cm absorption ( see table [ t1 ] ) , all of which occur at redshifts below @xmath17 , although there are a near equal number of non - detections also below this redshift . @xcite therefore advocate a scenario where low redshift dlas have a mix of low ( 21-cm detections ) and high ( 21-cm non - detections ) spin temperatures , with the high redshift absorbers having exclusively high spin temperatures . however , in a previous paper @xcite , we find evidence that the importance of the covering factor is underestimated and the common practice of setting @xmath18 could possibly have the effect of assigning artificially high spin temperatures to dlas , particularly those not detected in 21-cm . furthermore , we found no statistical difference in the spin temperature / covering factor ratio between the low and high redshift samples , although the larger absorbing galaxies ( spirals ) group together at low values of @xmath19 and @xmath20 . since the ratio of spin temperature / covering factor is @xmath21 ( equation [ enew ] ) , we have taken into account the total column density of the absorber and integrated optical depth of the 21-cm absorption ( incorporating the radio flux ) . this suggests that the difference between the detections and non - detections is due to some other effect , a possibility which we investigate in this paper . we note that the cut - off of the 21-cm detections , @xmath22 , is close to the atmospheric cut - off of the lyman band at @xmath23 and over the range from which the mgii 2796/2803 doublet may be observed by ground - based telescopes ( @xmath24 ) . indeed only 4 of the 17 21-cm detections occur in dlas originally identified through the lyman- line , cf . 13 of the 18 non - detections . . [ cols="<,^,>,^,^,^,^,>",options="header " , ] notes : @xmath25just prior to this paper being accepted , a detection of 21-cm absorption in this dla was published @xcite . this was previously flagged a non - detection by @xcite , although the data and subsequent limit were poor . @xmath263c196 , @xmath273c286 , @xmath283c216 , @xmath2921-cm absorption searched at @xmath30 , although the 5 mhz bandwidth used should cover @xmath31 @xcite , @xmath323c446 . + references : @xmath33@xcite , @xmath34@xcite , @xmath35@xcite , @xmath36@xcite , @xmath37@xcite , @xmath38@xcite , @xmath39@xcite , @xmath40@xcite , @xmath41@xcite , @xmath42@xcite , @xmath43@xcite , @xmath44@xcite , @xmath45@xcite , @xmath46@xcite , @xmath47@xcite , @xmath48@xcite , @xmath49@xcite , @xmath50@xcite , @xmath51@xcite , @xmath52@xcite , @xmath53@xcite , @xmath54@xcite , @xmath55@xcite , @xmath56@xcite , @xmath57 @xcite , @xmath58@xcite , @xmath59@xcite , @xmath60@xcite , @xmath61@xcite , @xmath62 @xcite , @xmath63@xcite . mgii selection gives rise to a range of absorbing galaxy types ( @xcite ) , and although most dlas discovered through the lyman-@xmath0 line have unidentified host types ( mainly due to the high redshift selection , e.g. table [ t1 ] ) , low redshift studies suggest that equal numbers of dwarfs and spirals should contribute to the dla population @xcite . in table [ t1 ] we see that the dlas detected in 21-cm absorption exhibit a variety of host galaxy types , although there is the strong preference for 21-cm absorption to occur in mgii selected sources . however , @xcite find that , while the 21-cm line strength appears correlated to the rest frame equivalent width of the mgii line , 21-cm absorption is perfectly detectable at low equivalent widths . furthermore , large equivalent widths do not necessarily ensure a detection of 21-cm absorption ( fig . [ n - w - det ] ) . since host type and mgii equivalent width seem incidental in determining whether 21-cm absorption is detected , we suggest that the spin temperature / covering factor ratios in the dlas searched for in 21-cm absorption are due to geometric effects introduced by the dla discovery method : in fig . [ distance ] we show the spin temperature / covering factor ratio against the ratio of the angular diameter distances to the absorber and background continuum . although the sizes and morphologies of the radio sources differ considerably ( table 2 of @xcite ) , for given background continuum size and 21-cm absorbing cross section , the covering factor is obviously larger for those absorbers , at least at low redshift ( fig . [ z - distance ] ) , which are located very much closer to us than the radio emitter . indeed we see that the dlas not detected in 21-cm have significantly larger angular diameter distance ratios than the detections , with the vast majority of these having ratios of @xmath64 ( fig . [ distance ] ) . these are 1157 + 014 ( ratio = 1.00 ) and 1328 + 307 ( ratio = 0.93 ) . at @xmath65 , the former is one of the highest redshifted 21-cm absorbers known and occults a radio source size of @xmath66 arc - secs @xcite and 1328 + 307 occults a core dominated source of 2.57 arc - secs @xcite . despite the pitfalls in assuming given absorber and emitter sizes , for the larger number of non - detections , the distribution does appear very skewed towards high fractional distances . ] . in order to demonstrate the differences in fractional distances between the 21-cm detections and non - detections , in fig . [ z - distance ] we show how the distance ratio is distributed with absorber redshift . we see that most of the 21-cm detections are located in the bottom left quadrant , defined here by @xmath67 and @xmath68 . this is the approximate redshift of the turnover in the angular diameter distance , where objects increase their angular size with redshift . it is also close to the redshift where the lyman-@xmath0 transition can be observed by ground based telescopes , thus giving the appearance that lyman-@xmath0 selected dlas are less likely to be detected , although , as we see from fig . [ z - distance ] , this is purely a consequence of the higher redshifts probed by this transition . using this partitioning , at @xmath69 , for ratios of @xmath70 , 21-cm absorption tends to be detected ( 11 out of 13 cases ) and for @xmath71 , 21-cm absorption tends to be undetected ( 8 out of 10 cases ) . within each range , if there is an equal likelihood of obtaining either a 21-cm detection or non - detection , the binomial probability of 11 or more out of 13 detections occurring in one bin , while 8 or more out of 12 non - detections occur in the other bin is just 0.06% . changing the redshift partition to @xmath72 gives a binomial probability of 0.25% and no redshift partition , i.e. @xmath73 detections in one bin with @xmath74 non - detections in the other bin , gives 0.03% . this leads to the hypothesis that high redshift ( @xmath75 ) dlas tend not to be detected in 21-cm absorption because @xmath76 for all of these systems . [ z - distance ] illustrates that a redshift distance ratio bias arises , since at @xmath77 the covering factor , see equation [ f ] . ] becomes effectively independent of distance and is thereby determined by the relative extents of the absorption cross section and continuum emission region only . at lower redshifts ( particularly @xmath78 ) the close - to - linear decrease of angular size with distance means that , for a given absorption cross section , low redshift systems can much more effectively cover the background emission . that is , above @xmath79 , 21-cm absorption searches are disadvantaged by the fact that in all cases @xmath76 . furthermore , the angular diameter redshift relationship dictates that the bottom right - hand quadrant of fig . [ z - distance ] is destined to always remain empty and the higher the value of @xmath80 , the lower @xmath81 must be in order to yield @xmath82 . using the luminosity distances , a similar distribution to fig . [ z - distance ] is seen , with the concentration of 21-cm detections occurring at @xmath83 and the majority of non - detections having luminosity distance ratios of @xmath84 . therefore , as well as affecting the effective coverage of the quasar s emission , the generally close dla qso proximity in the high redshift sample ( fig . [ distance ] ) could have implications for the spin temperature of the 21-cm absorbing gas in the dla through increased incident 21-cm flux . we now investigate this possibility as well as attempting to quantify the effect of the proximity bias on the covering factors . the high incidence of 21-cm non - detections with large fractional distances raises an interesting possibility : in order to explain the decrease in the number density of lyman-@xmath0 absorbers as @xmath85 , against the general increase in the number density with redshift , @xcite invoke a `` proximity effect '' , where at close to @xmath80 the absorber is subject to a high ionising flux from the quasar it occults , thus reducing the number of lyman-@xmath0 clouds observed . in light of the large number of 21-cm non - detections located relatively close to the background radio source , an analogy of the proximity effect may be at play , where a high 21-cm flux is maintaining a higher population in the upper hyperfine level @xcite . this would decrease the observed 21-cm optical depth through an increase in stimulated emission ( equation [ enew ] ) . the intrinsic luminosity of the quasar at the rest frame emission frequency , @xmath86 , is @xmath87 , where @xmath88 is the luminosity distance to the quasar , @xmath89 is the observed flux density ( given in table 1 of @xcite ) and @xmath90 is the k - correction @xcite . furthermore , the 1420 mhz flux density at the absorber is @xmath91 , where @xmath92 is the luminosity distance between the absorber and the quasar . combining this with the previous equation gives @xmath93 since the observed frequency is given by both @xmath94 and @xmath95 , the continuum emission frequency in the rest frame of the quasar is given by @xmath96 [ where @xmath97 mhz in the rest frame of the absorber ] . from this , the redshift of the quasar in the rest frame of the absorber is given by @xmath98 which we use to determine @xmath99 . in fig . [ 1216 ] we show the observational results of the 21-cm searches ( the spin temperature / covering factor ratio ) against the 21-cm flux density calculated at a distance @xmath99 from the quasar . from this we see that below a flux density of @xmath100 jy at the absorber , there is no overwhelming difference in the 21-cm detections and non - detections , although the latter do tend to be more slightly numerous above @xmath101 jy , as well as being dominant at @xmath102 jy . however , the numbers are small and it appears as though increased flux densities due to close proximity to the background source is not the dominant cause of the non - detection of 21-cm absorption . since it appears that the non - detections are not due to spin temperatures being raised by the quasar flux , we now focus on the how the covering factor varies with quasar proximity . as usual , we can not separate out the relative contributions from the spin temperature and the covering factor , although we can define , using the small angle approximation . ] , the covering factor as @xmath103 where @xmath104 is the 21-cm absorbing cross section and @xmath105 is the radio source size as determined from high resolution observations ( see table 2 of @xcite ) . it should be borne in mind that these are usually measured at frequencies which are several times higher than the redshifted 21-cm ( 1420 mhz ) line . in addition to assuming that these provide an accurate indicator of the radio source size at the absorption frequency , equation [ f ] also assumes that the emission is uniform over the extent of this radio emission . if these assumptions are reasonable , substituting equation [ f ] into equation [ enew ] allows us to plot a covering factor `` free '' version of fig . [ distance ] , which we show in fig [ toverr - ratio ] . in the plot , like figures 4 and 5 of @xcite , we see a clear distinction between the distribution of the spirals and the more compact galaxies ( more evident in fig . [ r - flux ] ) . this may indicate that , at least at low redshift ( where @xmath106 ) , each group has similar ratios ( @xmath107 k kpc@xmath108 for the compact galaxies and @xmath109 k kpc@xmath108 for the spirals ) , with the absorbing cloud size making a large contribution in the very different values between these two groups : a span of @xmath110 orders of magnitude seems unlikely through spin temperature alone , although for a given temperature , a span of only @xmath111 dex is required in the radius of the absorbing region . furthermore , figure 4 of @xcite shows that the radio sources of @xmath112 tend to be adequately covered by the compact galaxies , whereas for the larger radio sources spirals are required . bearing in mind that two of the @xmath113 values at @xmath114 ( @xmath115 ) are lower limits , in fig . [ toverr - ratio ] there may be a trend for @xmath116 to decrease as the dla qso distance ( angular diameter & luminosity ) closes , for 21-cm absorption detected in non - spirals . presuming that the spin temperature does not decrease with proximity to the quasar ( contrary to what we would expect ) , this may suggest a selection effect where only large 21-cm absorbers are detected close to the background continuum , implying that self shielding against high fluxes are important , where the effectiveness of this scales with cloud size ( fig . [ r - flux ] ) . this , however , relies upon the aforementioned assumptions regarding the radio sources and many more detections would be required to adequately test this hypothesis . we note with interest , that the 21-cm detection located furthest to the bottom right in fig . [ toverr - ratio ] ( absorber i d unknown at @xmath117 ) , which has the very low value of @xmath118 k kpc@xmath108 ( @xmath119 kpc k@xmath120 , fig . [ r - flux ] ) , is due to 0458020 , where a large absorbing cross section of @xmath121 kpc is deduced from vla and vlbi observations ( @xcite ) . regarding the detectability of 21-cm absorption in dlas , we have found : * in general , the non - detections of 21-cm absorption in dlas have been searched as deeply as the detections , meaning that the ratio of spin temperature / covering factor does not differ significantly between the two samples . * there is an apparent bias for 21-cm absorption to be detected in dlas originally discovered through the mgii doublet rather than the lyman- line . this , however , is superficial and merely reflects the true bias introduced by the redshift distribution of the dlas : at @xmath77 , the absorbers are effectively at the same ( angular ) distances as the background quasars . after ruling out the possibility that the closer proximity of the undetected dlas to the background quasars significantly raises the spin temperatures above those of the detections , we believe that the non - detections are due to low covering factors , the result of the flattening of the angular diameter distance at @xmath122 . since dlas are not detected in 21-cm absorption at @xmath123 ( table [ t1 ] ) . ] , @xcite suggested that the non - detections are due to the high redshift ( @xmath124 ) dlas having exclusively high spin temperatures and the presence of both 21-cm detections and non - detections at low redshift is attributed to a mix of spin temperatures . however , the distribution of 21-cm detections and non - detections closely follows that of low and high angular diameter distance ratios , respectively , so that high redshift dlas have exclusively high ratios , whereas @xmath125 systems exhibit a mix of ratios . this geometric effect means that absorbers at high redshift will always cover the background quasar much less effectively than at low redshift and the degeneracy between spin temperature and covering factor may only ever be resolved by targetted searches for 21-cm absorption in high redshift dlas towards very compact radio sources . we would like to thank matthew whiting and michael murphy for their advice as well as the referee , emma ryan - weber , for her prompt , detailed and very helpful review in addition to her subsequent feedback . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this research has also made use of nasa s astrophysics data system bibliographic services .

in this paper we investigate the possible reasons why 21-cm absorption in damped lyman-@xmath0 systems ( dlas ) has only been detected at low redshift : to date , no 21-cm absorption has yet been detected at @xmath1 and at redshifts less than this , there is a mix of detections and non - detections in the dlas searched . this has been attributed to the morphologies of the galaxies hosting the dlas , where at low redshift the dlas comprise of both large and compact galaxies , which are believed to have low and high spin temperatures , respectively . likewise , at high redshift the dla population is believed to consist exclusively of compact galaxies of high spin temperature @xcite . however , in a previous paper @xcite we found that by not assuming or assigning an , often uncertain , value for the coverage of the radio continuum source by the 21-cm absorbing gas , that there is generally no difference in the spin temperature / covering factor ratio between the 21-cm detections and non - detections or between the low and high redshift samples . furthermore , only one of the 18 non - detections has a known host morphology , thus making any link between morphology and 21-cm detectability highly speculative . we suggest that the lack of 21-cm absorption detections at high redshift arises from the fact that these dlas are at similar angular diameter distances to the background quasars ( i.e. the distance ratios are always close to unity ) : above @xmath2 the covering factor becomes largely independent of the dla qso distance , making the high redshift absorbers much less effective at covering the background continuum emission . at low redshift , small distance ratios are strongly favoured by the 21-cm detections , whereas large ratios are favoured by the non - detections . this mix of distance ratios gives the observed mix of detections and non - detections at @xmath3 . in addition to the predominance of large distance ratios and non - detections at high redshift , this strongly suggests that the observed distribution of 21-cm absorption in dlas is dominated by geometric effects . [ firstpage ] quasars : absorption lines cosmology : observations cosmology : early universe galaxies : ism

neutrino astronomy is now an emerging field and entails the need to have improved flux estimates as well as a good understanding of relevant detector capabilities for all flavor neutrinos , particularly in light of the recent growing experimental support for flavor oscillations @xcite . several high - energy cosmic neutrino ( @xmath1gev ) detectors based on under water / ice muon detection are now at proposal or construction stages @xcite and alternative techniques for @xmath2 gev neutrinos are being considered through coherent radio @xcite or acoustic @xcite pulses as well as through horizontal air shower measurements with conventional arrays @xcite or with fluorescent light either from the ground @xcite or from orbiting detectors @xcite . a number of astrophysical high - energy neutrino sources , such as active galactic nuclei ( agn ) , have been discussed in the literature and predicted to produce fluxes that could be detected in some of these detectors @xcite . in the event of a successful detection of high - energy astrophysical neutrinos the range of parameter space for flavor oscillations that can be tested could be considerably enhanced provided a flavor identification can be done . cosmic tau neutrinos are possibly the easiest flavor to identify above 10@xmath3 gev . two ideas have already been put forward based on the short decay lifetime of the @xmath4 produced in charged current interactions . there is a suggestion of measuring 10@xmath3 gev @xmath5 flux through double shower ( _ double bang _ ) events in under water / ice erenkov telescopes @xcite . a more recent suggestion is to detect a small pile up of upgoing @xmath6-like events in the 10@xmath7 10@xmath8 gev range with a fairly flat zenith angle dependence @xcite . on the other hand the lpm effect , that lengthens electromagnetic showers in water and ice , has been suggested to separate electron neutrino charged current interactions from the rest . this could be done with erenkov light detectors or with the radio technique for energies above @xmath9 gev @xcite . it is conceivable that a combination of several of such techniques will allow the establishment of neutrino flavor ratios at energies above @xmath10 gev . we will concentrate on cosmic tau neutrino detection in this article . specifically , we discuss in some detail , the prospects for detection of high - energy cosmic tau neutrinos originating from the cores of agn . we consider vacuum flavor oscillations as an example to illustrate the possibility offered by detectors in construction to distinguish between different neutrino flavors . for absolute event rates we use upper limit flux calculations as an example @xcite and consider @xmath11km@xmath12 detector sizes which are now being planned @xcite . both the energy ranges of interest and the relative numbers of @xmath4- and @xmath6-like event rates are however independent of the assumed normalization of the neutrino fluxes . we show that for the chosen neutrino flux , a km@xmath13 size surface area under ice / water erenkov light neutrino detector may be able to either set useful upper limits or may obtain first examples of the high - energy tau neutrinos originating from this cosmologically distant astrophysical source . the plan of the paper is as follows : in section ii , after a brief discussion of intrinsic production mechanisms of high - energy muon and tau neutrinos in agns , we estimate the relevant vacuum flavor oscillation probability . in section iii , we discuss in some detail the detection technique making use of the double shower structure of the tau neutrino charged current interactions and calculate the expected event rates for a typical km@xmath12 surface size under water / ice detector . in section iv , we summarize our results . agns are the brightest objects in the sky and high - energy photons reaching tens of thousands of gev have been observed from them . this is commonly interpreted as an indication that some kind of fermi acceleration is taking place . in conventional models , electrons are the particles that get accelerated . it has been argued that if fermi mechanisms are able to accelerate electrons in these objects , protons could also be accelerated by them . in proton acceleration models , the photons arise from neutral pion decays either in @xmath14 or @xmath15 collisions . if this is true , electron and muon neutrinos are also expected to be produced at similar flux levels from charged pion decays . neutrino detection can provide the signature for proton acceleration in agn and it is one of the main goals of neutrino detectors in construction and design stages . for an update review of various @xmath16 and @xmath17 flux estimates from agns in @xmath18 and @xmath15 collisions ( as well as from some other interesting astrophysical / cosmological sites ) , see @xcite . we will here explore the expected tau neutrino fluxes intrinsically produced in the collisions and how these fluxes can vary in a possible neutrino oscillation scenario . in @xmath18 collisions , high - energy @xmath16 and @xmath17 are mainly produced through the resonant reaction @xmath19 . the same collisions will give rise to a greatly suppressed high - energy @xmath5 ( and @xmath20 ) flux mainly through the reaction @xmath21 . the production cross - section for @xmath22 is essentially up to three orders of magnitude lower than that of @xmath23 production for the relevant center of mass energy scale . moreover the branching ratio of @xmath24 to decay eventually into @xmath5 ( @xmath20 ) is approximately two orders of magnitude lower than for @xmath23 to subsequently decay into @xmath16 and @xmath17 through @xmath25 . these two suppression factors along with the relevant kinematic limits give approximately the ratio of intrinsic fluxes of tau neutrinos and muon neutrinos as : @xmath26 . in @xmath15 collisions , the @xmath5 flux may be obtained through @xmath27 . the relatively small cross - section for @xmath22 production together with the low branching ratio into @xmath5 implies that the @xmath5 flux in @xmath15 collisions is also suppressed up to @xmath28 orders of magnitude relative to @xmath16 and/or @xmath17 fluxes . the situation is quite similar to the prompt atmospheric @xmath5 flux calculation ; the result is basically a rescaling of the prompt @xmath17 flux from the decay of charmed @xmath29 s and results in a negligibly small @xmath5 flux for the energies under discussion @xcite . in proton acceleration models the intrinsically produced tau neutrino flux is thus expected to be very small , typically a factor between @xmath30 and @xmath31 relative to electron and muon neutrino fluxes @xcite . however , recent experimental measurements of atmospheric neutrinos suggest that neutrinos could just have vacuum flavor oscillations and the tau neutrino flux would be dramatically enhanced . it has been pointed out that there are no matter effects for high - energy cosmic tau neutrinos originating from cores of agns primarily because of relatively small matter density in the vicinity of core of the agn for all relevant @xmath32 @xcite . we will restrict the following discussion to vacuum oscillations between two flavors , @xmath17 and @xmath5 , for simplicity . the flavor precession probability for non vanishing vacuum mixing angle is obtained from the effective hamiltonian matrix in the two flavor basis @xmath33 : @xmath34 where @xmath35 with @xmath36 and @xmath37 the neutrino energy , leading to the well known result : @xmath38 if we take the values of @xmath39 and @xmath32 suggested by recent superkamiokande data ( @xmath40 ev@xmath13 ) @xcite and @xmath41 100 mpc ( 1 pc @xmath42 cm ) as a representative distance between the agn and our galaxy , then the above rapidly oscillating probability averages out to @xmath43 for all relevant neutrino energies to be considered for detection . very similar fluxes of muon and tau neutrinos would thus be expected . let us further note that after averaging the @xmath44 given by eq . ( 2 ) is independent of not only @xmath37 but also @xmath32 and thus leads to a constant suppression of high - energy cosmic muon neutrino flux . the deficit measured by superkamiokande in atmospheric muon neutrino flux may currently be explained either through @xmath45 or through @xmath46 , where @xmath47 is a sterile neutrino is now being disfavoured @xcite . ] . in the first case and for high - energy neutrinos originating at cosmological distances , the ratio @xmath48 is close to 1/2 . therefore , a ratio different from 1/2 excludes this possibility . high - energy cosmic tau neutrino detection could be achieved by making use of the characteristic double shower events @xcite or by the pileup effect expected as they travel through the earth @xcite . such events could be seen in conventional neutrino telescopes and in principle also with other alternative techniques that have been proposed . we will discuss in some detail the possibility of detecting double shower events for conventional underground telescopes by estimating rates using the the fluxes of ref . @xcite and the oscillation probability addressed in the previous section . this is intended to provide a reference calculation . the downgoing cosmic tau neutrinos reaching close to the surface of the detector may undergo a charged current deep inelastic scattering with nuclei inside / near the detector and produce a tau lepton in addition to a hadronic shower . this tau lepton traverses a distance , on average proportional to its energy , before it decays back into a tau neutrino and a second shower most often induced by decaying hadrons . the second shower is expected to carry about twice as much energy as the first and such double shower signals are commonly referred to as double bangs . as tau leptons are not expected to have further relevant interactions ( with high - energy loss ) in their decay timescale , the two showers should be separated by a _ clean _ @xmath6-like track @xcite . we are going to restrict our estimate to downgoing neutrinos as at these energies tau neutrinos that go through the earth interact . effectively the process of interaction and tau decays can be regarded as an energy degradation to the range @xmath49 gev @xcite . unfortunately the two shower signature will be difficult to be resolved below @xmath50gev ( see below ) . for downgoing cosmic tau neutrinos , the _ double bang _ event rate in water / ice is estimated using @xcite @xmath51 where @xmath52 is the area of the neutrino telescope and @xmath53 gives the probability that a tau neutrino of energy @xmath37 produces two contained and separable showers with the tau lepton energy greater than @xmath54 . it is given by : @xmath55 \frac{\mbox{d}\sigma^{cc}(e , y)}{\mbox{d}y},\ ] ] where @xmath56 is the avogadro s number , @xmath57 is the density of the detector medium , @xmath58 is the charged current @xmath59 differential cross - section , @xmath60 is the fraction of neutrino energy that is transferred to the hadron in the laboratory frame . @xmath29 is the detector length scale which we fix to be @xmath11 km and the tau lepton range @xmath61 which must be contained within @xmath29 is given by : @xmath62 in eq . ( 5 ) , @xmath63 is the lifetime and @xmath64 is the mass of the high - energy tau lepton . the lower limit of integration in eq . ( 3 ) is @xmath65 . we take @xmath66 greater than @xmath67 gev because at this energy the tau lepton range ( separation between the two showers ) in water is @xmath68 100 m which allows a clear separation between the two showers . the upper limit of integration in eq . ( 3 ) is taken to be @xmath69 gev as for energies above it the tau lepton range exceeds the telescope size ( see fig . 1 ) . finally in eq . ( 3 ) , d@xmath70d@xmath37 is the differential high - energy tau neutrino flux and is obtained by multiplying @xmath71 given by eq . ( 2 ) with d@xmath72/d@xmath37 for @xmath73 taken from @xcite . in fig . 2 , we depict downgoing differential event rates for double shower events using the parton distributions mrs r@xmath74 from @xcite for km@xmath13 under water neutrino telescopes as an example . we have checked that other modern parton distributions give quite similar events rates and are therefore not depicted here . we have taken into account the fact that @xmath75 of the times the tau decay does not induce any shower . for comparison we also plot the @xmath76-like event rate induced by muon neutrinos . note that these @xmath77 and the tau neutrino interactions in which the tau lepton decays outside the detector volume have identical ( @xmath76-like ) experimental signature . the signature of double shower events depends on the detector capabilities for shower identification and energy resolution and difficulties can be envisaged . we have used 100 m as the minimum distance to resolve two showers , what is quite conservative in view of typical spacing between optical modules in an under water / ice detector . in fig . 3 , we show the dependence of shower size and shower separation on neutrino energy @xmath37 in ice and in water for which we have used the parametrization of @xcite . as shower size is basically proportional to energy , the size of the second shower is on average a factor of 2 higher than the first one ( see also @xcite ) . this value results by taking into account the relevant kinematics of the allowed decay channels and the corresponding branching ratios and using the average energy transfer @xmath78 . the @xmath60 distribution and decay kinematics will lead to a spread in this ratio . while @xmath79 enhances the energy ratio of the second and first showers to a value of about 6 , for @xmath60 values higher than 0.4 the ratio of the two shower sizes starts to be lower than unity obscuring the tau neutrino signature . another relevant point is the evaluation of the backgrounds , a double shower signature not induced by a tau neutrino . as it was discussed in @xcite such probability is very small and should not affect the detection of the high - energy cosmic tau neutrino . also one should take into account the possibility that the muon component of a single cascade induced from a muon or an electron neutrino charged / neutral current interaction can be confused with the second shower of the tau lepton decay . however , in this case the size of the second shower is smaller than the first one which should be sufficient to distinguish it from a tau neutrino event . thus , the selection criteria of amplitude of second shower greater than the first one typically by a factor of @xmath68 2 , depending on @xmath60 value , essentially makes the observation background free . the high - energy neutrino telescopes have quite small double shower event rates ( yr@xmath80sr@xmath80 ) due to small high - energy intrinsic tau neutrinos flux , thus any observed change in this situation may provide indirect evidence of neutrino mass . a corresponding comparable change in this situation is currently not expected from variations of astrophysical model inputs . the almost simultaneous measurement of the two showers may provide useful information on the incident neutrino energy as well as the @xmath60 distribution . summarizing , in the context of relevant backgrounds , we envisage essentially the simultaneous presence of two types of events ( with different topologies ) serving as background for the tau neutrino induced contained but separable double showers connected by a @xmath6-like track such that the amplitude of the second shower is typically 2 times the first one . the first type of background events are due to relatively long ( @xmath68 10 km ) range muons passing through the detector identified as @xmath6-like tracks . their estimated number is given by the upper slanted curve in fig . the second type of background is the single showers due to charged / neutral current interactions . these may be estimated as 1/10 of the continuous @xmath6-like tracks . thus , the signature of the tau neutrinos as emphasized earlier remained distinct from these two type of backgrounds . we emphasize that within the respective energy window , the essential factor in prospective detection of contained but separable double shower events connected by a @xmath6-like track as a signature of tau neutrinos is the _ difference _ in the incident tau neutrino energy dependences on spread and separation of the two showers . this _ difference _ is also clearly crucial for separating the tau neutrino events from the ( relatively abundent ) @xmath6-like events . the intrinsic fluxes of the high - energy cosmic neutrinos originating from proton acceleration in cores of agns are estimated to have typically the following ratios : @xmath81 . thus , if an enhanced @xmath82 ratio ( as compared to no precession situation ) is observed _ correlated _ to the direction of source for high - energy cosmic neutrinos , then it may be an evidence for vacuum flavor oscillations of neutrinos induced by non zero vacuum mixing angle depending on the finer details of the relevant high - energy cosmic neutrino spectra . for vacuum flavor oscillations of high - energy cosmic neutrinos , the relevant range of neutrino mixing parameters are : @xmath83 ev@xmath13 with @xmath84 . we have identified the incident tau neutrino energy range and the relevant neutrino mixing parameters which may give rise to high - energy cosmic tau neutrino induced downward contained but separable double shower events . for @xmath85 , a km@xmath13 detector may be able to obtain first examples of downgoing high - energy cosmic tau neutrinos through contained but separable double shower events or may at least provide some useful relevant upper limits . the authors acknowledge the financial support from xunta de galicia ( xuga-20602b98 ) and cicyt ( aen96 - 1773 ) . h. a. also thanks agencia espaola de cooperacin internacional ( aeci ) and japan society for the promotion of science ( jsps ) for financial support . g. raffelt in proceedings of _ 1998 summer school in high - energy physics and cosmology _ , edited by g. senjanovic and a. yu . smirnov , ictp , trieste ( italy ) , june / july 1998 ( to be published , hep - ph/9902271 ) and references cited therein . see , for instance , l. moscoso , in _ sixth international workshop on topics in astroparticle and underground physics _ ( taup 99 ) , paris ( france ) , september 1999 [ to appear in its proceedings , edited by m. froissart , j. dumarchez and d. vignaud ; preprint dapnia - spp-00 - 01 ( january 2000 ) ] . see , for a latest discussion , j. alvarez - muiz and e. zas , phys . b * 434 * , 396 ( 1998 ) . l. g. dedenko _ et al_. , in proceedings of _ 25th international cosmic ray conference _ , edited by m. s. potgieter , b. c. raubenheimer and d. j. van der walt , durban , south africa , p. 89 ( vol . 7 ) . k. s. capelle , j.w . cronin , g. parente and e. zas , astropart . * 8 * , 321 ( 1998 ) . s. c. corbato _ et al_. , nucl . b ( proc . suppl . ) * 28 * , 36 ( 1992 ) . n. hayashida _ et al_. ( telescope array collaboration ) , astro - ph/9804043 . see , for instance , g. domokos and s. kovesi - domokos , hep - ph/9801362 ; hep - ph/9805221 . see also , d. fargion , astro - ph/0002453 and references cited therein . thomas k. gaisser , francis halzen and todor stanev , phys . rep . * 258 * , 173 ( 1995 ) ; * 271 * , 355(e ) ( 1996 ) . j. g. learned and s. pakvasa , astropart . * 3 * , 267 ( 1995 ) . f. halzen and d. saltzberg , phys . * 81 * , 4305 ( 1998 ) . for some recent discussions , see , s. i. dutta , m. h. reno and i. sarcevic , hep - ph/0005310 ; j. alvarez - muiz , f. halzen and d. w. hooper , astro - ph/0006027 . j. alvarez - muiz , r. a. v ' azquez and e. zas , phys . rev . d * 61 * , 023001 ( 2000 ) . a. p. szabo and r. j. protheroe , astropart . * 2 * , 375 ( 1994 ) . see , for instance , r. j. protheroe , nucl . b ( proc . suppl . ) * 77 * , 465 ( 1999 ) and references cited therein . f. halzen , b. keszthelyi and e. zas , phys . d * 52 * , 3239 ( 1995 ) ; l. pasquali and m. h. reno , phys . d * 59 * , 093003 ( 1999 ) . m. c. gonzalez - garcia and j. j. gomez - cadenas , phys . d * 55 * , 1297 ( 1997 ) . a more detailed numerical study supports this estimate ; h. athar , r. a. v ' azquez and e. zas ( in preparation ) . see , for example , h. athar , m. jeabek and o. yasuda , hep - ph/0005104 and references cited therein . y. fukuda _ et al_. , phys . lett . * 81 * , 1562 ( 1998 ) ; phys . b * 433 * , 9 ( 1998 ) ; * 436 * , 33 ( 1998 ) ; * 467 * , 185 ( 1999 ) . m. nakahata , talk given at _ sixth international workshop on topics in astroparticle and underground physics _ ( taup 99 ) , paris , france ( to appear in its proceedings , edited by m. froissart , j. dumarchez and d. vignaud ) . r. gandhi , c. quigg , m. h. reno , and i. sarcevic , astropart . phys . * 5 * , 81 ( 1996 ) ; phys . d * 58 * , 093009 ( 1998 ) . a. d. martins , r. g. roberts and w. j. stirling , phys . b * 387 * , 419 ( 1996 ) . t. k. gaisser , _ cosmic rays and particle physics _ ( cambridge university press , cambridge , 1990 ) and references cited therein .

we study prospects for the observations of high - energy cosmic tau neutrinos ( @xmath0 gev ) originating from proton acceleration in the cores of active galactic nuclei . we consider the possibility that vacuum flavor neutrino oscillations induce a tau to muon neutrino flux ratio greatly exceeding the rather small value expected from intrinsic production . the criterias and event rates for under water / ice light erenkov neutrino telescopes are given by considering the possible detection of downgoing high - energy cosmic tau neutrinos through characteristic double shower events .

astronomers and planetary scientists are just beginning to understand the atmospheres of the short period giant planets known as `` hot jupiters '' or `` pegasi planets . '' a key subset of these planets are those that transit the disk of their parent stars , which make them well - suited for follow - up studies . the most well studied of these transiting hot jupiters is the first to be discovered , hd 209458b @xcite , a 0.69 @xmath1 planet that orbits its sun - like parent star at a distance of 0.045 au . currently , considerable work on hot jupiters is occuring on both the observational and theoretical fronts . in the past few years , several groups have computed dynamical atmosphere models for hd 209458b in an effort to understand the structure , winds , and temperature contrasts of the planet s atmosphere @xcite . if the planet has been tidally de - spun and become locked to its parent star , dynamical models are surely needed to understand the extent to which absorbed stellar energy is transported onto the planet s permanent night side . with the launch of the _ spitzer space telescope _ , there is now a platform that is well - suited for observations of the thermal emission from hot jupiter planets . _ spitzer _ observations spanning the time of the planet s secondary eclipse ( when the planet passes behind its parent star ) have been published for hd 209458b at 24 @xmath0 m @xcite , tres-1 at 4.5 and 8.0 @xmath0 m @xcite , and hd 189733b at 16@xmath0 m @xcite . in all cases , the observed quantity is the planet - to - star flux ratio in _ infared array camera ( irac ) , infrared spectrograph ( irs ) , or multiband imaging photometer for _ spitzer _ ( mips ) bands . the dual tres-1 observations are especially interesting because they allow for a determination of the planet s mid - infrared spectral slope . several efforts have also been made from the ground to observe the secondary eclipse of hd 209458b . although no detections have been made , some important , occasionallly overlooked upper limits at k and l band have been obtained . these include @xcite around 2.3 @xmath0 m , @xcite in k band , and @xcite at l band . these ground - based observations constrain the flux emitted by the planet in spectral bands where water vapor opacity is expected to be minimal ; therefore , emitted flux should be high . this influx of data has spurred a new generation of radiative - convective equilibrium models , whose resulting infrared spectra can be compared with data @xcite . see @xcite and @xcite for reviews . the majority of these models are one - dimensional . authors weight the incident stellar flux by 1/4 , to simulate planet - wide average conditions , or by 1/2 , to simulate day - side average conditions ( with a cold night side ) . @xcite have investigated two - dimensional models with axial symmetry around the planet s substellar - antistellar axis and computed infrared spectra as a function of orbital phase . @xcite have extended one - dimensional models by adding heat transport due to a simple parametrization of winds to generate longitude - dependent temperature maps , but they did not compute disk averaged spectra for these models . very recently @xcite have also investigated spectra and light curves of planets with various day - night effective temperature differences , assuming 1d profiles for each hemisphere . these one- and two - dimensional radiative - convective equilibrium models have had some success in matching _ spitzer _ observations , but @xcite and @xcite have shown that ground - based data for hd 209458b do not indicate promiment flux peaks at 2.3 and 3.8 @xmath0 m , which solar composition models predict . the various dynamical models for hd 209458b @xcite are quite varied in their treatment of the planet s atmosphere . we will not review them here , as that is not the focus of this paper . in general , temperature contrasts in the visible atmosphere are expected to be somewhere between 300 and 1000 k. what has been somewhat lacking for these dynamical models are clear observational signatures , which would in principle be testable with _ spitzer _ or other telescopes . the purpose of this paper is to remedy that situation . here we generate infrared spectra and light curves as a function of orbital phase for the @xcite ( hereafter : cs06 ) dynamical simulation . we present the first spectra generated for three - dimensional models of the atmosphere of hd 209458b . here we take the first step towards understanding the effects of atmospheric dynamics on the infrared spectra of hot jupiters . a consistent treatment of coupled atmospheric dynamics , non - equilibrium chemistry , and radiative transfer would be a considerable task . in a coupled scheme , given a three - dimensional _ p t _ grid at a given time step , with corresponding chemical mixing ratios , the radiative transfer scheme would solve for the upward and downward fluxes in each layer . these fluxes would be wavelength dependent and would differ from layer to layer . the thermodynamical heating / cooling rate , which is the vertical divergence of the net flux , would then be calculated . the dynamics scheme would then use this heating rate , together with the velocities and _ p t _ profiles in the grid at the previous time step , to a calculate the chemical abundances , velocities , and _ _ profiles on the grid at the new time step . the process steps forward in time , as the radiative transfer solver again finds the new heating / cooling rates . the emergent spectrum of the planet could be found at any stage . in our work presented here , we performed a simplified calculation that contains some aspects of what will eventually be included in a fully consistent treatment . our input pressure - temperature ( _ p t _ ) map is from cs06 . as the dynamical simulations are described in depth in @xcite and cs06 , we will only give an overview here . the cs06 model employs the aries / geos dynamical core , version 2 ( agdc2 ; @xcite ) . the agdc2 solves the primitive equations of dynamical meteorology , which are the foundation of numerous climate and numerical weather prediction models @xcite . the primitive equations simplify the navier - stokes equations of fluid mechanics by assuming hydrostatic balance of each vertical column of atmosphere . forcing is due to incident flux from the parent star , through a newtonian radiative process described in cs06 . the cs06 model is forced from the one - dimensional radiative - convective equilibrium profile of @xcite , which assumes globally averaged planetary conditions . for their simulations , cs06 take the top layer of the model to be 1 mbar . the model atmosphere spans @xmath215 pressure scale heights between the input top layer and the bottom boundary at 3 kbar . cs06 use 40 layers evenly spaced in log pressure . a _ p profile is generated at locations evenly spaced in longitude ( in 5 degree increments ) and latitude ( in 4 degree increments ) . the 72 longitude and 44 latitude points create 3168 _ p t _ profiles . in we show the cs06 grid at 3 pressure levels , near @xmath22 , 20 , and 200 mbar . previous work has shown these levels likely bracket the pressures of interest for forming mid - infrared spectra @xcite . all three panels of use the same brightness scale for easy comparison between pressure levels . at the 2 mbar level , where radiative time constants are short @xcite , the atmosphere responds quickly to incident radiation . winds , though reaching a speed of up to 8 km s@xmath3 , are not fast enough to lead to significant deviations from a static atmosphere , which implies that the hottest regions remain at the substellar point . the arrows indicate a wind pattern that attempts to carry energy radially away from the hot spot . at this pressure , the atmosphere somewhat resembles the two - dimensional radially symmetric static atmosphere of @xcite , who found a hot spot at the substellar point and a uniformly decreasing temperature as radial distance from this point increased . the night side appears nearly uniform , and colder . at the 25 mbar level , it is clear that a west - to - east circulation pattern has emerged at the equator , and the center of the planet s warm region has been blown downstream by @xmath235 degrees . the wind from the west dominates over the predominantly radially outward wind seen at the 2 mbar level . day - night temperature contrasts are not as large at this pressure as they are at 2 mbar . at the 220 mbar level , the center of the hot spot has been blown downstream by @xmath260 degrees . this jet extends from the equator to the mid - latitudes ; the gas in the jet is warmer than gas to the north or south . the radiative time constants become longer the deeper one goes into the atmosphere @xcite . hence , winds are better able to redistribute energy , leading to weaker temperature contrasts , which can not simply be characterized as day - night . " unlike other published models , we stress that the spectra generated here are from a dynamical atmosphere model that is _ not _ in radiative - convective equilibrium . each of the 3168 _ p t _ profiles from cs06 have , without modification ( aside from interpolation onto a different pressure grid ) , been run through our radiative transfer solver . no iteration is done to achieve radiative equilibrium . the equation of radiative transfer is solved with the two - stream source function technique described in @xcite . this is the same infrared radiative transfer scheme used in @xcite , @xcite and m. s. marley et al . ( in prep . ) . we ignore contributions due to scattered stellar photons , as discussed below . at a given orbital position , the cs06 map in longitude is re - mapped into an apparent longitude ( as seen from earth ) , while the latitude remains unchanged . see for a diagram . here we ignore the 3.4 degrees that the orbit differs from being exactly edge - on @xcite . at a given orbital angle @xmath4for each patch of the planet the cosine of the angle @xmath5 from the sub - observer point , @xmath0 , is calculated . this @xmath0 is consistently included when solving the radiative transfer , which means that the effects of limb darkening ( or brightening ) are automatically incorporated . we interpolate in a pressure - temperature - abundance grid @xcite , such that any given point in the three - dimensional model has the atomic and molecular abundances that are consistent with that point s pressure and temperature . at any given time , the one - half of the planet that is not visible is not included in the radiative transfer . the 1584 visible points , at which the emergent specific intensity ( erg s@xmath3 @xmath6 hz@xmath3 sr@xmath3 ) is calculated , are then weighted by the apparent visible area of their respective patches . these intensities are summed up to give the total emergent flux density ( erg s@xmath3 @xmath6 hz@xmath3 ) from the planet . the spectra generated by the patch - by - patch version of the code were tested against spectra from one - dimensional gray atmospheres and our previously published hot jupiter profiles . emergent spectra are calculated from 0.26 to 325 @xmath0 m , but since we ignore the contribution due to scattered stellar flux , the spectra at the very shortest wavelengths have little meaning . however , for the radiative - equilibrium hd 209458b model published in @xcite , they found that scattered stellar flux is greater than thermal emission only at wavelengths less than @xmath20.68 @xmath0 m . we expect that a considerable amount of visible " light that may eventually be seen from hot jupiters is due to thermal emission . we note that by 1 @xmath0 m thermal emission is 100 times greater than scattered flux . here we will present spectra for wavelengths from 1 to 30 @xmath0 m . we note that the radiative transfer at every point is solved in the plane - parallel approximation . while this treatment is sufficient for our purposes , we wish to point out two drawbacks . the first is that , near the limb of the planet , we will tend to overestimate the path lengths of photons emerging from the atmosphere , as the curvature of the atmosphere is neglected . the second issue also occurs near the limb . we can not treat photons whose path , in a completely correct treatment , would start in one column but emerge from an adjoining column . however , atmospheric properties in any two adjoining columns are in general quite similar . the former issue , that of the plane - parallel approximation , is likely more important , and should be addressed at a later time , when data precision warrants it . here we note that in our tests 95% of planetary flux emerges from within 75 degrees of the sub - observer point , such that these limb effects will have little effect on the disk - summed spectra and light curves that we present . when generating our model spectra , we use the elemental abundance data of @xcite and chemical equilibrium compositions computed with the condor code , as described in @xcite , @xcite , and @xcite . in 2.3 we will discuss deviations from equilibrium chemistry in the atmosphere of hd 209458b , as calculated by cs06 . at this time , we ignore photochemistry , which has been shown by @xcite to be reasonable at @xmath7 mbar . for the most part , the infrared spectra of hot jupiters are sensitive to opacity at @xmath8 mbar @xcite , so equilibrium chemistry calculations are probably sufficient . as discussed in @xcite , we maintain a large and constantly updated opacity database , which is described in detail in r. s. freedman & k. lodders ( in prep . ) . the pressure , temperature , composition , and wavelength - dependent opacity is tabulated beforehand using the correlated - k method @xcite in 196 wavelength interval bins . the resulting spectra are therefore of low resolution . however , low resolution is suitable for the task at hand , as we are interested in band - integrated fluxes and the radiative transfer must be solved at 1584 locations on the planet at many ( here , 36 ) orbital phases . in @xcite , which focused on examining asymmetrical secondary eclipse light curves caused by dynamical redistribution of flux , we investigated the effects of limb darkening for hd 209458b for the cs06 map . these effects are hard to disentangle from general brightness variations due to temperature differences generated by dynamics . limb darkening in a particular wavelength would be manifested as a brightness temperature that decreases towards the limb relative to a brightness temperature map computed assuming normal incidence at every point . on the planet s day side , where the _ p t _ profiles are somewhat isothermal , limb darkening is not expected to be significant . indeed , only at angles greater than @xmath280 degrees from the subsolar point was day - side limb darkening as large as 100200 k calculated . as noted , due to our plane - parallel approximation , this is likely to be somewhat of an overestimation . before calculating spectra for this dynamic atmosphere , it is worthwhile to step back and look at the atmospheric _ p t _ profiles with an eye towards understanding the effects that clouds and chemistry may have on the emergent spectra . in , we plot a random sampling of the 3168 _ p t _ profiles and compare them to cloud condensation and chemical equilibrium boundaries . it should be noted that there is a greater density of profiles on the left side of the plot . these are profiles from the relatively uniform and cool night hemisphere . also of note is that at pressures less than @xmath2200 mbar , the warmer profiles are fairly isothermal , with quite shallow temperature gradients . the cs06 profiles are nearly everywhere cooler than the condensation curves of iron and mg - silicates . these clouds will form , but for these profiles , cloud bases would lie deep in the atmosphere at pressures greater than 1 kbar , as this is the highest pressure at which the profiles cross the condensation curves of these species . the opacity from such deep clouds would have _ no effect _ on the spectrum of hd 209458b . the temperatures of the cs06 simulation are computed as departures from the equilibrium temperature profile of @xcite , as described in cs06 . if a warmer base profile had been selected , such as @xcite or @xcite ( which are 100 - 300 k warmer at our pressures of interest ) , these curves could be shifted to the right by 100 - 300 degrees . @xcite provide a graphical comparison of profiles computed by @xcite , @xcite , and @xcite under similar assumptions concerning redistribution of stellar flux and atmospheric abundances ; they find differences of up to 300 k at 100 mbar . these differences can probably be attributed to different opacity databases , molecular abundances , radiative transfer methods , and perhaps incident stellar fluxes . for the @xcite profile , the major heating species are h@xmath9o bands from 1 to 3 @xmath0 m and neutral atomic na and k , which absorb strongly in the optical . the major cooling species are co , at 5 @xmath0 m , and h@xmath9o in bands at 3 and 4 - 10 @xmath0 m , although for a colder profile , such as cs06 night side profiles , ch@xmath10 would also cool across these wavelengths . cs06 chose the @xcite profile because iro et al . also computed atmospheric radiative time constants , which other authors have not done to date . here , the computed night side profiles cross the condensation curve of na@xmath9s at low pressure . @xcite and @xcite pointed out that the transit detection of weak neutral atomic na absorption by @xcite could be explained if a large fraction of na was tied up in this condensate . if the predictions of equilibrium chemistry hold for this atmosphere , then it would appear that at pressures up to 100 mbar , ch@xmath10 would be the main carbon carrier on the planet s night - side , while co would be the main carrier on the day side . this would lead to dramatically different spectra for these hemispheres , especially at near - infrared and mid - infrared wavelengths . however , equilibrium chemical abundances would be expected only if chemical timescales are faster than any mixing timescales . the relative abundances of co and ch@xmath10 can be driven out of equilibrium if mixing timescales due to dynamical winds are faster than the timescale for relaxation to equilibrium of the ( net ) reaction @xmath11 the time constant for chemical relaxation toward equilibrium is a strong function of temperature and pressure ; it is short in the deep interior and extremely long in the observable atmosphere . deviations from ch@xmath10/co equilibrium are due to the long timescale for conversion of co to ch@xmath10 , and are well known in the atmospheres of the jovian planets @xcite and brown dwarfs @xcite , where vertical mixing can be due to convection and/or eddy diffusion . cs06 follow timescales for ch@xmath10/co chemistry taken from @xcite and find that vertical winds of 5 - 10 m sec@xmath3 push this system out of equilibrium at pressures of interest for the formation of infrared spectra . they find that the quench level , where the mixing timescale and chemistry timescale are equal , is @xmath21 - 10 bar , which is below the visible atmosphere . above this quench level , the mole fractions of co and ch@xmath10 are essentially constant . they find that the ch@xmath10/co ratio becomes homogenized at pressures less than 1 bar _ everywhere _ on the planet . for their nomical models this ch@xmath10/co ratio is @xmath20.014 , meaning that the majority of carbon is indeed in co. however , a non - negligible fraction of carbon remains in ch@xmath10 on both the day and the night hemispheres . we note that cs06 ignore possible effects due to photochemistry on the co and ch@xmath10 mixing ratios . given the few explorations into hot jupiter carbon and oxygen photochemistry to date @xcite , it is unclear how important photochemistry will be in determining the abundances of these species at pressures of tens to hundreds of millibar . cs06 did explore other effects such at atmospheric metallicity and temperature . if the atmosphere is greater than [ m / h]=0.0 , or if the atmosphere is hotter , the ch@xmath10/co ratio would be even smaller . see cs06 for additional discussion . in our spectral calculations , we find that significant differences arise depending on our treatment of chemistry . we will therefore investigate the effects of a few chemistry cases , as explained below , and shown in table 1 . we label our equilibrium chemistry trial `` case 0 , '' as it is the standard case . for `` case 1 , '' we fix the ch@xmath10/co ratio at 0.014 ( as found by cs06 ) at all temperatures and pressures along the profiles . consistently incorporating the increasing ch@xmath10/co ratio at depth ( @xmath12 bar ) would be difficult with previously tabulated opacities , and would have little to no effect on the emergent spectra . recall that 23% of available oxygen is lost to the formation of mg - silicate clouds @xcite , which in this model have cloud bases near 1 kbar . the remaining oxygen is almost entirely found in co and h@xmath9o . as the ch@xmath10 and co abundances are fixed , for case 1 we will also fix the abundance of h@xmath9o ; the mixing ratios of ch@xmath10 , co and h@xmath9o are consistent with the amount of available oxygen at t@xmath2 1200 k and pressures of tens of millibars . we also briefly consider a `` case 2 , '' in which the ch@xmath10/co ratio has been further reduced , in an ad - hoc manner , which could be due to quenching of the abundances at a hotter temperature or the photochemical destruction of ch@xmath10 . the ch@xmath10/co ratio is reduced by nearly a factor of 500 , to @xmath13 . the mixing ratios of co and h@xmath9o are nearly the same as in case 1 , although the slightly increased co abundance ( due to the drop in the abundance of ch@xmath10 and conservation of carbon atoms ) uses up some oxygen at the expense of h@xmath9o . this case is valuable for comparison purposes because it shows the spectral effects of a negligible ch@xmath10 abundance . in all cases , the mixing ratios of all other chemical species are given by equilibrium values . however , only ch@xmath10 , co , and h@xmath9o have a discernable impact on the spectra . in we plot mixing ratios of ch@xmath10 , co , and h@xmath9o predicted from equilibrium chemistry along three _ profiles in the atmosphere of hd 209458b . in gray are the mixing ratios for the one - dimensional profile of @xcite . the abundance of ch@xmath10 is negligible . in thin black and thick black are the mixing ratios at the sub - stellar point and the anti - stellar point , respectively , of the cs06 simulation . the dominant carbon carrier is clearly co , rather than ch@xmath10 , except at @xmath14100 mbar for the anti - stellar point . most of the nightside has a chemistry profile similar to the anti - stellar point , and hence , strong ch@xmath10 absorption will be seen . at the top of the plot , arrows indicate the assigned mixing ratios of cases 1 and 2 for these three molecules . in 3 we make quantitative comparisons to current ground - based and space - based infrared data . while we find that the agreement is good for the model presented here , we wish to stress that this study is exploratory . this is the first study that quantitatively explores the influence of atmospheric dynamics on the emergent spectra of a hot jupiter atmosphere . the greatest uncertainty likely lies in the calculation of heating / cooling rates with a simple newtonian cooling scheme in the dynamical model , as discussed in @xcite and cs06 . another issue is that cs06 calculate temperature deviations from a one dimensional radiative - convective profile published in @xcite . however , our emergent spectrum is calculated using our radiative transfer solver , which is a different code , and we use different abundances @xcite than these authors @xcite . however , we find that when solving the radiative transfer for the @xcite profile we obtain a @xmath15 that differs by only 1% from their value . in their non - equilibrium chemistry work , cs06 chose the abundances of @xcite , while here we use @xcite , in order for the most natural comparison with our previous work @xcite obviously no choice would be fully self consistent . however , the choice of elemental abundances is certainly a smaller concern than the current debate concerning the correct chemical timescale for conversion of co to ch@xmath10 in planetary atmospheres ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in addion , cs06 have shown that a 300 k decrease in temperature , for instance , leads to a 20-fold increase in ch@xmath10/co ( see 5.3 ) , so temperature uncertainties likely swamp any abundance issues . we stress that our focus in the following sections is on highlighting the spectral differences between a model that accounts for dynamical redistribution of energy around the planet and one that does not . indeed the magnitude of the spectral effect suggests that additional , more internally self consistent work , is clearly appropriate . we now turn to the predicted infrared spectrum of our dynamical model atmosphere . as has been shown by many authors since @xcite , the infrared spectra of hot jupiters are believed to be carved predominantly by absorption by h@xmath9o , co , and , if temperatures are cool enough , ch@xmath10 . in general , absorption features of hot jupiters are predicted to be shallower than brown dwarfs of similar effective temperature and abundances . this is predicated on hot jupiters having a significantly shallower atmospheric temperature gradient , which is due to the intense external irradiation by the parent star . shows absorption cross sections per molecule for ch@xmath10 , co , and h@xmath9o from 1 to 30 @xmath0 m . to avoid clutter on our spectral plots , we will not label absorption features , so referring to will be helpful . our first spectral calculation is for case 0 chemistry , as a function of orbital phase . this is shown in . we show the emitted spectrum at six orbital locations : zero ( during the transit ) , 60 , 120 , 180 ( during the secondary eclipse ) , 240 , and 300 degrees , where the degrees are orbital degrees after transit . at zero degrees ( the black spectrum ) , we see the full night side of the planet . the temperature gradient is fairly steep , leading to deep absorption features . most carbon is in the form of ch@xmath10 , leading to strong methane absorption in bands centered on 1.6 , 2.2 , 3.3 , and 7.8 @xmath0 m . absorption due to h@xmath9o at many wavelengths is also strong . absorption due to co at 2.3 and 4.5 @xmath0 m is quite weak , due to its small mixing ratio . sixty degrees later ( the red spectrum ) , the hemisphere we see is 2/3 night and 1/3 day . the part of the day hemisphere we see has the hot west - to - east jet coming towards us . this will later be an interesting point of comparison with the 300 orbital degree spectrum ( in magenta ) , which is also 2/3 night . here all absorption features are muted because , due to the jet shown in , more than @xmath21/3 of the planet is showing the nearly isothermal profiles representative of the day side . these profiles lead to a blackbody - like spectrum . at 120 orbital degrees ( in green ) , we are seeing 2/3 day side , and that part of the day side is the hottest , due to the wind from the west blowing the hot spot downstream . the absorption features have become very weak . the peak in total infrared flux actually occurs 153 orbital degrees after transit . during the secondary eclipse , 27 degrees later , the planet is less luminous . during the secondary eclipse ( in dark blue ) the planetary spectrum is essentially featureless long - ward of 1.8 @xmath0 m . the fully illuminated hemisphere possesses profiles that are nearly isothermal in our pressure range of interest . intriguingly , this leads to a spectrum that is very similar to a blackbody . for comparison , we plot a dashed black curve that is the thermal emission of a 1330 k blackbody . it plots behind the 180 degree spectrum for all @xmath16 @xmath0 m . as we move to orbital phases after the secondary eclipse , we see that the spectra are not symmetric with their reflections on the other half of the orbit . the light blue spectrum is significantly lower than the green one , even though at both points we are seeing 2/3 day and 1/3 night . this is again due to the west - to - east jet . at 240 orbital degrees , the hottest part of the day side is not in view . at 300 orbital degrees ( in magenta ) 1/3 of our view is the coolest part of the day side , along with 2/3 of the night side . the spectrum is quite similar to what was seen during the transit . the planet appears least luminous 31 orbital degrees before transit . since cs06 find significant deviations from equilibrium carbon chemistry , it is useful to examine spectra with non - equilibrium chemistry we now investigate the spectra using cases 1 and 2 . shows comparison spectra at two orbital phases : zero and 180 orbital degrees , which is during transit and secondary eclipse , respectively . on the day side , we see that the emergent spectrum is essentially exactly the same , independent of chemistry . because the _ p t _ profiles are nearly isothermal , essentially no information about the chemistry can be gleaned from the spectrum . for comparison with our previous work , in , we plot with a dotted line the cloudless planet - wide average spectrum of hd 209458b from @xcite , which shows absorption due to h@xmath9o and co and strong flux peaks between these absorption features . we will now examine the small inset in . this is a zoom - in of the secondary - eclipse spectra near 2.2 @xmath0 m . the three thick horizontal bars connected by dashes make up an interesting observational one - sigma upper limit that was first reported in @xcite . these data were analyzed again and presented in @xcite . the measurement was a relative one ; only the vertical distance between the central band and two side bands is important : they can be moved up or down as a group . specifically , the flux in the central band can not exceed the combined flux in the two adjacent bands by an amount greater than vertical distance shown ( which is @xmath17 erg s@xmath3 @xmath6 hz@xmath3 ) . our dynamical day - side spectra , which are nearly featureless due to the nearly isothermal atmosphere , easily meet this constraint . we note that an upper limit obtained by @xcite in k band , which covers a similar wavelength range , has an error bar that is too large to distinguish among published models . in contrast to the models presented here that include dynamics , published 1d models to date all predict large flux differences between 2.2 @xmath0 m and the surrounding spectral regions when abundances are solar . the 1d models with solar abundances presented in @xcite , for example , were not able to meet the 2.2-@xmath0 m flux - difference constraint , because the h@xmath9o bands adjoining the 2.2-@xmath0 m flux peak were too deep . @xcite suggested that if c / o @xmath18 1 , then little h@xmath9o would exist in the atmosphere , and the observational constraint could be met . here , we propose instead that dynamics produces a relatively isothermal dayside temperature , in which case the constraint can easily be met even with solar c / o . nevertheless , the observational constraint is not yet firm enough to fully rule out the 1d radiative - equilibrium models at solar abundances . the @xcite solar - abundance models , for example , marginally meet the current observational flux - difference constraint . ( fortney et al . 2005 find a flux difference of @xmath19 erg s@xmath3 @xmath6 hz@xmath3 . ) the cause for the discrepancy between the 1d models in @xcite and @xcite remains unclear but could occur if the _ p t _ profiles in seager et al . were steeper than those in fortney et al . , leading to deeper bands in the former study . more precise flux - difference observations like that of @xcite during secondary eclipse would help confirm or rule out the 1d solar - abundance models . even a factor of two decrease in the observational upper limit between the `` in '' and `` out '' bands could rule out the solar - abundance @xcite model , while supporting the isothermal - dayside models ( e.g. , cs06 ) and the c / o @xmath18 1 models with weak water bands . alternatively , the detection of a small flux peak at 2.2 @xmath0 m would lend support to the 1d models with c / o @xmath20 1 , while constraining the temperature gradient on the planet s dayside . the fact that both the solar - abundance cs06 models ( presented here ) and the radiative - equilibrium 1d models with c / o @xmath18 1 produce almost no flux difference between 2.2 @xmath0 m and the surrounding continuum implies that the 2.2-@xmath0 m band can not be used to distinguish between these alternatives . instead , additional constraints at other wavelengths will be necessary to discriminate between them . measurements of flux differences surrounding co ( rather than h@xmath9o ) bands could provide such a test . the radiative - equilibrium c / o @xmath18 1 models , although lacking water bands , would presumably have strong co bands and would hence predict a strong flux difference between the center of a co band and the surrounding continuum . on the other hand , because the dayside atmosphere is nearly isothermal in the cs06 models , these circulation - altered models would predict little flux difference between the co band and the surrounding regions even in the presence of large quantities of co. an observation of minimal flux difference across co bands as well as h@xmath9o bands would support isothermal - dayside models like cs06 while arguing against the radiative - equilibrium 1d models with steep temperature gradients . although observations of tres-1 may not be directly applicable to hd 209458b , as tres-1 may have a @xmath15 300 k cooler , there is an important issue to note for our discussion here . the models of @xcite , @xcite , and @xcite show a mid - infrared spectral slope that is bluer than observed from 4.5 to 8.0 @xmath0 m by @xcite for tres-1 , although the model of @xcite appears to be consistent with this slope . @xcite and @xcite have discussed that models with an atmospheric temperature inversion would give infrared spectra with a redder spectral slope , due to molecular emission features , in better agreement with observations . the cs06 model of the day side of hd 209458b , which lacks the strong negative temperature gradient of these radiative equilibrium models , also leads to a redder mid - infrared spectral slope . on the planet s night side , we see significant spectral differences between the three chemistry cases . in cases 1 and 2 , the ch@xmath10 mixing ratio is constrained to a small abundance , weakening these absorption features . absorption due to ch@xmath10 , co , and h@xmath9o is readily seen in case 1 . case 2 shows essentially the same co and h@xmath9o absorption , but ch@xmath10 absorption is no longer seen . the transition , as a function of orbital phase from deep to essentially nonexistent absorption features in cases 1 and 2 , are similar to what was seen in case 0 . in the interest of conciseness , and because case 2 is ad - hoc , hereafter we concentrate on cases 0 and 1 . the spectra for cases 1 and 2 are essentially the same , except in regions of ch@xmath10 absorption ; we will highlight these differences when necessary . it is important to remember that in case 1 and case 2 chemistry , the mixing ratios of our principal absorbers , ch@xmath10 , co , and h@xmath9o are fixed . therefore , changes in the spectra with orbital phase are only due to changes in the planetary _ p _ profiles on the visible disk , due to the rotation of the planet through its orbit . one can integrate the spectrum of the planet s visible hemisphere over all wavelengths , as a function of orbital phase , to determine the apparent luminosity of the planet at all phases . here we divide out @xmath21 to calculate the apparent effective temperature ( @xmath15 ) of the visible hemisphere . this is plotted in for cases 0 and 1 . we can clearly see that in both cases , the time of maximum flux precedes the time of secondary eclipse by @xmath227 degrees , or 6.3 hours . since the spectra of the two cases overlap around the time of secondary eclipse , so do the plots of @xmath15 . at other orbital phases , the case 0 curve always plots lower . the largest effect is before the time of transit . the effect is tied to the ch@xmath10 abundance . if ch@xmath10 is able to attain a large mixing ratio , it leads to an atmosphere that has a higher opacity , meaning one can not see as deeply into the atmosphere . one then reaches an optical depth of @xmath21 higher in the atmosphere , which is significantly cooler for night - side _ profiles , leading to a lower @xmath15 . the point of minimum planetary flux precedes the transit by 31 to 37 degrees , depending on the chemistry . in their ( very similar ) previous dynamical model , @xcite predicted a time of maximum planetary flux of 60 degrees ( or 14 hours ) before secondary eclipse . the large timing difference between that work and this one is due almost entirely to the choice of photospheric pressure " made in @xcite . they chose a pressure of 220 mbar in their work , which is deeper than the mean " photosphere that we find here . at higher pressure , the radiative timescales are longer , such that winds are able to blow the atmosphere s hottest point farther downstream . @xcite and @xcite previously discussed how their prediction varied as a function of the chosen photospheric pressure . for the foreseeable future , the most precise data for understanding the atmospheres of hot jupiters will come from the _ spitzer space telescope_. for hd 209458b , only an observation at 24 @xmath0 m ( the shortest wavelength mips band ) , has been published . it seems likely that observations in all four irac bands ( 3.6 , 4.5 , 5.8 , and 8.0 @xmath0 m ) , as well as irs at 16 @xmath0 m , will be obtained within the next year or two . as such , we have integrated the spectra of our planet models and a @xcite model of star hd 209458 over the _ spitzer _ bands in order to generate planet - to - star flux ratios as a function of orbital phase . these are plotted for cases 0 and 1 in . the stellar model fits the stellar parameters derived in @xcite and is the same model used in @xcite . the behavior of the planet - to - star flux ratios is quite interesting . while the @xmath15 of the planet was found to reach a maximum 27 degrees before secondary eclipse , the behavior in individual bands is more varied . for instance , in both chemistry cases , the planetary flux in the 24 @xmath0 m band peaks only 15 degrees before secondary eclipse . this is because the `` photospheric pressure '' is at a lower pressure in this band than the planet s `` mean photospheric pressure . '' as previously discussed , at lower pressures , the radiative time constants are shorter , and the atmosphere is able to more quickly adjust to changes in incident flux . at higher pressure , winds are better able to blow the planet s hot spot downstream . one should keep in mind that the light curves generated are a function of the dynamical calculation and the radiative transfer . it is the radiative transfer calculation that determines how deeply into the atmosphere ( and therefore , to what temperature ) we see . the 3.6 @xmath0 m band peaks earliest , @xmath227 degrees before transit , as the @xmath15 does as well . this band shows a 15-fold variation in flux ( peak to trough ) as a function of orbital phase for case 0 because it encompasses a significant ch@xmath10 absorption feature that waxes and wanes . since the abundances of ch@xmath10 and co are not free to vary in case 1 , the flux ratios in this case do not show the large amplitudes found in some bands in case 0 . the dotted lines in the case 1 panels are for case 2 in the 3.6 and 8.0 @xmath0 m bands , where ch@xmath10 absorption occurs . the flux variation in these bands is even further reduced as ch@xmath10 absorption is not seen . ( see . ) at 24 @xmath0 m , our model is 1.6 sigma higher than the secondary eclipse data point published by @xcite , indicating that the planet may be dimmer at 180 orbital degrees than we predict . in all bands , differences in chemistry between the two cases have essentially no effect on the timing of the maxima in planetary flux . however , since the night side is much more sensitive to chemistry , the minima can vary by as much as 20 degrees in bands that are sensitive to ch@xmath10 absorption . the results of ground - based observations of flux from hot jupiters have been mixed . all searches for visible light have yielded only upper limits @xcite , which have ruled out some models with bright reflecting clouds . in the near infrared , specifically for hd 209458b , @xcite have constrained molecular bands of ch@xmath10 and h@xmath9o . the constraint on emission at 2.3 @xmath0 m between h@xmath9o absorption features was shown in . the predicted planet - to - star flux ratio really does not become favorable until wavelengths longer than 3 @xmath0 m , which may continue to challenge observers . in , we plot planet - to - star flux ratios in the mauna kea observatory ( mko ) h ( @xmath21.6 @xmath0 m ) , k ( @xmath22.2 @xmath0 m ) , l@xmath22 ( @xmath23.8 @xmath0 m ) , and m@xmath22 ( @xmath24.7 @xmath0 m ) bands . since the wavelength ranges of the l@xmath22 and m@xmath22 bands have significant overlap with the irac 3.6 and 4.5 @xmath0 m bands , the predicted ratios are quite similar . in dark blue , we show the l@xmath22 band upper limit of -0.0007@xmath230.0014 from @xcite . the model is just above the 1 sigma error bar.m , within the standard l@xmath22 band . our calculated planet - to - star flux ratio at secondary eclipse increases by 4% when using this narrow band , compared to standard l@xmath22 . since this is a small correction , our conclusions are unchanged . ] this is a better match than is attained with one dimensional radiative equilibrium models @xcite , which predict a flux peak just short of 4 @xmath0 m , as shown in . the @xcite hd 209458b radiative equilibrium model , which has somewhat muted flux peaks compared with similar models by other authors , has a planet - to - star flux ratio of 0.00114 in l@xmath22 band . in dotted blue in the case 1 panels is our l@xmath22 band ( which encompasses ch@xmath10 absorption ) prediction for case 2 chemistry . looking at shorter wavelengths , in the h and k bands , the planet is predicted to be dimmer , while the star is brighter , leading to low flux ratios . the peak emission in these bands does occur earlier than in the _ spitzer _ bands . for instance , the h band peak is 42 orbital degrees before secondary eclipse . for the cs06 dynamical model of the atmosphere of hd 209458b , we find that the apparent @xmath24 of the visible hemisphere is strongly variable , with a maximum of 1390 k and a minimum of 915 k for equilibrium chemistry and 1025 k for ( the probably more realistic case of ) disequilibrium chemistry . this leads to a luminosity of the visible planetary hemisphere that varies by factors of 5.3 and 3.4 , respectively for these two cases . for the one dimensional @xcite planet - wide profile , we derive a @xmath25 k , 1% less than found by the authors . the day - side @xmath24 for the dynamical model is not significantly larger than this planet - wide @xmath15 . as an additional comparison , we calculate the mean of the planetary luminosity over the entire orbit and then convert to @xmath15 to find a mean @xmath15 for cases 0 and 1 . this gives @xmath15s of 1195 k and 1227 k , respectively , showing that the cs06 model , at the pressures levels that radiate to space , is as a whole somewhat colder than the one dimensial profile of @xcite . this is likely a consequence of the radiative forcing scheme employed in cs06 , which will be reinvestigated when models that consistently couple radiative transfer and dynamics are developed . the @xmath2400 k @xmath15 contrasts found are predominantly due to changes in the temperature structure of the hemisphere that is visible as function of orbital phase . in addition , atmospheric opacity makes an important contribution when the ch@xmath10/co ratio is free to vary with pressure and temperature . when the ch@xmath10 mixing ratio is below that predicted by equilibrium chemistry , this leads to a lower opacity atmosphere , for a given _ t _ profile . while the few individual bands of co are somewhat stronger than those of ch@xmath10 , ch@xmath10 absorption across the planet s broad the 2 - 10 @xmath0 m flux peak dominates over the two bands of co. one is able to see more deeply into a ch@xmath10-depleted atmosphere , leading to a higher @xmath15 . currently , the only published secondary eclipse datum from _ spitzer _ for hd 209458b is the 24 @xmath0 m detection of @xcite . the models presented here have a planet - to - star flux ratio during secondary eclipse that is 1.6 sigma larger than this observational data point . together with our excellent agreement with the ground - based data at 2.3 @xmath0 m from @xcite and @xcite , and our 1.1 sigma difference with the 3.8 @xmath0 m data from @xcite , we regard this as excellent agreement significantly better than has been previously obtained with one dimensional radiative equilibrium models . it is interesting to discuss a few issues that arise if the flux ratios are indeed 10 - 25% less than we calculate here , as indicated by the l@xmath22 and 24 @xmath0 m band data . for instance , perhaps day - night temperature contrasts in the atmosphere are not as large as predicted by cs06 , leading to smaller deviations from a `` planet - wide '' @xmath26k . this may involve radiative time constants that are significantly longer than predicted by @xcite , winds faster than predicted by cs06 , or both . it would be worthwhile for other groups that possess hot jupiter radiative transfer codes to compute radiative time constants at these temperatures and pressures . this is an area that we will pursue in the near future . another possibility for a smaller planet - to - star flux ratio during secondary eclipse would be if the planet had a larger bond albedo than calculations currently indicate . this would mean less absorbed stellar flux and correspondingly lower temperatures everywhere on the planet . cooler temperatures everywhere on the planet would lead to chemical abundances that differ from our previous cases . more ch@xmath10 would form , at the expense of co , which would also lead to a slightly higher mixing ratio for h@xmath9o , which shares oxygen with co. hot jupiters are believed to have very low bond albedos on the order of 90% or more of incident stellar flux is expected to be absorbed . in @xcite , we found that our one - dimensional model atmosphere for hd 209458b scattered only 8% of incident flux . for tres-1 , this was 6% . to date , there is at least a hint that the bond albedo of tres-1 may have been underestimated . @xcite , under the assumption that the planet emits as a blackbody , determined a bond albedo of @xmath27 from their _ spitzer _ irac observations at 4.5 and 8.0 @xmath0 m . perhaps hot jupiters are not quite as hot as had been previously thought . however , this determination should be considered very preliminary at this time . @xcite have recently observed the _ transit _ of hd 209458b at 24 @xmath0 m , as well . from an observed transit depth of @xmath28 , they determined the radius of the planet in this band to be 1.26 @xmath29 @xmath30 , which includes uncertainties in the stellar radius . our model predicts a change in the apparent planet - to - star flux ratio of @xmath20.00008 during the 20 orbital degrees of the @xcite transit observations , @xmath24 times smaller than their uncertainty , and hence too small to have been detected . as was seen in and [ figure : mko ] , the change in planetary flux as a function of orbital phase is not as pronounced near the time of transit ( and secondary eclipse ) as it is at other phases . to illustrate the sensitivity of our results to temperature changes , we have computed spectra as a function of orbital phase , using equilibrium ( case 0 ) chemistry for two additional models . these are additional dynamical models described in cs06 , in which the base _ p t _ profile of @xcite has been increased or decreased by 300 k , with correspondingly warmer or colder night sides . the full dynamical simulations have been run again with these parameters . the resulting light curves , in _ spitzer _ bands , are shown in a , and c. for the `` cold '' ( -300 k ) model ch@xmath10 is the dominant carbon carrier on the night hemisphere , and co on the day hemisphere , leading to large flux variation in excess of that found for our nominal cs06 simulation , as shown in . for the `` hot '' ( + 300 k ) model , on both the day and night hemispheres , co is the dominant carbon carrier . methane absorption features are very weak on the night side , leading to flux variations in every band no larger than a factor of 2.7 from peak to trough . this model is somewhat similar to our earlier case 2 , but at warmer temperatures , as at all phases co is the dominant carbon carrier . fluxes are everywhere greater in the hot model than the nominal model , and everywhere less in the cold model than in the nominal model . this is is due to the differing atmospheric temperatures . the cold model best fits the @xcite datum at 24 @xmath0 m . in addition , the phase of maximum and minimum flux in a given band can change by up to @xmath210 orbital degrees between these simulations , due both to differing chemistry and atmospheric dynamics . as discussed in @xcite and cs06 , while these models all possess similarly strong east - to - west jets , the dynamical atmospheres differ slightly in detail . cs06 also examined non - equilibrium ch@xmath10/co chemistry for these models . for the `` cold '' ( -300 k ) model , non - equilibrium chemistry was important , and the ch@xmath10/co ratio at @xmath31 bar became homogenized at 0.20 around the entire planet . this ratio was 0.014 for the nominal case 1 decribed earlier . for the cold model equilibrium chemistry would predict a night side dominated by ch@xmath10 and a day side dominated by co. for the `` hot '' model ( + 300 k ) both equilibrium and non - equilibrium chemistry predicts that co is the dominant carbon carrier on both the day and side hemispheres . d shows our computed flux ratios for non - equilibrium chemistry for the cold case . as was shown previously for our nominal case , flux variation is smaller with non - equilibrium chemistry , because the mixing ratios of ch@xmath10 , co , and h@xmath9o are the same on the night and day hemispheres . again , because of the nearly isothermal temperature structure of the day - side , chemical abundances have little effect on the day - side planet - to - star flux ratios , which are nearly equal for these two chemistry cases . these additional cases further highlight the importance of ch@xmath10/co chemistry in the computation of infrared light curves . infrared fluxes as a function of orbital phase are sensitive to the temperatures of the hemisphere facing the observer , as well as the abundances of important molecular absorbers . for planetary _ p _ profiles that cross important ch@xmath10/co chemical boundaries , as most hot jupiters surely do , whether or not these species are found in their equilibrium mixing ratios has a major impact on resulting infrared flux in _ spitzer _ bands , especially on the night side . from the size of the error bar from the @xcite 24 @xmath0 m observations , it is clear that , could this instrument stability be sustained over the course of tens of hours of observations , the change in flux over time presented here could be detected . one - half of an orbital period for hd 209458b is 42 hours . if flux in the 8.0 @xmath0 m band for hd 209458b is higher than models predict , as was the case for tres-1 @xcite , then this would be an attractive band as well , as the error bars should be smaller . what might one hope to see with sustained observations ? if the peak in infrared flux does occur only @xmath225 - 30 orbital degrees before secondary eclipse , this would be difficulty to discern . however , the detection of any sort of ramp up in flux from the time of transit to secondary eclipse would give us important information on the day - night temperature contrast . the recently discovered transiting planet hd 189733b @xcite would likely be an even more attractive target , as the planet - star flux ratios are likely twice as large @xcite , and the orbital period is @xmath240% shorter . we predict secondary eclipse planet - to - star flux ratios for this system in @xcite ; our calculation at 16 @xmath0 m is a good match to the published observation of @xcite . a clear prediction from our calculations here is that when one uses realistic non - equilibrium chemistry calculations , the change in planetary flux as a function of orbital phase is greatly reduced , relative to equilibrium chemistry calculations , because the atmosphere s composition is fixed . for the _ spitzer _ bands , for case 0 , the maximum flux variation is in the 3.6 @xmath0 m band , which varies by a factor of 15 from peak to trough . the minimum variation is a factor of 2.2 , in the 16 @xmath0 m band . for case 1 , this variation drops significantly , and the maximum variation is a factor of 3.7 , in the 5.8 @xmath0 m band , and the minimum is 2.0 , again in the 16 @xmath0 m band . we suggest that the 5.8 and 8.0 @xmath0 m bands may be the best _ spitzer _ bands in which to search for flux variations as a function of orbital phase , as these bands combine a high planet - to - star flux ratio and the sensitivity and stability of the irac detectors . if the day - side thermal emission of hot jupiters is similar to a blackbody , as we find here , problems arise with the notion that the emission can be used to characterize the atmospheric chemistry from secondary eclipse observations . absorption features due to ch@xmath10 , co , and h@xmath9o would be nonexistent or extremely weak . observations at other orbital phases would then take on additional importance . given the significant spectral differences between our model and radiative - equilibrium models , it is clear that more work in this area is certainly warranted . we note that blackbody - like hot jupiter emission can simultaneously explain all observations to date . first is the secondary eclipse observation at 24 @xmath0 m by @xcite , a clear detection . second is the very low flux ratio upper limit in l@xmath22 band by @xcite . third is the 2.3 @xmath0 m relative flux observation of @xcite , which most one - dimensional models can not fit @xcite . fourth is the set of observations of tres-1 by @xcite , who found an infrared spectral slope from 4.5 to 8 @xmath0 m that was redder than that found by most one - dimensional models @xcite . additional observations of these planets , especially in the _ spitzer _ 3.6 @xmath0 m band , which catches much of the predicted 4 @xmath0 m flux peak , would strengthen or refute this argument and provide a critical test for the cs06 dynamical simulation . @xcite posited , perhaps with a wink , that a blackbody - like spectrum would be more consistent with observations to date than any published hot jupiter model . due to the dynamically altered temperature structure of the atmosphere of hd 209458b , we find that this could be the reality . + + we acknowledge support from nasa postdoctoral program ( npp ) and spitzer space telescope fellowships ( j. j. f. ) , nasa gsrp fellowship ngt5 - 50462 ( c. s. c. ) , nsf grant ast-0307664 ( a. p. s. ) , and nasa grants nag2 - 6007 and nag5 - 8919 ( m. s. m. ) . , d. , richardson , l. j. , seager , s. , & harrington , j. 2006 , tenth anniversary of 51 peg - b : status of and prospects for hot jupiter studies ( paris , france : frontier group , 2006 , eds . l. arnold , f. bouchy , c. moutou , 218 - 225 )

we explore the infrared spectrum of a three - dimensional dynamical model of planet hd 209458b as a function of orbital phase . the dynamical model predicts day - side atmospheric pressure - temperature profiles that are much more isothermal at pressures less than 1 bar than one - dimensional radiative - convective models have found . the resulting day - side thermal spectra are very similar to a blackbody , and only weak water absorption features are seen at short wavelengths . the dayside emission is consequently in significantly better agreement with ground - based and space - based secondary eclipse data than any previous models , which predict strong flux peaks and deep absorption features . at other orbital phases , absorption due to carbon monoxide and methane is also predicted . we compute the spectra under two treatments of atmospheric chemistry : one uses the predictions of equilibrium chemistry , and the other uses non - equilibrium chemistry , which ties the timescales of methane and carbon monoxide chemistry to dynamical timescales . as a function of orbital phase , we predict planet - to - star flux ratios for standard infrared bands and all _ spitzer space telescope _ bands . in _ spitzer _ bands , we predict 2-fold to 15-fold variation in planetary flux as a function of orbital phase with equilibrium chemistry , and 2-fold to 4-fold variation with non - equilibrium chemistry . variation is generally more pronounced in bands from 3 - 10 @xmath0 m than at longer wavelengths . the orbital phase of maximum thermal emission in infrared bands is 1545 orbital degrees before the time of secondary eclipse . changes in flux as a function of orbital phase for hd 209458b should be observable with _ spitzer _ , given the previously acheived observational error bars .

collisionless magnetized plasmas are described by the kinetic ( vlasov - maxwell ) equations and are characterized by high dimensionality , anisotropy and a wide variety of spatial and temporal scales @xcite , thus requiring the use of sophisticated numerical techniques to capture accurately their rich non - linear behavior . in general terms , there are three broad classes of methods devoted to the numerical solution of the kinetic equations . dimensional phase space via macro - particles that evolve according to newton s equations in the self - consistent electromagnetic field @xcite . pic is the most widely used method in the plasma physics community because of its robustness and relative simplicity . the well known statistical noise associated with the macro - particles implies that pic is really effective for problems where a low signal - to - noise ratio is acceptable . the eulerian - vlasov methods discretize the phase space with a six dimensional computational mesh . as such they are immune to statistical noise but they require significant computational resources and this is perhaps why their application has been mostly limited to problems with reduced dimensionality . for reference , storing a field in double precision on a mesh with @xmath1 cells requires about @xmath2 terabytes of memory . a third class of methods , called transform methods , is spectral and is based on an expansion of the velocity part of the distribution function in basis functions ( typically fourier or hermite ) , leading to a truncated set of moment equations for the expansion coefficients @xcite . similarly to eulerian - vlasov methods , transform methods might be resource - intensive if the convergence of the expansion series is slow . in recent years there seems to be a renewed interest in hermite - based spectral methods . some reasons for this can be attributed to the advances in high performance computing and to the importance of simpler , reduced kinetic models in elucidating aspects of the complex dynamics of magnetized plasmas @xcite . another reason is that ( some form of ) the hermite basis can unify fluid ( macroscopic ) and kinetic ( microscopic ) behavior into one framework @xcite . thus , it naturally enables the fluid / kinetic coupling that might be the ( inevitable ) solution to the multiscale problem of computational plasma physics and is a very active area of research @xcite . the hermite basis is defined by the hermite polynomials with a maxwellian weight and is therefore closely linked to maxwellian distribution functions . two kinds of basis have been proposed in the literature ( differing in regard to the details of the maxwellian weight ) : symmetrically- and asymmetrically - weighted @xcite . the former features @xmath0-stability but conservation laws for total mass , momentum and energy are achieved only in limited cases ( i.e. , they depend on the parity of the total number of hermite modes , on the presence of a velocity shift in the hermite basis , ... ) . the latter features exact conservation laws in the discrete and the connection between the low - order moments and typical fluid moments , but @xmath0-stability is not guaranteed @xcite . earlier works pointed out that a proper choice of the velocity shift and the scaling of the maxwellian weight ( free parameters of the method ) is important to improve the convergence properties of the series @xcite . indeed , the optimization of the hermite basis is a crucial aspect of the method , which however at this point does not yet have a definitive solution . one could of course envision a different spectral approach which considers a full polynomial expansion without any weight or free parameter . while any connection with maxwellians is lost , such expansion could be of interest in presence of strong non - maxwellian behavior and eliminates the optimization problem . the legendre polynomials are a natural candidate in this case , because of their orthogonality properties . they are normally applied in some preferred coordinate system ( for instance spherical geometry ) to expose quantities like angles that are defined on a bounded domain . indeed , legendre expansions are very popular in neutron transport @xcite and some application in kinetic plasma physics can be found for electron transport described by the boltzmann equation @xcite . surprisingly , however , we have not found any example in the context of collisionless kinetic theory and in particular for the vlasov - poisson system . the main contribution of the present paper is the formulation , development and successful testing of a spectral method for the one dimensional vlasov - poisson model of a plasma based on a legendre polynomial expansion of the velocity part of the plasma distribution function . the expansion is applied directly in the velocity domain , which is assumed to be finite . it is shown that the legendre expansion features many of the properties of the asymmetrically - weighted hermite expansion : the structure of the equations is similar , the low - order moments correspond to the typical moments of a fluid , and conservation laws for the total mass , momentum and energy ( in weak form , as defined in sec . 4 ) can be proven . it also features properties of the symmetrically - weighted hermite expansion : @xmath0-stability is also achieved by introducing a penalty on the boundary conditions in weak form . this strategy is inspired by the simultaneous approximation strategy ( sat ) technique @xcite . the paper is organized as follows . in sec . [ sec : vlasov ] the vlasov - poisson equations for a plasma are introduced together with the spectral discretization : the velocity part of the distribution function is expanded in legendre polynomials while the spatial part is expressed in terms of a fourier series . the time discretization is handled via a second - order accurate crank - nicolson scheme . in sec . [ sec : l2:stability ] the sat technique is used to enforce the @xmath0-stability of the numerical scheme . in sec . [ sec : conservation : laws ] conservation laws for the total mass , momentum and energy are derived theoretically . numerical experiments on standard benchmark tests ( i.e. , landau damping , two - stream instabilities and ion acoustic wave ) are performed in sec . [ sec : numerical ] , proving numerically the stability of the method and the validity of the conservation laws . conclusions are drawn in sec . [ sec : conclusions ] . we consider the vlasov - poisson model for a collisionless plasma of electrons ( labeled `` @xmath3 '' ) and singly charged ions ( `` @xmath4 '' ) evolving under the action of the self - consistent electric field @xmath5 . the behavior of each particle species @xmath6 with mass @xmath7 and charge @xmath8 is described at any time @xmath9 in the phase space domain @xmath10\times[{{v}_a},{{v}_b}]$ ] by the distribution function @xmath11 , which evolves according to the _ vlasov equation _ : @xmath12 we assume the physical space to be periodic in @xmath13 , so that no boundary condition for @xmath14 is necessary at @xmath15 and @xmath16 , and that suitable boundary conditions , e.g. , @xmath17 , are provided for @xmath14 at the velocity boundaries @xmath18 and @xmath19 for any time @xmath9 and any spatial position @xmath20 $ ] . we also assume that an initial solution @xmath21 is given at the initial time @xmath22 . [ remark : zero : velocity : bcs ] if the initial solution @xmath23 has a compact support in the phase space domain @xmath10\times[{{v}_a},{{v}_b}]$ ] , then @xmath14 has also a compact support at any time @xmath24 . moreover , the size of the support may increase in time in a controlled way , cf . @xcite . in such a case , it holds that @xmath17 until the size of the support equals the size of the velocity domain . this condition can be used to determine the final time at which a plasma simulation based on this numerical model is valid . in the vlasov - poisson system , the electric field @xmath25 is the solution of the _ poisson equation _ : @xmath26 where @xmath27 is the dielectric constant and @xmath28 is the _ total charge density _ of the plasma . by taking the time derivative of the poisson equation and using the continuity equation @xmath29 where @xmath30 is the _ total current density _ defined as @xmath31 we obtain the _ ampere equation _ @xmath32 where @xmath33 is a suitable constant factor . the ampere equation can be used with the vlasov equation instead of the poisson equation to obtain the vlasov - ampere formulation . in the continuum setting , the two formulations are equivalent in the one - dimensional electrostatic case without any external electric field as the one considered in this work . consider the infinite set of _ legendre polynomials _ @xmath34 , which are recursively defined for @xmath35 $ ] by ( * ? ? ? * chapters 8 , 22 ) : @xmath36 and normalized as follows @xmath37 we remap the legendre polynomials onto the velocity range @xmath38 $ ] through the linear transformation @xmath39 . let @xmath40 be the inverse mapping from @xmath38 $ ] to @xmath41 $ ] . the @xmath42-th legendre polynomial is given by @xmath43 , where the scaling factor in front of @xmath44 is chosen to satisfy the orthogonality relation : @xmath45 the first derivative of the legendre polynomials is given by @xmath46,\end{aligned}\ ] ] where @xmath47 is a switch that takes value @xmath48 if @xmath49 is even , and @xmath50 if @xmath49 is odd . using the chain rule and adjusting the normalization factor , we obtain the first derivative of the translated and rescaled legendre polynomials : @xmath51 the recursion relations that are used to expand the vlasov equation on the legendre basis are reported in appendix a for completeness . consider the spectral decomposition of the distribution function @xmath14 on the basis of legendre polynomials given by @xmath52 in the vlasov equation . the boundary conditions @xmath17 are not exactly satisfied since they are imposed in weak form and the polynomials @xmath53 are not zero at the velocity boundaries . a possible way to circumvent this issue is to consider the modified basis functions given by @xmath54 for @xmath55 . from the properties of the legendre polynomials , it readily follows that @xmath56 for each @xmath42 and the expansion of @xmath14 on this set of functions will automatically satisfied the homogeneous conditions at the boundary of the velocity range . nonetheless , we verified numerically in the first stages of this work that this approach may yield an unstable method and the numerical instability can not be fixed as there is no mechanism that allows us to control the growth of the absolute value of the legendre coefficients @xmath57 . another possible choice is to consider @xmath58 for even @xmath59 and @xmath60 for odd @xmath61 . although we have not implemented this second basis , a common characteristic of these choices is the loss of orthogonality , which we suspect may influence negatively the stability properties of the method . the alternative approach that we consider hereafter is to integrate by parts the velocity term in the vlasov equations . this strategy allows us to set the boundary conditions in weak form , and , then , to introduce a penalty term to enforce the @xmath0 stability of the method through the boundary conditions ( see section [ sec : l2:stability ] ) . to this end , we substitute into , we multiply the resulting equation by @xmath62 and integrate over @xmath38 $ ] . then , we use the recursion formulas - and the orthogonality relation and we obtain the following system of partial differential equations for the legendre coefficients @xmath63 : @xmath64}_{{{v}_a}}^{{{v}_b } } \right ) = 0 \quad\textrm{for~}n\geq 0 , \label{eq : legendre : system}\end{aligned}\ ] ] where conventionally @xmath65 , @xmath66 \displaystyle\frac{{{v}_b}-{{v}_a}}{2}\,\frac{n}{\sqrt{(2n+1)(2n-1 ) } } & \textrm{for~}n\geq 1 , \end{cases } \label{eq : legendre : sigma : def}\end{aligned}\ ] ] and @xmath67}_{{{v}_a}}^{{{v}_b } } = \frac{{f^s}(x,{{v}_b},t)\phi_n({{v}_b})-{f^s}(x,{{v}_a},t)\phi_n({{v}_a})}{{{v}_b}-{{v}_a } } , \label{eq : delta_v : def}\end{aligned}\ ] ] is the boundary term resulting from an integration by parts of the integral term that involves the velocity derivative . the derivation of the coefficients @xmath68 and @xmath69 can be found in appendix a. if the distribution @xmath11 has compact support in @xmath70 , the homogeneous boundary conditions at @xmath18 and @xmath19 are imposed in weak form by assuming that @xmath71}_{{{v}_a}}^{{{v}_b}}$ ] in is zero . however , since this term plays a major role in establishing the conservation laws and ensuring the @xmath0 stability of the discretization method , we will consider it in all the further developments and in the analysis of the next sections . we truncate the spectral expansion of @xmath14 after the first @xmath72 _ legendre modes _ by assuming that @xmath73 for @xmath74 and we approximate the distribution function by the finite summation : @xmath75 the evolution of each coefficient @xmath57 with @xmath76 is still given by . to ease the notation , we will drop the subindex @xmath77 in @xmath78 by tacitly assuming that all the quantities containing @xmath14 are indeed numerical approximations dependent on the first @xmath72 modes of the truncated series . let @xmath79 be the vector that contains all the coefficients @xmath57 for @xmath80 $ ] , i.e. , @xmath81 , and @xmath82 the vector containing the values of the legendre shape functions evaluated at @xmath83 . it holds that @xmath84 . system can be rewritten in the non - conservative vector form : @xmath85}_{{{v}_a}}^{{{v}_b } } \right ) = 0 , \label{eq : vlasov : legendre : non - conservation}\end{aligned}\ ] ] where @xmath86 ( { \mathbbm{b}}{\mathbf{c}^s})_{n } & = \sum_{i=0}^{n-1}\sigma_{n , i}{{c}^s}_{i } , \label{eq : legendre : matb : def}\end{aligned}\ ] ] and @xmath87}_{{{v}_a}}^{{{v}_b}}\big)_{n}={\delta_v\big [ { f^s}\phi_{n } \big]}_{{{v}_a}}^{{{v}_b}}$ ] . since @xmath88 is a constant matrix , it follows that @xmath89 therefore , system also admits the conservative form : @xmath90}_{{{v}_a}}^{{{v}_b } } \right ) = 0 , \label{eq : vlasov : legendre : conservation}\end{aligned}\ ] ] where @xmath88 is a real and symmetric matrix with real eigenvalues and eigenvectors . we expand each legendre coefficient @xmath63 on the first @xmath91 functions of the fourier basis @xmath92 ( for @xmath93 $ ] ) as follows @xmath94 where each coefficient @xmath95 is a complex function of time @xmath96 . the fourier basis functions satisfy the orthogonality relation @xmath97 substituting in and using , we derive the system for the coefficients @xmath98 , which reads as : @xmath99}_{{{v}_a}}^{{{v}_b } } \right ) \right]_{k } = 0 , \label{eq : legendre : fourier : system}\end{aligned}\ ] ] for @xmath100 and @xmath101 , and where @xmath102 denotes the convolution integral and @xmath103_{k}$ ] denotes the @xmath104 mode of the fourier expansion of the argument inside the square brackets . explicit formulas for these quantities are given below . we also recall that if @xmath105 and @xmath106 are two given real functions of @xmath20 $ ] and @xmath107 and @xmath108 the coefficients of their fourier expansion on the basis functions @xmath109 , then the @xmath110-th fourier mode of the convolution product @xmath111 is given by @xmath112_k = \sum_{k'=-{n_f}}^{{n_f}}g_{k ' } h_{k - k'}$ ] . the poisson equation for the electric field is similarly transformed by using and the fourier expansion of the electric field @xmath113 into to obtain @xmath114 for @xmath115 the equation above becomes @xmath116 which , according to the hypothesis of neutrality of the plasma , expresses the fact that the total charge in the system is zero . this implies that we can set the @xmath48-th fourier mode of the electric field to zero , i.e. , @xmath117 for convenience of notation , we introduce the vector @xmath118 that contains the @xmath110-th fourier coefficients @xmath119 for all the legendre modes @xmath80 $ ] , i.e. , @xmath120 . system can be rewritten in the vector form : @xmath121}_{{{v}_a}}^{{{v}_b } } \right)\right]_k = 0 , \label{eq : legendre : fourier : compact}\end{aligned}\ ] ] where @xmath122 ( { \mathbbm{b}}{\mathbf{c}^{s}_k})_{n } & = \sum_{i=0}^{n-1}\sigma_{n , i}{{c}^s}_{i , k } \label{eq : legendre : fourier : matb : def}\\[0.5em ] \big({\delta_v\big [ { f^s}{\bm\phi } \big]}_{{{v}_a}}^{{{v}_b}}\big)_{n } & = { \delta_v\big [ { f^s}\phi_{n } \big]}_{{{v}_a}}^{{{v}_b } } = \frac{1}{{{v}_b}-{{v}_a}}\big({f^s}(x,{{v}_b},t)\phi_{n}({{v}_b } ) - { f^s}(x,{{v}_a},t)\phi_{n}({{v}_a})\big).\end{aligned}\ ] ] note the vector expressions : @xmath123_{k } & = \sum_{k'=-{n_f}}^{{n_f}}{e}_{k'}\big[{\mathbbm{b}}{\mathbf{c}}\big]_{k - k'}\\[0.5em ] \big[{e}{\star}{\delta_v\big [ { f^s}{\bm\phi } \big]}_{{{v}_a}}^{{{v}_b}}\big]_{k } & = \sum_{k'=-{n_f}}^{{n_f}}{e}_{k ' } \sum_{n'=0}^{{n_l}-1}{{c}^s}_{n',k - k'}(t)\big(\phi_{n'}({{v}_b}){\bm\phi({{v}_b})}-\phi_{n'}({{v}_a}){\bm\phi({{v}_a})}\big)\end{aligned}\ ] ] and for the @xmath42-th legendre components : @xmath123_{n , k } & = \sum_{k'=-{n_f}}^{{n_f}}{e}_{k'}\sum_{i=0}^{n-1}\sigma_{n , i}{{c}^s}_{i , k - k'}(t)\\[0.5em ] \big[{e}{\star}{\delta_v\big [ { f^s}{\bm\phi } \big]}_{{{v}_a}}^{{{v}_b}}\big]_{n , k } & = \sum_{k'=-{n_f}}^{{n_f}}{e}_{k ' } \sum_{n'=0}^{{n_l}-1}{{c}^s}_{n',k - k'}(t)\big(\phi_{n'}({{v}_b})\phi_n({{v}_b})-\phi_{n'}({{v}_a})\phi_n({{v}_a})\big).\end{aligned}\ ] ] consider the current density of species @xmath124 given by . we apply the legendre decomposition , the fourier decomposition and we use to obtain the legendre - fourier representation of the total current density : @xmath125 taking the derivative in time of and using with @xmath126 , we obtain the fourier representation of ampere s equation : @xmath127 & = -({{v}_b}-{{v}_a})l\,\sum_{s\in\{e , i\}}{q^s}\left ( \left(\frac{2\pi{i}}{l}k\right)\big(\sigma_1{{c}^s}_{1,k}+\overline{\sigma}{{c}^s}_{0,k}\big ) + \frac{{q^s}}{{m^s}}\left[{e}{\star}{\delta_v\big [ { f^s}\phi_0 \big]}_{{{v}_a}}^{{{v}_b}}\right]_{k } \right ) . \label{eq : legendre : fourier : ampere:00}\end{aligned}\ ] ] for @xmath128 and using definition we reformulate ampere s equation as @xmath129 where @xmath130}_{{{v}_a}}^{{{v}_b}}\right]_k . \label{eq : legendre : fouriere : ampere : boundary : condition}\end{aligned}\ ] ] for @xmath115 , the fourier decomposition of ampere s equation gives the consistency condition @xmath131 , the zero - th fourier mode of the total current density @xmath132 . to control the filamentation effect , we modify system by introducing the artificial collisional operator @xmath133 in the right - hand side @xcite : @xmath121}_{{{v}_a}}^{{{v}_b } } \right)\right]_k = \mathcal{c}({\mathbf{c}^{s}_k } ) . \label{eq : legendre : fourier : compact : collisional}\end{aligned}\ ] ] consider the diagonal matrix @xmath134 whose @xmath42-th diagonal entry is given by : @xmath135 and where @xmath136 is an artificial diffusion coefficient whose value can be different from species to species . then , the collisional term is given by @xmath137 . the effect of this operator is to damp the highest - modes of the legendre expansion , thus reducing the filamentation and avoiding recurrence effects . this operator is designed to be zero for @xmath138 , in order not to have any influence on the conservation properties of the method . let @xmath139 be the time step , @xmath140 the time index , and each quantity superscripted by @xmath140 as taken at time @xmath141 , e.g. , @xmath142 , @xmath143 @xmath144 , etc . we advance the legendre - fourier coefficients @xmath95 in time by the crank - nicolson time marching scheme @xcite . omitting the superscript `` @xmath124 '' in @xmath145 and @xmath146 to ease the notation , vlasov equation for each species and any legendre - fourier coefficient becomes : @xmath147 & \qquad -\frac{{q^s}}{4{m^s } } \left [ \big({e^{\tau+1}}+{e^{\tau}}\big){\star}\left ( \sum_{i=0}^{n-1}\sigma_{n , i}\big({c^{\tau+1}}_{i}+{c^{\tau}}_{i}\big ) -\gamma^s{\delta_v\big [ \big({f^{\tau+1}}+{f^{\tau}}\big)\phi_n \big]}_{{{v}_a}}^{{{v}_b } } \right ) \right]_{k } = \mathcal{c}\left(\frac{1}{2}{c^{\tau+1}}_{n , k } + \frac{1}{2}{c^{\tau}}_{n , k}\right ) . \label{eq : legendre : fourier : system : time}\end{aligned}\ ] ] equation provides an implicit and non - linear system for the legendre - fourier coefficients @xmath95 as each electric field mode @xmath148 for @xmath149 depends on the unknown coefficient @xmath150 that must be evaluated at the same time @xmath151 . in practice , we apply a jacobian - free newton - krylov solver @xcite to search for the minimizer of the residual given by . consider the difference of the fourier representation of poisson s equation at times @xmath152 and @xmath151 @xmath153 by setting @xmath126 in , recalling that @xmath154 and noting that the collisional term does not give any contribution , we find that @xmath155 & -\frac{{q^s}}{4{m^s } } \left [ \big({e^{\tau+1}}+{e^{\tau}}\big){\star}\gamma^s{\delta_v\big [ \big({f^{\tau+1}}+{f^{\tau}}\big ) \big]}_{{{v}_a}}^{{{v}_b } } \right]_{k}=0 . \label{eq : ampere : discrete:05}\end{aligned}\ ] ] using in yields the discrete analog of ampere s equation that is consistent with the full crank - nicolson based discretization of the vlasov - poisson system : @xmath156 where we have introduced the explicit symbol @xmath157}_{{{v}_a}}^{{{v}_b } } \right]_{k } \qquad \textrm{for~}k\neq 0 \label{eq : full : discrete : ampere : bc}\end{aligned}\ ] ] to denote the boundary terms related to the behavior of all the distribution functions of the plasma species at the boundaries of the velocity domain . in section [ sec : conservation : laws ] we make use of and to characterize the conservation of the total energy . the distribution function @xmath14 solving the vlasov equation satisfies the so - called @xmath158-stability property for @xmath159 . to see this , just multiply equation by @xmath160 and integrate over the phase space domain @xmath10\times[{{v}_a},{{v}_b}]$ ] . assuming that the velocity range is sufficiently large for having @xmath17 , a simple calculation shows that @xmath161 . this property is particularly useful for @xmath162 , which implies the _ @xmath0 stability _ of the method ( sometimes called also `` energy stability '' in the literature ) . to derive a relation for the @xmath0 stability of the legendre - fourier method , we need the result stated by the following lemma . the proof of the lemma requires a few lengthy calculations and is reported , for the sake of completeness , in appendix c. [ lemma : l2stab : useful ] let @xmath118 be the vector containing the legendre coefficients of the @xmath110-th fourier mode of the distribution function @xmath14 , @xmath163 the electric field and @xmath164 the matrix of coefficients defined in . then , it holds that : @xmath165_k = \left[{e}{\star}{\delta_v\big [ ( { f^s})^2 \big]}_{{{v}_a}}^{{{v}_b}}\right]_{0 } = \sum_{k=-{n_f}}^{{n_f}}({\mathbf{c}^{s}_k})^{\dagger}\big[{e}{\star}{\delta_v\big [ { f^s}\bm\phi \big]}_{{{v}_a}}^{{{v}_b}}\big]_k , \label{eq : legendre : fourier : l2stab : useful } \end{aligned}\ ] ] where @xmath166_{0}$ ] denotes the zero - th fourier mode of the argument inside the brackets , and @xmath167 denotes the conjugate transpose . all terms in are real numbers . the @xmath0 stability of the legendre - fourier method depends on the behavior of the distribution function @xmath14 at the boundaries @xmath18 and @xmath19 . this result is stated by the following theorem . [ theo : l2stab ] the coefficients of the legendre - fourier decomposition have the property that : @xmath168}_{{{v}_a}}^{{{v}_b}}\right]_{0 } -2\sum_{n=0}^{{n_l}-1}{\left|d^{s}_{n}\right|}\sum_{k=-{n_f}}^{{n_f}}{\left|{{c}^s}_{n , k}(t)\right|}^2 . \label{eq : l2stab } \end{aligned}\ ] ] _ proof_. multiply from the left by @xmath169 , the conjugate transpose of @xmath118 , to obtain : @xmath170}_{{{v}_a}}^{{{v}_b } } \right)\right]_k = ( { \mathbf{c}^{s}_k})^{\dagger}{\mathbbm{d}}^{s}_{\nu}{\mathbf{c}^{s}_k}. \ ] ] add to this equation its conjugate transpose . matrix @xmath88 is real and symmetric , @xmath171 is real and the spatial term cancels out from the equation . summing over the fourier index @xmath110 we end up with : @xmath172}_{{{v}_a}}^{{{v}_b } } \right)\ , \right]_{k } + 2\textsf{re}\big(({\mathbf{c}^{s}_k})^{\dagger}{\mathbbm{d}}^{s}_{\nu}{\mathbf{c}^{s}_k}\big ) . \label{eq : legendre : fourier : l2stab : thm:10}\end{aligned}\ ] ] since @xmath173 is a diagonal matrix with negative real entries @xmath174 and @xmath175 is a real quantity it holds that @xmath176 the assertion of the theorem follows by applying the result of lemma [ lemma : l2stab : useful ] . if @xmath17 at the velocity boundaries ( see remark [ remark : zero : velocity : bcs ] ) , then at any instant @xmath24 the time derivative in is negative due to the collisional term and we have that @xmath177 note that in absence of the collisional term ( take @xmath178 in ) the time derivative is exactly zero and @xmath179 is constant . we refer to this property as _ the @xmath0 stability _ because the orthogonality of the legendre and fourier basis functions implies that @xmath180 ( see appendix b ) , from which we immediately find the @xmath0 stability of the distribution function @xmath11 . however , @xmath181 and @xmath182 can be different than zero and in general they are non zero since the legendre polynomials are globally defined on the whole domain and are non zero at the velocity boundaries . if the right - hand side of becomes positive , the collisional term may be not enough to control the other term in the right - hand side of . therefore , the method may become unstable and the time integration of @xmath14 is arrested . according to @xcite we can enforce the stability of the method by introducing the boundary conditions @xmath17 in weak form in the right - hand side of system through the penalty coefficient @xmath183 . to this end , we modify system as follows : @xmath184}_{{{v}_a}}^{{{v}_b } } \right)\right]_k = { \mathbbm{d}}_{\nu}{\mathbf{c}^{s}_k}. \label{eq : legendre : fourier : modified}\end{aligned}\ ] ] by suitably choosing the value of the penalty we minimize or set equal to zero the term in the right - hand side of that may cause the numerical instability . this result is presented in the following theorem . [ theo : l2-stab : modified ] the modified form of the legendre - fourier method for solving the vlasov - poisson system is @xmath0-stable for @xmath185 and any @xmath186 . the coefficients of the legendre - fourier decomposition have the property that : @xmath187 _ proof_. repeating the proof of theorem [ theo : l2stab ] yields : @xmath188}_{{{v}_a}}^{{{v}_b } } \right)\ , \right]_{k}\nonumber\\[0.5em ] & \ , -2\sum_{n=0}^{{n_l}-1}{\left|d^{s}_{n}\right|}\sum_{k=-{n_f}}^{{n_f}}{\left|{{c}^s}_{n , k}(t)\right|}^2 . \label{eq : stability : time : variation}\end{aligned}\ ] ] due to , the first term of the right - hand side of is zero ( when the coefficient that multiplies @xmath183 is non zero ) by setting @xmath189_{k } } { \sum_{k=-{n_f}}^{{n_f}}({\mathbf{c}^{s}_k})^{\dagger}\left[\,e{\star}{\delta_v\big [ { f^s}{\bm\phi } \big]}_{{{v}_a}}^{{{v}_b}}\right]_{k } } = \frac{1}{2}.\end{aligned}\ ] ] the assertion of the theorem is then proved by noting that any choice of @xmath186 in the collisional term makes the time derivative non - positive . the coefficient @xmath183 in affects also the first three moment equations and eventually perturbs the conservation properties of the vlasov - poisson system . we may overcome this issue by considering the modified system @xmath190}_{{{v}_a}}^{{{v}_b } } \right)\right]_k = { \mathbbm{d}}_{\nu}{\mathbf{c}^{s}_k } , \label{eq : legendre : fourier : modified : b } \end{aligned}\ ] ] where the penalty @xmath183 is introduced through the diagonal matrix @xmath191 and does not change the conservation properties of the method . the penalty @xmath183 can be determined at any time cycle by the formula : @xmath192_{k } } { \sum_{n=3}^{{n_l}-1}\sum_{k=-{n_f}}^{{n_f}}\overline{c}^{s}_{nk}\left[\,e{\star}{\delta_v\big [ { f^s}{\bm\phi } \big]}_{{{v}_a}}^{{{v}_b}}\right]_{n , k } } , \end{aligned}\ ] ] where @xmath193 is the conjugate of @xmath119 , and the result of theorem [ theo : l2-stab : modified ] still holds . alternatively , we can apply @xmath194 to all the legendre modes except the first three , i.e. , for @xmath138 . this option is simpler to implement and computationally less expensive , but may not fix the stability issue of the method completely . instead of equation , it holds that @xmath195}_{{{v}_a}}^{{{v}_b}}\right]_{n , k } -2\sum_{n=0}^{{n_l}-1}{\left|d^{s}_{n}\right|}\sum_{k=-{n_f}}^{{n_f}}{\left|{{c}^s}_{n , k}(t)\right|}^2 , \label{eq : stability : time : variation : gr2 } \end{aligned}\ ] ] and the first term in the right - hand side may still be a source of instability if it has the wrong sign . nonetheless , if the dissipative effect of the collisional term in is strong enough the scheme will remain stable . we investigated the effectiveness of this latter strategy in the numerical experiments of section [ sec : numerical ] . the vlasov - poisson model in the continuum setting is characterized by the exact conservation of mass , momentum and energy . the spectral discretization that is proposed in the previous section reproduces these conservation laws in the discrete setting . it turns out that the discrete analogs of the conservation of mass , momentum and energy depends on the variation in time of the legendre - fourier coefficients @xmath98 for @xmath138 and @xmath115 , i.e. , @xmath196 , @xmath197 , and @xmath198 . the contribution of the second term in is zero when @xmath115 and the transformed equation for the coefficients @xmath199 ( including the stabilization factor @xmath183 of section [ sec : l2:stability ] ) becomes : @xmath200}_{{{v}_a}}^{{{v}_b}}\right)\right]_{0}. \label{eq : legendre : fourier : reduced}\end{aligned}\ ] ] in particular , we have : @xmath201}_{{{v}_a}}^{{{v}_b}}\right]_{0 } , \label{eq : legendre : fourier : dert_c00}\\[1.em ] \textrm{for~}n=1 , k=0:\qquad & \frac{d{{c}^s}_{1,0}}{{dt } } = \frac{{q^s}}{{m^s}}\left[{e}{\star}\big(\sigma_{1,0}{{c}^s}_{0}-\gamma^s{\delta_v\big [ { f^s}\phi_1 \big]}_{{{v}_a}}^{{{v}_b}}\big)\right]_{0 } , \label{eq : legendre : fourier : dert_c10}\\[1.em ] \textrm{for~}n=2 , k=0:\qquad & \frac{d{{c}^s}_{2,0}}{{dt } } = \frac{{q^s}}{{m^s}}\left[{e}{\star}\big(\sigma_{2,1}{{c}^s}_{1}-\gamma^s{\delta_v\big [ { f^s}\phi_2 \big]}_{{{v}_a}}^{{{v}_b}}\big)\right]_{0}. \label{eq : legendre : fourier : dert_c20}\end{aligned}\ ] ] to derive the conservation laws for mass , momentum and energy for the fully discrete approximation , we note that the analog of equation for @xmath115 becomes : @xmath202 & -\gamma^s{\delta_v\big [ \big({f^s}(\cdot,{v},t^{\tau+1})+{f^s}(\cdot,{v},t^{\tau})\big)\phi_n \big]}_{{{v}_a}}^{{{v}_b } } \bigg)\bigg]_{0 } \label{eq : legendre : fourier : reduced : full}\end{aligned}\ ] ] as the collisional term is zero , and where @xmath203 and @xmath204 are the electric field and the distribution function , respectively , as functions of @xmath13 for a given value of @xmath70 and @xmath96 . by setting @xmath138 in we can also derive the analog of equations - for the fully discrete approximation , which we omit . in the following developments we consider the boundary term : @xmath205}_{{{v}_a}}^{{{v}_b } } \right]_{0}.\end{aligned}\ ] ] note that @xmath206 when @xmath207 for @xmath208 . using the legendre - fourier expansion of @xmath14 and the orthogonality relations and , the total mass of the species @xmath124 is given by @xmath209 by taking the time derivative of eq and using it follows that @xmath210}_{{{v}_a}}^{{{v}_b}}\right]_{0}. \label{eq : legendre : fourier : m3}\end{aligned}\ ] ] the conservation of the total mass per species includes a boundary term that is zero if @xmath17 ( see remark [ remark : zero : velocity : bcs ] ) . from and using with @xmath126 , we derive the conservation of the total mass per species in the full discrete model : @xmath211 equation states that the mass variation between times @xmath151 and @xmath152 is balanced by the boundary term in the right - hand side . the total momentum of the plasma is defined as @xmath212 where @xmath213 is the total momentum of the species @xmath124 . introducing the legendre - fourier expansion of @xmath14 , using the integrated recursive formula , orthogonality relations and , and mass equation yield @xmath214 taking the time derivative of equation and using it follows that @xmath215 & = \sum_{s\in\{e , i\ } } { q^s}({{v}_b}-{{v}_a})l\,\left ( \sigma_{1}\sigma_{1,0}\,\big[{e}{\star}{{c}^s}_0\big]_{0 } - \gamma^s \left[\,{e}{\star}\left ( \overline{\sigma}{\delta_v\big [ { f^s}\phi_0 \big]}_{{{v}_a}}^{{{v}_b } } + \sigma_{1}{\delta_v\big [ { f^s}\phi_1 \big]}_{{{v}_a}}^{{{v}_b}}\right)\,\right]_{0 } \right ) . \label{eq : legendre : fourier : p3}\end{aligned}\ ] ] using the poisson equation the first term in the last right - hand side is zero because the summation on the convolution index is on a symmetric range of indices and the argument of the summation is anti - symmetric : @xmath216_{0 } & = ( { { v}_b}-{{v}_a})l\,\sum_{k=-{n_f}}^{{n_f}}{e}_{k}(t)\,\sum_{s\in\{e , i\}}{q^s}{{c}^s}_{0,-k}(t ) \nonumber\\[0.5em ] & = -2\pi{i}\epsilon_0\,\sum_{k=-{n_f}}^{{n_f}}k{e}_{k}(t){e}_{-k}(t ) = 0.\end{aligned}\ ] ] consequently , equation becomes : @xmath217}_{{{v}_a}}^{{{v}_b}}+\sigma_{1}{\delta_v\big [ { f^s}\phi_1 \big]}_{{{v}_a}}^{{{v}_b } } \right ) \ , \right]_{0}. \label{eq : legendre : fourier : p4}\end{aligned}\ ] ] the conservation of the total momentum includes a boundary term that is zero if @xmath17 ( see remark [ remark : zero : velocity : bcs ] ) . from and using with @xmath218 , we derive the variation of momentum per species @xmath124 between times @xmath152 and @xmath151 : @xmath219 & = \frac{{q^s}}{4}({{v}_b}-{{v}_a})l\,\sigma_{1}\delta t\ , \bigg[\big({e}(\cdot , t^{\tau+1})+{e}(\cdot , t^{\tau})\big){\star}\big(\sigma_{1,0}\big({{c}^s}_{0}(t^{\tau+1})+{{c}^s}_{0}(t^{\tau})\big)\big)\bigg]_{0 } \nonumber\\[0.5em ] & \quad+\delta t\left ( \sigma_{1}\mathcal{b}^{s;\tau,\tau+1}_{1,0 } + \overline{\sigma}\mathcal{b}^{s;\tau,\tau+1}_{0,0 } \right).\end{aligned}\ ] ] furthermore , summing over all the species , taking the zero - th fourier mode of the convolution product , and using the poisson equation yield : @xmath220_{0 } \nonumber\\ & \qquad\qquad= \sum_{k=-{n_f}}^{{n_f}}\big(e(\cdot , t^{\tau+1})+e(\cdot , t^{\tau})\big)_{k } \sum_{s\in\{e , i\}}{q^s}\,({{v}_b}-{{v}_a})l\,\big({{c}^s}_{0}(t^{\tau+1})+{{c}^s}_{0}(t^{\tau})\big)_{-k } \nonumber\\ & \qquad\qquad= -2\pi i\epsilon_0 \sum_{k=-{n_f}}^{{n_f}}k \big(e(\cdot , t^{\tau+1})+e(\cdot , t^{\tau})\big)_{k } \big(e(\cdot , t^{\tau+1})+e(\cdot , t^{\tau})\big)_{-k } = 0.\end{aligned}\ ] ] therefore , in the full discrete model the _ conservation of the total momentum _ holds in the form : @xmath221 which states that the variation of the total momentum between times @xmath152 and @xmath151 is balanced by the boundary terms in the right - hand side of . the total energy of the plasma is defined as @xmath222 where @xmath223 and @xmath224 are the kinetic energy of the species @xmath124 and the potential energy at time @xmath96 , respectively . introducing the legendre - fourier expansion of @xmath14 and using the orthogonality relations and , the kinetic energy of species @xmath124 is reformulated as : @xmath225 we take the derivative in time of @xmath223 and use - to obtain @xmath226 & = \frac{{q^s}}{2}({{v}_b}-{{v}_a})l\,\left[{e}{\star}\big ( \sigma_{1}\sigma_{2}\sigma_{2,1}\,{{c}^s}_{1 } + 2\overline{\sigma}\sigma_{1}\sigma_{1,0}\,{{c}^s}_{0 } \big)\right]_{0 } + \mathcal{b}^s_{kin } \label{eq : kinetic : energy:00}\end{aligned}\ ] ] where we introduced the `` kinetic '' boundary term per species @xmath124 : @xmath227}_{{{v}_a}}^{{{v}_b } } + 2\sigma_1\overline{\sigma}{\delta_v\big [ { f^s}\phi_1 \big]}_{{{v}_a}}^{{{v}_b } } + ( \sigma_1 ^ 2+\sigma_0 ^ 2+\overline{\sigma}^2){\delta_v\big [ { f^s}\phi_0 \big]}_{{{v}_a}}^{{{v}_b } } \,\big)\right]_{0}.\end{aligned}\ ] ] as @xmath228 , and applying to , we obtain : @xmath229_{0 } + \mathcal{b}^{s}_{kin } = \big[{e}{\star}{j^s}\big]_{0 } + \mathcal{b}^{s}_{kin}. \label{eq : legendre : fourier : e3}\end{aligned}\ ] ] using , the orthogonality relation and the convolution notation , the potential energy of the electric field is given by : @xmath230_{0}. \label{eq : epot : fourier}\end{aligned}\ ] ] then , we take the time derivative of the equation above , use ampere s equation and note that @xmath231_0=0 $ ] as the average of @xmath5 on @xmath10 $ ] is zero to obtain : @xmath232_{0 } = -\bigg[{e}{\star}\bigg ( \sum_{s\in\{e , i\}}\big({j^s}+\gamma^sq^s\big ) + c_{a } \bigg ) \bigg]_{0 } = -\bigg[{e}{\star}\sum_{s\in\{e , i\}}{j^s}\bigg]_{0 } + \mathcal{b}_{pot}\end{aligned}\ ] ] where , after expanding the convolution product , we introduced the symbol @xmath233 for the `` potential '' boundary term , @xmath234 being the boundary term defined in ampere s equation . adding the total kinetic energy for all species and the potential energy gives : @xmath235 the conservation of the total energy includes a boundary term that is zero if @xmath17 ( see remark [ remark : zero : velocity : bcs ] ) . from , the variation of the kinetic energy @xmath223 between times @xmath152 and @xmath151 reads as : @xmath236 & \qquad + 2\sigma_{1}\overline{\sigma}\,\big({{c}^s}_{1,0}(t^{\tau+1 } ) - { { c}^s}_{1,0}(t^{\tau})\big ) + \big ( \sigma_{1}^2+\sigma_{0}^2+\overline{\sigma}^2 \big)\,\big({{c}^s}_{0,0}(t^{\tau+1 } ) - { { c}^s}_{0,0}(t^{\tau})\big ) \big).\end{aligned}\ ] ] using with @xmath237 yields : @xmath238 & + 2\sigma_{10}\sigma_{1}\overline{\sigma}\big({{c}^s}_{0}(t^{\tau+1})+{{c}^s}_{0}(t^{\tau})\big ) \big ) \bigg]_{0 } + \delta t\mathcal{b}^{s;\tau,\tau+1}_{kin},\end{aligned}\ ] ] where @xmath239 noting that @xmath240 , using the definition of the convolution product @xmath102 , the fourier decomposition of the electric field and the legendre coefficients , and the definition of the fourier coefficients of the current density @xmath241 given in yield : @xmath242 from , the variation of the potential energy between times @xmath151 and @xmath152 is given by : @xmath243_{0 } - \frac{\epsilon_0}{2 } \big[{e}(\cdot , t^{\tau}){\star}{e}(\cdot , t^{\tau})\big]_{0 } \nonumber\\[0.5em ] & = \frac{\epsilon_0}{2}\big [ \big({e}(\cdot , t^{\tau+1})-{e}(\cdot , t^{\tau})\big){\star}\big({e}(\cdot , t^{\tau+1})+{e}(\cdot,\tau)\big ) \big]_{0 } \nonumber\\[0.5em ] & = \frac{\epsilon_0}{2 } \sum_{k=-{n_f}}^{{n_f } } \big({e}^{\tau+1}_{k}-{e}^{\tau}_{k}\big)\ , \big({e}^{\tau+1}_{-k}+{e}^{\tau}_{-k}\big).\end{aligned}\ ] ] using the discrete analog of ampere s equation given by and yields : @xmath244 where @xmath245 finally , we add the kinetic energy terms for @xmath6 in and the potential energy to find the relation expressing the _ total energy conservation _ for the full discrete approximation : @xmath246 equation states that the variation of the total energy between times @xmath152 and @xmath151 is balanced by the proper combination of kinetic and potential boundary terms in the right - hand side and expresses the _ conservation of the total energy _ for the full discretization of the vlasov - poisson system . in this section we assess the computational performance of the legendre - fourier method by solving the landau damping , two - stream instability and ion acoustic wave problems . these test cases are classical problems in plasma physics and are routinely used to benchmark kinetic codes . in our numerical experiments , we are mainly interested in showing the conservation properties of the method , i.e. , the discrepancy between the initial value of mass , momentum and energy , and their value at successive instants in time during the simulation . we also investigate the stability of the method , i.e. , how the @xmath0-norm of the distribution function defined as in changes during the time evolution of the system . the penalty @xmath247 is applied to all legendre modes except the first three and the stability of the legendre - fourier method is ensured by the artificial collisional term when @xmath248 . this strategy , which is discussed at the end of section [ sec : l2:stability ] , is very effective in providing a stable method with good conservation properties . in the two - stream instability problem , we also investigate the effect of applying penalty @xmath247 on all the moment equations on the conservation of the total energy . in the first two test problems , the ions constitute a fixed background with density @xmath249 . we also introduce the following normalization : time is normalized on the electron plasma frequency @xmath250 ; position @xmath13 on the electron debye length @xmath251 ; velocity @xmath83 on the electron thermal velocity @xmath252 where @xmath110 is the boltzmann constant , @xmath253 the electron temperature and @xmath254 the electron mass ; the electric field @xmath5 on @xmath255 , where @xmath3 is the elementary charge ; species densities on a reference density @xmath256 ; and the species distribution function on @xmath257 . landau damping is a classical kinetic effect in warm plasmas , due to particles in resonance with an initial wave perturbation . this interaction leads to an exponential decay of the electric field perturbation . this problem is particularly challenging for kinetic codes because of the continuous filamentation in velocity space , which is a characteristic feature of the collision - less plasma described by the vlasov equation . filamentation is controlled by the artificial collisional operator introduced in . the initial distribution of the electrons is given by @xmath258 , \label{finit}\ ] ] with @xmath259 and @xmath260 . the legendre - fourier expansion of eq . ( [ finit ] ) implies that the modes @xmath261 , @xmath262 and @xmath263 are excited at @xmath22 . in this test case , the final simulation time is @xmath264 with time step @xmath265 , @xmath266 legendre modes and @xmath267 fourier modes . the domain of integration is set to @xmath268 , @xmath269 . figure [ fig : ld:00 ] shows the first mode of the electric field @xmath270 versus time for two different values of the stabilization parameter ( @xmath271 ) and the collisional frequency ( @xmath272 ) . for all cases the damping rate is in good agreement with the landau damping theory , which predicts @xmath273 . one can also notice that for all cases the simulation is stable , regardless of the value of @xmath247 , and that @xmath247 does not really affect much the dynamics . as expected , when @xmath274 the system exhibits recursive behavior . the collisional operator with @xmath248 is however sufficient to remove the recurrence effect and @xmath270 stabilizes around @xmath275 for @xmath276 . figure [ fig : ld:02 ] ( left ) shows the time evolution of @xmath277 , which is normalized to its value at time @xmath22 , for the same cases of fig . [ fig : ld:00 ] . according to , this quantity is computed as @xmath278 when @xmath274 , the @xmath0 norm of @xmath279 is constant on the scale of the plot and the boundary term in has a rather negligible effect . instead , when @xmath248 , the @xmath0 norm of @xmath279 decreases with an almost constant slope since the collisional term in is dominant . figure [ fig : ld:02 ] ( right ) shows that theorem [ theo : l2stab ] [ equation ] is indeed satisfied numerically . in fig . [ fig : ld:02 ] ( right ) the time derivative is computed by central finite differences . figure [ fig : ld:04 ] shows the time evolution of the maximum value of the distribution function at the boundary of the system @xmath280 : @xmath281 , with the same format of fig . [ fig : ld:00 ] . one can notice the beneficial effect of the collisional operator : when @xmath274 there is a sharp increase of @xmath282 around @xmath283 , while for @xmath248 it holds that @xmath282 approximately @xmath284 throughout the whole simulation . finally , the legendre - fourier method presented in this work provides exact conservation laws . the relative discrepancy of the mass , defined as @xmath285 , and the discrepancy of momentum , defined as @xmath286 , are _ exactly zero _ at any discrete time step @xmath287 in our double precision implementation and are therefore not shown . the relative discrepancy of the total energy , defined as @xmath288 is shown in figure [ fig : ld:06 ] and is smaller than @xmath289 . the two - stream instability is excited when the distribution function of a species consists of two populations of particles streaming in opposite directions with a large enough relative drift velocity . we initialize the electron distribution function with two counter - streaming maxwellians with equal temperature : @xmath290 \,\left[1+\varepsilon\cos\big(\frac{2\pi}{l}kx\big)\right ] \label{eq : two - stream : init - sol}\end{aligned}\ ] ] where @xmath291 is the drift velocity . for this test case , we have chosen the following parameters : @xmath292 , @xmath293 , @xmath260 , @xmath259 . we integrate the vlasov - poisson system by using the time step @xmath294 , @xmath266 legendre modes , and @xmath267 fourier modes . the domain of integration in phase space is set to @xmath295 , @xmath269 for all the calculations shown in figures [ fig:2s:00]-[fig:2s:05 ] , while in figure [ fig : two - stream : fs : phase - space ] we show the distribution function of electrons that is computed for three different combinations of @xmath72 and velocity range @xmath38 $ ] . this example was also considered with similar input parameters as in ref . in @xcite , where the vlasov equation was discretized using @xmath296 hermite modes . the electron distribution function was inizialized by combining two drifting maxwellians centered at two different velocities , each expanded in the hermite basis . since the discretization was based on the asymmetrically weighted hermite basis functions @xcite , the two maxwellians were completely described by setting only the first mode of each expansion . the remaining modes were needed to describe the non - maxwellian evolution of the solution . when using the legendre - fourier discretization proposed in this work , there is no correspondence between the first mode and the maxwellian distribution . thus , in order to have sufficient accuracy , the spectral expansion requires to consider all the polynomial modes from the beginning . in figure [ fig:2s:00 ] we show the first fourier mode of the electric field @xmath297 versus time for the four combinations of @xmath298 and @xmath272 . the initial part of the dynamics is the same for the four curves and one can see the development of the two - stream instability . the slope of the numerical curves matches well the theoretical slope predicted by the linear theory , which is shown as a dashed line in the plot . when @xmath299 , the two curves for @xmath300 stop at @xmath301 ( slightly prior to the end of the linear phase ) because of the development of a numerical instability . when @xmath302 and @xmath248 the scheme is numerically stable and reaches the final time of the simulation , @xmath303 , without problems . instead , the case @xmath274 stops converging at around @xmath304 because of problems related to the behavior of @xmath279 at the boundary ( as documented below ) . figure [ fig:2s:02 ] ( left ) shows the time evolution of the @xmath0 norm of the distribution function @xmath279 normalized with respect to initial value according to the cases presented in fig . [ fig:2s:00 ] . figure [ fig:2s:02 ] ( right ) shows a zoom around @xmath50 . one can clearly see that the @xmath0 norm of @xmath279 grows unboundedly when @xmath299 , indicating that the first term on the right hand side of equation provides a positive feedback that is not even compensated by the collisional term when @xmath248 . hence , the scheme is numerically unstable . when @xmath302 the scheme is numerically stable . indeed , by applying @xmath247 to all the moment equation , we have verified numerically that the @xmath0 norm of @xmath279 is constant in time for @xmath274 and damps for @xmath248 as predicted by theorem @xmath305 , cf . equation . if @xmath247 is applied to all the legendre modes _ except the first three _ we obtain the behavior shown in figure [ fig:2s:02 ] , where a slow growth of the @xmath0 norm of @xmath279 is visible for @xmath274 . figure [ fig:2s:03 ] shows the numerical representation of equation , where the time derivative is approximated by central finite differences for the case @xmath302 and @xmath248 . from this figure , we deduce that theorem [ theo : l2stab ] and equation , are verified numerically to a good degree of accuracy . the behavior of the maximum value of the distribution function on the domain boundary , @xmath306 , is shown in fig . [ fig:2s:04 ] . as expected , for @xmath299 the simulation is unstable and @xmath279 grows unbounded on the boundary . the stabilization provided by @xmath302 is effective and limits the value of @xmath307 there . however , when @xmath274 one can see that @xmath306 still grows sizably and becomes of order unity ( i.e. of the same order of the initial distribution function ) at around @xmath308 . clearly this signals that the simulation is not accurate anymore . when @xmath248 , on the other hand , @xmath306 remains reasonably small throughout the simulation . in figures [ fig:2s:05 ] and [ fig:2s:09-all ] we show the variation in time of momentum ( left plot , @xmath286 although @xmath309 in this case ) and relative variation in time of total energy ( right plot , @xmath288 ) with respect to the initial value . as for all the previous figures , the plots shown in figure [ fig:2s:05 ] are obtained by applying the penalty @xmath247 to all the moment equations except the first three . in this case , total momentum and total energy , as well as mass which is not shown , are conserved extremely well in the simulations , as predicted by the analysis of sections [ sec : conservation : laws ] and [ sec : time : integration ] . instead , the results of figure [ fig:2s:09-all ] are obtained by applying penalty @xmath247 to all the moment equations . in this case , the total momentum variation that is visible is of the order of magnitude of @xmath275 and total energy variation is of the order of magnitude of @xmath310 . these results are still in accord with the analysis of sections [ sec : conservation : laws ] because we know from sections [ subsect : momentum : conservation ] and [ subsect : energy : conservation ] that both momentum and energy variation contain boundary terms that are not included in this diagnostics . it is worth noting that these boundary terms explicitly contain @xmath247 , and are zero if @xmath299 in their expression . also note that in the two - stream instability problem , momentum is symmetric and that these results show that the symmetry of the problem is not violated by the legendre - fourier method . in figure [ fig : two - stream : fs : phase - space ] we show the electron distribution function in phase space that is computed by using three different combinations of @xmath72 , the number of legendre modes , and velocity range @xmath38 $ ] for @xmath311 and @xmath312 . in particular , the plots on top are obtained by using @xmath313 and integrating over the velocity range @xmath314 $ ] ; the plots in the middle are obtained by using @xmath315 and the velocity range @xmath314 $ ] ; the plots on bottom are obtained by using @xmath315 and the velocity range @xmath316 $ ] . the plots on the left show the distribution function at @xmath317 , the plots on the right at @xmath318 . the resolution of @xmath279 clearly depends on the combination that is chosen : it improves by increasing @xmath72 in a fixed velocity range and it worsen by increasing the domain size with a fixed @xmath72 . last , we consider the evolution of an ion acoustic wave . this is a truly multiscale example , occurring on the slow time scales associated with the ions but where the electron motion concurs in defining the properties of the wave . following @xcite , we initialize a perturbation in the ion distribution function at t=0 @xmath319\ ] ] while the electrons are maxwellian and unperturbed @xmath320 other parameters are @xmath321 , @xmath322 , @xmath323 , @xmath324 , @xmath325 , @xmath326 , @xmath302 , while @xmath139 and @xmath327 are varied parametrically . although we only present results with a smaller perturbation @xmath324 , we have also tried larger perturbations and essentially successfully reproduced the results of ref . @xcite for @xmath328 . figure [ fig : ia:00 ] shows the amplitude of the electric field for the first fourier mode initially excited at @xmath22 . four curves are plotted , corresponding to @xmath329 and @xmath330 . the initial evolution of the system is the same for all the curves and one can see some electron oscillations . however , when @xmath178 the simulations are corrupted by a large amount of unphysical oscillations ( quite irrespective of @xmath139 ) . when @xmath331 , on the other hand , the ion acoustic wave signal is recovered well : the period of @xmath270 obtained from the simulations is @xmath332 , in good agreement with the theoretical value of @xmath333 . we note that the curves obtained with @xmath331 and @xmath265 and @xmath334 are virtually indistinguishable , showing the ability of our numerical scheme to step over the faster frequency in the system , the electron plasma frequency , without any sign of numerical instability . we have also performed simulations with larger @xmath139 ( up to @xmath335 , not shown ) . the ion acoustic wave becomes progressively less accurate but , as expected , there is no sign of numerical instabilities . figure [ fig : ia:01 ] shows the time evolution of the @xmath0 norm of the distribution function normalized as in for the four simulations of fig . [ fig : ia:00 ] . as for the landau damping case , when @xmath178 the @xmath0 norm of the distribution function is flat ( on the scale of the plot ) , indicating a minimal contribution of the boundary terms in . when @xmath331 , the @xmath0 norm of the distribution function decreases in time due to the dominant contribution of the collisional term . figure [ fig : ia:02 ] shows the maximum of @xmath336 on the boundaries of the velocity space , with the same format of fig . [ fig : ia:00 ] . although @xmath282 remains fairly small for all the cases , once again one can see the beneficial effect of the collisional operator : for @xmath331 it holds that @xmath282 is more than an order of magnitude smaller than for @xmath178 . finally , fig . [ fig : ia:03 ] shows the time evolution of the total momentum and the relative variation of the total energy for the simulations with @xmath331 and @xmath265 and @xmath334 ( total mass is not shown since it is conserved exactly ) . in general , as expected , both quantities are conserved well . one can notice that the error in the total momentum is controlled by the time step , while this is not the case for the total energy . in this paper a spectral method for the numerical solution of the vlasov - poisson equations of a plasma has been presented . the plasma distribution function is decomposed in legendre polynomials applied directly on a finite domain in velocity space . the resulting set of moment equations is further discretized spatially by a fourier decomposition ( periodic boundary conditions are assumed ) and in time by a fully - implicit , second order accurate crank - nicolson scheme . a collisional term is also considered in the discrete model to control the filamentation effect , but does not affect the conservation properties of the method . a jacobian - free newton - krylov method ( with the gmres solver for the inner linear iterations ) is used to solve the discrete non - linear equations . the most significant aspects of our work are three . first , the method is formulated in such a way that the boundary conditions in velocity space ( @xmath337 at the boundary of the velocity domain ) are applied in weak form . that is , they are not enforced exactly through an expansion basis obtained by a linear combination of the legendre polynomials . instead , the boundary conditions are satisfied approximately via an integration by parts once the vlasov equation is projected onto the legendre basis functions ( see sec . [ sec : vlasov ] ) . second , introducing a penalty on the weak form of the boundary conditions allows the formulation of the numerical scheme to be @xmath0-stable . third , the numerical scheme features conservation laws for total mass , momentum and energy in weak form . the numerical experiments performed in sec . [ sec : numerical ] on landau damping , two - stream instability and ion acoustic wave test cases confirm both the stability of the method and the validity of the conservation laws . lin - landau - cropped - e1_all_gr2 ( 48,-4 ) * time * ( -5,36 ) @xmath338 lin - landau - cropped - c2_all_gr2 ( 48,-6)*time * ( -8,21 ) * @xmath0 norm of @xmath14 * lin - landau - cropped - dc2dt_alpha05_nu1_gr2 ( 48,-6)*time * ( -8,13 ) * variation of @xmath339 * lin - landau - cropped - maxfbc_all_gr2 ( 48,-5 ) * time * ( -5,29 ) * max(@xmath340 ) * lin - landau - cropped - wtot_alpha05_nu1_gr2 ( 48,-5 ) * time * ( -5,15 ) * variation of total energy * two - stream - cropped - e1_all_gr2_large ( 48,-6)*time * ( -5,36 ) @xmath338 two - stream - cropped - e1_all_gr2_small ( 48,-6)*time * ( -5,36 ) @xmath338 two - stream - cropped - c2_all_gr2_large ( 48,-6)*time * ( -8,21 ) * @xmath0 norm of @xmath14 * two - stream - cropped - c2_all_gr2_small ( 48,-6)*time * ( -8,21 ) * @xmath0 norm of @xmath14 * two - stream - cropped - dc2dt_alpha05_nu1_gr2 ( 48,-6 ) * time * ( -8,20 ) * variation of @xmath339 * two - stream - cropped - maxfbc_all_gr2 ( 48,-5 ) * time * ( -5,29 ) * max(@xmath340 ) * two - stream - cropped - mom_alpha05_nu1_gr2 ( 48,-6)*time * ( -8,0 ) * variation of total momentum * two - stream - cropped - wtot_alpha05_nu1_gr2 ( 48,-6)*time * ( -8,6 ) * variation of total energy * two - stream - cropped - mom_alpha05_nu1_all ( 48,-6)*time * ( -8,0 ) * variation of total momentum * ./two - stream - cropped - wtot_alpha05_nu1_all ( 48,-6)*time * ( -8,6 ) * variation of total energy * [ cols="^,^ " , ] ion - acoustic - cropped - e1_gr2_all.pdf ( 48,-4 ) * time * ( -5,36 ) @xmath338 ion - acoustic - cropped - c2_gr2_all.pdf ( 48,-6 ) * time * ( -8,24 ) * @xmath0 norm of @xmath14 * ion - acoustic - cropped - maxfbc_gr2_all.pdf ( 48,-5 ) * time * ( -5,29 ) * max(@xmath341 ) * ion - acoustic - cropped - mom_gr2_nu05.pdf ( 48,-6)*time * ( -8,0 ) * variation of total momentum * ion - acoustic - cropped - wtot_gr2_nu05.pdf ( 48,-6)*time * ( -8,6 ) * variation of total energy * the authors gratefully acknowledge discussions with c. la cognata and j. nordstrom ( university of linkoping , sweden ) ; and l. chacn and d. moulton ( los alamos national laboratory ) ; d. funaro ( university of modena and reggio emilia , italy ) . this work was partially funded by the laboratory directed research and development program ( ldrd ) , under the auspices of the national nuclear security administration of the u.s . department of energy by los alamos national laboratory , operated by los alamos national security llc under contract de - ac52 - 06na25396 . the content of this article is also published in the journal paper of reference @xcite . 10 m. abramowitz and i. stegun . . dover publications , 1965 . t. p. armstrong , r. c. harding , g. knorr , and d. montgomery . solution of vlasov s equation by transform methods . in m. rotenberg b. alder , s. fernbach , editor , _ methods in computational physics . plasma physics _ , volume 9:30 . academic press , new york , london , 1970 . c. k. birdsall and a. b. langdon . . taylor & francis , 2004 . e. camporeale , g. l. delzanno , b. k. bergen , and j. d. moulton . on the velocity space discretization for the vlasov - 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( submitted ) . s. wollman . existence and uniqueness theory of the vlasov - poisson system with application to the problem with cylindrical symmetry . , 90(1):138170 , 1982 . consider the set of legendre polynomials @xmath342 that are recursively defined in @xmath41 $ ] by . the two following recursion formulas hold : @xmath343 { v}^2\phi_{n}({v } ) & = \sigma_{n+2}\sigma_{n+1}\,\phi_{n+2 } ( { v } ) + 2\sigma_{n+1}\overline{\sigma}\,\phi_{n+1 } ( { v } ) + \big ( \sigma_{n+1}^2+\sigma_{n}^2+\overline{\sigma}^2 \big)\,\phi_{n } ( { v } ) \nonumber\\[0.5em]&\quad + 2\sigma_{n}\overline{\sigma}\,\phi_{n-1 } ( { v } ) + \sigma_{n}\sigma_{n-1}\,\phi_{n-2 } ( { v } ) , \label{eq : recursion : formula : b } \end{aligned}\ ] ] where @xmath68 and @xmath69 are defined in . to prove , note that the left - hand side term and the two right - hand side terms of the recursion formula for @xmath344 can be rewritten as @xmath345 ( n+1 ) l_{n+1}(s ) & = \frac{n+1}{\sqrt{2(n+1)+1 } } \sqrt{2(n+1)+1}\,l_{n+1 } ( s({v } ) ) = \frac{n+1}{\sqrt{2(n+1)+1}}\phi_{n+1}({v } ) \\[0.5em ] n l_{n-1}(s ) & = \frac{n}{\sqrt{2(n-1)+1 } } \sqrt{2(n-1)+1}\,l_{n-1 } ( s({v } ) ) = \frac{n}{\sqrt{2(n-1)+1}}\,\phi_{n-1 } ( { v})\end{aligned}\ ] ] collecting together and rearranging the three terms yields : @xmath346 + \frac{{{v}_a}+{{v}_b}}{2}\,\phi_{n}({v}),\end{aligned}\ ] ] which has the same form as where @xmath68 and @xmath69 can be readily determined by comparison . to prove just consider @xmath347 and apply twice . moreover , a straightforward calculation yields @xmath348 and in particular we have that @xmath228 . integrating @xmath53 , @xmath349 , @xmath350 and using - give other three useful recurrence formulas : @xmath351 \int_{{{v}_a}}^{{{v}_b}}{v}\phi_{n}({v})\d{v}&= ( { { v}_b}-{{v}_a})\big ( \sigma_{1}\delta_{n,1 } + \overline{\sigma}\delta_{n,0 } \big ) \label{eq : intg : vf}\\[0.5em ] \int_{{{v}_a}}^{{{v}_b}}{v}^2\phi_{n}({v})\d{v}&= ( { { v}_b}-{{v}_a})\big ( + \big ( \sigma_{1}^2+\overline{\sigma}^2 \big)\,\delta_{n,0 } + 2\sigma_{1}\overline{\sigma}\,\delta_{n,1 } + \sigma_{2}\sigma_{1}\,\delta_{n,2 } \big ) . \label{eq : intg : v2f } \end{aligned}\ ] ] all these three relations follows by noting that @xmath352 and applying the orthogonality property . relation is obvious . to derive and we also note that we can remove the terms containing @xmath353 and @xmath354 since @xmath55 . moreover , we can substitute @xmath138 in the @xmath355-coefficients of @xmath356 , @xmath357 , and @xmath358 , and note that the effect of @xmath357 and @xmath358 is respectively equivalent to @xmath359 and @xmath360 . finally , we note that @xmath361 . relation follows from @xmath362 & = ( { { v}_b}-{{v}_a})\big ( \sigma_{n+1}\delta_{n+1,0 } + \sigma_{n}\delta_{n-1,0 } + \overline{\sigma}\delta_{n,0 } \big).\end{aligned}\ ] ] relation follows from @xmath363 @xmath364 & = ( { { v}_b}-{{v}_a})\big ( \sigma_{n+2}\sigma_{n+1}\,\delta_{n+2,0 } + 2\sigma_{n+1}\overline{\sigma}\,\delta_{n+1,0 } + \big ( \sigma_{n+1}^2+\sigma_{n}^2+\overline{\sigma}^2 \big)\,\delta_{n,0 } \\[0.5em ] & \qquad + 2\sigma_{n}\overline{\sigma}\,\delta_{n+1,0}+\,\delta_{n-1,0 } + \sigma_{n}\sigma_{n-1}\,\delta_{n-2,0 } \big).\end{aligned}\ ] ] the proof of equation starts by applying expansion , legendre orthogonality property , expansion and fourier orthogonality property : @xmath365 & \qquad= ( { { v}_b}-{{v}_a})\sum_{m , n=0}^{{n_l}-1}\int_{0}^{l}{{c}^s}_{m}(x , t){{{c}^s}_{n}}(x , t)\delta_{m , n}{dx}= ( { { v}_b}-{{v}_a})\sum_{n=0}^{{n_l}-1}\int_{0}^{l}{\left|{{{c}^s}_{n}}(x , t)\right|}^2{dx}\\[0.5em ] & \qquad= ( { { v}_b}-{{v}_a})\sum_{n=0}^{{n_l}-1}\sum_{k , k'=-{n_f}}^{{n_f}}\big({{c}^s}_{n , k}(t)\big)^{\dagger}{{c}^s}_{n , k'}(t)\int_{0}^{l}\psi_{-k}(x)\psi_{k}(x){dx}\\[0.5em ] & \qquad= ( { { v}_b}-{{v}_a})l\,\sum_{n=0}^{{n_l}-1}\sum_{k , k'=-{n_f}}^{{n_f}}\big({{c}^s}_{n , k}(t)\big)^{\dagger}{{c}^s}_{n , k'}(t)\delta_{-k+k',0 } = ( { { v}_b}-{{v}_a})l\,\sum_{n=0}^{{n_l}-1}\sum_{k=-{n_f}}^{{n_f}}{\left|{{c}^s}_{n , k}(t)\right|}^2.\end{aligned}\ ] ] to prove the left - most equality in , we first note that : @xmath366_k = l\sum_{n=0}^{{n_l}-1}\sum_{k=-{m_k}}^{{m_k}}({{c}^s}_{n , k})^{\dagger}\big[{e}{\star}({\mathbbm{b}}{\mathbf{c}^s})_{n}\big]_k \label{eq : lemma : proof:10:a}\end{aligned}\ ] ] using the definition of the discrete fourier expansion of the electric field @xmath163 , the legendre coefficients @xmath63 , and @xmath367 , we obtain : @xmath368_k = l\sum_{k , k'=-{m_k}}^{{m_k}}({{c}^s}_{n , k})^{\dagger}{e}_{k'}\big({\mathbbm{b}}{\mathbf{c}^s}\big)_{n , k - k'}\nonumber\\[0.5em ] & \qquad\qquad= \sum_{k , k',k''=-{m_k}}^{{m_k}}({{c}^s}_{n , k})^{\dagger}{e}_{k'}\big({\mathbbm{b}}{\mathbf{c}^s}\big)_{n , k''}\,l\delta_{-k+k'+k'',0}\nonumber\\[0.5em ] & \qquad\qquad= \int_{0}^{l } \left(\sum_{k=-{m_k}}^{{m_k } } ( { { c}^s}_{n , k})^{\dagger}\psi_{-k}(x)\right ) \left(\sum_{k'=-{m_k}}^{{m_k}}{e}_{k'}\psi_{k'}(x)\right ) \left(\sum_{k''=-{m_k}}^{{m_k}}\big({\mathbbm{b}}{\mathbf{c}^s}\big)_{n , k''}\psi_{k''}(x)\right ) \,{dx}\nonumber\\[0.5em ] & \qquad\qquad= \int_{0}^{l}{{c}^s}_{n}(x , t){e}(x , t)\big({\mathbbm{b}}{\mathbf{c}^s}\big)_{n}. \label{eq : lemma : proof:10}\end{aligned}\ ] ] then , we note that : @xmath369 } \qquad\qquad\qquad\qquad \\[1.75em ] \end{array}\end{aligned}\ ] ] @xmath370}\\[1.5em ] & \qquad\displaystyle=\frac{1}{{{v}_b}-{{v}_a}}\sum_{n=0}^{{n_l}-1}\sum_{i=0}^{n-1}\sigma_{n , i}{{{c}^s}_{n}}(x , t)\int_{{{v}_a}}^{{{v}_b}}{f^s}(x,{v},t)\phi_{i}({v})\d{v}&\qquad\mbox{\big[use derivative formula~\eqref{eq : legendre : first : derivative}\big]}\\[1.5em ] & \qquad\displaystyle=\frac{1}{{{v}_b}-{{v}_a}}\sum_{n=0}^{{n_l}-1}{{{c}^s}_{n}}(x , t)\int_{{{v}_a}}^{{{v}_b}}{f^s}(x,{v},t)\frac{d\phi_{n}({v})}{d{v}}\d{v}&\qquad\mbox{\big[use again decomposition~\eqref{eq : legendre : decomposition}\big]}\\[1.5em ] & \qquad\displaystyle=\frac{1}{{{v}_b}-{{v}_a}}\int_{{{v}_a}}^{{{v}_b}}{f^s}(x,{v},t)\frac{\partial{f^s}(x,{v},t)}{\partial{v}}\d{v}&\qquad\mbox{\big[use the definition of the derivative\big]}\\[1.5em ] & \qquad\displaystyle = \frac{1}{2({{v}_b}-{{v}_a})}\int_{{{v}_a}}^{{{v}_b}}\frac{\partial({f^s})^2}{\partial{v}}\d{v}= \displaystyle\frac{1}{2}{\delta_v\big [ ( { f^s})^2 \big]}_{{{v}_a}}^{{{v}_b } } \end{array}\end{aligned}\ ] ] using the last relation above in and tranforming back in fourier space yield : @xmath366_k & = \sum_{n=0}^{{n_l}-1}\int_{0}^{l}{{c}^s}_{n}(x , t){e}(x , t)\big({\mathbbm{b}}{\mathbf{c}^s}\big)_{n } = \frac{1}{2}\int_{0}^{l}{e}(x , t){\delta_v\big [ ( { f^s}(x,{v},t))^2 \big]}_{{{v}_a}}^{{{v}_b}}\,{dx}\nonumber\\[0.5em ] & = \frac{1}{2}l\left[{e}{\star}{\delta_v\big [ ( { f^s}(x,{v},t))^2 \big]}_{{{v}_a}}^{{{v}_b}}\right]_0 , \label{eq : lemma : proof:15}\end{aligned}\ ] ] where @xmath166_{0}$ ] denotes the zero - th fourier mode and which is the first equality in . applying again the definition of the discrete fourier transform , the right - most equality in is proved as follows : @xmath371_{{{v}_a}}^{{{v}_b}}\big]_k = l\sum_{k=-{m_k}}^{{m_k}}\sum_{n=0}^{{n_l}-1}({{c}^s}_{n , k})^{\dagger}\big[{e}{\star}\delta_{{v}}\big[{f^s}\phi_n\big]_{{{v}_a}}^{{{v}_b}}\big]_k \nonumber\\[0.5em ] & \qquad = l\sum_{n=0}^{{n_l}-1}\sum_{k , k'=-{m_k}}^{{m_k}}({{c}^s}_{n , k})^{\dagger}{e}_{k'}\big(\delta_{{v}}\big[{f^s}\phi_n\big]_{{{v}_a}}^{{{v}_b}}\big)_{k - k ' } \nonumber\\[0.5em ] & \qquad=\sum_{n=0}^{{n_l}-1}\sum_{k , k',k''=-{m_k}}^{{m_k}}({{c}^s}_{n , k})^{\dagger}{e}_{k'}\big(\delta_{{v}}\big[{f^s}\phi_n\big]_{{{v}_a}}^{{{v}_b}}\big)_{k '' } \,l\delta_{-k+k'+k'',0}\nonumber\\[0.5em ] & \qquad=\sum_{n=0}^{{n_l}-1}\int_{0}^{l } \left(\sum_{k=-{m_k}}^{{m_k}}({{c}^s}_{n , k})^{\dagger}\psi_{-k}(x)\right ) \left(\sum_{k'=-{m_k}}^{{m_k}}{e}_{k'}\psi_{k'}(x)\right ) \left(\sum_{k''=-{m_k}}^{{m_k}}\big(\delta_{{v}}\big[{f^s}\phi_n\big]_{{{v}_a}}^{{{v}_b}}\big)_{k''}\psi_{k''}(x)\right ) \,{dx}\nonumber\\[0.5em ] & \qquad = \sum_{n=0}^{{n_l}-1}\int_{0}^{l}{{c}^s}_{n}(x , t){e}(x , t)\delta_{{v}}\big[{f^s}(x,{v},t)\phi_n({v})\big]_{{{v}_a}}^{{{v}_b}}{dx}\nonumber\\[0.5em ] & \qquad = \int_{0}^{l}{e}(x , t)\delta_{{v}}\big[{f^s}(x,{v},t)\,\sum_{n=0}^{{n_l}-1}{{c}^s}_{n}(x , t)\phi_n({v})\big]_{{{v}_a}}^{{{v}_b}}{dx}\nonumber\\[0.5em ] & \qquad = \int_{0}^{l}{e}(x , t)\delta_{{v}}\big[{f^s}(x,{v},t)^2\big]_{{{v}_a}}^{{{v}_b}}{dx}=l\left[{e}{\star}\delta_{{v}}\big[({f^s})^2\big]_{{{v}_a}}^{{{v}_b}}\right]_{0}. \label{eq : lemma : proof:20}\end{aligned}\ ] ] the three members of are real numbers since intermediate steps in the previous developments are formed by real quantities .

we present the design and implementation of an @xmath0-stable spectral method for the discretization of the vlasov - poisson model of a collisionless plasma in one space and velocity dimension . the velocity and space dependence of the vlasov equation are resolved through a truncated spectral expansion based on legendre and fourier basis functions , respectively . the poisson equation , which is coupled to the vlasov equation , is also resolved through a fourier expansion . the resulting system of ordinary differential equation is discretized by the implicit second - order accurate crank - nicolson time discretization . the non - linear dependence between the vlasov and poisson equations is iteratively solved at any time cycle by a jacobian - free newton - krylov method . in this work we analyze the structure of the main conservation laws of the resulting legendre - fourier model , e.g. , mass , momentum , and energy , and prove that they are exactly satisfied in the semi - discrete and discrete setting . the @xmath0-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term . the impact of the penalty term on the conservation properties is investigated theoretically and numerically . an implementation of the penalty term that does not affect the conservation of mass , momentum and energy , is also proposed and studied . a collisional term is introduced in the discrete model to control the filamentation effect , but does not affect the conservation properties of the system . numerical results on a set of standard test problems illustrate the performance of the method . vlasov - poisson , legendre - fourier discretization , conservation laws stability

as it is well known , see e.g. @xcite , cosmic microwave background ( cmb ) anisotropies can be thought of as fluctuations @xmath0 around the mean black body temperature @xmath1 k of the cosmological radiation . cmb anisotropies in a particular direction @xmath2 appear to us as a line of sight integral on the temperature fluctuations @xmath3 carried by cmb photons last scattered at a distance @xmath4 from us , and weighted with the last scattering probability @xmath5 : @xmath6 the last scattering probability @xmath5 is fixed by cosmological recombination history and it turns out to be a narrow peak around a decoupling redshift @xmath7 with mean and dispersion given by @xmath8 corresponding to physical distances @xmath9 therefore , cmb anisotropies can be thought of as a snapshot of the cosmic thermodynamical temperature in the early universe . their dependence on the line of sight is usually described through an expansion into spherical harmonics @xmath10 the anisotropy two point correlation function , obtained averaging the product of fluctuations coming from all pairs of directions separated by an angle @xmath11 , can be expanded into legendre polynomials @xmath12 : @xmath13 the relation of @xmath14s with @xmath15 coefficients is @xmath16 at high multipoles , @xmath17 , the legendre polynomials have a sharp peak at @xmath18 degrees ; as a consequence , a @xmath14 coefficient quantifies the anisotropy power on the same angular scale . moreover , taking into account that cmb anisotropies come essentially from a narrow spherical shell in redshift as in eq.([rdec ] ) , also known as last scattering surface , @xmath14 probes perturbations on a cosmological scale @xmath19 represented as in figure [ f1 ] . given that the cosmological horizon at decoupling subtends roughly one degree on the sky , this scale separates the sub - horizon from super - horizon regimes . after the first discovery of large scale cmb anisotropies by cobe @xcite , a breakthrough on the sub - degree structure of this signal is underway . data from the two balloon - borne experiments boomerang and maxima @xcite and the ground based interferometer dasi @xcite gave strong evidence of the presence of a peak at angular scales corresponding to a degree , as well as important indications for the existence of other peaks on smaller scales . forthcoming data from satellites map ( http://map.nasa.gsfc.gov ) and planck ( http://astro.estec.esa.nl/planck ) , will reveal cmb acoustic oscillations on the whole sky . in this paper we give a brief review of the most important physical mechanisms responsible for the formation of cmb acoustic peaks on sub - degree angular scales , together with some application of present data to constrain cosmological models . in section ii we put cmb anisotropies in the context of cosmological perturbation theory . in section iii we describe the phenomenology of acoustic peaks . finally in section iv we show an example of the impact of cmb on cosmology , briefly describing how these data can constrain cosmologies with dark energy . linear cosmological perturbation theory describes small fluctuations around background quantities in cosmology . a few years after the first historical works @xcite , a general treatment of cosmological perturbations has been written @xcite and extended to the more general context of scalar - tensor theories of gravity @xcite . recent works @xcite focus on cmb anisotropies , giving a complete description of their theoretical and phenomenological aspects . even if we can give here only the very basic details of this theory it is useful to put cmb anisotropies into their context . we restrict our analysis to flat cosmologies . -.5 in on cosmological scales , the line element @xmath20 is described by a perturbed friedmann robertson walker ( frw ) metric tensor @xmath21\ , \ ] ] where @xmath22 is the scale factor describing cosmic expansion and @xmath23 represents the conformal time , defined in terms of the usual time by @xmath24 ; @xmath25 represents he background minkowski metric . the linear fluctuations @xmath26 are conveniently decomposed into the three different components @xmath27 transforming as scalar , vector and tensor quantities under spatial rotations , respectively @xcite . scalar metric fluctuations represent , in a newtonian fashion , scalar quantities like a gravitational potential ; vectors modes represent vorticity while tensors represent cosmological gravitational waves . we restrict our analysis to scalar perturbations only . linearity introduces a gauge freedom so that equations of motion and perturbed quantities can have different expressions in different frames separated by a linear coordinate transformation . in other words not all the elements of @xmath28 are independent , but some of them can be set to zero via a proper gauge choice . here we fix the conformal newtonian gauge for which the metric fluctuations appear isotropic with respect to the cosmic expansion : the non - zero metric perturbations of scalar type are @xmath29 on the other hand , linearity allows to analyze cosmological perturbation evolution in fourier space , since fourier modes do not mix at a linear level . unless otherwise specified , we write equations in the fourier space in the following . correspondingly to the metric fluctuations , the stress energy tensor @xmath30 is perturbed : for what concerns scalar perturbations , any cosmological component @xmath31 admits energy density , velocity and pressure perturbations due to isotropic and anisotropic stress @xcite : @xmath32 the above quantities fully describe scalar perturbations for non - relativistic species , for which velocity perturbations are enough to describe their peculiar motion with respect to the comoving expansion . relativistic species move at the speed of light and are characterized by a propagation direction @xmath2 which needs to be properly treated . the dependence on the propagation direction of the thermodynamical temperature of cmb photons is a key aspect of cmb perturbation phenomenology . each fourier component at wavevector @xmath33 , describing the spatial dependence , is expanded in the fourier space , then the dependence on @xmath2 is described through a spherical harmonic expansion taking the direction in the fourier space , @xmath34 , as polar axis ; for scalar perturbations , only legendre polynomials are necessary @xcite : @xmath35 monopole , dipole and quadrupole in the above expansion are related to density , velocity and stress perturbations of cmb radiation . in newtonian gauge these relations , for density and velocity , take the form @xmath36 where the first one recalls the stephan - boltzmann law @xmath37 . as we expose in the next section , most of the cmb phenomenology derives from the behavior of the monopole term . unperturbed einstein equations link the einstein gravitational tensor @xmath38 to the stress energy tensor and describe the scale factor evolution . conservation equations @xmath39 complete the evolution system . in the same way , perturbed einstein and conservation equations describe perturbation evolution : @xmath40 we do not write here the form of the above equations for all cosmological species . in the next section we write only the ones which are relevant to give a simple understanding of the phenomenology of cmb anisotropies in terms of the initial conditions which are supposed to be fixed in the early universe . the dynamics of the cmb thermodynamical temperature fluctuations is dictated by the thomson scattering cross section . for each fourier mode , evolution equations for each multipole defined as in ( [ dttl ] ) can be expanded in power of the ratio between the wavevector amplitude @xmath41 and the differential optical depth for thomson scattering @xmath42 , which corresponds to the inverse of the photon mean free path . dynamics is frozen to the initial conditions for scales which are larger than cosmological horizon scale @xmath43 and the photon mean free path , @xmath44 , @xmath45 ; generically , initial conditions are such that only the lowest multipoles are different from zero . after the horizon crossing these multipoles evolve giving rise to acoustic oscillations but do not transmit power to the higher multipoles until decoupling : at that time , the mean free path for photons increases rapidly up to the cosmological horizon and the oscillations are transmitted also to higher multipoles @xcite ; therefore , the decoupled photons carry the snapshot of acoustic oscillations on sub - horizon scales at decoupling . projected on the last scattering surface , the horizon corresponds to a degree in the sky , so that acoustic oscillations are mapped by sub - degree cmb anisotropies . but different directions ( top ) and their mean quadratic power ( middle ) ; acoustic peaks in the sky signal in typical adiabatic cosmological models ( bottom ) , see text . ] for our purposes here , the study of the evolution of the zero - point fluctuation @xmath46 is enough . at the lowest order in @xmath47 , after horizon crossing , neglecting the cosmological friction and the time derivatives of gravitational potentials , the zero - point temperature fluctuation obeys the simple equation @xmath48 which merely represents an harmonic oscillator with radiation sound velocity @xmath49 , forced by the cosmological gravitational potential @xmath50 @xcite . in the limit in which the latter is constant , the solution , which is valid after the horizon crossing time @xmath51 , can be found analytically and describes most of the cmb acoustic oscillation phenomenology . in the simplest inflationary scenario ( see e.g. @xcite ) with adiabatic initial conditions , the curvature is perturbed with the same power on all cosmological scales at the horizon crossing . it can be seen @xcite that curvature perturbation is closely related to the cosmological gravitational potential @xmath50 . writing the generic fourier mode as its module multiplied by its phase , @xmath52 , inflationary initial conditions generate an initial gaussian spectrum where the first term has in mean the same amplitude for all modes at the horizon crossing , while the phase @xmath53 is random . at the horizon crossing the initial conditions for thermodynamical temperature fluctuations are simply related to the gravitational potential as @xmath54 so that the solution to ( [ dtt0 ] ) takes the simple form @xmath55 & \times & \nonumber\\ \times\cos\left[{k(\eta -\eta_{hc})\over\sqrt{3}}\right]-\psi\ & , & \label{dtt0adiaaini}\end{aligned}\ ] ] representing oscillation occurring for a given scale @xmath41 ; as it is schematically sketched in figure [ f2 ] , @xmath0 fluctuations for fourier wavevectors with different direction but same amplitude @xmath41 have random phases so that their mean is zero but have the same zeros ( top panel ) , and their root mean square power presents coherent acoustic peaks ( middle panel ) located at @xmath56 , with @xmath57 integer . in the bottom panel , we show the sky signal of a typical cosmological model having adiabatic initial conditions . the highest peak corresponds to scales crossing the horizon just at decoupling , and occur at a multipole @xmath58 corresponding precisely to the angle subtended by the horizon at last scattering . the second peak at @xmath59 corresponds to scales that were in horizon crossing slightly before decoupling and that at the time of decoupling , when the cmb snapshot is taken , were in the maximum of their second oscillation . in the same way , the third peak corresponds to scales in horizon crossing even before , being in the maximum of their third oscillation at the moment of the snapshot . the series of peaks continues at higher multipoles , with decreasing amplitude because of diffusion damping . as we mentioned in the introduction , present data are strongly supporting this scenario , having revealed the first peak with very high confidence level and significant indications for a second and a third peak in the spectrum . competing models for cosmological structure formation , see e.g. @xcite and references therein , predict markedly different spectra . isocurvature models are generally characterized by a non - zero entropy perturbation between different species , keeping the curvature unperturbed ; consequently , at horizon crossing the zero - point temperature fluctuation is zero , but not its time derivative , resulting in a sine time dependence instead of a cosine like in adiabatic models ( [ dtt0adiaaini ] ) , with a consequent shift of acoustic peaks by @xmath60 with respect to the adiabatic case . coherent acoustic peaks are generally absent in non - gaussian models like cosmological defects , because at horizon crossing both @xmath46 and @xmath61 can be different from zero , in a way which is different for each fourier mode , thus destroying coherence . in the next section we conclude this paper , giving a worked example on how the evidence for acoustic peaks in the cmb spectrum and their sensitivity to the main cosmological parameters can be used to constrain cosmological models . ( top ) and @xmath62 ( bottom ) showing preference for quintessence models with @xmath63.,title="fig : " ] ( top ) and @xmath62 ( bottom ) showing preference for quintessence models with @xmath63.,title="fig : " ] as it is well known , cmb data are a powerful tool to constrain cosmological models , either because of the sensitivity on the most important cosmological parameters , either because they represent the universe as it was before non - linear structure formation , thus being relatively simple to be read and understood . on the other hand , cmb alone can not fix all cosmological parameters , either because of internal degeneracies @xcite , either because of the high number , about 10 , of parameters to be investigated ; independent datasets , mostly from large scale structure ( lss ) @xcite and type ia supernovae @xcite are needed in order to go deep in precision cosmology . however , under reasonable hypothesis , even the present cmb data , which of course do not reach the precision of the satellites map and planck , can be used to derive interesting constraints on most important cosmological parameters . here we give an example of this , constraining a dynamical vacuum energy in flat cosmologies with the present cmb data . the dark energy , also known as quintessence , occupies a central position in modern cosmology after that at least three independent evidences , lss , supernovae and cmb , gave indications that almost @xmath64 of the cosmological energy density today resides in a sort of vacuum energy which is responsible for the cosmic acceleration today @xcite . to explain these observations a cosmological component with negative equation of state is necessary . dark energy is described as a self - interacting scalar field rolling on its potential which admits dynamical trajectories , tracking solutions , in which its equation of state @xmath62 can take values in the range [ -0.5,-1 ] , where the last value recovers the ordinary cosmological constant ; these trajectories have been proved to exist both in ordinary and scalar - tensor cosmology ( see e.g. @xcite and references therein ) . the angle @xmath65 subtended by the horizon at decoupling is sensitive to the dark energy equation of state @xmath62 ; indeed it is the ratio between the comoving value of hubble horizon at decoupling , which is almost insensitive to values of @xmath62 relevant for cosmic acceleration today , and the comoving distance of the last scattering surface from us , which is @xmath66^{1/2}\ & , & \label{taudec}\end{aligned}\ ] ] where @xmath67 , @xmath68 and @xmath69 represent matter , curvature and quintessence present energy density , respectively . therefore the angle subtended by the horizon at decoupling is essentially inversely proportional to @xmath70 . as it is evident , the expression above is degenerate in the sense that a given @xmath70 can be made of different combinations of the parameters entering into the integral . however , one should remember that this is not the only effect of dark energy on cmb ; the change in the equation of state at low redshift enhance the cmb power on low multipoles @xmath71 and breaks the degeneracy between @xmath69 and @xmath62 ( see e.g. @xcite and references therein ) . more serious is the degeneracy of dark energy with closed cosmological models with @xmath72 . in our recent work @xcite we fit cmb data @xcite gridding several cosmological parameters as the baryon amount , cosmological gravitational waves and scalar spectral index , in addition to quintessence parameters @xmath69 and @xmath62 , by assuming a number of priors including flatness @xmath73 . interestingly , we find a preference of these data for dark energy models with respect to ordinary cosmological constant . this is shown in figure [ f3 ] , representing the likelihood for @xmath69 , peaking at @xmath64 , and equation of state , peaking at @xmath63 . this effect can be understood as follows : best fits obtained in the original works on the cmb data @xcite slightly prefer closed cosmological models , even if flatness is well within errors . as it is evident from the expression ( [ taudec ] ) , dark energy models with @xmath74 induce a term which is similar to the one of closed models . therefore , since we are assuming perfectly flat cosmologies , the best fit peak on dynamical vacuum energy models simply because they produce a similar geometrical effect on the angle subtended by the horizon at decoupling . future data will help to test more deeply this interesting result . 9 a. liddle and d.h . lyth ( eds . ) , cosmological inflation and large scale structure , cambridge university press , 2000 . k.m . gorski , astrophys.j.s . 114 ( 1998 ) 1 . c. b. netterfield et al . , submitted to astrophys.j ( 2001 ) , preprint astro - ph/0104460 . lee et al . , submitted to astrophys.j.lett . ( 2001 ) , preprint astro - ph/0104459 . n.w . halverson et al . , submitted to astrophys.j . ( 2001 ) , preprint astro - ph/0104489 p.j.e . peebles and j.t . yu , astrophys.j . 162 ( 1970 ) 815 . bardeen , phys.rev.d22 ( 1980 ) 1882 . i. kodama and m. sasaki , progr . of theor.phys.supp , 78 ( 1984 ) , 1 . hwang , astrophys.j . 375 ( 1991 ) 443 . ma and e. bertschinger astrophys.j . 455 ( 1995 ) 7 . w. hu , u. seljak , m. white , m. zaldarriaga , phys.rev.d56 ( 1997 ) 596 . g. efstathiou , submitted to mnras ( 2001 ) , preprint astro - ph/0109151 . peacock et at . , nature 410 ( 2001 ) 169 s. perlmutter et at . , astrophys.j . 517 ( 1999 ) 565 ; a. riess et al . , astron.j . 116 ( 1998 ) 1009 . c. baccigalupi , s. matarrese , f. perrotta phys.rev.d62 ( 2000 ) 123510 . c. baccigalupi , a. balbi , s. matarrese , f. perrotta , n. vittorio , submitted to phys.rev.d ( 2001 ) , preprint astro - ph/0109097 .

we give a brief review of the physics of acoustic oscillations in cosmic microwave background ( cmb ) anisotropies . as an example of the impact of their detection in cosmology , we show how the present data on cmb angular power spectrum on sub - degree scales can be used to constrain dark energy cosmological models .

affleck - dine mechanism @xcite is a promising candidate for baryogenesis in supersymmetric ( susy ) theories due to its consistency with the observational bound on reheating temperature which avoids the gravitino problem @xcite . in the affleck - dine mechanism , baryon asymmetry is generated through the dynamics in the phase direction of a complex scalar field which carries nonzero baryon charge @xcite . the scalar field is called affleck - dine field . affleck - dine field is spatially homogeneous when it starts to oscillate , but spatial inhomogeneities due to quantum fluctuations grows exponentially into non - topological solitons , which are called q - balls @xcite . q - ball is a spherical condensate of a scalar field and is defined as a solution in a global u(1 ) theory which minimizes energy of the system with its charge fixed @xcite . q - ball is known to decay into quarks or leptons , so that the final baryon number is carried by quarks produced through the decay of q - balls . however , in gauge mediated susy breaking models , a baryonic q - ball , with sufficiently large charge , can be stable against decay into nuclei when energy per charge of q - ball is smaller than the proton mass @xcite . on the other hand , for a leptonic q - ball , there exist decay channels into leptons . in this case , baryon number of the universe is generated from lepton asymmetry through sphaleron effect , i.e. leptogenesis . we focus on q - balls that carry both baryon and lepton charges . in fact , these q - balls can be formed when we consider the affleck - dine baryogenesis with @xmath6 flat direction , for instance . in this case , it is possible that its lepton component can decay into leptons while its baryon component can not decay into baryons . this implies that the difference between the baryonic component and the leptonic component in decay rate may induce an electric charge . therefore , through the decay of the leptonic components only , an electric charge is induced even if the neutral q - ball was formed at the beginning . the electric charge is expected to make differences in the experimental signatures of the relic q - balls . for instance , the neutral q - balls can be detected by super - kamiokande @xcite , which probes the absorption of protons , but this detector is not suited for detection of the charged q - balls since the charged q - balls can not absorb the protons due to the electrical repulsion . the charged q - balls are likely to behave as some kind of nuclei , and are known to be detectable by such detectors as macro @xcite and the observational bounds on mass and flux of the relic charged q - balls are obtained @xcite . in fact , electrically charged q - ball has been studied in the literature and is called gauged q - ball @xcite . however , the previous works studied gauged q - balls in one scalar field theories , so that their results can not be applied to q - balls generated after the affleck - dine baryogenesis . there is a previous work which also discussed the evolution of q - balls in two scalar model @xcite , in which the gauge field was neglected . however , in the course of decay , it is expected that an electrical repulsion arises , so that the gauge field must be taken into account . in this paper , we consider a simplified model where there are two complex scalar fields and u(1 ) gauge field . one of the complex fields carries baryon charge and positive electric charge , while the other one carries lepton charge and negative electric charge . our main purpose is to demonstrate that gauged q - balls may be realized in our universe through the decay process of the lepton component , even if q - balls are initially neutral . we find stable solutions for different baryon and lepton charges , taking into account the effect of the gauge field . if we suppose that only the leptonic component decays off , a sequence of solutions with @xmath7 represents the decay process . we examine quantitatively whether such a process is energetically allowed . in the following sections , we first review the main properties of global q - balls and gauged q - balls , i.e. neutral and charged q - balls respectively . subsequently , in order to discuss the evolution of a q - ball which is formed from the flat direction , we consider a two scalar model and find sequences of gauged q - ball solutions with @xmath8 . we calculate the total energy of q - balls and gauge fields and examine whether the leptonic decay is energetically favorable . finally , we give our conclusions in sec . [ sec : conc ] . in this section , we briefly review the main properties of a global q - ball , which is a stable configuration of a complex scalar field with a fixed conserved charge . consider a theory of a complex scalar field with a global u(1 ) charge . the lagrangian is written as @xmath9 where @xmath10 is a scalar potential and is normalized so that @xmath11 . let us renormalize the field for later convenience as @xmath12 the global u(1 ) charge density @xmath13 is given by @xmath14 q - ball is defined as a solution which minimizes energy of the system with its charge fixed . using the lagrange multiplier method , we only need to minimize the following function : @xmath15\end{aligned}\ ] ] where @xmath16 is given by @xmath17}.\end{aligned}\ ] ] equation ( [ eq : energy ] ) is rewritten as @xmath18+\omega q,\\ & v_\omega(\phi)\equiv v(\phi)-\frac12\omega^2|\phi|^2.\end{aligned}\ ] ] by minimizing the first term of eq . ( [ eq : en2 ] ) , we can derive the time dependence of the solution as @xmath19 moreover , it is known that the solution which minimizes the second term of eq . ( [ eq : en2 ] ) is real and spatially symmetric @xcite . therefore , the radial direction @xmath20 is a solution of the following equation : @xmath21=0.\end{aligned}\ ] ] here , in order to avoid a singularity at @xmath22 and to find a specially localized solution , we set boundary conditions as @xmath23 now let us find the condition that there exists a solution of eq . ( [ eq : eqmq ] ) . first , we redefine the spatial coordinate and the potential as @xmath24 the equation of motion then becomes @xmath25 and we see that it is equivalent to the equation of motion of a classical point particle under the potential @xmath26 . the boundary conditions eq . ( [ eq : bc ] ) imply that the classical particle is initially at rest and converges toward the origin asymptotically . in order that this kind of motion is possible , the potential @xmath26 should be in an appropriate shape as shown in fig . [ fig : syuqball ] . depending on @xmath27 . ( i ) : @xmath28 , ( ii ) : @xmath29 , ( iii ) : @xmath30 . the potential of type ( ii ) is appropriate for the existence of the solution . ] quantitatively , @xmath26 must satisfy the following conditions : @xmath31>0,\end{aligned}\ ] ] @xmath32 the first condition is necessary in order for the particle to possess the potential energy enough to reach the origin . the second means that the particle must be subjected to a backward force near the origin and is also necessary in order for the particle to stop at the origin , not to roll over it . the conditions above are rewritten as the conditions to @xmath27 as @xmath33.\end{aligned}\ ] ] furthermore , it generally holds that @xmath34 for any q - ball solution , which can be easily shown by taking a variation of energy in the following way : @xmath35}\nonumber\\ & = \omega\int{d^3x\left[\delta\omega\phi^2 + 2\omega\phi\delta\phi\right]}\nonumber\\ & = \omega\delta q,\end{aligned}\ ] ] where we used the equation of motion eq . ( [ eq : eqmq ] ) . therefore , the condition eq . ( [ eq : omc ] ) is rewritten as @xmath36 it is noted that the second inequality indicates that a q - ball with a charge @xmath37 is energetically favorable compared to a q - ball with a charge @xmath38 and a particle , which is consistent with the definition of q - ball solution . here , let us identify the scalar field @xmath39 as a d- and f- flat direction in gauge mediated susy breaking models . in this case , the potential of the flat direction is mainly given by its soft mass term . however , the soft mass is suppressed for energy scale larger than messenger scale and the potential becomes flat . if we approximate the potential as @xmath40 , there exists an analytic solution @xcite given by @xmath41 0,&r > r\end{array}\right.\\\end{aligned}\ ] ] whose energy is then written as @xmath42 then , from @xmath43 , which is proven above , @xmath44 this indicates that for a large charge , @xmath45 may become small enough . therefore , for a baryonic q - ball with a large baryon number , it may hold that @xmath46 where @xmath47 ( @xmath48 ) is the proton mass . this means that the q - ball is stable against the decay into protons . on the other hand , if q - ball carries a lepton number , it can decay into leptons . this is the motivation of our assumption in sec . [ sec : qtwo ] that baryonic component of q - ball is stable while leptonic component can decay into leptons . in this section , we consider a complex scalar field that is charged under u(1 ) gauge symmetry . although this toy model is not motivated in susy theories , we investigate it to see the effect of gauge force on q - balls . in this case , we must solve the equation of motion for the gauge field @xmath49 as well as the complex scalar field . the spatially localized configuration in this theory is called gauged q - ball . the lagrangian is written as @xmath50 where @xmath10 is a scalar potential . we parameterize the scalar field in the same way as in the case of global q - ball : @xmath51 for the gauge field , we adopt the following parameterization @xcite . @xmath52 the first indicates that we are searching the spatially symmetric solution and the second implies that we are assuming the absence of a magnetic field , which means in turn the absence of an electric current . the equations of motion are then given by @xmath53 where we redefine the gauge field to absorb @xmath27 as @xmath54 . note that @xmath55 is gauge invariant . we set boundary conditions as @xmath56 to avoid singularities at @xmath22 . note that when @xmath57 as @xmath58 , the gauge field asymptotes to a certain constant as @xmath58 by eq . ( [ eq : eomgaugedqball ] ) . thus the boundary condition @xmath59 is just a definition of @xmath27 . note that eq . ( [ eq : eomgaugedqball ] ) can be rewritten as @xmath60 this implies that if @xmath61 , then @xmath62 becomes positive for @xmath63 , so that @xmath55 increases , while in the opposite case @xmath64 , @xmath55 decreases . in either case , @xmath65 always increases . here , let us consider an analogy in a similar way as considered in the previous section ( fig . [ fig : gaugedpoten2 ] ) . ( [ eq : eomf ] ) is analogous to the equation of motion for a classical particle which is subjected to the potential @xmath66 . as mentioned above , @xmath65 always increases , so that the effective mass term @xmath67 increases with time . from this we can derive some important properties of gauged q - ball . since the effective mass increases , it is energetically possible that the particle reaches the origin even if it moves away from the origin at the beginning , which means that there exists radially non - monotonic solution . we can interpret this kind of solution as the result of the scalar field being pushed outward due to the electrical repulsion . we show both kinds of solutions for gauge mediation - like model in fig . [ fig : gauex ] , where we approximate the potential as @xmath68 . indeed the non - monotonic solutions arise for charges larger than those of the monotonic ones . and @xmath69 . ] and @xmath69 . ] , and where @xmath70 it holds that @xmath71 for gauged q - ball as well , whose proof is similar to that for global q - ball @xcite . the energy of the q - ball is given by @xmath72,\end{aligned}\ ] ] whose variation leads to @xmath73\nonumber\\ & = \int d^3x\left[2\phi\delta\phi ( \omega - ea_0)^2+(\omega - ea_0)\delta ( \omega - ea_0)\phi^2-a_0\delta\delta a_0\right]\nonumber\\ & = \int d^3x\left[(\omega - ea_0)\delta q - a_0\delta\delta a_0\right]\nonumber\\ & = \omega\delta q-\int d^3x\left[ea_0\delta q+a_0\delta\delta a_0\right]\nonumber\\ & = \omega\delta q.\end{aligned}\ ] ] here we used eqs . ( [ eq : eomf ] ) and ( [ eq : eomgaugedqball ] ) , and the charge of the q - ball is given by @xmath74 the charge of a gauged q - ball has an upper limit , above which there is no localized solution . this is contrast to the case of global q - balls , where q - ball solutions exist for arbitrarily large @xmath37 . the blue circles in fig . [ fig:1sdedq1 ] are the upper limits on charges , and the results can be fitted as @xmath75 which is shown by the blue line . for a global q - ball , the condition @xmath76 was necessary for the existence of solution ( see eq . ( [ eq : omc ] ) ) . however , for a gauged q - ball , it is possible that @xmath77 which implies a q - ball with a charge @xmath38 and a particle at infinity are energetically favorable compared to a q - ball with a charge @xmath37 . this behavior arises when the charge is large enough and therefore may be interpreted as a result of electrical repulsion . even so , a gauged q - ball may exist as itself because a q - ball with a charge @xmath37 is still energetically favorable compared to a q - ball with a charge @xmath38 and a particle near the surface just after emission , due to the large coulomb potential . thus , gauged q - ball is expected to be a metastable solution if eq . ( [ eq : metas ] ) is satisfied . the red circles in fig . [ fig:1sdedq1 ] demonstrates when @xmath45 becomes @xmath78 . the results can be fitted as @xmath79 this dependence can be explained in the following way . if we approximate the energy of an emitted particle @xmath45 as the energy with electricity switched off , plus coulomb energy , @xmath80 where we used the analytic expression in the previous section with @xmath81 , @xmath82 . the charge of q - ball at which @xmath83 is thus given by @xmath84 or @xmath85 where we used @xmath86 and set @xmath87 into @xmath88 , whose value is appropriate for the solutions we are dealing with . this estimation roughly explains our numerical solutions of fig . [ fig:1sdedq1 ] . since our main interest is in the evolution of global q - balls formed from the flat direction which is responsible for the affleck - dine mechanism , we must consider the case of several numbers of scalar fields coupled with a gauge field . here we consider the simplest case in which the flat direction consists of two scalar fields which carry baryon and lepton numbers respectively , and the gauge field is abelian . the electric charges must be opposite since the flat direction is neutral . the lagrangian is then written as @xmath89 and baryon and lepton charges are @xmath90 where @xmath91 and @xmath92 are baryon and lepton number densities . since we assign the positive and negative electric charges for b and l components , respectively , the total electric charge is given by @xmath93 we find stable solutions and calculate their energies , and examine if leptonic decay is energetically allowed . first , we adopt the same parameterization as before : @xmath94 the equations of motion then become @xmath95 with the boundary conditions given by @xmath96 we prove here @xmath97 for later use , which is analogous to @xmath43 for 1-scalar gauged q - ball . the energy of the system is @xmath98,\end{aligned}\ ] ] and its variation with respect to @xmath99 , @xmath100 and @xmath101 is given by @xmath102\nonumber\\ & + \int d^3x\left[\phi_1\delta\phi_1(\omega_1-ea_0)^2+\phi_1 ^ 2(\omega_1-ea_0)\delta(\omega_1-ea_0)\right]\nonumber\\ & + \int d^3x\left[\phi_2\delta\phi_2(\omega_2+ea_0)^2+\phi_2 ^ 2(\omega_2+ea_0)\delta(\omega_2+ea_0)\right]\nonumber\\ & = \int d^3x\left[(\omega_1-ea_0)\delta b+(\omega_2+ea_0)\delta l - a_0\delta\delta a_0\right]\nonumber\\ & = \omega_1\delta b+\omega_2\delta l+\int d^3xa_0\left[e(-\delta b+\delta l)-\delta\delta a_0\right]\nonumber\\ & = \omega_1\delta b+\omega_2\delta l,\end{aligned}\ ] ] where we used eqs . ( [ eq : eom2s ] ) , ( [ eq : e25 ] ) , and ( [ eq : eom2s3 ] ) . the proof can be easily generalized into the case of arbitrary number of scalar fields . we assume that the decay of the q - ball always takes place if energetically allowed and that the evolution of the q - ball can be approximated as a sequence of gauged q - ball solutions . then the decay of the leptonic component only can be represented by a sequence of gauged q - ball solutions with the same baryon number arranged in descending order of lepton number , which is expected to decrease due to the decay . the results for @xmath103 and @xmath104 are shown in fig . [ fig : pptf ] and in fig . [ fig : pptf2 ] respectively , where we also used the following approximate form of the gauge mediation potential : @xmath105 here , we include the d - term potential , which arises since the d - flat condition @xmath106 is not valid anymore , and in addition , we assume @xmath107 . we see that as the leptonic component decays , the gauge field , or the coulomb potential arises . therefore , it is expected that through decay process , the initially formed neutral q - ball may evolve into the charged or gauged q - ball , which means that the charged q - balls may emerge in our universe . next , if we look at the energy , we see that in fig . [ fig : pptf ] , the energy decreases along the decay , which indicates that the particle which comes out from the q - ball has positive energy . this means that the q - ball emits free ( i.e. unbounded to the q - ball ) particles until the leptonic component completely vanishes . whereas in fig . [ fig : pptf2 ] , the energy starts to increase in the middle of the decay , which means that the energy of the emitted particle becomes negative . this in turn means that the particle starts to be bound to the q - ball . this may be understood in the context of quantum mechanics of many - body system as a cloudy bound state which is analogous to that of an atomic system . we also see a slight decrease in energy near the end of the decay , which means that the emitted particle becomes free again . this is interpreted as follows . as shown in fig . [ fig : pptf2 ] , the leptonic component concentrates at the center while the baryonic one locates outside away from the center . since the baryonic component is far enough from the surface of the leptonic component , from which the particle is emitted , the particle is initially accelerated outward enough to escape from the q - ball eventually . , @xmath69 and @xmath103 . ] , @xmath69 and @xmath103 . ] , @xmath69 and @xmath103 . ] , @xmath69 and @xmath103 . ] , @xmath69 and @xmath104 . ] , @xmath69 and @xmath104 . ] , @xmath69 and @xmath104 . ] , @xmath69 and @xmath104 . ] , @xmath69 and @xmath104 . ] , @xmath69 and @xmath104 . ] the black circles are the solutions with @xmath108 and the black squares are the solutions with @xmath109 . ] when @xmath110 . ] . ] we show the solutions with various baryon numbers in fig . [ fig : kekka ] . the black circles indicates the solutions with @xmath108 , which are realized at the q - ball formation after the affleck - dine mechanism , except for ref . @xcite . ] . from there , the energy decreases along with @xmath1 , which means that free particles are emitted and the q - ball becomes electrically charged . the emission continues until the leptonic component vanishes for @xmath111 . however , for @xmath112 , the energy starts to increase in the middle of the decay , which means that the particle starts to form a cloud of bounded leptonic particles . in fig . [ fig:2sdedl ] , we show the electric charge of q - ball at which @xmath110 . from the figure , @xmath110 occurs for @xmath113 and it is independent of baryon charge . this can be understood in the following way . when we approximate the energy of an emitted leptonic particle @xmath114 as the energy with electricity switched off plus coulomb energy , @xmath115 where we used the analytic expression with @xmath116 . equating @xmath114 to zero , we obtain @xmath117 this roughly explains fig . [ fig:2sdedl ] . as in the case of 1-scalar gauged q - ball , it is possible that @xmath118 due to the electrical repulsion . in order to check this , we only need to examine @xmath119 of each solution since @xmath120 as proven above . we illustrate where @xmath121 by black squares in fig . [ fig : kekka ] . we plot the electric charge @xmath37 at which @xmath122 becomes @xmath78 for each @xmath0 in fig . [ fig:2sdedb ] . the results can be fitted as @xmath123 this can be explained as follows . in the same way as deriving eq . ( [ eq : llo ] ) , the condition @xmath109 is written as @xmath124 thus , we obtain @xmath125 which leads to @xmath126 where we used the same approximation as before . this roughly explains our numerical results . the decay into protons is also expected to occur due to the electrical repulsion , even if we assume that the initial neutral q - ball is stable against it . if this happens , baryonic component also decays off , and therefore charged q - balls can not be left in the universe . however , if the leptonic decay stops before the electric charge becomes large enough for the baryonic decay to occur , the evolution stops and charged q - balls may survive . one way to stop the leptonic decay is that the leptonic cloud is close enough to the surface of the q - ball so that the particle can not come out . we derive the condition that this happens before the baryonic decay occurs . let us suppose that the emitted leptons are electrons and roughly estimate the size of the leptonic cloud as bohr radius . then the condition that the cloud radius becomes equal to the size of the q - ball is written as@xmath127 this must happen for @xmath128 that is , before the baryonic decay starts to occur , where we have used the analysis parallel to that below eq . ( [ eq : dedbph ] ) . therefore , the condition that the leptonic decay stops before baryonic decay occurs is given by @xmath129 this implies that if the baryon number is large enough , the leptonic decay may stop before the baryonic decay takes place , so that the charged q - ball may survive as a relic in the universe . we see that for @xmath130 bohr radius is already smaller than the q - ball size when the cloud is about to be formed , which means that the evolution stops without forming the cloud . thus , more accurately , if @xmath131 then the charged q - balls are expected to survive with a cloud surrounding it , and if @xmath132 then the charged q - balls are expected to survive without the cloud , the baryonic decay starts to occur when @xmath133 , and this is larger than @xmath134 , which is when the evolution is expected to stop . thus , there is no need to worry that the baryonic decay occurs before the evolution stops . ] . finally , the dashed line in fig . [ fig : kekka ] illustrates @xmath135 for each @xmath0 , above which q - ball solutions can not exist . this results from electrical repulsion due to the gauge field . in this paper , we considered gauged q - balls in the two scalar model in order to discuss the evolution of neutral q - balls which are formed from the flat direction in the affleck - dine mechanism and the possibility of realization of gauged q - balls during their evolution . we approximated the evolution as a sequence of charged q - ball solutions , which is implied from the situation that only the leptonic component decays off . as a result , a coulomb potential arises and the q - ball becomes electrically charged as expected . in other words , it is energetically favored for leptonic decay to occur . however , since there is an upper bound on charge of gauged q - ball , the amount of decay is limited as well , which we examined quantitatively . in addition , if the baryon number of the initially formed q - ball is large enough , the electric charge of the q - ball grows enough so that the particle which is emitted from the q - ball is bound to it . the baryonic decay is also expected to occur by virtue of the electrical repulsion , which leads to the vanishing of the charged q - balls in the universe . however , it is expected that if the leptonic cloud is close enough to the surface of the q - ball , the leptonic decay can stop before the electric charge becomes large enough for the baryonic decay to occur , so that the evolution stops and charged q - balls can survive . we roughly estimated when this can happen , and as a consequence , we found that there exists a lower bound on the baryon number . suppose that dark matter consists of q - balls . since a q - ball is known to absorb protons and emit pions , the neutral q - balls can be detected by super - kamiokande @xcite or icecube @xcite , while these detectors are not suited for detection of the charged q - balls since charged q - balls can not absorb the protons due to the electrical repulsion . thus , the electric charge is expected to make differences in the experimental signatures of the relic q - balls . the charged q - balls are expected to behave as some kind of nuclei , and are known to be detectable by such detectors as macro @xcite and the observational bounds on mass and flux of the relic charged q - balls are obtained @xcite . the emitted particles are also expected to contribute to the energy components of the universe and leptonic particles must satisfy the observational bounds on neutrino component from wmap @xcite , which must be verified in future work . although we assumed that the decay occurs if energetically allowed , we did not specify which kind of decay we are considering , for instance species of particle which is emitted , an actual decay rate etc . these informations must be added when we consider electrodynamical effects and actually solve time development of q - ball . lastly , here we are considering the simplest model of a flat direction , but in reality , we may need to consider more complex flat directions , possibly in nonabelian gauge theories as well . this work is supported by grant - in - aid for scientific research from the ministry of education , science , sports , and culture ( mext ) , japan , no . 25400248 ( m.k . ) , world premier international research center initiative ( wpi initiative ) , mext , japan , and the program for the leading graduate schools , mext , japan ( m.y . ) . m.y . acknowledges the support by jsps research fellowships for young scientists , no.25.8715 . 90 i. affleck and m. dine , nucl . b249 , 361 ( 1985 ) . m. kawasaki , k. kohri , and t. moroi , a. yotsuyanagi , phys . d78 , 065011 ( 2008 ) . t. gherghetta , c. kolda , and s. martin , nucl . b468 , 37 ( 1996 ) . a. kusenko and m. shaposhnikov , phys . b418 , 46 ( 1998 ) . s. kasuya and m. kawasaki , phys . d61 , 04130 ( 2000 ) . s. kasuya and m. kawasaki , phys . d62 , 023512 ( 2000 ) . s. coleman , nucl . b262 , 263 ( 1985 ) . g. dvali , a. kusenko , and m. shaposhnikov , phys . b417 , 99 ( 1998 ) . a. kusenko , v. kuzmin , m. e. shaposhnikov and p. g. tinyakov , phys . 80 , 3185 ( 1998 ) , [ hep - 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it is known that after affleck - dine baryogenesis , spatial inhomogeneities of affleck - dine field grow into non - topological solitons called q - balls . in gauge mediated susy breaking models , sufficiently large q - balls with baryon charge are stable while q - balls with lepton charge can always decay into leptons . for a q - ball that carries nonzero @xmath0 and @xmath1 charges , the difference between the baryonic component and the leptonic component in decay rate may induce nonzero electric charge on the q - ball . this implies that charged q - ball , also called gauged q - ball , may emerge in our universe . in this paper , we investigate two complex scalar fields , a baryonic scalar field and a leptonic one , in an abelian gauge theory . we find stable solutions of gauged q - balls for different baryon and lepton charges . those solutions shows that a coulomb potential arises and the q - ball becomes electrically charged as expected . it is energetically favored that some amount of leptonic component decays , but there is an upper bound on its amount due to the coulomb force . the baryonic decay also becomes possible by virtue of electrical repulsion and we find the condition to suppress it so that the charged q - balls can survive in the universe . ipmu 15 - 0069 1.2 cm * charged q - balls in gauge mediated susy breaking models * 1.8 cm jeong - pyong hong@xmath2 , masahiro kawasaki@xmath3 , and masaki yamada@xmath3 0.4 cm _ @xmath4institute for cosmic ray research , the university of tokyo , 5 - 1 - 5 kashiwanoha , kashiwa , chiba 277 - 8582 , japan _ + _ @xmath5kavli ipmu ( wpi ) , utias , the university of tokyo , 5 - 1 - 5 kashiwanoha , kashiwa , 277 - 8583 , japan _

since the discovery of dwarfs of spectral type later than m as companions to nearby stars ( becklin & zuckerman 1988 ; nakajima et al . 1995 ) major observational and theoretical progress has been made , thanks to sensitive new wide - area surveys at optical ( 0.41.0 @xmath12 m ) and infrared ( 1.02.5 @xmath12 m ) wavelengths and models of atmospheres of dwarfs with effective temperatures between those of the coolest stars and the giant planets ( see the reviews by chabrier & baraffe 2000 ; burrows et al . 2001 ) . two new spectral classes have been identified later than type m : the l dwarfs , characterized by the disappearance of gas - phase tio and vo , and the t dwarfs , characterized by methane band absorption in the @xmath13 and @xmath5 spectral regions ( martn et al . 1997 , 1999b ; kirkpatrick et al . 1999 , 2000 ; strauss et al . 1999 ; leggett et al . 2000 , 2002b ; geballe et al . 2002 , hereafter g02 ; burgasser et al . 2002a ; hawley et al . most l dwarfs and all t dwarfs are brown dwarfs . these objects are of interest because they occupy the mass range between that of stars and giant planets ; because many of them are likely to have the intrinsic properties of giant planets , which at present can not be directly observed ; and because they allow the investigation of the initial mass function to substellar masses . field l and t dwarfs have been discovered in large numbers in recent sky surveys : the deep near infrared survey ( denis , epchtein 1997 ) ; the two micron all sky survey ( 2mass , skrutskie et al . 1997 ; beichman et al . 1998 ) , and the optical sloan digital sky survey sdss ( york et al . 2000 ) . including objects described in the present paper , there are now about 280 l dwarfs and 58 t dwarf systems published ( e.g. delfosse et al . 1997 , 1999 ; kirkpatrick et al . 1999 , 2000 ; burgasser et al . 2002a , 2003e ; g02 ) . this large sample has been used to establish a complete spectral sequence from l0 to t9 ( g02 ; burgasser et al . 2002a ; mclean et al . 2003 ; present paper ) . unlike stars , brown dwarfs lack a sustained source of thermonuclear energy , and hence cool continuously , passing through the l and t stages , with their initial spectral types depending on their masses . their observational properties are thus a function not only of mass and metallicity , but also of age . not all dwarfs of a given spectral type or effective temperature are identical ; they have different gravities and different colors , the latter likely due to differing amounts of particulate matter in the atmosphere . for example , in mid to late l dwarfs there is a large scatter in the @xmath0 colors and apparently no one - to - one correspondence between effective temperature and spectral type ( leggett et al . 2002a ; golimowski et al . 2004 ) . such considerations drive searches for and measurements of additional dwarfs , in order to more fully characterize their atmospheres . 2mass and sdss have been highly complementary in the discovery of l and t dwarfs . most of the flux from late - type dwarfs is emitted longward of 1 @xmath12 m , and the @xmath14 and @xmath4 colors of m and l dwarfs become redder with decreasing effective temperature , allowing the identification by 2mass of large numbers of l dwarfs ( kirkpatrick et al . 1999 , 2000 ) . however , in the transition from l to t , @xmath15 absorption appears in the @xmath13 and @xmath5 regions ( and @xmath16 absorption predominantly at @xmath5 ) , strengthening with later spectral type and causing the t dwarfs to become increasingly blue in their @xmath0 colors . the @xmath0 colors of early t dwarfs are similar to those of the common k and m stars , making their identification in 2mass very difficult . however in the sdss filters the dwarfs simply become redder and thus the early t dwarfs have been found primarily in sdss imaging ( leggett et al . 2000 ) . only three objects with spectral types between t0 and t3.5 have been identified from sources other than the sdss martn et al . ( 2001 ) in an optical and near - infrared imaging survey of the @xmath17 orionis cluster tentatively classified one member as t0 , liu et al . ( 2002 ) found a distant field t3t4 dwarf in a deep @xmath18 survey , and mccaughrean et al . ( 2003 ) found that @xmath19 indi b , discovered in a high proper motion optical survey , is a binary consisting of a t1 and t6 pair ( see also scholz et al . 2003 , smith et al . 2003 and volk et al . 2003 ) . to date , all t dwarfs later than t7 have been found in the 2mass database ( burgasser et al . 2002a ) but we expect such objects to be found in the sdss imaging data as sky coverage is increased . in this paper we present near - infrared photometry and spectra of new and previously reported l and t dwarfs ( including 14 new t dwarfs ) and compare the colors and spectra with predictions from state of the art model atmospheres with and without clouds . the new objects observed are described in the next section , and the new @xmath0 photometric and spectroscopic observations are described in 3 , where we also derive spectral types . 4 presents colors , spectral types and absolute magnitudes for the entire body of near - infrared data on l and t dwarfs which we have accumulated to date . in 5 , we compare these data to model atmospheres . the conclusions are given in 6 . the general goals of our observational efforts are to identify samples of late l and t dwarfs which are at least representative and ideally complete , and to measure their spectral and photometric characteristics . to this end , we have observed new candidate very cool dwarfs selected from the photometric observations of the sdss . we have also re - observed some previously published sdss l and t dwarfs for which our observations are incomplete or suspect , and have added some 2mass dwarfs where they complement our sample . previous observations of the 2mass objects are discussed by kirkpatrick et al . ( 1999 , 2000 ) , reid et al . ( 2001 ) , burgasser et al . ( 2002a , b ; 2003b , c ) and dahn et al . ( 2002 ) . table 1 lists the names and sdss @xmath20 photometry of 51 confirmed sdss dwarfs , most of them previously unpublished . a small number of these dwarfs are also detected in the sdss @xmath21 band , and these magnitudes are presented later in the paper . as noted in table 1 , a few of these objects are described by hawley et al . ( 2002 ) , who present optical spectral types of a large sample of m , l and t dwarfs observed by sdss . one object , 2mass j090837.97@xmath1503208.0 , was identified as an l dwarf by cruz et al . ( 2003 ) from the 2mass database while this paper was in preparation . finding charts from the sdss @xmath22-band imaging are presented in figure 1 for all 51 objects in table 1 as charts have not been previously published for any of the objects . following the iau convention , the sdss names are based on the j2000 coordinates at the epoch of the initial observation and will be abbreviated in the text when individual objects are discussed ; thus sdss j003259.36@xmath1141036.6 will be called by the shortened name sdss j0032@xmath11410 . a similar convention is used for 2mass objects , whose full coordinate names are given later ( in table 9 ) . note that these faint stars and brown dwarfs are nearby and likely to have significant proper motions , so the names do not necessarily provide accurate coordinates for later epochs . a future paper will discuss the measured sdss positions and proper motions , since many of the objects were measured at more than one epoch . the sdss @xmath23 and @xmath22 magnitudes are @xmath24 magnitudes , for which the zeropoint in all bands is 3631 jy ( oke & gunn 1983 ; fukugita et al . 1996 ) . the @xmath25 magnitudes used at the united kingdom infrared telescope ( ukirt ) and discussed in this paper are on the @xmath26(vega)@xmath27 0.0 system . the sdss magnitudes are modified to be asinh magnitudes , identical to logarithmic magnitudes for high signal - to - noise ratio measurements ( @xmath28 ) and linear with flux below this ( lupton , gunn & szalay 1999 ) . for the observations discussed herein , zero flux density corresponds to @xmath29 25.1 , @xmath30 24.4 and @xmath31 22.8 . the sdss camera ( gunn et al . 1998 ) scans the sky and produces near - simultaneous ccd images in five filters covering the optical bands @xmath32 and @xmath22 ( centered at 3551 , 4686 , 6166 , 7480 and 8932 . ) the imaging data are reduced through a set of automated software pipelines . the photometric pipeline photo ( lupton et al . 2002 ) corrects the data , finds and measures objects , and applies photometric and astrometric calibrations . the photometric calibration is provided via a network of standard stars ( hogg et al . 2001 ; smith et al . 2002 ) , and the astrometric calibration via matches to standard astrometric catalogues ( pier et al . the photometry is accurate to about 2% in @xmath33 and @xmath23 and to about 3% in @xmath34 and @xmath22 for objects brighter than about 20 and 19 respectively , while the astrometric accuracy is better than @xmath35 ( r.m.s . ) in each coordinate . the result is a catalogue of objects with magnitudes in five bands , positions , and shape parameters ( e.g. abazajian et al . 2003 and references therein ) . all l and t dwarfs are undetected in sdss @xmath34 and @xmath10 , and all save the brightest are undetected in @xmath21 . almost all late l and t dwarfs have @xmath36 2 ; thus candidate field brown dwarfs are selected from the sdss photometry to be very red . given the extremely red colors and low luminosities of these dwarfs , they are often detected only in the @xmath22 band . objects this red are rare ; at the sdss magnitude limits they are late l and t dwarfs , quasars at redshift greater than 5.7 ( fan et al . 2001 , 2003 ) or very rare unusual broad absorption line quasars ( see hall et al . 2002 ) , and their surface density is smaller than about one per 50 square degrees . as a result , @xmath22-band only detections are overwhelmed by data artifacts , in particular `` cosmic rays '' in the @xmath22 detectors . the winnowing of `` objects '' to find those that are real is an exhaustive process , consisting of careful analysis of the image to reject cosmic rays ( which usually have imprints smaller than the point spread function ) , re - observation with another telescope at @xmath22 band ( usually the arc 3.5 m telescope at apache point observatory ) , and comparison with @xmath7-band observations from 2mass or elsewhere . @xmath7-band photometry also allows a first judgment as to whether an object is a brown dwarf or a high redshift quasar . these procedures are described in detail by fan et al . ( 2001 , 2003 ) . the total area searched to date for candidate very red dwarfs in sdss , including objects in our previous papers and in table 1 , is 2870 square degrees , so the surface density of t dwarfs found by sdss is approximately 1 per 100 square degrees . the regions of the sky searched are shown by fan et al . ( 2003 ) . some of the objects in the final sample presented in 4 have been found to be close binaries , usually from @xmath37 imaging . such objects will be designated by `` ab '' attached to the names , since the photometric and spectroscopic observations measure the total flux of both members . these known binaries are denis - p j0205@xmath21159ab ( koerner et al . 1999 ; leggett et al . 2001 ; bouy et al . 2003 ) ; denis - p j1228@xmath21547ab ( koerner et al . 1999 ; martn et al . 1999a ; bouy et al . 2003 ) ; 2mass j0746@xmath12000ab and 2mass j0850@xmath11057ab ( reid et al . 2001 ; bouy et al . 2003 ) ; 2mass j1225@xmath22739ab and 2mass j1534@xmath22952ab ( burgasser et al . 2003d ) ; and 2mass j1553@xmath11532ab ( a. burgasser , private communication , 2003 ) . not all of the objects in the sample have been imaged at high angular resolution , however , and there are likely to be more binaries among them . table 2 gives new @xmath38 photometry and table 3 new @xmath0 photometry . the central wavelengths of the filter passbands are 0.95 , 1.25 , 1.64 and 2.2 @xmath12 m ; more details of the filters and calibration are given by leggett et al . ( 2002a ) . all data were obtained on ukirt . all @xmath38-band data were obtained with ukirt s fast - track imager ( ufti , roche et al . 2003 ) on the dates shown in table 2 . the @xmath0 data were taken with the mauna kea consortium filter set ( mko - nir ) on the dates and with the cameras as listed in table 3 . three cameras were used for the @xmath0 observations the infrared camera ( ircam , mclean et al . 1986 ) , ufti , and the ukirt imager - spectrometer ( uist , ramsay - howat et al . ufti contains a hawaii 1024@xmath391024 hgcdte detector and has a plate scale of 0@xmath40091 pixel@xmath41 . ircam contains an sbrc 256@xmath39256 insb detector and has a plate scale of 0@xmath40081 pixel@xmath41 . uist contains an aladdin 1024@xmath391024 insb detector and has a choice of plate scales , either 0@xmath40061 pixel@xmath41 or 0@xmath40120 pixel@xmath41 . the 0@xmath40120 pixel@xmath41 plate scale was used with uist for these @xmath0 observations . readout of the full ircam array was employed , but 512@xmath39512 subarrays were used with ufti and uist to reduce overheads ( i.e. to increase efficiency ) . the fields of view were thus 20@xmath407 , 46@xmath406 and 61@xmath404 for ircam , ufti , and uist respectively . individual exposure times were usually 250 seconds at @xmath38 and 60 seconds at each of @xmath0 . observations were made in a three or five position dither pattern at @xmath38 , and with five or nine dither positions at @xmath0 . the @xmath0 data were calibrated using the ukirt faint standards of hawarden et al . ( 2001 ) translated onto the mko - nir system using as yet unpublished observations carried out at ukirt as part of an observatory project to provide calibrators in the mko - nir system . the @xmath38 data were calibrated using unpublished ukirt observations . these calibration data are currently available via the ukirt web pages . we have investigated the effects of the different optical elements , their coatings and the detector anti - reflection coatings on the @xmath0 photometric systems of the three cameras . synthesizing @xmath0 for l and t dwarfs using flux - calibrated spectra ( see g02 ) shows that the differences at @xmath7 are about 0.009 magnitudes for l to early t types , and 0.013 magnitudes for late t dwarfs . at @xmath13 the differences are around 0.001 magnitudes for objects of both l and t spectral type . at @xmath5 , the difference is 0.001 magnitudes for objects of type l , 0.003 magnitudes for objects of type early t , and 0.010 magnitudes for late t dwarfs . in all cases this is significantly less than the measurement error , so that the data from the three cameras are effectively on the same photometric system , defined by the mko - nir filter set . transformations between this filter set and other widely - used @xmath0 filter sets ( e.g. the 2mass system ) are described by stephens & leggett ( 2004 ) . table 4 lists the instrument configurations for the new spectroscopic observations . all spectra were obtained at ukirt using either the cooled grating spectrometer ( cgs4 , wright et al . 1993 ) or uist . cgs4 has a sbrc 256@xmath39256 insb detector with 0@xmath406 pixels . in uist s spectroscopy mode the aladdin array has 0@xmath4012 pixels . individual exposure times were typically 120 seconds for the cgs4-z and cgs4-j settings , 60 seconds for cgs4-h and cgs4-k and 120 seconds for uist - hk . the targets were nodded 712 arcseconds along the slit . a- or early - to - mid f - type bright stars were used as calibrators to remove the effects of the terrestrial atmosphere , with h i recombination lines in their spectra removed artificially prior to ratioing . both instruments have lamps that provide accurate flatfielding and wavelength calibration . a log of the measured spectra is given in table 5 . we concentrated on obtaining spectra in the @xmath13- and @xmath5-bands , because indices in these wavelength regions can be used for a wide range of types ( g02 ) . table 6 gives the derived spectral indices and the mean implied type on the g02 scheme ; the individual classifications are rounded off to the nearest 0.5 of a subclass but the mean type is derived from the unrounded values . errors are given for those dwarfs with multiple indices which show a scatter larger than the estimated classification uncertainty of 0.5 subclasses . the reader is referred to g02 for examples of the spectral sequences and line identifications . spectra and photometry from this and our previous papers are available on request or from our l and t dwarf web pageskl / ltdata.html ] . note that unlike g02 we do not incorporate red `` pc3 '' and `` color - d '' spectral indices for the current sample . these indices can be used for classifying dwarfs of spectral type l6 and earlier ( g02 ) ; however , in this paper we present near - infrared data only . a future paper will examine red spectra where available for the sample , and investigate the wavelength - dependent effects of cloud condensation ( see discussion later in 5.2 ) . table 6 contains 14 new t dwarfs , including one sdss object optically classified as l8 by hawley et al . the total number of t dwarfs presently known is 58 , four of which are close binaries . the distribution of new sdss t dwarfs is : two t0 , three t1t1.5 , two t2 , two t4.5t5 , two t5.5 , two t6 , and one t7 . the last of these , sdss j1758@xmath14633 , is the latest - type dwarf found to date in the sdss . nine l8l9.5 dwarfs have been identified from new infrared spectra . together with the seven new early - t dwarfs , they significantly increase the number of known dwarfs in the l t transition region . finally , the near - infrared photometry for sdss j1649@xmath13842 and the photometry and spectroscopy for sdss j0747@xmath12937 show that they are m dwarfs . the ch@xmath42-k index for the latter ( see table 6 ) is at the limit of the g02 scheme and is very uncertain . these m dwarfs are not discussed further . we have not obtained infrared spectra for six of the 51 objects listed in table 1 . the spectrum of 2mass j0415@xmath20935 has significantly deeper @xmath43 and @xmath15 bands than previously known t8 dwarfs and we classify it as t9 ; it is the latest spectral type dwarf presently known . figure 2 shows the @xmath13- and @xmath5-band spectra of sdss j1758@xmath14833 ( t7 ) and 2mass j0415@xmath20935 ( t9 ) , in addition to our previously - published spectra of the t6 dwarf sdss j1624@xmath10029 ( strauss et al . 1999 ) and the t8 dwarf gl 570d ( geballe et al . the steady increase in the depths of the @xmath43 and @xmath44 bands from t6 to t9 can be seen . figure 2 suggests that there is room for one more t type which would have essentially zero flux at 1.45 @xmath12 m , 1.7 @xmath12 m , and 2.25 @xmath12 m . provisional indices for the end of the t sequence are given in table 7 . according to the models of burrows , sudarsky & lunine ( 2003 ) , @xmath45 is expected to be detectable in the @xmath13 and @xmath5 spectral regions ( at @xmath11 1.5 @xmath12 m , 1.95 @xmath12 m and 2.95 @xmath12 m ) for @xmath46 600 k , and its presence may mark the transition to the spectral type after t although new non - equilibrium chemistry models suggest that the abundance of @xmath45 may be reduced ( saumon et al . 2003 ) . as we discuss in our companion paper ( golimowski et al . 2004 ) , the effective temperature of 2mass j0415 - 0935 is @xmath11700 k , too warm for @xmath45 absorption . there is no sign of @xmath45 absorption in the spectrum ( figure 2 ) , or in any other @xmath47 spectra we have obtained to date . we have compiled a large sample of l and t dwarfs for further study by combining the new data presented in 3 with our previously published work ( strauss et al . 1999 ; fan et al . 2000 ; tsvetanov et al . 2000 ; leggett et al . 2000 , 2001 , 2002a , b ; geballe et al . 2001 , 2002 ) . this final sample consists of 63 spectroscopically confirmed l dwarfs ( 59 of which have infrared spectra ) , six other possible l dwarfs measured photometrically only , and 42 spectroscopically confirmed t dwarfs . the distances to 45 of these l and t dwarfs are known by virtue of recent parallax measurements or because they are companions to nearby stars with accurately - measured parallaxes , either from the ground ( van altena et al . 1995 ) or from @xmath48 ( esa 1997 , perryman et al . 1997 ) . since the discovery of isolated l and t dwarfs , much effort has been devoted to the measurement of accurate parallaxes , both at optical wavelengths ( tinney et al . 1995 , dahn et al . 2002 ) and recently at near - infrared wavelengths ( tinney et al . 2003 ; vrba et al . 2004 ) . table 8 presents available parallaxes of l and t dwarfs for which we have obtained ukirt data , together with the derived @xmath5-band luminosities on the mko system . some of the parallaxes are weighted mean values from more than one source , as noted in table 8 . the errors for @xmath49 given in the table are the combined errors in the parallax and in the photometry . table 9 summarizes our final sample of l and t dwarfs . column 1 gives full coordinate names for the sdss and 2mass dwarfs ; these are listed as footnotes for dwarfs discovered in other work ( e.g. for kelu1 , ruiz et al . column 2 lists spectral types from the following sources : this paper ( using spectra described in table 6 or presented by burgasser et al . 2002a ) ; g02 ; kirkpatrick et al . ( 1999 , 2000 ) ; and burgasser et al . ( 2002a ) . the uncertainty in the assigned type is given if there are multiple infrared spectral indices which deviate by more than the estimated classification uncertainty of 0.5 subclasses . the spectral type is also flagged if the infrared and optical types ( kirkpatrick et al . 1999 , 2000 ; hawley et al . 2002 ; cruz et al . 2003 ) differ by more than 1.0 subclass . discrepant indices , either between the optical and infrared or even within the infrared range , are a sign that the spectra are sampling very different regions of the atmosphere , as we discuss later in 5.2 and 5.7.4 . column 3 lists @xmath50 , derived from our @xmath7 measurements and the parallaxes in table 8 . the next two columns list the sdss @xmath51 and @xmath22 . these values are given only for those objects for which @xmath52 or @xmath53 0.2 magnitudes . they are based on the most recent sdss reductions ( photo @xmath54 , july 2003 ) and may differ slightly from previously published values . sdss photometry for non - sdss dwarfs , where available , are included in table 9 . the remaining columns list @xmath55 , @xmath56 , @xmath7 , @xmath57 , @xmath14 and @xmath4 . some near - infrared magnitudes and colors are synthesized from the spectra , as noted in the table . a small number of dwarfs have detectable sdss @xmath21-band fluxes ; those dwarfs for which @xmath58 mag are given in table 10 . note again that the sdss @xmath21 , @xmath23 and @xmath22 measurements are on the ab system , while the other magnitudes are on the vega@xmath270 system . @xmath38 is on the ukirt ufti photometric system while @xmath7 , @xmath13 and @xmath5 are on the mko - nir system . for most of the objects in the sample , we have only single measurements of @xmath0 . however , we have obtained repeat photometry for a few objects with unusual colors . for each of the following dwarfs two measurements were obtained that agree to within the observational uncertainties , and the results are simply averaged in table 9 : 2mass j0036@xmath11821 ( observed on 20001205 and 20021207 ) ; sdss j0107@xmath10041 ( 19991017 , 20010124 ) ; sdss j0830@xmath14828 ( 20001119 , 20001206 ) ; sdss j0931@xmath10327 ( 20020217 , 20030104 ) ; sdss j1104@xmath15548 ( 20020109 , 20031204 ) and sdss j1331@xmath20116 ( 20020109 , 20020620 ) . we have discarded data obtained on 20000314 ( leggett et al . 2002a ) for two objects . the @xmath7 measurements for sdss j1314@xmath20008 appear to be spuriously bright ; we have combined the @xmath13 and @xmath5 data from that night with the @xmath0 data presented in table 3 to produce the colors given in table 9 . all of the @xmath0 data taken on that same night for sdss j1326@xmath20038 appear to be discrepant and we have averaged data from 20010124 and 20030129 in table 9 . finally , data for two of the redder mid l dwarfs ( 2mass j0028@xmath11501 and sdss j2249@xmath10044 ) suggest that they may be variable at the 5%10% level , which is not unexpected , given published detections of variability of l dwarfs and the possibility that non - uniform clouds exist in their atmospheres ( e.g. gelino et al . 2002 ; enoch et al . 2003 , and the discussion in 5.7 ) . including data taken with ircam on 20001119 that were discarded by leggett et al . ( 2002a ) , four measurements exist for 2mass j0028@xmath11501 and three for sdss j2249@xmath10044 . the former object shows variations at @xmath0 of about 0.05 mag , while the latter object varies by about 0.1 mag at each of @xmath0 . weighted means are given for these two objects in table 9 and the individual datasets are listed in the footnotes . as effective temperatures of dwarfs cool to those of the late m dwarfs and below , two chemical changes occur in their photospheres that strongly impact their emergent spectral energy distributions . the first to occur , for late m dwarfs , is the appearance of corundum ( @xmath59 grains within the photosphere ( jones & tsuji 1997 ) and the formation of condensate clouds . at the even lower effective temperatures of the l dwarfs iron and silicate are the most important condensates . the effect of the clouds is to weaken or veil the molecular absorption bands and to redden the @xmath0 colors of l dwarfs ( see e.g. ackerman & marley 2001 , allard et al . 2001 , marley et al . 2002 , tsuji & nakajima 2003 ) . the extent of these effects depends upon the number , size , and vertical distribution of the condensates ; for spectral modeling these parameters must either be computed from a model or somehow specified . ackerman & marley ( 2001 ) developed a one - dimensional model of mixing and sedimentation for this purpose . in their model upward vertical mixing of gas and condensate replaces condensates that fall through the cloud base , while far above the cloud base sedimentation efficiently cleanses the atmosphere of condensates . tsuji & nakajima ( 2003 ) model the cloud by specifying the temperature range within which the condensates are found . they describe their limits to be the points at which the atmosphere is cool enough for condensation but hot enough that the condensates are small enough to remain suspended in the atmosphere and are less prone to sedimentation . the temperature domain in which the cloud is formed depends on the details of the model used , but is around @xmath1115001700 k ( ackerman & marley 2001 ) or @xmath1118002000 k ( tsuji & nakajima 2003 ) . in the t dwarfs the cloud layer lies near the base or below the wavelength - dependent photosphere and plays a smaller role in determining the observed flux distribution . the second and later chemical change that occurs in these high - pressure , low - temperature atmospheres is the formation of additional molecular species in the photosphere , most importantly ch@xmath42 . co and h@xmath3o are abundant in m dwarf atmospheres , but by mid - l the abundance of ch@xmath42 becomes significant at the expense of co ( noll et al . 2000 ) . at moderate spectral resolution ch@xmath42 absorption is not seen in the near - infrared until temperatures drop to those of the late l dwarfs , at which point @xmath5-band ch@xmath42 features are detectable , and at t0 ( by definition , g02 ) ch@xmath42 absorptions are seen at both @xmath13 and @xmath5 . the increasing ch@xmath42 absorption largely accounts for the increasingly blue @xmath0 colors of the t dwarfs with later spectral type , more than compensating for the reddening due to the decreasing effective temperature . for dwarfs of type t5 and later , pressure - induced h@xmath3 becomes a significant opacity source . this opacity depresses the flux in the @xmath5 band , and to a lesser extent the @xmath13 band , and also contributes to the blue near - infrared colors . for a useful summary of the important molecular species and the wavelength ranges in which they are observed see figure 15 of burrows et al . ( 2001 ) . spectral classification schemes for l and t dwarfs have been developed using both the red and the near - infrared spectral regions . in the late 1990s kirkpatrick et al . ( 1999 ) and martn et al . ( 1999b ) developed schemes using the strengths of various absorption features and pseudo - continuum slopes seen in optical spectra to classify the l dwarfs . a few years later , burgasser et al . ( 2002a ) and g02 presented schemes using the strengths of the near - infrared molecular absorption bands to classify the t dwarfs , and in the case of g02 , l dwarfs also . while the burgasser et al . ( 2002a ) and g02 schemes for t dwarfs give results in very close agreement , the g02 scheme for l dwarfs can give results that differ by as much as 2.5 subclasses from those given by the kirkpatrick et al . ( 1999 ) scheme , suggesting that there are significant differences in the optical and infrared classification of l dwarfs . dwarfs in our sample whose optical and infrared spectral classes differ by more than one subclass are identified in table 9 . some of the scatter is due to the small differences in the infrared indices from one subtype to the next , combined with measurement errors . however , the differences in optical and infrared spectral types are not entirely random . where there are differences , g02 generally assign earlier spectral types to those objects that are redder than average in @xmath57 , and later types to those which are bluer ( stephens 2001 , figure 7.5 ) . models of l and t dwarf atmospheres give some insight into the variations seen among the spectral indices . generally speaking , the spectra of l and t dwarfs are less sensitive to the effects of cloud decks in the 0.71.0 @xmath12 m spectral region than at wavelengths longer than 1@xmath12 m . this is because , for effective temperatures corresponding to the earliest l types , optical depth unity in the far - red is reached below the cloud deck , but as the cloud is still fairly optically thin it does not substantially influence the red spectrum . at lower effective temperatures , the clouds place a `` floor '' on the region from which the emergent flux arises , but because of the large opacity in the far - red due to initially refractory diatomics and water and later to k i and na i resonance line absorption , most of the outgoing red flux arises from above the cloud decks . thus for dwarfs with effective temperature cooler than about 1800 k ( types @xmath11l3 and later , leggett et al . 2002a , b ; golimowski et al . 2004 ) slight changes in the cloud deck optical depth have little effect on the emergent red spectrum . in the near - infrared ( particularly the @xmath38 and the @xmath7 bands ) , the windows between the molecular bands of water and other opacity sources allow flux to emerge from very deep in the atmosphere . in these regions the opacity floor imposed by the clouds substantially alters the depth to which one can see into the atmosphere ( see figure 7 of ackerman & marley 2001 and figure 4 of marley et al . thus for dwarfs with @xmath60 in the range from about 1800 to 1500 k ( roughly l3 to l7 ) , slight changes in the cloud profile substantially alter the near - infrared spectrum . this proposed atmospheric structure implies that spectral typing schemes for mid to late l dwarfs that rely on far - red spectra ( e.g. kirkpatrick et al . 1999 ) tend to be less sensitive to the vertical distribution of condensates in the atmosphere than schemes that rely upon near - infrared spectra or spectral indices ( g02 ) . stephens ( 2001 , 2003 ) considered the boundaries of the regions employed in the g02 spectral typing system and found that the flux in the g02 bandpasses usually originates from within the cloud decks . for the earliest l dwarfs , cloud opacity is not significant , but for the mid l dwarfs , the g02 1.5 @xmath12 m water index can be a more sensitive indicator of _ cloud optical depth _ than of _ effective temperature_. at the same time , the 2.2 @xmath12 m methane index is more sensitive to @xmath60 than to cloud properties , since this spectral region is more opaque and optical depth unity is reached higher in the atmosphere . however , the classification system of g02 relies heavily on the 1.5 @xmath12 m index , as it is the only infrared index that covers the entire l spectral type range . this index undergoes a much larger change through the l sequence than does the other useful infrared index , ch@xmath42k . note that g02 do not claim that defining the spectral type is equivalent to measuring the effective temperature ; they use a simple classification scheme based on spectral appearance in which the effects of gravity and clouds are not separated from those of temperature . this larger sample of dwarfs also suggests some inconsistency in the infrared classification of late l dwarfs . while the g02 scheme provides excellent internal consistency for t dwarfs , the h@xmath3o 1.5 @xmath12 m index tends to give a later spectral type than does the ch@xmath42 2.2 @xmath12 m index for dwarfs in the range l5l9.5 . this tendency was not apparent in the smaller g02 sample ; the results for the present larger sample suggest that an adjustment of the flux ratio definitions as a function of spectral type for h@xmath3o 1.5 @xmath12 m and ch@xmath42 2.2 @xmath12 m in the l5l9.5 range could give better internal consistency . figure 3 plots , for the final sample presented in table 9 , various colors against spectral type ( determined by the g02 scheme apart from four l dwarfs classified optically , see table 9 ) ; typical error bars are shown . only those dwarfs with types determined from their spectra are shown . clouds strongly affect both colors and spectral types of mid l dwarfs classified from the near - infrared indices , as discussed in 5.1 and 5.2 . this effect can be seen in figure 3 , where the spread in the @xmath38 through @xmath5 colors is greatest from l3 to l7.5 , just when the clouds in the detectable atmosphere are expected to be most optically thick . the overall conclusion is that color can not be used as an ( infrared ) spectral type indicator for l dwarfs . for t dwarfs , @xmath55 and @xmath14 appear to be reasonable indicators of type . given improved sensitivity , @xmath51 may also be a useful t - type indicator . note , however , that @xmath51 is expected to turn blueward at @xmath61 600 k as the na , k , and other alkalis condense into solids and the opacity of the na and k lines falls ( burrows et al . 2002 ; marley et al . the late t dwarfs show significant scatter in their @xmath4 and @xmath57 colors . this can be understood in terms of the onset of pressure - induced h@xmath3 absorption , which is very gravity sensitive ( borysow , jorgensen & zheng , 1997 ) . as described in 5.6 , we can interpret the observed spread in @xmath4 as a range in surface gravity for the field t dwarf population . the @xmath56 colors of the mid t dwarfs also show considerable scatter ( figure 3 ) . this is an intriguing result , because the @xmath38 and @xmath7 bands are the most transparent windows into these atmospheres , with the @xmath38 band being the clearer of the two . as such , these bands are particularly sensitive to any deep variations in opacity between atmospheres , such as might be related to the upper reaches of any remaining deep silicate cloud ( see figure 7 of ackerman & marley 2001 ) . the variation may thus be due to differences in the process(es ) responsible for the removal of condensates at the l to t transition . since the @xmath38 band is sensitive to the far wings of the optical na - d and k i resonance lines ( burrows & volobuyev 2003 ) these variations might arise from differences in the removal of gaseous na and k , due to gravity or metallicity effects . more detailed modeling of the l to t dwarf transition and the removal of atmospheric condensates is required to account for these observations . in the remainder of this paper we compare observed colors with two varieties of models from marley et al . ( 2002 ) . in the first type of model , condensate opacity is ignored , although condensation chemistry is accounted for in chemical equilibrium and molecular opacities . we term these the `` cloud - free '' models . in the second type the effects of condensate opacity are computed using the ackerman & marley ( 2001 ) cloud model . in these `` cloudy '' models the efficiency of condensate sedimentation is parameterized by @xmath62 " to describe the efficiency of condensate sedimentation in a brown dwarf atmosphere . strictly speaking , `` rain '' is falling water , and this term has now been replaced by @xmath62 . ] . when the sedimentation efficiency is high ( large @xmath62 ) both the optical depth and vertical extent of the cloud are small . in the extreme case of no condensate sedimentation @xmath63 . figure 4 shows @xmath14 plotted against @xmath4 for the l dwarfs in the sample , where spectral subclass ranges are indicated by different symbols . overlaid are cloudy model sequences with @xmath64 and @xmath62 values of 3 and 5 . the @xmath65 models match the data well , although a shift in modeled @xmath14 color of about 0.15 mag would encompass many more of the data points . the discrepancy is likely attributable to the modeled tio bands at @xmath7 being too deep , causing the @xmath7-band magnitudes to be too faint and the @xmath14 model color to be too red . whether this is a shortcoming in the chemical equilibrium calculation or the molecular opacities themselves is as yet unclear . log @xmath664 models make the @xmath4 colors bluer , in better agreement with the data , but this is an unlikely gravity for these field dwarfs . ( burrows et al . 1997 show that if @xmath60=15002200 k and age=15 gyr then log @xmath675.0 . ) the detailed distribution of most of the data points in this color - color space is challenging to interpret as both the models and the observations show that @xmath14 and @xmath4 first become redder and then bluer with falling @xmath68 . thus the colors double back on themselves . despite the scattered distribution , some extreme objects stand out . the l7.5 dwarf 2mass j2244@xmath12043 and the l5.5 dwarf sdss j0107@xmath10041 are quite red in both colors . this may imply that their condensate cloud decks are more optically thick than average , which could arise from either less efficient sedimentation ( @xmath69 ) or higher metallicity . our sample also includes four late - type l dwarfs sdss j0805@xmath14812 , sdss j0931@xmath10327 , sdss j1104@xmath15548 and sdss j1331@xmath20116 that are unusually blue for their spectral types . as shown in figure 3 , they are bluer than average at @xmath14 by about 0.2 mag and at @xmath4 by about 0.1 mag . applying the shift to the models described above of about @xmath70 in @xmath14 , figure 4 suggests that these dwarfs are better described by the @xmath71 models , i.e. the sedimentation efficiency is high and the cloud optical depth is small . the spectral indices for all four of these objects show a large range the @xmath13-band indices imply a late type of l9 to t1 while the @xmath5-band index gives an earlier type of l5.57.5 ( the @xmath7-band index only implies a type earlier than t0 ) . our spectra show that they have enhanced feh , k i and @xmath43 absorption ( although the spectrum of sdss j1104@xmath15548 is noisy ) . figure 5 shows the @xmath7-band spectra for sdss j0805@xmath14812 , sdss j0931@xmath10327 and sdss j1331@xmath20116 bracketed by more typical l5.5 and l9 dwarfs . gorlova et al . ( 2003 ) and mclean et al . ( 2003 ) show that the equivalent widths of the @xmath7-band feh and k i features peak at spectral types around l3 and then become smaller as the fe condenses into grains and k is lost to kcl . the strengths of these features in sdss j0805@xmath14812 , sdss j0931@xmath10327 and sdss j1331@xmath20116 are similar to those of the early l types , while their h@xmath3o bands are more typical of the latest l dwarfs , supporting the interpretation that we are looking deep into unusually condensate - free atmospheres . the possibility of low metallicity should also be considered . burgasser et al . ( 2003a ) identified a late l dwarf ( 2mass j05325346@xmath18246465 ) whose extremely blue near - infrared colors are similar to those of the mid t types . this high velocity dwarf appears to be an extremely metal - poor halo subdwarf with strong feh features as well as h@xmath3 absorption which depresses the @xmath13 and @xmath5 band fluxes . cruz et al . ( 2003 ) identify two early l dwarfs ( 2mass j1300425@xmath1191235 and 2mass j172139@xmath1334415 ) that are bluer than average at @xmath57 by about 0.2 mag . as condensate clouds are optically thin for early l dwarfs , and these dwarfs have significant proper motion , they may also be part of a low - metallicity population . the @xmath4 and @xmath57 colors of t5t9 dwarfs are scattered ( figure 3 ) , even though the @xmath13- and @xmath5-band indices of g02 yield consistent classifications . figure 6 shows @xmath14 against @xmath4 for the t dwarfs in the sample with sequences from the cloud - free and cloudy @xmath71 models by marley et al . ( 2002 ) overlaid . the synthetic @xmath4 colors for the late t dwarfs , and the range in color over a plausible range of gravities of log @xmath725.5 , reproduce the observed colors extremely well . gl 570 d provides a further test for the model / data correspondence shown in figure 6 . geballe et al . ( 2001 ) fit models to the absolute luminosity of gl 570 d and used age constraints to find @xmath73824 k and a surface gravity in the range 5.005.27 . this gravity range is consistent with that implied by figure 6 . the effective temperature implied by figure 6 , however , is high by about @xmath74 . although geballe et al . ( 2001 ) found that their best fitting models generally reproduced the @xmath0 spectrum of gl 570d quite well , there were notable discrepancies . in particular the notorious inadequacy of the @xmath13-band methane opacity database and the tendency of all clear - atmosphere models to overestimate the water - band depths limit the fidelity of the fit . these deficiencies both result in the best - fitting temperature contours in figure 6 being somewhat too warm . further , the trends shown in figure 6 may break down for lower temperatures . one interesting challenge is the t9 dwarf 2mass j0415@xmath20935 , the latest and coolest t dwarf currently known , with @xmath75700 k ( golimowski et al . 2004 , vrba et al . 2004 ) . figure 6 , 8 and 9 show that instead of being bluer in @xmath14 , @xmath4 and @xmath57 than gl 570d , it is _ redder _ , by 0.2 magnitudes , in @xmath57 . while models by marley et al . ( 2002 ) and burrows et al . ( 2003 ) predict that , indeed , the coolest dwarfs will become redder in @xmath57 with falling @xmath60 and the onset of water cloud formation , this happens only for models with @xmath61 500 k unless the brown dwarf is older than 7 gyr and more massive than 40 @xmath9 . the condensation of the alkalis into their solid chloride forms may also lead to redder colors at these kind of temperatures ( lodders 1999 , marley 2000 , burrows et al . 2003 ) . despite these discrepancies , the overall trends seen in figure 6 can be understood in the context of our presently limited understanding of brown dwarf atmospheres . at the effective temperatures of late t dwarfs , the @xmath5-band flux is very sensitive to gravity . this is because the opacity of pressure induced @xmath16 absorption is proportional to the square of the local gas number density . higher gravity objects of a given @xmath60 tend to be cooler at a given pressure than lower gravity dwarfs , and thus have denser , more opaque , atmospheres . hence high gravity objects tend to be dimmer at @xmath5 and bluer in @xmath4 than comparable lower gravity objects . 2mass j0937@xmath12931 shows a particularly depressed @xmath5-band flux , standing out in figure 6 , presumably due to strong @xmath16 opacity ( burgasser et al . 2002a ) . structural models imply an upper limit to @xmath76 of 5.5 for brown dwarfs ( e.g. burrows 1997 ) and therefore figure 6 suggests that 2mass j0937@xmath12931 may be both a high gravity _ and _ a low metallicity dwarf , as also suggested by burgasser et al . ( 2003b ) . while @xmath16 opacity is also enhanced by decreasing metallicity ( e.g. saumon et al . 1994 , borysow et al . 1997 ) , variations in metallicity are less likely than variations in gravity for this sample of local field brown dwarfs . @xmath4 appears to be an easy to obtain and straightforward indicator of gravity for late t dwarfs . this is significant as , for a field sample with a likely range in age of 15 gyr ( dahn et al . 2002 estimate 24 gyr based on kinematic arguments ) , gravity corresponds directly to mass . this tight relationship is due to the small dependence of radius on age or mass for brown dwarfs older than about 200 myr ( burrows et al . figure 9 of burrows et al . ( 1997 ) shows that log @xmath72 implies a mass of 15 @xmath9 , log @xmath77 a mass of 35 @xmath9 , and log @xmath78 a mass of 75 @xmath79 . we list the @xmath4 implied surface gravities for the later t dwarfs in table 11 . recent investigations of spectroscopic gravity indicators for l and t dwarfs include those by lucas et al . ( 2001 ) , burgasser et al . ( 2003b ) , gorlova et al . ( 2003 ) , martn & zapatero osorio ( 2003 ) and mcgovern et al . figure 2 of martn & zapatero osorio shows synthetic spectra for @xmath801000 k from cond models by allard et al . the models imply that the lines of k i at 1.243 and 1.254 @xmath12 m become weaker with increasing gravity for t dwarfs in an atmosphere is proportional to @xmath81 , where @xmath10 is the gravity . in a higher gravity atmosphere an outside observer must , all else being equal , look to higher pressure to observe the same column of absorber as in a lower gravity atmosphere . the calculations of lodders ( 1999 ) show that with rising pressure at a fixed temperature chemical equilibrium increasingly favors kcl over k. thus in a higher gravity atmosphere the total column of potassium above the floor set by the continuum opacity is less than in a lower gravity model . the sensitivity to gravity should be greater in later t ( lower @xmath60 ) atmospheres since the line forming region ( roughly 1000 and 1500 k in the 1.15 and @xmath82 regions respectively ) falls closer to the chemical equilibrium boundary than for earlier t dwarfs ( see figure 2 of lodders 1999 ) . in addition higher pressures produce greater line broadening , thus decreasing the line depth . ] . figure 7 shows @xmath7-band spectra of three t6(@xmath83 ) dwarfs ( sdss j1110@xmath10116 , 2mass j2339@xmath113 and 2mass j0937@xmath12931 ) and two t8 dwarfs ( 2mass j0727@xmath11710 and gl 570d ) . these dwarfs span a range in @xmath4 color and are identified in figure 6 . it can be seen that as @xmath4 increases from top to bottom in figure 7 , the k i lines strengthen , supporting the interpretation of increasing @xmath4 as being due to decreasing gravity . the increase in k i equivalent widths for these dwarfs is confirmed by the higher resolution nirspec data of mclean et al . ( 2003 ) . comparison of our figure 7 with figure 2 of martn & zapatero osorio ( 2003 ) shows that the depths of the k i lines seen in 2mass j0937@xmath12931 are similar to the model predictions for log @xmath845.5 , and that the lines seen in sdss j1110@xmath10116 are almost as strong as the synthetic spectrum with log @xmath843.5 . figure 6 and the models of marley et al . ( 2002 ) imply that sdss j1110@xmath10116 has log @xmath10 between 4.0 and 4.5 . there is as yet no direct measurement of the effective temperature of this dwarf , but the effective temperatures of other t6 dwarfs are in the range 9001075 k ( golimowski et al . 2004 ) . if sdss j1110@xmath10116 has @xmath85 and @xmath86 , the evolutionary models of burrows et al . ( 2003 , their figure 1 ) imply that it is a 10 @xmath79 brown dwarf with an age of about 1@xmath3910@xmath87 years , i.e. similar to that of the pleiades cluster . however we argue in 5.7.4 that one might shift the model contours on figure 6 up and to the right to bring them into better agreement with observed @xmath57 and measured effective temperatures . in that case sdss j1110@xmath10116 would have a somewhat larger gravity , mass , and age . we thus adopt a more conservative estimated mass of 10 15@xmath88 and an age of 1 3 @xmath89 years . we note also that the candidate young - cluster t dwarf s ori 70 has an unusually red @xmath90 color ( zapatero osorio et al . 2002 ) apparently consistent with @xmath91 . it has been apparent since the earliest discoveries of l and t dwarfs that l - type objects evolve into t - type objects as they cool . with decreasing effective temperatures the condensation level for the principle l dwarf condensates ( iron and silicates ) falls progressively deeper in the atmosphere . unless upward mixing is very efficient , the clouds will eventually disappear beneath the photosphere , whose location is strongly wavelength - dependent for these objects . both the general evolutionary cooling trend and the removal of condensates produce lower atmospheric temperatures ; under these conditions the equilibrium chemistry rapidly begins to favor ch@xmath42 over co as the dominant c - bearing species . with less photospheric condensates to veil the molecular bands and the growing importance of ch@xmath42 opacity in the @xmath5-band , the objects turn to the blue in @xmath57 . marley ( 2000 ) employed a simple , one scale - height thick cloud layer to demonstrate that the sinking of a finite cloud deck explains the red to blue transition in @xmath57 . this was subsequently confirmed by models employing more elaborate cloud models ( marley et al . 2002 ; tsuji 2002 ; allard et al . 2003 ) . the absolute magnitudes presented here can be used to better understand this behavior . figures 8 and 9 show , respectively , absolute @xmath7 and @xmath5 magnitude against spectral type and @xmath57 color . a fifth order polynomial fit is shown to absolute magnitude against spectral type and the coefficients of the fit are given in table 12 . known binaries were removed from the sample before fitting the data ; the mean scatter around the fit is 0.4 magnitudes for m@xmath92 and 0.3 magnitudes for m@xmath93 . as noted by burgasser et al . ( 2002b ) these data suggest that the l to t dwarf transition may be more complex than implied by the picture of a continuously sinking cloud . the most notable discrepancy between the simple picture and the data shown in these figures is the brightening seen at @xmath7-band as the objects transition from l to t. in this section we summarize various suggested mechanisms to explain this behavior and compare their predictions to the photometric data presented here . a finite - thickness cloud deck forming progressively lower in the atmosphere will eventually disappear from sight . in the opposite extreme , dust that is well mixed through the entire observable atmosphere , as in the dusty models of the lyon group ( e.g. allard et al . 2001 ) , will by definition never disappear . models of such objects show that they simply become progressively redder as they cool and thus , due to veiling of the changing molecular bands , never exhibit an l to t transition . although models with finite - thickness cloud decks do move from red , l - like colors to blue , t - like colors , they tend to do so relatively slowly . this is because the cloud always has a finite thickness and thus does not disappear from a given bandpass instantaneously . during the time the cloud is departing , say from @xmath7-band visibility , the overall atmosphere is continuing to cool and become fainter . thus in a color magnitude diagram ( figures 8 and 9 ) models with finite - thickness clouds that are opaque enough to reach the colors of the latest l dwarfs ( @xmath94 ) tend to leisurely turn to the blue as they cool and so reach the colors of the early t dwarfs at too faint magnitudes . the @xmath95 model in figure 8 is an example . tsuji & nakajima ( 2003 ) proposed an interesting solution to this problem . they found a family of models with relatively thin cloud decks in which the turnoff from red to blue in @xmath57 was a function of gravity . in these models low - gravity 10 @xmath9 objects depart from what might be called the `` l - type cooling sequence '' ( the progressive reddening in @xmath57 with later spectral type ) and turn from red to blue at a point almost 2 magnitudes brighter than high - gravity 70 @xmath9 objects . a similar , though less extreme , bright turn off is seen in the @xmath96 family of models in our figures 8 and 9 . tsuji & nakajima then suggest that there is not a single evolutionary path in which objects first fade at @xmath7 band as they get redder and then brighten as they turn blue . rather they propose that the brighter transition t dwarfs are low mass objects that cooled to mid - l type and then turned from red to blue colors around @xmath97 . dimmer transition objects would represent intermediate - mass brown dwarfs that turned off the l cooling sequence at a later l type and redder @xmath57 , and and the latest ls would represent the highest mass objects . this model makes a number of interesting predictions . first , the @xmath50 vs. @xmath57 phase space between the l and t dwarfs should eventually be found to be fairly evenly populated both at brighter and fainter magnitudes than is shown by the transition objects detected to date . second , bright early t dwarfs , like sdss j1021@xmath20304 ( t3 ) or 2mass j0559@xmath21404 ( t4.5 ) should be fairly low mass objects while the latest and reddest l dwarfs , like 2mass j1632@xmath11904 ( l7.5 ) , should be fairly high mass objects . emerging gravity indicators should be able to test this hypothesis . plotted in the right panels of figures 8 and 9 are model sequences from marley et al . both cloud - free and cloudy models are shown , the latter with sedimentation parameters @xmath65 and 5 . for the @xmath71 and the no - cloud models , 3 gravities are shown ( @xmath98 , 5 , and 5.5 ) . only @xmath99 is shown for @xmath65 . model effective temperatures are given on the right axis of figure 9 . the general agreement between the observed l colors and the @xmath65 models seen in figures 4 , 8 and 9 , suggests that the @xmath0 colors of the l dwarfs can be explained by a uniform global cloud model . the cloud s vertical extent and optical depth are limited by sedimentation . models with much less or much more efficient sedimentation would be generally be too red or too blue , respectively , than most of the l dwarf population . however the existence of a few dwarfs that are redder and a few that are bluer than most ( 5.5 ) implies that about 10% of the l dwarf population would require more extreme models . these variations could arise from either metallicity or sedimentation efficiency differences between objects . however , the @xmath95 models turn too slowly to the blue and reach the colors of the bluest t dwarfs at @xmath7 band magnitudes that are too faint . faced with this difficulty of finite cloud layers taking too long to disappear in models like those of tsuji & nakajima ( 2003 ) , burgasser et al . ( 2002b ) propose a different mechanism for the l - to - t transition . drawing on a suggestion from ackerman & marley ( 2001 ) , burgasser et al . ( 2002b ) propose that the l to t transition region is marked by the appearance of holes in the global cloud deck , not unlike those seen in the `` 5-@xmath100 hot spots '' on jupiter . deeply - seated flux , particularly in the clear @xmath38 and @xmath7 windows would then pour out of these holes , pushing the disk - integrated color to the blue . indeed burgasser et al . ( 2002b ) found that a relatively small fraction of holes would appreciably move a late l - type object towards the blue in @xmath57 . they argued that such a mechanism would explain the brightening observed at @xmath38 and @xmath7 , and not at other bands , across the transition , and also the observed resurgence in feh absorption from the latest ls to the early to mid ts . to illustrate the effect such holes might have , we have joined with a dotted line the magnitude : color values for the cloudy and cloud - free models at @xmath101 k in both figures 8 and 9 . the agreement between the datapoints for t1 to t3 dwarfs and the cloudy to cloud - free interpolations in figures 8 and 9 is generally good . the parameters @xmath101 k and @xmath102 4.55.5 appear to bracket the known transition objects in these plots , in fair agreement with golimowski et al . ( 2004 ) who show that there is an apparent plateau at @xmath103 1450 k for types l7 to t4 . the cloud clearing model simply posits that at a given effective temperature the global cloud deck begins to break up , perhaps because it has settled sufficiently deeply into the convection zone that it becomes subject to the global circulation pattern . the observed constancy of @xmath60 across the transition is not required by this hypothesis , although it does raise problems for the tsuji & nakajima ( 2003 ) suggestion of continuous cooling across the transition . if the clearing does happen over a narrow temperature range , with a large spectroscopic change occurring over a small change in temperature and luminosity as the brown dwarf cools , we would not expect to discover many early t dwarfs . however about one - third of our t dwarf sample is made up of types t0t3.5 , with a possible dearth of t34 types ( figure 3 ) . a study of the spectral type distribution in an sdss magnitude - limited sample will be the subject of a future paper ( collinge et al . 2002 ) . rather than relying on spatial inhomogeneities , figures 8 and 9 suggests a third possibility for the l to t transition , which we term the `` sudden downpour '' model . the @xmath95 models do a reasonably good job of reproducing the colors of the latest l dwarfs . like the thin tsuji & nakajima ( 2003 ) cloud , models with more efficient sedimentation ( larger @xmath62 ) turn off the l dwarf cooling track sooner ( at brighter magnitudes ) than seems to be consistent with the available data . however , one might argue that l dwarfs first cool at essentially constant @xmath104 , then at around @xmath105 , @xmath62 begins to gradually increase from @xmath106 to infinity at roughly fixed effective temperature . this rapid increase in the efficiency of sedimentation would , in essence , produce a torrential rain of condensed iron and silicate grains . unlike the tsuji & nakajima ( 2003 ) mechanism this would begin at the late l spectral type for all masses . t1 to t4 dwarfs would represent different stages of this cloud thinning process . figures 8 and 9 shows where the @xmath71 models would lie , for example . once grains are essentially completely removed from the atmosphere the object would continue to evolve and cool . a useful diagnostic for evaluating the various transition models may be gravity . the sudden downpour mechanism , for example , would predict that the t3.5 dwarf sdss j1750@xmath11759 ( see figure 8) would have @xmath107 . evolution tracks for the patchy cloud model curve downward at blue @xmath57 compared to the straight lines shown in the figure and thus this model would predict a smaller gravity , say @xmath108 . on the other hand , tsuji & nakajima ( 2003 ) would predict that since this relative bright t dwarf has already made the transition to blue @xmath57 , it must be relatively low in mass and have a substantially lower gravity , say @xmath109 . there are also gravity tests among the late l dwarfs , although these are more subtle since all of the models would predict that the faintest and reddest l dwarfs will have progressively higher masses . tsuji & nakajima ( 2003 ) predict that the lowest mass dwarfs turn to the blue relatively early . the downpour model also predicts that lower masses turn blueward earlier than higher masses as can be seen by studying the tracks for the various gravities in the @xmath71 case . however this turn off happens at later spectral types than in the tsuji & nakajima model and is accompanied by a subsequent brightening in @xmath50 . for example the l7 dwarf labeled in figure 8 could have @xmath110 under the cloud clearing and downpour models , but tsuji & nakajima ( 2003 ) would predict a higher minimum gravity , likely around 5.3 or so . certainly there are hints in figure 8 that there is a width in @xmath50 to the transition and this will facilitate such tests . there are also other diagnostics to consider . the patchy - cloud model straightforwardly accounts for the resurgence seen in feh absorption across the transition ( burgasser et al . 2002b ) , and it is not clear that other models can account for this . clearly more modeling of all mechanisms must be completed to better define gravity and other diagnostics of the transition mechanism . unfortunately gravity indicators among the late l and early t field dwarfs are elusive and it may be difficult to use them to definitively test the models . ( one might expect that the li test ( martn , rebolo & magazzu 1994 ) could be used to identify high gravity dwarfs since only brown dwarfs more massive than 0.06 m@xmath111 will have burned li during their evolution . as a practical matter , however , the li line at 0.6708 @xmath12 m in late l and early t dwarfs is detectable only with the largest telescopes due to the lack of continuum flux in this region . also , in t dwarf atmospheres licl and lioh become the dominant li bearing molecule ( lodders 1999 ) and these species are currently undetectable . ) self consistent evolutionary models be developed as well as mechanisms to explain the onset of either patchiness or varying @xmath62 at a particular @xmath60 . we plan to more fully explore these issues in a future paper . discrepant objects to note in figures 8 and 9 are kelu-1 ( l3 ) , sdss j0423@xmath20414 ( t0 ) and 2mass j0415@xmath20935 ( t9 ) . 2mass j0415@xmath20935 is significantly redder in @xmath57 than the models would predict , as discussed in 5.6 . kelu-1 and sdss j0423@xmath20414 are both superluminous by about 0.75 magnitudes , suggesting that they may be pairs of identical dwarfs in unresolved binary systems . kelu-1 has been imaged by @xmath37 and by keck , with no evidence of duplicity found ( martn et al . 1999a ; koerner et al . 1999 ) , while sdss j0423@xmath20414 has not been imaged at high resolution to our knowledge . cruz et al . ( 2003 ) classify this dwarf as l7.5 using red spectra ; however , ch@xmath42 bands are clearly seen in the g02 spectrum ( figure 3 in g02 ) , and the bolometric correction ( i.e. ratio of @xmath5-band flux to total luminosity ) is more compatible with an infrared classification of t than l ( golimowski et al . although it has been suggested that this object consists of a late l and early t close pair ( burgasser et al . 2003b ) a discrepancy between the optical and infrared types is not unexpected . as described in 5.2 , different wavelength regions probe different levels of the photosphere . it is likely that the optically derived spectral types are more representative of effective temperature , and as golimowski et al . ( 2004 ) show that there is an apparent plateau at @xmath103 1450 k for types l7 to t4 we would expect optical types to be earlier than infrared types for the l / t transition objects . the earlier optical classification is in fact observed for six l8 to t0 dwarfs in our sample , as indicated in table 9 . although the clear atmosphere models do a good job of reproducing the colors of the later t dwarfs in the @xmath112 diagram ( figure 6 ) , they tend to predict bluer @xmath57 colors for these objects than is observed ( figures 8 and 9 ) . this seems to be a generic problem with clear models ( see also tsuji & nakajima 2003 ) . the @xmath57 result implies that the model tracks shown in figure 6 should slide up and to the right , consistent with the suggestion in 5.6 that the temperature contours are too warm . because of the overall shape of the model contours the quality of the fit in @xmath112 would remain about the same . the discussion of the gravity signature seen in @xmath4 is still qualitatively valid , although sdss j1110@xmath10116 would be expected to have a somewhat higher gravity . finally , we consider detectability limits for sdss using figures 3 , 8 and 9 . many sdss t dwarfs have to be selected as @xmath22-only objects and they are either not detected or only barely detected at @xmath23 ( see table 1 ) . the nominal ( 5@xmath17 , better than @xmath113 seeing ) @xmath22 detection limit is 20.8 , and the faintest sdss t dwarf discovered to date is close to this limit , at @xmath22 = 20.4 . since the colors of late ( e.g. t8 ) dwarfs are @xmath1143.8 , the corresponding @xmath7 limit is @xmath11516.6 ; table 9 therefore indicates that sdss should be able to detect t9 dwarfs to 10 pc . the latest sdss dwarf discovered to date is a t7 , but we anticipate that some later types will be found . we have presented new near - infrared photometry and spectroscopy for cool dwarfs from two sources : new very red objects from sdss and known l and t dwarfs from sdss and 2mass . we have obtained new @xmath0 photometry for 71 l and t dwarfs ( 53 from sdss and 18 from 2mass ) , @xmath38 photometry for 7 2mass objects , and spectroscopy of 56 l and t dwarfs ( 45 from sdss and 11 from 2mass ) . the spectral types have been obtained using the classification scheme of g02 which uses four molecular band indices at @xmath7 , @xmath13 and @xmath5 . the combined data from this and our previous papers are analyzed . absolute magnitudes are available for 45 late - type dwarfs thanks to recent parallax measurements . the relationships among color , absolute magnitude and spectral type are compared with model atmospheres with and without clouds . the major results and conclusions are as follows . * of the 44 new sdss targets for which infrared spectra were obtained , one is an m dwarf , six are l dwarfs previously reported by hawley et al . ( 2002 ) and cruz et al . ( 2003 ) which are also classified as l in the g02 scheme , 23 are new l dwarfs , and 14 are new t dwarfs ( one of which was classified as late l from optical spectra by hawley et al . the new t dwarf sample consists of seven t0t2 and seven t4.57 types , and we also identify nine l dwarfs with type l8 and later . these observations add significantly to the sample of l / t transition objects , and bring to 58 the total number of published t dwarf systems . * we provide provisional indices on the g02 scheme for the end of the t spectral sequence . the spectral type of 2mass j0415@xmath20935 is t9 ; it is currently the coolest known dwarf , with an effective temperature of @xmath11 700 k ( golimowski et al . 2004 , vrba et al . * as recognized previously , the relatively muted colors of the l dwarfs ( compared to models in which there is no sedimentation ) imply that silicate and iron cloud optical depths are limited by condensate sedimentation . * as noted in previous work , the @xmath0 colors of mid to late l dwarfs show a large scatter within a given spectral type . the @xmath0 colors are reasonably reproduced by models which incorporate cloud formation with a modest range of condensate sedimentation efficiencies ( or equivalently cloud optical depth ) . about 10% of the dwarfs in our sample seem to either be substantially bluer or redder than a modest range in @xmath62 , of about 3 to 4 , would predict . this suggests that cloud properties are generally similar , but can differ , among l dwarfs . * the reddest and bluest l dwarfs show a scatter in the infrared spectral indices , and there can also be significant differences between the spectral types determined using optical and near - infrared spectra . the differences between indices can be understood in terms of the depths probed by the different wavelength regions . * the near - infrared colors of t dwarfs become rapidly bluer towards later types . however , beyond about t5 , these colors , especially @xmath4 , show large scatter . this is correlated with the equivalent width of the @xmath7-band k i doublet absorption , strongly suggesting that the @xmath4 color is gravity - dependent . model atmospheres show that the @xmath5-band flux is depressed by @xmath16 absorption and that at a given effective temperature @xmath4 becomes bluer with increasing gravity . as intermediate age t dwarfs all have essentially the same radius , @xmath4 is a good indicator of mass for field t dwarfs of a given spectral type . * the implied masses of almost all of the t dwarfs in the sample are 15 75 @xmath9 . the t5.5 dwarf sdss j1110@xmath10116 appears to have a particularly low mass ( 10 15 @xmath9 ) with an inferred age of about 1 3@xmath89 years . * the absolute magnitude spectral type relationship for l and t dwarfs shows a mostly steady decline towards later spectral type , from @xmath50 @xmath11 11.5 , @xmath49 @xmath11 10.5 at l0 to @xmath50 @xmath11 16.5 , @xmath49 @xmath11 17 at t9 . there is a peak or plateau , depending on wavelength , in absolute magnitude between types l7 and t4 ( where @xmath1031450 k , golimowski et al . 2004 ) . models ( e.g. burgasser et al . 2002b ) which invoke an onset of some type of modification to the vertical or horizontal extent of the cloud at a fixed @xmath60 seem to better explain this observation than models which assume continuous sinking of a cloud that is spatially and vertically uniform over time . additional observational tests of the various cloud disruption possibilities are needed . we are most grateful to the staff at ukirt for their assistance in obtaining the data presented in this paper . some data were obtained through the ukirt service programme . ukirt is operated by the joint astronomy centre on behalf of the u. k. particle physics and astronomy research council . dag thanks the center for astrophysical sciences at johns hopkins university for its moral and financial support of this work . grk is grateful for support to princeton university and to nasa via grants nag5 - 8083 and nag5 - 11094 . msm acknowledges support from nasa grants nag2 - 6007 and nag5 - 8919 and nsf grant ast 00 - 86288 . trg s research is supported by the gemini observatory , which is operated by the association of universities for research in astronomy on behalf of the international gemini partnership of argentina , australia , brazil , canada , chile , the united kingdom , and the united states of america . 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.a . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is * http://www.sdss.org*. the sdss is managed by the astrophysical research consortium ( arc ) for the participating institutions . the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , los alamos national laboratory , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , university of pittsburgh , princeton university , the united states naval observatory , and the university of washington . collinge , m. j. , knapp , g. r. , fan , x. , lupton , r. h. , narayanan , v. , strauss , m. a. , gunn , j. e. , schlegel , d. j. , ivesi , . , rockosi , c. m. , geballe , t. r. , leggett , s. k. , golimowski , d. , & hawley , s. l. 2002 , , 201 , 1603 dahn , c. c. , harris , h. c. , vrba , f. j. , guetter , h. h. , canzian , b. , henden , a. a. , levine , s. e. , luginbuhl , c. b. , monet , a. k. b. , monet , d. g. , pier , j. r. , stone , r. c. , walker , r. l. , burgasser , a. j. , gizis , j. e. , kirkpatrick , j. d. , liebert , j. , & reid , i. n. 2002 , , 124 , 1170 delfosse , x. , 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der marel , h. , mignard , f. , murray , c. a. , le poole , r. s. , schrijver , h. , turon , c. , arenou , f. , froeschl m. , petersen , c. s. 1997 , , 323 , l49 smith , j. a. , tucker , d. l. , kent , s. m. , richmond , m. w. , fukugita , m. , ichikawa , t. , ichikawa , s. , jorgensen , a. m. , uomoto , a. , gunn , j. e. , hamabe , m. , watanabe , m. , tolea , a. , henden , a. , annis , j. , pier , j. r. , mckay , t. a. , brinkmann , j. , chen , b. , holtzman , j. , shimasaku , k. , york , d. g. 2002 , , 123 , 2121 strauss , m. a. , fan , x. , gunn , j. e. , leggett , s. k. , geballe , t. r. , pier , j. r. , lupton , r. h. , knapp , g. r. , annis , j. , brinkmann , j. , crocker , j. h. , csabai , i. , fukugita , m. , golimowski , d. a. , harris , f. h. , hennessy , g. s. , hindsley , r. b. , ivesi , . , kent , s. , lamb , d. q. , munn , j. a. , newberg , h. j. , rechenmacher , r. , schneider , d. p. , smith , j. a. , stoughton , c. , tucker , d. l. , waddell , p. , york , d. g. 1999 , , 522 , l61 tsvetanov , z. i. , golimowski , d. a. , zheng , w. , geballe , t. r. , leggett , s. k. , ford , h. c. , davidsen , a. f. , uomoto , a. , fan , x. , knapp , g. r. , strauss , m. a. , brinkmann , j. , lamb , d. q. , newberg , h. j. , rechenmacher , r. , schneider , d. p. , york , d. g. , lupton , r. h. , pier , j. r. , annis , j. , csabai , i. , hindsley , r. b. , ivesi , . , munn , j. a. , thakar , a. r. , waddell , p. 2000 , , 531 , l61 vrba , f. j. , henden , a. a. , luginbuhl , c. b. , guetter , h. h. , munn , j. a. , canzian , b. , burgasser , a. j. , kirkpatrick , j. d . , fan , x. , geballe , t. r. , golimowski , d. a. , knapp , g. r. , leggett , s. k. , schneider , d. p. , & brinkmann , j. 2004 , aj , in press sdss j000013.54@xmath1255418.6 & 25.59 & 0.56 & 18.48 & 0.04 sdss j000112.18@xmath1153535.5 & 20.29 & 0.04 & 18.55 & 0.03 sdss j001608.44@xmath2004302.3 & 21.11 & 0.11 & 19.34 & 0.07 sdss j012743.50@xmath1135420.9 & 22.27 & 0.17 & 19.62 & 0.07 sdss j020333.26@xmath2010812.5 & 22.76 & 0.36 & 20.36 & 0.15 sdss j020735.60@xmath1135556.3 & 19.84 & 0.03 & 18.06 & 0.02 sdss j035448.73@xmath2002742.1 & 21.63 & 0.10 & 19.61 & 0.07 sdss j040100.96@xmath2060933.0 & 22.93 & 0.49 & 20.19 & 0.20 sdss j074007.30@xmath1200921.9 & 22.91 & 0.34 & 19.78 & 0.08 sdss j074149.15@xmath1235127.5 & 24.88 & 0.39 & 19.65 & 0.05 sdss j074201.41@xmath1205520.5 & 23.85 & 0.75 & 19.28 & 0.05 sdss j074719.71@xmath1293748.6 & 21.64 & 0.10 & 19.91 & 0.08 sdss j075515.26@xmath1293445.4 & 21.73 & 0.18 & 19.40 & 0.07 sdss j075656.54@xmath1231458.5 & 22.89 & 0.31 & 19.82 & 0.07 sdss j075840.33@xmath1324723.4 & 21.92 & 0.13 & 17.96 & 0.03 sdss j080531.80@xmath1481233.0 & 19.82 & 0.05 & 17.62 & 0.03 sdss j080959.01@xmath1443422.2 & 21.82 & 0.16 & 19.28 & 0.06 sdss j083048.80@xmath1012831.1 & 24.91 & 0.52 & 19.59 & 0.08 sdss j083120.81@xmath1304417.1 & 21.45 & 0.11 & 19.68 & 0.10 sdss j085234.90@xmath1472035.0 & 21.99 & 0.15 & 18.90 & 0.05 2mass j090837.97@xmath1503208.0 & 20.06 & 0.03 & 17.22 & 0.02 sdss j093109.56@xmath1032732.5 & 22.00 & 0.15 & 19.28 & 0.05 sdss j100401.41@xmath1005354.9 & 21.81 & 0.18 & 19.76 & 0.11 sdss j103026.78@xmath1021306.4 & 23.43 & 0.61 & 19.94 & 0.11 sdss j104409.43@xmath1042937.6 & 21.67 & 0.08 & 18.73 & 0.03 sdss j104625.76@xmath1042441.0 & 22.38 & 0.16 & 19.74 & 0.07 sdss j110454.25@xmath1554841.4 & 22.95 & 0.34 & 19.94 & 0.09 sdss j112615.25@xmath1012048.2 & 22.28 & 0.25 & 19.79 & 0.12sdss j115553.86@xmath1055957.5 & 21.26 & 0.12 & 18.45 & 0.04 sdss j115700.50@xmath1061105.2 & 24.12 & 0.67 & 20.20 & 0.11 sdss j120747.17@xmath1024424.8 & 21.47 & 0.11 & 18.41 & 0.04 sdss j121001.96@xmath1030739.2 & 22.25 & 0.26 & 19.79 & 0.10 sdss j123147.39@xmath1084730.7 & 22.79 & 0.30 & 18.94 & 0.04 sdss j133148.90@xmath2011651.4 & 20.56 & 0.08 & 18.14 & 0.04 sdss j135923.99@xmath1472843.2 & 23.15 & 0.37 & 19.76 & 0.09 sdss j140814.74@xmath1053952.9 & 20.48 & 0.05 & 18.71 & 0.05 sdss j143211.74@xmath2005900.8 & 22.00 & 0.23 & 19.62 & 0.10 sdss j143535.70@xmath2004347.0 & 20.86 & 0.06 & 19.02 & 0.04 sdss j144016.20@xmath1002638.9 & 20.65 & 0.05 & 18.75 & 0.03 sdss j151603.03@xmath1025928.9 & 22.25 & 0.26 & 19.89 & 0.12 sdss j152103.24@xmath1013142.7 & 24.46 & 0.57 & 19.57 & 0.06 sdss j152531.32@xmath1581053.1 & 22.05 & 0.16 & 19.75 & 0.09 sdss j161626.46@xmath1221859.2 & 23.07 & 0.26 & 20.33 & 0.10 sdss j162441.00@xmath1444145.8 & 23.17 & 0.40 & 20.07 & 0.11 sdss j163030.35@xmath1434404.0 & 22.10 & 0.14 & 19.45 & 0.05 sdss j163239.34@xmath1415004.3 & 23.29 & 0.37 & 20.35 & 0.11 sdss j164939.34@xmath1384248.7 & 22.36 & 0.31 & 20.05 & 0.10 sdss j175024.01@xmath1422237.8 & 24.07 & 0.96 & 19.38 & 0.09 sdss j175805.46@xmath1463311.9 & 24.18 & 0.57 & 19.67 & 0.07 sdss j204749.61@xmath2071818.3 & 23.87 & 0.78 & 19.74 & 0.10 sdss j212413.89@xmath1010000.3 & 23.77 & 0.54 & 19.71 & 0.12 lrc 2mass j0030@xmath21450 & 18.13 & 20021106 2mass j0243@xmath22453 & 16.98 & 20021106 2mass j0415@xmath20935 & 17.30 & 20021106 2mass j0727@xmath11710 & 17.17 & 20021106 2mass j0825@xmath12115 & 16.62 & 20021106 2mass j0937@xmath12931 & 16.01 & 20021106 2mass j2356@xmath21553 & 17.60 & 20021022 lrrrrrrcc 2mass j0028@xmath11501 & 16.45 & 0.03 & 15.42 & 0.03 & 14.51 & 0.03 & 20011124 & ircam 2mass j0028@xmath11501 & 16.44 & 0.03 & 15.44 & 0.03 & 14.53 & 0.03 & 20020108 & ufti2mass j0030@xmath21450 & 16.39 & 0.03 & 15.37 & 0.03 & 14.49 & 0.03 & 20011124 & ircam2mass j0036@xmath11821 & 12.29 & 0.03 & 11.65 & 0.03 & 11.05 & 0.03 & 20021207 & uist2mass j0243@xmath22453 & 15.13 & 0.03 & 15.39 & 0.03 & 15.34 & 0.03 & 20011124 & ircam2mass j0415@xmath20935 & 15.32 & 0.03 & 15.70 & 0.03 & 15.83 & 0.03 & 20010829 & ufti2mass j0727@xmath11710 & 15.19 & 0.03 & 15.67 & 0.03 & 15.69 & 0.03 & 20011125 & ircam2mass j0755@xmath12212 & 15.46 & 0.03 & 15.70 & 0.03 & 15.86 & 0.03 & 20020109 & ufti2mass j0801@xmath14628 & 16.21 & 0.03 & 15.31 & 0.03 & 14.58 & 0.03 & 20020109 & ufti2mass j0908@xmath15032 & 14.40 & 0.03 & 13.54 & 0.03 & 12.89 & 0.03 & 20020108 & ufti2mass j0937@xmath12931 & 14.29 & 0.03 & 14.67 & 0.03 & 15.39 & 0.06 & 20011124 & ircam2mass j1503@xmath12525 & 13.55 & 0.03 & 13.90 & 0.03 & 13.99 & 0.03 & 20030104 & uist 2mass j1534@xmath22952ab & 14.60 & 0.03 & 14.74 & 0.03 & 14.91 & 0.03 & 20020715 & ircam2mass j1553@xmath11532ab & 15.34 & 0.03 & 15.76 & 0.03 & 15.95 & 0.03 & 20020109 & ufti2mass j2224@xmath20158 & 13.89 & 0.03 & 12.84 & 0.03 & 11.98 & 0.03 & 20020620 & ufti2mass j2244@xmath12043 & 16.33 & 0.03 & 15.06 & 0.03 & 13.90 & 0.03 & 20020620 & ufti2mass j2254@xmath13123 & 15.01 & 0.03 & 14.95 & 0.03 & 15.03 & 0.03 & 20010829 & ufti2mass j2339@xmath11352 & 15.81 & 0.03 & 16.00 & 0.03 & 16.17 & 0.03 & 20010829 & ufti2mass j2356@xmath21553 & 15.48 & 0.03 & 15.70 & 0.03 & 15.73 & 0.03 & 20011124 & ircamsdss j0000@xmath12554 & 14.73 & 0.05 & 14.74 & 0.03 & 14.82 & 0.03 & 20031207 & ufti sdss j0001@xmath11535 & 15.29 & 0.03 & 14.40 & 0.03 & 13.52 & 0.05 & 20031207 & ufti sdss j0016@xmath20043 & 16.34 & 0.05 & 15.34 & 0.05 & 14.52 & 0.03 & 20031207 & ufti sdss j0127@xmath11354 & 16.71 & 0.03 & 15.84 & 0.03 & 15.09 & 0.03 & 20020108 & uftisdss j0203@xmath20108 & 17.83 & 0.05 & 16.87 & 0.03 & 16.18 & 0.03 & 20030104 & uistsdss j0207@xmath11355 & 15.27 & 0.03 & 14.45 & 0.03 & 13.81 & 0.03 & 20020109 & uftisdss j0354@xmath20027 & 17.14 & 0.03 & 16.46 & 0.03 & 15.95 & 0.03 & 20030104 & uistsdss j0401@xmath20609 & 17.38 & 0.03 & 16.39 & 0.03 & 15.71 & 0.03 & 20030129 & uftisdss j0740@xmath12009 & 16.67 & 0.03 & 15.82 & 0.03 & 15.11 & 0.03 & 20030104 & uistsdss j0741@xmath12351 & 15.87 & 0.03 & 16.12 & 0.05 & 16.12 & 0.05 & 20020217 & uftisdss j0742@xmath12055 & 15.60 & 0.03 & 15.95 & 0.03 & 16.06 & 0.03 & 20030104 & uistsdss j0747@xmath12937 & 17.87 & 0.05 & 17.28 & 0.05 & 16.93 & 0.05 & 20020108 & uftisdss j0755@xmath12934 & 16.71 & 0.03 & 15.94 & 0.03 & 15.32 & 0.03 & 20020109 & uftisdss j0756@xmath12314 & 16.80 & 0.03 & 15.82 & 0.03 & 15.00 & 0.03 & 20030104 & uistsdss j0758@xmath13247 & 14.78 & 0.03 & 14.21 & 0.03 & 13.87 & 0.03 & 20020217 & uftisdss j0805@xmath14812 & 14.61 & 0.03 & 14.01 & 0.03 & 13.51 & 0.03 & 20020109 & uftisdss j0809@xmath14434 & 16.37 & 0.03 & 15.25 & 0.03 & 14.31 & 0.03 & 20020109 & uftisdss j0830@xmath10128 & 15.99 & 0.03 & 16.17 & 0.03 & 16.38 & 0.05 & 20020108 & uftisdss j0831@xmath13044 & 17.45 & 0.05 & 16.89 & 0.05 & 16.35 & 0.05 & 20030105 & uistsdss j0852@xmath14720 & 16.13 & 0.03 & 15.21 & 0.03 & 14.62 & 0.03 & 20020108 & uftisdss j0931@xmath10327 & 16.60 & 0.05 & 16.09 & 0.05 & 15.53 & 0.05 & 20020217 & uftisdss j0931@xmath10327 & 16.62 & 0.03 & 16.11 & 0.03 & 15.63 & 0.03 & 20030104 & uistsdss j1004@xmath10053 & 17.40 & 0.05 & 16.82 & 0.05 & 16.24 & 0.05 & 20020620 & uftisdss j1030@xmath10213 & 17.10 & 0.05 & 16.27 & 0.05 & 15.67 & 0.05 & 20020108 & uftisdss j1044@xmath10429 & 15.84 & 0.03 & 14.97 & 0.03 & 14.32 & 0.03 & 20020109 & uftisdss j1046@xmath10424 & 16.97 & 0.03 & 16.03 & 0.03 & 15.35 & 0.03 & 20020620 & uftisdss j1104@xmath15548 & 17.31 & 0.05 & 16.71 & 0.05 & 16.31 & 0.05 & 20020109 & uftisdss j1104@xmath15548 & 17.26 & 0.05 & 16.75 & 0.03 & 16.38 & 0.09 & 20021204 & uistsdss j1126@xmath10120 & 16.68 & 0.03 & 15.81 & 0.03 & 15.04 & 0.03 & 20020620 & uftisdss j1155@xmath10559 & 15.63 & 0.03 & 14.74 & 0.03 & 14.09 & 0.03 & 20020109 & uftisdss j1157@xmath10611 & 17.09 & 0.05 & 16.45 & 0.05 & 16.00 & 0.05 & 20010530 & uftisdss j1207@xmath10244 & 15.38 & 0.03 & 14.63 & 0.03 & 14.16 & 0.03 & 20020108 & uftisdss j1210@xmath10307 & 17.27 & 0.05 & 16.58 & 0.03 & 16.03 & 0.03 & 20020620 & uftisdss j1231@xmath10847 & 15.14 & 0.03 & 15.40 & 0.03 & 15.46 & 0.03 & 20020620 & uftisdss j1314@xmath20008 & 16.54 & 0.03 & 15.86 & 0.03 & 15.32 & 0.03 & 20020109 & uftisdss j1326@xmath20038 & 16.22 & 0.03 & 15.11 & 0.03 & 14.16 & 0.03 & 20030129 & uftisdss j1331@xmath20116 & 15.34 & 0.03 & 14.67 & 0.03 & 14.09 & 0.03 & 20020109 & uftisdss j1331@xmath20116 & 15.30 & 0.03 & 14.63 & 0.03 & 14.04 & 0.03 & 20020620 & uftisdss j1359@xmath14728 & 16.95 & 0.03 & 16.00 & 0.03 & 15.34 & 0.03 & 20020620 & uftisdss j1408@xmath10539 & 16.49 & 0.03 & 15.95 & 0.03 & 15.49 & 0.03 & 20030104 & uistsdss j1432@xmath20059 & 16.99 & 0.05 & 16.14 & 0.05 & 15.45 & 0.05 & 20010530 & uftisdss j1435@xmath20043 & 16.41 & 0.03 & 15.68 & 0.03 & 15.12 & 0.03 & 20020620 & uftisdss j1440@xmath10026 & 15.93 & 0.03 & 15.23 & 0.03 & 14.64 & 0.03 & 20020620 & uftisdss j1516@xmath10259 & 16.88 & 0.05 & 16.07 & 0.05 & 15.35 & 0.05 & 20010530 & uftisdss j1521@xmath10131 & 16.06 & 0.03 & 15.63 & 0.03 & 15.48 & 0.03 & 20020620 & uftisdss j1525@xmath15810 & 16.90 & 0.05 & 16.13 & 0.05 & 15.43 & 0.05 & 20020109 & uftisdss j1616@xmath12218 & 17.53 & 0.03 & 16.50 & 0.03 & 15.65 & 0.03 & 20030618 & uistsdss j1624@xmath14441 & 17.56 & 0.05 & 16.88 & 0.05 & 16.46 & 0.05 & 20010720 & uftisdss j1630@xmath14344 & 16.48 & 0.03 & 15.51 & 0.03 & 14.70 & 0.03 & 20020620 & uftisdss j1632@xmath14150 & 16.87 & 0.05 & 16.42 & 0.08 & 16.19 & 0.08 & 20020620 & uftisdss j1649@xmath13842 & 17.78 & 0.10 & 17.35 & 0.10 & 16.71 & 0.10 & 20010720 & uftisdss j1649@xmath13842 & 17.79 & 0.10 & 17.26 & 0.10 & 16.78 & 0.10 & 20020620 & uftisdss j1750@xmath14222 & 16.12 & 0.03 & 15.57 & 0.03 & 15.31 & 0.03 & 20020620 & uftisdss j1758@xmath14633 & 15.86 & 0.03 & 16.20 & 0.03 & 16.12 & 0.03 & 20020620 & uftisdss j2047@xmath20718 & 16.70 & 0.03 & 15.88 & 0.03 & 15.34 & 0.03 & 20011125 & ircamsdss j2124@xmath10100 & 15.88 & 0.03 & 16.12 & 0.03 & 16.07 & 0.03 & 20031208 & ufti sdss j2249@xmath10044 & 16.29 & 0.03 & 15.30 & 0.03 & 14.38 & 0.03 & 20020108 & ufti lccccc 2mass j0415@xmath20935 & 20021214 & 20021213 & 20020110 & 20020111 & 2mass j0727@xmath11710 & & 20021212 & & & 20021212 2mass j0929@xmath13429 & & & 20011029 & & 2mass j0908@xmath15032 & 20031208 & 20021212 & 20020110 & 20020111 & 2mass j0937@xmath12931 & & 20020112 & 20020110 & 20020111 & 2mass j1439@xmath11929 & & 20020625 & 20020619 & 20020622 & 2mass j1507@xmath21627 & & 20020625 & 20020624 & 20020624 & 2mass j2224@xmath20158 & & 20020714 & 20020618 & 20020622 & 2mass j2244@xmath12043 & & 20021214 & 20020618,21 & 20020618,22,23 & 2mass j2254@xmath13123 & 20030812 & 20020714 & 20020109 & 20020112 & 2mass j2339@xmath11352 & & 20021214 & 20020625 & 20020625 & sdss j0000@xmath12554 & & 20031208 & 20031207 & 20031207 & sdss j0001@xmath11535 & & & & & 20031120 sdss j0016@xmath20043 & & & & & 20031120 sdss j0127@xmath11354 & & & 20020112 & 20020112 & sdss j0203@xmath20108 & & & & & 20030923 sdss j0207@xmath11355 & & & 20020111 & 20020112 & 20030105 sdss j0354@xmath20027 & & & & & 20030105 sdss j0401@xmath20609 & & & & & 20030923 sdss j0741@xmath12351 & & 20030924 & & & 20021130 sdss j0742@xmath12055 & & 20030925 & 20030228 & 20030228 & sdss j0747@xmath12937 & & & & & 20021214 sdss j0755@xmath12934 & & & 20020110 & 20020111 & 20021212 sdss j0758@xmath13247 & 20031208 & 20020225 & 20020225 & 20020225 & sdss j0805@xmath14812 & 20031207 & 20031207 & & & 20030321 sdss j0830@xmath10128 & & & 20020110 & & sdss j0852@xmath14720 & & 20031124 & 20020110 & 20020111 & sdss j0931@xmath10327 & & 20031207 & & & 20021212 sdss j1004@xmath10053 & & & & & 20021214 sdss j1030@xmath10213 & & 20031207 & 20020110 & 20020111 & sdss j1044@xmath10429 & & & 20020111 & 20020111 & sdss j1046@xmath10424 & & & & 20020624 & sdss j1104@xmath15548 & & & & & 20021213 sdss j1110@xmath10116 & & & & 20020624 & sdss j1126@xmath10120 & & & & 20020622 & sdss j1155@xmath10559 & & & 20020111 & 20020112 & sdss j1157@xmath10611 & & 20031208 & 20020110 & 20020112 & sdss j1207@xmath10244 & 20040110 & 20020714 & 20020112 & 20020112 & sdss j1210@xmath10307 & & & & & 20021213 sdss j1231@xmath10847 & & 20020625 & 20020619 & 20020622 & sdss j1314@xmath20008 & & & & 20020623 & sdss j1331@xmath20116 & & 20031207 & 20020112 & 20020622 & 20021213 sdss j1359@xmath14728 & & & 20020625 & 20020623 & sdss j1432@xmath20059 & & & 20020621 & 20020622 & sdss j1435@xmath20043 & & & 20020624 & & sdss j1440@xmath10026 & & & 20020624 & & sdss j1516@xmath10259 & & & 20020619 & 20020622 & sdss j1521@xmath10131 & & 20030809 & 20020619 & 20020625 & sdss j1525@xmath15810 & & & 20020621 & 20020623,25 & sdss j1616@xmath12218 & & & & & 20030618 sdss j1630@xmath14344 & & & 20020624 & 20020622 & sdss j1632@xmath14150 & & 20030821 & 20020624,20020714 & 20020625 & sdss j1750@xmath14222 & & 20030812 & 20020621 & 20020622 & sdss j1758@xmath14633 & & 20030812 & 20020618 & 20020622,23 & sdss j2047@xmath20718 & & 20030814 & 20020718 & 20020807 & sdss j2124@xmath10100 & & & 20031208 & & lrlrlrlrll 2mass j0415@xmath20935 & 26.35 & t9.5 & 18.66 & t9 & 9.99 & t8.5 & 25.26 & t9 & t92mass j0727@xmath11700 & 10.69 & t7.5 & 11.69 & t8.5 & 6.09 & t7.5 & 16.26 & t8 & t82mass j0908@xmath15032 & 1.41&[@xmath116t0 ] & 1.92 & l9.5 & 1.04 & t0 & 1.12 & l7.5 & l9@xmath1171.02mass j0929@xmath13429 & & & 1.72 & l7.5 & 0.96 & [ @xmath116t0 ] & & & l7.5 2mass j0937@xmath12931 & 6.05 & t6.5 & 6.85 & t6.5 & 3.70 & t6.5 & 4.39 & t5 & t62mass j1439@xmath11929 & 1.10 & [ @xmath116t0 ] & 1.30 & l1 & 0.96&[@xmath116t0 ] & 0.95&[@xmath116l3 ] & l12mass j1507@xmath21627 & 1.31 & [ @xmath116t0 ] & 1.59 & l5.5 & 0.98 & [ @xmath116t0 ] & 1.03 & l5.5 & l5.52mass j2224@xmath20158 & 1.37 & [ @xmath116t0 ] & 1.48 & l3 & 0.97 & [ @xmath116t0 ] & 0.95 & l4 & l3.52mass j2244@xmath12043 & 1.62 & t0 & 1.71 & l7.5 & 0.91 & [ @xmath116t0 ] & 0.99 & l4.5 & l7.5@xmath1172.02mass j2254@xmath13123 & 2.46 & t3.5 & 3.69 & t3.5 & 1.83 & t4.5 & 3.36 & t4 & t42mass j2339@xmath11352 & 4.08 & t5 & 7.26 & t6.5 & 2.35 & t5.5 & 5.03 & t5 & t5.5sdss j0000@xmath12554 & 3.03 & t4.5 & 4.11 & t4 & 1.84 & t4.5 & 3.70 & t4.5 & t4.5 sdss j0001@xmath11535 & & & 1.59 & l5.5 & 0.95 & [ @xmath116t0 ] & 0.92 & l3 & l4@xmath1171.0sdss j0016@xmath20043 & & & 1.61 & l5.5 & 0.85 & [ @xmath116t0 ] & 1.03 & l5.5 & l5.5sdss j0127@xmath11354 & & & 1.58 & l5 & 0.96&[@xmath116t0 ] & 1.09 & l7 & l6@xmath1171.0sdss j0203@xmath20108 & & & 2.00 & l9.5 & 1.04 & t0 & 1.21 & l9 & l9.5sdss j0207@xmath11355 & & & 1.37 & l2 & 0.98&[@xmath116t0 ] & 0.97 & l4.5 & l3@xmath1171.5sdss j0354@xmath20027 & & & 1.31 & l1 & 1.00&[@xmath116t0 ] & 0.93 & l3 & l2@xmath1171.0sdss j0401@xmath20609 & & & 1.43 & l2.5 & 0.98 & [ @xmath116t0 ] & 1.01 & l5 & l4@xmath1171.5sdss j0741@xmath12351 & 4.56 & t5.5 & 4.65 & t4.5 & 2.71 & t5.5 & 5.50 & t5.5 & t5.5sdss j0742@xmath12055 & 3.88 & t5 & 4.78 & t5 & 2.74 & t5.5 & 5.10 & t5.5 & t5sdss j0747@xmath12937 & & & 1.07 & m6 & 0.94&[@xmath116t0 ] & 0.98 & [ l4.5 ] & late msdss j0755@xmath12934 & & & 1.41 & l2.5 & 0.99&[@xmath116t0 ] & 0.97 & l4.5 & l3.5@xmath1171.0sdss j0758@xmath13247 & 2.47 & t3.5 & 2.67 & t2 & 1.12 & t1 & 1.59 & t1.5 & t2@xmath1171.0sdss j0805@xmath14812 & 1.38 & [ @xmath116t0 ] & 2.01 & l9.5 & 1.06 & t0.5 & 1.07 & l6.5 & l9@xmath1171.5sdss j0830@xmath10128 & & & 6.25 & t6 & 2.44 & t5.5 & & & t5.5sdss j0852@xmath14720 & 1.44 & [ @xmath116t0 ] & 2.14 & t0.5 & 1.07 & t0.5 & 1.15 & l8 & l9.5@xmath1171.0sdss j0931@xmath10327 & 1.46 & [ @xmath116t0 ] & 1.89 & l9 & 0.90&[@xmath116t0 ] & 1.06 & l6 & l7.5@xmath1171.5sdss j1004@xmath10053 & & & 1.29 & l1 & 0.92&[@xmath116t0 ] & 0.92 & l3 & l2@xmath1171.0sdss j1030@xmath10213 & 1.48 & [ @xmath116t0 ] & 2.24 & t1 & 1.06 & t0.5 & 1.14 & l8 & l9.5@xmath1171.0sdss j1044@xmath10429 & & & 1.64 & l6.5 & 0.98 & [ @xmath116t0 ] & 1.11 & l7 & l7sdss j1046@xmath10424 & & & & & & & 1.06 & l6 & l6sdss j1104@xmath15548 & & & 1.98 & l9.5 & 1.09 & t1 & 1.17 & l7.5 & l9.5@xmath1171.5sdss j1110@xmath10116 & 5.07 & t6 & 5.37 & t5.5 & 2.47 & t5.5 & 6.28 & t6 & t5.5sdss j1126@xmath10120 & & & & & & & 1.05 & l6 & l6sdss j1155@xmath10559 & & & 1.74 & l8 & 0.97&[@xmath116t0 ] & 1.09 & l7 & l7.5sdss j1157@xmath10611 & 1.78 & t1 & 3.00 & t2.5 & 1.14 & t1.5 & 1.45 & t1 & t1.5sdss j1207@xmath10244 & 1.64 & t0 & 2.15 & t0.5 & 1.06 & t0.5 & 1.24 & l9.5 & t0sdss j1210@xmath10307 & & & 1.28 & l0.5 & 0.94 & [ @xmath116t0 ] & 0.91 & l2.5 & l1.5@xmath1171.0sdss j1231@xmath10847 & 5.54 & t6 & 6.51 & t6 & 2.94 & t6 & 5.21 & t5.5 & t6sdss j1314@xmath20008 & & & 1.39 & l2 & 1.00&[@xmath116t0 ] & 0.98 & l4.5 & l3.5@xmath1171.5sdss j1331@xmath20116 & 1.40 & [ @xmath116t0 ] & 2.12,2.19 & t0,t0.5 & 1.00,0.96 & [ @xmath116t0 ] & 1.03,1.05 & l5.5,l6 & l8@xmath1172.5sdss j1359@xmath14727 & & & 1.77 & l8 & 1.00&[@xmath116t0 ] & 1.18 & l8.5 & l8.5sdss j1432@xmath20059 & & & 1.53 & l4 & 0.95 & [ @xmath116t0 ] & 1.01 & l5 & l4.5sdss j1435@xmath20043 & & & 1.43 & l2.5 & 0.97 & [ @xmath116t0 ] & & & l2.5 sdss j1440@xmath10026 & & & 1.32 & l1 & 0.98 & [ @xmath116t0 ] & & & l1sdss j1516@xmath10259 & & & 2.11 & t0 & 1.14 & t1.5 & 1.16 & l8 & t0@xmath1171.5sdss j1521@xmath10131 & 2.07 & t2 & 3.07 & t2.5 & 1.10 & t1 & 2.01 & t2.5 & t2sdss j1525@xmath15810 & & & 1.70 & l7.5 & & & 1.03 & l5.5 & l6.5@xmath1171.0sdss j1616@xmath12218 & & & 1.46 & l3 & 0.95&[@xmath116t0 ] & 1.08 & l6.5 & l5@xmath1172.0sdss j1630@xmath14344 & & & 1.76 & l8 & 1.01&[@xmath116t0 ] & 1.03 & l5.5 & l7@xmath1171.5sdss j1632@xmath14150 & 1.59 & t0 & 2.13 & t0 & 1.15 & t1.5 & 1.81 & t2 & t1@xmath1171.0sdss j1750@xmath14222 & 1.75 & t1 & 2.21 & t0.5 & 1.09 & t1 & 1.69 & t1.5 & t1 sdss j1758@xmath14633 & 8.63 & t7 & 9.08 & t7.5 & 4.16 & t6.5 & 10.23 & t7 & t7sdss j2047@xmath20718 & 1.42 & [ @xmath116t0 ] & 1.85 & l9 & 1.08 & t0.5 & 1.18 & l8.5 & l9.5@xmath1171.0sdss j2124@xmath10100 & & & 6.42 & t6 & 2.40 & t5.5 & & & t6 2mass j0030@xmath21450 & 37.4 & 4.5 & 12.36 & 0.26 & l7 & v03 2mass j0036@xmath11821 & 114.2 & 0.8 & 11.33 & 0.03 & l4 & d02 2mass j0328@xmath12302 & 33.1 & 4.2 & 12.47 & 0.28 & l9.5 & v032mass j0345@xmath12540 & 37.1 & 0.5 & 10.51 & 0.04 & l1 & d02 2mass j0415@xmath20935 & 174.3 & 2.8 & 17.04 & 0.04 & t9 & v032mass j0559@xmath20559 & 96.7 & 1.0 & 13.66 & 0.04 & t4.5 & d02,v032mass j0727@xmath11710 & 110.1 & 2.3 & 15.90 & 0.05 & t8 & v032mass j0746@xmath12000ab & 81.9 & 0.3 & 10.00 & 0.03 & l1 & d02 2mass j0825@xmath12115 & 94.2 & 0.9 & 12.80 & 0.04 & l6 & d02,v032mass j0850@xmath11057ab & 33.8 & 2.7 & 12.00 & 0.18 & l6 & d02,v032mass j0937@xmath12931 & 162.8 & 3.9 & 16.45 & 0.08 & t6 & v032mass j1047@xmath12124 & 98.8 & 3.3 & 16.17 & 0.08 & t6.5 & v03,t032mass j1217@xmath20311 & 93.2 & 2.1 & 15.77 & 0.06 & t8 & v03,t03 2mass j1225@xmath22739ab & 74.8 & 2.0 & 14.65 & 0.07 & t6 & v03,t03 2mass j1439@xmath11929 & 69.6 & 0.5 & 10.68 & 0.03 & l1 & d022mass j1507@xmath21627 & 136.4 & 0.6 & 11.96 & 0.03 & l5.5 & d022mass j1523@xmath13014 & 54.4 & 1.1 & 13.03 & 0.06 & l8 & hip , ypc , v032mass j1534@xmath22952ab & 73.6 & 1.2 & 14.24 & 0.06 & t5.5 & t032mass j1632@xmath11904 & 65.0 & 1.8 & 13.03 & 0.07 & l7.5 & d02,v032mass j2224@xmath20158 & 87.0 & 0.9 & 11.68 & 0.04 & l3.5 & d02,v032mass j2356@xmath21553 & 69.0 & 3.4 & 14.92 & 0.11 & t6 & v03 denis - p j0205.4@xmath21159ab & 50.6 & 1.5 & 11.51 & 0.07 & l5.5 & d02denis - p j1058.7@xmath21548 & 57.7 & 1.0 & 11.36 & 0.05 & l3 & d02 denis - p j1228.2@xmath215ab & 49.4 & 1.9 & 11.18 & 0.09 & l6 & d02 gd 165b & 31.7 & 2.5 & 11.59 & 0.17 & l3 & ypc gl 229b & 173.2 & 1.1 & 15.55 & 0.03 & t6 & hip , ypcgl 570d & 170.2 & 1.4 & 16.67 & 0.04 & t8 & hip , ypckelu-1 & 53.6 & 2.0 & 10.43 & 0.09 & l3 & d02 lhs 102b & 104.7 & 11.4 & 11.46 & 0.24 & l4.5 & ypc sdss j0032@xmath11410 & 30.1 & 5.2 & 12.39 & 0.37 & l8 & v03 sdss j0107@xmath10041 & 64.1 & 4.5 & 12.61 & 0.16 & l5.5 & v03 sdss j0151@xmath11244 & 46.7 & 3.4 & 13.53 & 0.16 & t1 & v03 sdss j0207@xmath10000 & 34.9 & 9.9 & 14.33 & 0.62 & t4.5 & v03sdss j0423@xmath20414 & 65.9 & 1.7 & 12.05 & 0.06 & t0 & v03sdss j0539@xmath20059 & 76.1 & 2.2 & 11.81 & 0.07 & l5 & v03sdss j0830@xmath14828 & 76.4 & 3.4 & 13.10 & 0.10 & l9 & v03sdss j0837@xmath20000 & 33.7 & 13.5 & 13.62 & 0.87 & t0.5 & v03sdss j1021@xmath20304 & 35.4 & 4.2 & 13.00 & 0.26 & t3 & v03,t03sdss j1254@xmath20122 & 74.0 & 1.6 & 13.22 & 0.06 & t2 & v03,t03sdss j1326@xmath20038 & 50.0 & 6.3 & 12.66 & 0.28 & l5.5 & v03sdss j1346@xmath20031 & 69.1 & 2.1 & 14.93 & 0.07 & t6 & v03,t03sdss j1435@xmath20043 & 16.1 & 5.8 & 11.15 & 0.78 & l2.5 & v03sdss j1446@xmath10024 & 45.5 & 3.2 & 12.09 & 0.16 & l5 & v03sdss j1624@xmath10029 & 90.7 & 1.0 & 15.40 & 0.04 & t6 & d02,v03,t03sdss j1750@xmath11759 & 36.2 & 4.5 & 13.82 & 0.27 & t3.5 & v03 lllrrrrrrrrr sdss j083120.81@xmath1304417.1 & [ m l2 ] & & 1.77 & 19.68 & 2.23 & & 17.45 & 1.10 & 0.56 & 0.54sdss j140814.74@xmath1053952.9 & [ m l2 ] & & 1.77 & 18.71 & 2.22 & & 16.49 & 1.00 & 0.54 & 0.46sdss j162441.00@xmath1444145.8 & [ m l3 ] & & & 20.07 & 2.51 & & 17.56 & 1.10 & 0.68 & 0.422mass j03454316@xmath12540233 & l1@xmath1171 & 11.69 & & & & & 13.84 & 1.18 & 0.63 & 0.54 2mass j07464256@xmath12000321ab & l1 & 11.21 & 1.80 & 14.30 & 2.66 & 1.47 & 11.64&1.21 & 0.63 & 0.582mass j14392836@xmath11929149 & l1 & 11.87 & & & & & 12.66 & 1.19 & 0.61 & 0.58 sdss j144016.20@xmath1002638.9 & l1 & & 1.90 & 18.75 & 2.82 & & 15.93 & 1.29 & 0.70 & 0.59sdss j121001.96@xmath1030739.2 & l1.5@xmath1171 & & & 19.79 & 2.52 & & 17.27 & 1.24 & 0.69 & 0.55sdss j035448.73@xmath2002742.1 & l2@xmath1171 & & 2.02 & 19.61 & 2.47 & & 17.14 & 1.19 & 0.68 & 0.51sdss j100401.41@xmath1005354.9 & l2@xmath1171 & & 2.05 & 19.76 & 2.36 & & 17.40 & 1.16 & 0.58 & 0.58sdss j143535.70@xmath2004347.0 & l2.5 & 12.44 & 1.84 & 19.02 & 2.61 & & 16.41 & 1.29 & 0.73 & 0.562mass j00283943@xmath11501418&l3 & & 2.19 & 19.52 & 3.03 & 2.01 & 16.49 & 1.95 & 1.01 & 0.94denis - p j1058.7@xmath21548 & l3 & 12.93 & & & & 1.64 & 14.12 & 1.57 & 0.84 & 0.74 gd 165b & l3@xmath1172 & 13.14 & & & & & 15.64 & 1.55 & 0.89 & 0.66 kelu1 & l3@xmath1171&11.88 & & & & 1.77 & 13.23 & 1.45 & 0.78 & 0.67sdss j020735.60@xmath1135556.3 & l3@xmath1171.5 & & 1.78 & 18.06 & 2.79 & & 15.27 & 1.46 & 0.82 & 0.642mass j22244381@xmath20158521 & l3.5 & 13.59 & & & & & 13.89 & 1.91 & 1.05 & 0.86 sdss j075515.26@xmath1293445.4 & l3.5@xmath1171 & & 2.33 & 19.40 & 2.69 & & 16.71 & 1.39 & 0.77 & 0.62sdss j131415.52@xmath2000848.1 & l3.5@xmath1171.5&&2.00 & 19.63 & 3.11 & & 16.52 & 1.22&0.68 & 0.55 2mass j00361617@xmath11821104 & l4@xmath1171&12.61 & & & & 1.81 & 12.30 & 1.26&0.66 & 0.60 sdss j000112.18@xmath1153535.5 & l4@xmath1171 & & 1.74 & 18.55 & 3.26 & & 15.29 & 1.77 & 0.89 & 0.88 sdss j040100.96@xmath2060933.0 & l4@xmath1171.5 & & & 20.19 & 2.81 & & 17.38 & 1.67 & 0.99 & 0.68 lhs 102b & l4.5 & 13.16 & & & & 1.61 & 13.06 & 1.70&0.92 & 0.78 sdss j143211.74@xmath2005900.8 & l4.5 & & & 19.62 & 2.63 & & 16.99 & 1.54 & 0.85 & 0.69 sdss j053951.99@xmath2005902.0 & l5 & 13.26 & 2.24 & 16.78 & 2.93 & 1.75 & 13.85 & 1.45&0.81 & 0.64 sdss j125737.26@xmath2011336.1 & l5 & 13.86&2.14 & 18.56 & 2.92 & & 15.64 & 1.58 & 0.96&0.62sdss j144600.60@xmath1002452.0 & l5 & 13.85 & 2.30 & 18.48 & 2.92 & & 15.56&1.76 & 0.97 & 0.79 sdss j161626.46@xmath1221859.2 & l5@xmath1172 & & & 20.33 & 2.80 & & 17.53 & 1.88 & 1.03 & 0.85 sdss j224953.45@xmath1004404.2&l5@xmath1171.5 & & 2.16 & 19.42 & 2.95 & 1.77 & 16.47&2.05 & 1.11 & 0.94 sdss j074007.30@xmath1200921.9 & [ l38 ] & & & 19.78 & 3.11 & & 16.67 & 1.56 & 0.85 & 0.71sdss j075656.54@xmath1231458.5&[l38 ] & & & 19.82 & 3.02 & & 16.80 & 1.80 & 0.98 & 0.822mass j15074769@xmath21627386 & l5.5 & 13.37 & & & & & 12.70 & 1.41&0.80 & 0.61denis - p j0205.4@xmath21159ab & l5.5@xmath1172 & 12.95 & & & & 1.58 & 14.43 & 1.44 & 0.82 & 0.62 sdss j001608.44@xmath2004302.3 & l5.5 & & 1.77 & 19.34 & 3.00 & & 16.34 & 1.82 & 1.00 & 0.82 sdss j010752.33@xmath1004156.1 & l5.5 & 14.78 & 2.83 & 18.70 & 2.95 & 1.58 & 15.75&2.17 & 1.19 & 0.98 sdssj132629.82@xmath2003831.5 & l5.5 & 14.70 & 2.63 & 19.05 & 2.84 & & 16.21&2.04 & 1.11 & 0.93denis - p j1228.2@xmath21547ab & l6@xmath1172 & 12.75 & & & & 1.73 & 14.28 & 1.57 & 0.88 & 0.692mass j08251968@xmath12115521 & l6 & 14.76 & & & & 1.73 & 14.89 & 1.96 & 1.08 & 0.88 2mass j08503593@xmath11057156ab & l6&13.85 & & & & 1.95 & 16.20&1.85 & 0.99 & 0.86 sdss j012743.50@xmath1135420.9 & l6@xmath1171 & & 2.65 & 19.62 & 2.91 & & 16.71 & 1.62 & 0.87 & 0.75 sdss j104625.76@xmath1042441.0 & l6 & & 2.64 & 19.74 & 2.77 & & 16.97 & 1.62 & 0.94 & 0.68 sdss j112615.25@xmath1012048.2 & l6 & & & 19.79 & 3.11 & & 16.68&1.64 & 0.87 & 0.77 sdss j080959.01@xmath1443422.2 & [ l58 ] & & 2.54 & 19.28 & 2.91 & & 16.37 & 2.06 & 1.12 & 0.94 2mass j08014056@xmath14628498 & l6.5 & & 2.50 & 18.78 & 2.57 & & 16.21 & 1.63 & 0.90 & 0.73sdss j023617.93@xmath1004855.0 & l6.5 & & 2.87 & 18.80 & 2.79 & 1.53 & 16.01&1.47 & 0.85 & 0.62sdss j152531.32@xmath1581053.1 & l6.5@xmath1171 & & & 19.75 & 2.85 & & 16.90 & 1.47 & 0.77 & 0.702mass j00303013@xmath21450333 & l7 & 14.26 & & & & 1.74 & 16.39 & 1.90 & 1.02 & 0.88sdss j104409.43@xmath1042937.6 & l7 & & 2.94 & 18.73 & 2.89 & & 15.84 & 1.52 & 0.87 & 0.65sdss j163030.53@xmath1434404.0 & l7@xmath1171.5 & & 2.65 & 19.45 & 2.97 & & 16.48 & 1.78 & 0.97 & 0.812mass j09293364@xmath13429527 & l7.5 & & & 19.52 & 2.83 & 1.69 & 16.69&1.96 & 1.07 & 0.89 2mass j16322911@xmath11904407 & l7.5 & 14.83 & 3.20 & 18.58 & 2.81 & 1.75 & 15.77&1.80 & 1.09 & 0.71 2mass j22443167@xmath12043433 & l7.5@xmath1172 & & & & & & 16.33 & 2.43 & 1.27 & 1.16sdss j093109.56@xmath1032732.5 & l7.5@xmath1171.5 & & 2.72 & 19.28 & 2.68 & & 16.60 & 1.07 & 0.51 & 0.56sdss j115553.86@xmath1055957.5 & l7.5 & & 2.81 & 18.45 & 2.82 & & 15.63&1.54 & 0.89 & 0.652mass j15232263@xmath13014562 & l8 & 14.63 & & & & 1.65 & 15.95&1.60 & 0.90 & 0.70sdss j003259.36@xmath1141036.6 & l8 & 13.98 & & 19.40 & 2.82 & 1.67 & 16.58&1.59 & 0.92 & 0.67sdss j085758.45@xmath1570851.4 & l8@xmath1171 & & 2.98 & 17.77 & 2.97 & 1.72 & 14.80 & 1.86 & 1.00 & 0.86 sdss j133148.90@xmath2011651.4 & l8@xmath1172.5 & & 2.42 & 18.14 & 2.82 & & 15.32&1.25 & 0.67 & 0.58sdss j135923.99@xmath1472843.2 & l8.5 & & & 19.76 & 2.81 & & 16.95&1.61 & 0.95 & 0.662mass j03105986@xmath11648155 & l9 & & & & & & 15.88 & 1.70 & 0.97 & 0.73 2mass j09083803@xmath15032088 & l9@xmath1171 & & 2.84 & 17.22 & 2.82 & & 14.40 & 1.51 & 0.86 & 0.65sdss j080531.80@xmath1481233.0 & l9@xmath1171.5 & & 2.20 & 17.62 & 3.01 & & 14.61&1.10 & 0.60 & 0.50sdss j083008.12@xmath1482847.4 & l9@xmath1171 & 14.64 & 3.14 & 18.08 & 2.86 & 1.64 & 15.22 & 1.54 & 0.82 & 0.72 2mass j03284265@xmath12302051 & l9.5 & 13.95 & & & & 1.71 & 16.35 & 1.48 & 0.88 & 0.60 sdss j020333.26@xmath2010812.5 & l9.5 & & & 20.36 & 2.53 & & 17.83 & 1.65 & 0.96 & 0.69sdss j085234.90@xmath1472035.0 & l9.5@xmath1171 & & 3.09 & 18.90 & 2.77 & & 16.13 & 1.51 & 0.92 & 0.59 sdss j103026.78@xmath1021306.4 & l9.5@xmath1171 & & & 19.94 & 2.84 & & 17.10 & 1.43 & 0.83 & 0.60sdss j110454.25@xmath1554841.4 & l9.5@xmath1171.5 & & & 19.94 & 2.66 & & 17.28 & 0.94 & 0.55 & 0.39 sdss j204749.61@xmath2071818.3 & l9.5@xmath1171 & & & 19.74 & 3.04 & & 16.70 & 1.36 & 0.82 & 0.54 sdss j042348.57@xmath2041403.5 & t0 & 13.39 & 2.89 & 17.29 & 2.99 & 1.68 & 14.30 & 1.34&0.79 & 0.55 sdss j120747.17@xmath1024424.8 & t0 & & 3.06 & 18.41 & 3.03 & & 15.38 & 1.22 & 0.75 & 0.47sdss j151603.03@xmath1025928.9 & t0@xmath1171.5 & & & 19.89 & 3.01 & & 16.88 & 1.53 & 0.81 & 0.72sdss j083717.21@xmath2000018.0 & t0.5 & 14.54 & & 20.06 & 3.16 & 1.69 & 16.90&0.92 & 0.69 & 0.23 sdss j015141.69@xmath1124429.6 & t1@xmath1171 & 14.60 & & 19.45 & 3.20 & 1.84 & 16.25&1.07 & 0.71 & 0.36 sdss j163239.34@xmath1415004.3 & t1@xmath1171 & & & 20.35 & 3.48 & & 16.87 & 0.68 & 0.45 & 0.23 sdss j175024.01@xmath1422237.8 & t1 & & & 19.38 & 3.26 & & 16.12&0.81 & 0.55 & 0.26 sdss j115700.50@xmath1061105.2 & t1.5 & & & 20.20 & 3.11 & & 17.09 & 1.09 & 0.64 & 0.45sdss j075840.33@xmath1324723.4 & t2@xmath1171 & & 3.96 & 17.96 & 3.18 & & 14.78 & 0.91 & 0.57 & 0.34sdss j125453.90@xmath2012247.4 & t2 & 14.00 & 4.22 & 18.03 & 3.37 & 1.74 & 14.66 & 0.82&0.53 & 0.29 sdss j152103.24@xmath1013142.7 & t2 & & & 19.57 & 3.51 & & 16.06&0.58 & 0.43 & 0.15sdss j102109.69@xmath2030420.1 & t3 & 13.62 & & 19.33 & 3.45 & 1.78 & 15.88 & 0.62&0.47 & 0.15sdss j175032.96@xmath1175903.9 & t3.5&13.94 & & 19.63 & 3.49 & & 16.14&0.12 & 0.20 & 0.08 2mass j22541892@xmath13123498 & t4 & & & & & & 15.01 & 0.02 & 0.06 & 0.08 2mass j05591914@xmath21404488 & t4.5 & 13.50 & & & & 1.98 & 13.57&0.16 & 0.07 & 0.09sdss j000013.54@xmath1255418.6 & t4.5 & & & 18.48 & 3.75 & & 14.73 & 0.09 & 0.01 & 0.08 sdss j020742.83@xmath1000056.2 & t4.5&14.34 & & 20.11 & 3.48 & 2.08 & 16.63&0.01 & 0.03 & 0.04 sdss j092615.38@xmath1584720.9 & t4.5 & & & 19.01 & 3.54 & & 15.47&0.03 & 0.05 & 0.08 2mass j07554795@xmath12212169 & t5 & & & 19.12 & 3.66 & & 15.46 & 0.40 & 0.24 & 0.16sdss j074201.41@xmath1205520.5 & t5 & & & 19.28 & 3.68 & & 15.60&0.46 & 0.35 & 0.112mass j15031961@xmath12525196 & t5.5 & & & & & & 13.55 & 0.44 & 0.35 & 0.092mass j15344984@xmath22952274ab & t5.5 & 13.93 & & & & & 14.60 & 0.31&0.14&0.172mass j23391025@xmath11352284 & t5.5 & & & & & & 15.81 & 0.36 & 0.19 & 0.17sdss j074149.15@xmath1235127.5 & t5.5 & & & 19.65 & 3.78 & & 15.87&0.35 & 0.25 & 0.10sdss j083048.80@xmath1012831.1 & t5.5 & & & 19.59 & 3.60 & & 15.99&0.39 & 0.18 & 0.21 sdss j111010.01@xmath1011613.1 & t5.5 & & & 19.64 & 3.52 & & 16.12&0.07 & 0.10 & 0.17 2mass j02431371@xmath22453298 & t6 & 14.99 & & & & 1.85 & 15.13 & 0.21 & 0.26 & 0.052mass j09373487@xmath12931409 & t6 & 15.35 & & & & 1.72 & 14.29 & 1.10 & 0.38 & 0.722mass j12255432@xmath22739466ab & t6 & 14.25 & & & & 1.89 & 14.88&0.40 & 0.29 & 0.11 2mass j23565477@xmath21553111 & t6 & 14.67 & & & & 2.12 & 15.48 & 0.25 & 0.22&0.03gl 229b & t6@xmath1171&15.20 & & & & 2.17 & 14.01 & 0.35&0.35 & 0.00sdss j123147.39@xmath1084730.7 & t6 & & & 18.94 & 3.80 & & 15.14&0.32 & 0.26 & 0.06sdss j134646.45@xmath2003150.4 & t6 & 14.69 & & 19.29 & 3.80&2.24 & 15.49 & 0.24 & 0.35 & 0.11sdss j162414.37@xmath1002915.6 & t6 & 14.99 & & 19.02 & 3.82&2.12 & 15.20 & 0.41 & 0.28 & 0.13 sdss j212413.89@xmath1010000.3 & t6 & & & 19.71 & 3.83 & & 15.88 & 0.19 & 0.24 & 0.05 2mass j10475385@xmath12124234 & t6.5 & 15.43 & & & & 1.93 & 15.46 & 0.74 & 0.37 & 0.37 sdss j175805.46@xmath1463311.9 & t7 & & & 19.67 & 3.81 & & 15.86 & 0.26 & 0.34 & 0.08 2mass j15530228@xmath11532369ab & t7.5 & & & & & & 15.34 & 0.60 & 0.42 & 0.18 2mass j07271824@xmath11710012 & t8 & 15.40 & & & & 1.98 & 15.19&0.50 & 0.48 & 0.02 2mass j12171110@xmath20311131 & t8 & 15.41 & & 19.18 & 3.62 & 2.00 & 15.56 & 0.36 & 0.42 & 0.06 gl 570d & t8 & 15.97 & & & & 1.92 & 14.82&0.70 & 0.46 & 0.24 2mass j04151954@xmath20935066 & t9 & 16.53 & & & & 1.98 & 15.32&0.51 & 0.38&0.13 2mass j0746@xmath12000ab & 18.65 & 0.012mass j0755@xmath12212 & 22.27 & 0.232mass j0908@xmath15032 & 22.43 & 0.17 sdss j0001@xmath11535 & 23.02 & 0.22 sdss j0207@xmath11355 & 22.37 & 0.14 sdss j0423@xmath20414 & 22.82 & 0.22 sdss j0539@xmath20059 & 21.42 & 0.08 sdss j0755@xmath12934 & 22.96 & 0.29 sdss j0805@xmath14812 & 22.79 & 0.27sdss j1257@xmath20113 & 22.68 & 0.27 sdss j1314@xmath20008 & 23.32 & 0.27sdss j1331@xmath20116 & 22.91 & 0.26 sdss j1440@xmath10026 & 22.92 & 0.19sdss j1446@xmath10024 & 23.20 & 0.23 lll 2mass j0243@xmath22453 & 2mass j0415@xmath20935 & 2mass j0755@xmath122122mass j0727@xmath11710 & 2mass j1225@xmath22739ab & 2mass j0937@xmath12931 2mass j1217@xmath20311 & 2mass j1503@xmath12525 & 2mass j1047@xmath12124 gl 229b & 2mass j1553@xmath11532ab & 2mass j1534@xmath22952absdss j1110@xmath10116 & 2mass j2356@xmath21553 & 2mass j2339@xmath11352 sdss j1346@xmath20031 & gl 570d & sdss j0830@xmath10128 sdss j1758@xmath14633 & sdss j0741@xmath12351 & sdss j2124@xmath10100 & sdss j0742@xmath12055 & & sdss j1231@xmath10847 & & sdss j1624@xmath10029 & cllllll m@xmath119 & 12.03 & @xmath26.278e@xmath120 & 4.500e@xmath120 & @xmath26.848e@xmath121 & 3.986e@xmath122 & @xmath27.923e@xmath123m@xmath124 & 10.93 & @xmath26.485e@xmath120 & 3.876e@xmath120 & @xmath25.819e@xmath121 & 3.524e@xmath122 & @xmath27.351e@xmath123

we present new @xmath0 photometry on the mko - nir system and @xmath0 spectroscopy for a large sample of l and t dwarfs . photometry has been obtained for 71 dwarfs and spectroscopy for 56 . the sample comprises newly identified very red objects from the sloan digital sky survey ( sdss ) and known dwarfs from the sdss and the two micron all sky survey ( 2mass ) . spectral classification has been carried out using four previously defined indices ( from geballe et al . 2002 , g02 ) that measure the strengths of the near infrared water and methane bands . we identify 9 new l89.5 dwarfs and 14 new t dwarfs from sdss , including the latest yet found by sdss , the t7 dwarf sdss j175805.46@xmath1463311.9 . we classify 2mass j04151954@xmath20935066 as t9 , the latest and coolest dwarf found to date . we combine the new results with our previously published data to produce a sample of 59 l dwarfs and 42 t dwarfs with imaging data on a single photometric system and with uniform spectroscopic classification . we compare the near - infrared colors and absolute magnitudes of brown dwarfs near the l t transition with predictions made by models of the distribution and evolution of photospheric condensates . there is some scatter in the g02 spectral indices for l dwarfs , suggesting that these indices are probing different levels of the atmosphere and are affected by the location of the condensate cloud layer . the near - infrared colors of the l dwarfs also show scatter within a given spectral type , which is likely due to variations in the altitudes , spatial distributions and thicknesses of the clouds . we have identified a small group of late l dwarfs that are relatively blue for their spectral type and that have enhanced feh , h@xmath3o and k i absorption , possibly due to an unusually small amount of condensates . the scatter seen in the @xmath4 color for late t dwarfs can be reproduced by models with a range in surface gravity . the variation is probably due to the effect on the @xmath5-band flux of pressure - induced @xmath6 opacity . the correlation of @xmath4 color with gravity is supported by the observed strengths of the @xmath7-band k i doublet . gravity is closely related to mass for field t dwarfs with ages @xmath8 yrs and the gravities implied by the @xmath4 colors indicate that the t dwarfs in our sample have masses in the range 15 75 @xmath9 . one of the sdss dwarfs , sdss j111010.01@xmath1011613.1 , is possibly a very low mass object , with log @xmath10 @xmath11 4.2 4.5 and mass @xmath11 10 15 @xmath9 .

transport through a chaotic cavity is usually studied through a scattering description . for a chaotic cavity attached to two leads with @xmath1 and @xmath2 channels respectively , the scattering matrix is an @xmath3 unitary matrix , where @xmath4 . it can be separated into transmission and reflection subblocks @xmath5 which encode the dynamics of the system and the relation between the incoming and outgoing wavefunctions in the leads . the unitarity of the scattering matrix @xmath6 leads to the following relations ( among others ) @xmath7 while the transport statistics themselves are related to the terms in involving the scattering matrix ( or its transmitting and reflecting subblocks ) and their transpose conjugate . for example , the conductance is proportional to the trace @xmath8 $ ] ( landauer - bttiker formula @xcite ) , while other physical properties are expressible through higher moments like @xmath8^n$ ] . there are two main approaches to studying the transport statistics in clean ballistic systems : a random matrix theory ( rmt ) approach , which argues that @xmath9 can be viewed as a random matrix from a suitable ensemble , and a semiclassical approach that approximates elements of the matrix @xmath9 by sums over open scattering trajectories through the cavity . it was shown by blmel and smilansky @xcite that the scattering matrix of a chaotic cavity is well modelled by the dyson s circular ensemble of random matrices of suitable symmetry . thus , transport properties of chaotic cavities are often treated by replacing the scattering matrix with a random one ( see @xcite for a review ) . the eigenvalues of the transmission matrix @xmath10 then follow a joint probability distribution , which depends on whether the system has time reversal symmetry or not , and from which transport moments and other quantities can be derived . though the conductance and its variance were known for arbitrary channel number @xcite , other quantities were limited to a diagrammatic expansions in inverse channel number , see @xcite . however , the rmt treatment has recently experienced a resurgence due to the connection to the selberg integral noticed in @xcite . following the semiclassical result for the shot noise @xcite , the authors of @xcite used recursion relations derived from the selberg integral to calculate the shot noise and then later all the various moments up to fourth order for arbitrary channel number @xcite . since then a range of transport quantities have been treated , for example the moments of the transmission eigenvalues for chaotic systems without time reversal symmetry ( the unitary random matrix ensemble ) @xcite and those with time reversal symmetry ( the orthogonal random matrix ensemble ) @xcite . for the unitary ensemble , the moments of the conductance itself were also obtained in @xcite and , using a different approach , in @xcite which was later extended to the moments of the shot noise @xcite . building again on the selberg integral approach , the moments of the conductance and shot noise have been derived for both symmetry classes @xcite . interestingly these results , though all exact for arbitrary channel number , are given by different combinatorial sums , and the question of how they are related to each other is still open in many cases . on the other hand , the semiclassical approach makes use of the following approximation for the scattering matrix elements @xcite @xmath11 which involves the open trajectories @xmath12 which start in channel @xmath13 and end in channel @xmath14 , with their action @xmath15 and stability amplitude @xmath16 . the prefactor also involves @xmath17 which is the average dwell time , or time the trajectory spends inside the cavity . for transport moments we consider quantities of the type @xmath18^{n } \right\rangle \sim \left\langle \frac{1}{{(n\td)}^{n } } \prod_{j=1}^{n } \sum_{{i_j , o_j } } \sum_{\substack{\gamma_j(i_j\to o_j ) \cr \gamma'_j ( i_{j+1}\to o_{j } ) } } a_{\gamma_j}a_{\gamma'_j}^ { * } \rme^{\frac{\rmi}{\hbar}(s_{\gamma_j}-s_{\gamma'_j } ) } \right\rangle,\ ] ] where the trace means we identify @xmath19 and where @xmath20 is either the transmitting or the reflecting subblock of the scattering matrix . the averaging is performed over a window of energies @xmath21 . the choice of the subblock @xmath20 affects the sums over the possible incoming and outgoing channels , but not the trajectory structure which involves @xmath22 classical trajectories connecting channels . of these , @xmath0 trajectories @xmath23 , @xmath24 , contribute with positive action while @xmath0 trajectories @xmath25 contribute with negative action . in the semiclassical limit of @xmath26 we require that these sums cancel on the scale of @xmath27 so that the corresponding trajectories can contribute consistently when we apply the averaging in . ( a ) the semiclassical trajectories for the second moment travel around a closed cycle . by collapsing the trajectories onto each other as in ( b ) we can create a small action difference and a trajectory quadruplet with a single encounter that contributes to the shot noise at leading order in inverse channel number . ( c ) for the conductance , a trajectory pair with a similar encounter provides the first subleading order correction for systems with time reversal symmetry . ] the main idea of the semiclassical treatment is that , in order to achieve a small action difference , the trajectories @xmath28 , must follow the path of trajectories @xmath29 most of the time , deviating only in small regions called _ encounters_. this is best illustrated with an example . in ( a ) the schematic depiction of the trajectories is shown for the case @xmath30 . we have 2 trajectories @xmath12 depicted by solid lines , @xmath31 and @xmath32 , and 2 trajectories @xmath33 depicted by dashed lines , @xmath34 and @xmath35 . ( b ) shows one possible configuration for achieving a small action difference : trajectory @xmath36 departs from the incoming channel @xmath37 following the path of trajectory @xmath38 . then , when the trajectories @xmath39 and @xmath38 come close to each other in phase space ( thus the term ` encounter ' ) , trajectory @xmath36 switches from following @xmath38 to following @xmath39 , before arriving at its destination channel @xmath40 . the trajectory @xmath41 does the opposite . the picture in ( b ) is referred to as a ` diagram ' ; it describes the topological configuration of the trajectories in question , while leaving out metric details . the task of semiclassical evaluation can therefore be divided into two parts : evaluation of the contribution of a given diagram by integrating over all possible trajectories of given structure and enumeration of all possible diagrams . historically , the semiclassical treatment started with the mean conductance @xmath42\right\rangle$ ] , involving a single trajectory and its partner . the leading order contribution comes from trajectory pairs that are identical the so - called diagonal approximation which was evaluated in @xcite . the first non - diagonal pair was treated in @xcite and involved a single encounter where one trajectory had a self - crossing while the partner avoided crossing as in ( c ) . such a pair can only exist when the system has time reversal symmetry and its contribution was shown to be one order higher in inverse channel number , @xmath43 , than the diagonal terms . the expansion to all orders in inverse channel number was then performed in @xcite for systems with and without time reversal symmetry by considering arbitrarily many encounters each involving arbitrarily many trajectory stretches . importantly , the work on the conductance @xcite showed that the semiclassical contribution of a diagram can be decomposed into a product over the constituent parts of the diagram , greatly simplifying the resulting sums . in fact for the second moment , the shot noise , all such diagrams were generated in @xcite and with them the full expansion in inverse channel number . this , along with the conductance variance and other transport correlation functions , as well as the semiclassical background was covered in detail in @xcite . however , the method for diagram enumeration considered in @xcite becomes unwieldy for higher moments , which encode finer transport statistics . to the leading order in @xmath43 , the higher moments were derived in @xcite . the semiclassical approach requires a large number of channels in each lead , @xmath44 , but to unambiguously separate the orders in inverse channel number one may additionally assume that both @xmath1 and @xmath2 are of the same order as @xmath45 . for example , the result of @xcite was in terms of the variable @xmath46 which should then be constant and introduce no further channel number scaling . we therefore make the same assumption in this article when describing the different orders in @xmath43 , though of course a different scaling , say keeping @xmath1 fixed so that @xmath47 , may simply lead to a mixing of the different ` orders ' without changing the individual results . the diagrams contributing at the leading order to the @xmath0-th moment were shown in @xcite to be trees . the tree expansions turned out to be very well suited to analysis of other interesting physical quantities , such as the statistics of the wigner delay times @xcite , which are a measure of the time spent in the scattering region , and the density of states in andreev billiards @xcite . if we imagine replacing the scattering leads by a superconductor we have a closed system called an andreev billiard . each time an electron inside the system hits the superconductor it is reflected as a hole retracing its path until it hits the superconductor and is retroreflected as an electron again . wave interference between these paths leads to significant effects , most notably a complete suppression of the density of states for a range of energies around the fermi energy . similarly , strong effects on the conductance ( of the order of the mean conductance ) can also be seen if we attach additional superconducting leads to our original chaotic cavity ( making a so - called andreev dot ) @xcite . the size of these effects make such systems especially interesting for a semiclassical treatment . but treating these effects effectively requires knowledge of _ all _ the higher moments , and gives us strong reason to go beyond low @xmath0 . one particular nicety of the semiclassical approach is that it can incorporate , in a natural way , the effect of the ehrenfest time . this is the time scale that governs the transition from classical dynamics to wave interference , which dominates when the ehrenfest time is small ( on the scale of the typical dwell time ) . for larger ehrenfest times , the competition between the different types of behaviour leads to quite striking features , like an additional gap both in the density of states of andreev billiards and in the probability distribution of the wigner delay times @xcite . semiclassically we can explicitly track the effect of the ehrenfest time all the way to the ` classical ' limit , which can only be achieved using rmt by postulating the ehrenfest time dependence of the scattering matrix . alongside the case of ballistic systems , the typical chaotic behaviour and transport statistics can also be induced by introducing disorder in the system . for weak disorder the transport properties coincide with those obtained from rmt , and one can also obtain the full counting statistics at leading order , as well as weak localisation corrections and universal conductance fluctuations , using circuit theory @xcite . if the disorder averaging is treated using diagrammatic perturbation theory ( see , for example , @xcite ) multiple scattering events can be summed , in the limit of weak disorder , as a ladder diagram known as a diffuson . this corresponds to the parts of semiclassical diagrams where the trajectories are nearly identical , as on the left of the encounter in ( c ) . the disordered systems counterpart of the loop on the right of the encounter in ( c ) , traversed by trajectories in opposite directions , is called a cooperon , while the encounter itself corresponds to a hikami box @xcite . although transport properties like the weak localisation diagram related to ( c ) and conductance and energy level fluctuations @xcite can be treated diagrammatically , usually powerful field - theoretic methods involving the nonlinear @xmath48 model are used ( see @xcite for an introduction ) . these methods can treat both weak and stronger disorder non - perturbatively , and by using supersymmetry @xcite a large range of transport and spectral properties can be obtained , for open and closed systems correspondingly . more importantly , the applicability of rmt for weakly disordered systems can be justified and rmt shown to be the zero - dimensional variant of the @xmath48 model @xcite . alongside the supersymmetric @xmath48 model , there is also the replica @xmath48 model which is particularly useful for perturbative expansions . this leads to a diagrammatic expansion , with diagrams that can be reinterpreted as correlated semiclassical trajectories @xcite . in fact this connection between semiclassical diagrams and disorder diagrams from the replica @xmath48 model lay behind the semiclassical treatment of energy level correlations in closed systems @xcite which in turn led to the semiclassical treatment of transport @xcite discussed above . to summarize , there are established semiclassical tools for the analysis of for small @xmath0 to all orders of @xmath43 and for all @xmath0 but only to the leading order of @xmath43 . it is the purpose of this article to start closing this gap . for all @xmath0 we derive the next two corrections for and related quantities . we show that the contributing diagrams can be generated by grafting trees onto the ` base diagrams ' , which can be obtained by ` cleaning ' the diagrams used in @xcite . we therefore first review the leading order tree recursions in before treating transport moments beyond the leading order in . we start by cleaning the diagram of ( c ) which gives the first subleading order orthogonal correction . grafting trees onto the base diagram leads to a generating function , which we apply to calculate the moments of the transmission and reflection eigenvalues . proceeding to the next order in @xmath43 we then treat the second subleading order diagrams for the unitary and the orthogonal case . for the moments of the reflection and transmission eigenvalues we find that our generating functions simplify and become rather straightforward . the graphical recursions we use provide a new insight into the leading order terms which is particularly useful for energy dependent correlation functions . such correlation functions are needed for a treatment of the density of states of andreev billiards , which we consider in where we find that the hard gap , previously found at leading order in @xmath43 , persists at least for the next two orders . also derivable from energy dependent correlation functions are the moments of the wigner delay times , treated in , and we find that the corrections at each order in @xmath43 are also generated by relatively simple functions . of course , the transport moments in are only one type of transport quantity , and we finally look at non - linear statistics in and see how their treatment follows naturally from the previous semiclassical considerations . we shall be comparing our semiclassical results with the prediction of rmt , where those predictions are available : previously ( of the quantities treated here ) only the moments of the transmission amplitudes for systems without time reversal symmetry have been given for an arbitrary number of channels @xcite . explicit results for systems with time reversal symmetry have just been derived @xcite and we were pleased that @xcite shared those results with us beforehand . the moments of the wigner delay times for both symmetry classes have also been obtained @xcite . of the recent rmt results , it is those concerned with the asymptotic expansion as the number of channels increases , currently to leading order @xcite , that particularly connect with the work here . semiclassically , without the equivalent of the selberg integral , we are still restricted to an expansion in inverse powers of the channel number , but as we shall see the semiclassical treatment leads to explicit and surprisingly simple generating functions at each order in inverse channel number . this simplicity until now remained hidden in the combinatorial sums of the rmt results and may suggest ways of simplifying those results and of highlighting the underlying combinatorial structure . the semiclassical treatment of the conductance beyond the diagonal contribution , starting @xcite with the trajectory pair depicted in ( c ) , required two main ingredients . the first was to estimate how often a trajectory would come close to itself and have a self - encounter . this is performed using the global ergodicity of the chaotic dynamics . the second was that , given such an encounter , we can use the local hyperbolicity of the motion to find the partner trajectory which reconnects the stretches of the original trajectory in a different way . then one can determine the action difference between the two trajectories and hence their contribution in the semiclassical limit . when treating diagrams with more numerous and more complicated encounters , the authors of @xcite showed that these two ingredients allowed them to express the total contribution as a product of integrals over the encounters and over the ` links ' , the trajectory stretches which connect the encounters together . performing these integrals then led to simple rules for the contributions of the constituent parts of any diagram , and essentially reduced the problem down to the combinatorial one of finding all the possible diagrams . for the first two transport moments , this was done @xcite by cutting open the periodic orbit pairs that contribute to spectral statistics @xcite . for the higher moments , as shown in @xcite , the diagrams that contribute at leading order in inverse channel number are rooted plane trees . the reason is simple : according to the semiclassical evaluation rules of @xcite , every encounter contributes a factor of @xmath49 while every link contributes a factor of @xmath43 . the leading order is thus achieved by a diagram with the minimal possible difference between the number of links ( edges ) and encounters ( internal vertices ) . it is a basic fact of graph theory that this difference is minimized by trees ; each independent cycle in a graph adds one to this difference . thus to go beyond the leading order one needs to consider diagrams with an increasing number of cycles . we will approach this task by describing the topology of the cycles using ` base diagrams ' graphs with no vertices of degree 1 or 2 and then grafting subtrees onto the base diagrams . adding a subtree does not change the order of the contribution in inverse channel number @xmath43 but adds more incoming and outgoing channels thus changing the order of the moment @xmath0 . because we will be joining the trees to existing structures , unlike the treatment in @xcite , here we do not root our trees in an incoming channel , but at an arbitrary point . these trees then correspond to the restricted trees in @xcite and will be referred to as ` subtrees ' . we also note that the generating function variables we use here have slightly different definitions than in @xcite . our present choice is more appropriate for the subleading orders and the different transport quantities considered . we now summarize the derivation of the subtree generating functions which were introduced in @xcite and further developed in @xcite . a subtree consists of a root , several vertices of even degree ( called ` nodes ' , they correspond to encounters between various trajectories ) and @xmath50 vertices of degree one ( called ` leaves ' , they correspond to incoming or outgoing channels ) . the leaves are labelled @xmath13 or @xmath14 alternatingly as we go around the tree anti - clockwise . there are two types of subtrees : the @xmath51-subtrees have leaves labelled @xmath52 , @xmath53 , @xmath54 , @xmath55 etc . the label @xmath56 would correspond to the root if we were to label it too . the @xmath57-subtrees have leaf labels @xmath53 , @xmath54 , @xmath55 , @xmath58 etc . the reference index @xmath59 depends on the location of the subtree on the diagram . it is possible that an encounter happens immediately as several trajectories enter the cavity from the lead or exit the cavity into the lead . to keep account of these situations , we say that an @xmath60-encounter ( node of degree @xmath61 ) may ` @xmath13-touch ' the lead if it is connected directly to @xmath60 incoming channels ( leaves with label @xmath13 ) and ` @xmath14-touch ' if connected to @xmath60 outgoing channels . when an encounter touches the lead , the edges connecting it to the lead get cut off and all the channels must coincide , although in the diagrams we keep short ` stubs ' to avoid changing the degree of the encounter vertex . we define the generating functions @xmath62 and @xmath63 which are counting @xmath51- and @xmath57-trees correspondingly . the meaning of the variables @xmath64 , @xmath65 and @xmath66 is as follows : * @xmath67 enumerate the @xmath60-encounters that do not touch the lead , * @xmath68 enumerate the @xmath60-encounters that @xmath13-touch the lead , * @xmath69 enumerate the @xmath60-encounters that @xmath14-touch the lead . for example , the coefficient of @xmath70 gives the number of trees with one 3-encounter ( a vertex of degree 6 ) and three 2-encounters ( vertices of degree @xmath71 ) , one of which @xmath13-touches the lead . an example of such a tree is given in ( a ) . we note that if an encounter may touch the lead , the generating function includes ( and sums ) both possibilities : touching and non - touching . for example , the left - most vertex of the tree in ( a ) may @xmath14-touch the lead , but this possibility is counted separately . in addition we will use several secondary parameters that will allow us to adapt the subtree generating functions to each of the four quantities considered in the paper . these parameters are : * @xmath72 is the semiclassical contribution of an edge ( link ) * @xmath73 and @xmath74 are the contributions of an incoming and an outgoing channel * @xmath48 is a special correction parameter for the situation when an @xmath14-touching node is directly connected to an @xmath13-channel ( @xmath75 everywhere apart from ) . the subtree shown in ( a ) is cut at its top node ( of degree 4 ) creating subtrees ( b)-(d ) . subtree ( c ) has the incoming and outgoing directions reversed . the lower vertex in ( a ) , and hence ( c ) , is @xmath13-touching the lead so that the channels @xmath76 and @xmath77 ( not shown ) coincide ( @xmath78 ) . this is represented by the short stubs , and the encounter now starts in the incoming lead.,scaledwidth=50.0% ] we obtain a recursion for the functions @xmath51 and @xmath57 by cutting the subtree at the top encounter node . if this node is of degree @xmath61 , this leads to @xmath79 further subtrees as illustrated in . assuming we started with an @xmath51-subtree , @xmath60 of the new subtrees also have type @xmath51 , while the remaining @xmath80 are @xmath57-subtrees . thus an @xmath51-subtree with an @xmath60-encounter at the top contributes @xmath81 to the generating function @xmath51 . additionally , we consider the possibility for the top node of an @xmath51-subtree to @xmath14-touch . in this case its odd - numbered further subtrees are empty stubs and the even - numbered subtrees are still arbitrary , leading to the contribution @xmath82 . here we have included a correction term @xmath48 which is used in to control the contribution of any @xmath57-subtree that consists of one edge and directly connects an incoming and outgoing channel and is set to 0 in the rest of the paper . we start our recursion relation at the value for an empty tree , which consists of a link ( with the factor @xmath72 ) and an outgoing channel ( providing a factor @xmath83 ) , @xmath84.\ ] ] the recursion is similar for @xmath57 , with the roles of @xmath13- and @xmath14-variables switched , @xmath85 .\ ] ] for the reflection into lead 1 we will consider the generating function @xmath86^n\right\rangle , \ ] ] where the power of @xmath87 counts the order of the moments . for the individual semiclassical diagrams we make use of the diagrammatic rules of @xcite , where each link contributes a factor of @xmath43 while each encounter provides the factor @xmath49 . each channel is in lead 1 , so can be chosen from the @xmath1 available and provides this factor . when an encounter starts ( or ends ) in the lead , all the incoming ( or outgoing ) channels must then coincide in the same channel , leading again to the factor @xmath1 . bearing in mind the meaning of the variables introduced above , we therefore have to make the following semiclassical substitutions : @xmath88 where we have introduced @xmath89 whose power counts the total number of channels and which allows us to keep track of the total contributions to different moments . the @xmath0-th moment involves @xmath22 channels so we have the relation @xmath90 . each channel factor @xmath91 then includes the factor @xmath89 , while the formula for @xmath92 in accounts for the fact that when an @xmath60-encounter enters the incoming channels we have @xmath60 channels coinciding but only a single channel factor . if we define @xmath93 , the subtree recursions and both become @xmath94 performing the sums ( where the terms @xmath51 and @xmath95 correspond to @xmath96 of the sums ) this is @xmath97 which can be written as the quadratic @xmath98 where @xmath99 and where we take the solution whose expansion agrees with the contributions of the semiclassical diagrams . for the transmission we treat the function @xmath100^n\right\rangle,\ ] ] and to distinguish it more clearly from the reflection we will call the corresponding subtree generating function @xmath101 here . for the transmission , the equations are a bit more complicated than for the reflection because @xmath102 in general . for the substitution we need @xmath103 where the only difference from is that the outgoing channels are now in lead 2 and can be chosen from the @xmath2 available . the contribution of the subtrees once summed becomes @xmath104 with @xmath105 and using that @xmath106 . likewise becomes @xmath107 where , as before @xmath108 . with @xmath109 we have @xmath110h + r^2\xi=0 , \ ] ] from which we can find the equations @xmath111 by a simple counting argument , the order of a diagram in terms of inverse channel number is the number of edges minus the number of vertices ( both leaves and nodes ) . thus a diagram contributes at the order @xmath112 , where @xmath113 is the number of the independent cycles in the diagram ( also known as the cyclomatic number or the first betti number , hence the notation ) . the leading contribution thus comes from tree diagrams which have @xmath114 and the next contribution comes from diagrams with one cycle . the correlated trajectory quadruplet in ( a ) which contributes to the second moment at leading order in inverse channel number can be redrawn as the ribbon tree in ( b ) by ` untwisting ' the encounter . the four trajectories themselves can be read off from the boundary walk shown . at subleading order in inverse channel number , we start with the correlated periodic orbit pair in ( c ) which can be represented as the graph in ( d ) with corresponding boundary walks . cutting the periodic orbit along the left link ( which is traversed in the same direction by the orbit and its partner ) creates the correlated trajectory pair in ( e ) which contributes to the first moment . changing this diagram into a graph we arrive at the structure in ( f ) which is a mbius strip with an empty subtree inside and outside the loop . the intertwined s s in diagrams ( d ) and ( f ) represent twists in the corresponding ribbon links . ] a diagram with one cycle can be thought of as a loop with trees grafted on it . but there is a twist . the reconstruction of the trajectories structure from a tree , see @xcite , was done by means of the boundary walk . it helps to visualise the edges of the tree as strips , a model that is called a _ fat _ or _ ribbon _ graph in combinatorics . this fixes the circular order of edges around each vertex and , going along the boundary , prescribes a unique way to continue the walk around a vertex ( see @xcite for an accessible introduction ) . the trajectories @xmath23 of equation are then read as the portions of the walk going from @xmath115 to @xmath116 . the trajectories @xmath117 , on the other hand , appear in reverse as portions of the walk going from @xmath116 to @xmath118 . for example , the diagram in ( a ) which contributes at leading order in @xmath43 can be redrawn as the tree in ( b ) with the corresponding boundary walk shown . the trace in means that the boundary walk is closed and the equality of total actions implies that each edge of the diagram is traversed twice ( once by @xmath12 and once by @xmath33 ) . this means that a valid diagram must have _ one face_. in particular , there must be a way for the walk to cross from inside to outside of the cycle of the first correction diagram . the diagram thus has the topology of a mbius strip with ( ribbon ) trees grafted on the edges . we will refer to the diagram without any trees ( mbius strip in this case ) as the _ base diagram _ or _ structure_. it is also beneficial to consult the full expansion in powers of the inverse channel number of the first two moments of the transmission eigenvalues @xcite and to draw the corresponding diagrams as ribbon graphs . the procedure of going from the closed periodic orbits to scattering trajectories and then to a graph is illustrated in for the first subleading order correction . removing the remaining subtrees from ( f ) leads to the base structure in ( a ) to which we can append subtrees to create valid diagrams like ( b ) whose boundary walk is depicted in ( c ) . as the base structure involves a loop which is traversed in opposite directions by the trajectory and its partner , all the diagrams created in this way can only exist in systems with time reversal symmetry ( corresponding to the orthogonal rmt ensemble ) . to obtain the base structure in ( a ) we can simply remove the empty subtrees of ( f ) . appending subtrees to ( a ) we can then create all the possible graphs , but for the graph to remain a mbius strip we need an odd number of odd nodes , as for example in the graph in ( b ) . we draw the boundary walk in ( c ) where we truncated the subtrees at their first node as they always have an odd number of leaves thereafter . the top left and bottom right nodes along the mbius strip in ( b ) or ( c ) may also enter the lead for reflection quantities . ] along the loop we can add subtrees at any point and to make a valid @xmath60-encounter we must add @xmath119 subtrees ( the remaining two stretches in the encounter belong to the loop itself ) . if the node has an odd number of trees both inside and outside the loop , we refer to it as an _ odd node_. it is easy to convince oneself that in order to have each stretch of the loop traversed once by a @xmath12-trajectory and once by a @xmath33-trajectory there must be an odd number of odd nodes around the loop . we start by evaluating the contribution of a node along the loop . for the node we include all possible sizes @xmath60 of the resulting encounter . adding the @xmath119 subtrees ( of which @xmath80 start with an incoming direction and @xmath80 with an outgoing direction ) there are @xmath79 ways of splitting them into groups inside or outside of the encounter . with the @xmath80 ways which result in an odd node we include the factor @xmath120 whose power will later count the total number of odd nodes around the loop . this leads to @xmath121 .\ ] ] the number of incoming and outgoing channels connected to the same node is equal [ see an example in ( b ) ] . an @xmath60-encounter can touch the lead only if every _ other _ edge connected to it is empty ( connected directly to a leaf ) . since for nodes on the loop we need to include the edges that belong to the loop itself and which can not be empty , we conclude that only odd nodes can possibly touch the lead . since in this case we need @xmath60 empty edges and we have @xmath80 edges of each type , touching the lead is possible only if the incoming and outgoing channels are in the same lead , as they are when we consider a reflection quantity . with @xmath122 trees on the inside of which @xmath59 must be empty ( and the remaining @xmath123 arbitrary ) and the remaining @xmath124 on the outside ( with @xmath125 empty and @xmath126 arbitrary ) and with @xmath127 for reflection quantities we add the following to the node contribution @xmath128 we then allow any number of nodes along the loop , though each time we add a new node it creates a new edge of the loop . because of the rotational symmetry , we divide by the number of nodes . in addition , there is a symmetry between the inside and the outside of the loop , leading to a factor of @xmath129 . the total contribution thus becomes @xmath130^{k}}{k } = -\frac{1}{2}\ln[1-y(a+b ) ] .\ ] ] finally , to ensure that we have an odd number of odd nodes along the loop , we set @xmath131 this function then generates all the diagrams with @xmath22 channels . we can now choose any of the leaves to be labelled @xmath132 , which fixes the numbering of all other leaves : they are numbered in order along the boundary walk . the freedom of choosing one of the leaves gives a factor of @xmath22 . to get this factor we differentiate the result with respect to @xmath89 and multiply by @xmath89 so that the power of @xmath89 still counts the total number of channels . thus we obtain the generating function @xmath133 for the transmission , using the semiclassical values of the variables in , we find that the node contribution in becomes @xmath134 as we do not allow the nodes to enter the leads ( as the incoming and outgoing channels are now in different leads ) we also have @xmath135 . note that the node contribution is given solely in terms of @xmath109 and the full contribution evaluates to @xmath136 putting in the correct explicit solution for @xmath137 from and transforming according to , we find the following generating function for the orthogonal correction to the moments of the transmission eigenvalues @xmath138 where we set @xmath90 to generate the moments as the @xmath0-th moment involves @xmath22 channels . this order correction was previously treated using a rmt diagrammatic expansion @xcite , and can be derived by performing an asymptotic expansion in inverse channel number of the rmt result for arbitrary channel number of @xcite . for the reflection we have @xmath139 and the node contributions in and are @xmath140 using relation we can rewrite @xmath141 as @xmath142 so that for the generating function we find @xmath143 where for the last term we simplified the numerator and denominator inside the logarithm by only keeping the remainder after polynomial division with respect to the quadratic for @xmath51 in . putting in the explicit solution from and following we obtain the rather simple generating function for the orthogonal correction to the moments of the reflection eigenvalues @xmath144 note that this result only depends on @xmath145 , which is not so obvious from and . however , the relations in and the fact that the trace of the identity matrix , being the respective number of channels , is only leading order in inverse channel number means that the dependence only on @xmath146 of the subleading transmission moments must be mirrored in the reflection moments . for the reflection into lead 2 we simply swap @xmath147 and @xmath148 , which clearly does not affect this order correction . we can continue using the ideas above to treat higher order corrections . in particular , for systems without time reversal symmetry the first correction occurs at the second subleading order in inverse channel number . the semiclassical diagrams for the conductance are given , for example , in @xcite and can be represented as the graph diagrams shown in figures [ unittrees](a ) and ( b ) . we note that this representation is not unique and is chosen for simplicity . it is also important to observe that , despite the twists , the corresponding ribbon graphs are orientable , _ i.e. _ have two surfaces ( unlike the mbius strip ) . it can be shown this is true in general : diagrams contributing to the unitary case are orientable . further , the diagrams contributing at this order have genus 1 , _ i.e. _ embeddable on a torus ( but not a sphere ) . this , too , can be shown to continue : the contribution to the order @xmath149 comes from diagrams of genus @xmath150 . from the diagrams in figures [ unittrees](a ) and ( b ) we can form the base structures by removing the channels and their links , see figures [ unittrees](c ) and ( d ) . a similar restriction to the one above still holds when appending subtrees to ensure that the resulting diagrams are permissible . namely , the total number of odd nodes and twists along every closed cycle in the diagram has to be even . we note that the definition of an odd node depends on the cycle : the left node of ( a ) is odd with respect to the cycles formed from the top and bottom arcs and is even relative to the cycle formed from the top arc and the middle edge . we remark that this rule was enforced for the mbius diagram as well . the first subleading order semiclassical diagrams for systems without time reversal symmetry . we start with the trajectory pairs which contribute to the conductance in ( a ) and ( b ) . removing the channels and their links we obtain the base structures ( c ) and ( d ) for this case.,scaledwidth=60.0% ] finally , we need to discuss the symmetries of each base diagram . the generators of the symmetry group of base diagram [ unittrees]c are the shift of the edge numbering and the reflection , giving a group of size 6 . the generators for base diagram [ unittrees]d are inside - outside mappings of the two edges , giving a group of size 4 . when we append subtrees along each edge of a diagram , whether we append @xmath51 or @xmath57-type subtrees depends in a complicated way on the types of subtrees appended along the other edges . therefore we restrict ourselves to the simpler situation where @xmath151 and only treat reflection quantities ( with @xmath75 ) . along the links connecting the nodes in the base diagrams we can append subtrees as before , but because the rotational symmetry is now broken we no longer divide by the number of nodes . the edge contributions can therefore be written as @xmath152^{k } = \frac{y}{1-y(a+b ) } , \ ] ] where @xmath153 and @xmath141 are as in and but with the simplification @xmath151 and @xmath75 . we also need to append subtrees to the nodes of the base structures and , finally , ensure that we have the correct number of objects ( odd nodes and twists ) around each closed cycle . to proceed , we number each of the regions around the nodes and label the closed cycles with greek letters as in figures [ unittrees](c ) and ( d ) . we start with ( c ) and use powers of @xmath154 , @xmath155 and @xmath156 to count the number of objects along the respective cycles . at the top node we can add subtrees in any region we like as long as we add an odd number in total to ensure that the top node becomes a valid @xmath60-encounter . an @xmath60-encounter involves @xmath61 stretches and we have 3 stretches already from the base structure . if we place @xmath157 subtrees in each region @xmath13 and use the power of @xmath158 to count the total number of subtrees added , we can write the contribution of the top node as @xmath159 with @xmath160 and where the number 3 in the subscript refers to the fact that the node in the base diagram starts with 3 stretches while the ` c ' refers to its label in . further , when we have an odd number of trees in each region , and when the odd numbered trees in each region are empty , then the top node can also enter the lead ( since we are considering reflection quantities ) . if we define @xmath161 then we have @xmath162 empty subtrees and @xmath163 arbitrary subtrees in each region . in total we would then add the contribution @xmath164 where @xmath165 and we simplified the powers of @xmath158 and @xmath120 as in the end we are only interested in whether they are odd or even and they are all odd here . finally to ensure that the total number of trees added is odd , we substitute @xmath166 the complete diagram in ( c ) is made up of two such nodes as well as three links . each of the links lies on two cycles so we can write the full contribution as @xmath167 where we divide by 6 to account for the symmetry of the base structure . here the number in the subscript now refers to the order of the contribution while the ` u ' in the superscript refers to the fact that these diagrams correspond to the unitary ensemble . then to ensure that the number of objects along each cycle is even we simply average @xmath168 for @xmath154 , @xmath155 and @xmath156 in turn . for the base structure in ( d ) we now have a single node and four regions . region 3 lies inside both cycles and each cycle starts with a single object inside ( and likewise outside ) which are the stretches of the other cycle leaving or entering the node . treating the node as above , we obtain the contributions @xmath169 with @xmath170 and @xmath171 where @xmath172 and with an odd number of trees in each region in this second case we are guaranteed to add an even number to each cycle and an even number overall . with four edges touching the node in the base structure we need to add an even number of subtrees in total to the node to make a valid @xmath60-encounter so the node contribution reduces to @xmath173 including the two edges we have a total contribution of @xmath174 where we divide by 4 to account for the symmetry of the diagram and the @xmath175 accounts for the fact that the cycles each start with a single object ( the original node , which is odd for both cycles ) . we likewise take the average @xmath176 for @xmath154 and @xmath155 in turn . when we put in the semiclassical substitutions from the formulae above can be summed and simplified . after applying the operator @xmath177 we find that first correction for the reflection for the unitary case ( adding the two base cases ) has the generating function @xmath178 by restricting ourselves above to the situation where @xmath151 we are not able to obtain the transmission directly , but we can instead find the likely transmission generating function using that @xmath179 @xmath180 the fact that we get such simple functions is a little surprising especially as the result from each base case is notably more complex . in fact this pattern can be seen to continue if we expand the rmt result as in [ higherreflection ] . the generating function in can also be obtained @xcite from their rmt result . when the system has time reversal symmetry , the edges and encounters can again be traversed in different directions by the trajectory set and their partners . for the conductance there are 7 further semiclassical diagrams at this order as depicted , for example in @xcite . when we remove the starting and end links to arrive at the base structures , we find that they reduce to the 4 base cases depicted in . we note that the additional diagrams are non - orientable when viewed as ribbon graphs . their groups of symmetry contain two elements each : reflection for diagrams [ ortho2trees](a)(c ) and inside - out flipping of both edges simultaneously for diagram [ ortho2trees](d ) . the additional 4 base structures that exist for systems with time reversal symmetry at the second subleading order in inverse channel number.,scaledwidth=70.0% ] figures [ ortho2trees](c ) and ( d ) are almost the same as figures [ unittrees](c ) and ( d ) , so we start with evaluating the contribution of ( a ) . although region 1 and 3 are spatially connected they differ as to where we append subtrees at the nodes . starting with the node on the left we therefore get the contributions @xmath181 with @xmath160 and @xmath182 again to ensure that an odd number of trees are appended , we substitute @xmath183 for the node on the right we obtain the contribution @xmath184 which is the same as @xmath185 but with @xmath154 swapped with @xmath155 ( and also @xmath186 with @xmath187 ) . along with the two nodes in ( a ) we have three links , two of which form cycles which already contain a single object ( a twist ) . the total contribution is then @xmath188 and to have an even number of objects along both cycles we average @xmath189 for @xmath154 and @xmath155 in turn . the node in the base structure in ( b ) provides the following contributions @xmath190 with @xmath170 and @xmath191 the total number of trees added must be even , leading to @xmath192 while with the two cycles ( which each start with a single object ) we have a total contribution of @xmath193 as before we take the average @xmath194 for @xmath154 and @xmath155 in turn . the difference of the structure in ( c ) from that in ( c ) is that now cycles @xmath195 and @xmath113 start with an odd number of objects . the contribution is then @xmath196 before averaging over the @xmath120 s in turn . only the @xmath195 cycle in the structure in ( d ) now starts with an odd number of objects so its contribution is @xmath197 and then we average over @xmath154 and @xmath155 in turn . summing over the four new base cases ( as well as the two that also exist without time reversal symmetry ) we obtain the second subleading correction for the orthogonal case @xmath198}{(1 - 4\xi s)^{\frac{5}{2 } } } .\ ] ] from this we can again find the likely generating function for the transmission @xmath199}{(1-s)^{\frac{3}{2}}(1-s+4\xi s)^{\frac{5}{2 } } } .\ ] ] the moments generated by can be proven @xcite to to agree with the moments obtained from an asymptotic expansion of their rmt result . semiclassically , time reversal symmetry allows more possible diagrams , so the results here are somewhat more complicated than the results and for systems without time reversal symmetry , but the rmt result @xcite for the orthogonal case is notably more complex than for the unitary case . the results here are therefore useful in simplifying asymptotic expansions of rmt moments . to obtain the leading order contributions @xcite , the start of the tree was fixed in the first incoming channel @xmath132 which allowed the top node to possibly @xmath13-touch . for example we simply place an incoming channel on top of the tree in ( a ) . if the odd numbered subtrees after the top node were empty then the top node could @xmath13-touch the lead , leading to the generating function @xcite @xmath200 here the terms involving @xmath48 derive from the fact that the section of the tree above the top node is actually an empty @xmath57 tree . however , using the ideas we developed for the subleading corrections we can imagine a way of generating the leading order trees without fixing any of the channels as a root . as we shall see this is particularly beneficial for calculating energy dependent correlation functions as in sections [ andreev ] and [ wigner ] . to start , we view a single point as the base structure for the leading order diagrams . we therefore obtain the leading order diagrams by joining subtrees to this point . to create a valid encounter we need to add @xmath61 subtrees ( with @xmath201 ) , @xmath60 of which are @xmath51-type subtrees starting from an incoming direction and the other @xmath60 are @xmath57-type subtrees . if all of the subtrees of a particular type are empty then the node created can touch the lead . we obtain the generating function @xmath202 where we divide by @xmath60 because of the rotational symmetry and by a further factor of 2 because of the additional possibility of swapping the incoming and outgoing channels . the last two terms in represent moving the @xmath60-encounter ( formed by appending the subtrees to the starting point ) into the incoming and outgoing channels . importantly we overcount the trees by the factor @xmath203 of their total number of encounters because , for a given resulting tree , any of the nodes could have been used as the base structure . on the other hand we can also construct trees by joining _ two _ subtrees together , one @xmath51-type and one @xmath57-type . after joining , the new vertex of degree two gets absorbed into the edge . a tree has exactly @xmath204 internal edges , therefore there are @xmath204 ways to obtain a given tree from joining two subtrees . the joining operation gives the contribution @xmath205 where we divide by 2 because of the symmetry of swapping the incoming and outgoing channels . in the first term in we subtract the empty tree from both @xmath51 quantities to ensure that they both include at least one node so that the edge formed is an internal one . the last term in is then to ensure that the diagram made of a single diagonal link with no encounters ( @xmath206 ) is included with the correct factor of @xmath207 . taking the difference between and then means that we count each tree exactly once . fittingly , for all the physical quantities we consider in this article equals minus the @xmath96 term in the sums in . this is natural because by joining two subtrees we essentially create a 1-encounter . we simplify the difference to @xmath208 now that no root is fixed we can make any channel to be the first incoming channel and this generating function indeed misses a factor @xmath22 compared to the generating function @xmath209 . this turns out to be very useful for the density of states of andreev billiards in and to recover @xmath209 we can use relation . the generating function for the reflection into lead 1 becomes @xmath210 taking the solution of or inverting and substituting into leads to the generating function for the reflection @xmath211 for the reflection into lead 2 we swap @xmath147 and @xmath148 . note again that when we take the explicit solutions to any of the generating functions we chose the solution whose expansion in @xmath89 agrees with the semiclassical diagrams . if , on the other hand , we start with , we obtain the generating function without the factor @xmath22 , @xmath212 to obtain the missing factor @xmath22 we substitute the correct solution of into and apply the operator @xmath177 , as in . simplifying the result we recover . for the leading order transmission moments we also start with so with we obtain @xcite @xmath213 the integrated generating function is @xmath214 as for the reflection , substituting the solutions of and transforming according to we recover . if we imagine merging the scattering leads and replacing them by a superconductor , then our chaotic cavity becomes an andreev billiard . using the scattering approach @xcite , the density of states ( normalised by its average ) of such a billiard can be written as @xcite @xmath215 in terms of energy - dependent correlation functions of the full scattering matrix @xmath216^n .\ ] ] here the energy difference is in units of the thouless energy @xmath217 ( which depends on the average classical dwell time @xmath17 ) and measured relative to the fermi energy @xmath21 . strictly speaking , for andreev billiards we should use @xmath218 ( matrix @xmath9 with complex - conjugated entries ) instead of the adjoint matrix @xmath219 but we will only consider systems with time reversal symmetry where @xmath9 is symmetric . with a superconductor at the lead , each time the particle ( electron or hole ) hits a channel it is retroreflected as the opposite particle ( hole or electron ) and semiclassically ( see @xcite for fuller details ) we traverse the partner trajectories in the opposite direction than for the reflection or transmission . for the leading order diagrams , this means that all the links ( and encounters ) are traversed in opposite directions by electrons and holes , so that if we break the time reversal symmetry , with a magnetic field say , then none of these diagrams are possible any longer . interestingly , at subleading order some diagrams are still allowed when the symmetry is completely broken , as for example the coherent backscattering contribution which comes from moving the node in ( e ) into the lead . we can consider the generating function @xmath220 which generates the required correlation functions . we note that the definition of @xmath221 here is marginally different than in @xcite . the semiclassical treatment there just requires us to make the substitutions @xmath222 where @xmath223 . as @xmath224 we have @xmath151 so becomes @xmath225 or @xmath226 which reduces to the cubic @xmath227 the leading order in inverse channel number was obtained semiclassically in @xcite , and for the energy dependent correlation functions we have @xmath228 setting @xmath229 , inverting and substituting into we obtain the cubic @xmath230 however , for the density of states the correlation function @xmath231 turns out to be more useful . indeed with @xmath232 and comparing with , we see that @xmath233 introducing @xmath234 this is precisely what is required for the density of states of andreev billiards in , as by setting @xmath235 we have @xmath236 performing the energy differential implicitly , we arrive at the cubic @xmath237{h_{0}}^2 + \left[s(1 - 2a)-(1-a)^{2}\right]{h_{0}}+s=0 .\ ] ] having the generating function @xmath238 therefore allows us easier access to the density of states than we had previously @xcite . to make the connection to the rmt treatment , we set @xmath235 and make the final substitution @xmath239}{2}\ ] ] so that the leading order contribution to the density of states is @xmath240 where @xmath241 satisfies the cubic @xmath242 as found previously using rmt @xcite . the result can be written explicitly as @xmath243 , \label{singledossol}\ ] ] where @xmath244^{\frac{1}{3 } } , \qquad d=\epsilon^4 + 44\epsilon^2 - 16 .\ ] ] note that the density of states is only non - zero when the discriminant @xmath245 is positive , which occurs when @xmath246 . with the techniques in this article we can go beyond this , and rmt , and see what happens at the next two orders in inverse channel number . for the first subleading order the even node contribution in becomes @xmath247 while the odd node contribution in is @xmath248 which we rewrite using . this simplifies the generating function , which reduces to @xmath249 we provide the further generating functions in [ moregenfuncts ] , but for the density of states we substitute @xmath250 in and obtain the cubic @xmath251 so that the first correction to the density of states is given by @xmath252 , \label{singledoscorrection}\ ] ] with @xmath253^{\frac{1}{3 } } .\end{aligned}\ ] ] as this involves the same discriminant as , the correction to the density of states is also only non - zero when @xmath246 , _ i.e. _ it has the same gap as the leading order term . the correction however is negative and has a singular peak from the discriminant in the denominator . repeating this procedure for the six base cases that contribute at the second subleading order we find @xmath254 , \label{singledossecondcorrection}\ ] ] with @xmath255^{\frac{1}{3 } } .\end{aligned}\ ] ] this correction is positive again and has a larger and steeper singular peak than the first order correction . to illustrate this we plot the leading order result , as well as the two corrections in ( a ) for @xmath256 , while in ( b ) we sequentially add the corrections to the leading order result , again for @xmath256 . ( a ) the leading order density of states ( dotted ) along with the first ( dashed ) and second ( solid ) correction for @xmath256 . ( b ) the leading order density of states ( dotted ) with the first ( dashed ) and then the second ( solid ) correction added for @xmath256 . ] the fact that the hard gap remains derives from the discriminant @xmath245 which is already present in @xmath51 ( at @xmath257 ) from . we could then expect that the gap is robust against further higher order corrections , seeing as the expressions always involve @xmath51 . someway above the gap we see that the corrections ( especially the second ) make little difference but it is the region directly above the gap which is particularly interesting . the expansion in inverse channel numbers is poorly ( if at all ) convergent but if the pattern of alternating singular peaks continues its sum ( or rather the exact result for finite channel number ) could take any value . in particular the gap could widen . though treating the density of states of andreev billiards semiclassically basically just involves using different values for the variables in the graphical recursions , on the rmt side it remains to be seen how one could use recent advances , like the selberg integral approach , to proceed beyond the leading order in inverse channel number . for the leading order @xcite a diagrammatic expansion was performed , but a result for an arbitrary number of channels would be especially welcome . it would determine whether the gap indeed persists and would clarify what happens to the density of states just above the current gap . another quantity related to the energy dependent correlation functions are the moments of the wigner delay times . to obtain them we define the correlation function @xmath258^n , \label{depseqn}\ ] ] where we subtract the identity matrix in the form of @xmath259 . the corresponding generating function is @xmath260 the moments of the wigner - smith matrix @xcite @xmath261 are then given by @xcite @xmath262^{n } = \frac{\td^{n}}{\rmi^{n}n!}\frac{\rmd^{n}}{\rmd \epsilon^{n } } d(\epsilon , n)\big\vert_{\epsilon=0 } , \ ] ] whose generating function we will denote @xmath263 the identity matrix in follows by removing the @xmath264 dependence of the scattering matrices . however , as the identity matrix only has diagonal elements we identify these elements as diagonal trajectory pairs that travel directly from incoming to outgoing channels . we considered these to be formed when we moved encounters into the outgoing channels ( equivalently we could use the incoming channels instead ) whenever we formed an empty @xmath57 subtree . the empty subtree is included in the general @xmath57 contribution and we included the contribution @xmath48 to allow us to change the effective value of @xmath57 in this situation . the empty @xmath57 subtrees consist of a single link and an incoming channel , producing the contribution @xmath265 . to mimic subtracting the identity matrix in we simply take away the value of this empty subtree at zero energy ( @xmath266 ) by setting @xmath267 along with the remaining semiclassical values in . including @xmath48 breaks the symmetry of @xmath51 and @xmath57 , so the subtree recursions and become @xmath268 and we find @xcite that @xmath51 satisfies the following cubic @xmath269 while @xmath57 is related by @xmath270 for the energy dependent correlation functions we have @xmath271 setting @xmath272 , inverting and substituting into we obtain the cubic @xmath273 for the @xmath0-th moment of the delay times we want the coefficient of the @xmath0-th power of @xmath274 which we can extract by transforming @xmath275 and then setting @xmath266 . this leads to a quadratic and the leading order moment generating function @xcite @xmath276 alternatively we can start with the generating function from @xmath277 \label{kmoteqn}\end{aligned}\ ] ] from which we can recover and hence by differentiating with respect to @xmath89 , multiplying by @xmath89 and using the result for @xmath278 from differentiating implicitly . for the first orthogonal correction we evaluate the contributions of the even and odd nodes around the mbius strip @xmath279 and @xmath280 which we again rewrite using and both contributions only depend on @xmath281 . putting these contributions into the generating function we again obtain @xmath282 as in but with different values for @xmath51 and @xmath57 as given in and . differentiating in line with and differentiating implicitly we arrive at the generating function , given as in [ moregenfuncts ] , which generates the orthogonal correction to the correlation functions @xmath283 . finally by transforming @xmath275 and setting @xmath266 we find the correction to the moments of the delay times to be @xmath284 since we only treated reflection quantities where @xmath151 for the next order corrections we can not obtain the corresponding generating functions of the moments of the delay times . instead we can generate the energy dependent correlation functions @xmath285 by expanding the generating function @xmath286 which can be found by treating the six base cases as in . expanding to finite order , we can then obtain the functions @xmath283 using the relation @xmath287 which follows from the binomial expansion of and where @xmath288 has no subleading order contribution . doing this only for the two base structures which exist without time reversal symmetry and plugging the resultant @xmath283 into we obtain the moments for the unitary case to low order . we find that the generating function @xmath289 fits with these moments , while if we treat all six base structures , the generating function @xmath290 fits the low moments for systems with time reversal symmetry . along with transport moments , we can also consider non - linear statistics such as the cross correlation between transport moments , generated by @xmath291}(s_1,s_2 ) = \sum_{n_1,n_2=1}^{\infty}s_{1}^{n_{1}}s_{2}^{n_{2 } } \left\langle\tr[x_{1}^{\dagger}x_{1}]^{n_1}\tr[x_{2}^{\dagger}x_{2}]^{n_2}\right\rangle , \ ] ] which involves two traces inside the energy average . semiclassically we then have an expression involving two trajectory sets that form two separate cycles , as in ( a ) . of course when we look for trajectory sets which lead to a small action difference we can have independent diagrams for each set , as in ( b ) . however , these are included in the individual moments treated previously , and when we remove them , @xmath292}(s_1,s_2 ) = p_{[x_1,x_2]}(s_1,s_2 ) - \sum_{n_1,n_2=1}^{\infty}s_{1}^{n_{1}}s_{2}^{n_{2 } } \left\langle\tr[x_{1}^{\dagger}x_{1}]^{n_1}\right\rangle \left\langle\tr[x_{2}^{\dagger}x_{2}]^{n_2}\right\rangle,\ ] ] we are left with trajectories that must interact , as in ( c ) . in rmt , this quantity is known as the ` connected ' part of the correlation function . ( a ) with two traces , the semiclassical trajectories separate into two closed cycles . when the two sets do not interact as in ( b ) we recreate terms from @xmath293^{n_1}\right\rangle \left\langle\tr[x_{2}^{\dagger}x_{2}]^{n_2}\right\rangle$ ] , but when they do interact as in ( c ) further diagrams are possible as in . ] it is interesting to consider the combinatorial interpretation of the interacting sets of trajectories . denote the incoming channels belonging to the first trace by @xmath115 , @xmath294 and the incoming channels from the second trace by @xmath295 , @xmath296 . input channels are mapped onto output channels by trajectories from @xmath20 and also by trajectories from @xmath297 . if we apply the first mapping followed by the inverse of the second mapping , we end up with the following transitions @xmath298 thus the overall result is a permutation , written in the cycle notation as @xmath299 . we can interpret each @xmath60-encounter as a cycle permuting @xmath60 labels of the corresponding @xmath20 trajectories . then an entire leading order diagram can be interpreted as a _ factorization _ of @xmath300 into smaller cycles ( _ c.f . _ @xcite ) . in the language of combinatorics , interacting trajectories correspond to a _ transitive factorization _ , leading order corresponds to the _ minimality _ condition and the fact that encounters on different ` branches ' have no ordering imposed on them corresponds to counting _ inequivalent factorizations _ ( up to a permutation of commuting factors ) . to summarize , the leading order interacting diagrams are in one - to - one correspondence with the minimal transitive inequivalent factorizations of a permutation into smaller cycles . this question has been studied combinatorially ( for factorizations into transpositions only ) for a permutation consisting of two cycles in @xcite and for three and more cycles ( these correspond to 3-point and higher cross correlations ) in @xcite . we note that the above combinatorial questions and the evaluation of transport properties are not completely equivalent problems . to evaluate transport properties we need to find additional information regarding the number of encounters touching the lead . on the other hand , we make substitutions or which significantly simplify the results . we now proceed to expand the contributions of the interacting sets of trajectories in inverse powers of the total channel number by describing the corresponding graphical representations . the base diagram for the leading order term is just a single loop , like in ( d ) except with no twist . without the twist , the two cycles of the permutation arise from the two walks on the inside and outside of the loop , see ( a ) . one requirement , to ensure a small action difference , is that the parts of the loop are traversed on either side by parts of trajectories that contribute actions with different signs in the semiclassical expression . in this means that the ( blue ) solid or dashed dotted lines on either side of the loop must partner ( red ) dashed or dotted lines on the other . without time reversal symmetry the parts of the loop must additionally be traversed in the same direction by trajectory stretches and their partners , so that at odd nodes ( those with an odd number of subtrees on each side of the loop ) there is unequal number of channels of a given type , as in ( a ) . with time reversal symmetry , parts of the loop may be traversed in any direction and we may also swap all the incoming and outgoing directions on one side ( the inside say ) of the loop as in ( b ) . example graphs , which contribute to the leading order in inverse channel number of @xmath301}(s_1,s_2)$ ] , made out of a single loop with subtrees attached at nodes . for a small action difference , the ( blue ) solid or dashed dotted lines must partner ( red ) dashed or dotted lines on the other side of the loop . ( a ) without time reversal symmetry the odd nodes have either incoming or outgoing channels on the outer subtrees on each side of the loop . ( b ) with time reversal symmetry we may also swap the incoming and outgoing channels inside the loop.,scaledwidth=70.0% ] with these restrictions we can start to append subtrees at nodes around the loop . we will use the tree function @xmath302 and generating variable @xmath303 for the subtrees outside the loop and @xmath304 and @xmath305 for those inside . at each node we can either add an even or odd number of subtrees on each side of the loop and we start with the contribution when we add an even number @xmath306 where @xmath307 . for systems without time reversal symmetry , with an odd number of subtrees on each side ( an odd node ) we have two possibilities as in ( a ) . the subtrees in the odd positions on both sides all connect ( first and last ) to either incoming or outgoing channels . with outgoing channels we have the contribution @xmath308 while with incoming channels we swap @xmath51 with @xmath57 @xmath309 as before it is possible that an odd node touches the lead when the odd positioned subtrees on both sides are empty . of course this also requires that the incoming or outgoing channels of the two quantities @xmath310 and @xmath311 originate or end in the same lead . when this is the case , we also have the following contribution @xmath312 where we needed to include the explicit @xmath303 and @xmath305 dependence of @xmath313 on the number of empty trees on each side of the loop . with incoming channels instead we again swap @xmath51 and @xmath57 @xmath314 we allow an arbitrary number , @xmath59 , of nodes along the loop , but the total number of odd nodes must be even . taking into account rotational symmetry , we get @xmath315,\ ] ] where @xmath316 is a contribution of one node . the node can either be even or odd of one of two types . since the number of odd @xmath13-nodes is equal to the number of odd @xmath14-nodes , we can write @xmath316 as @xmath317 to ensure that we indeed have an even number of odd nodes , we set @xmath318 leading to @xmath319.\ ] ] for the additional contribution , in the case of systems with time reversal symmetry , from the diagrams like ( b ) we swap the incoming and outgoing channels inside the loop so that we now have the contributions @xmath320 and @xmath321 and correspondingly @xmath322.\ ] ] there is also an additional freedom of placing the label @xmath132 on any leaf outside , giving a factor of @xmath323 . once @xmath132 has been placed the type ( in- or out- ) of the leaves inside is fixed and the freedom of placing the label @xmath324 inside produces only a factor of @xmath325 . we obtain these factors by differentiating with respect to the variables @xmath303 and @xmath305 , @xmath326 we further note that the differentiation ensures that there are at least two channels both inside and outside the loop . using the appropriate semiclassical values of the variables @xmath327 , @xmath328 and @xmath72 as well as the corresponding subtree contributions , we find the following generating functions @xmath329,1}^{\u}(s_1,s_2 ) = \tilde{p}_{[r_1,r_2],1}^{\u}(s_1,s_2 ) = \frac { s_1s_2}{2\left(s_1-s_2\right)^{2 } } \left[\frac{1 - 2\xi\left(s_1+s_2\right ) } { \sqrt { 1 - 4\xi s_1}\sqrt { 1 - 4\xi s_2 } } - 1\right ] , \ ] ] with @xmath330 and @xmath331 and twice this result for the orthogonal case with time reversal symmetry . even though for the autocorrelation @xmath332,1}^{\u}$ ] we can always move the odd nodes into the lead while for the cross correlation @xmath333,1}^{\u}$ ] we can not , this is somehow compensated for by the different subtree contributions and both give the same result . for the transmission autocorrelation we can move the odd nodes into the lead only for the unitary diagrams , leading to the generating function @xmath334,1}^{\u}(s_1,s_2 ) = \frac { s_1s_2}{2\left(s_1-s_2\right)^{2}}\left[\frac{1+\left(2\xi-1\right ) \left(s_1+s_2\right ) + \left(1 - 4\xi \right)s_1s_2 } { ( 1-s_1 ) \sqrt { 1+\frac{4\xi s_1}{1-s_1}}(1-s_2)\sqrt { 1+\frac{4\xi s_2}{1-s_2}}}-1\right],\ ] ] and we still obtain twice this for the orthogonal result . finally for the cross correlation between the reflection and transmission we have @xmath335,1}^{\u}(s_1,s_2 ) = \frac { s_1s_2}{2\left(s_1-s_1s_2+s_2\right)^{2 } } \left[1-\frac{1-s_2 + 2\xi \left(s_2-s_1+s_1s_2\right ) } { \sqrt { 1 - 4\xi s_1}(1-s_2)\sqrt { 1+\frac{4\xi s_2}{1-s_2}}}\right],\ ] ] and twice this for the orthogonal case . note that the above results remain unchanged if we swap @xmath336 and @xmath337 as this just means swapping @xmath147 and @xmath148 in the semiclassical contributions which does not change @xmath146 . from these results we can obtain the corresponding @xmath338}$ ] up to the first subleading order by including the first three orders in inverse channel number of the moments corresponding to @xmath310 multiplied by the moments corresponding to @xmath311 . if we also include @xmath339 terms ( which are just the number of channels in the respective lead ) with those moments then we obtain the @xmath340 and @xmath341 terms in . this then allows us to check that expansions of the various transport correlation functions indeed fulfil the unitarity conditions in . note that if we assume _ a priori _ that the unitarity is preserved by the semiclassical approximation , any one of equations implies all others . we can continue this process and look at the base structures like in figures [ unittrees ] and [ ortho2trees ] but which separate into two cycles . in fact the possibilities are almost the same as in but with one twist more or fewer as depicted in . these can also exist only for systems with time reversal symmetry and we can treat them in a similar way as before , but with the modifications above . the base structures which break into two cycles ( for systems with time reversal symmetry ) at the second subleading order in inverse channel number.,scaledwidth=70.0% ] again the types of subtrees at each node depends on the nodes elsewhere in the diagram , so we restrict out attention to the simpler case of the reflection where @xmath151 . because of , we have @xmath342^{n } - n_1 = \sum_{k=1}^n ( -1)^k { n\choose k } \tr[t^{\dagger}t]^{k}= \tr[r_{2}^{\dagger}r_{2}]^{n } - n_2 , \ ] ] so that @xmath332 } = \tilde{p}_{[r_1,r_2]}$ ] where the reflection autocorrelation is equal to the reflection cross correlation . however for the cross correlation , as the channels of the two reflections are in different leads , the nodes which lie on both cycles can not enter the lead , and this further simplifies the calculation . the edges which travel through both cycles then provide factors @xmath343 while the edges which only pass through one cycle provide factors @xmath344 as before , but with the semiclassical values corresponding to the reflection into lead 1 or lead 2 as appropriate . we will denote this correspondence by a subscript in the following . the treatment of the diagrams is very similar to that in so we merely highlight the steps here . but first we discuss the symmetry factors . because of time - reversal symmetry , we can put the first incoming channels on both faces on any leaf , leading to the differential operator @xmath345 . unlike the diagrams in , the ` inside ' and ` outside ' faces are , in general , not related by symmetry , and we should consider both putting @xmath302-trees on the ` outside ' and on the ` inside ' . for brevity we will only list the former contributions . finally , the symmetry groups of diagrams [ ortho2corr](c ) and [ ortho2corr](d ) have order @xmath346 and @xmath71 correspondingly and we will divide their contributions by the appropriate factor . starting with the diagram in ( a ) , for the node on the left , which can not enter the lead , we have @xmath347 with an odd number of trees appended @xmath348 for the node on the right we have @xmath184 as before but with semiclassical values corresponding to the reflection into lead 1 . to ensure a valid semiclassical diagram we still need each cycle to contain an even number of objects , so we have @xmath349 this is then averaged @xmath350 for @xmath154 and @xmath155 in turn . finally , we add the contribution where we place @xmath302 subtrees along the inside and @xmath304 subtrees along the outside . for the reflection cross correlation this reduces to swapping @xmath303 with @xmath305 and swapping @xmath147 with @xmath351 . the node in ( b ) gives @xmath352 with an even number of subtrees in total @xmath353 the total contribution is then @xmath354 averaged over @xmath154 and @xmath155 in turn and we again add the result where we swap the trees on the inside and the outside . for the structure in ( c ) the nodes provide @xmath355 with @xmath356 so that the contribution is then @xmath357 before averaging over the @xmath120 s in turn . likewise we include the contribution where we swap the subtrees on the inside with those on the outside . finally the node in ( d ) provides @xmath358 with @xmath359 the total contribution is then @xmath360 averaged over @xmath154 and @xmath155 in turn . we also include the contribution where we swap the subtrees on the inside and outside . summing the four diagrams , we eventually arrive at the generating function @xmath361,2}^{\o}(s_1,s_2)&=&\left[s_1+s_2 - 2s_1s_2 -2\xi \left(4s_1 ^ 2+s_2 ^ 2 + 3s_1s_2-s_1 ^ 3 - 5s_1 ^ 2s_2 - 2s_1s_2 ^ 2\right)\right . \nonumber \\ \fl & & \qquad \left . { } + 8s_1 ^ 2\xi^2\left(s_1 + 3s_2 - 3s_1s_2-s_2 ^ 2\right)\right]\nonumber \\ \fl & & \times \frac{s_1s_2}{(s_1-s_2)^{3}(1 - 4\xi s_1)^2\sqrt{1 - 4\xi s_2 } } + \left(s_1 \leftrightarrow s_2\right),\end{aligned}\ ] ] where @xmath362 means we add the result with @xmath363 and @xmath364 swapped . we could check that the results in this section all agree with the first four moments calculated from the arbitrary channel rmt results of @xcite . we described a method for the semiclassical calculation of the expansion of several transport statistics asymptotically in the inverse channel number @xmath43 . the calculation is performed by grafting trees onto the base structures with a low number of cycles , and relies on the fact that attaching trees does not change the order in inverse channel number . instead , the trees add more incoming and outgoing channels and so increase the order of the moment . with graphical recursions , this allows us to generate all the moments at a given order in inverse channel number , which we performed up to the third order . the terms we considered suggest the following observations about the ribbon graphs that arise as the contributing diagrams * absence of time reversal symmetry results in graphs being orientable ; both orientable and non - orientable graphs contribute to the calculation with time reversal symmetry , * the order ( in @xmath43 ) of a contribution is reflected in the genus of the corresponding graph , * linear moments ( with one trace ) result in graphs with one face , while non - linear moments with @xmath365 traces will require considering graphs with @xmath365 faces . the above general observations suggest that a complete expansion in @xmath43 should be both feasible and interesting to specialists working in algebra and combinatorics . we find that the semiclassical contribution of individual base diagrams depends significantly on the global structure of the diagram . this is in contrast to the expansions of the first two moments performed in @xcite , where the contribution factorized into a product over the vertices of the diagram and the problem was thus reduced to a combinatorial enumeration . the latter was achieved through finding recursion relations which connect different diagrams , and similar ideas could well be useful for the base structures we need for all moments . to illustrate the scale of the problem of going to higher order in @xmath43 , for the unitary case there are 1848 base diagrams at the next contributing order . as they can involve more than two nodes , they can no longer be derived by cleaning the corresponding semiclassical conductance diagrams as was the case for the orders treated in this article . however , in the end all these diagrams would probably lead to the generating functions and , highlighting the scale of the simplifications that take place . our results fully agree with the predictions of rmt theory ( as far as those are available @xcite ) , and importantly are given in terms of very simple generating functions . this would suggest that extending the types of asymptotic analyses of @xcite beyond the leading order , ( as is currently being performed @xcite ) , one could also expect to see simplifications of the rmt results . for the unitary case , where several different formulae are known for the moments of the transmission eigenvalues @xcite , this analysis and the semiclassical endpoints could shed light on the combinatorial relationships between the different approaches . because of the connection between rmt and weakly disordered systems , we can expect that our results also apply to such systems . likewise , with the close correspondence between semiclassical and disorder diagrams @xcite , one might hope to find similar graphical recursions in a perturbative expansion of the appropriate nonlinear @xmath48 model . if we were to consider non - linear statistics involving three traces ( with their mean parts removed ) , the leading contribution would come from the diagrams like in figures [ ortho2trees ] and [ ortho2corr ] which split into three cycles , _ i.e. _ diagrams ( a)(c ) without any twists . considering quantities with @xmath365 traces we would then need to treat base diagrams related to those which contribute to order @xmath366 ( and higher ) for the linear statistics , and so we immediately run into the considerations and difficulties described above . curiously though the moments of the conductance and the shot noise themselves can be efficiently treated using rmt @xcite . semiclassically , the @xmath365-th moment of these quantities corresponds to having exactly 2 or 4 channels along each of the @xmath365 cycles , and the rmt results might then provide a pathway for generating such semiclassical diagrams . this could in turn be useful for generating and treating the corresponding base diagrams for the linear statistics . the methods described in this article were also used to treat the density of states of andreev billiards . replacing the normal conducting leads of the chaotic cavity by a superconductor produces strong effects like the complete suppression of the density of states around the fermi energy @xcite . being interested in the density of states one must evaluate moments of all orders , something that our methods are particularly geared towards . going beyond leading order in inverse channel number we could show that this gap persists for the next two orders , and that the behaviour of the density of states slightly above the gap is not determined by just these terms in the expansion . because of the superconductor , one not only needs to know all the moments but also all the higher orders in inverse channel number . a result for arbitrary channel number would therefore be particularly welcome for such systems , which leads to the question of how to adapt the recent rmt advances to tackle this problem . similarly , chaotic cavities with additional superconductors attached ( andreev dots ) also exhibit significant effects due to the presence of the superconductors and also require one to be able to treat what would correspond to all the moments of usual transport quantities . for example , at leading order in inverse channel number , the conductance through a normal chaotic cavity requires just the diagonal pair of trajectories , while for the conductance through an andreev dot one needs full tree recursions @xcite . the treatment is actually similar to the edges in the base diagrams here , but with the added ingredient of having two ( or more ) different species of subtree . one can then see that treating the transport moments of andreev dots requires an extra layer of complexity compared to normal chaotic cavities . the results in this article are all for the case in which the leads are perfectly coupled to the chaotic cavity , rather than for the more general and experimentally relevant case of non - ideal coupling . this is typically modelled by introducing tunnel barriers into the leads with some probability to backscatter when entering ( or leaving ) the cavity . semiclassically , along with affecting the contributions of the channels and modifying the survival probability and hence contributions of the links and the correlated trajectory stretches inside the encounters , the main change is that a wealth of new diagrams become possible @xcite . specifically , encounters may now partially touch the leads and have some of their links backreflected at the tunnel barrier while the rest tunnel through to enter or exit the system . in principle , these possibilities would become extra terms in the tree and graphical recursions in this article , but so far these types of diagrams have only been treated semiclassically for the lowest moments @xcite . however , from a rmt viewpoint at leading order in inverse channel number and with the same tunnelling probability for each channel , the non - ideal contacts just increase the order of the generating functions by one , for example for both the density of states of andreev billiards @xcite and the moments of the wigner delay times @xcite . finally one can wonder whether the effect of the ehrenfest time can be incorporated into the graphical recursions developed here . for the leading order in inverse channel number , the effect could be included @xcite in the tree recursions . first , the trees are related to each other through a continuous deformation , for example by giving the nodes a certain size ( actually , the ehrenfest time itself ) and allowing them to slide into each other . second , this is then partitioned in a particular way so that one can extract the ehrenfest time dependence efficiently . each partition and hence the sum of all diagrams leads to the same simple ehrenfest time dependence at leading order in inverse channel number @xcite and this pattern and treatment seems to also hold at the first subleading order @xcite . whether this continues to higher order and nonlinear statistics , which start to include the complications of periodic orbit encounters , is an intriguing question . using the rmt result from @xcite we can compute the moments up to finite order and expand them in powers of the inverse channel number . looking at the patterns for the reflection in , and , we can expect that each order in the inverse channel number just increases the powers in the denominators ( the square root comes simply from the subtrees ) . in fact we find that the first several subleading orders can be written as @xmath367 where @xmath368 is the vector @xmath369 , @xmath370 is an @xmath371 matrix and @xmath372 is the row vector @xmath373 . the first few values of @xmath374 are @xmath375 @xmath376 @xmath377 @xmath378 similarly we can write the transmission as @xmath379 where @xmath380 is the vector @xmath381 and @xmath372 is the row vector @xmath373 . the first few values of the matrix @xmath382 are @xmath383 @xmath384 @xmath385 @xmath386 similar patterns hold for the moments of the delay times for the unitary case , and the results are actually simpler than for the transmission and reflection since there is one parameter fewer . the likely generating functions can be found by expanding the rmt result of @xcite and fitting to the behaviour of and . for the energy dependent correlation functions , we find that the generating function @xmath387 satisfies the cubic @xmath388^{2}(ng_{1})^{3 } \nonumber \\ \fl & & { } + 2\left[(a-1)^3+s^2(a+1)^3 \right]\left[4(1-a)^3+s(a^4 - 20a^2 - 8)+4s^2(a+1)^3\right](ng_{1})^{2 } \nonumber \\ \fl & & { } + \left[(a-1)^6 - 4s(a+1)(a-1)^4 + 3s^2(a-1)(a+1)(a^4 - 8a^2 - 2)\right . \fl & & { } \qquad \left . + 4s^3(a-1)(a+1)^4+s^4(a+1)^6\right]ng_{1 } \nonumber \\ \fl & & { } + a^2s\left[(a-1)^3+s^2(a+1)^3\right ] , \label{gortheqn}\end{aligned}\ ] ] while the energy differentiated generating function instead satisfies @xmath389^{2}(nh_{1})^{3 } \nonumber \\ \fl & & { } + 2a\left[(4-a)(a-1)^2 - 2s(5a^2 + 4)+s^2(a+4)(a+1)^2 \right]\nonumber \\ \fl & & { } \qquad \times \left[4(1-a)^3+s(a^4 - 20a^2 - 8)+4s^2(a+1)^3\right](nh_{1})^{2 } \nonumber \\ \fl & & { } + \left[(a-4)^2(a-1)^4 + 4s(a-1)^2(6a^3 - 21a^2 + 4a-16)\right . \fl & & { } \qquad + s^2(-3a^6 + 156a^4 + 102a^2 + 96 ) \nonumber \\ \fl & & { } \qquad \left . -4s^3(a+1)^2(6a^3 - 21a^2 + 4a-16)+s^4(a+4)^2(a+1)^4\right]nh_{1 } \nonumber \\ \fl & & { } + as(s-1)\left[(a+2)(a-4)^2+s(a-2)(a+4)^2\right ] . \label{hortheqn}\end{aligned}\ ] ] for the moments of the delay times we have the first subleading order generating function @xmath390^{2}(nl_{1})^{3 } \nonumber \\ \fl & & { } + 2(s+1)\left[(a-1)^3 + 2s(a-1)^3 + 2s^2a(a^2 + 3)\right ] \nonumber \\ \fl & & { } \qquad \times \left[4(1-a)^3+sa(a^3 - 8a^2 + 4a-24)+s^2a^2(a^2 + 4)\right](nl_{1})^{2 } \nonumber \\ \fl & & { } + \left[(a-1)^6 + 4sa(a-3)(a-1)^4 + 3s^2a^2(a-1)(3a^3 - 13a^2 + 20a-28)\right . \fl & & { } \qquad \left . + 2s^3a^2(a-1)(5a^3 - 11a^2 + 16a-32)^4+s^4a^2(5a^4 + 27a^2 + 32)\right]nl_{1 } \nonumber \\ \fl & & { } + sa^2\left[(a-1)^3 + 2s(a-1)^3 + 2s^2a(a^2 + 3)\right ] . \label{motortheqn}\end{aligned}\ ] ]

for chaotic cavities with scattering leads attached , transport properties can be approximated in terms of the classical trajectories which enter and exit the system . with a semiclassical treatment involving fine correlations between such trajectories we develop a diagrammatic technique to calculate the moments of various transport quantities . namely , we find the moments of the transmission and reflection eigenvalues for systems with and without time reversal symmetry . we also derive related quantities involving an energy dependence : the moments of the wigner delay times and the density of states of chaotic andreev billiards , where we find that the gap in the density persists when subleading corrections are included . finally , we show how to adapt our techniques to non - linear statistics by calculating the correlation between transport moments . in each setting , the answer for the @xmath0-th moment is obtained for arbitrary @xmath0 ( in the form of a moment generating function ) and for up to the three leading orders in terms of the inverse channel number . our results suggest patterns which should hold for further corrections and by matching with the low order moments available from random matrix theory we derive likely higher order generating functions .

the detailed understanding of the behavior of strongly interacting matter under extreme conditions of temperature and/or density has become an issue of great interest in recent years . unfortunately , even if a significant progress has been made on the development of ab initio calculations such as lattice qcd , these are not yet able to provide a full understanding of the qcd phase diagram and the related hadron properties , due to the well - known difficulties of dealing with small current quark masses and finite chemical potentials . thus it is important to develop effective models that show consistency with lattice results and can be extrapolated into regions not accessible by lattice calculation techniques . recently , models in which quark fields interact via local four point vertices and where the polyakov loop is introduced to account for the confinement - deconfinement phase transition ( so - called polyakov - nambu - jona - lasinio ( pnjl ) models @xcite ) have received considerable attention . here , we consider a non - local extension of these pnjl models , which includes terms leading to the quark wave function renormalization . two different parameterizations are used : an exponential form , and a parametrization based on a fit to the mass and renormalization function obtained in lattice calculations . in the context of this type of model the properties of the vacuum and meson sectors at @xmath0 have been studied in ref.@xcite . this contribution is organized as follows . in sec . 2 we introduce the model lagrangian and its parameterizations . in sec . 3 we present and discuss our results for the behavior of some thermodynamical properties and the corresponding phase diagrams . finally , in sec . 4 our main conclusions are summarized . we consider here a nonlocal su(2 ) chiral quark model which includes quark couplings to the color gauge fields . the corresponding euclidean effective action is given by @xmath1 \psi(x ) \ ! - \ ! \frac{g_{s}}{2 } \big [ j_{a}(x)j_{a}(x)\ ! - \ ! j_{p}(x ) j_{p}(x ) \big ] \ ! + \ ! { \cal u}\,(\phi[a(x)])\right\ } , \label{action}\end{aligned}\ ] ] where @xmath2 is the @xmath3 fermion doublet @xmath4 , and @xmath5 is the current quark mass matrix , in what follows we consider isospin symmetry , that is @xmath6 . the fermion kinetic term includes a covariant derivative @xmath7 , where @xmath8 are color gauge fields . the nonlocal currents @xmath9 are given by @xmath10 here , @xmath11 and @xmath12 . the functions @xmath13 and @xmath14 in eq.([currents ] ) , are nonlocal covariant form factors characterizing the corresponding interactions . the scalar - isoscalar component of the @xmath15 current will generate the momentum dependent quark mass in the quark propagator , while the `` momentum '' current , @xmath16 will be responsible for a momentum dependent wave function renormalization of this propagator . to proceed it is convenient to perform a standard bosonization of the theory . thus , we introduce the bosonic fields @xmath17 and @xmath18 , and integrate out the quark fields . in what follows , we work within the mean - field approximation ( mfa ) , in which these bosonic fields are replaced by their vacuum expectation values @xmath19 and @xmath20 . next , we extend the so obtained bosonized effective mfa action to finite temperature @xmath21 and chemical potential @xmath22 using the matsubara formalism . concerning the gluon fields we will assume that they provide a constant background color field @xmath23 , where @xmath24 are the su(3 ) color gauge fields . then the traced polyakov loop , which is taken as order parameter of confinement , is given by @xmath25 , where @xmath26 , @xmath27 . we will work in the so - called polyakov gauge , in which the matrix @xmath28 is given a diagonal representation @xmath29 , which leaves only two independent variables , @xmath30 and @xmath31 . owing to the charge conjugation properties of the qcd lagrangian @xcite , the mean field value of the polyakov loop field @xmath32 is expected to be a real quantity . in addition , we assume as usual that @xmath30 and @xmath31 are real - valued fields @xcite , this implies that @xmath33 , then @xmath34/3 $ ] . within this framework the mean field thermodynamical potential @xmath35 results @xmath36 + \frac{\bar \sigma_1 ^ 2}{2\,g_s } + \frac{\kappa_p^2\ \bar \sigma_2 ^ 2}{2\,g_s } + { \cal{u}}(\bar \phi , t ) \ , \label{granp_reg}\end{aligned}\ ] ] here , the shorthand notation @xmath37 has been used , and @xmath38 and @xmath39 are given by @xmath40 \qquad , \qquad z(p ) = \left [ 1 - \bar{\sigma}_{2 } \ f(p ) \right]^{-1 } \ , \label{mz}\end{aligned}\ ] ] where @xmath41 and @xmath42 are the fourier transform of @xmath13 and @xmath14 , respectively . in addition , we have defined @xmath43 ^ 2 + { \vec{p}}\ \ ! ^2 \ , \ ] ] where the quantities @xmath44 are given by the relation @xmath45 . namely , @xmath46 with @xmath47 for @xmath48 respectively . at this stage we need to specify the explicit form of the polyakov loop effective potential . here , we used the fit to qcd lattice results proposed in ref . @xcite . @xmath35 turns out to be divergent and , thus , needs to be regularized . for this purpose we use the same prescription as in ref . namely @xmath49 where @xmath50 is obtained from the first term in eq.([granp_reg ] ) by setting @xmath51 and @xmath52 is the regularized expression for the quark thermodynamical potential in the absence of fermion interactions , @xmath53 \ , \label{freeomegareg}\ ] ] with @xmath54 . finally , note that in eq.([omegareg ] ) we have included a constant @xmath55 which is fixed by the condition that @xmath56 vanishes at @xmath0 . the mean field values @xmath57 and @xmath58 at a given temperature or chemical potential , are obtained from a set of three coupled `` gap '' equations . this set of equations follows from the minimization of the regularized thermodynamical potential , that is @xmath59 once the mean field values are obtained , the @xmath60 behavior of other relevant quantities can be determined . in order to fully specify the model under consideration we have to fix the model parameters as well as the form factors @xmath61 and @xmath62 which characterize the non - local interactions . following ref.@xcite , we consider two different type of functional dependencies for these form factors . the first one corresponds to the often used exponential forms , @xmath63 \qquad , \qquad f(q)= \mbox{exp}[-q^{2}/\lambda_{1}^{2 } ] \ . \label{regulators}\ ] ] note that the range ( in momentum space ) of the nonlocality in each channel is determined by the parameters @xmath64 and @xmath65 , respectively . fixing the @xmath0 values of @xmath66 and chiral quark condensate to reasonable values @xmath67 mev and @xmath68 mev the rest of the parameters are determined so as to reproduce the empirical values @xmath69 mev and @xmath70 mev , and @xmath71 which is within the range of values suggested by recent lattice calculations@xcite . in what follows this choice of model parameters and form factors will be referred as parametrization s1 . the second type of form factor functional forms we consider is given by @xmath72 where @xmath73^{-1 } \qquad ; \qquad f_{z}(q ) = \left [ 1 + \left ( q^{2}/\lambda_{1}^{2}\right ) \right]^{-5/2 } \ . \label{parametrization_set2}\ ] ] as shown in ref.@xcite , with a convenient choice of parameters one can very well reproduce the momentum dependence of mass and the renormalization function obtained in a landau gauge lattice calculation as well as the physical values of @xmath74 and @xmath75 . in what follows this parametrization will be referred as s2 . finally , in order to compare with previous studies where the wavefunction renormalization of the quark propagator has been ignored we consider a third parametrization ( s3 ) . in such case we take @xmath39 = 1 ( setting @xmath42 = 0 ) and exponential parametrization for @xmath41 . the values of the model parameters for each of the chosen parameterizations are summarized in table i. [ c]ccccc & & s1 & s2 & s3 + @xmath76 & mev & 5.70 & 2.37 & 5.78 + @xmath77 & & 32.030 & 20.818 & 20.650 + @xmath78 & mev & 814.42 & 850.00 & 752.20 + @xmath79 & gev & 4.180 & 6.034 & @xmath80 + @xmath81 & mev & 1034.5 & 1400.0 & @xmath80 + [ tab1 ] we start by analyzing the behavior of some mean field quantities as functions of @xmath21 and @xmath22 . since the results obtained for our three different parameterizations are qualitatively quite similar we only present explicitly those corresponding to the parametrization s1 . , @xmath82 and @xmath83 as functions of @xmath21 for low ( left ) , high ( right ) and cep ( central ) chemical potentials . note that the scale to the left corresponds to that of @xmath84 while that to the right to @xmath82 and @xmath32 . since @xmath82 turns out to be negative we plot @xmath85 . ] they are given in fig.1 where we plot @xmath86 , @xmath82 and @xmath32 as functions of @xmath21 for some values of the chemical potential . 1a shows that at @xmath87 there is a certain value of @xmath21 at which @xmath84 drops rapidly signalling the existence of a chiral symmetry restoration crossover transition , its position being determined by the peak of the chiral susceptibility . at basically the same temperature the polyakov loop @xmath32 increases which can be interpreted as the onset of the deconfinement transition . as @xmath22 increases there is a certain value of @xmath88 above which the transition starts to be discontinuous . at this precise chemical potential the transition is of second order . this situation is illustrated in fig.1b . the corresponding values @xmath89 define the position of the so - called `` critical end point '' . as displays in fig.1c , for @xmath90 the transition becomes discontinuous , i.e. of first order . finally , for chemical potentials above @xmath91 mev the system is in the chirally restored phase for all values of the temperature . it is important to note that although @xmath82 appears to be rather constant in fig.1 , at higher values of @xmath21 it does go to zero as expected . concerning the deconfinement transition we see that as @xmath22 increases there appears a region where system remains in its confined phase ( signalled by @xmath32 smaller than @xmath92 ) even though chiral symmetry has been restored . this corresponds to the recently proposed quarkyonic phase@xcite . the phase diagrams corresponding to our three different parameterizations are given in fig.[phase diagrams ] . here the dotted line corresponds to the line of crossover chiral transition while the full line to the line of first order chiral transition . the dashed lines correspond to the deconfinement transition ( the lower and upper lines correspond to @xmath93 and @xmath94 , respectively ) . comparing those of s1 and s3 we see that the main effect of the wave function renormalization term is to shift the location of the cep towards lower values of @xmath21 and higher values @xmath22 . concerning the lattice adjusted parametrization s2 we observe that it leads to even lower values of @xmath95 and higher values @xmath96 . and @xmath97 , respectively ) . a non - local extension of the pnjl model momentum which leads to momentum dependent quark mass and wave function renormalization has been studied . this model provides a simultaneous description for the deconfinement and chiral phase transition . non - local interactions have been described by considering both a set of exponential form factors , and a set of form factors obtained from a fit to the mass and renormalization functions obtained in lattice calculations . the resulting phase diagrams turn out to be qualitative similar , the position of the critical end point being the feature which depends more crucially on each particular parametrization . we would like to thank the members of the organizing committee for their warm hospitality during the workshop . this work has been supported in part by anpcyt ( argentina ) , under grant pict07 03 - 00818 . s. roessner , c. ratti and w. weise , phys . d * 75 * ( 2007 ) 034007 . s. noguera , n. n. scoccola , phys . d * 78 * ( 2008 ) 114002 . a. dumitru , r. d. pisarski and d. zschiesche , phys . d * 72 * ( 2005 ) 065008 . d. gomez dumm and n. n. scoccola , phys . c * 72 * ( 2005 ) 014909 . m. b. parappilly et al , phys . d * 73 * , 054504 ( 2006 ) .

we analyze the chiral restoration and deconfinement transitions in the framework of a non - local chiral quark model which includes terms leading to the quark wave function renormalization , and takes care of the effect of gauge interactions by coupling the quarks with the polyakov loop . non - local interactions are described by considering both a set of exponential form factors , and a set of form factors obtained from a fit to the mass and renormalization functions obtained in lattice calculations .

qubit is the information stored in the quantum state of a two - level system , routinely used as the smallest unit of information processed in the quantum circuit model of quantum computation @xcite . in order to construct a universal computational gate set , single - qubit rotations , about at least two distinct rotational axes are required as well as a two - qubit gate , e.g. , the cnot gate . single - qubit rotation gates , such as hadamard and pauli x , y , and z gates have been implemented on numerous physical systems , including photons @xcite , ions @xcite , atoms @xcite , molecules @xcite , quantum dots @xcite , and superconducting qubits @xcite . many single - qubit rotations in a sequence can also be performed with _ a single arbitrary rotation gate _ , which simplifies otherwise complex physical implementation of many distinct rotations in a unified fashion . an arbitrary rotation ( of rotation angle @xmath0 and rotational axis @xmath1 ) can be constructed with a minimum of three rotations that correspond to the set of euler angle rotations : for example , the three rotations in the best - known zyz - decomposition are given by @xmath2 where @xmath3 represents a rotational transformation , and @xmath1 and @xmath0 are respectively given as a function of three rotation angles @xmath4 , @xmath5 , and @xmath6 @xcite . in an optical implementation of two - level system dynamics , z - rotations use either a time - evolution or a far - detuned excitation @xcite , and x or y - rotations a resonant area - pulse interaction , both of which and their combinations require a precise control of the relative phase and timing among the constituent pulsed interactions . in this paper , we show that an arbitrary rotation can be , alternatively , performed with a single laser - pulse , when the pulse is programmed to be _ a chirped pulse with a temporal hole_. as to be discussed in the rest of the paper , a single laser pulse with the given pulse shape implements zyz - decomposed rotations all at once , where the temporal hole in the middle of a chirped pulse induces a strong non - adiabatic evolution , which is a y - rotation , amid an otherwise monotonic adiabatic evolution , a z - rotation , due to the chirped pulse . the predicted behavior of the zyz - decomposition is to be experimentally verified with cold atomic qubits and as - programmed femtosecond laser pulses . we consider the dynamics of a two - level atom , driven by a chirped laser pulse with a temporal hole . the electric field of the pulse , where both the main pulse and the hole are assumed to be of gaussian pulse shape , is given by @xmath7 where @xmath8 and @xmath9 are respectively the widths of the main pulse and the hole , @xmath10 ( @xmath11 ) is the depth of the hole , @xmath12 is the linear chirp parameter , and @xmath13 is the carrier phase . the contribution of the carrier phase is a simple z - rotation , _ i.e. _ @xmath14 , so we will first consider the @xmath15 case . when the state vectors are defined by @xmath16 and @xmath17 ( of respective energies @xmath18 and @xmath19 ) , the interaction hamiltonian in the adiabatic basis @xcite , after the rotating wave approximation , is given by @xmath20,\ ] ] where @xmath21 are the eigenvalues in the bare basis , for the rabi frequency @xmath22 and the instantaneous detuning @xmath23 , and @xmath24 is the adiabatic mixing angle defined by @xmath25 for @xmath26 . however , with eq . , the phase of the state diverges at @xmath27 , so we use an additional transformation @xmath28 with @xmath29 $ ] to remove this rapidly oscillating phase . the resulting hamiltonian is given in the interaction picture by @xmath30.\ ] ] figure 1 shows the behavior of the mixing angle @xmath24 , compared with the rabi frequency @xmath22 for various hole depth @xmath10 ( left panel ) , and the corresponding bloch vector evolution ( right panel ) . the pulse without a hole in fig . [ fig1](a ) shows a slow change in @xmath24 and relatively large @xmath22 , suggesting that the adiabatic condition , @xmath31 , is satisfied in all time . so , a pulse without a hole induces an adiabatic evolution , _ i.e. _ a z - rotation in the adiabatic basis , as depicted in fig . [ fig1](d ) . on the other hand , the pulses with a hole in figs . [ fig1](b ) and [ fig1](c ) exhibit an abrupt change in @xmath24 near @xmath32 . therefore , the overall dynamics may be decomposed to three sub - dynamics in different time zones : @xmath33 , @xmath34 , and @xmath35 , as clearly shown in figs . [ fig1](e ) and [ fig1](f ) . in the central time zone ( @xmath34 ) , the hole makes @xmath22 small and a rapid change in @xmath24 occurs . since the hamiltonian is dominated by the non - adiabatic coupling in the off - diagonal components , it is approximately given by @xmath36,\ ] ] which corresponds to a y - rotation with a rotation angle @xmath37.\end{aligned}\ ] ] in both side regions ( @xmath33 and @xmath35 ) , z - rotations occur due to the adiabatic evolution of the chirped pulse . the rotation angles are respectively given by @xmath38 dt \\ \phi_2 & = & \int_{\tau_h}^{\infty } \left [ |\delta(t)| - \sqrt{\delta^2(t)+\omega^2(t)}\right ] dt,\end{aligned}\ ] ] and when there is no detuning @xcite , @xmath4 equals @xmath5 due to the symmetry . as a result , the total time - evolution , including the z - rotation due to the carrier phase @xmath14 , is given by @xmath39 , \label{eq.rrr}\end{aligned}\ ] ] which corresponds to zyz rotations . when the qubit starts from an initial state defined by @xmath40 it can go , through the interaction with the as - programmed laser pulses , to an arbitrary position on the bloch sphere , as shown in fig . [ fig2](a ) . . the pulse carrier - envelope phase @xmath13 and the equivalent pulse - area @xmath41 @xcite were varied , while @xmath42 ps , @xmath43 rad / ps@xmath44 , @xmath45 , and @xmath46 were kept constant . note that the shaded regions near the poles are physically unfeasible , requiring extreme conditions . ( b ) estimated z- and y - rotation angles , @xmath5 and @xmath6 , spanned by varrying @xmath41 and @xmath47 ( detuning ) @xcite , while @xmath48 ps , @xmath49 rad / ps@xmath44 , @xmath45 and @xmath50 were fixed.,scaledwidth=40.0% ] while this is only true for certain initial qubit states , as @xmath51 and @xmath6 in eq . are not fully independent , the rotations can be made arbitrary : by using the detuning and pulse area in eq . , the full range range @xmath52 for @xmath53 and @xmath54 for @xmath6 are completely spanned as in fig . [ fig2](b ) , ensuring the given zyz rotations to be arbitrary . in order to verify the zyz rotations , of the chirped pulse with a temporal hole , we performed a proof - of - principle experiment with cold atomic qubits and as - programmed femtosecond laser pulses ( see fig . [ fig3 ] ) . the detail of our laser experimental setup was described in our previous work @xcite . briefly , we used amplified optical pulses from a ti : sapphire mode - locked laser . initial pulses were produced at a repetition rate of 1 khz from the laser , wavelength - centered at the resonance wavelength 795 nm of the rubidium transition from 5s@xmath55 to 5p@xmath55 . the spectral bandwidth was 2.5 thz in gaussian width , equivalent to a pulse duration of 212 fs ( fwhm ) for a transform - limited gaussian pulse . the pulses were then shaped with an acousto - optic pulse programming device ( aopdf , dazzler from fastlite ) @xcite . the two - level system was formed with the ground and excited states , @xmath56s@xmath55 and @xmath57p@xmath55 , of atomic rubidium ( @xmath58rb ) and the atoms were held in a magneto - optical trap @xcite . the inhomogeneity of the laser - atom interaction @xcite , due to the spatial intensity profile of the laser , was minimized by reducing the size of the atom cloud 2.3 times smaller than the the laser beam size . the size of the atom cloud was 250 @xmath59 m ( fwhm ) . the control experiment was conducted in three steps : initialization , qubit rotation , and detection . the atoms were first excited by a @xmath60-area pulse to initialize the atoms in the superposition state @xmath61 defined in eq . . then , the chirped pulse with a temporal hole performed a rotation of the atomic state , @xmath62 . lastly , atoms in the excited state were detected through ionization , using a frequency - doubled split - off of an un - shaped laser pulse and a micro - channel plate ( mcp ) detector . the laser pulses for the initialization and qubit rotation were programmed , with a fixed relative phase , by the aopdf . in the frequency domain , the combined field is given by @xmath63 where @xmath64 is the @xmath60-area pulse , @xmath65 is the chirped pulse with a temporal hole , and @xmath13 is the carrier phase of the rotation pulse relative to the initialization pulse . the total energy of these two pulses was up to 20 @xmath59j and the energy of each pulse was pre - calibrated through cross - correlation measurements . the chirp parameter for the control pulse was fixed at @xmath66 rad / ps@xmath44 , which corresponses to frequency chirp of 60,000 fs@xmath44 in the spectral domain . figure [ fig4 ] shows a comparison between experimental and theoretical results . when atoms , in the initial superposition state @xmath61 in eq . , undergo the rotation , given in eq . , the excited - state probability is given by @xmath67 \label{eqevol2}.\end{aligned}\ ] ] the resulting behavior of @xmath68 is an oscillatory function , of which the amplitude and phase are determined by @xmath6 and @xmath69 . in fig . [ fig4](a ) , the measured probability is plotted as a function of the equivalent ( peak ) pulse - area @xcite and the carrier phase @xmath13 . the result strongly agrees with the calculation in fig . [ fig4](b ) , performed with the corresponding time - domain schrdinger equation ( tdse ) . each point in figs . [ fig4](a ) and [ fig4](b ) corresponds to a distinct bloch vector evolution . a few characteristic trajectories are shown in figs . [ fig4](c , d,@xmath70,h ) ( see the figure caption for more detail ) . along the dashed lines in figs . [ fig4](a ) and [ fig4](b ) , data points are extracted and compared in fig . [ fig4](i ) , where the excited - state probabilities , @xmath71 , are plotted as a function of @xmath13 at fixed @xmath6 and @xmath4 . the change of the peak oscillation point in fig . [ fig4](i ) is related to the @xmath72-dependence of @xmath4 as in eq . ; @xmath4 is a monotonically decreasing function of @xmath72 , so the peaks in fig . [ fig4](i ) shift to the upper right corner as @xmath72 increases . also , the change in the oscillation amplitude is related to the @xmath72-dependence of @xmath6 . as the electric - field amplitude @xmath72 increases , so does the rotation angle @xmath6 of the y - rotation ; however , it is up to a certain maximum @xmath72 , at above of which the dynamics involved with the hole gradually becomes adiabatic . such behavior of @xmath6 is clearly demonstrated in fig . [ fig4](i ) , where the oscillation amplitude given by @xmath73 in eq . reaches maximal , along the line marked by , and decreases as @xmath72 increases . therefore , the expected behavior @xmath4 and @xmath6 in eq . is clearly observed in the experimental results . in summary , we proposed and demonstrated the use of hybrid adiabatic and non - adiabatic interaction for single laser - pulse implementation of arbitrary qubit rotations . the chirped optical pulse with a temporal hole induced zyz - decomposed rotations of atomic qubits all at once , in which the temporal hole caused a non - adiabatic evolution amid an otherwise monotonic adiabatic evolution due to the chirped pulse . the proof - of - principle experimental verification of the given laser - atom interaction was performed with programmed femtosecond laser pulses and cold atoms . the result suggests that laser pulse - shape programming may be useful in quantum computation through concatenating gate operations in a quantum circuit . f. h. l. koppens , c. buizert , k. j. tielrooij , i. t. vink , k. c. nowack , t. meunier , l. p. kouwenhoven , and l. m. k. vandersypen , `` driven coherent oscillations of a single electron spin in a quantum dot , '' nature * 442 * , 766 ( 2006 ) . e. lucero , m. hofheinz , m. ansmann , r. c. bialczak , n. katz , m. neeley , a. d. oconnell , h. wang , a. n. cleland , and j. m. martinis , high - fidelity gates in a single josephson qubit , " phys . rev 100 * , 247001 ( 2008 ) .

arbitrary rotation of a qubit can be performed with a three - pulse sequence ; for example , zyz rotations . however , this requires precise control of the relative phase and timing between the pulses , making it technically challenging in optical implementation in a short time scale . here we show any zyz rotations can be implemented with a single laser - pulse , that is _ a chirped pulse with a temporal hole_. the hole of this shaped pulse induces a non - adiabatic interaction in the middle of the adiabatic evolution of the chirped pulse , converting the central part of an otherwise simple z - rotation to a y rotation , constructing zyz rotations . the result of our experiment performed with shaped femtosecond laser pulses and cold rubidium atoms shows strong agreement with the theory .

let @xmath10 be an @xmath0 binary matrix , and write @xmath11 for the support of its @xmath12th column ( that is , the locations of the @xmath13s ) . then @xmath1 is said to be _ @xmath3-separable _ if the sets @xmath4 are all distinct over all sets @xmath14 of cardinality @xmath3 ( see definition [ defsep ] , to come ) . separable matrices were first introduced by erds and moser in 1970 @xcite and have since been studied in different contexts , including coding theory , combinatorics and , as we discuss later , group testing , where they play a very important role . separable matrices are often studied through the slightly stronger concept of _ disjunct matrices _ ( see definition [ disj ] ) . disjunct matrices were first introduced by kautz and singleton @xcite and , just like separable matrices , they have been extensively studied in coding theory , combinatorics and group testing @xcite . a central question in the study of both separable and disjunct matrices is the following : given @xmath15 and @xmath3 , how large must @xmath7 be for there to exist either an @xmath0 @xmath3-separable or disjunct matrix ? in this paper , we investigate the asymptotics for separability as @xmath16 , where @xmath3 may grow with @xmath15 . a simple counting bound ( theorem [ counting ] ) shows that @xmath17 rows are required . disappointingly , when @xmath18 this bound is not tight , and we require roughly a factor of @xmath3 more than this , as in fact it has been shown @xcite that @xmath19 is needed . this lower bound is motivated by the connection between disjunctness and separability , as we discuss in section [ sepsec ] . notice that when @xmath3 grows linearly with @xmath15 , taking the identity matrix is order optimal for this reason , we consider only @xmath18 in this paper . in order to meet the lower bound @xmath20 , we consider a relaxation of the requirement of @xmath3-separability to _ almost @xmath3-separability_. roughly speaking , a matrix is _ almost @xmath3-separable _ if the sets @xmath4 are ` usually ' distinct see definition [ defalmsep ] for a formal definition . our main result shows that it is possible to achieve almost separability with only @xmath21 rows ( theorem [ mainthm ] ) . when @xmath9 , for any @xmath22 $ ] , this is order - optimal to the counting bound . however , we also aim to get best possible constants for @xmath7 - a goal motivated by the study of the rate of group testing algorithms . group testing is an old and well - studied search problem , first considered by dorfman @xcite , where the goal is to recover a sparse subset of @xmath3 _ defective _ elements spread among @xmath15 otherwise identical items . instead of testing each item for defectiveness individually , classic group testing algorithms test items in batches . in the noiseless binary model we consider , tests can only reveal whether a given set contains at least one defective ( a positive test ) or no defectives ( a negative test ) . the connection between separable matrices and nonadaptive group testing is well - known , and we discuss it in section [ gtsec ] . for the moment , we just observe that a sequence of tests designed a priori ( _ nonadaptive _ group testing ) has a natural binary - matrix representation : each length-@xmath15 row represents a test , with entries being @xmath13 if the corresponding item is being included in the test . a matrix being @xmath3-separable is equivalent to having zero probability of error for nonadaptive group testing , while a matrix being almost @xmath3-separable is equivalent to having a small probability of error . the ` arbitrarily small probability of error ' criterion we consider here is the same as that in shannon s theory of channel coding . with this comparison in mind , we consider the concept of _ rate _ of group testing ( definition [ defrate ] ) for @xmath23 defective items in a population of size @xmath15 , which can be thought of as the amount of information conveyed by each test . using a separable matrix with @xmath24 rows leads to a group testing rate of @xmath25 . however , using an almost separable matrix with @xmath26 rows gives a strictly positive rate , with the rate depending on the contant implied by the big-@xmath27 . hence , here we are interested in getting good constants for @xmath7 , not only in order - wise results . in theorem [ ratethm ] , we show that our results meet previous results for the limiting regime where @xmath3 is fixed as @xmath16 , and improves over the previous best known bounds for larger values of the _ sparsity parameter _ @xmath28 $ ] in the @xmath29 regime frequently considered in the group testing literature . we begin by recalling the definition of a separable matrix . [ defsep ] given an @xmath0 binary matrix @xmath30 , we shall write @xmath31 for the _ support _ of column @xmath12 and for @xmath32 also write @xmath33 for the support of a disjunction of columns . the matrix @xmath1 is called _ @xmath3-separable _ matrix if the for all sets @xmath5 of size @xmath3 , there is no other set @xmath34 also of size @xmath3 with @xmath35 = @xmath36 . the case @xmath37 is trivial , so we assume @xmath38 throughout . we shall also assume @xmath39 , which will be no restriction in the limiting regimes we study . the following counting bound is described by chen and hwang as `` simple - minded '' @xcite . [ counting ] let @xmath40 be the smallest @xmath7 such that an @xmath0 @xmath3-separable matrix exists . then @xmath41 clearly @xmath42 where @xmath43 denotes the power set . hence for @xmath1 to be @xmath3-separable we require @xmath44 , and taking logarithms gives the result . using the lower bound of @xmath45 ( which we shall use many times in this paper ) , we see that a @xmath3-separable matrix must have at least @xmath46 rows . as we anticipated , separable matrices are tightly related to another class of matrices , namely that of disjunct matrices . [ disj ] with the notation of definition [ defsep ] , @xmath1 is _ @xmath3-disjunct _ if for all sets @xmath5 of cardinality @xmath47 , there does not exist @xmath48 such that @xmath49 . in the language of set systems , a matrix @xmath1 being @xmath3-seperable is equivalent to the family @xmath50 being @xmath3-union - free , and @xmath1 being @xmath3-disjunct is equivalent to @xmath50 being @xmath3-cover - free . it s easy to see that @xmath3-disjunctness implies @xmath3-separability ( see , for example , @xcite , ( * ? ? ? * section 7.2 ) , or the special case @xmath51 of lemma [ implies ] below ) . on the other hand , chen and hwang ( * ? ? ? * theorem 2 ) have shown that it is possible to construct a @xmath3-disjunct matrix from a @xmath52-separable matrix by adding at most one row to it , which means that disjunct and separable matrices share the same order - wise asymptotics . dyachkov and rykov have quantified these asymptotics by showing that @xmath19 rows are necessary for a matrix to be @xmath3-disjunct @xcite similar results appear elsewhere @xcite @xcite ( * ? ? ? * theorem 7.2.14 ) . this means that it is not possible to create a @xmath3-separable matrix with @xmath8 rows . as disjunctness is a stronger ( and , in some ways , simpler ) property than separability , efforts to derive upper bounds on @xmath7 for separable matrices have often proceeded via the construction of disjunct matrices . in their seminal paper @xcite , kautz and singleton give a probabilistic existence theorem for @xmath3-disjunct matrices with @xmath53 rows . in the group testing literature there exist explicit constructions of testing schemes with @xmath54 rows , see for example porat and rothschild @xcite . since separable matrices can not meet the counting bound , it would be of interest if a matrix could be close to being separable using only @xmath21 rows . such a matrix would be order - optimal . with this in mind , we define the concept of an _ almost separable _ matrix in a similar manner to defintion [ defsep ] . [ defalmsep ] with the notation of definition [ defsep ] , @xmath10 is _ @xmath55-almost @xmath3-separable _ if for at most @xmath56 sets @xmath5 of size @xmath3 does there exist another set @xmath34 of size @xmath3 with @xmath57 . an analogous definition is present in for example @xcite , where almost separable matrices are called _ weakly separating designs_. note that setting @xmath51 gives the definition of a separable matrix . the main result of this paper is to show the existence of @xmath55-almost @xmath3-separable matrices with @xmath8 rows ( see theorem [ mainthm ] below ) . we also examine the implicit constants for the case when @xmath23 grows polynomially in @xmath15 . malyutov @xcite effectively showed that @xmath55-almost @xmath3-separable matrices exist with @xmath58 rows in the regime where @xmath3 is fixed as @xmath16 . this is a special case of a more general result malyutov proved using an information theoretic argument this and similar work is reviewed in @xcite . seb showed effectively the same result @xcite , again for fixed @xmath3 , by analysing a concrete bound on the probability that there are two different sets of size @xmath3 whose disjunctions coincide we follow a similar route here later . the same result for @xmath3 fixed and @xmath59 was rediscovered by zhigljavsky ( * ? ? ? * theorem 5.5 ) . although technically different from seb s argument , zhigljavsky s proof is morally similar : given two sets @xmath5 and @xmath34 of @xmath3 columns each , zhigljavsky counts how many rows it is possible to construct that would produce the same value for both @xmath36 and @xmath35 . he calls this number a r ' enyi coefficient and only considers designs with fixed- or bounded - size tests . our result improves on these by allowing @xmath3 to vary arbitrarily with @xmath15 , subject to @xmath18 . in our discussion of group testing in section [ gtsec ] we show how , in some regimes , this work also improves on recent results on nonadaptive group testing giving bounds of the form @xmath8 . the definition of a disjunct matrix ( defintion [ disj ] ) can similarly be weakened to give an _ almost disjunct matrix_. ( this definition also appears in @xcite and , previously , in @xcite . ) with the notation of definition [ defsep ] , @xmath1 is _ @xmath55-almost @xmath3-disjunct _ if for at most @xmath56 sets @xmath5 of size @xmath3 does there exist a column @xmath48 with @xmath49 . note again that @xmath51 corresponds to a disjunct matrix . unsurprisingly , almost disjunctness implies almost separability . [ implies ] let @xmath1 be an @xmath55-almost @xmath3-disjunct matrix . then @xmath1 is @xmath55-almost @xmath3-separable ( with the same @xmath55 and @xmath3 ) . we prove the contrapositive . suppose @xmath1 is not @xmath55-almost @xmath3-separable . then there are more than @xmath56 sets of size @xmath3 breaking separability . let @xmath5 be one of these sets , so there is another set @xmath34 of size @xmath3 with @xmath60 . letting @xmath61 , we have @xmath49 , breaking disjunctness . hence there are more than @xmath56 sets breaking disjunctness , and @xmath1 is not @xmath55-almost @xmath3-disjunct . mazumdar @xcite shows that there exist almost @xmath3-disjunct matrices with @xmath62 rows in the regime @xmath63 , @xmath64 , which is the same as that we consider for group testing . mazumdar s construction is similar to those of kautz and singleton @xcite and porat and rothschild @xcite . in particular , @xcite shows how to build fully disjunct matrices with @xmath65 rows by mapping the symbols of a @xmath66-ary reed - solomon code to unit - weight binary vectors of length @xmath66 , while @xcite improves on this scheme by replacing the rs code with a linear @xmath66-ary code achieving the gilbert - varshamov bound . this produces fully disjunct matrices with @xmath67 rows . this improves on the @xmath68 required for full disjunctness or separability , while being less good than the @xmath21 we achieve for almost separability here . our main result is then the following . [ mainthm ] for any sequence @xmath69 and @xmath70 , there exist an @xmath55-almost @xmath3-separable matrix with @xmath8 rows . more precisely , for @xmath71 $ ] , define @xmath72 } \max\left\{m_1(n , k,\alpha),m_2(n , k,\alpha)\right\}. \notag \end{aligned}\ ] ] then for any @xmath73 , for @xmath15 sufficiently large , and @xmath74 , there exists and @xmath75 @xmath55-almost @xmath3-separable matrix . consider the special case @xmath76 . it is possible to see that @xmath77 dominates , and hence that there exist almost separable matrices with @xmath78 rows . note that this is sufficient to show the @xmath8 result and comes with a slightly easier proof than the general case ( see below ) . this bound also meets the malyutov seb result of @xmath79 for @xmath3 constant . however , it is possible to get slightly better constants for most @xmath80 by allowing different values of @xmath81 . in particular , @xmath77 with @xmath82 gives the best result in many regimes . in section [ gtsec ] we discuss the constants in more detail in the regime @xmath83 for @xmath84 . ( the reader may wish to skip ahead to figure [ rategraph ] , to get a feeling for this result . ) our proof gives a randomised construction where the matrix is chosen to have entries sampled from iid bernoulli random variables ; we discuss this in the next section . we proceed to prove theorem [ mainthm ] as follows . fix @xmath15 and @xmath3 . we will choose @xmath1 to be an @xmath0 matrix ( where @xmath7 will be determined later ) with each entry independently @xmath13 with probability @xmath85 and @xmath86 with probability @xmath87 , for some @xmath85 also to be chosen later . we aim to show that there is a choice of @xmath7 and @xmath85 so that , with positive probability , @xmath1 is @xmath55-almost @xmath3-separable , and hence that such a matrix exists . the following bound will be important , and is fairly well known see for example seb @xcite , who analyses its asymptotics for fixed @xmath3 as @xmath16 . let @xmath1 be a randomly chosen matrix in @xmath88 with each entry independently @xmath89 with probability @xmath85 . for any set @xmath5 of size @xmath39 , then @xmath90 say that an _ overlap _ occurs if there exists @xmath34 with @xmath91 and @xmath57 . take two distinct sets @xmath92 , both of size @xmath3 , that have @xmath93 elements in common . then a row @xmath94 of @xmath1 could distinguish between @xmath5 and @xmath34 in two ways : either we have @xmath95 while @xmath96 , or the other way round : @xmath97 while @xmath98 . if the entries of the row @xmath99 are iid bernoulli@xmath100 , these two events each occur with probability @xmath101 . hence , row @xmath94 fails to distinguish between @xmath5 and @xmath34 with probability @xmath102 . since the rows of @xmath1 are iid , the whole matrix fails to distinguish between @xmath5 and @xmath34 with probability @xmath103 . the result then follows by a union bound over @xmath34 , noting that the number of sets of size @xmath3 sharing @xmath104 elements with @xmath5 is precisely @xmath105 . the main work in this paper is a careful asymptotic analysis of the overlap probability , showing for which @xmath7 it can be made arbitrarily small . [ hardlem ] for every sequence @xmath69 , @xmath106 , there exists @xmath107 so that if @xmath108 and @xmath109 , with @xmath40 as in theorem [ mainthm ] , then @xmath110 . we first prove that it suffices to have @xmath111 , with @xmath112 . this is simpler to prove than the full result and illustrates the main techniques . here , we take @xmath113 , as does seb @xcite , so that @xmath114 . this is a special case of the general value of @xmath85 used in the appendix , @xmath115 , by taking @xmath76 . note that , in group testing parlance , this is the value of @xmath85 that gives a @xmath116 chance of a test being positive . the bound then becomes @xmath117 it will be convenient to write @xmath118 for the number of nonoverlapping items , to get @xmath119 when @xmath120 , then the terms in the above sum are decreasing since @xmath121 for @xmath122 and @xmath123 . thus , the probability of an overlap can be estimated by the largest term with @xmath124 which , for fixed @xmath125 , can be made arbitrarily small for @xmath15 sufficiently large . further , since @xmath126 , we see that @xmath127 . we can get the more general result that it suffices to have @xmath109 , with @xmath40 as in , by instead taking @xmath128 , and then optimising over @xmath81 . the analysis is very similar to that above , but somewhat more longwinded . the interested reader is directed to the appendix for the details . proving our main result is now straightforward . choose the matrix @xmath1 at random as above , with @xmath7 and @xmath15 chosen as in lemma [ hardlem ] so that the overlap probability is at most @xmath129 . write @xmath130 for the number of sets @xmath5 of size @xmath3 that experience an overlap . it is clear @xmath1 will be @xmath55-almost @xmath3-separable provided that @xmath131 . then we have @xmath132 by the markov inequality . but this expectation is , by lemma [ hardlem ] @xmath133 hence , our random @xmath1 is @xmath55-almost @xmath3-separable with probability at least @xmath134 , so such matrices must exist . in this section , we show how the use of almost separable matrices can give new results on the rate of nonadaptive group testing . as we outlined in the introduction , in a nonadaptive group testing procedure we aim to find a subset @xmath5 of @xmath3 defective items within a population of @xmath15 identical items . we use @xmath7 pooled tests . recall that the outcome of a test @xmath94 is positive if one or more of the defective items is in the test pool , and negative if none of them are . we summarise our testing procedure by a matrix @xmath135 , where @xmath136 denotes that item @xmath12 is in the pool for test @xmath94 , and @xmath137 denotes that it is not . recalling the notation of definition 1 , the set of positive tests for a defective set @xmath5 is precisely @xmath36 . the aim is , given the outcomes @xmath36 and the matrix @xmath1 , to identify the defective set @xmath5 . clearly if there is no other @xmath34 with @xmath60 , then we can find @xmath5 ( at least theoretically : for study of practical algorithms for this , see , for example , @xcite ) . conversely , if there is an @xmath34 with @xmath60 , then our error probability is at least @xmath134 . a comprehensive survey of combinatorial group testing is given in @xcite . likewise , the study of nondeterministic testing schemes is addressed in the field of probabilistic group testing see for example @xcite and references therein . the derivation of both non - constructive results and practical algorithms has been addressed in different contexts , including combinatorial @xcite , probabilistic @xcite and information - theoretic @xcite scenarios . the connection between separable matrices and nonadaptive group testing is well explored . in particular , if there are known to be exactly @xmath3 defective items , then a testing matrix will allow us to find the defective set with certainty if and only if it is @xmath3-separable . the advantages of using what we call almost separability for group testing in the fixed-@xmath3 regime have also been discussed in @xcite . while separable matrices allow detection with zero probability of error , the study of group testing within the scope of information theory and the need for efficient algorithms generated an interest in nonadaptive group testing with low but not necessarily zero probability of error , a situation which has gained considerable attention @xcite . here the probability of error is defined as an average over all possible defective sets of size @xmath3 ; that is , @xmath138 baldassini , johnson and aldridge @xcite introduced a concept of the _ rate _ of group testing to quantify how well a group testing design works . ( an earlier definition of rate for the fixed @xmath3 regime had been introduced by malyutov @xcite . ) the rate is the ratio of the number of tests to the counting bound @xmath139 . if we interpret the counting bound as a binary labelling of all possible defective sets of size @xmath3 , the rate can be considered as the number of bits learned per test by the group testing procedure . [ defrate ] consider a group testing problem with @xmath15 items of which @xmath3 are defective . a design with @xmath7 tests is said to have _ rate _ @xmath140 . given a sequence of group testing problems for @xmath15 items of which @xmath80 are defective , a rate @xmath141 is said to be _ achievable _ for a design @xmath1 if , for any @xmath70 , the design finds the defective set with error probability at most @xmath55 with rate at least @xmath141 for @xmath15 sufficiently large . we follow baldassini et al . @xcite and study achievable rates in regimes where @xmath142 for different values of the sparsity parameter @xmath22 $ ] . note from the above that using a @xmath3-separable matrix with @xmath143 + @xmath144 tests gives rate @xmath25 for all values of @xmath145 . as far as we are aware , the best known rate for nonadaptive group testing until now is achieved by the ` dd ` algorithm of aldridge , baldassini and johnson @xcite , which has a lower bound on the maximum achievable rate of @xmath146 together with the malytuov seb result that @xmath147 can be achieved in the fixed-@xmath3 regime . baldassini , johnson and aldridge @xcite also showed that for adaptive group testing , the generalized binary splitting algorithm of hwang @xcite gives a rate of @xmath89 ( the best possible ) for all @xmath22 $ ] . from theorem [ mainthm ] , we know that using an @xmath55-almost @xmath3-separating matrix will find the defective set with error probability at most @xmath55 , since the sets @xmath5 without overlaps can by definition be recovered with certainty . hence , the number of rows of the almost separating matrix gives bounds on the rate . therefore , using our above results , we have the following : . ] [ ratethm ] for @xmath22 $ ] and @xmath83 , the maximum achievable rate of nonadaptive group testing with @xmath15 items of which @xmath3 are defective is bounded below by @xmath148 } \min \left\ { 2\alpha { \mathrm{e}}^{-\alpha } \frac{\beta}{2 - \beta } , -\ln\left(1 - 2{\mathrm{e}}^{-\alpha } + 2{\mathrm{e}}^{-2\alpha}\right ) \right\ } .\ ] ] figure [ rategraph ] illustrates the result of theorem [ ratethm ] . note that our result improves over the best known result for @xmath149 , and meets the malyutov seb point as @xmath150 . following directly from theorem [ mainthm ] and the definition of rate , we have @xmath151 } \min \big\ { -\ln\left(1 - 2{\mathrm{e}}^{-\alpha } + 2 { \mathrm{e}}^{-\alpha(1 + 1/k)}\right ) k \frac{\beta}{2 - \beta } , \\ -\ln\left(1 - 2{\mathrm{e}}^{-\alpha } + 2{\mathrm{e}}^{-2\alpha}\right ) \big\ } , \end{gathered}\ ] ] noting that , when @xmath152 , @xmath153 when @xmath154 , the second term is the minimum . when @xmath145 , since we have that @xmath155 , we can take limits in the first minimand . we have @xmath156 the result follows . , showing theorem [ ratethm ] for different values of @xmath81 and the approximation of corollary [ cor : beta-1].,scaledwidth=95.0% ] note that our ` simpler ' result with @xmath76 gives a bound almost as good the general case , namely @xmath157 in particular , this choice of @xmath76 is optimal at @xmath154 . note also that for all but the sparsest cases , we get the bound by taking @xmath82 . specifically , for @xmath158 , where @xmath159 the best value of the bound is @xmath160 for @xmath161 , the optimal rate is given as the maximum in , and the optimal @xmath81 is that which achieves the maximum . it s easy to see for @xmath162 that the maximum over @xmath81 is achieved when the two terms in the minimum are equal , and this is simple to solve numerically . however , here we also provide some closed form approximations to this which could be useful . [ cor : beta-1 ] for @xmath163 and @xmath83 , the maximum achievable rate of nonadaptive group testing with @xmath15 items , of which @xmath3 are defective , is bounded from below by @xmath164 this is illustrated in figure [ graph2 ] . from this , we see that the bound of corollary [ cor : beta-1 ] is very good for @xmath165 , but that when @xmath166 is not much above @xmath167 , then the bound of simply @xmath82 is better . hence , setting @xmath168 and taking @xmath167 as above , we get the following bound : [ cor : beta - cases ] for @xmath84 and @xmath83 , the maximum achievable rate of nonadaptive group testing with @xmath15 items , of which @xmath3 are defective , is bounded from below by @xmath169 the proofs of these statements can be found in appendix b. we have explored the asymptotics of almost separability and we have shown that almost separable matrices exist with @xmath21 rows . furthermore , we have proved that the use of almost separable matrice can improve the lower bounds on the rate of nonadaptive group testing in the very sparse regime . several interesting questions , however , remain still open , and provide scope for future research . most notably , while we have given new achievable rates , the maximum rate of nonadative group testing is still unknown . in particular , we know of no upper bounds beyond the trivial counting bound . as discussed in section 2 , chen and hwang @xcite have proved that disjunct and separable matrices share the same asymptotics by showing how to construct a @xmath3-disjunct matrix out of a @xmath52-separable matrices by adding at most one row to it . unlike its inverse ( disjunctness implying separability ) , this statement does nt naturally carry through to the case of almost separability / disjunctness . another problem is to extend the existing results to other regimes than the @xmath83 for @xmath22 $ ] considered here . of particular interest is the case where @xmath170 grows like a constant proportion of @xmath15 , as in recent work by wadayama @xcite . note that the counting bound now gives a lower bound of @xmath171 , while , for coupon - collector reasons , the iid random approach here inevitably leads to the suboptimal @xmath172 . we now show the full result of lemma [ hardlem ] . we use the same random construction as the special case described in section [ secproof ] , but now take @xmath173 , so @xmath174 , where @xmath81 is a parameter to be chosen later ( simply taking @xmath76 as in section [ secproof ] gives @xmath175 ) . within the group testing literature , different values of @xmath85 have also been considered . for example , the value @xmath176 ( which gives an average of one defective per test ) has been considered before by many authors @xcite , while sejdinovic and johnson @xcite consider the more general @xmath177 for noisy group testing . the same value can be obtained asymptotically in this context , as @xmath178 if @xmath155 as @xmath16 . we wish to find values of @xmath7 such that @xmath179 can be made arbitrarily small . it will be convenient to write @xmath180 allowing us to rewrite the bound as @xmath181 as before , it will be more convenient to deal with @xmath182 , which gives @xmath183 now , we expand out @xmath184 in using the binomial theorem and reverse the order of summation to get @xmath185 consider the inner sum of . it is possible to approximate it by its largest term , which will depend on the value of @xmath94 . to start with , the following bound holds : @xmath186 note that for any @xmath187 , the function @xmath188 attains its maximum at @xmath189 ; and further is increasing for @xmath190 and decreasing for @xmath191 . in , the maximum corresponds to @xmath192 . now , @xmath193 when @xmath194 or , since @xmath174 , @xmath195 then , in light of the above , we will split between the three cases : first , @xmath196 ; second , @xmath197 ; and third , @xmath198 . for the first case , @xmath196 , the maximum of is attained at @xmath199 , giving the bound @xmath200 summing over this range for @xmath94 yields @xmath201 provided that @xmath202 for some @xmath125 , then this can be made arbitrarily small for @xmath15 sufficiently large . for the second case , @xmath197 , the maximum is attained at @xmath192 , giving the bound @xmath203 then we have that @xmath204 where we called @xmath205 , and we have used that @xmath206 . then as long as @xmath207 we have by the azuma hoeffding inequality that @xmath208 given , this can be made arbitrarily small for @xmath15 sufficiently large . we can rewrite as @xmath209 now for the final case , when @xmath198 . note that for @xmath198 , @xmath210 hence @xmath211 . then , splitting up @xmath212 , @xmath213 , and @xmath214 , and noting that @xmath215 , we have @xmath216 thus , @xmath217 to make this small requires @xmath218 in order to compare the condition in to and , note that for any @xmath219 , @xmath220 the above inequality can be seen , for example , since for each @xmath221 , the function @xmath222 is concave for @xmath223 $ ] with @xmath224 . thus , since @xmath225 , then @xmath226 thus , condition is always stronger than and one can see that when @xmath3 tends to infinity , the two conditions are asymptotically equal . hence from , , and our requirements are @xmath227 from the above , we can optimise this result over @xmath81 . noting that @xmath228 is minimised at @xmath76 and @xmath77 is minimised at @xmath82 , it is sufficient to just consider @xmath71 $ ] . this proves lemma [ hardlem ] . here we give the proofs of corollaries [ cor : beta-1 ] and [ cor : beta - cases ] . in order to simplify some of the expressions that follow , define @xmath229 and @xmath230 . then , for @xmath71 $ ] we have @xmath231 $ ] and as @xmath166 tends to @xmath89 , @xmath232 tends to @xmath25 . further , the expressions in theorem [ ratethm ] can be simplified as @xmath233 and @xmath234 consider now the inequality from in the case @xmath250 . note that for all @xmath240 $ ] , @xmath251 the above inequality can be seen to be true since it holds for @xmath252 and @xmath253 and @xmath254 is concave . thus , in order to prove the inequality in , it suffices to show that for @xmath255 $ ] , @xmath256 again , the inequality in equation can be seen to be true since it holds for @xmath257 and @xmath253 and the function @xmath258 is concave for @xmath240 $ ] . in corollaries [ cor : beta-1 ] and [ cor : beta - cases ] , a better bound for the case @xmath264 can be obtained by substituting in theorem [ ratethm ] , @xmath81 chosen so that @xmath265 but the expression obtained does not seem simpler than statement of theorem [ ratethm ] itself . m malyutov . search for sparse active inputs : a review . in h aydinian , f cicalese , and c deppe ( eds ) , _ information theory , combinatorics and search theory _ lecture notes in computer science , * 7777 * , springer , 609647 , 2013 . e porat and a rothschild . explicit non - adaptive combinatorial group testing schemes . in l aceto , i damgard , la goldberg , mm halldorsson , a ingolfsdottir and i walukiewicz ( eds ) , _ icalp 2008 _ , lecture notes in computer science , * 5125 * , 748759 , 2008 . d sejdinovic and ot johnson . note on noisy group testing : asymptotic bounds and belief propagation reconstruction . _ proceedings of the 48th annual allerton conference on communication , control and computing _ , 9981003 , 2010 .

an @xmath0 matrix @xmath1 with column supports @xmath2 is @xmath3-separable if the disjunctions @xmath4 are all distinct over all sets @xmath5 of cardinality @xmath3 . while a simple counting bound shows that @xmath6 rows are required for a separable matrix to exist , in fact it is necessary for @xmath7 to be about a factor of @xmath3 more than this . in this paper , we consider a weaker definition of ` almost @xmath3-separability ' , which requires that the disjunctions are ` mostly distinct ' . we show using a random construction that these matrices exist with @xmath8 rows , which is optimal for @xmath9 . further , by calculating explicit constants , we show how almost separable matrices give new bounds on the rate of nonadaptive group testing .

in this supplementary material we give some details on how the series @xmath10 is obtained . a section is devoted to the 3d ferromagnetic heisenberg model on the bcc lattice where we show that even if the present method has not been adapted for a critical phase transition , it nevertheless give some evidence of a critical temperature within 10% of the correct result . ht - series are given in the last section . we consider @xmath27 spins on a given lattice in an external constant magnetic field @xmath29 . the hamiltonian reads : @xmath107 where @xmath32 is a spin hamiltonian depending on a few exchange parameters , @xmath108 and @xmath35 . the partition function is @xmath109 , with @xmath110 and the free energy , the energy and the entropy per spin are obtained from @xmath111 the ht - series of @xmath112 as a function of @xmath42 and @xmath12 is written as @xmath113 where @xmath114 are homogeneous polynomials of the exchange parameters . in the following , @xmath12 is a parameter and @xmath115 is evaluated numerically as a polynomial in @xmath42 , as well as all other functions . thus we have @xmath116 where @xmath72 is the order of ht - series and @xmath117 as we can define @xmath32 such as the mean energy at infinite temperature is 0 . the series @xmath118 is obtained by eliminating @xmath42 between @xmath119 and @xmath120 : @xmath121 we get for the first term @xmath122 , @xmath123 and @xmath124l_2 ^ 2 \!-\!l_6l_2 ^ 3.\end{aligned}\ ] ] an algorithm to get the general term reads : 1 . store the polynomial @xmath119 whose coefficients are @xmath125 for @xmath126 ( eq.[eq - def - ebeta - ht ] ) 2 . compute and store the polynomials @xmath127 up to order @xmath72 ( @xmath127 starts at order @xmath128 ) . 3 . for @xmath129 from 2 to @xmath72 compute @xmath130 by solving the triangular set of linear equations : @xmath131_i \right)\end{aligned}\ ] ] where @xmath132_i$ ] is the coefficient of order @xmath133 in @xmath127 . this part is an unpublished work done more than ten years ago with g. misguich . in this section , we show that the analysis of the function @xmath134 of a 3d ferromagnetic spin model provides some evidence of a critical behavior . we recall that the first derivative of @xmath134 is positive @xmath135 its second derivative is negative , and the specific heat is given by ( see ref.14 of main article ) @xmath136 suppose that @xmath134 is almost linear on an interval @xmath137 $ ] . then @xmath138 is almost constant , and @xmath139 is very small and may vanish . this results in a sharp peak in @xmath3 , and a divergence if @xmath140 vanishes . this is exactly what we find in the case of the three dimensional heisenberg model on the bcc lattice ( @xmath141 ) . in fig.[bcc - ferro]-left , the dashed lines represent the direct pade approximants of @xmath118 at order @xmath142 . we see that in the interval @xmath143 $ ] the function is rather flat . building the specific heat from those pades approximants and eq.[eq - def - cv - from - se ] , leads to a divergence of @xmath3 around @xmath144 , a temperature much larger than @xmath145 [ j. oitmaa and e. bornilla , phys . b p14228 ( 1996 ) ] . applying the method described in the main article with a low temperature behavior @xmath146 and with the knowledge of the exact ground state energy @xmath147 , we build the smooth pade functions @xmath148 ( full lines of fig.[bcc - ferro]-left ) . fig.[bcc - ferro]-right shows the corresponding specific heat , in log - scale , where we see sharp peaks at temperatures about 10% lower than @xmath106 . here we have a good example of the power of working with the function @xmath134 . the constrains imposed on @xmath134 are so strong that even if we miss some feature ( here a singularity of @xmath149 at some critical value @xmath150 ) , we get a clear signal of its presence . in that sense , a careful analysis of the function @xmath134 may be sufficient to predict new physics . improving the present method to account properly for the singularity at @xmath106 is an interesting prospect . in this section we give the coefficients of the polynomials for various models . the hamiltonians are of the form @xmath151 first , second and third neighbor ( across the hexagon ) model on the kagome lattice . the polynomials are @xmath158 for example , for @xmath159 , the coefficients are : @xmath160 , @xmath161 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , @xmath166 , @xmath167 , @xmath168 , @xmath169 .

we explain how and why all thermodynamic properties of spin systems can be computed in one and two dimensions in the whole range of temperatures overcoming the divergence towards zero temperature of the standard high temperature series expansions ( hte ) . the method relies on an approximation of the entropy versus energy ( microcanonical potential function ) on the whole range of energies . the success is related to the intrinsic physical constraints on the entropy function and a careful treatment of the boundary behaviors . this method is benchmarked against two one - dimensional solvable models : the ising model in longitudinal field and the xy model in a transverse field . with ten terms in the hte , we find a spin susceptibility within a few % of the exact results in the whole range of temperature . the method is then applied to two two - dimensional models : the supposed - to - be gapped heisenberg model and the @xmath0-@xmath1-@xmath2 model on the kagome lattice . recent years have seen an outburst of magnetic materials that might be realistic candidates for the long searched spin liquids @xcite . in this quickly maturing field , it is now highly desirable to compare the experimental properties of these new states of matter with theory . modelization of the magnetic interactions in a mott insulator is a challenge . first principle calculations of the magnetic interactions are delicate@xcite . in a pragmatic approach , the experimentally measured specific heat @xmath3 and/or uniform spin susceptibility @xmath4 can be compared against high temperature series expansions ( hte ) of spin models.@xcite this simple trail is not sufficient for frustrated magnets . in fact hte diverge at low temperature and increasing the length of the series and/or using pad approximants do not help much to reach useful information at temperatures lower than the main interaction . but , as was noticed at the early years of this quest @xcite , the interesting physics in frustrated systems appear in a range of temperatures at least an order of magnitude lower than the main coupling , and specially for competing interactions where the temperature range available with the raw hte is quite insufficient . many mathematical methods have been tried to reach low temperature properties of magnets . for example , biased differential approximants have been successful to account for the nel ground states of the heisenberg model on the square and the triangular lattice.@xcite for spin liquids presently under investigation other tools are needed . in refs.@xcite a different approach , based on the use of sum rules , was proposed to compute specific heat at zero magnetic field . unfortunately in real materials , phonon and magnetic contributions to the specific heat are often mixed and extracting the magnetic contribution is delicate . on the contrary , the experimental information on the magnetic susceptibility obtained by squid or nmr measurements is free of these uncertainties and it would be extremely valuable to have a way to use it . the extension of refs.@xcite to spin - susceptibility calculation did not seem a priori possible as its success was thought to be related to the existence of sum rules constraining the specific heat , sum rules which do not have equivalent for the spin - susceptibility . in fact deep physical reasons imply that refs.@xcite regularization and interpolation procedure is more powerful than expected . from a conceptual point of view the first key point is the move from an expansion of the free energy @xmath5 as a function of the temperature @xmath6 , to a expansion of the entropy @xmath7 as a function of the internal energy @xmath8 . elementary statistical mechanics tells us that these two descriptions ( canonical versus microcanonical ensembles ) are indeed equivalent and that all thermodynamic quantities can be computed at the thermodynamic limit in any of them . the major drawback of the standard use of truncated hte is the intrinsic divergence arising in the low temperature free energy : trying to extend its range of validity towards @xmath9 is thus extremely difficult . the choice to build a reasonable approximation of the entropy versus energy @xmath10 ( and by extension of @xmath11 , where @xmath12 is the external magnetic field ) is more efficient because : @xmath13 @xmath10 is defined on the finite interval from the ground state energy @xmath14 , to @xmath15 ( the average energy reached by the system for @xmath16 ) , and its boundary values are known : @xmath17 and @xmath18 . @xmath19 the series expansion of @xmath10 at @xmath15 can be exactly deduced from the knowledge of the free energy hte . @xmath20 in the absence of phase transition ( one- or two - dimensional behavior ) the function @xmath21 is an infinitely derivable function on @xmath22e_0,e_\infty]$ ] , monotonously increasing ( @xmath23 ) and concave ( second derivatives of @xmath11 negative because of stability conditions of the thermodynamical equilibrium ) @xmath24 the correct behavior of @xmath25 at @xmath14 can be determined from the qualitative knowledge ( or prediction ) of the first excitations ( see below ) . the interpolation of @xmath10 between @xmath14 and @xmath15 is thus very strongly constrained by physical considerations . all these thermodynamic conditions do exist in a canonical description of statistical mechanics and they should equally constrain physical expansions of the free energy versus temperature and their extrapolation . their implementation in the computation is never done ( it would be difficult , if not impossible to do ) whereas it is very simple in the present approach . in this paper , we show that this approach in the microcanonical ensemble allows the construction of @xmath11 and as a consequence of all thermodynamic properties at all temperatures in zero and moderate magnetic fields . we concentrate on the spin susceptibility @xmath26 and test the method on two cases where the exact function is known : the gapped one - dimensional ising model and the gapless one - dimensional xy - model . then , we address two open problems : the antiferromagnetic heisenberg model on the kagome lattice ( supposed to be gapped@xcite ) and the supposed - to - be gapless cuboc2 spin liquid phase in the @xmath0-@xmath1-@xmath2 model on the same lattice.@xcite we consider a system of @xmath27 spin-@xmath28 in a constant magnetic field @xmath29 in the @xmath30-direction . the hamiltonian reads : @xmath31 where @xmath32 is a spin hamiltonian , @xmath33 the total spin along @xmath29 , @xmath34 and @xmath35 . in the following , @xmath12 will be considered as a parameter . the free energy per spin @xmath36 reads : @xmath37 at fixed @xmath12 , the entropy per spin @xmath38 and the energy per spin @xmath39 are given by @xmath40 from the series expansion of @xmath41 in @xmath6 and @xmath12 , we first compute the ht - series of @xmath36 at fixed @xmath12 , and from now , each function is evaluated at this @xmath12 . the ht series @xmath39 and @xmath38 are deduced , and elimination of @xmath42 between them leads to the series expansion ( se ) @xmath43 of @xmath44 around @xmath15 ( see supplementary material ) . the next step consists to extrapolate @xmath43 down to @xmath45 . in the absence of a phase transition @xmath44 is indeed analytic on @xmath22 \eg , 0]$ ] , but singular at the boundary @xmath46 , as @xmath47 , when @xmath48 . the key point introduced in @xcite is then to build from @xmath44 a function @xmath49 defined on @xmath50 $ ] removing this singularity . this can be achieved by noting that two main kinds of leading singularities are met . if the system is gapless with a specific heat @xmath51 , then @xmath52 , and we choose @xmath53 if the system is gapped , with @xmath54 , then the singularity of @xmath55 , and we can choose @xmath56 where the @xmath57 denotes the derivation with respect to @xmath8 . @xmath49 is a smooth function on @xmath50 $ ] easier to extrapolate . this is done as follows : from the series expansion of @xmath43 at @xmath15 , we deduce the series expansion of @xmath58 and build the pad approximants ( pa ) @xmath59 . the inversion of eq.[eq : gegapless ] or [ eq : gegapped ] gives for each pa a function @xmath60@xcite by construction this method preserves both the exact information coming from the high temperature series and the supposed - to - be correct behavior at @xmath46 . _ at this stage any unphysical pa , i.e. not verifying @xmath61 , @xmath62 and @xmath63 is discarded . _ the method is considered as successful when most of the _ physical _ pas coincide for @xmath64 $ ] . this criterium is a way to select the more robust approximation , and extract the more plausible information from the restricted amount of data : it is a _ soft _ measurement of the self - consistency of this approach ( see ref.@xcite for example ) . heuristically , we noticed that spoiling the appropriate regularization at @xmath46 prevents the obtention of many coincident pas and gives erratic results when increasing the length of the input ht series . in order to evaluate the magnetic susceptibility @xmath26 , we need @xmath65 . using @xmath66= @xmath67= @xmath42 , we compute @xmath68 from @xmath69 and @xmath70 . @xmath4 is given by @xmath71 where the second derivative of @xmath36 with respect to @xmath12 is obtained by finite differences of the same pa at different @xmath12 . the results presented here have been obtained from series expansion of @xmath41 at order 4 in @xmath12 , this limits the range of applicability to small magnetic fields ( not a conceptual limit just a current technical one ) . and the various approximations . ht ( dotted line ) stands for the hte at order 12 , pa ( dashed line ) stands for the [ 6 - 6]-pad approximant , the other curves stand for the present method using hte at various order @xmath72 . , title="fig : " ] and the various approximations . ht ( dotted line ) stands for the hte at order 12 , pa ( dashed line ) stands for the [ 6 - 6]-pad approximant , the other curves stand for the present method using hte at various order @xmath72 . , title="fig : " ] + _ gapped systems : the longitudinal spin susceptibility of the 1d - ising model at @xmath73 . _ + the hamiltonian is @xmath74 . we use the regularizing function defined in eq.[eq : gegapped ] . with the exact value @xmath75 , and an hte at order 4 only , @xmath4 is already reproduced within 1% . increasing the order of the series decreases both the maximum error and the range of temperature where the errors are non negligible . as in most cases @xmath76 is unknown , we have also considered it as a free parameter to check the method . @xmath14 is then adjusted by maximizing the number of @xmath77 . this criterium is very accurate and gives @xmath78 $ ] . the error on @xmath4 for the 12-order ht - series is less than @xmath79 ( see fig.[fig - ising1d - xs ] ) , and we find a gap to be the exact value 1 , within an error of @xmath80 . for the transverse spin susceptibility of 1d - xy model , where @xmath76 is left free to adjust.,title="fig : " ] for the transverse spin susceptibility of 1d - xy model , where @xmath76 is left free to adjust.,title="fig : " ] _ gapless model : the transverse spin susceptibility of the 1d xy - model . _ + the hamiltonian reads @xmath81 . exact results have been obtained by katsura.@xcite the specific heat is linear in @xmath6 at low temperature , thus the singularity of @xmath44 around @xmath46 is regularized through eq.[eq : gegapless ] . using the exact value of @xmath46 leads to the exact value of @xmath82 and values of @xmath4 within an error of less than @xmath83 in the whole range of temperature for a 12-term ht - series . leaving @xmath46 as a free parameter , the errors do not exceed a few percents in the whole range of temperature ( see fig.[fig : xye0free ] ) . and specific heat @xmath84 of the antiferromagnetic heisenberg model on the kagome lattice . shown in the figures are the ht series expansion to order 17 ( green dotted lines ) , the best pad approximant of this simple series ( magenta dotted line ) , and the results of the present interpolation ( full lines ) . the sensitivity of the interpolation to the ground - state energy @xmath14 is displayed on both quantities . the full red curves are associated to the best commonly admitted value of @xmath14 ( see text ) . [ fig : kag ] , title="fig:",scaledwidth=35.0% ] and specific heat @xmath84 of the antiferromagnetic heisenberg model on the kagome lattice . shown in the figures are the ht series expansion to order 17 ( green dotted lines ) , the best pad approximant of this simple series ( magenta dotted line ) , and the results of the present interpolation ( full lines ) . the sensitivity of the interpolation to the ground - state energy @xmath14 is displayed on both quantities . the full red curves are associated to the best commonly admitted value of @xmath14 ( see text ) . [ fig : kag ] , title="fig:",scaledwidth=35.0% ] _ antiferromagnetic heisenberg model on the kagome lattice_. + the spin-1/2 antiferromagnetic heisenberg model on the kagome lattice , @xmath85 , is a quintessential example of the effects of both geometric frustration and quantum fluctuations pushed to their limit . after many studies , decisive progresses in 2d dmrg have ascertain the value of the ground - state energy of this system , @xmath86 , and estimate the spin gap of the order of 0.13(1).@xcite early hte of eltsner et young @xcite , extended in this work to order 17 give a first idea of the ht behavior of the thermodynamical quantities . the hte diverges around @xmath87 and the pad approximants of this series diverge below @xmath88 ( see fig.[fig : kag ] ) . with the hypothesis of a gapped system , @xmath89 is built using eq.([eq : gegapped ] ) . results displayed in fig.[fig : kag ] are obtained by fixing @xmath14 to its best known value ( @xmath90 ) ( red curve ) and to two extreme values which differ by 5 standard deviations from the the present best dmrg estimate . for a given value of @xmath14 , the differences between the various @xmath91 are less than the thickness of the lines . for this range of ground state energies , we find a gap 0.03(1 ) , significantly smaller than the gap obtained in the dmrg approach . compared to exact diagonalisations ( ed ) on 18 and 24-spin samples@xcite , we find , at the thermodynamic limit , a smaller value for the maximum of @xmath4 ( @xmath92 ) . in ed , the spin - spin correlations are indeed overemphasized by the very small lengths of the samples . the existence of a low temperature shoulder in the specific heat is confirmed . unfortunately , comparison with experimental date of herbertsmithite can not be done because of sizable dzyaloshinskii - moriya interactions that change the low energy spectrum of excitations and probably close the gap.@xcite _ spin susceptibility of kapellasite_. + -@xmath1-@xmath2 model for different values of the magnetic field 0 , 1 , 2 , 3 t with @xmath93 $ ] and the new fit ( see text ) . [ fig : kapellasite ] , scaledwidth=35.0% ] this material is in a mott phase and its properties are analyzed using a spin-1/2 hamiltonian on the kagome lattice:@xcite @xmath94 where @xmath2 is the third - neighbor exchange energy across the hexagon . the best set of parameters , obtained from a fit of the spin susceptibility down to @xmath95 , reads @xmath96 , @xmath97 , and @xmath98.@xcite the low - temperature specific heat is experimentally known to be @xmath99 , we then use eq.[eq : gegapless ] to regularize @xmath10 and adjust @xmath46 . the full curves of fig.[fig : kapellasite ] are obtained for the above - mentioned best set . as expected they indeed agree with experiments down to 17.5k . we see an increasing disagreement with experimental data when going to lower temperatures . part of the disagreement can be assigned to the magnetic field , which has a large effect in this system with competing interactions ( fig.[fig : kapellasite ] ) . up to now , the hte for this model are available to order 4 in @xmath12 only , which limits the evaluation of @xmath100 to @xmath101 , that is a magnetic field less than 3 tesla , while experimental data are at 5 tesla . nevertheless , with a small change of the coupling constants @xmath102k @xmath103k and @xmath104k , and 3 tesla magnetic field , we can fit almost all experimental data . a small disagreement persists at the lowest temperature , where the magnetic field effects are the most important . the uncertainties on the set of parameters is considerably reduced with this present method because we use experimental data at all temperatures and include the effect of the magnetic field . for kapellasite , experimental data at lower field and/or longer series in @xmath12 will help to achieve a better determination of the parameters . in this paper we have proposed a method to extend the hte of the spin susceptibility down to @xmath9 , based on a reconstruction of the entropy versus energy per spin . we have checked the method against gapless and gapped exact models : the largest deviations from the exact results are of the order of @xmath105 or better with an original ht series expansion of ten terms . we have applied this method to open problems on the kagome lattice . being not limited by finite size effects , we believe in the accuracy of the present method compared to that of exact diagonalisations , specially at low temperature . we have equally shown that this approach can be used to compute the spin susceptibility in a finite magnetic field , which is a major point for the comparison between models and experimental squid data.@xcite the method is general in its principle and can be applied straightforwardly to other models , whatever the size of the spin , as much as the high temperature expansion of the free energy per spin is available.@xcite this opens a large range of interesting studies such as herbertsmithite confronted to heisenberg model with dzaloshinskyi - moryia interactions , volborthite with spatially anisotropic couplings , spin ices ... @xcite a further conceptual question has not been studied in the present work : is this approach able to deal with critical phase transitions ? this might be possible for temperatures larger than @xmath106 in as much as the correct diverging behavior at @xmath106 is taking into account , but this is probably more delicate than the present work as these divergences are singularities in the derivatives of @xmath10 ( see supplementary material ) . building of a suitable regularization function and benchmarking the method is a new subject in itself , beyond the scope of this letter . we thank g. misguich for fruitful discussions . we acknowledge the support of the french ministry of research through the anr grant `` spinliq '' . 23ifxundefined [ 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.91.107001 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.077204 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.107204 [ * * , ( ) ] http://dx.doi.org/10.1038/nphys942 [ * * , ( ) ] http://dx.doi.org/10.1038/ncomms1274 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.109.037208 [ * * , ( ) ] http://dx.doi.org/10.1038/nature11659 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.157202 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.106403 [ * * , ( ) ] link:\doibase 10.1103/physrevb.88.075106 [ * * , ( ) ] link:\doibase 10.1103/physrevb.89.014415 [ * * , ( ) ] in @noop _ _ ( ) link:\doibase 10.1103/physrevb.58.11115 [ * * , ( ) ] link:\doibase 10.1103/physrevb.63.134409 [ * * , ( ) ] http://link.aps.org/abstract/prb/v71/e014417 [ * * , ( ) ] link:\doibase 10.1126/science.1201080 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.109.067201 [ * * , ( ) ] link:\doibase 10.1103/physrevb.87.155107 [ * * , ( ) ] @xmath59 being a rational function of @xmath8 , @xmath69 is evaluated as a function of @xmath8 also . for gapped systems , @xmath59 is first decomposed in simple fractions and then integrated on @xmath8 . link:\doibase 10.1103/physrev.127.1508 [ * * , ( ) ] link:\doibase 10.1103/physrevb.50.6871 [ * * , ( ) ] @noop * * , ( ) http://link.aps.org/doi/10.1103/physrevlett.101.026405 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.140405 [ * * , ( ) ] e. kermarrec , a. zorko , f. bert , r.h . colman , b. koteswararao , f. bouquet , p. bonville , a. hillier , a. amato , j. van tol , a. ozarowski , a.s . wills , and p. mendels , submitted to phys . rev . b. additional data can be found on the website : http://www.lptmc.jussieu.fr/lptmcdata/spin/htseries . ( see also supplementary material ) .

in the last two decades , our knowledge on neutrinos has been greatly improved by a number of elegant neutrino oscillation experiments @xcite . now , we are convinced that neutrinos are massive , and they can transform from one flavor to another when propagating in vacuum or in matter . the lepton flavor mixing phenomenon can be described by a @xmath8 unitary matrix @xmath9 , namely the leptonic mixing matrix , which is conventionally parametrized through three mixing angles @xmath10 , @xmath0 and @xmath11 , as well as three cp - violating phases @xmath1 , @xmath12 and @xmath13 , viz . , @xmath14 where @xmath15 and @xmath16 for @xmath17 . note that @xmath18 is a diagonal matrix with @xmath12 and @xmath13 being two majorana - type cp - violating phases if neutrinos are majorana particles , while @xmath19 if neutrinos are dirac particles . current experimental data indicate that the three leptonic mixing angles are @xmath20 , @xmath21 and @xmath22 . two independent neutrino mass - squared differences are found to be @xmath23 and @xmath24 . the latest global - fit results of neutrino parameters are shown in table i. however , we are still unclear whether the neutrino mass ordering is normal ( i.e. , @xmath25 ) or inverted ( i.e. , @xmath26 ) , and the leptonic dirac cp - violating phase @xmath1 remains experimentally unconstrained . the recent results from daya bay @xcite and reno @xcite reactor neutrino experiments have established that @xmath27 , which is rather large . hence , it is quite promising to determine the leptonic dirac cp - violating phase @xmath1 by comparing the oscillation probabilities of neutrinos and antineutrinos in future long - baseline neutrino oscillation experiments @xcite . in addition , the @xmath28-scale neutrino telescopes ( e.g. , icecube and km3net ) could provide us with useful and complementary information about leptonic cp violation by precisely measuring the flavor composition of ultrahigh - energy astrophysical neutrinos @xcite . if the deep core of the icecube detector is made denser to lower the energy threshold down to a few @xmath29 , such as the proposal pingu @xcite , a large amount of atmospheric neutrino events can be collected and used to determine the neutrino mass hierarchy and perhaps the leptonic cp - violating phase @xcite . on the other hand , a lot of neutrino mass models based on discrete flavor symmetries or phenomenological assumptions have recently been proposed to describe the observed leptonic mixing pattern , in particular a relatively large @xmath0 . interestingly , the leptonic cp - violating phase @xmath1 has been predicted in some models to be rather large ( e.g. , @xmath30 ) or even maximal ( i.e. , @xmath2 ) @xcite . in other models , leptonic cp violation is shown to be absent , namely @xmath31 or @xmath3 @xcite . it is worthwhile to mention that the latest global - fit analyses of neutrino oscillation experiments yield @xmath32 @xcite and @xmath7 @xcite , although the @xmath33 errors are still quite large . for the normal mass hierarchy and @xmath34 for the inverted mass hierarchy by another global - fit group @xcite however , there is no constraint on @xmath1 within the @xmath33 range . ] therefore , we have already obtained some preliminary information on the leptonic cp - violating phase @xmath1 from the global - fit analyses . .the best - fit values and 1@xmath13 ranges of the neutrino parameters from the latest global - fit analyses of neutrino oscillation experiments , where the normal neutrino mass hierarchy is assumed . [ cols="^,^,^,^",options="header " , ] cccc @xmath35 & @xmath36 & @xmath37 & @xmath38 + @xmath39 & @xmath40 & @xmath41 & @xmath42 + & @xmath43 & @xmath44 & @xmath45 + & @xmath46 & & + & @xmath47 & @xmath48 & @xmath49 + & @xmath50 & & + + @xmath51 & @xmath36 & @xmath37 & @xmath38 + @xmath39 & @xmath52 & @xmath52 & @xmath52 + @xmath53 & @xmath54 & @xmath55 & @xmath56 + @xmath57 & @xmath58 & @xmath59 & @xmath60 + since the rge s of neutrino mass matrix @xmath61 in the sm and uedm , or @xmath62 in the mssm , are given by the same formula in eq . ( 3 ) , the evolution of neutrino mass eigenvalues and leptonic mixing parameters can be figured out in the same way . in flavor basis , where the yukawa coupling matrix of the charged leptons is diagonal , namely @xmath63 , @xmath64 can be diagonalized by the leptonic mixing matrix @xmath9 , namely @xmath65 . generally speaking , an arbitrary @xmath8 unitary matrix @xmath66 can be factorized as @xmath67 , where @xmath68 and @xmath18 are pure phase matrices , while the unitary matrix @xmath69 consists of three mixing angles @xmath10 , @xmath0 , @xmath11 and the dirac cp - violating phase @xmath1 [ cf . although the phases @xmath70 ( for @xmath71 ) are unphysical and can be removed by rephasing the charged - lepton fields , we will keep them in the derivation of the rge s for neutrino masses and leptonic mixing parameters . since @xmath72 , we take into account the dominant contribution from the tau - lepton yukawa coupling to the rge of @xmath64 . following ref . @xcite , one obtains @xmath73 where @xmath74 and @xmath75 should bear the corresponding superscripts when eq . ( 4 ) is applied to a specific model . given @xmath76 ( for @xmath77 ) , we observe that eq . ( 4 ) determines the evolution of absolute neutrino masses . moreover , it is straightforward to find that @xmath78 , @xmath12 , @xmath13 , and @xmath70 ( for @xmath79 and @xmath77 ) have to fulfill the following equations : @xmath80 + \sum_\alpha 0 \ ; , \nonumber \\ & & { \rm im}\left[(u^\dagger \dot{u})^{}_{22}\right ] + \sum_\alpha \nonumber \\ & & { \rm im}\left[(u^\dagger \dot{u})^{}_{33}\right ] + \sum_\alpha @xmath1 could be significant . to next - to - leading order , eq . ( 7 ) approximates to @xmath96\right\ } \ ; , % ( 8)\ ] ] where we have taken @xmath97 as the absolute neutrino mass and ignored the difference between @xmath98 and @xmath99 . some comments are in order : * in general , the evolution of @xmath1 is dominated by the leading - order term @xmath100 on the right - hand side of eq . ( 7 ) . at higher order , if the terms suppressed by @xmath101 are taken into account , then those by @xmath102 should also be kept for consistency , since they are of the same order of magnitude , as we have done in eq . ( 8) . the relative error in eq . ( 7 ) is at the level of latexmath:[$s^{}_{13 } best - fit values of @xmath0 and neutrino mass - squared differences . * it is evident from eq . ( 7 ) that the evolution of @xmath1 is entangled with that of three mixing angles and two majorana cp - violating phases . in particular , it depends crucially on the majorana phases @xmath12 and @xmath13 . it has been found that the dirac cp - violating phase @xmath1 can be radiatively generated from @xmath12 and @xmath13 , even if the initial value of @xmath1 is vanishing @xcite . on the other hand , the rg evolution of @xmath1 becomes negligible when @xmath104 , while the mixing angle @xmath10 is quite sensitive to the rg effect in this case . * the rge s of @xmath1 in the sm , the mssm , and the uedm are given by the same formula in eq . ( 7 ) , but with different values of the coefficient @xmath75 . we have @xmath105 in the sm , while @xmath106 in the mssm and @xmath107 in the uedm , respectively . therefore , given the same majorana cp - violating phases and leptonic mixing angles , the evolution of @xmath1 in the mssm will be in the direction opposite to that in the sm and the uedm . finally , we observe from eq . ( 5 ) that the identity @xmath108 holds in the standard parametrization of @xmath69 . the proof is as follows . given a general non - singular matrix @xmath109 , whose elements are functions of the running parameter @xmath84 , one can prove that @xmath110/{\rm d}t = { \rm det}(x ) \cdot { \rm tr}[x^{-1 } ( { \rm d}x/{\rm d}t)]$ ] . if we take @xmath109 to be a unitary matrix @xmath69 with @xmath111 and @xmath112 , then @xmath113 can be obtained . this observation together with eq . ( 5 ) leads to the identity @xmath114 . however , this identity depends on the specific parametrization of @xmath69 . for instance , if @xmath115 with @xmath116 being the dirac cp - violating phase , then we have @xmath117 , as shown in ref . @xcite . we proceed in this subsection with the numerical solution to the rge of the leptonic dirac cp - violating phase @xmath1 . since the evolution of @xmath1 in the sm is negligible even in the case of a nearly - degenerate neutrino mass spectrum , we consider only the mssm and the uedm . note that no approximations to the rge of @xmath1 will be made in our numerical calculations . our numerical results are shown in fig . 1 , and the main points are summarized as follows . in the mssm , we have taken two typical values of @xmath118 and @xmath119 for illustration . in both cases , the absolute neutrino mass @xmath120 is assumed , which is consistent with the cosmological bound @xmath121 ( @xmath122 c.l . ) from the wmap collaboration @xcite . for the initial values of @xmath1 at the electroweak scale , we have chosen @xmath2 , @xmath3 , and @xmath4 as typical examples . since the tau - lepton yukawa coupling is given by @xmath123 in the mssm , the evolution of @xmath1 should be significantly enhanced for a large value of @xmath5 , as shown in the upper plots of fig . 1 . for @xmath119 , the rg running of @xmath1 is quite significant . in particular , even if @xmath124 is found at the low - energy scale , namely , there is no cp - violating effect in neutrino oscillation experiments , the maximal cp - violating phase @xmath125 or @xmath4 can be achieved at the cutoff scale @xmath126 . in other words , one can change from the scenario with a zero cp - violating phase to that with a maximal cp - violating phase , or vice versa . for @xmath118 , the radiative correction to @xmath1 is at most @xmath127 even at @xmath126 . in the uedm , we have input two different values of the absolute neutrino mass @xmath120 and @xmath128 . as shown in the lower plots of fig . 1 , @xmath1 is rather stable against radiative corrections for @xmath120 . even for @xmath128 , which is marginally in tension with the cosmological bound , the relative change of @xmath1 at the cutoff scale @xmath129 is not larger than @xmath127 . the cutoff scale @xmath129 in the uedm has been chosen to avoid the landau pole , where the higgs mass is @xmath130 and @xmath131 with @xmath132 being the radius of the compactified extra dimension . since the valid energy range in the uedm is much smaller than that in the mssm , the rg running does not develop as much . however , it should be noted that the rg running in uedm is actually in the form of a power law , and thus can be more significant than in the sm and in the mssm . for majorana neutrinos in the mssm ( upper plots ) and in the uedm ( lower plots ) . the initial values @xmath133 , @xmath134 , and @xmath135 are assumed , while the majorana cp - violating phases @xmath12 and @xmath13 are marginalized . the values of @xmath10 , @xmath0 , @xmath11 and @xmath136 , @xmath98 in the @xmath33 ranges from the global - fit analysis ( for @xmath25 ) have been used as input @xcite.,scaledwidth=90.0% ] it should also be noted that the majorana cp - violating phases @xmath12 and @xmath13 have been marginalized over the range @xmath137 in our numerical results . if the specific values of @xmath12 and @xmath13 are chosen , the variation of @xmath1 will be even smaller . therefore , we conclude that the leptonic dirac cp - violating phase @xmath1 is stable against radiative corrections in all the models under consideration , except for the mssm with a large value of @xmath138 . in comparison , the dirac cp - violating phase in the quark sector is stable even in the mssm with a large value of @xmath138 , since the quark mass spectrum is strongly hierarchical . ( upper plots ) and the jarlskog invariant @xmath139 ( lower plots ) for majorana neutrinos at @xmath33 c.l . with @xmath140 ( dark red or dark gray ) and @xmath141 ( light red or gray ) in the mssm . the result of @xmath139 in the mssm with @xmath142 is also given in the lower plots ( yellow or light gray ) . the global - fit data from ref . @xcite are adopted for the left column , while that from ref . @xcite for the right column.,scaledwidth=90.0% ] now , we turn to the rg running behavior of @xmath1 by taking the global - fit results @xmath6 @xcite and @xmath143 @xcite as input . since the present uncertainty is large , we will choose the @xmath33 range for illustration . in the upper plots of fig . 2 , the allowed regions of @xmath1 at the superhigh - energy scale have been given in the mssm . in the case of @xmath119 , one can observe that @xmath1 is almost arbitrary within @xmath144 due to the large uncertainty of the input , so any predictions for @xmath1 from a high - energy flavor model could be made consistent with the low - energy observations by the rg running . this is true for the global - fit results from both groups @xcite . in reality , any observable effects of cp violation should be related to the jarlskog invariant @xmath145 . therefore , we also show the rg running of @xmath139 in the mssm for @xmath146 , @xmath147 , @xmath148 , in the lower plots of fig . it can be observed that @xmath139 at a superhigh - energy scale could be quite different from that at the low - energy scale , in particular for @xmath149 and @xmath142 . the possibility for neutrinos to be dirac particles has never been experimentally excluded . moreover , it has been shown that the leptogenesis mechanism responsible for the matter - antimatter asymmetry in our universe also works well in a different way for dirac neutrinos @xcite . hence , we assume neutrinos to be dirac particles , and give them masses through the coupling to the higgs doublet @xmath150 with @xmath151 being the neutrino yukawa coupling matrix . it is convenient to write the rge s of dirac neutrino parameters as @xcite @xmath152 \ ; , % ( 9)\ ] ] where @xmath153 has been defined . the rge s of @xmath64 in the sm and the mssm take the same form in eq . ( 8) , but with different coefficients @xmath154 and @xmath155 , as given in appendix b. since the beta function for dirac neutrino yukawa couplings is currently not available in the uedm , we consider only the sm and the mssm . similarly , as in the majorana neutrino case , we find the rge for the leptonic dirac cp - violating phase @xmath1 in the case of dirac neutrinos @xmath156 \ ; , % ( 10)\ ] ] where @xmath157 has been defined . the relative error in the above equation is at the level of @xmath158 . it is worth mentioning that the last term in eq . ( 10 ) is comparable in magnitude to the second term , since the suppression by a factor of @xmath159 is compensated by the enhancement from @xmath160 . some general comments are in order : * the evolution of @xmath1 is proportional to @xmath161 at all orders , so @xmath1 will be kept unchanged by the rg running if @xmath162 , namely , @xmath163 or @xmath124 . in other words , if leptonic cp violation is absent at low energies , it will never be generated by rg running . this is quite different from the majorana case , where @xmath1 can be radiatively generated via the non - vanishing majorana cp - violating phases even if @xmath31 or @xmath124 has been used as an initial condition . * two qualitative differences between the sm and the mssm should be noted . first , the tau - yukawa coupling @xmath164 in the mssm is significantly enhanced for a large value of @xmath5 . hence , the rg effect is more remarkable than that in the sm . second , the coefficient @xmath155 takes opposite signs in the sm and in the mssm , indicating the evolution of @xmath1 in opposite directions in these two models . for dirac neutrinos in the mssm for @xmath118 ( left plot ) and @xmath119 ( right plot ) . the initial values @xmath133 and @xmath135 are assumed , and the values of @xmath10 , @xmath0 , @xmath11 and @xmath136 , @xmath98 in the @xmath33 ranges from the global - fit analysis ( for @xmath25 ) have been used as input @xcite.,scaledwidth=95.0% ] ( upper and middle plots ) and the jarlskog invariant @xmath165 ( lower plots ) for dirac neutrinos at @xmath33 c.l . with @xmath140 ( dark red or dark gray ) , @xmath141 ( light red or gray ) and @xmath142 ( yellow or light gray ) in the mssm . the absolute neutrino mass @xmath120 has been assumed . the global - fit data from ref . @xcite are adopted for the left column , while that from ref . @xcite for the right column.,scaledwidth=90.0% ] to illustrate the rg running behavior of @xmath1 in the dirac neutrino case , we have shown in fig . 3 two typical examples in the mssm . in both examples , the initial values of @xmath1 have been taken to be @xmath166 and @xmath4 , and the absolute neutrino mass is @xmath120 . the left plot is for @xmath118 , while the right for @xmath119 . note that the beta function of @xmath1 is proportional to @xmath167 in eq . ( 9 ) , where @xmath168 in the mssm . therefore , @xmath1 increases for @xmath169 , while it decreases for @xmath2 , as the energy scale evolves towards higher energies . this feature can be clearly observed in fig . 3 . furthermore , the variation of @xmath1 at any energy scale is quite small , compared to that in the case of majorana neutrinos , where the arbitrary majorana cp - violating phases play an important role in the evolution of @xmath1 . as we have already mentioned , @xmath1 will be kept unchanged if the initial values lead to @xmath162 , so the trivial cases of @xmath170 and @xmath124 have not been considered . now , we continue with the global - fit results of @xmath1 in refs . @xcite as initial values . the rg running of @xmath1 in the mssm for @xmath171 and @xmath142 have been shown in the upper and middle plots of fig . 4 , respectively . as before , the absolute neutrino mass @xmath172 is assumed . in the former case , the rg running effects are insignificant , which is in accordance with the results in fig . 3 . in the latter case , however , it is interesting to note that a wide range of values @xmath173 $ ] can not be reached at the superhigh - energy scale @xmath174 , no matter what initial value of @xmath1 is chosen . the reason for this behavior is that the mixing angle @xmath0 is approaching zero around @xmath175 . in the limit of an extremely small value of @xmath0 , eq . ( 8) can be written as @xmath176 where @xmath168 has been chosen for the mssm . therefore , the rg running of @xmath1 will be rapidly accelerated around @xmath175 to the large - value region for @xmath177 ( i.e. , @xmath178 ) , while to the small - value region for @xmath179 ( i.e. , @xmath180 ) . this observation applies also to any initial value of @xmath1 . in fact , we have numerically checked the whole parameter region of @xmath181 at low energies , and found that only @xmath182 $ ] and @xmath183 can be reached at high energies . however , the exact allowed range of @xmath1 at high - energy scales really depends on the initial values of @xmath1 and three mixing angles . for @xmath124 , the rg running of @xmath1 will be absent , but @xmath0 becomes negative above @xmath184 , so we have to redefine @xmath185 to make @xmath0 positive , leading to @xmath31 or @xmath186 at high - energy scales . in the lower plots of fig . 4 , the evolution of the jarlskog invariant @xmath139 is shown . unlike the dirac cp - violating phase @xmath1 itself , the physical observable @xmath165 evolves smoothly over the whole range of energy scales , as it should . for @xmath142 , the value of @xmath187 can initially be as large as @xmath188 , it becomes vanishingly small at @xmath174 . one reason for this is that @xmath1 shrinks into a small region around @xmath52 or @xmath186 at the high - energy scale , as indicated in the middle plots of fig . 4 . obviously , the evolution of the three mixing angles is also relevant here . in secs . ii and iii , we have examined the rg running behaviors of the leptonic dirac cp - violating phase @xmath1 in the cases of majorana neutrinos and dirac neutrinos , respectively . now , we compare these two cases and summarize the main differences : * in the majorana case , the two majorana cp - violating phases are playing a crucial role in the rg running of @xmath1 . one can start from a cp - conserving scenario with @xmath31 or @xmath3 at the low - energy scale , and end up with a cp - violating scenario even with @xmath2 or @xmath4 . in the dirac case , the evolution of @xmath1 is proportional to @xmath161 , so the cp conservation at the low - energy scale definitely implies that cp violation is absent at a superhigh - energy scale . * the mixing angle @xmath0 could approach zero at some high - energy scale @xmath189 in both cases if a large value of @xmath5 is assumed in the mssm . on the other hand , there exist in the rge s of @xmath1 some terms inversely proportional to @xmath91 . therefore , the rg running behavior of @xmath1 will be dramatically changed around @xmath189 . given the global - fit values of @xmath1 within the @xmath33 range , it turns out that @xmath1 could be arbitrary at the high - energy scale in the majorana case due to the marginalization over @xmath12 and @xmath13 . in the dirac case , @xmath1 is found to be in two narrow ranges @xmath190 $ ] or @xmath183 in the mssm with @xmath142 . however , if a concrete mass model for majorana neutrinos or dirac neutrinos is assumed , the rg running of @xmath1 may depend on the model details . in particular , when new particles or interactions come into play at some intermediate energy scale , the rge s of the neutrino parameters are completely changed @xcite . hence , we have assumed that this is not the case in the previous discussions , at least below the cutoff scale . as we have mentioned before , many flavor symmetry models , which are intended for describing the observed leptonic mixing angles , predict the leptonic dirac cp - violating phase @xmath1 . for instance , it has been shown in ref . @xcite that @xmath191 ( or @xmath192 ) and @xmath193 ( or @xmath194 ) for different breaking patterns of the @xmath195 flavor symmetry in the type - i seesaw model , where three heavy right - handed neutrino singlets are introduced to realize the dimension - five weinberg operator . if the vacuum alignment problem is further solved in the framework of supersymmetry , significant radiative corrections to these theoretical predictions of @xmath1 could be possible . thus , the leptonic dirac cp - violating phase to be measured in neutrino oscillation experiments is related by the rg running to the theoretical prediction at the seesaw scale . on the other hand , the cp - violating and out - of - equilibrium decays of the heavy right - handed neutrinos can generate the lepton number asymmetry in the early universe , which will be converted into the baryon number asymmetry via the sm sphaleron processes . in this case , the leptonic cp violation in neutrino oscillations can be associated with the matter - antimatter asymmetry in our universe . thanks to the recent measurements of @xmath0 in the daya bay and reno experiments , the discovery of cp violation in neutrino oscillation experiments seems to be promising if the leptonic cp violation really exists and the leptonic dirac cp - violating phase @xmath1 happens to be far away from @xmath52 or @xmath3 . on the other hand , we have already had a preliminary result for the leptonic cp - violating phase @xmath1 from the global - fit analysis of all kinds of neutrino oscillation experiments , namely @xmath32 @xcite and @xmath7 @xcite . therefore , we are well motivated to study the rg running of @xmath1 from the low - energy scale to a superhigh - energy scale , where a unified model for fermion masses , flavor mixing , and cp violation is expected . in the case of majorana neutrinos , we have introduced the dimension - five weinberg operator to account for neutrino masses . the rge of @xmath1 has been derived analytically in great detail for the sm , the mssm , and the uedm , and a self - consistent approximation to it has been given as well . by a self - consistent approximation , we mean that the rge of @xmath1 has been expanded in terms of @xmath92 and @xmath159 , and all the terms of the same order of magnitude should be preserved . it turns out that @xmath1 is rather stable against radiative corrections in all these models , except for the case of a large @xmath5 in the mssm ( e.g. , @xmath119 together with a nearly degenerate neutrino mass spectrum ) . in this case , the majorana cp - violating phases play an important role in the evolution of @xmath1 such that a maximal phase @xmath2 or @xmath4 can be radiatively generated at a superhigh - energy scale even if @xmath124 ( i.e. , no cp - violating effects in neutrino oscillation experiments ) at the low - energy scale . the evolution of @xmath1 and the jarlskog invariant @xmath139 have been illustrated by taking the @xmath33 global - fit results of @xmath1 as input . in the case of dirac neutrinos , we have derived the rge of @xmath1 in the sm and mssm , and the self - consistent approximation to it has been made . note that a nearly degenerate neutrino mass spectrum and the absolute neutrino mass @xmath120 are assumed in our analysis . the rg running effect of @xmath1 can be neglected in the sm and in the mssm with a small @xmath5 ( e.g. , @xmath196 ) . however , @xmath1 can be modified by more than @xmath197 for @xmath119 . the evolution of @xmath1 and the jarlskog invariant @xmath139 have been examined by inputting the @xmath33 global - fit results of @xmath1 . in the case of @xmath142 , @xmath1 in the range of @xmath198 $ ] is found to be unreachable at @xmath174 , since the mixing angle @xmath0 approaches zero at some intermediate scale ( e.g. , @xmath175 ) , which forces @xmath1 to be in a large - value region for @xmath178 or a small - value region for @xmath180 . at the same time , the jarlskog invariant @xmath139 becomes vanishingly small at a superhigh - energy scale . as we already know some information and will soon learn more about the leptonic dirac cp - violating phase @xmath1 , it is thus meaningful to see how large it will be at a superhigh - energy scale . at such an energy scale , the leptonic dirac cp - violating phase might be related to the quark dirac cp - violating phase in a unified flavor model , or to the generation of matter - antimatter asymmetry in our universe via the leptogenesis mechanism . in any case , the precise determination of @xmath1 in the ongoing and upcoming neutrino oscillation experiments or at a future neutrino factory will shed light on the flavor dynamics at a high - energy scale . h.z . would like to thank for financial support the gran gustafsson foundation , and for the hospitality the kth royal institute of technology , where part of this work was performed . this work was supported by the swedish research council ( vetenskapsrdet ) , contract no . 621 - 2011 - 3985 ( t.o . ) , the max planck society through the strategic innovation fund in the project manitop ( h.z . ) , and the gran gustafsson foundation ( s.z . ) . in the sm extended with the dimension - five weinberg operator , the rge for @xmath64 has already been given in eq . ( 3 ) , while those for the yukawa coupling matrices @xmath199 of charged fermions ( i.e. , @xmath200 for charged leptons , @xmath201 for up - type quarks and @xmath202 for down - type quarks ) can be written as @xmath203 y^{}_l \ ; , \nonumber \\ 16\pi^2 \frac{{\rm d}y^{}_{\rm u}}{{\rm d}t } & = & \left [ \alpha^{\rm sm}_{\rm u } + c^{\rm sm}_{\rm u , u } \left(y^{}_{\rm u } y^\dagger_{\rm u } \right ) + c^{\rm sm}_{\rm u , d } \left(y^{}_{\rm d } y^\dagger_{\rm d}\right ) \right ] y^{}_{\rm u } \ ; , \nonumber \\ 16\pi^2 \frac{{\rm d}y^{}_{\rm d}}{{\rm d}t } & = & \left [ \alpha^{\rm sm}_{\rm d } + c^{\rm sm}_{\rm d , u } \left(y^{}_{\rm u } y^\dagger_{\rm u } \right ) + c^{\rm sm}_{\rm d , d } \left(y^{}_{\rm d } y^\dagger_{\rm d}\right ) \right ] y^{}_{\rm d } \ ; . % ( a1)\end{aligned}\ ] ] the relevant coefficients in eqs . ( 3 ) and ( a1 ) are @xmath204 , @xmath205 , and @xmath206 with @xmath207 $ ] . the rge s for the @xmath208 gauge couplings @xmath209 , @xmath210 , and @xmath211 are given by @xmath212 with @xmath213 . the quartic coupling @xmath214 of the higgs field appears in the rge of @xmath64 , which affects the evolution of absolute neutrino masses . it should satisfy the following rge @xmath215 \ ; . % ( a4)\end{aligned}\ ] ] it is worth mentioning that if the experimental uncertainties of the top quark mass @xmath216 and the strong coupling @xmath217 are taken into account , the sm vacuum could be stable up to the planck scale @xmath218 @xcite , even for a higgs mass @xmath219 indicated by the recent results of the atlas and cms experiments . in the mssm , the rge s in eqs . ( 3 ) and ( a1 ) are still applicable , but the relevant flavor - universal coefficients are as follows : @xmath220 , @xmath221 , and @xmath222 \ ; , \nonumber \\ \alpha^{\rm mssm}_{\rm u } & = & -\frac{13}{15 } g^2_1 - 3 g^2_2 - \frac{16}{3 } g^2_3 + 36~{\rm tr } \left ( y^{}_{\rm u}y^\dagger_{\rm u } \right ) \ ; , \nonumber \\ \alpha^{\rm mssm}_{\rm d } & = & -\frac{7}{15 } g^2_1 - 3g^2_2 - \frac{16}{3 } g^2_3 + { \rm tr}\left [ 3\left(y^{}_{\rm d}y^\dagger_{\rm d}\right ) + \left(y^{}_l y^\dagger_l\right)\right ] \ ; . % ( a5)\end{aligned}\ ] ] the rge s for the gauge couplings are given in eq . ( a3 ) , but with @xmath223 in the beta functions . as we can see from the rge of @xmath64 , the running neutrino parameters are determined by the charged - lepton yukawa coupling matrix @xmath224 , especially the tau - lepton yukawa coupling @xmath225 , which could significantly be enhanced for a large value of @xmath226 . such a unique feature can make the rg running of leptonic mixing parameters remarkable in the mssm . in the uedm , all the sm fields are promoted to a higher - dimensional spacetime , so every sm particle is accompanied by a tower of kaluza klein ( kk ) modes @xcite . in the simplest uedm with only one extra spatial dimension , which is compactified on an @xmath227 orbifold with radius @xmath132 , the kk parity defined as @xmath228 for the @xmath229-th kk mode is conserved after compactification . the mass scale of the first excited kk mode , i.e. , @xmath230 , has been constrained to be larger than about @xmath231 . if we extend the uedm by an effective operator @xmath232 to accommodate majorana neutrino masses , just as in eq . ( 1 ) , then the effective majorana neutrino mass matrix after electroweak symmetry breaking is @xmath61 with @xmath233 . the rge of @xmath64 now receives contributions from the kk modes , which are excited at the energy scale of interest . more explicitly , the rge s for @xmath64 and the yukawa coupling matrices of the charged fermions are also given by eqs . ( 3 ) and ( a1 ) , but with the following coefficients @xcite @xmath234 and @xmath235 with @xmath236 " being any relevant subscript . note that @xmath237 counts the number of excited kk modes for a given energy scale @xmath53 . in addition , the coefficients in the beta functions of gauge couplings turn out to be @xmath238 finally , the rge for the quartic higgs coupling @xmath214 is quite relevant in the uedm , as in the sm case . it has been found to be @xcite @xmath239 \ ; . % ( a8)\end{aligned}\ ] ] if the sm is extended with three right - handed neutrino singlets , then neutrinos acquire dirac masses in the same way as the charged leptons and quarks do . at one - loop level , the rge s of the fermion yukawa coupling matrices read @xcite @xmath240 y^{}_\nu \ ; , \nonumber \\ 16\pi^2 \frac{{\rm d}y^{}_l}{{\rm d}t } & = & \left[\alpha^{\rm sm}_l + c^{\rm sm}_{l,\nu } \left(y^{}_\nu y^\dagger_\nu\right ) + c^{\rm sm}_{l , l } \left(y^{}_l y^\dagger_l\right)\right ] y^{}_l \ ; , \nonumber \\ 16\pi^2 \frac{{\rm d}y^{}_{\rm u}}{{\rm d}t } & = & \left [ \alpha^{\rm sm}_{\rm u } + c^{\rm sm}_{\rm u , u } \left(y^{}_{\rm u } y^\dagger_{\rm u } \right ) + c^{\rm sm}_{\rm u , d } \left(y^{}_{\rm d } y^\dagger_{\rm d}\right ) \right ] y^{}_{\rm u } \ ; , \nonumber \\ 16\pi^2 \frac{{\rm d}y^{}_{\rm d}}{{\rm d}t } & = & \left [ \alpha^{\rm sm}_{\rm d } + c^{\rm sm}_{\rm d , u } \left(y^{}_{\rm u } y^\dagger_{\rm u } \right ) + c^{\rm sm}_{\rm d , d } \left(y^{}_{\rm d } y^\dagger_{\rm d}\right ) \right ] y^{}_{\rm d } \ ; , % ( b1)\end{aligned}\ ] ] where @xmath241 ( for @xmath242 ) and @xmath243 ( for @xmath244 ) , and @xmath245 with @xmath246 $ ] . the rge s of fermion yukawa coupling matrices are the same as in eq . ( b1 ) for the mssm , but with different coefficients , namely @xmath247 ( for @xmath242 ) and @xmath248 ( for @xmath244 ) , and @xmath249 \ ; , \nonumber \\ mssm}_l & = & -\frac{9}{5 } g^2_1 - 3 g^2_2 + { \rm tr}\left[3\left(y^{}_l y^\dagger_l\right ) + \left(y^{}_\nu y^\dagger_\nu\right)\right ] \ ; , \nonumber \\ \alpha^{\rm mssm}_{\rm u } & = & -\frac{13}{15 } g^2_1 - 3 g^2_2 - \frac{16}{3 } g^2_3 + { \rm tr}\left[3\left(y^{}_{\rm u } y^\dagger_{\rm u}\right ) + \left(y^{}_\nu y^\dagger_\nu\right)\right ] \ ; , \nonumber \\ \alpha^{\rm mssm}_{\rm d } & = & -\frac{7}{15 } g^2_1 - 3 g^2_2 - \frac{16}{3 } g^2_3 + { \rm tr}\left[3\left(y^{}_{\rm d } y^\dagger_{\rm d}\right ) + \left(y^{}_l y^\dagger_l\right)\right ] \;. % ( b3)\end{aligned}\ ] ] the rge s of three gauge couplings @xmath211 , @xmath210 , and @xmath209 are the same as those in the case of majorana neutrinos [ see eq . 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since the smallest leptonic mixing angle @xmath0 has been measured to be relatively large , it is quite promising to constrain or determine the leptonic dirac cp - violating phase @xmath1 in future neutrino oscillation experiments . given some typical values of @xmath2 , @xmath3 , and @xmath4 at the low - energy scale , as well as current experimental results of the other neutrino parameters , we perform a systematic study of the radiative corrections to @xmath1 by using the one - loop renormalization group equations in the minimal supersymmetric standard model and the universal extra - dimensional model . it turns out that @xmath1 is rather stable against radiative corrections in both models , except for the minimal supersymmetric standard model with a very large value of @xmath5 . both cases of majorana and dirac neutrinos are discussed . in addition , we use the preliminary indication of @xmath6 or @xmath7 from the latest global - fit analyses of data from neutrino oscillation experiments to illustrate how it will be modified by radiative corrections .

few - particle systems with resonant interactions are universal in the sense that their properties are independent of the short - range interaction details . well known example is the efimov effect , where in the unitary limit , characterized by the two - particle scattering length @xmath0 , an infinite number of weakly bound trimers with zero spin and positive parity ( @xmath1 ) may exist @xcite . furthermore , refs . @xcite predicted the existence of two @xmath1 tetramers for each efimov trimer . the two lowest tetramers , i.e. , the ones associated with the trimer ground state , are true bound states and have already been studied extensively @xcite ; however , they may be affected significantly by the finite - range corrections such that even some contradictions between refs.@xcite and @xcite exist . in contrast , all other tetramers lie above the lowest particle - trimer threshold and therefore have finite width and lifetime . thus , although a number of sophisticated numerical methods @xcite is available for the four - boson bound states , not all of them can be applied to a rigorous study of higher tetramers ; a proper treatment of the continuum is needed . however , in this case the technical difficulties in describing the scattering processes involving very weakly bound dimers and trimers in the universal regime may limit the accuracy of the coordinate - space methods @xcite . alternative calculations using the momentum - space framework have been recently performed by us for the atom - trimer @xcite and dimer - dimer @xcite scattering . the description is based on the exact four - particle alt , grassberger , and sandhas ( ags ) equations @xcite for the transition operators . the numerical technique , with some important modifications , is taken over from the four - nucleon scattering calculations @xcite . in this work it will be used to determine the universal properties of unstable tetramers . while their positions and limits of existence have already been calculated using coordinate - space methods @xcite , in our momentum - space framework we are able to achieve the universal limit with much higher accuracy , revealing in some cases quite drastic differences as compared to the predictions of refs . furthermore , we obtain results for the widths of the tetramers . in sec . [ sec:4bse ] we describe the employed four - boson scattering equations and the technical framework . in sec . [ sec : res ] we present results for tetramer properties and their effect on the atom - trimer and dimer - dimer scattering observables ; we also compare our predictions with those by other authors . we summarize in sec . [ sec : sum ] . an exact description of the four - particle scattering can be given by the faddeev - yakubovsky equations @xcite for the wave - function components or by the equivalent alt , grassberger , and sandhas ( ags ) equations @xcite for the transition operators ; the latter are more convenient to solve in the momentum - space framework preferred by us . the number of independent transition operators ( wave - function components ) is significantly reduced in the case of four identical particles where there are only two distinct two - cluster partitions , one of @xmath2 type and one of @xmath3 type . we choose those partitions to be ( 12,3)4 and ( 12)(34 ) and denote them in the following by @xmath4 and @xmath5 , respectively . the corresponding transition operators @xmath6 for the system of four identical bosons obey symmetrized ags equations [ eq : u ] @xmath7 here @xmath8 is the free green s function of the four - particle system with energy @xmath9 and kinetic energy operator @xmath10 , the two - particle transition matrix @xmath11 acting within the pair ( 12 ) is derived from the corresponding potential @xmath12 using the lippmann - schwinger equation @xmath13 and the symmetrized operators for the 1 + 3 and 2 + 2 subsystems are obtained from the integral equations @xmath14 the employed basis states have to be symmetric under exchange of two particles in subsystem ( 12 ) for @xmath2 partition and in ( 12 ) and ( 34 ) for @xmath3 partition . the correct symmetry of the four - boson system is ensured by the operators @xmath15 , @xmath16 , and @xmath17 where @xmath18 is the permutation operator of particles @xmath19 and @xmath20 . all observables for two - cluster reactions are determined by the transition amplitudes @xmath21 obtained @xcite as on - shell matrix elements of the ags operators ; the weight factors @xmath22 with values @xmath23 , @xmath24 , and @xmath25 arise due to the symmetrization @xcite . the matrix elements are calculated between the faddeev components @xmath26 of the corresponding initial / final atom - trimer or dimer - dimer states @xmath27 . the calculation of scattering observables is done at real energies @xmath28 where @xmath29 is the binding energy of the initial @xmath30th state in the @xmath31 channel , @xmath32 is the corresponding relative two - cluster momentum , and @xmath33 the reduced two - cluster mass . however , in the spin / parity @xmath1 states at complex energy values @xmath34 , corresponding to each unstable tetramer with energy @xmath35 ( relative to the four - body breakup threshold ) and width @xmath36 , the ags transition operators have simple poles , i.e. , the energy - dependence of @xmath6 at @xmath37 can be given by @xmath38 the unstable bound state ( ubs ) pole in the complex energy plane is located in one of the unphysical sheets that is adjacent to the physical sheet @xcite . the ubs therefore affects the physical observables leading to resonant effects in the four - boson collisions . as a consequence , the properties of those unstable tetramers can be extracted from the behavior of the four - boson scattering amplitudes or observables in the region @xmath39 where the series is approximated very well by few terms with @xmath40 or 1 . we solve the ags equations in the momentum - space partial - wave framework with two different types of basis states as explained in refs . @xcite . in this representation the ags equations for each total angular momentum @xmath41 become a system of coupled integral equations in three continuous variables , the magnitudes of the jacobi momenta @xmath42 , @xmath43 , and @xmath44 for the relative motion in the 1 + 1 , 2 + 1 , and 3 + 1 ( 1 + 1 , 1 + 1 , and 2 + 2 ) subsystems of the 3 + 1 ( 2 + 2 ) configurations , respectively . although such equations can be solved as done in refs . @xcite for the four - nucleon scattering , the technical implementation is highly demanding . on the other hand , in this work we are interested in the universal properties of the four - boson system that must be independent of the short - range interaction details and therefore we can choose the most convenient form of the potential . the practical solution simplifies considerably by using a separable two - boson potential @xmath45 . in this case the ags equations can be reduced to a system of integral equations with only two variables , @xmath43 and @xmath44 ; the details are given in ref . @xcite . the four - boson reactions from which we extract the tetramer properties involve at most one dimer - dimer channel but several ( up to five in the present calculations ) atom - trimer channels with the binding energies differing by many orders of magnitude . as pointed out in ref . @xcite , this leads to additional difficulties that are very hard to overcome in the coordinate - space approaches but can be resolved reliably in our momentum - space framework : we discretize the integrals using gaussian quadrature rules and use momentum grids of correspondingly broad range ; each subsystem bound state pole of @xmath46 is isolated in a different subinterval when performing the integration over @xmath44 @xcite . the discretization of integrals in the ags equations leads to a system of linear algebraic equations whose solution is described in refs . the interaction model is taken over from ref . @xcite , i.e. , we use a rank-1 separable potential limited to the @xmath47 state with the form factor @xmath48e^{-(k_x/\lambda)^2}\ ] ] and the strength @xmath49 constrained to reproduce the given value of the scattering length @xmath19 for two particles of mass @xmath50 . the rank-1 potential supports at most one two - boson bound state , i.e. , there are no deeply bound dimers . the results will be presented as dimensionless ratios that are independent of the used @xmath51 and @xmath50 values in the universal limit . to demonstrate that our results are indeed independent of the details of the short - range interaction , we use two very different form factors with @xmath52 and @xmath53 . we start by presenting the results in the unitary limit @xmath54 where the dimer binding energy @xmath55 vanishes but an infinite number of the trimers exists with a geometric spectrum of binding energies @xmath56 , i.e. , @xmath57 . this number was predicted analytically by efimov @xcite but our numerical calculations reproduce it very well for highly excited trimers , i.e. , for @xmath30 large enough such that the finite - range corrections become negligible . with @xmath58 we achieve at least six digit accuracy as demonstrated in ref . @xcite for both choices of the form factor ( [ eq : gsep ] ) . in contrast , significant deviations were found for the ground states , e.g. , @xmath59 and 2126 with @xmath60 and @xmath61 , respectively ; this is caused by a very different short - range behavior of the two used models . in our nomenclature we characterize the tetramers by two integers @xmath62 where @xmath30 refers to the associated trimer and @xmath63 ( 2 ) for a deeper ( shallower ) tetramer . our preliminary predictions for the tetramer positions @xmath64 and widths @xmath65 , i.e. , their relation to the associated trimer binding energy @xmath66 , were given already in ref . @xcite . here the study of tetramer properties is improved and extended . first we investigate the convergence of the results with respect to the number of included partial waves determined by the parameter @xmath67 such that @xmath68 . example results for @xmath69 and @xmath60 are collected in table [ tab : res - l ] . the convergence is quite fast but the nonzero angular momentum states for the 2 + 1 and 3 + 1 subsystems can not be neglected . our previous results of ref . @xcite obtained with @xmath70 are already well converged , the inclusion of @xmath71 states yields only tiny corrections . furthermore , we note that the contributions of even @xmath72 are attractive while those of odd are repulsive . * 5c @xmath67 & @xmath73 & @xmath74 & @xmath75 & @xmath76 + 0 & 4.6754 & 0.01422 & 1.00404 & @xmath77 + 1 & 4.6056 & 0.01474 & 1.00215 & @xmath78 + 2 & 4.6108 & 0.01485 & 1.00228 & @xmath79 + 3 & 4.6102 & 0.01484 & 1.00227 & @xmath79 + in table [ tab : res - n ] we present our results for the positions @xmath64 and widths @xmath65 of the tetramer pairs up to @xmath80 ; they were obtained with @xmath81 . the ratios @xmath82 and @xmath83 for both choices of the potential form factor converge towards universal values [ eq : res ] @xmath84 as @xmath30 increases . however , significant potential - dependent deviations due to finite - range effects can be seen for @xmath82 and @xmath83 at @xmath85 and @xmath86 , respectively . with @xmath53 , where the @xmath87 trimer is a non - efimov - like state @xcite , the @xmath88 tetramer is even absent . including a strong repulsive three - body force would decrease the binding energies and increase the size of the states and thereby speedup the @xmath30-convergence @xcite but the ground state calculations ( @xmath87 ) would be insufficient anyway since @xmath87 does nt account for the inelastic collisions and finite width . this also explains why the convergence for @xmath65 is slower than for @xmath64 . * 5c @xmath30 & @xmath89 & @xmath90 & @xmath91 & @xmath92 + 0 & 5.6402 & & 1.04185 & + 1 & 4.5169 & 0.03363 & 1.00105 & @xmath93 + 2 & 4.6035 & 0.01366 & 1.00216 & @xmath94 + 3 & 4.6098 & 0.01471 & 1.00226 & @xmath95 + 4 & 4.6102 & 0.01484 & 1.00227 & @xmath79 + 5 & 4.6102 & 0.01483 & 1.00227 & @xmath79 + 0 & 3.2192 & & & + 1 & 4.9923 & 0.01360 & 1.00996 & @xmath96 + 2 & 4.6108 & 0.02084 & 1.00227 & @xmath97 + 3 & 4.6098 & 0.01493 & 1.00226 & @xmath98 + 4 & 4.6102 & 0.01483 & 1.00227 & @xmath79 + 5 & 4.6102 & 0.01483 & 1.00227 & @xmath79 + ( color online ) elastic and inelastic cross sections for the atom scattering from the @xmath30th trimer in the vicinity of the @xmath99th tetramer.,title="fig : " ] ( color online ) elastic and inelastic cross sections for the atom scattering from the @xmath30th trimer in the vicinity of the @xmath99th tetramer.,title="fig : " ] next we study the effect of unstable tetramers on the elastic and inelastic cross sections @xmath100 in atom - trimer collisions ; @xmath30 and @xmath101 characterize the trimer state in the initial and final channel , respectively . to form dimensionless ratios for each trimer we introduce the length scale @xmath102 . as already found in ref . @xcite , for sufficiently large @xmath30 and @xmath101 the ratios @xmath103 depend only on @xmath104 but not on the employed potential . furthermore , for the inelastic cross sections ( @xmath105 ) an additional relation @xmath106 was established @xcite . thus , in the universal limit the atom scattering from the @xmath30th trimer can be fully characterized by only two cross sections , the elastic one @xmath107 and the leading inelastic one @xmath108 . in the case of @xmath107 the atom - trimer @xmath109- and @xmath110-wave contributions calculated in ref . @xcite have to be added ; they are negligible for @xmath108 . in fig . [ fig : rs - cs ] we study the behavior of the elastic and inelastic atom - trimer cross sections in the vicinity of the @xmath99th tetramer ; we use @xmath60 and @xmath111 such that the finite - range effects are negligible . we use the energy variable @xmath112 that measures the distance to the tetramer position . despite very different tetramer widths , the behavior of the cross section as function of @xmath113 is very similar for @xmath63 and 2 . although @xmath114 has pole in all open channels @xmath115 , the elastic and inelastic cross sections @xmath107 and @xmath108 have characteristic resonance peaks only in the case of @xmath116 where they increase by a factor of 5 . for @xmath117 a minimum is seen close to @xmath118 which becomes less and less pronounced as the difference @xmath119 increases ; we therefore show only @xmath120 results where in the minimum the elastic ( inelastic ) cross section is decreased by 5% ( 10% ) . in accordance with this behavior the phase shift ( not shown here ) increases by @xmath121 only for @xmath122 while local minima take place for @xmath117 . trimers and the associated tetramers exist in a certain regime of large finite @xmath123 . in fig . [ fig : ba ] we show the tetramer positions @xmath64 as functions of @xmath19 ; we include also the binding energies in the atom - trimer and dimer - dimer channels , @xmath66 and @xmath124 . as a reference point for @xmath19 we choose the intersection of the dimer - dimer and the @xmath30th atom - trimer thresholds , i.e. , @xmath125 at @xmath126 . all the binding energies are normalized by the @xmath30th trimer binding energy in the unitary limit @xmath127 . in fig . [ fig : ba ] we show @xmath128 only at @xmath129 ; at @xmath130 the shallow tetramer lies very close the atom - trimer threshold and exhibits a nontrivial behavior that will be presented separately in the next subsection . the @xmath19-evolution of the width @xmath131 of the deeper tetramer is shown in fig . [ fig : g1 ] . ( color online ) tetramer , trimer and dimer binding energies as functions of the two - boson scattering length . ] ( color online ) the width of the deeper tetramer as a function of the two - boson scattering length . ] as can be seen in fig . [ fig : ba ] , on the side of negative @xmath19 the trimers ( tetramers ) emerge at the three ( four ) free atom threshold with zero energy . we denote by @xmath132 and @xmath133 the specific negative values of @xmath19 where @xmath134 and @xmath135 , respectively . in an ultracold atomic gases these @xmath19 values would correspond to a resonant enhancement of the three or four - atom recombination process @xcite . on the side of positive @xmath19 the trimers decay via the atom - dimer threshold , i.e. , @xmath136 at @xmath137 ; this situation is outside the range of fig . [ fig : ba ] since @xmath138 @xcite . the tetramers decay via the dimer - dimer threshold , i.e. , @xmath139 and @xmath140 at @xmath141 , leading to a resonant behavior of the dimer - dimer scattering length @xmath142 shown for @xmath63 in fig . [ fig : add1 ] . the consequence of this phenomenon in an ultracold gas of dimers is a resonant enhancement of the trimer creation and dimer - dimer relaxation processes @xcite , the zero - temperature rate being @xmath143 . ( color online ) dimer - dimer scattering length as a function of the atom - atom scattering length in the vicinity of the @xmath63 tetramer intersection with the dimer - dimer threshold . ] all those special values of @xmath19 are related in a universal way provided @xmath30 is sufficiently large . with the present interaction models the universal limit is reached with high accuracy at @xmath69 as can be seen in numerous examples for tetramer properties given in table [ tab : res - n ] and for various atom - trimer scattering observables in ref . in particular , the convergence for the tetramer intersections with the dimer - dimer threshold is demonstrated in ref . @xcite , yielding the ratios [ eq : a - dd ] @xmath144 where the uncertainties are estimated by comparing the predictions obtained with different @xmath30 and @xmath145 . for the intersection of the tetramers with the four free atom threshold we get [ eq : a-0 ] @xmath146 furthermore , semi - analytical results @xcite are available for some quantities of the three - boson nature , namely , @xmath147 and @xmath148 for @xmath149 . they agree well with our numerical predictions [ eq : a-3 ] @xmath150 thereby confirming their reliability . since the shallow tetramer lies very close to the associated atom - trimer threshold , we use a different representation to show the @xmath19-evolution of its position , namely , we consider its relative distance to the atom - trimer threshold @xmath151 . together with the width it is presented in fig . [ fig : bg2 ] in the whole region of its existence . for both tetramers the widths @xmath65 remain finite at the respective @xmath152 where @xmath64 vanish ; the @xmath63 case is shown in fig . [ fig : g1 ] . the shallow tetramer detaches most from the atom - trimer threshold around @xmath153 where @xmath66 almost vanishes . most remarkably , at two special positive values of @xmath154 the shallow tetramer intersects the atom - trimer threshold , i.e. , when moving away from the unitary limit it first decays at @xmath155 , then reappears at @xmath156 , and finally decays via the dimer - dimer threshold at @xmath157 . in other words , the shallow tetramer in a particular regime @xmath158 becomes an inelastic virtual state ( ivs ) @xcite with @xmath159 instead of an ubs with positive width . the intersection points @xmath160 correspond to @xmath161 and are universal , i.e. , [ eq : a - ivs ] @xmath162 in the vicinity of @xmath154 the tetramer position @xmath163 while otherwise @xmath164 . thus , it may seem that the tetramer ivs is mostly below the atom - trimer threshold . however , one has to keep in mind the changed sign of @xmath165 for ivs that implies the change of the energy sheet . the ivs corresponds to the pole of the transition operators @xmath6 in the complex energy plane on one of the nonphysical sheets that is , in contrast to the one of ubs , more distant from the physical sheet @xcite . for this reason the ivs affects the physical scattering observables in a completely different way as compared to ubs : the elastic and inelastic cross sections around @xmath166 for ivs , i.e. , for @xmath158 show no resonant peaks that were seen in fig . [ fig : rs - cs ] for the ubs . thus , the parameters of the ivs can not be extracted using eq . . however , an approximate procedure based on the atom - trimer scattering length and effective range parameter as described in ref . @xcite was applied to obtain the ivs results in fig . [ fig : bg2 ] . the ivs pole only affects the physical observables when it is located extremely close to the atom - trimer threshold . in that case the cross sections and phase shift have a cusp exactly at the atom - trimer threshold , i.e. , at @xmath167 , but the cusp disappears rapidly with increasing @xmath168 and @xmath169 ; an examples can be found in ref . we note that the tetramers become ivs also after crossing the dimer - dimer threshold , i.e. , at @xmath170 . . we used @xmath171 at @xmath172 where the @xmath30th trimer does nt exist . ] the most prominent effect of the tetramer ubs - ivs conversions through the atom - trimer ( dimer - dimer ) threshold is a resonant enhancement of the atom - trimer scattering length @xmath173 around @xmath154 ( dimer - dimer scattering length @xmath142 around @xmath141 ) ; thus , at the corresponding @xmath19 values @xmath173 exhibits qualitatively the same behavior as shown for @xmath142 in fig . [ fig : add1 ] . the regime around @xmath160 is not yet explored experimentally . however , in an ultracold mixture of atoms and excited trimers tuning the atom - atom scattering length to the values close to @xmath160 would lead to a resonantly increased rate of the atom - trimer relaxation whose zero - temperature limit is @xmath174 . of course , in real experiments the resonance positions may deviate from the universal values ( [ eq : a - ivs ] ) due to finite - range effects . a number of numerical techniques is available for the four - boson bound state calculations , both in the momentum @xcite and coordinate space @xcite . some of them @xcite , neglecting the finite width of the higher tetramers , were used to find their energies and threshold intersection points . the results of refs . @xcite for @xmath82 at unitarity , @xmath175 , and @xmath176 agree with ours within few percents . those calculations @xcite were limited to @xmath85 or 2 where the finite - range effects could not be entirely neglected and the convergence with @xmath30 was not better than few percents ; this is consistent with our findings as can be seen in table [ tab : res - n ] , albeit with different potentials . we demonstrated that higher @xmath30 are needed to achieve the universal limit accurately ; this is technically very demanding , especially in the coordinate - space framework . however , in contrast to our work , ref . @xcite has not predicted the shallow tetramer intersections with the atom - trimer threshold at @xmath160 . it appears that the results of refs . @xcite are quite poorly converged for the fine - scale quantities @xmath177 at unitarity and @xmath178 . for the former our five best - converged results are 0.00227 within 0.5% according to table [ tab : res - n ] while the three best - converged results of ref . @xcite are 0.006 , 0.03 , and 0.001 . for @xmath178 the prediction of ref . @xcite , 0.019 , overestimates our well - converged result 0.00053 by a factor of 35 . both these deviations indicate that refs . @xcite strongly overestimate the distance of the shallow tetramer from the atom - trimer threshold such that they never cross each other . we note that the nonuniversal @xmath88 tetramer of our work is also bound considerably stronger than the universal ones and therefore does nt intersect the atom - trimer threshold . we studied universal bosonic tetramers that are unstable bound states in the continuum and strongly affect collisions in the four - boson system . we extracted the tetramer properties such as positions , widths , and limits of existence from the behavior of the atom - trimer and dimer - dimer scattering observables . these collision processes were described using exact four - particle scattering equations for the transition operators that were solved in the momentum - space framework with high precision . a rigorous treatment of the four - boson continuum enabled us to determine the widths of the tetramers that were out of reach in previos works . furthermore , we accurately achieved the universal limit by considering reactions involving high excited trimers where the finite - range effects are negligible . in this respect our results are much better converged than those of previous works @xcite where only the tetramer positions and existence limits have been calculated . while the agreement between our predictions and those of refs . @xcite is reasonable for the more tightly bound tetramer , there are drastic differences for the shallow one . we demonstrate that changing the two - boson scattering length the shallow tetramer intersects the atom - trimer threshold twice and in a special regime becomes an inelastic virtual state ; these ubs - ivs conversions lead to resonant effects in ultracold atom - trimer collisions . ferlaino , f. , knoop , s. , mark , m. , berninger , m. , schbel , h. , ngerl , h .- c . , grimm , r. : collisions between tunable halo dimers : exploring an elementary four - body process with identical bosons . * 101 * , 023201 ( 2008 ) . grassberger , p. , sandhas , w. : systematical treatment of the non - relativistic n - particle scattering problem . b2 * , 181 ( 1967 ) ; e. o. alt , p. grassberger , and w. sandhas , jinr report no . e4 - 6688 ( 1972 ) .

the system of four identical bosons is studied using momentum - space equations for the four - particle transition operators . positions , widths and existence limits of universal unstable tetramers are determined with high accuracy . their effect on the atom - trimer and dimer - dimer scattering observables is discussed . we show that a universal shallow tetramer intersects the atom - trimer threshold twice leading to resonant effects in ultracold atom - trimer collisions .

one of the first discrete models of reaching consensus ( decentralized coordination ) was proposed by degroot @xcite . suppose that @xmath1 is the vector of initial opinions of the members of a group and @xmath2 is the vector of their opinions after the @xmath3th step of coordination . in accordance with degroot s model , @xmath4 where @xmath5 is a row stochastic influence matrix whose entry @xmath6 specifies the degree of influence of agent @xmath7 on the opinion of agent @xmath8 thereby , _ a consensus is _ [ asymptotically ] _ reached _ if @xmath9 for some @xmath10 and all @xmath11 degroot states that a consensus is reached for any initial opinions if and only if the matrix @xmath12 exists and all rows of @xmath13 are identical , which is equivalent to the regularity ] of @xmath5 . if @xmath5 is not regular , then the opinions do not generally tend to agreement . yet , a consensus can be reached if the vector of initial opinions belongs to a certain subspace . in @xcite , we give a characterization of this subspace and propose the _ method of orthogonal projection _ which generalizes degroot s method . it is shown that in the method of orthogonal projection , as well as in degroot s method , no nonbasic agent can affect the final result . in this paper , we show that the result of any degroot s procedure with a strong communication digraph as well as the result of any procedure of orthogonal projection without nonbasic agents can be represented by degroot s procedure whose communication digraph is a hamiltonian cycle with loops . in this cycle , the weight of each arc ( which is not a loop ) is inversely proportional to the influence of the agent the arc leads to . with a stochastic influence matrix @xmath5 in degroot s model we associate the _ communication digraph _ @xmath14 with vertex set @xmath15 @xmath14 has a @xmath16 arc with weight @xmath17 whenever @xmath18 ( i.e. , whenever agent @xmath7 influences agent @xmath19 ) . thus , arcs in @xmath14 are oriented _ in the direction of influence _ ; the weight of an arc is the power of influence . the _ kirchhoff matrix _ ( see @xcite ) @xmath20 of digraph @xmath14 is defined as follows : if @xmath21 then @xmath22 whenever @xmath14 has the @xmath16 arc and @xmath23 otherwise ; @xmath24 @xmath25 . @xmath26 will denote the identity matrix of appropriate dimension . by virtue of the above definitions , for the digraph @xmath14 associated with @xmath5 we have any maximal by inclusion strong ( i.e. , with mutually reachable vertices ) subgraph of a digraph is called a _ strong component _ ( or a _ bicomponent _ ) of this digraph . basic bicomponent _ is a bicomponent such that the digraph has no arcs coming into this bicomponent from outside . vertices belonging and not belonging to basic bicomponents are called _ basic _ and _ nonbasic _ , respectively . similarly , we call an agent _ basic_/_nonbasic _ whenever the vertex representing this agent is _ basic_/_nonbasic_. let @xmath27 and @xmath28 be the number of basic vertices and the number of basic bicomponents in @xmath29 respectively . if a consensus in degroot s model is reached and vertex @xmath7 is nonbasic , then , as stated in @xcite , column @xmath7 in the limiting matrix @xmath13 is zero and the initial opinion of agent @xmath7 does not affect the resulting opinion . now we present the results from algebraic graph theory used in this paper . a longer list of results useful for decentralized control is given in @xcite . if the sequence of powers @xmath30 of a stochastic matrix @xmath5 has a limit @xmath31 then where @xmath32 is the normalized matrix of maximum out - forests of the corresponding weighted digraph @xmath14 ( a corollary of the matrix tree theorem for markov chains @xcite ) . @xmath13 is the eigenprojection corresponding to @xmath33 ( principal idempotent ) of @xmath34 and where @xmath28 is the number of basic bicomponents in @xmath14 ( * ? ? ? * proposition 11 ) . by @xmath35 where @xmath36 is the dimension of the kernel ( the _ nullity _ ) of @xmath37 finally , by ( * ? ? ? * proposition 12 ) , @xmath38 where @xmath39 ( the index of @xmath34 ) is the order of the largest jordan block of @xmath34 corresponding to the zero eigenvalue . this implies that where @xmath40 is the multiplicity of @xmath33 as an eigenvalue of @xmath34 . as noted above , degroot s method with matrix @xmath5 leads to a consensus for any initial opinions if and only if there exists a limiting matrix @xmath12 and all its rows are identical . the equality of the rows of @xmath13 implies that @xmath41 for some probability vector ( the components are non - negative and sum to @xmath42 ) @xmath43 where @xmath44 in this case , the consensus @xmath45 is expressed by the inner product of the vectors @xmath46 and @xmath47 : where @xmath48 is the resulting vector of opinions , @xmath46 is the final weight distribution of the degroot algorithm , and @xmath49 is the consensus . a probability vector @xmath46 is called a _ stationary vector _ of a stochastic matrix @xmath5 if it is a left eigenvector of @xmath5 corresponding to the eigenvalue@xmath42 : @xmath50 . obviously , this condition is satisfied for the vector @xmath46 in the representation @xmath41 of @xmath31 provided that the convergence of degroot s method is guaranteed by the regularity of @xmath51 by theorem 3 in @xcite , if for any vector of initial opinions @xmath52 degroot s method converges to the consensus @xmath53 then @xmath46 is a _ unique _ stationary vector of @xmath5 . let us formulate a criterion of convergence in degroot s model in terms of the communication digraph @xmath54 the equality of the rows of @xmath13 is equivalent to @xmath55 therefore , owing to , when the sequence @xmath30 converges , consensus is reached for any initial opinions if and only if the communication digraph @xmath14 corresponding to @xmath5 has a single basic bicomponent ( @xmath56 ) . consequently , provided that the sequence @xmath30 converges , @xmath56 is equivalent to the regularity of @xmath5 . in turn , by , this is the case if and only if @xmath33 is a simple eigenvalue of @xmath37 finally , @xmath56 if and only if @xmath14 has a spanning _ out - tree _ ( also called _ arborescence _ and _ branching _ ) ( * ? ? ? * proposition6 ) . in this case ( see ) , @xmath57 is the normalized matrix of spanning out - trees : where @xmath58 is the total weight of @xmath14 s spanning out - trees rooted at @xmath59 and @xmath60 is the total weight of all spanning out - trees of @xmath54 a survey of some results on degroot s iterative pooling model and its generalizations can be found in @xcite . note that one of the new applications of degroot s model is information control in social networks @xcite . as shown in ( * ? ? ? * section6 ) , the final result of the method of orthogonal projection can be represented by a weight vector @xmath61 : the inner product of @xmath61 and the vector of initial opinions gives the consensus : @xmath62 . this is analogous to the representation of the result of a convergent degroot s method : @xmath41 and @xmath63 ( see ) . on the other hand , given a probability vector @xmath43 it is easy to construct a weighted communication digraph generating @xmath46 as the final weight distribution ( the stationary vector of @xmath5 ) of degroot s method . in this section , we show that all positive weight distributions on the set of agents opinions are generated by a rather narrow class of digraphs , namely , hamiltonian cycles of the form @xmath64 with loops , where the vertices are denoted by @xmath65 by @xmath66 where @xmath32 is the normalized matrix of maximum out - forests of the weighted digraph @xmath14 corresponding to @xmath67 thus , given @xmath43 a necessary and sufficient condition of the fulfilment of @xmath41 is a hamiltonian cycle is a strong digraph , so its maximum out - forests are precisely spanning out - trees ( see ) . recall that the normalized matrix of spanning out - trees ( which in this case coincides with @xmath32 ) is the matrix whose @xmath68-entry is @xmath69 @xmath70 . adding loops does not change the set of out - trees . thus , to solve the problem , i.e. , to implement a given probability vector @xmath71 as the final weight distribution of degroot s method with a communication digraph in the form if a hamiltonian cycle with loops , it is sufficient to construct a weighted cycle whose vector @xmath72 is proportional to @xmath73 indeed , in this case @xmath74 coincides with @xmath43 which guarantees . such a cycle can be constructed by means of the following lemma . [ 250810th2 ] for any positive vector @xmath75 there exists a unique weighted hamiltonian cycle of the form @xmath76 whose vector @xmath77 of total weights of out - trees coincides with @xmath78 . the weight of the arc entering vertex @xmath3 in this cycle is @xmath79{\prod_{i=1}^{n\phantom{i}}q_i},\,$ ] @xmath80 the proof of lemma[250810th2 ] is given in the appendix . now we apply lemma[250810th2 ] to solve our problem . [ p_picy ] for any positive probability vector @xmath81 there exists a family of weighted hamiltonian cycles of the form @xmath82 with loops such that using each of them as the communication digraph in degroot s method implements the consensus @xmath49 for any vector of initial opinions @xmath47 . the weight of the arc entering vertex @xmath3 in such a cycle is proportional to @xmath83 as stated above , any communication digraph whose vector @xmath84 of the weights of out - trees is proportional to @xmath46 implements @xmath46 as the final weight distribution of degroot s method . therefore , to prove proposition[p_picy ] , it is sufficient to observe that by lemma[250810th2 ] , a hamiltonian cycle with loops and any given order of visiting vertices can be taken as such a digraph . the only thing that should be taken care of is that this cycle must be a _ communication _ digraph , i.e. , the matrix @xmath85 ( see ) corresponding to it must be stochastic . since @xmath34 has zero row sums and nonpositive off - diagonal entries , @xmath85 is stochastic if and only if the diagonal entries of @xmath34 are less than or equal to @xmath86 for a hamiltonian cycle with loops , this condition is satisfied if and only if all its arc weights do not exceed @xmath86 [ ex_ham2 ] consider the influence matrix @xmath87 and the corresponding communication digraph ( fig.[f_011211]a , which does not show loops for simplicity ) . proposition[p_picy ] enables one to construct another communication digraph , in the form of a hamiltonian cycle with loops , that shares the final consensus obtained through degroot s method with the given digraph for any initial opinions . observe that @xmath5 is regular and its stationary vector is : @xmath88 as stated in section[s_deso ] , @xmath89 now we construct a hamiltonian cycle @xmath90 with loops such that @xmath91 . proposition[p_picy ] implies that the vector of arc weights indexed by the vertices of @xmath90 can be @xmath92 where @xmath93 taking @xmath94 we obtain @xmath95 in this communication digraph ( which is shown in fig.[f_011211]b ) , loops with weights @xmath96 and @xmath97 must be attached to vertices @xmath42 and @xmath98 respectively . it follows from proposition[250810th2 ] that if the weights of arcs entering each vertex are preserved , then the order of vertices in the cycle can be arbitrary . in particular , the cycle in fig.[f_011211]c is also suitable . thus , for each cycle with loops of this type , degroot s method leads to the same final consensus procedure ( determined by the weight distribution ) as for the original communication digraph . since in proposition[p_picy ] , @xmath99 is the weight of the original opinion of @xmath19th agent in the final consensus ( the `` influence '' of the @xmath19th agent ) , it is worth noting that for communication digraphs in the form of hamiltonian cycles with loops , this weight is inversely proportional to the weight of the arc entering vertex @xmath8 thus , in the upshot , the most powerful agent is the one least subject to the influence of the previous agent in the cycle rather than the one maximally affecting the next agent . finally , proposition[p_picy ] enables one , for any given coordination procedure based on the method of orthogonal projection @xcite , to construct a degroot algorithm ( with the communication digraph in the form of a hamiltonian cycle ) that leads to the same consensus for any initial opinions . it is shown that any convergent discrete iterative pooling procedure taking into account the opinions of all agents with positive weights can be approximated ( in terms of achieving the same end result ) by degroot s procedure whose communication digraph is a hamiltonian cycle with loops . the weight of the arc entering vertex @xmath19 in this cycle is inversely proportional to the influence of agent @xmath100 while the order of visiting vertices can be arbitrary . for any vertex in a hamiltonian cycle , there is exactly one tree outgoing from this vertex . that tree includes all arcs of the cycle except for the arc entering this vertex . hence the elements of the vector @xmath72 of total weights of out - trees are given by the equations where @xmath101 is the weight of the arc entering vertex @xmath102 consequently , @xmath103{\prod_{i=1}^n\prod_{j\ne i}x_j } = \frac{\sqrt[n-1]{\prod_{i=1}^{n\phantom{i}}t_i}}{t_k},\quad k=\1n.\end{gathered}\ ] ] < ? 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( but not for the journal `` computational linguistics'')</summary > < published>2013 - 11 - 20t21:00:00z</published > < updated>2015 - 01 - 22t22:19:51 + 00:00</updated > < rights license=``http://creativecommons.org / licenses / by - sa/3.0/''>this work is licensed under a creative commons attribution - sharealike 3.0 license</rights > < /info > < macro name=``author '' > < names variable=``author '' > < name form=``long '' and=``text '' delimiter= `` , ' ' / > < substitute > < text value=``''/ > < /substitute > < /names > < /macro > < macro name=``author - short '' > < names variable=``author '' > < name form=``short '' and=``text '' delimiter= `` , ' ' / > < substitute > < names variable=``editor '' > < name form=``short '' and=``text '' delimiter= `` , ' ' / > < /names > < text value=``mis''/ > < /substitute > < /names > < /macro > < macro name=``editor '' > < names variable=``editor '' > < name form=``long '' and=``text '' delimiter= `` , ' ' / > < label prefix= `` , '' form=``long''/ > < /names > < /macro > < macro name=``author - or - editor '' > < names variable=``author '' > < name form=``long '' and=``text '' delimiter= `` , ' ' / > < substitute > < text macro=``editor''/ > < text value=``''/ > < /substitute > < /names > < /macro > < macro name=``year - date '' > < choose > < if variable=``issued '' > < date variable=``issued '' > < date - part name=``year''/ > < /date > < /if > < else > < text value= `` ' ' / > < /else > < /choose > < /macro > < macro name=``month '' > < date variable=``issued '' > < date - part name=``month''/ > < /date > < /macro > < macro name=``edition '' > < number variable=``edition '' form=``ordinal''/ > < text term=``edition '' prefix= `` ' ' / > < /macro > < macro name=``volume - or - number '' > < choose > < if variable=``volume '' > < group delimiter= `` '' > < label variable=``volume''/ > < number variable=``volume''/ > < /group > < /if > < else - if variable=``number '' > < group delimiter= `` '' > < text value=``number''/ > < number variable=``number''/ > < /group > < /else - if > < else - if variable=``issue '' > < group delimiter= `` '' > < text value=``number''/ > < number variable=``issue''/ > < /group > < /else - if > < /choose > < /macro > < macro name=``event - or - publisher - place '' > < choose > < if variable=``event - place '' > < text variable=``event - place''/ > < /if > < else > < text variable=``publisher - place''/ > < /else > < /choose > < /macro > < macro name=``thesis - type '' > < choose > < if variable=``genre '' > < text variable=``genre''/ > < /if > < else > < text value=``ph.d . thesis''/ > < /else > < /choose > < /macro > < macro name=``volume - and - collection - title '' > < choose > < if variable=``volume collection - title '' match=``all '' > < group delimiter= `` '' > < text term=``volume''/ > < number variable=``volume''/ > < text value=``of''/ > < text variable=``collection - title '' font - style=``italic''/ > < /group > < /if > < else - if variable=``volume '' > < group delimiter= `` '' > < text term=``volume''/ > < number variable=``volume''/ > < /group > < /else - if > < ! todo : else - if for issue or number > < else > < text variable=``collection - title''/ > < /else > < /choose > < /macro > < macro name=``technical - report '' > < choose > < if variable=``number issue '' match=``any '' > < group delimiter= `` '' > < ! uppercased `` report '' > < text value=``technical report''/ > < choose > < if variable=``number '' > < text variable=``number''/ > < /if > < else > < text variable=``issue''/ > < /else > < /choose > < /group > < /if > < else > < ! lowercased `` report '' > < text value=``technical report''/ > < /else > < /choose > < /macro > < citation et - al - min=``3 '' et - al - use - first=``1 '' disambiguate - add - year - suffix=``true '' > < ! no sorting for citation > < layout prefix= `` ( '' suffix= `` ) '' delimiter= < text macro=``author - short''/ > < text macro=``year - date '' prefix= `` , ' ' / > < /layout > < /citation > < bibliography et - al - min=``20 '' et - al - use - first=``19 '' > < sort > < key macro=``author - or - editor''/ > < key macro=``year - date''/ > < key variable=``title''/ > < /sort > < layout > < ! > < choose > < if type=``book '' > < text macro=``author - or - editor '' suffix=``.''/ > < /if > < else > < text macro=``author '' suffix=``.''/ > < /else > < /choose > < ! year > < date variable=``issued '' prefix= `` '' suffix= `` . '' > < date - part name=``year''/ > < /date > < ! title and other information > < choose > < ! corresponds to `` article '' in bibtex > < if type=``article - journal '' > < text variable=``title '' prefix= `` '' suffix=``.''/ > < group prefix= `` '' suffix= `` . '' > < text variable=``container - title '' font - style=``italic''/ > < text variable=``volume '' prefix= `` , ' ' / > < text variable=``issue '' prefix= `` ( '' suffix=``)''/ > < text variable=``page '' prefix=``:''/ > < text macro=``month '' prefix= `` , ' ' / > < /group > < /if > < ! corresponds to `` proceedings '' , `` manual '' , `` book '' , and `` periodical '' in bibtex > < else - if type=``book '' > < text variable=``title '' prefix= `` '' suffix= `` . '' font - style=``italic''/ > < group delimiter= `` , '' suffix= `` . '' > < text macro=``volume - and - collection - title''/ > < /group > < group delimiter= `` , '' prefix= `` '' suffix= `` . 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'' > < text macro=``technical - report''/ > < text variable=``publisher''/ > < text variable=``publisher - place''/ > < text macro=``month''/ > < /group > < /else - if > < ! corresponds to `` phdthesis '' and `` masterthesis '' in bibtex > < else - if type=``thesis '' > < text variable=``title '' prefix= `` '' suffix= `` . '' font - style=``italic''/ > < group delimiter= `` , '' prefix= `` '' suffix= `` . '' if you want to output text other than `` ph.d thesis '' , specify the `` genre '' field to an appropriate value like `` master s thesis''. > < text macro=``thesis - type''/ > < text variable=``publisher''/ > < text variable=``publisher - place''/ > < text macro=``month''/ > < /group > < /else - if > < ! misc . > < else > < text variable=``title '' prefix= `` '' suffix=``.''/ > < text macro=``month '' prefix= `` '' suffix=``.''/ > < /else > < /choose > > < text variable=``note '' prefix= `` '' suffix=``.''/ > < /layout > < /bibliography >

we show that any discrete opinion pooling procedure with positive weights can be asymptotically approximated by degroot s procedure whose communication digraph is a hamiltonian cycle with loops . in this cycle , the weight of each arc ( which is not a loop ) is inversely proportional to the influence of the agent the arc leads to . n1, ,n # 1 # 1@xmath0 0 [ 1 ] # 1

here , we compute quantum friction on a moving two - level atom by solving the equation of motion for the atomic dipole in perturbation theory . the dynamics of a two - state system can be derived from the hamiltonian @xmath144 where @xmath145 and @xmath6 ; together with @xmath146 they satisfy the algebra of pauli matrices . @xmath147 is the free electromagnetic field hamiltonian . the internal state dynamics is given by @xmath148 this is a nonlinear equation that does not allow for an exact solution , and in the following we solve it using a perturbative scheme in powers of the dipole coupling @xmath5 . the computation of the quantum frictional force requires the evaluation of the two - time correlator tensor @xmath149 . to second - order it can be evaluated from the pauli matrices free evolution , and at zero temperature and for a ground state atom @xmath150 , resulting in a frictional force that is exponentially suppressed in @xmath104 . to compute the frictional force to fourth - order one needs to evaluate @xmath151 at second order . to this end we first insert in equation the formal solution for the dynamics of @xmath152 , and then replace the exact field @xmath153 by its free evolution @xmath154 . at second order we obtain @xmath155 multiplying this equation from the right by @xmath156 , averaging on the initial factorized state , taking the infinite time limit , and finally fourier transforming the resulting equation , we can write the power spectrum to fourth order in the dipole coupling as ( for simplicity , we omit the green tensor s dependence on the position of the atom ) @xmath157 where @xmath158^{-1}$ ] . the functions @xmath159 and @xmath160 are even in @xmath88 and give the second order atomic ( velocity dependent ) frequency shift and decay rate . one can see that the dynamic power spectrum is symmetric and real . we now study the low velocity expansion of the quantum frictional force on the two - level atom . we start by noting that only the symmetric part of the green tensor contributes to ( [ tsa_s ] ) , and since @xmath161 , the power spectrum ( [ tsa_s ] ) is even in @xmath81 . therefore , the power spectrum can be expanded as @xmath162 , where the double prime denotes second derivative with respect to velocity . defining @xmath163 , we obtain low velocity expansion of the power spectrum of the two - level atom ( valid to fourth order in the dipole coupling ) @xmath164 + { \cal o}(v_x^4 ) . \label{tsaspectrumlowv}\ ] ] note that it has the same form as the low velocity expansion of the harmonic oscillator model ( see eq . ( 11 ) of the main text ) . in this case , however , the function @xmath165 is connected with the fourth order perturbative expression of the imaginary part of the polarizability while @xmath166^ { * } $ ] ( the double prime denotes second derivative with respect to velocity ) , and @xmath167 . we now use ( [ tsaspectrumlowv ] ) in the expression for the quantum friction force ( see eq.(6 ) of the main paper ) @xmath73 .\end{gathered}\ ] ] we need to expand the integrals ( @xmath168 ) @xmath169 , \\ i_2 = \frac{v_x^2}{2 } \int_0^{k_x v_x } d\omega { \rm tr } \left [ \underline{\tilde\eta}(k_x v_x - \omega;0 ) \cdot \underline{g}_i(\mathbf{k},\omega ) \right],\end{gathered}\ ] ] to lowest order in @xmath67 . using that @xmath170 and @xmath171 are odd in @xmath88 , it follows that @xmath172 and results in exactly the same quantum friction force as in eq.(7 ) of the main text . using that also @xmath173 is odd in @xmath88 , it follows that @xmath174 . hence , the stationary quantum frictional on a moving two - level atom scales as the cubic power of its velocity to leading order , as shown in the main paper using an alternative method based on the fluctuation - dissipation theorem . 1 . theoretical division , ms b213 , los alamos national laboratory , los alamos , new mexico 87545 , usa 2 . max - born - institut , 12489 berlin , germany 3 . center for nonlinear studies , los alamos national laboratory , los alamos , new mexico 87545 , usa 4 . department of applied physics , yale university , new haven , connecticut 06511 , usa

we use general concepts of statistical mechanics to compute the quantum frictional force on an atom moving at constant velocity above a planar surface . we derive the zero - temperature frictional force using a non - equilibrium fluctuation - dissipation relation , and show that in the large - time , steady - state regime quantum friction scales as the cubic power of the atom s velocity . we also discuss how approaches based on wigner - weisskopf and quantum regression approximations fail to predict the correct steady - state zero temperature frictional force , mainly due to the low frequency nature of quantum friction . a remarkable example of fluctuation - induced interactions is quantum friction , the drag force experienced between two bodies in relative motion in vacuum , associated with the energy and momentum transfer from one body to the other mediated by the quantum electromagnetic field . radiation mediated friction is deeply rooted in the foundations of quantum mechanics and it was already discussed by einstein in his seminal 1917 paper on black body spectrum @xcite . quantum friction has recently attracted attention in the context of macroscopic bodies and atoms in linear @xcite or rotational @xcite motion above a surface , coulomb drag in electron transport phenomena @xcite , and as the dissipative counterpart of the dynamical casimir effect @xcite . several authors @xcite have obtained quite diverse results for the atom - surface drag at zero temperature , making different predictions as to its dependence on the velocity of the atom and the atom - surface separation . here we revisit the problem of quantum friction using general concepts of quantum statistical mechanics . we derive a quantum non - equilibrium fluctuation - dissipation theorem ( fdt ) for an atom in steady - state motion above a surface and compare its predictions with the quantum regression theorem ( qrt ) . we first consider the prototype problem of a static atom above a planar material surface at zero temperature . the atom is described by an electric dipole operator @xmath0 located at position @xmath1 . in a simple two - state system model ( ground @xmath2 and excited state @xmath3 ) the atomic electric dipole operator is given by @xmath4 , where @xmath5 is the ( real ) dipole vector and @xmath6 describes the internal degrees of freedom @xcite ( the generalization to multi - level atoms is straightforward @xcite ) . alternatively , in a model of the atom as a harmonic oscillator , @xmath7 , where @xmath8 is a dimensionless position operator @xcite . at any given time @xmath9 , the force on the atom normal to the surface is given by @xmath10 . from the maxwell equations the electric field operator can be written as @xmath11 , where @xmath12 is the electric green tensor of the surface ( the subscripts @xmath13 and @xmath14 will denote real and imaginary part ) , and @xmath15 denotes the positive - frequency solution for the electric field in the absence of the atom . we will assume that the initial atom + field / matter state is factorizable , @xmath16 , with the joint field / matter subsystem in its vacuum state . using normal ordering the force can be written as @xmath17 denotes expectation value over the initial state . note that in this equation @xmath18 represents the exact dynamics of the dipole operator , including back action from the field / matter . the two - time correlation tensor @xmath19 will be a key quantity in what follows . for the equilibrium problem being considered , the stationary ( @xmath20 ) density matrix of the coupled atom - field - matter system has the kubo - martin - schwinger ( kms ) form , @xmath21 ( @xmath22 is the inverse temperature and @xmath23 is the system s hamiltonian ) ; at zero temperature @xmath24 corresponds to the ground state of the whole system . hence , in the stationary state the two - time correlation tensor tends to @xmath25 , and the zero - temperature fdt @xcite relates the corresponding power spectrum @xmath26 with the atom s polarizability tensor @xmath27 \hat{\rho}_{\rm kms } \}$ ] @xmath28 where @xmath29 is the step function and @xmath30 is the fourier transform of @xmath31 . equation is valid for the two previous models for the atom , since the equilibrium fdt holds not only for linear but also for non - linear systems @xcite , including an atom treated using a ( nonlinear ) two- or multi - level model . this can be seen in the following derivation of the fdt , showing its validity for an arbitrary ( time independent ) system hamiltonian @xmath23 @xcite . let @xmath32 and @xmath33 be two observables , and define @xmath34 and @xmath35 \rangle$ ] . then @xmath36 . for @xmath37 , it follows that @xmath38 $ ] . using the equilibrium kms condition @xmath39 @xcite , we have @xmath40 , \ ] ] which reduces to in our case . both for the oscillator and the two - level atom , @xmath41 and @xmath42 are symmetric tensors , and therefore the power spectrum @xmath43 is real . note that @xmath44 is the non - perturbative polarizability that depends on the optical properties of the surrounding field , the atom , and the surface , and is a function of the atom s position @xmath1 ( omitted in the following for simplicity ) . taking the large - time limit of ( [ cpforce ] ) and using the fdt , one obtains the ( non - perturbative and non - markovian ) casimir - polder force @xcite @xmath45 another commonly used fluctuation relation is the regression theorem @xcite and its generalization to the quantum case , known as the quantum regression hypothesis ( sometimes called theorem " ) given by the lax formula @xcite . the quantum regression theorem ( qrt ) is approximate , valid only in the weak system - bath coupling limit and near a resonance ( see , for example , @xcite ) . although successfully used in quantum optics within its range of validity , the qrt is known to fail whenever non - markovian and off - resonance effects play an important role @xcite : the broadband nature of fluctuation - induced interactions suggests that its use in this context is therefore questionable . within the qrt the two - time dipole correlation tensor for a two - state atom or a harmonic oscillator for @xmath46 is given by @xmath47 , where @xmath48 and @xmath49 are the atomic transition frequency and dissipation rate , respectively . using this expression in ( [ cpforce ] ) and taking the large time limit one obtains a casimir - polder force of the same form as ( [ cp - exact ] ) , but with @xmath50 replaced by @xmath51/2 $ ] , where @xmath52 $ ] is the generalized ground state atomic polarizabilty @xcite . the qrt fails to give the expression ( [ cp - exact ] ) predicted by the fdt and the exact solution for the harmonic oscillator model @xcite , which coincides with and reduces to the well - known lifshitz formula . the mathematical reason for this discrepancy lies in the distinct large - time behavior of the correlation tensor @xmath53 . while the qrt predicts an exponential decay , the exact fdt results in a power - law decay for large times @xmath54 ( and agrees with the qrt only for @xmath55 ) . for example , in the large time limit , @xmath56 for @xmath57 ( ohmic dissipation ) . only in the weak coupling limit ( @xmath58 ) , corresponding to a second - order perturbative calculation in powers of the coupling strengths @xmath5 , does the qrt coincide with the fdt . a related phenomenon takes place in the spontaneous decay of an excited atom in vacuum , which in the wigner - weisskopf approximation is predicted to be exponential , but has large - time power - law corrections @xcite . the previous analysis shows that , beyond the weak coupling regime , the correct large time behavior of the two - time correlation tensor strongly affects the steady state casimir - polder force in ( [ cpforce ] ) . we show now that similar considerations also apply to the non - equilibrium situation of an atom moving parallel ( along the @xmath59-direction ) to a flat semi - infinite ( @xmath60 ) bulk ( fig.[friction ] ) . as before , we model the atom by an electric dipole operator and treat its center - of - mass coordinate @xmath61 semiclassically . the quantum frictional force is given by @xmath62 , where the expectation value is taken with respect to an initial uncorrelated atom+field / matter state in which the field / matter is in its vacuum state @xcite . the @xmath59-dynamics is governed by @xmath63 , where @xmath64 is an external classical force on the atom that drives it from the initial rest state at @xmath65 to a steady - state at time @xmath66 after which the atom moves at constant velocity @xmath67 above the surface , @xmath68 . in the large - time limit , the stationary frictional force is given by @xmath69 \right\}. \label{fxsteady}\end{gathered}\ ] ] here @xmath70 is the two - time correlation tensor in the non - equilibrium stationary state @xmath24 of the coupled moving atom plus field / matter . note that it depends on the velocity of the atom , which is denoted by the @xmath67 dependency after the semi - colon in the expression above . once more , we emphasize that @xmath71 contains the exact dynamics of the moving atomic dipole , i.e. including the backaction from the field / matter . there is an extensive literature on non - equilibrium fluctuation theorems , trying to generalize fundamental equilibrium results such as the fluctuation - dissipation theorem to non - equilibrium steady - state configurations ( see , for example , @xcite ) . one of the challenges is to identify the form of the non - equilibrium stationary density matrix , which is no longer described by a kms state but is model - dependent . despite this limitation , we will show that it is still possible to draw general conclusions about the frictional force in the low velocity limit . in analogy to the static case , we define a power spectrum @xmath72 , which is again a real and symmetric tensor since in our description @xmath41 is symmetric . using the symmetry properties of the green tensor @xmath12 for the homogeneous planar surface ( see @xcite , for example ) , ( [ fxsteady ] ) can be re - written as @xmath73 .\end{gathered}\ ] ] note that in this expression the power spectrum @xmath74 depends on the wave vector only through the doppler shifted frequency @xmath75 . the friction is the momentum transfer @xmath76 to the atom weighted by its doppler - shifted power spectrum and by the electromagnetic density of states , all integrated over frequency and momentum . as expected , the force vanishes for @xmath77 : since @xmath74 is symmetric only the symmetric part of @xmath78 ( even in @xmath79 @xcite ) is relevant . the integral then vanishes for parity reasons . generally one is interested in computing @xmath80 to leading - order in @xmath81 . for this , however , one needs to know the expression for @xmath82 , which in general is not available ( see , however , the harmonic oscillator model below ) . nevertheless , even without this knowledge , it is possible to prove that at zero temperature and in the stationary limit ( @xmath20 ) there are no linear in @xmath67 terms in the friction force , independently of the model for the atom s polarizability . indeed , terms proportional to @xmath67 could only arise either from @xmath83 or from @xmath84 . the contribution of the former term cancels again for parity reasons upon integration over @xmath85 . the latter term , corresponding to a stationary state @xmath24 in which the atom is static , can be evaluated using the equilibrium fdt ( [ fdt ] ) , i.e. @xmath86 . because of the motion - induced doppler - shift , only frequency modes @xmath87 contribute , implying that very low frequencies are relevant at small velocities . since the atomic polarizability and the green tensor are susceptibilities , they satisfy the crossing relation and their imaginary parts , being odd in @xmath88 , vanish at @xmath89 in our case @xcite . an expansion for small @xmath81 leads then to @xmath90 \nonumber \\ & \approx & - \frac{45 \hbar v_x^3}{256 \pi^2 \epsilon_{0 } z_a^7 } \alpha'_i(z_a,0 ) \delta'_i(0 ) , \label{friction - exact}\end{aligned}\ ] ] where in the first line we omitted to write the @xmath91 dependency of the green tensor at coincidence . in the second line we have considered the low - frequency ( near - field ) form of the green tensor for a dielectric semi - space described by a complex permittivity @xmath92 ( @xmath93 in the vacuum permittivity ) , with @xmath94/[\epsilon(\omega)+1]$ ] , and we have used @xmath95 ( we have reintroduced @xmath91 to underscore the dependency of the dressed polarizability on the position of the atom ) . the above argument proves that , within our description for the atom , the lowest - order expansion in velocity of the zero - temperature , stationary frictional force on an atom moving above a planar surface is at least cubic in @xmath67 . in principle , however , in addition to that in there could be other @xmath96 contributions to the frictional force arising from @xmath67-derivatives of @xmath97 . also , when either of the @xmath88-derivatives of the two tensors in ( [ friction - exact ] ) vanish at @xmath89 , higher - order terms in @xmath67 must be considered . regarding the dependency of the stationary frictional force ( [ friction - exact ] ) on the atom - surface separation , it must be emphasized that the @xmath98 scaling arises solely from the @xmath91-dependency of the green tensor . in addition , as explained above , the power spectrum @xmath74 and the polarizability @xmath42 implicitly depend on @xmath91 via the exact dynamics of the coupled atom - field / matter system . in particular , these quantities are related to the atomic decay , which at short distances and to lowest order in perturbation theory scales as @xmath99 , leading in ( [ friction - exact ] ) to a total @xmath100 dependency of the frictional force . for systems with intrinsic dissipation ( e.g. gold nanoparticles ) the radiation - induced damping is generally negligible and the frictional force has therefore a milder dependency on separation @xcite . in contrast to the fdt , the qrt predicts that for slow velocities the quantum frictional force is linear in @xmath67 . as shown above , such a dependency results in principle from contributions of @xmath101 in ( [ friction2 ] ) . using the qrt expression for the two - time correlation tensor in the static case , @xmath102 , and taking the @xmath20 limit one obtains indeed at the leading order expansion @xmath103 ^ 2 } { \rm tr } [ \underline{g}_{i}(\mathbf{k},z_a , z_a , \omega ) ] , \label{qrt - friction}\end{aligned}\ ] ] where , for simplicity , we assumed that the atom is isotropic @xcite . as for the static casimir - polder force , the quantum regression hypothesis fails to give the correct quantum frictional force . note , however , that once again both the fdt and the qrt give the same quantum frictional force in the limit @xmath58 , consistent with the observation before that the quantum regression hypothesis coincides with the exact fluctuation - dissipation theorem for systems near equilibrium in the weak coupling limit . in this limit , the resulting force is exponentially suppressed in @xmath104 @xcite . linear response relations in fluctuational electrodynamics , based on equilibrium fluctuations , can also be employed to study quantum friction for small perturbations around the equilibrium state @xcite . in agreement with our analysis , at zero temperature the linear - in - velocity frictional force vanishes . however , far from equilibrium situations require fully non - equilibrium fluctuation relations . the previous derivation uses general principles based on the fluctuation - dissipation theorem in non - equilibrium settings . in the following , we present an alternative derivation that does not resort to the fdt , and compute quantum friction for the harmonic oscillator model by directly solving the equations of motion for the atomic dipole in the stationary limit ( in the supplemental material we present a similar derivation for the two - state system ) . the dynamics of the dipole operator for the moving harmonic oscillator atom can be solved for exactly . its equation of motion , including the back reaction of the electromagnetic field , is given by @xmath105 . splitting the solution to maxwell s equations for the total field @xmath106 as a sum of free ( @xmath107 , homogeneous solution ) and source ( @xmath108 , particular solution ) parts and taking the fourier transform , the equation of motion can be re - written as @xmath109 \hat{q}(\omega ) = \frac{2\omega_{a}}{\hbar } \int \frac{d^2{\bf k}}{(2 \pi)^2 } { \bf d } \cdot \hat{\bf e}_0({\bf k},z_a,\omega+k_x v_x ) e^{i ( k_x x_a + k_y y_a)}$ ] . the polarizability of the moving oscillator is then given by @xmath110^{-1}$ ] , where we have omitted to write the @xmath91 dependency of the green tensor . the dynamic power spectrum @xmath111 is computed starting from @xmath112 and using that @xmath113 . the resulting exact expression for the zero - temperature case is @xmath114 where the current " @xmath115 is given by @xmath116 \nonumber \\ & & \times \underline{\alpha}(\omega;v_x ) \cdot \underline{g}_i({\bf k},\omega+k_x v_x ) \cdot \underline{\alpha}^*(\omega;v_x ) .\end{aligned}\ ] ] generalized fdt relations for non - equilibrium , stationary classical systems @xcite have the same structure as ( [ non - eq - fdt ] ) . since only the symmetric part of the green tensor contributes to @xmath117 , from the previous expressions for the polarizability and the current @xmath115 , we can deduce that the power spectrum is even in @xmath81 . using the identity @xmath118 , we rewrite the power spectrum as @xmath119 . an expansion at low velocity takes the form @xmath120+\mathcal{o}(v_{x}^{4 } ) . \label{sexpansion}\ ] ] here we have defined @xmath121^{*}$ ] ( the double prime denotes second derivative with respect to velocity ) , and @xmath122 . the tensor @xmath123 vanishes at @xmath124 because it is a sum of terms proportional either to the imaginary part of the green tensor or to its second derivative . using in one can verify that to leading order in @xmath67 the quantum frictional force for the harmonic oscillator model is exactly given by , and the next order is proportional to @xmath125 ( see supplemental material ) . our result for the @xmath96 dependence of the quantum friction force on a moving atom contrasts with some previous works in the literature that predicted a zero - temperature frictional force linear in @xmath67 . in @xcite the atom was modeled as a multi - level system and the dipole correlation function in ( [ fxsteady ] ) was computed using qrt , which lead to a stationary friction force linear in velocity ( [ qrt - friction ] ) . calculations of quantum friction based on qrt , wigner - weisskopf , or markovian approximations encompass an exponential - only decay of the dipole correlation tensor , which is valid for times @xmath126 . importantly , they miss the power - law decay at larger times @xmath127 which strongly affects the low - frequency behavior of the spectrum . the discussion after ( [ friction2 ] ) shows that , in the stationary case , quantum friction is a low - frequency phenomenon ( see also the paragraph after ) . therefore , it is not surprising that the above mentioned approximations fail to predict the correct stationary behavior and lead to a different dependence of the force on the atom s velocity . on the other hand , in @xcite the atom was modeled as a harmonic oscillator and , by calculating the power dissipated by the atom into pairs of surface plasmons using an approach based on standard perturbation theory , a linear - in - velocity frictional force similar to @xcite was obtained ( within the same approximations an identical result is obtained for a two - level atom ) . this time - dependent perturbative approach assumed that the atom remains in its bare ground state , and is valid for times not too long for which decays are still exponential . in contrast , our previous calculation shows that in the large - time , non - equilibrium steady - state the quantum frictional force becomes cubic in velocity . due to the weak nature of quantum friction , its experimental detection is challenging . indeed , in the near field our result ( [ friction - exact ] ) takes the form @xmath128 where @xmath129 is surface s electrical resistivity and @xmath130 the static atomic polarizability . as an example , for a ground state @xmath131rb atom ( @xmath132 @xcite ) flying at @xmath133 m/s at a distance @xmath134 nm above a silicon semi - space ( @xmath135 ) , the zero temperature drag force is @xmath136n . nevertheless , new experimental setups ( e.g. new materials @xcite and/or new geometries @xcite ) and techniques ( e.g. atom - interferometry ) could make it accessible in the near future . in summary , we have studied quantum friction using general concepts of quantum statistical mechanics . we have derived a generalized non - equilibrium fluctuation - dissipation relation for an atom in steady motion above a surface , and shown that at low speeds the quantum frictional force is cubic in velocity . the analysis can be extended to include thermal fluctuations . in the high - temperature ( classical ) limit ( @xmath137 @xcite ) , however , quantum regression agrees with the fdt @xcite , and the resulting frictional force scales linearly with velocity . a study similar to the one present here can be performed for the case of macroscopic bodies in relative motion @xcite ) . finally , we would like to stress that our discussion of the implications and limitations of the use of fluctuation relations in calculations of equilibrium and non - equilibrium atom - surface interactions can potentially impact a broad range of fields such as atom interferometry and atom - 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we thank both j. stirling and a. vogt for forwarding their numerical codes for the gluino parton densities , as well as r. harris and i. bertram for discussions on the cdf and d0 dijet data sample and providing us with the detailed numerical results from their respective experiments . # 1 # 2 # 3 mod . # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 nucl . # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . rep . * # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . * # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 phys . # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 rev . # 1 * , # 2 ( # 3 ) # 1 # 2 # 3 z. phys . * # 1 * , # 2 ( # 3 ) p.n . burrows , presented at _ 3rd international symposium on radiative corrections _ , cracow , poland , august 1996 , hep - ex/9612007 ; s. bethke , presented at _ qcd euroconference 96 _ , montpellier , france , july 1996 , hep - ex/9609014 . c. albajar , ( ua1 collaboration ) , b198 261 1987 ; r.m . barnett , in _ 4th international conference on physics beyond the standard model _ , lake tahoe , ca , december 1994 , ed . gunion , ( world scientific , singapore 1995 ) ; r.m . barnett , h.e . haber , and g.l . kane , 54 1983 1985 ; v. barger , d33 57 1986 . dishaw , ( na3 collaboration ) , 85b 142 1979 ; f. bergsma , ( charm collaboration ) , 121b 429 1983 ; r.c . ball , 53 1314 1984 ; wa66 collaboration , 160b 212 1985 ; t. akesson , ( helios collaboration ) c52 219 1991 ; i.f . albuquerque , ( e761 collaboration ) , hep - ex/9604002 . r. munoz - tapia and w.j . stirling , d49 3763 1994 ; a. de gouvea and h. murayama , hep - ph/9606449 ; s. moretti , r. munoz - tapia , and j.b . tausk , hep - ph/9609206 ; s. moretti , r. munoz - tapia , and k. odagiri , hep - ph/9609235 , hep - ph/9609295 ; delphi collaboration , contributed to the _ 28th international conference on high energy physics _ , warsaw , poland , july 1996 , delphi-96 - 68-conf-2 ; r. akers , ( opal collaboration ) , c65 367 1995 , and * c68 * , 555 ( 1995 ) . f. abe , ( cdf collaboration ) , hep - ex/9609011 . r. harris , in _ proceedings of the 10th topical workshop on proton - antiproton collider physics _ , batavia , il 1995 , ed . r. raja and j. yoh ; we thank r. harris for giving us the updated version of this analysis .

the effects of light , long - lived gluinos on @xmath0 processes at hadron colliders are examined . such particles can mediate single squark resonant production via @xmath1 which would significantly modify the dijet data sample . we find that squark masses in the range @xmath2 gev are excluded for gluino masses of @xmath3 gev from existing ua2 and tevatron data on dijet bump searches and angular distributions . run ii of the tevatron has the capability of excluding this scenario for squark masses up to @xmath4 tev . psfig.sty -0.5 in 6.5 in 8.50 in -0.5 in 4.0=0.5 in # 1_.25ex / plus0pt minus0pt # 1#20.7ex # 1@xmath5 # 1@xmath6 # 1 # 1 # 1[#1 ] * constraints on light gluinos from tevatron dijet data * + .3 cm stanford linear accelerator center , stanford ca 94309 , usa + .3 cm and + .3 cm michael a. doncheski + .3 cm ottawa carleton institute for physics , carleton university , ottawa , canada + department of physics , pennsylvania state university , mont alto , pa 17237 , usa + .3 cm supersymmetry is a compelling candidate for physics beyond the standard model ( sm ) and has engrossed both the theoretical and experimental communities . most of the attention has been focused on the minimal version of supersymmetry ( mssm ) , however , many other incarnations of supersymmetry could exist . in most cases these non - minimal models can significantly alter supersymmetric phenomenology and the associated search strategies , and hence all consequences of such models must be examined before regions of supersymmetric parameter space can be positively excluded . here , we examine one such non - minimal case : the light gluino scenario . in some models it is natural@xcite for gluinos to be much lighter than , , squarks if they acquire their masses radiatively . while several experiments presently cast doubt on the existence of the low mass gluino window ( @xmath7 gev ) , it has yet to be conclusively ruled out ( or verified ) . in fact , the experimental bounds on this possibility are surprisingly spotty and controversial as evidenced by the continual debate in the literature@xcite . it is thus imperative to examine all implications of this hypothesis in order to quell this dispute . in this work , we investigate an additional data sample which provides strong constraints on the light gluino scenario , namely @xmath0 processes at high energy hadron colliders . the window for a very light gluino was pointed out@xcite many years ago and its effects have since been analyzed in a variety of processes . a resurgence of interest in this scenario surfaced with the relatively recent observation@xcite that an apparent discrepancy between the value of @xmath8 measured from jet production at sld and lep and that discerned from low energy data is resolved by the slower running of @xmath8 in the presence of light gluinos . however , recent compilations@xcite of various determinations of @xmath8 no longer show evidence of such a discrepancy , within the errors , but also claim that the precision of each individual measurement is such that any anomalous effect up to the @xmath9 level may not be perceived . the most noticeable consequence of this model is that the standard signals for gluino and squark production are modified in the presence of light gluinos . the bounds on the gluino mass , @xmath10 gev from the tevatron@xcite ( with the range being due to the assumed relative sizes of the squark and gluino masses ) , are invalidated in this case as they depend on the fact that the is short - lived and decays with the characteristic missing energy signature . thus to be light , gluinos must be long - lived and appear to hadronize as jets . since they are unable to appear as free particles , light gluinos will indeed form hadrons , with the bound states having longer lifetimes , and fragment in such a way as to mimic jets in a high energy detector@xcite . if kinematically allowed , the gluino hadrons will eventually decay into a final state containing jets @xmath11 , where @xmath12 is the lightest neutralino . the crucial ingredient for detection is then the ability of the final state @xmath12 to pass the detector s missing energy cuts , which depends , amongst other things , on how the hadron fragments . it has been estimated@xcite that for @xmath13 gev the would have been detected at ua1 . however , as the gluino mass decreases , the missing energy signal disappears altogether . standard squark searches are also nullified in this model as now the primary decay is @xmath14 , which again , escapes searches based on missing energy . in this case , the squark mass bounds are reduced to @xmath15 , with the mass constraint being extended to @xmath16 gev from precision electroweak measurements at slc / lep@xcite . we expect lep ii to strengthen the squark mass bound to @xmath17 gev . we now discuss the results from a variety of light gluino searches . at present , the least controversial bound on light gluinos is from a search by cusb@xcite for radiative @xmath18 decays into bound states of gluinos . they exclude the mass range @xmath19 gev ( regardless of the gluino lifetime ) , where the lower limit is approximate due to questions@xcite concerning the validity of perturbative qcd in this regime . argus@xcite looked for secondary vertices from @xmath20 with subsequent decay of the gluino bound states and constrained a small region in the gluino mass - lifetime parameter space ; these results , however , also suffer@xcite from perturbative qcd uncertainties as well as those from fragmentation effects . beam dump experiments@xcite have looked for secondary vertices from the decay of hadrons and appear to disfavor light gluinos for restricted regions of the gluino lifetime , but these results depend on ( i ) assumptions on the production cross sections of the gluino hadrons , ( ii ) the value of the squark mass ( iii ) the interactions of the lightest color - singlet supersymmetric particle , @xmath12 , with the detector , and ( iv ) fragmentation effects and decay models . searches for new neutral particles at fermilab exclude@xcite @xmath21 gev for lifetimes in excess of @xmath22 s. jet angular distributions of decays of the @xmath23 into four jets and precision measurements of the qcd structure constants @xmath24 and @xmath25 have been shown to be particularly sensitive to the existence of light gluinos@xcite , but critically depend@xcite on currently uncalculated higher order qcd corrections and hence no firm conclusions can presently be drawn . the @xmath23 boson can decay into 2 gluinos , however the branching fraction is small@xcite ( @xmath26 ) , and would be hidden underneath ordinary qcd events . the detection of light gluinos at hera , through their effect on deep inelastic structure functions@xcite or via their production in the @xmath27 jet photoproduction cross section@xcite , have also been shown to be difficult . in this study , we examine the effects of light , long - lived gluinos on dijet production in hadronic collisions . one would expect the influence of light s to be large in such processes since they contribute at leading order in perturbation theory . it has been shown@xcite , however , that competing effects tend to suppress their impact on the single jet inclusive @xmath28 spectrum . nonetheless , we find that the influence of resonant squark production from the subprocess @xmath1 should not be neglected as it greatly modifies the dijet mass spectrum and places strong constraints on the light gluino window . our conclusions avoid some of the aforementioned difficulties in constraining this scenario , as non - perturbative qcd effects are negligible at the energies considered here and our results are insensitive to a long lifetime . the essential ingredients of this model for our analysis are ( i ) the evolution of @xmath8 is modified by the inclusion of light gluinos in the qcd @xmath29 function , ( ii ) long - lived gluinos in the final state hadronize as jets , and ( iii ) light gluinos contribute a non - negligible partonic content of the proton . this introduces several new @xmath0 parton scattering processes , as well as modifying the altarelli - parisi evolution of the parton densities . global fits of structure functions which include a light gluino distribution have been performed@xcite , and it has been found that the nlo parton distributions are roughly three ( five ) times larger than that of the strange quark at large ( small ) @xmath30 for very light gluinos , @xmath31 gev , and carry @xmath9 of the proton s momentum fraction at large @xmath32 for @xmath33 gev . we now proceed with our calculation . all @xmath0 subprocesses have been evaluated ; they naturally fall into three categories , ( i ) those of the sm , @xmath34 , @xmath35 , @xmath36 , and @xmath37 , ( ii ) all sm initiated @xmath0 processes with final state gluinos , @xmath38 , and ( iii ) all gluino initiated processes , @xmath39 , @xmath40 , and @xmath41 . note that resonant squark production appears in the latter set . higher order @xmath42 processes , including the new reactions@xcite which produce @xmath43jet and thus yield 3 jet final states once the squark decays , have not been included . the mass of the light gluino has also been neglected in the evaluation of the subprocess cross sections as the results should not be sensitive to at the energy scales considered here . the parton distributions@xcite of r " uckl and vogt have been used for @xmath44 gev and those of roberts and stirling for @xmath33 gev . these values of the gluino mass avoid all of the experimental constraints detailed above . the change in the evolution of @xmath8 has been taken into account by fixing @xmath45 to the world average value@xcite and then running it to the relevant scale using the appropriate 2-loop @xmath29 functions . we note that the 3-loop light @xmath29 functions have only recently been determined@xcite . in evaluating the squark resonance contribution to the cross section , we have used the narrow width approximation , which is valid for @xmath46 and hence is reliable in this case . we have included a 10% contribution to the squark width for potential non - dijet decays , , @xmath47 . this is conservative as dijet decays will be by far the dominant mode . the 10% figure should cover the additional weak decays @xmath48 and @xmath49 , whichever are kinematically allowed , as they are expected to have small branching fractions of order @xmath50 each and hence are suppressed compared to the dijet mode . we note that monojet signals from squark production in this scenario have been previously analyzed@xcite . we have also assumed that there are 5 degenerate squarks , with equal masses for the left- and right - handed states . our results are not dependent on this assumption , however , as the contribution of each squark flavor to the resonance peak is weighted by the corresponding quark s parton density . hence this supposition does not simply result in an overall multiplicative factor to the cross section . in fact , the charm and bottom squarks have essentially negligible contributions to the resonance peak . experimentally , the dijet system consists of the two jets with the highest transverse momentum in the event . in all cases , except where noted , we apply the cuts used by the cdf collaboration@xcite in their dijet analyses . this corresponds to @xmath51 gev , @xmath52 , where @xmath53 are the pseudorapidities of the two leading jets , and @xmath54 , with @xmath55 being the parton - parton scattering angle in the center of mass frame . following cdf , we evaluate these processes at the scale @xmath56 . in fig . 1 we display the dijet invariant mass and single jet inclusive @xmath57 distributions for the cases of @xmath58 and 500 gev , taking @xmath59 and 5 gev corresponding to the dotted , dashed , and solid curves , respectively . in the case of the @xmath57 distributions , we assume @xmath60 , @xmath61 , and no angular cuts are applied . we see that the resonance peaks stand out for all values of the gluino mass . note the degradation of the cross section as the mass increases . we now evaluate the dijet resonance cross section and compare it to searches for dijet mass peaks from the single production of new particles performed by hadron collider experiments@xcite . figure 2 presents the single squark production cross section in the dijet channel as a function of the squark mass for various values of . also displayed in the figure ( dotted curve ) is the upper limit on the production of dijet resonances at ( a ) ua2@xcite at @xmath62 c.l . , as well as both ( b ) cdf@xcite and ( c ) d0@xcite at the @xmath63 c.l . in the d0 case the applied cuts are somewhat different than those employed by cdf : @xmath64 and @xmath65 . we see that the three experiments combine to exclude substantial regions of the light gluino parameter space . the ranges of the squark masses which are ruled out for each value of are summarized in table [ bumptab ] . we do not expect the bounds to drastically improve as @xmath66 as the squark resonance cross section is not appreciably changing as the gluino mass decreases ( once @xmath31 gev ) as shown in fig . 2 . a short analysis shows that the cross section for massless gluinos is approximately 1.3(1.6 ) times larger than that for the case of @xmath67 gev at low(high ) dijet invariant masses . .the squark mass regions in gev excluded by the searches for dijet resonances by the ua2 ( at @xmath62 c.l . ) , cdf and d0 ( at @xmath63 c.l . ) collaborations for an assumed gluino mass . [ cols="^,^,^,^",options="header " , ] dijet angular distributions are a well known test of qcd and probe of new physics and have recently been measured at the tevatron@xcite . ordinary qcd processes have large @xmath68- and @xmath69-channel poles and are thus peaked in the forward direction , whereas , resonant squark production in the light gluino model will have a flat distribution due to the spin-0 nature of the squark . a convenient angular variable to use is @xmath70 . for the case of @xmath0 parton scattering , this is related to the center of mass scattering angle as @xmath71 . @xmath72 then corresponds to @xmath73 . as is well - known@xcite , the advantage of the @xmath74 variable is that it removes the apparent singularities associated with the @xmath75 and @xmath76channel poles present in qcd . thus @xmath77 shows greater sensitivity to new physics which does not possess such poles than does @xmath78 . to show the influence of the production of squark resonances on this distribution we display in fig . 3 the ratio of @xmath77 calculated in the light gluino model to that of the sm , , @xmath79 , for three dijet invariant mass bins ( as chosen by cdf@xcite ) assuming a resonance lies within each bin . in calculating the sm distributions , we employed the mrsa@xmath80 parton densities@xcite . in all cases , we see that squark production leads to an enhancement in the distribution at low values of @xmath74 compared to the sm . this would result in an increase in the dijet rate near @xmath81 . comparison with the corresponding figures presented by cdf@xcite shows that this rise in @xmath77 would be easily observable so that squarks with the masses chosen here could be excluded . we now make this procedure more rigorous in order to determine if the angular distributions can extend the excluded regions listed in table [ bumptab ] . following the procedure used by cdf@xcite , we employ the variable @xmath82 , which is the ratio of the number of dijet events in the two ranges of @xmath74 , for the five mass bins @xmath83 , @xmath84 , @xmath85 , @xmath86 , and @xmath87 gev . this variable has the advantages that it is not very sensitive to variations in the parton densities , to the choice of renormalization scale ( , @xmath56 versus @xmath88 ) , or to next - to - leading order qcd corrections , and that it characterizes the shape of the angular distribution in a mass bin with a single number . we have incorporated the systematic errors , as determined by cdf , as well as the statistical errors in our analysis . the systematic errors are highly correlated , and we have reconstructed the full covariance matrix according to the prescription in ref . @xcite . we then calculate @xmath89 in each @xmath88 bin with @xmath59 and 5 gev for squark masses in the range @xmath90 gev , and perform a fit to the cdf results using their data and correlation matrix . following the usual @xmath91 analysis procedure , we find the minimum value of @xmath91 for a given value of and then determine the excluded range of by examining the @xmath91 distribution as a function of the squark mass . for definiteness we perform a lo calculation taking the scale @xmath92 . our results are presented in fig . 4 for each assumed value of the gluino mass . note that the @xmath91 minima are generally found in the limit of very large squark masses . in all cases the @xmath91 distributions display a similar shape with 5 peaks which are associated with the 5 mass bins used by cdf and are due to the fact that the greatest sensitivity to a squark resonance occurs when it coincides in mass with the lower end of a given bin , , when the squark cross section is maximum . to be more specific , when is light ( @xmath93 gev ) and outside the dijet mass region examined by cdf , the @xmath91 is small but increases as the squark mass gets closer to the edge of lowest mass bin and then peaks once the bin is entered . the sensitivity then decreases as approaches the high end of the mass bin . as the value of rises there is a general loss in sensitivity due to decrease in statistics and the corresponding increase in the size of the errors . this analysis excludes at the @xmath63 c.l . the ranges @xmath94 , @xmath95 , and @xmath96 gev for = 0.4 , 1.3 , and 5 gev , respectively . it thus both extends and complements the constraints obtained from the dijet peak searches . here , we might expect improvements on these constraints for @xmath66 due to the increased enhancement in @xmath97 at @xmath98 . combining these results with the bounds from the resonance searches excludes squark masses in the range @xmath2 gev for gluino masses of @xmath3 gev . in summary , we have examined the constraints on models with light gluinos by using both the cross section and angular distribution for dijet events observed at hadron colliders . the critical observation is that a light gluino can act as a partonic component of the proton thus leading to the resonant production @xmath1 , provided the is sufficiently light . from our analysis , it would appear that the survival of the light gluino case requires either a light in the @xmath99 gev range , or a heavy with @xmath100 gev . from studies of the physics capabilities at run ii of the tevatron@xcite , we anticipate that this future data will be able to exclude or verify this model for squark masses up to @xmath4 tev . high energy hadron colliders may thus provide the best testing ground for this scenario .

turbulent flows in many different physical and engineering applications have a reynolds number so high that a direct numerical simulation of the navier - stokes equations ( dns ) is not feasible . the large - eddy simulation ( les ) is a method in which the large scales of turbulence only are directly solved while the effects of the small - scale motions are modelled . the mass , momentum and energy equations are filtered in space in order to obtain the governing equation for the large scale motions . the momentum and energy transport at the large - scale level due to the unresolved scales is represented by the so - called subgrid terms . standard models for such terms , as , for example , the widely used smagorinsky model , are based on the assumption that the unresolved scales are present in the whole domain and that turbulence is in equilibrium at subgrid scales ( see , e.g. , @xcite ) . this hypothesis can be questionable in free , transitional and highly compressible turbulent flows where subgrid scales , that is fluctuations on a scale smaller than the space filter size , are not simultaneously present in the whole domain . in such situations , subgrid models such as smagorinsky s overestimate the energy flow toward subgrid scales and , from the point of view of the large , resolved , scales , they appear as over - dissipative by exceedingly damping the large - scale motion . for instance , simulation of astrophysical jets could suffer from such limitation . in this regard , any improvement of the les methodology is opportune . astrophysical flows occur in very large sets of spatial scales and velocities , are highly compressible ( mach number up to @xmath1 ) and have a reynolds number which can exceed @xmath2 , so that only the largest scales of the flow can be resolved even by the largest simulation in the foreseeable future . as a consequence , today , in this field , les appears as a feasible simulation methods able to predict the unsteady system behaviour . we have recently proposed a simple method to localize the regions where the flow is underresolved @xcite . the criterion is based on the introduction of a local functional of vorticity and velocity gradients . the regions where the fluctuations are unresolved are located by means of the scalar probe function @xcite which is based on the vortical stretching - tilting sensor : @xmath3 where @xmath4 is the velocity vector , @xmath5 is the vorticity vector and the overbar indicates the statistical average . function ( [ f - dnsfluctuation ] ) is a normalized scalar form of the vortex - stretching term that represents the inertial generation of three dimensional vortical small scales inside the vorticity equation . when the flow is three dimensional and rich in small scales @xmath0 is necessarily different from zero , while , on the other hand , it is instead equal to zero in a two - dimensional vortical flow where the vortical stretching is absent . the mean flow is subtracted from the velocity and vorticity fields in order to consider the fluctuating part of the field only . . the initial velocity field is a laminar parallel flow , with an initial mach number equal to 5 , perturbed by eight waves which have an amplitude equal to 5% of the axis jet velocity and a wavelength from @xmath6 to @xmath7 times @xmath8 . ] of the probability that @xmath9 ( see equation [ f - dnsfluctuation ] ) and thus the probability that subgrid terms are introduced in the selective les balance equation by the localization procedure . the lines represent the points where the longitudinal velocity @xmath10 is constant , where @xmath11 is the jet axis mean velocity . all data in this figure have been computed averaging on lines parallel to the jet axis . a three - dimensional animation which shows the time evolution of the underresolved regions where the subgrid terms are introduced in the selective les can be seen in the supplemental material . ] a priori test of the spatial distribution of functional test have been performed by computing the statistical distribution of @xmath0 in a fully resolved turbulent fluctuation field ( dns of a homogeneous and isotropic turbulent flow ( @xmath12 , @xmath13 , data from @xcite ) ) and in some unresolved instances obtained by filtering this dns field on coarser grids ( from @xmath14 to @xmath15 ) . it has been shown @xcite that the probability that @xmath0 assumes values larger than a given threshold @xmath16 is always higher in the filtered fields and increases when the resolution is reduced . the difference between the probabilities in fully resolved and in filtered turbulence is maximum when @xmath16 is in the range @xmath17 $ ] for all resolutions . in such a range the probability @xmath18 that @xmath0 is larger than @xmath16 in the less resolved field is about twice the probability in the dns field . furthermore , beyond this range this probability normalized over that of resolved dns fields it is gradually increasing becoming infinitely larger . from that it is possible to introduce a threshold @xmath16 on the values of @xmath0 , such that , when @xmath0 assumes larger values the field could be considered locally unresolved and should benefit from the local activation of the large eddy simulation method ( les ) by inserting a subgrid scale term in the motion equation . the values of this threshold is arbitrary , as there is no sharp cut , but it can be reasonably chosen as the one which gives the maximum difference between the probability @xmath18 in the resolved and unresolved fields . this leads to @xmath19 . furthermore , it should be noted that the morkovin hypothesis , stating that the compressibility effects do not have much influence on the turbulence dynamics , apart from varying the local fluid properties @xcite , allows to apply the same value of the threshold in compressible and incompressible flows . such value of the threshold has been used to investigate the presence of regions with anomalously high values of the functional @xmath0 , by performing a set of a priori tests on existing euler simulations of the temporal evolution of a perturbed cylindrical hypersonic light jet with an initial mach number equal to 5 and ten times lighter than the surrounding external ambient @xcite . when the effect of the introduction of subgrid scale terms in the transport equation is extrapolated from those a priori tests , they positively compare with experimental results and show the convenience of the use of such a procedure @xcite . in this paper we present large - eddy simulations of this temporal evolving jet , where the subgrid terms are selectively introduced in the transport equations by means of the local stretching criterion @xcite . the aim is not to model a specific jet , but instead to understand , from a physical point of view , the differences introduced by the presence of sub - grid terms in the under - resolved simulations of hypersonic jets . our localization procedure selects the regions where subgrid terms are applied and , as such , its effect could be considered equivalent to a model coefficient modulation , as the one obtained by the dynamic procedure @xcite or by the use of improved eddy viscosity smagorinsky - like models like vreman s model @xcite , which gives a low eddy viscosity in non turbulent regions of the flow . however , it operates differently because it is completely uncoupled from the subgrid scale model used as , unlike the common practical implementations of the dynamic procedure , does not require ensemble averaging to prevent unstable eddy viscosity . other alternatives , such as the approximate deconvolution model @xcite , are more complicated than the present selective procedure because involve filter inversion and the use of a dynamic relaxation term . the computational overload of the selective filtering is modest and can make les an affordable alternative to a higher resolution inviscid simulation : the selective les increases the computing time of about one - third with respect to an euler simulation , while the doubling of the resolution can increase the computing time by a factor of sixteen . c + c + c + we have simulated the temporal evolution of a three dimensional jet in a parallelepiped domain with periodicity conditions along the longitudinal direction . the flow is governed by the ideal fluid equations ( mono - atomic gas flow ) for mass , momentum , and energy conservation . the beam is considered thermally confined by the external medium , and the initial pressure is set uniform in the entire domain . in the astrophysical context , this formulation is usually considered to approximate the temporal hydrodynamic evolution inside a spatial window of interstellar jets , which are highly compressible collimated jets characterized by reynolds numbers of the order @xmath20 . see for example , the herbig - haro jets hh24 , hh34 and hh47 @xcite . we do not consider the effect of the radiative cooling , which can change the jet dynamics substantially ( see , e.g. @xcite ) . the transient evolution includes basically two principal mechanism , the growth and evolution of internal shocks and the dynamics of the mixing process originated by the nonlinear development of the kelvin helmholtz instability . the analysis is carried out through hydro - dynamical simulations by considering only a fraction of the beam which is far from its base and head . due to the use of the periodic boundary conditions , the jet material is continually processed by the earlier evolution because of the multiple transits though the computational domain . in this way , the focus is put on the instability evolution and on the interaction between the jet and the external medium , rather than an analysis of the global evolution of the jet . it is known that the numerical solution of a system of ideal conservation laws ( such as the euler equations ) actually produces the equivalent solution of another modified system with additional diffusion terms . with the discretizations used in this study it possible to estimate _ a posteriori _ that the numerical viscosity implies an actual reynolds number of about @xmath21 . in such a situation it is clear that the addition into the governing equations of the diffusive - dissipative terms relevant to a reynolds number in the range @xmath20 would be meaningless . the formulation used is thus the following : @xmath22 & = & { \frac{\partial } { \partial x_i}}h(f_{\rm les}-t_\omega)q_i^{sgs}\nonumber\\ & & \label{eq.ene } \ ] ] where the field variables @xmath23 , @xmath24 and @xmath25 and @xmath26 are the filtered pressure , density , velocity , and total energy respectively . the ratio of specific heats @xmath27 is equal to @xmath28 . here @xmath29 and @xmath30 are the subgrid stress tensor and total enthalpy flow , respectively . function @xmath31 is the heaviside step function , thus the subgrid scale fluxes are applied only in the regions where @xmath32 . the threshold @xmath16 is here taken equal to 0.4 , which is the value for which the maximum difference between the probability density function @xmath33 between the filtered and unfiltered turbulence was observed @xcite . sensor @xmath0 , as defined in ( [ f - dnsfluctuation ] ) , does not depend on the subgrid model used and on the kind of discretization used to actually solve the filtered transport equations . in principle , it can be coupled with any subgrid model and any numerical scheme . we have chosen to implement the standard smagorinsky model as subgrid model , @xmath34 @xmath35 where @xmath36 is the rate of strain tensor and @xmath37 its norm . constant @xmath38 has been set equal to 0.1 , which is the standard value used in the les of shear flows , and @xmath39 , the turbulent prandtl number , is taken equal to 1 . the initial flow configuration is an axially symmetric cylindrical jet in a parallelepiped domain , described by a cartesian coordinate system @xmath40 . the initial jet velocity is along the @xmath41-direction ; its symmetry axis is defined by @xmath42 . the interface between the jet and the surrounding ambient medium is described by a smooth velocity and density transition in order to avoid the spurious oscillations that can be introduced by a sharp discontinuity . the longitudinal velocity profile is thus initialized as @xmath43 where @xmath44 is the distance from the jet axis , @xmath45 is the jet radius and @xmath11 the jet velocity . @xmath46 is a smoothing parameter which has been set equal to 4 . the same smoothing has been used for the initial density distribution , @xmath47 where @xmath48 is the initial density inside the jet ambient and @xmath49 is the ratio between the ambient density at infinity to that of on the jet axis . a value of @xmath49 larger than one implies that the jet is lighter than the external medium . the mean pressure is set to a uniform value @xmath50 , that is , we are considering a situation where there is initially a pressure equilibrium between the jet and the surrounding environment . this initial mean profile is perturbed at @xmath51 by adding longitudinal disturbances on the transversal velocity components whose amplitude is 5% of the jet velocity and whose wavenumber is up to eight times the fundamental wavenumber @xmath52 , @xmath53 @xmath54 with @xmath55 random phase shifts , so that even the perturbation with the shortest wavelength is , initially , fully resolved . the integration domain is @xmath56 , @xmath57 and @xmath58 , with @xmath59 and @xmath60 . we have used periodic boundary conditions in the longitudinal @xmath41 direction , while free flow conditions are used in the lateral directions . a scheme of the initial flow configuration used in the simulations is shown in figure [ fig.schema ] . as function of the distance @xmath61 from the axis of the jet . all averages have been computed as space averages on cylinders at constant @xmath62.,title="fig : " ] + as function of the distance @xmath61 from the axis of the jet . all averages have been computed as space averages on cylinders at constant @xmath62.,title="fig:"]-1.0 mm in the following , all data have been made dimensionless by expressing lengths in units of the initial jet radius @xmath45 , times in units of the sound crossing time of the radius @xmath63 , where @xmath64 is the reference sound velocity of the initial conditions , velocities in units of @xmath65 ( thus dimensionless velocities coincide with the initial mach number ) , densities in units of @xmath48 and pressures in units of @xmath50 . equations ( [ eq.cont]-[eq.ene ] ) have been solved , in cartesian geometry , using an extension of the pluto code @xcite , which is a godunov - type code that supplies a series of high - resolution shock - capturing schemes @xcite that are particularly suitable for the present application , because of their low numerical dissipation . in fact , as pointed out by @xcite , a high numerical viscosity can overwhelms the subgrid - scale terms effects . the code has been extended by adding the subgrid fluxes and the computation of the functional @xmath0 which allows to perform the selective large - eddy simulation . for this application , a third order accurate in space and second order in time piecewise - parabolic - method ( ppm ) has been chosen . we have performed three simulations of a jet with an initial mach number equal to 5 and a density ratio @xmath49 equal to 10 . the density ratio is an important parameter in such flow configuration , as it has been shown that it has a strong influence on the temporal evolution and on the flow entrainment as it has been shown by numerical simulations and laboratory experiments @xcite . the selective les of the jet has been carried out on a @xmath66 uniform grid . a uniform grids avoids the need to cope with the non - commutation terms in the governing equations ( see , e.g. @xcite ) . moreover , three additional simulations were performed for comparison : a standard non selective les where the subgrid model was introduced in the whole domain , which is obtained by forcing @xmath67 in equations ( [ eq.cont]-[eq.ene ] ) , and two euler simulations , which formally can be obtained by putting @xmath68 , one with the same resolution of the large eddy simulations and one which uses a finer grid ( @xmath69 ) . [ cols="^,^ " , ] , defined as the distance between the jet axis and the position where the normalized mean velocity @xmath70 is equal to 0.5 ; ( b ) density thickness @xmath71 , defined as the distance between the jet axis and the position where the mean density is equal to the average between the jet axis density and the external ambient density.,title="fig : " ] + , defined as the distance between the jet axis and the position where the normalized mean velocity @xmath70 is equal to 0.5 ; ( b ) density thickness @xmath71 , defined as the distance between the jet axis and the position where the mean density is equal to the average between the jet axis density and the external ambient density.,title="fig : " ] - in this work we show that the _ selective _ large eddy simulation , which is based on the use of a scalar probe function @xmath0 a function of the magnitude of the local stretching - tilting term of the vorticity equation can be conveniently applied to the simulation of time evolving compressible jets . in the present simulation , the probe function @xmath0 has been coupled with the standard smagorinsky sub - grid model . however , it should be noted that the probe function @xmath0 can be used together with any model because @xmath0 simply acts as an independent switch for the introduction of a sub - grid model . the main results is that even a simple model can give acceptable results when selectively used together with a sub - grid scale localization procedure . in fact , the comparison among the four kinds of simulations ( selective les , standard les , low and high resolution pseudo euler direct numerical simulations ) here carried out shows that this method can improve the dynamical properties of the simulated field . in particular , the selective les hugely improves the spectral distribution of energy and density over the resolved scales , the enstrophy radial distribution and the mean velocity ( up to the 200% ) and density profiles ( up to the 100% ) with respect to the standard les . furthermore , this method avoids the artificial over - damping of the unstable modes at the jet border which in the standard large eddy simulation inhibits the jet lateral growth . in comparison with an euler simulation which uses the same resolution , the selective les clearly improves the flow prediction when the field is reach in small scales ( up to the 50% on the momentum and 4% on the density fields ) . if , as in the example here shown , the kinetic energy in the small scales is not steady in the mean and decays , the improvement due to the use of the selective les in the long term reduces . thus , in flow simulations where the small scales are transient in time these two methods asymptotically offer same results . and internal energy @xmath72 in the computational domain , high resolution pseudo - dns simulation . the kinetic energy has been decomposed in the sum of the energy of the mean flow @xmath73 and of the energy of the fluctuations @xmath74 . the tilde denotes density weighted favre average : @xmath75 . all values have been normalized by the initial energy @xmath76 . the evolution of the kinetic energy of the fluctuations determines the extension of the underresolved regions where subgrid terms must be introduced in the governing equations , see the movie in the supplementary material visible online . ] in synthesis , the selective les explicitly introduces the sub - grid flows of momentum and energy in the governing equations in the regions of the flow where turbulence is physically present . in this way , one does not rely on the numerical diffusion to mimic the overall behaviour of all unresolved scales . this is a positive feature , since the numerical diffusion depends on the algorithm used and on the grid spacing and can not be conveniently controlled . the computing time of the selective les is about one third larger than that of the low resolution euler simulation and seven times smaller than the one of the higher resolution euler simulation . therefore , a selective les could be more convenient than a better resolved euler simulation . because of this properties , given the modest computational burden brought to the simulation , the application of the selective procedure to the simulation of complex flows in particular highly compressible free flows as , for instance , astrophysical jets seems promising . 99 d.tordella , m.iovieno , s.massaglia , `` small scale localization in turbulent flows . a priori tests applied to a possible large eddy simulation of compressible turbulent flows '' , _ comp . phys . comm . _ * 176*(8 ) , 539549 ( 2007 ) . d.k.lilly `` the representation of small - scale turbulence in numerical simulation experiments '' , _ proc . ibm scientific computing symp . on environmental sciences _ , yorktown heights , new york , ed . goldstine , ibm form no . 3201951 ( 1967 ) . s.stolz , n.a.adams , l.kleiser , `` the approximate deconvolution model for the large - eddy simulation of compressible flows and its application to shock - turbulent boundary layer interaction '' , _ phys.fluids _ * 10 * , 29853001 , ( 2001 ) . b.reipurth , j.bally , `` herbig - haro flows : probes of early stellar evolution '' , _ annual review of astronomy and astrophysics _ * 39 * , 403455 ( 2001 ) . j.m.stone , p.e.hardee and j.xu , `` the stability of radiatively cooled jets in three dimensions '' , _ astrophysical journal _ * 543*(1 ) , 161167 ( 1997 ) . p.rossi , g.bodo , s.massaglia , a.ferrari , `` evolution of kelvin - helmholtz instabilities in radiative jets .2 . shock structure and entrainment properties '' , _ astronomy and astrophysics _ * 321*(2 ) , 672684 ( 1997 ) . a.mignone , g.bodo , s.massaglia , t.matsakos , o.tesileanu , c.zanni and a.ferrari , `` pluto : a numerical code for computational astrophysics '' , _ astr . j. supplement series _ * 170*(1 ) , 228242 ( 2007 ) , and http://plutocode.to.astro.it . m.iovieno , d.tordella , `` variable scale filtered navier - stokes equations : a new procedure to deal with the associated commutation error '' , _ phys . fluids _ * 15*(7 ) , 19261936 ( 2003 ) . m. micono , g. bodo , s. massaglia , p. rossi , a. ferrari , r. rosner , `` kelvin - helmholtz instabilities in three dimensional radiative jets '' , _ astronomy and astrophys . _ , * 360 * , 795808 ( 2000 ) . g.bodo , s.massaglia , p.rossi , r.rosner , a.malagoli , a.ferrari , `` the long - term evolution and mixing properties of high mach number hydrodynamic jets '' , _ astronomy and astrophysics _ * 303*(1 ) , 281298 ( 1995 ) .

a new method for the localization of the regions where small scale turbulent fluctuations are present in hypersonic flows is applied to the large - eddy simulation ( les ) of a compressible turbulent jet with an initial mach number equal to 5 . the localization method used is called selective les and is based on the exploitation of a scalar probe function @xmath0 which represents the magnitude of the _ stretching - tilting _ term of the vorticity equation normalized with the enstrophy @xcite . for a fully developed turbulent field of fluctuations , statistical analysis shows that the probability that @xmath0 is larger than 2 is almost zero , and , for any given threshold , it is larger if the flow is under - resolved . by computing the spatial field of @xmath0 in each instantaneous realization of the simulation it is possible to locate the regions where the magnitude of the normalized vortical stretching - tilting is anomalously high . the sub - grid model is then introduced into the governing equations in such regions only . the results of the selective les simulation are compared with those of a standard les , where the sub - grid terms are used in the whole domain , and with those of a standard euler simulation with the same resolution . the comparison is carried out by assuming as reference field a higher resolution euler simulation of the same jet . it is shown that the _ selective _ les modifies the dynamic properties of the flow to a lesser extent with respect to the classical les . in particular , the prediction of the enstrophy , mean velocity and density distributions and of the energy and density spectra are substantially improved . small scale , turbulence , localization , large - eddy simulation , astrophysical jets 47.27.ep , 47.27.wg , 47.40.ki , 97.21.+a , 98.38.fs

for the description of pll - based circuits a physical model in the signals space and a mathematical model in the signal s phase space are used @xcite . the equations describing the model of pll - based circuits in the signals space are difficult for the study , since that equations are nonautonomous ( see , e.g. , @xcite ) . by contrast , the equations of model in the signal s phase space are autonomous @xcite , what simplifies the study of pll - based circuits . the application of averaging methods @xcite allows one to reduce the model of pll - based circuits in the signals space to the model in the signal s phase space ( see , e.g. , @xcite . consider a model of pll - based circuits in the signal s phase space ( see fig . [ ris : pllbased ] ) . a reference oscillator ( input ) and a voltage - controlled oscillator ( vco ) generate phases @xmath0 and @xmath1 , respectively . the frequency of reference signal usually assumed to be constant : @xmath2 the phases @xmath0 and @xmath1 enter the inputs of the phase detector ( pd ) . the output of the phase detector in the signal s phase space is called a phase detector characteristic and has the form @xmath3 the maximum absolute value of pd output @xmath4 is called a phase detector gain ( see , e.g. , @xcite ) . the periodic function @xmath5 depends on difference @xmath6 ( which is called a phase error and denoted by @xmath7 ) . the pd characteristic depends on the design of pll - based circuit and the signal waveforms @xmath8 of input and @xmath9 of vco . in the present work a sinusoidal pd characteristic with @xmath10 is considered ( which corresponds , e.g. , to the classical pll with @xmath11 and @xmath12 ) . the output of phase detector is processed by filter . further we consider the active pi filter ( see , e.g. , @xcite ) with transfer function @xmath13 , @xmath14 , @xmath15 . the considered filter can be described as @xmath16 where @xmath17 is the filter state . the output of filter @xmath18 is used as a control signal for vco : @xmath19 where @xmath20 is the vco free - running frequency and @xmath21 is the vco gain coefficient . relations ( [ eq : input ] ) , ( [ eq : filterorig ] ) , and ( [ eq : vco ] ) result in autonomous system of differential equations @xmath22 denote the difference of the reference frequency and the vco free - running frequency @xmath23 by @xmath24 . by the linear transformation @xmath25 we have @xmath26 where @xmath27 is the loop gain . for signal waveforms listed in table [ table : pdgains ] , relations ( [ sys : pllsys ] ) describe the models of the classical pll and two - phase pll in the signal s phase space . the models of classical costas loop and two - phase costas loop in the signal s phase space can be described by relations similar to ( [ sys : pllsys ] ) ( pd characteristic of the circuits usually is a @xmath28-periodic function , and the approaches presented in this paper can be applied to these circuits as well ) ( see , e.g. , @xcite ) . + @xmath29 & + @xmath30 & + @xmath29 & + @xmath31 & + @xmath32 , \\ 1-\frac{2}{\pi}\theta_1 , \theta_1 \in \left[\pi ; 2\pi \right ] \end{cases}$ ] & + @xmath33 & + + @xmath34 & + @xmath30 & + by the transformation @xmath35 ( [ sys : pllsys ] ) is not changed . this property allows one to use the concept of frequency deviation @xmath36 and consider ( [ sys : pllsys ] ) with @xmath37 only . the state of pll - based circuits for which the vco frequency is adjusted to the reference frequency of input is called a locked state . the locked states correspond to the locally asymptotically stable equilibria of ( [ sys : pllsys ] ) , which can be found from the relations @xmath38 here @xmath39 depends on @xmath24 and further is denoted by @xmath40 . since ( [ sys : pllsys ] ) is @xmath41-periodic in @xmath42 , we can consider ( [ sys : pllsys ] ) in a @xmath41-interval of @xmath42 , @xmath43 $ ] . in interval @xmath43 $ ] there exist two equilibria : @xmath44 and @xmath45 . to define type of the equilibria let us write out corresponding characteristic polynomials and find the eigenvalues : @xmath46 + @xmath47 denote the stable equilibrium as @xmath48 and the unstable equilibrium as @xmath49 thus , for any arbitrary @xmath24 the equilibria @xmath50 are locally asymptotically stable . hence , the locked states of ( [ sys : pllsys ] ) are given by equilibria @xmath51 . the remaining equilibria @xmath52 are unstable saddle equilibria . in order to consider the lock - in range of pll - based circuits let us discuss the global asymptotic stability . _ if for a certain @xmath24 any solution of ( [ sys : pllsys ] ) tends to an equilibrium , then the system with such @xmath24 is called globally asymptotically stable _ ( see , e.g. , @xcite ) . to prove the global asymptotic stability of ( [ sys : pllsys ] ) two approaches can be applied : the phase plane analysis @xcite and construction of the lyapunov functions @xcite . by methods of the phase plane analysis , in @xcite the global asymptotic stability of ( [ sys : pllsys ] ) for any @xmath24 is stated . however , to complete rigorously the proof given in @xcite , the additional explanations are required ( i.e. , the absence of heteroclinic trajectory and limit cycles of the first kind ( see fig . [ ris : cyclesphaseplane ] ) is needed to be explained ; e.g. , for the case of lead - lag filter a number of works @xcite is devoted to the study of these periodic trajectories ) . to overcome these difficulties , the methods of the lyapunov functions construction can be applied . the modifications of the classical global stability criteria for cylindrical phase space are developed in @xcite . the global asymptotic stability of ( [ sys : pllsys ] ) for any @xmath24 can be using the lyapunov function @xmath53 since the considered model of pll - based circuits in the signal s phase space is globally asymptotically stable , it achieves locked state for any initial vco phase @xmath54 and filter state @xmath55 . however , the phase error @xmath42 may substantially increase during the acquisition process . in order to consider the property of the model to synchronize without undesired growth of the phase error @xmath42 , a lock - in range concept was introduced in @xcite : `` _ if , for some reason , the frequency difference between input and vco is less than the loop bandwidth , the loop will lock up almost instantaneously without slipping cycles . the maximum frequency difference for which this fast acquisition is possible is called the lock - in frequency _ '' . the lock - in range concept is widely used in engineering literature on the pll - based circuits study ( see , e.g. , @xcite ) . remark , that it is said that cycle slipping occurs if ( see , e.g. , @xcite ) @xmath56 for ( [ sys : pllsys ] ) with fixed @xmath24 a domain of loop states for which the synchronization without cycle slipping occurs is called the lock - in domain @xmath57 ( see fig . [ ris : cycleslipping ] ) . however , in general , even for zero frequency deviation ( @xmath58 ) and a sufficiently large initial state of filter ( @xmath55 ) , cycle slipping may take place , thus in 1979 gardner wrote : _ `` there is no natural way to define exactly any unique lock - in frequency '' _ and _ `` despite its vague reality , lock - in range is a useful concept '' _ @xcite . to overcome the stated problem , in @xcite the rigorous mathematical definition of a lock - in range is suggested : @xcite _ the lock - in range of model ( [ sys : pllsys ] ) is a range @xmath59 such that for each frequency deviation @xmath60 the model ( [ sys : pllsys ] ) is globally asymptotically stable and the following domain @xmath61 contains all corresponding equilibria @xmath62 [ def : lockin ] _ for model ( [ sys : pllsys ] ) each lock - in domain from intersection @xmath63 is bounded by the separatrices of saddle equilibria @xmath64 and vertical lines @xmath65 . thus , the behavior of separatrices on the phase plane is the key to the lock - in range study ( see fig . [ ris : lockinphaseplane ] ) . consider an approach to the lock - in range computation of ( [ sys : pllsys ] ) , based on the phase plane analysis . to compute the lock - in range of ( [ sys : pllsys ] ) we need to consider the behavior of the lower separatrix @xmath66 , which tends to the saddle point @xmath67 as @xmath68 ( by the symmetry of the lower and the upper half - planes , the consideration of the upper separatrix is also possible ) . the parameter @xmath24 shifts the phase plane vertically . to check this , we use a linear transformation @xmath69 . thus , to compute the lock - in range of ( [ sys : pllsys ] ) , we need to find @xmath70 ( where @xmath71 is called a lock - in frequency ) such that ( see fig . [ ris : lockinphaseplane ] ) @xmath72 of ( [ sys : pllsys ] ) . ] by ( [ eq : lockinrelation ] ) , we obtain an exact formula for the lock - in frequency @xmath71 : @xmath73 for various @xmath74 , @xmath75 , @xmath76.,scaledwidth=90.0% ] numerical simulations are used to compute the lock - in range of ( [ sys : pllsys ] ) applying ( [ eq : lockinrelationnew ] ) . the separatrix @xmath77 is numerically integrated and the corresponding @xmath71 is approximated . the obtained numerical results can be illustrated by a diagram ( see fig . [ ris : lock - indiagram ] ) . note that ( [ sys : pllsys ] ) depends on the value of two coefficients @xmath78 and @xmath76 . in fig . [ ris : lock - indiagram ] , choosing x - axis as @xmath78 , we can plot a single curve for every fixed value of @xmath76 . the results of numerical simulations show that for sufficiently large @xmath78 , the value of @xmath71 grows almost proportionally to @xmath78 . hence , @xmath79 is almost constant for sufficiently large @xmath78 and in fig . [ ris : lock - indiagram ] the y - axis can be chosen as @xmath80 . to obtain the lock - in frequency @xmath71 for fixed @xmath75 , @xmath76 , and @xmath74 using fig . [ ris : lock - indiagram ] , we consider the curve corresponding to the chosen @xmath76 . next , for x - value equal @xmath78 we get the y - value of the curve . finally , we multiply the y - value by @xmath78 ( see fig . [ ris : lock - indiagramexample ] ) . .,scaledwidth=90.0% ] consider an analytical approach to the lock - in range estimation . main stages of the approach are presented in subsection [ subsec : analyticalapproach ] . consider an active pi filter with small parameter @xmath81 ( see , e.g. , @xcite ) . the consideration of ( [ sys : pllsys ] ) with such active pi filter allows us to estimate the lower separatrix @xmath77 and the lock - in range . for this purpose the approximations of separatrix @xmath77 in interval @xmath82 are used . the separatrix @xmath77 , which is a solution of ( [ sys : pllsys ] ) , can be expanded in a taylor series in variable @xmath83 ( since the parameter @xmath83 is considered as a variable , the separatrix @xmath84 depends on it ) . the first - order approximation of the lower separatrix @xmath85 has the form @xmath86 the second - order approximation of @xmath85 has the form @xmath87 for approximations ( [ sys : sfirstapprox ] ) , ( [ sys : ssecondapprox ] ) of separatrix @xmath85 the following relations are valid : @xmath88 for @xmath89 the relations ( [ sys : sfirstapprox ] ) , ( [ sys : ssecondapprox ] ) take the following values : @xmath90 using relation ( [ eq : lockinrelationnew ] ) the lock - in frequency @xmath71 is approximated as follows : @xmath91 @xmath92 for various @xmath74 , @xmath75.,scaledwidth=70.0% ] for fixed @xmath93 the three curves are shown in fig . [ ris : approxexample ] . the values of @xmath71 ( the blue curve , which is obtained numerically using relation ( [ eq : lockinrelationnew ] ) ) are estimated from below by ( [ eq : lockinfirstapprox ] ) and from above by ( [ eq : lockinsecondapprox ] ) ( the red and green curves correspondingly ) . since the lock - in frequency @xmath71 is approximated under the condition of small parameter @xmath83 , the estimates ( [ eq : lockinfirstapprox ] ) and ( [ eq : lockinsecondapprox ] ) give less precise result in the case of large @xmath94 . an another characteristic related to the cycle slipping effect is the pull - out frequency @xmath95 ( see , e.g. , @xcite . in @xcite the pull - out frequency is defined as a frequency - step limit , _ `` below which the loop does not skip cycles but remains in lock''_. however , in general case of filter ( see , e.g. , @xcite ) the pull - out frequency may depend on the value of @xmath24 . however , in the case of active pi filter , the pull - out frequency can be defined and approximated ( see , e.g. , @xcite ) , since the parameter @xmath24 only shifts the phase plane vertically . the pull - out frequency can be found as follows ( see fig . [ ris : pulloutphaseplane ] ) : @xmath96 ) equals to pull - out frequency @xmath97 . ] in fig . [ ris : comparison ] the estimates from @xcite are compared with estimates based on ( [ eq : lockinfirstapprox ] ) and ( [ eq : lockinsecondapprox ] ) . the pull - out frequency estimate , which is obtained according to fig . [ ris : lock - indiagram ] and ( [ eq : pulloutrelation ] ) , is drawn in blue color . analytical estimates based on ( [ eq : lockinfirstapprox ] ) , ( [ eq : lockinsecondapprox ] ) , and ( [ eq : pulloutrelation ] ) are drawn in red and green colors correspondingly . the black curve is the estimate of the pull - out frequency from @xcite . the dashed curve corresponds to the empirical estimate @xmath98 presented in @xcite . for @xmath94 not very large the relation ( [ eq : lockinsecondapprox ] ) is the most precise estimate compared to the presented ones . in the present work models of the pll - based circuits in the signal s phase space are described . the lock - in range of pll - based circuits with sinusoidal pd characteristic and active pi filter is considered . the rigorous definition of the lock - in range is discussed , and relation ( [ eq : lockinrelationnew ] ) for the lock - in range computation is derived . for the lock - in range estimation two approaches numerical and analytical are presented . the methods are based on the integration of phase trajectories . in subsection [ subsec : analyticalapproach ] the numerical estimates are verified by analytical estimates , which are obtained under the condition of small parameter . let us write out ( [ sys : pllsys ] ) in a different form with @xmath99 and @xmath100 : @xmath101 in virtue of @xmath41-periodicity of ( [ eq : apppllsys ] ) in variable @xmath42 , phase trajectories of ( [ eq : apppllsys ] ) coincides for each interval @xmath104 $ ] , @xmath105 . thus , one can study ( [ eq : apppllsys ] ) in interval @xmath43 $ ] only . in interval @xmath43 $ ] there exist two equilibria @xmath107 and @xmath108 . to define type of the equilibria points let us write out corresponding characteristic polynomials and find the eigenvalues : @xmath109 + @xmath110 thus , equilibrium @xmath111 is a stable node , a stable degenerated node , or a stable focus ( that depends on the sign of @xmath112 ) . equilibrium @xmath113 is a saddle point for all @xmath114 , @xmath115 , @xmath116 . moreover , in virtue of periodicity each equilibrium @xmath117 is a saddle point , and each equilibrium @xmath118 is a stable equilibrium of the same type as @xmath111 . note also that equilibria @xmath119 of ( [ eq : apppllsys ] ) and corresponding equilibria @xmath120 of ( [ app : pllsysorig ] ) are of the same type , and related as follows : @xmath121 let us consider the following differential equation : @xmath122 the right side of equation ( [ eq : plleqphaseplane ] ) is discontinuous in each point of line @xmath123 . this line is an isocline line of vertical angular inclination of ( [ eq : plleqphaseplane ] ) @xcite . equation ( [ eq : plleqphaseplane ] ) is equivalent to ( [ eq : apppllsys ] ) in the upper and the lower open half planes of the phase plane . let the solutions @xmath124 of equation ( [ eq : plleqphaseplane ] ) be considered as functions of two variables @xmath42 , @xmath125 . consider the solution of differential equation ( [ eq : plleqphaseplane ] ) , which range of values lies in the upper open half plane of its phase plane . right side of equation ( [ eq : plleqphaseplane ] ) in the upper open half plane is function of class @xmath126 for @xmath127 arbitrary large . solutions of the cauchy problem with initial conditions @xmath128 , @xmath129 ( which solutions are on the upper half plane ) are also of class @xmath126 on their domain of existence for @xmath127 arbitrary large @xcite . let us study the separatrix @xmath130 in interval @xmath131 , which tends to saddle point @xmath132 and is situated in its second quadrant . separatrix @xmath130 is the solution of the corresponding cauchy problem for equation ( [ eq : plleqphaseplane ] ) . the separatrix @xmath130 is of class @xmath126 on its domain of existence for @xmath127 arbitrary large . consider separatrix @xmath130 as a taylor series in variable @xmath125 in the neighborhood of @xmath133 : @xmath134 let us denote @xmath135 @xmath136 as the @xmath137-th approximation of @xmath130 in variable @xmath125 : @xmath138 the taylor remainder is denoted as follows : @xmath139 for the convergent taylor series its remainder @xmath140 for each point @xmath141 of interval @xmath131 . separatrix @xmath130 satisfies the following relation , which follows from ( [ eq : plleqphaseplane ] ) : @xmath142 @xmath143 @xmath144 let us represent @xmath130 as taylor series ( [ eq : tailors ] ) in relation ( [ eq : sineqint ] ) . @xmath145 @xmath146 @xmath147 let us write out the corresponding members of ( [ eq : taylorineq ] ) for each @xmath148 , @xmath149 . + for @xmath150 : @xmath151 for @xmath152 : @xmath153 for @xmath154 : @xmath155 let us consequently find @xmath156 , @xmath157 , @xmath158 using relations ( [ eq : approxeps0 ] ) , ( [ eq : approxeps1 ] ) and ( [ eq : approxeps2 ] ) . begin with evaluation of @xmath156 : @xmath159 according to ( [ rel : approxeps0 ] ) @xmath160 using equation ( [ eq : approxeps1 ] ) and relations ( [ rel : approxeps0 ] ) evaluate @xmath157 : @xmath161 @xmath162 let us evaluate the integral @xmath163 in the interval @xmath164 using the following substitutions : @xmath165 hence , an expression for @xmath157 in interval @xmath164 is obtained : @xmath167 moreover , @xmath168 to shorten the further evaluation of @xmath169 , write out @xmath170 in equivalent form ( in interval @xmath164 ) . @xmath171 @xmath172 @xmath173 @xmath174 hence , @xmath187 , @xmath170 , @xmath169 are evaluated ( equations ( [ rel : approxeps0 ] ) , ( [ rel : approxeps1 ] ) and ( [ rel : approxeps2 ] ) , correspondingly ) . i. e. the first and the second approximations @xmath188 , @xmath189 of separatrix @xmath130 are found . furthermore , using ( [ rel0:approxeps0 ] ) , ( [ rel0:approxeps1 ] ) and ( [ rel0:approxeps2 ] ) the following relations are valid : @xmath190 this work was supported by the russian scientific foundation and saint - petersburg state university . the authors would like to thank roland e. best , the founder of the best engineering company , oberwil , switzerland and the author of the bestseller on pll - based circuits @xcite for valuable discussion . alexandrov , n.v . kuznetsov , g.a . leonov , and s.m . best s conjecture on pull - in range of two - phase costas loop . in _ 2014 6th international congress on ultra modern telecommunications and control systems and workshops ( icumt ) _ , volume 2015-january , pages 7882 . ieee , 2014 . doi : 10.1109/icumt.2014.7002082 . best , n.v . kuznetsov , g.a . leonov , m.v . yuldashev , and r.v . simulation of analog costas loop circuits . _ international journal of automation and computing _ , 110 ( 6):0 571579 , 2014 . 10.1007/s11633 - 014 - 0846-x . best , n.v . kuznetsov , o.a . kuznetsova , g.a . leonov , m.v . yuldashev , and r.v . yuldashev . a short survey on nonlinear models of the classic costas loop : rigorous derivation and limitations of the classic analysis . in _ proceedings of the american control conference _ , pages 12961302 . ieee , 2015 . doi : 10.1109/acc.2015.7170912 . 7170912 , http://arxiv.org/pdf/1505.04288v1.pdf . gelig , g.a . leonov , and v.a . _ stability of nonlinear systems with nonunique equilibrium ( in russian)_. nauka , 1978 . ( english transl : stability of stationary sets in control systems with discontinuous nonlinearities , 2004 , world scientific ) . kuznetsov , o.a . kuznetsova , g.a . leonov , p. neittaanmaki , m.v . yuldashev , and r.v . . limitations of the classical phase - locked loop analysis . _ proceedings - ieee international symposium on circuits and systems _ , 2015-july:0 533536 , 2015 . doi : http://dx.doi.org/10.1109/iscas.2015.7168688 . kuznetsov , g.a . leonov , s.m . seledzgi , m.v . yuldashev , and r.v . elegant analytic computation of phase detector characteristic for non - sinusoidal signals . _ ifac - papersonline _ , 480 ( 11):0 960963 , 2015 . doi : http://dx.doi.org/10.1016/j.ifacol.2015.09.316 . kuznetsov , g.a . leonov , m.v . yuldashev , and r.v . rigorous mathematical definitions of the hold - in and pull - in ranges for phase - locked loops . _ ifac - papersonline _ , 480 ( 11):0 710713 , 2015 . doi : http://dx.doi.org/10.1016/j.ifacol.2015.09.272 . leonov , n.v . kuznetsov , m.v . yuldahsev , and r.v . analytical method for computation of phase - detector characteristic . _ ieee transactions on circuits and systems - ii : express briefs _ , 590 ( 10):0 633647 , 2012 . doi : 10.1109/tcsii.2012.2213362 . leonov , n.v . kuznetsov , m.v . yuldashev , and r.v . nonlinear dynamical model of costas loop and an approach to the analysis of its stability in the large . _ signal processing _ , 108:0 124135 , 2015 . doi : 10.1016/j.sigpro.2014.08.033 . leonov , n.v . kuznetsov , m.v . yuldashev , and r.v . hold - in , pull - in , and lock - in ranges of pll circuits : rigorous mathematical definitions and limitations of classical theory . _ ieee transactions on circuits and systems i : regular papers _ , 620 ( 10):0 24542464 , 2015 . doi : http://dx.doi.org/10.1109/tcsi.2015.2476295 . v. smirnova , a. proskurnikov , and n. utina . problem of cycle - slipping for infinite dimensional systems with mimo nonlinearities . in _ ultra modern telecommunications and control systems and workshops ( icumt ) , 2014 6th international congress on _ , pages 590595 . ieee , 2014 .

in the present work pll - based circuits with sinusoidal phase detector characteristic and active proportionally - integrating ( pi ) filter are considered . the notion of lock - in range an important characteristic of pll - based circuits , which corresponds to the synchronization without cycle slipping , is studied . for the lock - in range a rigorous mathematical definition is discussed . numerical and analytical estimates for the lock - in range are obtained . phase - locked loop , nonlinear analysis , pll , two - phase pll , lock - in range , gardner s problem on unique lock - in frequency , pull - out frequency

a @xmath1-coalescent is a stochastic model for the genealogy of a ( haploid ) population of @xmath1 individuals backward in time . in this model , the individuals of the population are identified with the integers of the set @xmath6 , and the @xmath1-coalescent takes its values in the partitions of @xmath6 . a partition of @xmath6 is composed of a certain number of blocks , between @xmath7 and @xmath1 . the initial state of the @xmath1-coalescent is the partition in @xmath1 blocks , that is the partition in singletons , @xmath8 any particular set of @xmath9 blocks then merges independently in one block at rate given by : @xmath10 } \lambda(dx ) x^{-2 } \ ; x^{k } \ ; ( 1-x)^{n - k},\ ] ] where @xmath11 is a probability measure on @xmath12 $ ] . after the first coalescence , again , any particular set of @xmath9 blocks merges independently in one block at rate given by ( [ ratecoalescent0 ] ) , with @xmath1 replaced by the current numbers of blocks . the procedure is then repeated , until the process terminates at the partition in one single block , @xmath13 _ the first motivation of this paper is to study the number of blocks involved in the last coalescence . _ the interpretation of the model is the following : two integers are in the same block of the partition at time @xmath14 in the @xmath1-coalescent if the corresponding individuals found their common ancestor at time @xmath15 backward in time . at the time of the last coalescence , all the individuals found their common ancestor . when @xmath16 , a construction of the @xmath1-coalescent is the following . we start with a poisson point measure on @xmath17,$ ] with intensity @xmath18 . to each atom @xmath19 of this random measure , we associate a random subset @xmath20 of the set of integers @xmath21 by sampling each integer independently with the same probability @xmath22 . then , the blocks with labels in @xmath23 at time @xmath24 coalesce in one block . first , among the ( possibly infinitely many ) atoms @xmath19 on a finite time interval , only a finite number give rise to an effective merge in the @xmath1-coalescent , and so we may distinguish a first coalescence , a second one , @xmath25 second , the construction requires to give labels to the blocks . while most definitions of the @xmath1-coalescent do not stress on the labelling of the blocks , the point of this paper is to emphasize a particular ordering . our choice is to label the blocks in the order of their smallest element . the @xmath1-coalescent is then described by the family @xmath26 of its ordered blocks at time @xmath15 : @xmath27 is the block containing @xmath7 , @xmath28 is the block containing the smallest integer not in @xmath27 ( if any ) @xmath25 the largest @xmath29 such that @xmath30 is non - empty is then the number of blocks in the @xmath1-coalescent , denoted by @xmath31 . for instance , for the @xmath32-coalescent depicted on figure [ picture00 ] , we have @xmath33 and : @xmath34 this leads to the following alternative description of the coalescent as a family of ( coalescing ) maps . for each @xmath35 , and @xmath14 , there exists a unique integer @xmath36 such that @xmath37 and we then set @xmath38 . the random map @xmath39 is non - increasing , starts at @xmath40 and terminates at @xmath7 . furthermore , if two functions @xmath41 started at different points meet , then they coincide at each further time . figure [ picture00 ] describes the collection of the maps @xmath41 started at @xmath42 . notice that a function @xmath41 not only decreases when the block labelled @xmath41 takes part to a coalescence , but also when at least two blocks with lower label take part to a coalescence . also , adding more and more functions @xmath43 starting at @xmath44 , @xmath45 , @xmath46 , @xmath25 allows to define the @xmath1-coalescents in a coupled way , hereafter referred to as the natural coupling . this coupling allows to define the coalescent started from an infinite number of blocks , simply called the coalescent in the following , or the @xmath0-coalescent if we need to stress on the measure @xmath11 . let us argue on our first motivation : when adding more and more functions @xmath43 , the block counting process @xmath47 of the @xmath1-coalescent evolves , and we may think of it as a wave moving to the right . the motion to the right , measured by the depth @xmath48 of the @xmath1-coalescent , is either a.s . bounded , or a.s . unbounded - we come back to this fact in subsection [ sub : keylemma ] . in both cases , we investigate the question of the existence of a limiting shape ( in distribution in the second case ) for the wave viewed from the right . it amounts to study the time - reversal of the block counting process for the coalescent started from @xmath1 blocks as @xmath5 : this is a slight elaboration on our first motivation . the depth of the @xmath1-coalescent @xmath49 corresponds to the first time the @xmath1 blocks have merged in @xmath7 block . in the aforementioned natural coupling , we may consider the random subset of the integers @xmath50 that will be called the set of records : the integer @xmath1 belongs to the set of records @xmath51 when the function @xmath41 started at @xmath1 reaches @xmath7 at some later time than the functions @xmath41 started at lower values @xmath29 for @xmath52 , thus establishing a new record . see figure [ picture00 ] for an illustration . since @xmath53 by definition , we have that @xmath54 a.s . in terms of population genetics , the label of an individual corresponds to a record if its addition in the sample modifies the most recent common ancestor of the sample . _ the study of the set of records is our second motivation . _ the paths @xmath41 started from @xmath55 are good candidates to time - reversal as we now explain . to make this time - reversal clear , we introduce , after donnelly and kurtz @xcite , the following model , called the lookdown model . it is essentially constructed by time - reversal of the coalescent represented as a family of coalescing maps . the model consists in a family , indexed by the time @xmath14 , of countably many individuals distinguished by their levels . the level of an individual is an integer , and the individual at time @xmath15 at level @xmath56 is denoted by @xmath57 . the genealogical relationships between individuals are described as follows . we start with the same poisson point measure on @xmath17,$ ] with intensity @xmath18 . a random set @xmath58 is associated with each atom @xmath19 by sampling independently each integer with the same probability @xmath22 . to each atom @xmath59 there corresponds a reproduction event and : * for @xmath36 in @xmath23 , the individual @xmath60 is a child of the individual @xmath61 . notice that @xmath23 , and therefore the set of children , is infinite . * the other lineages are shifted upwards , keeping the order they had before the birth event : if @xmath62 , the individual @xmath63 is the ( unique ) child of the individual @xmath64 , where @xmath65 , see figure [ picture01 ] . this defines a countable infinite population . the ancestral lineage of an arbitrary individual @xmath57 is the line composed of the individuals @xmath66 where @xmath67 is the level of the ancestor of the individual @xmath57 at time @xmath68 . figure [ picture01 ] displays the collection of the ancestral lineages . the map @xmath69 has the same dynamic as the ( restriction of the ) map @xmath70 in the coalescent model . more is true : for each @xmath71 , the backward genealogy of the individuals at the @xmath1 lowest levels at time @xmath15 gives rise to a process valued in the partitions of @xmath72 : @xmath29 and @xmath36 are in the same block at time @xmath68 , @xmath73 , if the individuals @xmath57 and @xmath60 share a common ancestor at time @xmath74 . this partition valued process has the law of ( the restriction to @xmath75 $ ] ) of a @xmath1-coalescent . a key ingredient in this paper is the following quantity @xmath76 defined within the lookdown model . instead of looking to the ancestors of an individual , the definition considers its offspring . the levels of the offspring at time @xmath14 of the individual @xmath77 , that is the individual at time @xmath78 at level @xmath79 , form a subset of @xmath21 , the minimal element of which we define to be @xmath80 . if the subset of @xmath21 is empty , then we set @xmath81 see the dotted line in figure [ picture01 ] for an illustration . the shift by @xmath7 in the definition is for technical reason . a direct , equivalent , but slightly more difficult definition of @xmath76 ( at least on a picture ) is the following : the levels of the offspring at time @xmath14 of the individual @xmath82 , that is the individual at time @xmath78 at level @xmath7 , form a subset of @xmath21 , the connected component including @xmath7 we denote by @xmath83 again , we set @xmath84 in case this subset is @xmath85 . the collection of the random variables @xmath86 builds a non - decreasing process called _ the fixation line_. when the fixation line @xmath87 reaches level @xmath1 , then the whole population of individuals at level @xmath88 consists of offspring of individual @xmath82 , an event called fixation in population genetics : this explains the name fixation line . the link between the fixation line and the set of records @xmath51 is as follows : for each @xmath14 , @xmath80 belongs to the set of records @xmath51 for the coalescent describing the backward genealogy of the individuals at time @xmath15 . see figure [ picture01 ] for an example . this paper shows some applications of the fixation line . + this is a review of the literature : the origin of the fixation line may be traced back to pfaffelhuber and wakolbinger @xcite in the kingman case @xmath89 . for general @xmath11 , it appears in labb @xcite and in @xcite . the originality of the present work is to study the fixation line not for itself , but for understanding better the coalescent . the coalescent we presented is the @xmath0-coalescent , it was introduced independently by pitman @xcite and sagitov @xcite , and the research area has been surveyed in berestycki @xcite and bertoin @xcite . we find the fixation line useful in studying two random quantities defined in the coalescent : the set of records @xmath51 and the number of blocks implied in the last coalescence of the @xmath1-coalescent . the set of records seems not to have been considered in the literature until now . an integer @xmath1 is a record when the corresponding external branch in the coalescent tree has depth equal to that of the @xmath1-coalescent tree . in that sense , our contribution may be seen as an atypical view on the intensively studied external branches , see caliebe et . @xcite , dhersin and mhle @xcite and the references therein . the numbers of blocks implied in the last coalescence , as well as the closely related hitting probabilities of the block counting processes , are quantities which relate to the coalescent tree near the root . this part of the tree is difficult to grasp from the standard construction of the coalescent . original techniques have been developed in the papers @xcite to circumvent this difficulty . among these papers , the ones with the closest objectives to ours are abraham and delmas @xcite and goldschmidt and martin @xcite , which both use connections to specific class of random trees . the article @xcite and mhle @xcite are concerned with the bolthausen - sznitman coalescent , which is the beta(@xmath4)-coalescent for @xmath90 , whereas @xcite deal with the beta(@xmath4)-coalescent for @xmath91 $ ] . we propose here a probabilistic approach based on the analysis of the fixation line . the main advantage with respect to previous methods is that it may be introduced for general @xmath0-coalescent , and conveniently specializes to the class of the beta(@xmath4)-coalescent for the whole family of parameters @xmath92 . short after the prepublication of this paper , mhle extended in @xcite the analytic method of @xcite to the beta(@xmath4)-coalescents , and obtained some of the results presented in this paper . we warmly invite the reader to consult this paper as a parallel reference . this is the organization of the paper : section [ sec1 ] contains a key lemma relating the depth of the @xmath1-coalescent to the hitting times of the fixation line , see lemma [ tau - alpha ] . also , the transition rates of the fixation line are computed in lemma [ prop : gamma ] for a general probability measure @xmath11 . they slightly differ from those of the block counting process , and they factorize in the special case of the beta(@xmath93-coalescents , as proved in lemma [ lem : factorize ] . two applications of the fixation line , associated with our two motivations , are detailed in section [ sec2 ] . first , we compute in subsection [ sec21 ] the probability for an integer to be in the random set of records @xmath51 . second , we present in subsection [ sec22 ] the problem of the construction of the coalescent from the root , and propose , as a first and partial answer , to characterize the time - reversal of the block counting process . a simple observation reduces this problem to the computation of the law of the number of blocks implied in the last coalescence . our main result , theorem [ prop : conv - delta ] , is a limit theorem for the law of the number of blocks implied in the last coalescence in @xmath1-beta(@xmath94)-coalescents . our second main result , corollary [ cor : hitting ] , reformulates theorem [ prop : conv - delta ] in term of hitting probabilities of an integer @xmath36 by the block counting process . we also give the asymptotics of these hitting probabilities as @xmath95 . last , we take the opportunity to present a third , probabilistic derivation of the depth of the bolthausen - sznitman coalescent . we did our best to produce a paper as much self contained as possible . _ assumption _ : we will assume throughout that the probability measure @xmath11 gives no mass to the singletons @xmath78 and @xmath7 , @xmath96 the assumption on @xmath97 allows to rely on the simple graphical construction mentioned in the introduction without struggling to include binary coagulations . this being said , most of the results are still valid for a probability measure @xmath11 with an atom at @xmath78 . the assumption on @xmath98 avoids an uninteresting case . we first come back to the definition of the fixation line and generalize it slightly by allowing the fixation line to be started at an arbitrary integer . fix an integer @xmath36 and consider the set of individuals @xmath99 at time @xmath78 at level @xmath29 , for @xmath100 . the offspring of this set of individuals at time @xmath14 forms a subset of @xmath21 in the lookdown model mentioned in the introduction , the connected component including @xmath7 we denote by @xmath101 with , again , @xmath102 if this set is @xmath85 . alternatively , the offspring at time @xmath15 of the _ single _ individual @xmath103 forms a subset of @xmath21 , the _ smallest _ element of which is @xmath104 . for @xmath36 and @xmath1 integers , we set @xmath105 to denote the ( partial ) depth of the @xmath1-coalescent and the hitting time of the fixation line respectively . we now state our key lemma . in fact , the whole paper may be seen as a digression on this relation . [ tau - alpha ] let @xmath106 . the two random variables @xmath107 and @xmath108 have the same distribution . fix @xmath109 . it is enough to observe that the events @xmath110 are equal in the coupling of a coalescent and of a fixation line provided by the lookdown model . more precisely , we compare the fixation line started at level @xmath36 at time @xmath78 and the coalescent describing the backward genealogy of the @xmath1 lowest level individuals at time @xmath15 . for the inclusion , observe that , if @xmath111 for the coalescent , then the fixation line started at @xmath36 at time @xmath78 has not reached @xmath1 at time @xmath15 , that is @xmath112 . for the reverse inclusion , if @xmath112 , then the coalescent presents more than @xmath36 blocks at time @xmath15 , that is time @xmath78 in the lookdown model , and @xmath111 . either the increasing sequence of the expected depth of the @xmath1-coalescent @xmath113 is bounded , or it goes to @xmath114 . in the first case , the coalescent is said to come down from infinity , whereas in the second case , it is said to stay infinite . the terminology is best explained by the following results of schweinsberg @xcite : for a coalescent which comes down from infinity , the increasing sequence @xmath115 is a.s . bounded , and the coalescent started from an infinite number of blocks presents a finite number of blocks at each positive time a.s . for a coalescent which stays infinite however , the increasing sequence @xmath115 a.s . goes to @xmath114 , the coalescent presents an infinite number of blocks at each non - negative time a.s . the link with the fixation line follows from lemma [ tau - alpha ] : the coalescent comes down from infinity when the increasing sequence @xmath116 is a.s . bounded , that is when the fixation line reaches @xmath114 in finite time a.s . also , the coalescent stays infinite when the increasing sequence @xmath116 goes to @xmath114 a.s . , that is when the the fixation line remains finite a.s . our next task is to determine the transition rates of the fixation line . [ prop : gamma ] for @xmath117 , the rate @xmath118 at which a fixation line @xmath119 goes from @xmath29 to @xmath36 is : @xmath120 } \lambda(dx ) x^{-2 } \ ; x^{j - i+1 } ( 1-x)^{i } , \ ; \ ; \ ; 1 \leq i < j < \infty.\ ] ] a fixation line jumps from @xmath29 to @xmath36 when , at the time of a reproduction event , @xmath121 levels exactly are chosen among the levels @xmath122 , and the level @xmath123 is not chosen . for a reproduction event with asymptotic frequency @xmath22 , this has probability @xmath124 for any unordered set with @xmath121 elements in @xmath125 . counting the number of such sets , and integrating with respect to the `` law '' of the asymptotic frequency @xmath22 gives the formula . the quantity @xmath118 should be compared with the rate @xmath126 at which the block counting process of the @xmath1-coalescent @xmath127 jumps from @xmath36 to @xmath29 : @xmath128 } \lambda(dx ) x^{-2 } \ ; x^{j - i+1 } ( 1-x)^{i-1 } , \ ; \ ; \ ; 1 \leq i < j < \infty.\ ] ] unlike the transitions of the block counting process , which involve ( a mixture of ) binomial distributions , the transitions of the fixation line involve ( a mixture of ) negative binomial distributions . the two quantities @xmath118 and @xmath126 differ in general , still we have the following relationship . [ lem : combi ] for each @xmath129 , the rate @xmath130 at which a fixation line jumps from @xmath29 to a level @xmath131 is equal to the rate @xmath132 at which the block counting process jumps from @xmath36 to @xmath133 blocks : @xmath134 a computational proof is given in the appendix . for another instance of such a duality relationship , abstracted at the level of measure valued process , we refer the reader to lemma 5 p. 282 of bertoin and le gall @xcite . the claim ( [ hypergeom2 ] ) may be justified directly as follows : a fixation line jumps from @xmath29 to a level @xmath135 when , at the time of a reproduction event , at least @xmath121 levels are chosen among the levels @xmath122 , without any condition on the level @xmath123 . the same event backward corresponds to a coalescence from @xmath36 blocks to @xmath133 blocks . setting @xmath136 in ( [ hypergeom2 ] ) , we obtain that the total rate at which the fixation lines jumps up from @xmath29 is equal to the total rate at which the block counting process jumps down from @xmath29 : @xmath137 two quantities that we simply denote by @xmath138 in the following . the beta(@xmath4 ) family of probability measures is given , for @xmath139 , by : @xmath140}(x ) \ ; dx .\ ] ] the associated beta(@xmath4)-coalescents interpolate between the star like coalescent , which corresponds to the limit case @xmath141 , the bolthausen - sznitman coalescent , @xmath142 , and the kingman coalescent , which corresponds to the limit case @xmath143 . example 15 in schweinsberg @xcite , or ( [ eq : depth ] ) and ( [ eq : depth2 ] ) in this paper , ensure that the beta(@xmath4)-coalescent comes down from infinity in the sense of ( [ cdi ] ) if and only if @xmath144 . [ lem : factorize ] when @xmath11 is given by _ ( [ betafamily ] ) _ for some @xmath92 , the jump rates @xmath145 of the fixation line @xmath119 factorize as follows : @xmath146 conversely , it is not difficult to show that the beta(@xmath4 ) family contains all the probability measures @xmath11 for which @xmath145 factorizes as a product of a function of @xmath29 and a function of @xmath36 . we stress that the transition rates @xmath147 of the block counting process of the beta(@xmath4)-coalescents do not enjoy such a factorization property . to sum up , adopting a backward viewpoint results in a seemingly anecdotic change of the exponent of @xmath148 in the rate ( [ def : gammaij ] ) with respect to the rate ( [ def : lambdaij ] ) , which in turn yields a factorization for beta(@xmath4)-coalescents . this factorization will be the key to exact computations . the claim follows from the following elementary calculation : @xmath149 let @xmath150 be the range of the fixation line started at @xmath36 . lemma [ lem : factorize ] entails that the law of the translated range @xmath151 does not depend on @xmath36 in the beta(@xmath4 ) case . we shall simply use @xmath152 to denote this random set . the set @xmath152 is the range of a renewal process , and we compute its renewal measure . we set : @xmath153 & \mbox { if } \alpha \in ( 0,2 ) \setminus \{1\}. \end{array } \right.\ ] ] when @xmath11 belongs to the _ beta(@xmath4 ) _ family given by _ ( [ betafamily ] ) _ for some @xmath92 , the generating function of the renewal measure is : @xmath154 the random set @xmath152 is a renewal point process on @xmath155 based on the interarrival measure : @xmath156 the measure @xmath157 is a probability measure , as confirmed by setting @xmath158 in the following computation of the generating function @xmath159 of @xmath157 . we first do the computation for @xmath160 : @xmath161 using the binomial theorem , we deduce that : @xmath162 = 1 + \frac{1}{(\alpha-1)s } \left [ ( 1-s)^{\alpha } - ( 1 - s ) \right].\\ \end{aligned}\ ] ] we now consider the case @xmath90 : @xmath163 using for the last equality that the primitive of @xmath164 null at @xmath78 is : @xmath165 we deduce the generating function @xmath166 of the renewal measure using the renewal property . let @xmath167 be the enumeration of the elements of @xmath152 in increasing order . we have : @xmath168 and the claim follows . in two particular cases , the renewal measure @xmath169 is explicit : in the case @xmath170 , we have : @xmath171 and in the case @xmath172 , we have : @xmath173 the measure @xmath157 given by ( [ eq : etabeta ] ) being a probability measure , we have , using ( [ factorize ] ) , that : @xmath174 where @xmath175 means that @xmath176 , using also the definition of @xmath138 ( short after ( [ hypergeom3 ] ) ) at the first equality . we also notice , for future use , that the transition rate from @xmath29 blocks to @xmath7 block satisfies : @xmath177 the first application of the fixation line consists in the computation of the probability for an integer @xmath29 to be a record . recall that the set @xmath51 of records is the set @xmath178 where the sequence @xmath179 is defined in the natural coupling of the @xmath1-coalescents , see the introduction for the definition of this coupling . the proposition uses the range @xmath180 of the fixation line started at @xmath7 . we stress the proposition is valid for a general probability measure @xmath11 . [ lem : record ] the marginal distribution of the set of records @xmath51 satisfies : @xmath181 since @xmath182 for @xmath183 an exponential random variable with parameter @xmath7 ( independent of @xmath184 , but we do not use this fact ) , we deduce that : @xmath185 using lemma [ tau - alpha ] for the second equality , and relation ( [ hypergeom3 ] ) for the last equality . let @xmath186 be a collection of independent exponential random variables with parameter @xmath7 , also independent of @xmath51 . iterating ( [ eq : record ] ) yields @xmath187 combining with the discussion on coalescents which come down from infinity following lemma [ eq : deftaualpha ] , we deduce that the cardinality of the set @xmath51 is a.s . infinite of a.s . it is infinite when the coalescent stays infinite , and finite when the coalescent comes down from infinity . now , proposition [ lem : record ] , ( [ eq : alphaonehalf ] ) and ( [ eq : sim - nalpha ] ) give the following expression for the record probabilities in the case @xmath170 : @xmath188 this result gains a clear interpretation in the representation of the @xmath1-beta(@xmath189)-coalescent found by abraham and delmas @xcite , which uses the pruning at nodes of a labelled binary tree with @xmath1 leaves . in case @xmath172 , we use ( [ eq : alphathreehalf ] ) instead of ( [ eq : alphaonehalf ] ) to obtain that : @xmath190 for general @xmath92 , we compute the generating function of the record probabilities . [ lem : record2 ] the marginal distribution of the set of records in _ beta(@xmath4)_-coalescents has the following generating function : @xmath191 in the bolthausen - sznitman case @xmath142 , and : @xmath192}\ ] ] in case @xmath193 . we do the following computation : @xmath194 using proposition [ lem : record ] , the definition of @xmath195 and formula ( [ eq : sim - nalpha ] ) at the first equality , the link between gamma and beta functions at the second equality as well as the fubini tonelli theorem . the claim now follows substituting @xmath196 by its value given in ( [ eq : pngenerayting ] ) , distinguishing wether @xmath160 or @xmath142 . [ cor : depth ] the depth @xmath49 of the _ @xmath1-beta(@xmath4)_-coalescent almost surely converges as @xmath5 in the natural coupling to a random variable @xmath197 with expectation : @xmath198}\ ] ] in case @xmath199 the almost sure and monotone convergence of @xmath49 in the natural coupling of the @xmath1-coalescents follows form the definition of the natural coupling . for the expectation : set @xmath158 in proposition [ lem : record2 ] , and use the first equality in ( [ eq : record2 ] ) : this gives a telescopic sum with sum @xmath200 . since the sequence @xmath201 has limit @xmath200 by monotone convergence , which is finite according to ( [ eq : depth ] ) , the beta(@xmath4)-coalescent for @xmath144 comes down from infinity . on the other hand , when @xmath202 $ ] , we have , using again proposition [ lem : record2 ] with @xmath158 , that : @xmath203 and the beta(@xmath4)-coalescent with @xmath202 $ ] therefore stays infinite . in this case , the suitably rescaled random variables @xmath204 , as @xmath5 , have been proved to converge in law . we refer to @xcite and the references therein for the last improvements . we consider the block counting process @xmath47 of the @xmath1-coalescent . its embedded markov chain starts at @xmath1 , and has transitions probabilities : @xmath205 with @xmath126 the transition rate from @xmath36 to @xmath29 blocks defined in ( [ def : lambdaij ] ) and @xmath206 . in this subsection , we consider the problem of the convergence of the time - reversal of this markov chain as the initial number of blocks @xmath1 goes to @xmath114 . unlike the transition probabilities of the original chain , the transition probabilities of the chain reversed in time depend on the starting point @xmath1 , and we shall use @xmath207 to denote the transition probability of the reversed chain from @xmath29 to @xmath36 , @xmath208 , when the original chain is starting at @xmath1 . this amounts to study the weak convergence of @xmath209 viewed as a family , indexed by the integers @xmath1 , of probability measures on @xmath85 , for each @xmath210 . the question for an arbitrary @xmath211 may be reduced to the case @xmath212 : if @xmath213 stands for the range of the block counting process of the @xmath1-coalescent in fact , we have the key equality : @xmath214 which entails that : @xmath215 the following proposition gives an expression of the distribution @xmath216 in term of quantities related to the fixation line . this is the essential conceptual step in the study of the last coalescence since the next steps , carried out in the case of the beta(@xmath4)-coalescent in the next subsection , reduce to the computation of @xmath217 . [ prop : lastco ] the distribution @xmath218 of the number of blocks involved in the last coalescence of a @xmath1-coalescent satisfies : @xmath219 , \ ; \ ; 2 \leq j \leq n.\ ] ] we compute : @xmath220,\ ] ] setting @xmath212 in ( [ eq : pij - n ] ) for the first equality , using the definition ( [ eq : pji ] ) of @xmath221 and the fact that the block counting process spends an exponential time at @xmath36 with parameter @xmath222 when @xmath223 for the second equality , and the pathwise relation @xmath224 at the last equality . we conclude using lemma ( [ tau - alpha ] ) . for each @xmath225 , we use @xmath226 to denote the increasing limit of @xmath227 . since the convergence is monotone , we have convergence of the expectations : @xmath228 keeping in mind that the quantity on the right - hand side may be infinite . in case the coalescent comes down from infinity , @xmath229 , and we have : @xmath230 for each @xmath231 . the convergence of @xmath232 for arbitrary @xmath233 follows from ( [ eq : pijn ] ) and ( [ eq : p1jn ] ) . a much more interesting case is when the coalescent stays infinite . then we can not directly conclude to the convergence of the difference of the expectations @xmath234 . setting @xmath235 for the range of the fixation line @xmath236 started at @xmath36 , we have : @xmath237 \ ; \frac{1}{\lambda_{i+1 } } \cdot\ ] ] using that the rate at which the fixation line leaves @xmath29 is @xmath138 . if the coalescent stays infinite , the serie with general term @xmath238 is easily seen to diverge : in fact , by the definition of the @xmath1-coalescent , we have @xmath239 and the left - hand side goes to @xmath114 a.s . for a coalescent which stays infinite . proving convergence in ( [ eq : problem ] ) as @xmath5 therefore requires a further study of @xmath240 $ ] , which is a difficult issue in general . the factorization property satisfied by the beta(@xmath4)-coalescents , see lemma [ factorize ] , allows to circumvent the difficulty . before stating our main theorem , it is perhaps opportune to recall the statement of the problem of the last coalescence in a self contained manner : the block counting process @xmath127 is a markov chain started at @xmath1 and a.s . absorbed at @xmath7 in finite time . in the case of the beta(@xmath4)-coalescents , we recall from ( [ def : lambdaij ] ) that the transitions rates of the block counting process from @xmath36 to @xmath29 are given by : @xmath241 what is the law of the last jump of @xmath127 as @xmath5 ? the following theorem answers the question . [ prop : conv - delta ] the distribution @xmath242 of the number of blocks implied in the last coalescence of the _ @xmath1-beta(@xmath4)_-coalescent weakly converges as @xmath5 towards a distribution @xmath243 with generating function : @xmath244 in the bolthausen - sznitman case @xmath142 , and : @xmath245\ ] ] in case @xmath160 . setting @xmath158 in formulas ( [ eq : generatingalphaeq1 ] ) and ( [ eq : generatingalphaneq1 ] ) allows to see that the distribution @xmath243 is a probability distribution . notice the convergence claim ( but not the limiting generating function ) for @xmath144 follows from the discussion on coalescents which come down from infinity in the previous subsection . proposition 1.5 of abraham and delmas @xcite establishes a similar claim ( including the limiting generating function ) for @xmath91 $ ] . the proof given there relies on a connection with the pruning of lvy trees . in the case @xmath142 , we deduce from ( [ eq : generatingalphaeq1 ] ) that : @xmath246 using the change of variable @xmath247 at the second equality , expanding @xmath248 and using the frullani integral at the third equality . this result has been obtained first by goldschmidt and martin @xcite , using a connection with the pruning of recursive trees . it is interesting to observe the diversity of the methods at work in @xcite and the present paper . recall @xmath152 denotes the range of the renewal point process on @xmath155 containing @xmath78 and with interarrival times with law @xmath157 given by ( [ eq : etabeta ] ) . we compute : @xmath249 beginning as in ( [ eq : problem ] ) for the first equality , using the definition of the translated range @xmath250 , independent of @xmath36 in the beta(@xmath4 ) setting , and then changing the index in the first sum at the second equality . bounding @xmath251 from above by @xmath7 , and using the positivity of @xmath252 , we obtain the following upper bound : @xmath253 the sequence @xmath254 is non - decreasing . therefore , the serie on the left - hand side of ( [ eq : bounded ] ) has non - negative terms . this serie is bounded , and therefore converges . also , the sequence @xmath254 goes to @xmath114 by ( [ eq : sim - nalpha ] ) . the second term in ( [ eq : sum ] ) then goes to @xmath78 . using ( [ eq : p1jn ] ) , we conclude that the limit as @xmath255 of the quantities @xmath256 exists , we denote it by @xmath257 . setting @xmath258 , we have that : @xmath259 setting the explicit values ( [ eq : sim - nalpha ] ) and ( [ eq : lambdaj1 ] ) of @xmath260 and @xmath261 gives : @xmath262\ ] ] in case @xmath142 , and : @xmath263 in case @xmath160 . recall the expression ( [ eq : pngenerayting0 ] ) for the generating function of the numbers @xmath264 . multiplying both sides of ( [ eq : pngenerayting0 ] ) by @xmath265 , integrating with respect to @xmath266 and using fubini - tonelli theorem , we deduce @xmath267 & = - \int_{(0,1 ) } ds \ ; \frac { s^{j-1 } } { \log(1-s)}\end{aligned}\ ] ] in case @xmath142 , and @xmath268 in case @xmath160 , using also the expression of the beta function in term of the gamma function . from the last four equations displayed , we obtain : @xmath269 in case @xmath142 , and @xmath270 in case @xmath271 which imply respectively ( [ eq : generatingalphaeq1 ] ) and ( [ eq : generatingalphaneq1 ] ) by multiplying by @xmath272 and summing over @xmath225 . [ cor : hitting ] the probability for an integer @xmath225 to be in the range @xmath273 of the block counting process of the _ @xmath1-beta(@xmath4)_-coalescent converges as @xmath5 and : @xmath274 in the bolthausen - sznitman case @xmath142 , and : @xmath275 in case @xmath276 . notice the integrands in both integral representations are non - negative whatever the value of @xmath277 . also , the bolthausen - sznitman case in corollary [ cor : hitting ] corresponds to the statement of theorem 1.1 of mhle @xcite , and the case @xmath160 answers a question posted in the same paper , see remark 3 . the question has also been independently answered by mhle in @xcite . this is a consequence of the equation ( [ eq : pij - n ] ) together with formula ( [ eq : alpha1 ] ) in the bolthausen - sznitman case @xmath142 , and together with formula ( [ eq : generatingalphaneq1-check ] ) in the case @xmath160 . in case @xmath144 , the coalescent comes down from infinity and it is possible to consider directly the range @xmath278 of the infinite coalescent : the range @xmath278 is the almost sure ( local ) limit of the @xmath273 in the sense that @xmath279 is the almost sure limit of @xmath280 . dominated convergence theorem then ensures that the right - hand side of ( [ eq : hitting ] ) corresponds to @xmath281 . we propose to write @xmath282 whatever the value of @xmath92 : this is however an abuse of notation since we do not give a meaning to @xmath278 when @xmath202 $ ] . [ cor : hit_asymptot ] the limiting probability for an integer @xmath225 to be in the range of the block counting process of the _ @xmath1-beta(@xmath4)_-coalescent satisfies : @xmath283 in case @xmath144 , and : @xmath284 in case @xmath285 . the asymptotics for @xmath144 have been previously derived in berestycki _ _ using a connection with @xmath277-stable branching processes , see theorem 1.8 of @xcite , and the bolthausen - sznitman case @xmath142 has been covered in mhle @xcite , see corollary 1.3 ; in this case , @xmath286 we first consider the case @xmath285 . estimating the left factor in ( [ eq : hitting ] ) is easy : @xmath287 for the remaining integral factor in ( [ eq : hitting ] ) , we write : @xmath288 for @xmath289 $ ] . then we decompose as follows : @xmath290.\ ] ] fix @xmath291 . from the continuity of @xmath292 at @xmath7 , there exists @xmath293 such that such @xmath294 for @xmath295 $ ] , and : @xmath296 for @xmath36 large enough . therefore , the left - hand side of ( [ eq : sim1 ] ) is equivalent to @xmath297 , and the claim follows in the case @xmath285 . for the case @xmath144 , it is more convenient to rewrite the integral factor as follows : @xmath298 then , we write : @xmath299 for @xmath300 $ ] this time . the same reasoning as before allows to conclude that : @xmath301 this term is compensated by the first factor in ( [ eq : hitting ] ) , and the claim follows for @xmath144 . the definition of the block counting process of the @xmath1-coalescent entails : @xmath302 taking the limit @xmath5 in this formula gives an alternative proof of corollary [ cor : depth ] on the expected depth of the beta(@xmath4)-coalescent for @xmath144 . we propose to investigate further the bolthausen - sznitman coalescent associated with @xmath304 } ( x ) \ ; dx$ ] . this coalescent stays infinite . in fact , it plays a special rle in the class of the beta(@xmath4)-coalescents , since it separates those coalescents which come down from infinity , @xmath305 , from those which stay infinite , @xmath306 . the case @xmath142 in ( [ factorize ] ) gives : @xmath307 therefore , the fixation line @xmath119 is a continuous time discrete state space branching process ( and this is the only coalescent for which it is the case ) . the reproduction law @xmath308 of the branching process is defined by : @xmath309 since a jump of @xmath310 for the fixation line corresponds to the arrival of @xmath36 children together with the death of the father . the generating function associated with the reproduction law @xmath308 is : @xmath311 the reproduction law @xmath308 has infinite mean , but the branching process is conservative , meaning it does not reach infinity in finite time . this agrees with our observation ( after lemma [ tau - alpha ] ) that the fixation line @xmath119 remains finite for coalescents which stay infinite , and the bolthausen - sznitman coalescent stays infinite . the rate of increase of @xmath119 is well - known , see grey @xcite for instance , we nevertheless include a proof for the ease of reference . in the bolthausen - sznitman case @xmath304 } ( x ) \ , dx$ ] , we have @xmath312 with @xmath183 an exponential random variable with parameter @xmath7 . notice this growth rate deviates from the exponential growth rate satisfied by supercritical branching processes with a finite mean reproduction law , cf . the seneta - heyde theorem . we begin with general considerations on continuous - time branching processes . the generating function @xmath313 of @xmath76 may be computed from the infinitesimal generating function @xmath314 using the partial differential equation : @xmath315 see harris @xcite on chapter v for instance . the function @xmath316 is a bijection from @xmath12 $ ] into itself , with inverse function @xmath317 . the process @xmath318 is markov and has constant expectation since : @xmath319 therefore it is a @xmath12$]-valued martingale , almost surely converging towards a limiting random variable @xmath320 as @xmath321 . at this point , we take advantage of the explicit formulas available in our case : @xmath322 which entails : @xmath323 using the dominated convergence theorem , we deduce that for each @xmath324 : @xmath325 this is possible only if @xmath320 is @xmath326-valued , equal to @xmath7 with probability @xmath68 . now , since @xmath327 is increasing in @xmath68 , there is a.s . a threshold random variable : @xmath328 , \lim_{t \to \infty } g_t(s)^{l_1(t ) } = 1\},\ ] ] which is uniformly distributed on @xmath12 $ ] since @xmath329 . then , we form the logarithm of the expression @xmath330 and use that @xmath331 is equivalent as @xmath321 to @xmath332 , iself equal to @xmath333 from the previous computation , to deduce that : @xmath334 set @xmath335 . the random variable @xmath320 is again uniformly distributed on @xmath12 $ ] . taking again logarithm , for @xmath336 , we have : @xmath337 now , the random variable @xmath338 is exponentially distributed with parameter @xmath7 . this concludes the proof . growth rate is the key to the following ( third ) proof of the depth of the bolthausen - sznitman coalescent , after the one by goldschmidt and martin @xcite based on a connection with recursive trees , and the one by freund and mhle @xcite based on the analysis of a recurrence equation . [ depthbs ] in the bolthausen - sznitman case @xmath304 } ( x ) \ , dx$ ] , we have the following convergence in distribution for the depth of the @xmath1-coalescent : @xmath339 where @xmath183 is an exponential random variable with parameter @xmath7 . the sequence @xmath115 evolves by independent exponential jumps at the moments of records in the natural coupling , see equation ( [ eq : record ] ) . the convergence in distribution can therefore not be reinforced in an a.s . convergence . from the a.s . growth rate ( [ asymptoticbs ] ) and the definition ( [ eq : deftaualpha ] ) of the hitting time @xmath340 , we deduce : @xmath341 using the definition of @xmath340 for the inequality and ( [ asymptoticbs ] ) for the almost sure convergence . therefore , @xmath342 similarly , taking the left limit at @xmath340 this time , @xmath343 and this implies @xmath344 this proves ( [ eq : depthbs ] ) with @xmath340 instead of @xmath204 and with a.s . convergence instead of weak convergence . we conclude using lemma [ tau - alpha ] . * acknowledgments*. the author is grateful to stephan gufler , gtz kersting , iulia stanciu , anton wakolbinger and linglong yuan for their interest in this work . this work was supported by the dfg priority programme spp 1590 `` probabilistic structures in evolution '' . 10 r. abraham and j .- f . delmas . . , 2013 r. abraham and j .- f . delmas . . , j. berestycki , n. berestycki , j. schweinsberg , et al . small - time behaviour of beta coalescents . , 44(2):214 , 2008 . n. berestycki . , volume 16 of _ ensaios matemticos [ mathematical surveys]_. sociedade brasileira de matemtica , rio de janeiro , 2009 . j. bertoin . , volume 102 of _ cambridge studies in advanced mathematics_. cambridge university press , cambridge , 2006 . j. bertoin and j .- f . le gall . stochastic flows associated to coalescent processes . , 126(2):261288 , 2003 . a. caliebe , r. neininger , m. krawczak , and u. rsler . on the length distribution of external branches in coalescence trees : genetic diversity within species . , 72(2):245252 , 2007 . j .- s . dhersin and m. mhle . on the external branches of coalescents with multiple collisions . , 18(40):111 , 2013 . donnelly and t. g. kurtz . particle representations for measure - valued population models . , 27(1):166205 , 1999 . f. freund and m. mhle . on the time back to the most recent common ancestor and the external branch length of the bolthausen - sznitman coalescent . , 15:387416 , 2009 . a. gnedin , a. iksanov , a. marynych , and m. mhle . on asymptotics of the beta - coalescents . . c. goldschmidt and j. b. martin . random recursive trees and the bolthausen - sznitman coalescent . , 10:no . 21 , 718745 ( electronic ) , 2005 . d. r. grey . almost sure convergence in markov branching processes with infinite mean . , 14(4):702716 , 1977 . t. e. harris . . die grundlehren der mathematischen wissenschaften , bd . springer - verlag , berlin , 1963 . o. hnard . change of measure in the lookdown particle system . , 123(6):20542083 , 2013 . g. kersting . the asymptotic distribution of the length of beta - coalescent trees . , 22(5):20862107 , 2012 . c. labb . from flows of lambda fleming - viot processes to lookdown processes via flows of partitions . , july 2011 . m. mhle . asymptotic hitting probabilities for the bolthausen - sznitman coalescent . . m. mhle . asymptotic hitting probabilities of beta coalescents . . p. pfaffelhuber and a. wakolbinger . the process of most recent common ancestors in an evolving coalescent . , 116(12):18361859 , 2006 . j. pitman . coalescents with multiple collisions . , 27(4):18701902 , 1999 . s. sagitov . the general coalescent with asynchronous mergers of ancestral lines . , 36(4):11161125 , 1999 . j. schweinsberg . a necessary and sufficient condition for the @xmath0-coalescent to come down from infinity . , 5:111 ( electronic ) , 2000 . we perform the following calculations : @xmath345 } \lambda(dx ) x^{-2 } \ ; x^{k+1 } ( 1-x)^i \\ & = \int_{[0,1 ] } \lambda(dx ) x^{-2 } \ ; \left [ \sum_{k \geq j - i } { k+i \choose k+1 } x^{k+1 } \right ] ( 1-x)^i \\ & = \int_{[0,1 ] } \lambda(dx ) x^{-2 } \ ; \left [ \frac{1}{(1-x)^i } - \sum_{0 \leq k \leq j - i } { k+i-1 \choose k } x^{k } \right ] ( 1-x)^i \\ & = \int_{[0,1 ] } \lambda(dx ) x^{-2 } \ ; \left [ 1 - \sum_{0 \leq k \leq j - i } { k+i-1 \choose k } x^{k } ( 1-x)^i \right ] \end{aligned}\ ] ] using the binomial theorem at the third equality . we also compute : @xmath346 } \lambda(dx ) x^{-2 } \ ; x^{k+1 } ( 1-x)^{j-(k+1 ) } \\ & = \int_{[0,1 ] } \lambda(dx ) x^{-2 } \ ; \sum_{j - i \leq k \leq j-1 } { j \choose k+1 } x^{k+1 } ( 1-x)^{j-(k+1 ) } \\ & = \int_{[0,1 ] } \lambda(dx ) x^{-2 } \ ; \left[1- \sum_{0 \leq k \leq j - i } { j \choose k } x^{k } ( 1-x)^{j - k } \right ] \\ \end{aligned}\ ] ] using the same theorem at the third equality again . it is enough to prove that the two integrands are equal , which amounts to verify : @xmath347 but setting @xmath348 in the right - hand side , we obtain : @xmath349 the claim therefore reduces to the following combinatorial statement : @xmath350 if @xmath351 however , we have : @xmath352 and ( [ eq : combinatorial ] ) reduces to : @xmath353 a simple identity ( also known as the vandermonde identity ) .

we define a markov process in a forward population model with backward genealogy given by the @xmath0-coalescent . this markov process , called the fixation line , is related to the block counting process through its hitting times . two applications are discussed . the probability that the @xmath1-coalescent is deeper than the ( @xmath2-@xmath3)-coalescent is studied . the distribution of the number of blocks in the last coalescence of @xmath1-beta(@xmath4)-coalescents is proved to converge as @xmath5 , and the generating function of the limiting random variable is computed . the probability for an integer to be in the range of the block counting process of @xmath1-beta(@xmath4)-coalescents is also given in the limit @xmath5 .

reflecting diffusion processes constitute an important class of stochastic processes that appear in various applications . we consider two particular examples of a _ diffusion ratchet _ variants of which were introduced in @xcite and @xcite motivated by protein transport across cell membranes . generally speaking a diffusion ratchet is a diffusion process reflected at a non - decreasing jump process . the name _ ratchet _ is justified by the fact that each jump prevents the diffusion from attaining lower values . in a sense a jump of the reflection boundary process can be thought of as a _ click _ of a ratchet . in @xcite a diffusion ratchet ( modelling a molecular motor ) in which a particle moves according to a brownian motion between equally spaced ( deterministic ) barriers is studied . the particle can cross such barriers from left to right but is reflected if it hits a barrier to its left . the two models that we consider in the present note are both generalisations of the diffusion ratchet studied in @xcite . in the model studied there a particle moves according to a reflecting brownian motion and the reflection boundary jumps a rate proportional to the distance between the particle and the current reflection boundary . at jump times the new reflection boundary is chosen uniformly between the old one and the position of the particle . let us now introduce the models that we consider here , then briefly explain the biological motivation and finally state our results . let @xmath5 be a time - homogeneous markov process starting in @xmath6 , for some @xmath7 . here @xmath8 is a diffusion process reflected at a non - decreasing jump process @xmath9 . given that @xmath10 is in @xmath11 at time @xmath12 , the reflection boundary process jumps at rate @xmath13 for some @xmath3 . if @xmath14 is a jump time of the reflection boundary then the new position is uniformly distributed on the interval @xmath15 $ ] . by this dynamics , @xmath16 for all @xmath17 , almost surely . in principle the above description works with any reflecting diffusion between the jumps . as mentioned earlier we study here two particular cases and to distinguish them we write @xmath10 and @xmath18 for the corresponding processes . 1 . if for @xmath19 the process @xmath20 is a brownian motion with negative infinitesimal drift @xmath21 , unit variance ( see section [ sec : rbmd ] ) and reflection boundary process @xmath9 then we refer to the process @xmath22 as _ the @xmath23-brownian ratchet_. 2 . if for @xmath19 the process @xmath24 is an ornstein - uhlenbeck process with infinitesimal drift @xmath25 , unit variance ( see section [ sec : refl - ou ] ) and reflection process @xmath26 then we refer to @xmath18 as _ the @xmath23-ornstein - uhlenbeck ratchet_. whenever we want to stress the dependence on the parameters , we write @xmath27 for the @xmath23-brownian ratchet and @xmath28 for the @xmath23-ornstein - uhlenbeck ratchet . ( 0,1.48 ) ( 0,1.62 ) ; ( -1.7,2.4 ) .. controls ( 0,1 ) and ( -0.6,0.5 ) .. ( -0.8,0.4 ) .. controls ( -1,0.3 ) and ( -1.6,0.2 ) .. ( -1.3,0.6 ) .. controls ( -1,1 ) and ( 0.2,1.7 ) .. ( -0.2,2 ) .. controls ( -0.6,2.3 ) and ( -1.1,2.6 ) .. ( -1,1.9 ) .. controls ( -0.9,1.5 ) and ( -0.2,1.65 ) .. ( 0,1.55 ) .. controls ( 0.2,1.45 ) and ( 0.6,0.7 ) .. ( 0.7,0.8 ) .. controls ( 0.8,0.9 ) and ( 0.8,2 ) .. ( 1,1.9 ) .. controls ( 1.2,1.8 ) and ( 1.35,1.6 ) .. ( 1.5,1.4 ) .. controls ( 1.65,1.2 ) and ( 1.7,2 ) .. ( 1.8,2.5 ) ; ( 0.2,0.3 ) circle ( 1.5pt ) ( 0.8,0.4 ) circle ( 1.5pt ) ( 1.1,2.5 ) circle ( 1.5pt ) ( 1.6,1 ) circle ( 1.5pt ) ( 0.3,2 ) circle ( 1.5pt ) ( 0.41,1.07 ) circle ( 1.5pt ) ( 1.07,1.85 ) circle ( 1.5pt ) ( 2.6,0.9 ) circle ( 1.5pt ) ( 2.3,2.7 ) circle ( 1.5pt ) ; ( 7,1.48 ) ( 7,0 ) ( 7,1.62 ) ; ( 5,1.55 ) ( 9.5,1.55 ) ; ( 9.5,1.80)(9.5,1.30 ) ; ( 8,1.80)(8,1.55 ) ; ( 8.05,1.75 ) ( 9.45,1.75 ) ; ( 7.05,1.35 ) ( 9.45,1.35 ) ; ( 5.05,1.75 ) (6.9,1.75 ) ; ( 8.8,1.95 ) node @xmath29 ( 8.2,1.1 ) node @xmath30 ( 5.7,1.95 ) node _ drift _ ; ( 9.2,0.3 ) circle ( 1.5pt ) ( 7.8,0.4 ) circle ( 1.5pt ) ( 8.1,2.5 ) circle ( 1.5pt ) ( 8.6,1 ) circle ( 1.5pt ) ( 8,1.55 ) circle ( 1.5pt ) ( 9,1.55 ) circle ( 1.5pt ) ( 9.3,2 ) circle ( 1.5pt ) ( 9.41,1.07 ) circle ( 1.5pt ) ( 7.5,2.75 ) circle ( 1.5pt ) ; inside a typical cell different proteins are involved in many processes . they usually need to be transported after or during the _ translation _ ( production ) to various locations at which they are required . depending on the protein and its functions there are different transport mechanisms . in the present paper we focus on the _ passive _ protein transport across membranes of e.g. endoplasmic reticulum ( er ) or mitochondria for which ratcheting models were introduced by @xcite and @xcite . the main idea in these models is that due to thermal fluctuations the protein moves , say inside and outside the er for definiteness , through a nanopore in the membrane according to a diffusion ; see the left part of figure [ fig : protein ] . inside the er , ratcheting molecules can bind to the protein at a certain rate . these ratcheting molecules are too big ( in our model they are actually infinitesimally small but one can imagine that binding of the molecules leads to a deformation of the protein at the ratcheting sites ) to pass through the nanopore and prevent the protein from diffusing outside the er , i.e. the protein performs a reflected diffusion with jumping reflection boundary which is due to binding of new ratcheting molecules . in the last two decades such models have been studied extensively in biology , physics as well as in mathematics . for a detailed overview of the recent literature and for more biological motivation we refer to @xcite and references therein . with this motivation in mind @xmath30 ( and @xmath31 ) can be interpreted as the length of the protein inside er at time @xmath12 and @xmath29 ( and @xmath32 ) as the distance between the `` head '' of the protein and the ratcheting molecule closest to the nanopore ; see the right part of figure [ fig : protein ] . since typically proteins have to be unfolded during translocation into er , the movement inside takes place against a force pointing outside which explains the locally negative drift of the ratchets . for both , the @xmath23-brownian ratchet and the @xmath23-ornstein - uhlenbeck ratchet we prove a law of large numbers as well as a central limit theorem . furthermore we compute the speed of the ratchets in terms of the airy @xmath33-function in the case of brownian ratchet and in terms of the tricomi confluent hypergeometric function in the case of ornstein - uhlenbeck ratchet . [ theoremrbm ] let @xmath34 be the @xmath23-brownian ratchet starting in @xmath35 with @xmath36 . if @xmath37 then @xmath38 where @xmath39 is the airy function . furthermore in the case @xmath40 there is @xmath41 such that @xmath42 here `` @xmath43 '' denotes convergence in distribution and @xmath44 is a standard gaussian random variable . note that though the result is formulated for @xmath19 for the proof we only need to consider the case @xmath45 . in the case @xmath46 the @xmath23-brownian ratchet as well as the@xmath23-ornstein - uhlenbeck ratchet reduce to the process studied in @xcite . [ theoremou ] assume @xmath47 and @xmath3 . let @xmath48 be the @xmath23-ornstein - uhlenbeck ratchet starting in @xmath35 for @xmath49 . for @xmath50 set @xmath51 where @xmath52 is the tricomi confluent hypergeometric function ( see for a definition ) . then @xmath53 furthermore in the case @xmath40 there is @xmath54 such that @xmath55 for a standard gaussian random variable @xmath44 . comparison of the speed of the ratchets for @xmath56 and @xmath57 $ ] . , scaledwidth=75.0% ] in figure [ fig : speed - compar ] we plot the speed of both ratchets in the interval @xmath57 $ ] in the case @xmath58 . the plots are based on numerical computations using mathematica . in the neighbourhood of zero , here approximately in the interval @xmath59 , the brownian ratchet is faster whereas outside that interval the ornstein - uhlenbeck ratchet has a higher speed . heuristically this can be explained : if @xmath60 are large and @xmath61 small then the drift of @xmath30 towards @xmath29 is smaller than that of @xmath31 towards @xmath32 . then in the brownian case the reflection boundary jumps on average `` earlier '' and `` higher '' than in the ornstein - uhlenbeck case . since both @xmath62 and @xmath63 are `` shortened '' at rate proportional to their values the described effect is not very pronounced and the speed of both ratchets is comparable in this region . on the other hand if @xmath61 is large and @xmath60 are close to zero then @xmath32 has a higher chance to jump `` earlier '' and `` higher '' than @xmath29 because the drift @xmath30 towards @xmath29 is constant and that of @xmath31 towards @xmath32 is proportional to their distance which is small in this case . note that the above heuristic arguments are similar in spirit to the following considerations in the case without jumping reflection boundaries ( @xmath64 , in that case the speed of both ratchets is zero ) . the invariant density of the reflected brownian motion with negative drift @xmath21 is @xmath65 , @xmath66 ( see e.g. * ? ? ? * ) and that of the reflected ornstein - uhlenbeck process with drift @xmath25 and unite variance is @xmath67 , @xmath50 ( this can be easily obtained from the invariant density of the ornstein - uhlenbeck process ) . for the expectations we have @xmath68 and @xmath69 . in particular , under the invariant distributions the expectation of the reflected brownian motion is larger than that of the reflected ornstein - uhlenbeck process for @xmath70 whereas for @xmath71 the opposite inequality holds . the rest of the paper is split in two sections in which theorem [ theoremrbm ] and theorem [ theoremou ] are proved . in section [ sec : gr - constr ] we deal with the brownian ratchet . first , in section [ sec : rbmd ] we recall an explicit construction of the reflecting brownian motion with drift . it will be used in section [ sec : gr - constr - app ] to give a graphical construction of the brownian ratchet . there we also prove a scaling property for the brownian ratchet and show that the graphical construction can also be used to construct a coupling of brownian ratchets with different initial conditions . between the jump times the brownian ratchet can be seen as a killed reflecting brownian motion with drift . for that reason in section [ sec : green ] we compute the corresponding green function and obtain several estimates on the moments of the killing time and the position at killing time . in section [ sec : invdist ] we study the markov chain of the increments of the brownian ratchet at jump times . we show that this markov chain possesses a unique invariant distribution and compute the expectations under this distribution . these will be used later to compute the speed of the ratchet explicitly . in section [ sec : reg - brr ] we define a regeneration structure for the brownian ratchet and show that the increment at these regeneration times have finite second moments . from that we obtain in section [ sec : proof12 ] the assertion of theorem [ theoremrbm ] . in section [ sec : ou - ratchet ] , which has a similar structure to section [ sec : gr - constr ] , we carry out the corresponding program for the ornstein - uhlenbeck ratchet . in this section we give a graphical construction of the brownian ratchet with negative local drift from which we deduce a scaling property and show that the construction allows to couple two brownian ratchets so that from some almost surely finite time on they have the same spatial as well as temporal increments . then we study the markov chain of the increments of the ratchet at the jump times of the boundary and show that it has a unique invariant distribution , which will allow to compute the speed of the ratchet explicitly . for the lln and clt we define regeneration times of the ratchet and show that the increments between these times have bounded second moments . before we start with the above schedule let us recall the definition and an explicit construction of the reflecting brownian motion with drift . though the definition given here is valid for any @xmath72 we will assume @xmath19 because the case @xmath73 is less interesting . for more information on reflecting brownian motion with drift we refer to e.g.@xcite . a _ reflecting brownian motion _ with infinitesimal drift @xmath21 started in @xmath50 , which we denote by @xmath74 , is a strong markov process with continuous paths ( i.e. a diffusion process ) associated with the infinitesimal operator @xmath75 acting on @xmath76 as follows : @xmath77 we shall omit the superscript @xmath78 and write @xmath79 whenever the initial value is not important . let us also recall from @xcite an explicit construction of @xmath74 that will be useful for our purposes . let @xmath80 be a standard brownian motion starting in @xmath81 . we define the brownian motion with drift @xmath61 , denoted by @xmath82 , and its running maximum , denoted by @xmath83 , by @xmath84 furthermore we define @xmath85 by @xmath86 then in ( * ? ? ? 2.1 ) it is shown that @xmath87 assume @xmath88 and let @xmath82 be as in . furthermore let @xmath89 be an independent poisson process on @xmath90 with intensity @xmath91 where @xmath92 is lebesgue measure on @xmath93 . we define a sequence of jump times @xmath94 and a sequence @xmath95 with @xmath96 as follows : @xmath97 given @xmath98 and @xmath99 for some @xmath100 we set @xmath101\times \{t\ } \ne \emptyset\bigr\}. \end{aligned}\ ] ] furthermore we let @xmath102 be the space component of the almost surely unique element of @xmath103\times \{\tau_n\}$ ] . for @xmath104 define @xmath105 finally we define @xmath106 and @xmath107 by setting @xmath108 note that @xmath109 is the `` running maximum '' process that jumps down to poisson points that are between the process itself and the brownian motion with drift . graphical construction of the brownian ratchet with locally negative drift.,scaledwidth=98.0% ] in the following lemma we verify that @xmath110 fits the description of @xmath23-brownian ratchet given in subsection [ sec : model ] . [ lem : gr - constr - br ] the process @xmath110 is @xmath23-brownian ratchet started in @xmath111 . by construction @xmath112 , so that @xmath113 is non - decreasing . furthermore , @xmath114 implies @xmath115 for all @xmath116 . between @xmath117 and @xmath118 the process @xmath119 is @xmath79 reflected at @xmath29 starting at time @xmath117 in @xmath120 thus , @xmath121 for all @xmath122 , and therefore the paths of @xmath119 are continuous . given the process up to time @xmath117 the jump rate of @xmath123 i.e. the rate of at which @xmath118 occurs is @xmath124 . then the reflection boundary jumps to @xmath125 . by homogeneity of the poisson process @xmath89 , @xmath126 is uniform on @xmath127 $ ] . thus , @xmath128 is uniform on @xmath129 $ ] because for some @xmath130)$ ] we have @xmath131 if we transform time and space in the graphical construction then we of course rescale the brownian motion with drift and transform the poisson process . there is only one such transformation that maps the poisson process @xmath132 to @xmath89 and the brownian motion with drift to a brownian motion with another drift . [ lem : scaling ] for @xmath40 and @xmath19 we have @xmath133 assume that we construct @xmath134 starting in @xmath135 using the poisson process @xmath132 and the brownian motion with drift @xmath136 . we define @xmath137 by @xmath138 on the one hand , rescaling space and time using @xmath139 we obtain the process on the right hand side of . on the other hand the rescaled graphical construction leads to the process on the left hand side of . for that we need to verify @xmath140 equation is clear . for we have using the scaling property of the brownian motion @xmath141 we now turn to the construction of a coupling of two @xmath23-brownian ratchets starting in @xmath142 . let @xmath143 and @xmath89 be as before , and let @xmath144 with @xmath145 without loss of generality . to construct the coupled brownian ratchet @xmath146 set @xmath147 , @xmath148 , @xmath149 and define as before the sequences @xmath150 , @xmath151 , @xmath152 and @xmath153 . furthermore define the corresponding processes @xmath109 , @xmath154 , @xmath155 and @xmath156 as in . define the coupling time by @xmath157 note that , since we use the same brownian motion and the same poisson process for both ratchets we have @xmath158 for all @xmath159 . thus , on the event @xmath160 , there are @xmath161 such that for @xmath162 we have almost surely @xmath163 the following lemma shows that @xmath164 is almost surely finite , i.e. , the coupling is successful . [ lem : coupling ] + for @xmath47 and @xmath165 we have @xmath166 \le e^{x(\mu -\sqrt{\mu^2 - 2\alpha } ) } . \end{aligned}\ ] ] at time @xmath164 either both @xmath109 and @xmath167 use the same point of the poisson process @xmath89 or the brownian motion @xmath82 touches the maximum of @xmath109 and @xmath167 which is @xmath109 by assumption @xmath168 . thus , we have @xmath169 for @xmath170 by construction @xmath171 can increase only after this time @xmath172 and decrease by jumping down when it uses points of the poisson process . ignoring this decrease by jumping down we obtain @xmath173 it is well known that ( see e.g. * ? ? ? * ) for @xmath174 we have @xmath175 = e^{x(\mu -\sqrt{\mu^2 - 2\alpha } ) } \end{aligned}\ ] ] and the result follows . since between the jumps the ratchet constructed in the previous section behaves as a killed reflected brownian motion we will need in the sequel some functionals of that process , such as expected killing time or expected position at killing . to this end we need to compute the corresponding green function . let us first give a short description on how the green function of a killed diffusion can be computed ; for details we refer to chapter ii in @xcite or chapter 4 in @xcite . [ rem : green - gener ] + let @xmath176 be a reflected diffusion process with killing on state - space @xmath177 associated with infinitesimal operator @xmath178 acting on @xmath179 as follows @xmath180 we consider in this paper the cases @xmath181 or @xmath182 and @xmath183 . since in both cases the killing time is almost surely finite the resulting diffusions are transient . the speed and the killing measures of @xmath184 are given by ( see e.g. * ? ? ? * ) @xmath185 where @xmath186 . let @xmath187 denote the transition density of @xmath184 with respect to the speed measure . then the green function of @xmath184 is defined by @xmath188 in the transient regular case ( the latter means here that every point in @xmath177 can be reached with positive probability starting from any other point ) the green function of @xmath184 is positive and finite . it is obtained in terms of two independent solutions @xmath189 and @xmath190 of the differential equation @xmath191 that are both unique up to a constant factor and satisfy the following conditions * @xmath189 is positive and strictly decreasing with @xmath192 as @xmath193 , * @xmath190 is positive and strictly increasing , * @xmath194 ( this condition is for the reflecting boundary ) . the wronskian , defined by @xmath195 is independent of @xmath78 . thus , the functions @xmath189 and @xmath190 can be chosen so that their wronskian equals one . then , the green function of @xmath184 is given by @xmath196 in our computations we will in principle not need the exact expressions for @xmath189 , @xmath190 and @xmath197 . in particular in the case of the ornstein - uhlenbeck process , where these solutions depend in a complicated manner on the model parameters , we will not compute the function @xmath190 explicitly . asymptotic bounds at infinity will suffice for our purposes . in the following remark we collect some properties of the airy functions that will be needed in the sequel . for further properties we refer to @xcite ( cf . also remark 5.2 in @xcite ) . [ rem : airy ] the airy functions @xmath33 and @xmath198 are two linearly independent solutions of the differential equation @xmath199 we will only need the properties of the airy functions on @xmath177 . on that domain the functions are positive , @xmath33 is decreasing with @xmath200 and @xmath198 is increasing with @xmath201 . the wronskian is independent of @xmath78 and is given by @xmath202 the integral of @xmath33 on @xmath203 is @xmath204 we will also need the function @xmath205 for fixed @xmath19 and @xmath206 we define a function @xmath207 by @xmath208 using , and positivity of @xmath33 and @xmath198 on @xmath177 we have for @xmath50 @xmath209 since the functions @xmath210 and @xmath33 are bounded on @xmath177 we may define @xmath211 [ rem : killed - rbm ] let @xmath212 denote a reflecting brownian motion with drift @xmath21 starting in @xmath50 ( see ) under the law @xmath213 and let @xmath214 denote the corresponding expectation . furthermore using an exponentially distributed rate @xmath215 random variable @xmath216 independent of @xmath217 we define the _ killing time _ by @xmath218 then _ reflecting brownian motion with infinitesimal drift @xmath21 killed at rate @xmath219 _ is defined as the process @xmath220 with @xmath221 for @xmath222 and @xmath223 for @xmath224 for some @xmath225 , often referred to as the _ cemetery state . _ the infinitesimal operator @xmath226 corresponding to @xmath227 acts on @xmath228 functions @xmath229 satisfying @xmath230 as follows @xmath231 in view of the scaling property it is enough to prove the results for the @xmath23-brownian ratchet in a particular case . in what follows we assume @xmath232 and fix @xmath19 . in this case the speed and killing measure corresponding to @xmath233 are given by @xmath234 + the [ lem : green - krbm ] green function of @xmath235 is given by @xmath236 where @xmath237 are defined by @xmath238 with @xmath239 as explained in remark [ rem : green - gener ] the green function @xmath197 is obtained in terms of solutions of @xmath240 where @xmath241 is defined in . the functions @xmath189 and @xmath190 defined in are two independent solutions of . in lemma [ lem : prop - phi - psi ] we show that they satisfy conditions ( i ) and ( ii ) from remark [ rem : green - gener ] , whereas ( iii ) holds by the choice of @xmath242 . it remains to show @xmath243 . using independence of the wronskian of @xmath78 and we obtain @xmath244 in particular , @xmath245 is also independent of @xmath61 . altogether the assertion of the lemma follows . let @xmath189 and @xmath190 be defined by . for [ lem : prop - phi - psi ] any @xmath19 the function @xmath189 is strictly decreasing and the function @xmath190 is strictly increasing in @xmath78 . properties of the airy function @xmath33 ( see remark [ rem : airy ] ) imply that for any @xmath19 the function @xmath189 is positive and that @xmath246 as @xmath247 . we will show that @xmath248 . to this end , it is enough to show that for @xmath19 , @xmath50 @xmath249 first we show that @xmath250 for @xmath251 . the assertion is true for @xmath46 and for @xmath252 we have @xmath253 . so if @xmath254 is positive on some interval then there is a local maximum in some @xmath255 such that on the one hand we have @xmath256 and on the other hand @xmath257 leading to a contradiction . now we fix @xmath19 and show @xmath258 for all @xmath50 . for @xmath259 it is true by the above argument . as @xmath260 we have @xmath261 . if @xmath262 has positive values in the interval @xmath263 then there is a local maximum @xmath264 such that @xmath265 and @xmath266 leading again to a contradiction . it remains to show that @xmath190 is increasing . by the choice of @xmath267 we have @xmath268 . let @xmath269 . for all @xmath270 and @xmath19 we have @xmath271 to see this note that , as we have shown above , @xmath272 and therefore @xmath273 the last inequality follows from the fact that @xmath198 and @xmath274 are increasing . we set @xmath275 and note that @xmath276 and @xmath277 solve the differential equation @xmath278 and that @xmath279 is up to a constant factor the unique decreasing solution of that equation satisfying @xmath280 as @xmath260 . furthermore we have @xmath281 and simple calculation shows @xmath282 [ rem : exp - kill ] + using the green function one can compute the mean killing time of the killed reflecting brownian motion starting in @xmath50 . it is given by @xmath283 & = \int_0^\infty g(x , y ) m(y ) \ , dy = \int_0^\infty g(x , y)2e^{-2\mu y } \ , dy \\ \intertext{which can be written as } \label{eq:42 } & = 2\left(\phi(x ) \int_0^x \psi(y)\,dy + \psi(x ) \int_x^\infty \phi(y)\ , dy \right ) . \end{aligned}\ ] ] furthermore the density of @xmath284 , i.e. the position at killing time is given by ( see * ? ? ? * ) @xmath285 [ lem : exp.m.killing ] + for @xmath286 and any @xmath50 we have @xmath287 < \infty . \end{aligned}\ ] ] set @xmath288 and recall the function @xmath207 ( for @xmath267 defined in ) and its bound @xmath289 in and . then follows from @xmath290 & = \int_0^\infty e^{\alpha y } g(x , y ) y e^{-2\mu y } \ , dy\\ & = \phi(x ) \int_0^x y \psi(y)e^{-(2\mu-\alpha ) y } \ , dy + \psi(x ) \int_x^\infty y \phi(y)e^{-(2\mu-\alpha ) y } \ , dy \\ & = e^{\mu x } \pi \bigl\{ai(\mu^2+x ) \int_0^x y e^{-(\mu-\alpha)y } ( bi(\mu^2+y ) + c ai(\mu^2+y ) ) \ , dy \\ & \qquad \quad + ( bi(\mu^2+x ) + c ai(\mu^2+x))\int_x^\infty y e^{-(\mu-\alpha)y } ai(\mu^2+y ) \ , dy \bigr\ } \\ & \le e^{\mu x } y^ * m ( x ) \le e^{\mu x } y^ * g^*. \end{aligned}\ ] ] [ lem:2nd.mom.kill.time ] + there is a positive finite constant @xmath291 such that for all @xmath50 @xmath292 < c^{{\textnormal}{kill}}. \end{aligned}\ ] ] since the killing time of the killed reflecting brownian motion starting in @xmath293 is bounded stochastically by the killing time of the killed reflecting brownian motion starting in @xmath81 , we have @xmath294 \le 1 + \mathbb e_0[\tau^2 ] . \end{aligned}\ ] ] by the kac s moment formula ( see e.g. * ? ? ? * ( 5 ) on p. 119 ) we have @xmath295 & = 2 \int_0^\infty g(0,x ) m(x ) \int_0^\infty g(x , y ) m(y ) \ , dy \ , dx . \\ \intertext{now using again \eqref{eq : def.m } and \eqref{eq : g - star } as in the proof of lemma~\ref{lem : exp.m.killing } we obtain } { \mathbb{e}}_0[\tau^2 ] & \le 4 g^ * \int_0^\infty g(0,x ) m(x ) e^{\mu x } \ , dx = 8 g^*\pi \psi(0 ) \int_0^\infty ai(\mu^2+x ) \,dx . \\\intertext{since $ ai$ is decreasing we obtain } { \mathbb{e}}_0[\tau^2 ] & \le 8 g^ * \pi \psi(0 ) \int_0^\infty ai(x ) \ , dx = \frac{8 g^ * \pi}{3}\psi(0 ) , \end{aligned}\ ] ] where the last equality follows from . [ lem : e_x < x+c ] + for any @xmath50 @xmath296\leq x + \phi(0 ) \psi(0 ) + \mu { \mathbb{e}}_0[\tau ] . \end{aligned}\ ] ] by and the definition of @xmath276 and @xmath297 in respectively we have @xmath298 & = \int_0^\infty y^2 g(x , y)e^{-2\mu y } \,dy\\ & = \phi(x ) \int_0^x y^2 \psi(y ) e^{-2\mu y } \,dy + \psi(x ) \int_x^\infty y^2 \phi(y ) e^{-2\mu y } \ , dy \\ & = \phi(x ) \int_0^x y^2 \psi(y ) \,dy + \psi(x ) \int_x^\infty y^2 \phi(y ) \ , dy . \end{aligned}\ ] ] now using the fact that @xmath276 and @xmath297 satisfy , integration by parts , and the fact that @xmath299 we arrive at @xmath298 & = \phi(x ) \int_0^x \left(y \psi''(y)+ 2 \mu y\psi'(y)\right ) \ , dy + \psi(x ) \int_x^\infty \left(y \phi''(y)+ 2 \mu y\phi'(y)\right)\ , dy \\ & = x \left[\phi(x)\psi'(x)- \psi(x)\phi'(x)\right ] \\ & \qquad \quad + \phi(x ) \left[-\psi(x)+ \psi(0 ) + 2 \mu x \psi(x ) - 2 \mu \int_0^x \psi(y)\,dy\right ] \\ & \qquad \qquad + \psi(x ) \left[\phi(x ) - 2 \mu x \phi(x ) - 2 \mu \int_x^\infty \phi(y)\,dy\right ] \\ \\ & = x+ \phi(x ) \psi(0 ) - 2 \mu\left(\phi(x ) \int_0^x \psi(y)\,dy + \psi(x ) \int_x^\infty \phi(y)\ , dy \right)\\ & = x+ \phi(x ) \psi(0 ) - \mu \int_0^\infty g(x , y)m(y ) dy \\ & = x+ \phi(x ) \psi(0 ) - \mu { \mathbb{e}}_x[\tau ] . \end{aligned}\ ] ] here the next to last equality follows from and from @xmath300 . since @xmath189 is decreasing and @xmath301 \leq { \mathbb{e}}_0[\tau]$ ] the assertion follows . in this subsection we consider the increments of the brownian ratchet at the jump times of the boundary process . we show that they constitute a markov chain with unique invariant distribution and compute the expected jump time and the expected killing position under the invariant distribution . [ eq : mkjt ] + let @xmath22 be @xmath23-brownian ratchet with sequence of jump times of @xmath123 given by @xmath150 . we define the markov chain @xmath302 of increments at jump times by @xmath303 since for any @xmath304 the law of @xmath305 depends on @xmath306 only through @xmath307 , @xmath308 is indeed a markov chain . [ prop : ex - uniq - inv ] there exists a unique invariant distribution of the markov chain @xmath309 . while uniqueness of an invariant distribution is guaranteed by the coupling result in lemma [ lem : coupling ] , to prove existence we need to show that the moments of @xmath310 are bounded for all @xmath311 . this implies then tightness of the sequence and also tightness of the cesro averages of the laws . weak limits of subsequences of the latter are invariant distributions of the markov chain . boundedness of the moments of @xmath312 follows from lemma [ lem:2nd.mom.kill.time ] and that of the moments of @xmath313 ( and @xmath314 since it has the same distribution as @xmath313 ) follows inductively from lemma [ lem : e_x < x+c ] . for details we refer to the proof of proposition 5.6 in @xcite . [ prop : exp - under - pi ] let @xmath315 be the invariant distribution of @xmath316 and let @xmath317 denote the expectation with respect to that distribution . then there is a constant @xmath318 so that @xmath319 & = -\frac1k(\mu ai(\mu^2)+ai'(\mu^2)),\\ \intertext{and } \label{eq:36 } { \mathbb{e}}_\nu[\eta_1 ] & = \frac { 2ai(\mu^2)}k . \end{aligned}\ ] ] let @xmath320 denote the density of @xmath315 with respect to lebesgue measure . let @xmath321 be killed reflecting brownian motion with drift starting in random value @xmath322 with increments independent of @xmath184 . furthermore let @xmath52 be uniformly distributed on @xmath323 . invariance of @xmath315 implies that for killing time @xmath14 of @xmath324 we have @xmath325 as in ( * ? ? ? * section 5.3 ) from that one can obtain the following recurrence equation @xmath326 and then compute @xmath327 where for the last equality we used , equations for @xmath328 , @xmath329 and . thus , @xmath330 integrating we obtain @xmath331 the integration constant is zero because from , and we see that @xmath332 . by the density @xmath320 must be strictly decreasing . up to a constant factor the positive decreasing solution of is @xmath279 and it follows that @xmath333 from it follows @xmath334= \int_0^\infty x f_\nu(x)\,dx & = \int_0^\infty ( f''_\nu(x ) + 2\mu f'_\nu(x))\ , dx \\ & = -f'_\nu(0)-2\mu f_\nu(0 ) = \frac\pi k(\mu ai(\mu^2)-ai'(\mu^2)-2\mu ai(\mu^2 ) ) \\ & = - \frac \pi k(\mu ai(\mu^2 ) + ai'(\mu^2 ) ) , \end{aligned}\ ] ] ( replace here @xmath335 by @xmath336 to get ) and @xmath337 & = 2\int_0^\infty f_\nu(x ) \int_0^\infty e^{-2 \mu y } g(x , y)\ , dy \ , dx \\ & = 2 \int_0^\infty f_\nu(x ) \bigl ( \phi(x ) \int_0^x \psi(y)\ , dy + \psi(x ) \int_x^\infty \phi(y ) \ , dy\bigl ) \ , dx . \\ \intertext{now using fubini 's theorem and then \eqref{eq:7 } we have } { \mathbb{e}}_\nu[\eta_1 ] & = 2 \int_0^\infty \bigl(\psi(y ) \int_y^\infty f_\nu(x)\phi(x ) \ , dx + \phi(y ) \int_0^y f_\nu(x ) \psi(x ) \ , dx\bigl ) \ , dy \\ & = - 2 \int_0^\infty f'_\nu(x)\ , dx = 2 f_\nu(0)= 2\frac{\phi(0 ) } k = 2\frac{\pi ai(\mu^2)}{k}. \end{aligned}\ ] ] again replacing @xmath335 by @xmath336 we get to . in this subsection we define a regeneration structure of the @xmath23-brownian ratchet and show that the second moments of the regeneration times and of the corresponding spatial increments are finite . [ def : cumu1 ] + given a brownian ratchet @xmath338 with @xmath339 , @xmath50 we define a sequence of regeneration times as follows @xmath340 then @xmath341 almost surely and we have @xmath342 in the case @xmath259 . furthermore the sequence @xmath343 is iid . we define @xmath344 then we have @xmath345 that is , @xmath30 is a type a cumulative process @xmath346 with remainder @xmath347 ( see * ? ? ? * ) . it is well known ( see e.g. @xcite cf . also remark 6.1 in @xcite ) that to prove the law of large numbers and the central limit theorem for @xmath348 we need to show that the second moments of @xmath349 and @xmath350 are bounded ( this is done in propositions [ prop : rho - bounds ] and [ prop : x - rho - bounds ] ) . then , to carry the result over to @xmath0 we have to prove that the remainder @xmath351 is asymptotically negligible ( this is done in proposition [ prop : asympa ] ) . [ prop : rho - bounds ] there exists a positive constant @xmath352 such that for all @xmath50 @xmath353 \le r^ * \ ; \text { and } \ ; \mathbb e_x [ ( \rho_1-\rho_0)^2 ] \le r^*. \end{aligned}\ ] ] in the case @xmath259 we have @xmath342 . consider the case @xmath270 . let @xmath354 be the hitting time of @xmath81 of the brownian motion with drift @xmath21 started in @xmath215 . this hitting time has exponential moments ( we used this fact already in the proof of lemma [ lem : coupling ] ) . furthermore set @xmath355 and note that @xmath356 , where @xmath357 is independent exponential random variable with rate @xmath358 , because as long as @xmath359 the rate at which the reflection boundary jumps into the interval @xmath360 $ ] ( and therefore @xmath172 occurs ) is @xmath358 . but @xmath172 also occurs if @xmath30 hits the interval @xmath361 $ ] . it follows that for any initial positions @xmath50 , @xmath362 is bounded stochastically by the sum of independent random variables @xmath357 and @xmath354 , both having exponential moments and not depending on @xmath78 . thus , @xmath363 $ ] is bounded by a constant not depending on @xmath78 . we write @xmath364 and argue that each of the terms in the brackets has bounded second moments . note that @xmath365 is the first jump time of the reflection boundary after @xmath362 . thus , the finiteness of its second moment follows from lemma [ lem:2nd.mom.kill.time ] and that the bound there , also does not depend on @xmath78 . the finiteness of the second moment of @xmath366 follows by the same argument as the finiteness of the second moment of @xmath362 . [ prop : x - rho - bounds ] there exists a positive constant @xmath367 so that for any @xmath50 @xmath368 \le r^ { * * } \ ; \text { and } \ ; { \mathbb{e}}_x [ ( x_{\rho_1}-x_{\rho_0})^2 ] \le r^{**}. \end{aligned}\ ] ] recall that before touching the reflection boundary the process @xmath30 behaves as a brownian motion with drift @xmath21 and is therefore bounded below by @xmath81 and above by a brownian motion without drift . applying the second wald identity and proposition [ prop : rho - bounds ] ( with @xmath352 from that proposition ) we get @xmath369 \le { \mathbb{e}}_x[b_{\rho_0}^2 ] = { \mathbb{e}}_x[\rho_0 ] \le \sqrt{{\mathbb{e}}_x[\rho_0 ^ 2 ] } \le \sqrt{r^*}. \end{aligned}\ ] ] now we write @xmath370 and note that as in the second moment of the first term is bounded by @xmath371 . the second moment of the second term is finite according to lemma [ lem : exp.m.killing ] with a bound independent of @xmath78 . taking @xmath372 to be the larger of these two bounds follows . [ prop : asympa ] we have @xmath373 we omit the proof here since it is almost the same as the proof of proposition 6.7 in @xcite where the corresponding result was shown for the brownian ratchet without drift . ( note that the definition of @xmath313 there should be @xmath374 } { \lvertx_t - x_{\rho_{n-1}}\rvert}$ ] . also the denominator in the last two displays of that proof should be @xmath375 instead of @xmath12 . ) here we only sketch the proof and refer for details to section 7 in @xcite . in the case @xmath64 , the law of large numbers in theorem [ theoremrbm ] holds since @xmath0 is then a reflecting brownian motion with negative drift @xmath21 bounded stochastically by a reflecting brownian motion without drift . therefore @xmath376 a.s . as @xmath377 . hence , we assume @xmath40 in the rest of the proof . we use the regeneration structure from definition [ def : cumu1 ] and set @xmath378 , \\ m & \coloneqq{\mathbb{e}}_x[x_{\rho_1}-x_{\rho_0 } ] , \\ \beta^2 & \coloneqq \text{var}_x\big[x_{\rho_1}-x_{\rho_0}-\frac{(\rho_1-\rho_0)m}r\big ] . \end{split}\end{aligned}\ ] ] here , @xmath379 and @xmath380 are independent of @xmath78 due to the regeneration structure . according to propositions [ prop : rho - bounds ] and [ prop : x - rho - bounds ] the temporal and spatial increments @xmath349 and @xmath350 have finite second moments . from that and proposition [ prop : asympa ] it follows @xmath381 furthermore , using the clt for cumulative processes ( see e.g. * ? ? ? * ; * ? ? ? * ) and proposition [ prop : asympa ] we obtain that for all @xmath382 @xmath383 where @xmath276 denotes the distribution function of the standard normal distribution , and @xmath380 and @xmath384 are as defined in . hence , the central limit theorem holds for @xmath385 . it remains to compute @xmath386 . to this end we use the ratio limit theorem for harris recurrent markov chains ( see e.g. * ? ? ? let @xmath315 denote the invariant distribution for @xmath387 . using the ratio limit theorem we obtain that @xmath388}{{\mathbb{e}}_\nu[\eta_1 ] } , \end{aligned}\ ] ] where for the second equality we have used proposition [ prop : asympa ] . we recall that @xmath389={\mathbb{e}}_\nu[y_1]$ ] . let @xmath390 , @xmath391 , denote the speed @xmath386 as a function of @xmath61 and @xmath392 . in the case @xmath393 we obtain from proposition [ prop : exp - under - pi ] @xmath394 } { { \mathbb{e}}_\nu[\tau_1 ] } = -\frac{\mu ai(\mu^2)+ai'(\mu^2)}{2ai(\mu^2 ) } = -\frac12\bigl(\frac{ai'(\mu^2)}{ai(\mu^2)}+\mu\bigr ) . \end{aligned}\ ] ] now let @xmath88 and @xmath395 be given . using the scaling property ( see lemma [ lem : scaling ] ) we obtain @xmath396 thus , we have @xmath397}{t } = ( 2\gamma)^{1/3}\lim_{t\to\infty } \frac{{\mathbb{e}}\left[x^{1/2,(2\gamma)^{-1/3}\mu}_{(2\gamma)^{2/3}t}\right]}{(2\gamma)^{2/3}t } \\ & = ( 2\gamma)^{1/3 } v\bigl((2\gamma)^{-1/3}\mu,\frac12\bigr ) \\ & = -\frac{(2\gamma)^{1/3}}{2 } \bigl(\frac{ai'((2\gamma)^{-2/3}\mu^2)}{ai((2\gamma)^{-2/3}\mu^2 ) } + ( 2\gamma)^{-1/3}\mu\bigr ) \end{aligned}\ ] ] which concludes the proof of theorem [ theoremrbm ] . in this section we carry out the same program for the ornstein - uhlenbeck ratchet as we did for the brownian ratchet . the arguments in many of the proofs here are similar to the corresponding proofs in the previous section . therefore some proofs in this section will be sketchy . let us first recall the definition of the reflecting ornstein - uhlenbeck ( ou ) process with infinitesimal drift @xmath25 and unit variance . suppose that @xmath398 is a standard brownian motion starting in @xmath81 and let @xmath7 . then using the representation of the ou process as a time changed brownian motion , we obtain that @xmath399 , defined by @xmath400 is a reflecting ou process with infinitesimal drift @xmath25 and unit variance starting in @xmath264 . it is a diffusion process on @xmath177 associated with infinitesimal operator @xmath401 acting on @xmath402 as follows : @xmath403 the graphical construction in the following definition ( see figure [ fig : ou - ratchet ] ) is different from the graphical construction of the brownian ratchet . here we use a family of independent brownian motions to construct reflected ou processes between the jumps of the ratchet . at any jump time a new reflected ou process starts in an initial value chosen uniformly between the state of the previous process and zero . then we stick this `` peaces '' together to obtain the ou ratchet . [ def : gr - constr - our ] let @xmath404 be a poisson process and let @xmath405 be independent standard brownian motions . we define a sequence of stopping times @xmath406 and a sequence of ornstein - uhlenbeck processes @xmath407 with @xmath408 reflecting at @xmath81 as follows : @xmath409 given @xmath410 and @xmath411 for some @xmath100 we set @xmath412\times \{t\ } \ne \emptyset\bigr\ } \end{aligned}\ ] ] and let @xmath413 be the space component of the almost surely unique element of @xmath414\times \{{\widehat}\tau_{n}\}$ ] . furthermore we set @xmath415 and for @xmath416 we define @xmath417 finally we set for @xmath418 @xmath419 note that by construction we have @xmath420 stochastically . the following lemma is the analogue of lemma [ lem : gr - constr - br ] . though the graphical construction there is somewhat different the proof is similar and will be omitted here . [ lem : gr - constr - our ] the process @xmath421 is a @xmath23-ornstein - uhlenbeck ratchet starting in @xmath35 . now we construct a coupling of two ornstein - uhlenbeck ratchets starting in @xmath422 , where we assume @xmath423 without loss of generality . let the poisson process @xmath89 and a sequence @xmath424 of independent standard brownian motions be given as before . to construct the coupling @xmath425 set @xmath426 , @xmath427 , @xmath428 and define the sequences @xmath429 and @xmath430 and the processes @xmath431 and @xmath432 as in definition [ def : gr - constr - our ] . furthermore define @xmath433 , @xmath434 , @xmath435 and @xmath436 as in . then define the coupling time by @xmath437 since we can use the same brownian motions and the same poisson process for both ratchets we have @xmath438 for all @xmath439 . thus , on the event @xmath440 , there are @xmath441 such that for @xmath162 @xmath442 the following lemma shows that the coupling is successful with probability one . [ lem : exp - mom - coupl - our ] for any @xmath47 there is @xmath443 so that @xmath444 < \infty . \end{aligned}\ ] ] we only need to consider the case @xmath445 . in that case @xmath446 for all @xmath12 . furthermore , from it follows that @xmath433 is stochastically dominated by the reflected ornstein - uhlenbeck process @xmath447 . thus , @xmath448 is stochastically bounded by the hitting time of @xmath81 , say @xmath449 , by the process @xmath447 . for @xmath449 we have ( see * ? ? ? * ) @xmath450 = \mathrm{erf}(x_0/\sqrt{2(e^{2\mu t } -1 ) } ) , \end{aligned}\ ] ] where @xmath451 . thus , as @xmath452 we obtain @xmath453 = \frac2{\sqrt\pi } \frac{x_0}{\sqrt{2(e^{2\mu t } -1 ) } } + o ( e^{-3\mu t } ) \le x_0 c e^{-\mu t } \end{aligned}\ ] ] for suitably chosen positive @xmath267 . in this subsection we carry out analogous computations to those in subsection [ sec : green ] ; recall in particular remark [ rem : green - gener ] . [ rem : killed - rou ] let @xmath454 denote reflecting ornstein - uhlenbeck process with infinitesimal drift @xmath25 and unit infinitesimal variance starting in @xmath50 and let @xmath455 denote the corresponding law on the paths space and @xmath456 the expectation under this law . furthermore using an exponentially distributed rate @xmath215 random variable @xmath216 independent of @xmath457 we define the _ killing time _ by @xmath458 the _ reflecting ornstein - uhlenbeck process killed at rate @xmath459 _ is defined as the process @xmath460 with @xmath461 for @xmath222 and @xmath462 for @xmath463 where @xmath225 is the _ cemetery state . _ the infinitesimal operator @xmath464 corresponding to @xmath465 acts on @xmath228 functions @xmath229 satisfying @xmath230 as follows @xmath466 the speed measure and the killing measure corresponding to the killed ornstein - uhlenbeck process are given by @xmath467 in the case of ornstein - uhlenbeck ratchet the _ confluent hypergeometric functions _ play a similar role as the airy functions in the case of brownian ratchet . in the following remark we collect some of their properties that will be needed in the sequel . we refer to ( * ? ? ? 13 ) for most of the properties and for more information on confluent hypergeometric functions . [ rem : confl - funct ] assume that @xmath468 and consider the kummer equation ( also known as the confluent hypergeometric equation ) @xmath469 note that solutions also exist in the case @xmath470 but some of relations that we are going to recall in this remark and use in the following may not hold in this case . _ standard solutions : _ one standard solution of that equation is given by the function @xmath471 which is defined by @xmath472 where @xmath473 is the pochhammer s symbol defined by @xmath474 and @xmath475 for @xmath100 . another standard solution is given by the function @xmath476 which can be defined ( in the case @xmath477 ) by @xmath478 where @xmath479 denotes the gamma function . if @xmath480 , then @xmath52 and @xmath207 are independent solutions of . for @xmath481 we have @xmath482 and the second summand on the right hand side of vanishes . thus , as is easily seen from and , in that case both @xmath52 and @xmath207 are polynomials which are equal up to a multiplicative constant . the system of independent solutions in the case @xmath481 is given by @xmath483 _ behaviour in the neighbourhood of zero : _ for any @xmath484 @xmath485 for @xmath486 @xmath487 for @xmath488 @xmath489 _ behaviour at infinity : _ for @xmath490 and @xmath491 @xmath492 here , as usual , we write @xmath493 if @xmath494 as @xmath495 . furthermore @xmath496 note also that @xmath52 is uniquely determined by this property . _ bounds for positive zeros of @xmath52 and @xmath207 : _ if @xmath497 then @xmath207 has no zeros on @xmath177 . let @xmath498 be the number of positive zeros of @xmath499 . first we note that using the kummer transformation @xmath500 we get @xmath501 if @xmath502 , @xmath503 and @xmath504 are non - integers , @xmath505 and @xmath506 then @xmath507 . let @xmath508 and @xmath509 . if @xmath264 is a positive zero of @xmath499 then by ( 2.19 ) in @xcite @xmath510 _ differentiation formulas : _ we have @xmath511 _ recurrence relation : _ there are many recurrence relations for @xmath52 and @xmath207 . we will need the following ( it follows from ( 13.4.25 ) in @xcite and ) @xmath512 by straightforward computation one can show that if a function @xmath513 is a solution of the kummer equation with @xmath514 and @xmath515 , then @xmath516 is a solution of @xmath517 as in the case of killed brownian motion with negative drift there are positive solutions @xmath518 and @xmath519 of on @xmath177 with wronskian @xmath520 such that @xmath521 then the green function of the killed ornstein - uhlenbeck process is given by @xmath522 we define the function @xmath523 by @xmath524 for general @xmath392 and @xmath61 the solutions of with @xmath514 and @xmath515 with @xmath514 are hard to deal with . thus , we will not formulate an analogue of lemma [ lem : green - krbm ] for the killed reflecting ornstein - uhlenbeck process . in the following lemma we will identify the decreasing solution @xmath518 . for @xmath519 we will assume that for any given @xmath525 there is a suitable linear combination , say @xmath526 , of @xmath527 if @xmath528 , so that the function @xmath529 defined by @xmath530 satisfies the condition in . [ lem : whphi - decr ] the function @xmath531 defined on @xmath177 by @xmath532 is a positive decreasing solution of with @xmath533 as @xmath260 . since @xmath518 is of the form it is a solution of . by it is clear that @xmath534 for large @xmath78 and @xmath535 as @xmath260 . if we show that @xmath536 for @xmath537 , i.e. that @xmath518 is convex , then it follows that @xmath518 is positive and decreasing on @xmath263 . we have @xmath538 where the last step follows by . again by it is clear that @xmath536 for large @xmath78 . thus , to show that @xmath539 for @xmath540 it is enough to show that this function has no positive zeros . using and the definition of @xmath207 one can easily compute that @xmath541 and @xmath542 . thus in the case @xmath543 we have @xmath544 for @xmath50 . let us now consider the case @xmath545 . then by the number of positive zeros of @xmath546 equals the number of positive zeros of @xmath547 . since @xmath548 by remark [ rem : confl - funct](iv ) , the function @xmath549 has no positive zeros , which implies that @xmath550 for @xmath540 . if @xmath551 then , by all positive zeros of @xmath552 are bounded by @xmath553 . since the set @xmath554 is empty , the function @xmath555 is positive on @xmath556 . in the following example we compute the function @xmath519 in a special case . assume @xmath557 , let @xmath518 be as in and define @xmath558 and @xmath559 by the choice of @xmath267 we have @xmath560 as required . furthermore , remark [ rem : confl - funct](iii ) implies that @xmath561 for @xmath562 . to show that @xmath519 satisfies the conditions from we need to show convexity of @xmath519 , i.e. positivity of @xmath563 on @xmath177 and @xmath564 . using we obtain @xmath565 by assumption we have @xmath566 and therefore remark [ rem : confl - funct](iv ) implies that the function @xmath567 has no zeros on @xmath263 . since by remark [ rem : confl - funct](iii ) this function is positive for large @xmath78 we obtain @xmath568 and @xmath569 for @xmath570 . in lemma [ lem : whphi - decr ] we have shown that @xmath555 is positive and @xmath571 is negative on @xmath263 . together with positivity of @xmath572 and @xmath573 it follows @xmath574 finally we have @xmath575 here @xmath576 is a constant independent of @xmath78 , and since @xmath577 , @xmath578 and @xmath579 are positive for large @xmath78 and @xmath580 is negative this constant must be positive . note also that , as is easily computed , @xmath581 . in order this to be one , we would need to normalize @xmath519 by that value . together with @xmath582 it follows @xmath564 as required . let @xmath519 and @xmath518 be as defined in and . for any positive @xmath392 and @xmath61 there are finite positive @xmath583 and @xmath584 such that @xmath585 assume first that @xmath586 . then two independent solutions of the equation are given by . in view of , , and it is clear that any linear combination of the functions from multiplied by @xmath587 is bounded on any interval of the form @xmath588 $ ] for @xmath589 . furthermore , as @xmath590 , it grows at most as a constant times @xmath591 from which the first part of the assertion follows . if @xmath528 , then two independent solutions of the equation are given by the functions in . using similar arguments as above one can show the first part of the assertion also in this case . the second part of the assertion follows from the definition of @xmath518 and . we set @xmath592 and note that @xmath593 and @xmath594 solve the differential equation @xmath595 from it follows @xmath596 furthermore , @xmath597 and @xmath598 [ rem : dens - kill - pos - ou ] + as in remark [ rem : exp - kill ] in the brownian case , the density of the position at killing time of the killed ornstein - uhlenbeck process starting in @xmath78 is given by @xmath599 the expected killing time is given by @xmath600 & = \int_0^\infty g(x , y ) m(y)\ , dy = \int_0^\infty g(x , y ) 2e^{-\mu y^2 } \ , dy \\ & = 2\left({\widehat}\phi(x ) \int_0^x { \widehat}\psi(y)\,dy + { \widehat}\psi(x ) \int_x^\infty { \widehat}\phi(y)\ , dy \right ) . \end{split } \end{aligned}\ ] ] [ lem : exp.m.killing.oup ] + for @xmath601 and any @xmath50 we have @xmath602 < \infty . \end{aligned}\ ] ] we have @xmath603 & = \int_0^\infty e^{\alpha y } g(x , y ) 2 \gamma y e^{-2\mu y^2 } dy \\ & = 2 \gamma \left[{\widehat}\phi(x ) \int_0^x y e^ { \alpha y}{\widehat}\psi(y ) \ , dy + { \widehat}\psi(x ) \int_x^\infty y e^{\alpha y } { \widehat}\phi(y ) \ , dy \right ] . \end{aligned}\ ] ] using and we see that the term in the brackets is bounded by @xmath604 \end{aligned}\ ] ] which itself can be bounded by @xmath605 for some constant @xmath606 independent of @xmath78 . [ lem : kil - pos - ou ] + there is a positive finite constant @xmath607 such that for all @xmath608 @xmath609 \le x+{\widehat}c . \end{aligned}\ ] ] we have @xmath610 & = 2 \gamma \int_0^\infty y^2 g(x , y ) e^{- \mu y^2 } \ , dy\\ & = { \widehat}{\phi}(x)\int_0^x 2 \gamma y^2 { \widehat}{\psi}(y ) \ , dy + { \widehat}\psi(x ) \int_x^\infty 2 \gamma y^2 { \widehat}\phi(y ) \ , dy . \end{aligned}\ ] ] let us first consider the two integrals . the functions @xmath593 and @xmath594 satisfy , which can be rewritten as @xmath611 using this ( twice in each computation ) and partial integration we obtain @xmath612 and @xmath613 now from @xmath614 , and it follows @xmath615&= ( x-\frac\mu\gamma ) \left({\widehat}{\phi}(x){\widehat}\psi'(x ) -{\widehat}\psi(x ) { \widehat}\phi ( x)\right ) + { \widehat}\phi(x ) { \widehat}\psi(0 ) + \frac\mu\gamma { \widehat}\phi(x ) { \widehat}\psi'(0 ) \\ & = x-\frac\mu\gamma + { \widehat}\phi(x ) { \widehat}\psi(0 ) \le x + \frac\mu\gamma + { \widehat}\phi(0 ) { \widehat}\psi(0 ) \end{aligned}\ ] ] where the last inequality follows because @xmath518 is decreasing . this concludes the proof . [ lem : kill - time - ou ] for all @xmath50 we have @xmath616 < \infty$ ] . as in the proof of lemma [ lem:2nd.mom.kill.time ] it is enough to show that @xmath617 $ ] is finite . we have @xmath618 & = 2 \int_0^\infty g(0,x ) m(x ) \int_0^\infty g(x , y ) m(y ) \ , dy \ , dx . \end{aligned}\ ] ] it follows that ( recall the definition of @xmath619 in ) @xmath620 now using an estimate for @xmath50 @xmath621 ( which can be deduced from @xcite ) , and we obtain after simple calculations @xmath622 for suitably chosen constant @xmath623 depending on @xmath392 and @xmath61 but independent of @xmath78 . putting this into equation we get @xmath624 \le 2 c^2(\gamma,\mu ) \end{aligned}\ ] ] and the proof is completed . also for the ornstein - uhlenbeck ratchet we consider the markov chain of increments at jump times . [ eq : mkjt - ou ] let @xmath625 be a @xmath23-ornstein - uhlenbeck ratchet with sequence of jump times of @xmath626 given by @xmath429 . at jump times we define the markov chain @xmath627 by @xmath628 now , using the moment bounds in lemma [ lem : exp - mom - coupl - our ] , lemma [ lem : kil - pos - ou ] and lemma [ lem : kill - time - ou ] the next result follows as in the case of the brownian ratchet . [ prop : ex - uniq - inv.oup ] there exists a unique invariant distribution of the markov chain @xmath629 . our next aim is to compute the moments under this invariant distribution . first we show that the invariant density of the @xmath630 component satisfies a differential equation . [ lem : our - density ] let @xmath315 be the invariant distribution of @xmath631 and let @xmath632 be the corresponding density . then @xmath632 is the unique positive decreasing solution of @xmath633 satisfying @xmath634 and @xmath635 . similar to the case of killed brownian motion with negative drift one can write down a recurrence equation for @xmath632 . we have @xmath636 the equations are analogous to the case of the brownian ratchet with negative drift . we only give some details on how we obtain . differentiating and using the fact that @xmath593 and @xmath594 solve we obtain @xmath637 thus , using , and we obtain @xmath638 which shows . now integrating @xmath639 we obtain @xmath640 here , the integration constant is zero because by and it follows that @xmath641 . by @xmath642 is decreasing . this concludes the proof . recall @xmath643 in . one can check that the general solution of is given by @xmath644 multiplied by a linear combination of @xmath645 if @xmath646 . in view of the modulus of any solution containing a non - zero proportion of @xmath207 in the above cases behaves ( up to a polynomial factor ) as @xmath647 as @xmath260 and therefore can not converge to @xmath81 for @xmath648 . thus , the density of the invariant distribution of the @xmath630 component of the markov chain @xmath649 is given by @xmath650 the following result shows that @xmath632 defined in satisfies conditions stated in lemma [ lem : our - density ] . we have 1 . @xmath651 is positive on @xmath177 , 2 . @xmath652 , 3 . @xmath653 , 4 . @xmath654 for @xmath540 . let @xmath655 be given . throughout the proof we write @xmath656 for @xmath651 . the assertions ( ii ) and ( iii ) hold because @xmath656 is a solution of and because by @xmath657 as @xmath260 . to show ( i ) we first observe that by @xmath658 for sufficiently large @xmath78 . thus , it is enough to show that @xmath656 has no positive zeros . by the kummer transformation the function @xmath656 has positive zeros if and only if the function @xmath659 has positive zeros which is not the case as we have seen in the proof of lemma [ lem : whphi - decr ] . to show ( iv ) we show that @xmath656 is convex , i.e. @xmath660 for @xmath540 . then ( iv ) follows from ( i ) and from @xmath661 as @xmath260 . for @xmath570 we have @xmath662 where the last equality follows from . now again by the kummer transformation @xmath663 is positive on @xmath263 if and only if the function @xmath664 is positive on that interval . this was shown in lemma [ lem : whphi - decr ] . the proof in the case of the ornstein - uhlenbeck ratchet is almost the same as in the case of the brownian ratchet with negative drift . that is , we can again define a sequence of regeneration times and show that the temporal and spatial increments of the ratchet between this regeneration times have finite second moments . all the ingredients needed for the proof of that have been provided in the previous subsections . we content ourselves with computation of the speed of the ratchet . to this end we need ( as in proposition [ prop : exp - under - pi ] ) to compute the expectation of @xmath665 and of @xmath666 under the invariant distribution @xmath315 . using we obtain @xmath667 & = \int_0^\infty x { \widehat}f_\nu ( x ) \ , dx = \frac1{2\gamma } \int_0^\infty \left({\widehat}f''_\nu(x ) + 2\mu x { \widehat}f_\nu'(x)\right ) \ , dx \\ & = \frac1{2\gamma } \left(- { \widehat}f'_\nu(0 ) - 2\mu \int_0^\infty { \widehat}f_\nu(x)\ , dx\right ) = \frac1{2\gamma } \left(- { \widehat}f'_\nu(0 ) - 2\mu\right ) . \end{aligned}\ ] ] furthermore , recalling , we have using fubini s theorem in the second equality and in the third @xmath668 & = 2 \int_0^\infty { \widehat}f_\nu(x ) \left({\widehat}\phi(x ) \int_0^x { \widehat}\psi(y)\,dy + { \widehat}\psi(x ) \int_x^\infty { \widehat}\phi(y)\ , dy \right ) \ , dx \\ & = 2 \int_0^\infty \left({\widehat}\psi(y ) \int_y^\infty { \widehat}f_\nu(x ) { \widehat}\phi(x ) \,dx + { \widehat}\phi(y ) \int_0^y { \widehat}f_\nu(x ) { \widehat}\psi(x ) \ , dx \right ) \ , dy \\ & = - \frac1\gamma \int_0^\infty { \widehat}f_\nu'(y ) \ , dy = \frac{{\widehat}f_\nu(0)}{\gamma}. \end{aligned}\ ] ] now the speed of the ornstein - uhlenbeck ratchet is given by @xmath669}{{\mathbb{e}}_\nu[{\widehat}\eta_1]}= - \frac{{\widehat}f'_\nu(0 ) + 2\mu}{2{\widehat}f_\nu(0 ) } = - \frac{h'_{\mu,\gamma}(0)}{2h_{\mu,\gamma}(0 ) } - \frac{\mu \int_0^\infty h_{\mu,\gamma}(x)\ , dx } { h_{\mu,\gamma}(0)}.\end{aligned}\ ] ] the authors would like to thank peter pfaffelhuber and martin kolb for fruitful discussions . a. depperschmidt , n. ketterer and p. pfaffelhuber . a brownian ratchet for protein translocation including dissociation of ratcheting sites ( 2012 ) . preprint available under ` http://de.arxiv.org/pdf/1107.5219v1 ` . l. gatteschi . new inequalities for the zeros of confluent hypergeometric functions . in _ asymptotic and computational analysis ( winnipeg , mb , 1989 ) _ , volume 124 of _ lecture notes in pure and appl . _ , pages 175192 . dekker , new york ( 1990 ) .

we consider two reflecting diffusion processes @xmath0 with a moving reflection boundary given by a non - decreasing pure jump markov process @xmath1 . between the jumps of the reflection boundary the diffusion part behaves as a reflecting brownian motion with negative drift or as a reflecting ornstein - uhlenbeck process . in both cases at rate @xmath2 for some @xmath3 the reflection boundary jumps to a new value chosen uniformly in @xmath4 $ ] . since after each jump of the reflection boundary the diffusions are reflected at a higher level we call the processes _ brownian ratchet _ and _ ornstein - uhlenbeck ratchet_. such diffusion ratchets are biologically motivated by passive protein transport across membranes . the processes considered here are generalisations of the brownian ratchet ( without drift ) studied in @xcite . for both processes we prove a law of large numbers , in particular each of the ratchets moves to infinity at a positive speed which can be computed explicitly , and a central limit theorem .

the second quantum revolution @xcite is dawning . the long - anticipated fundamental advantages brought about by quantum technologies in applications such as secure communication , precise sensing and metrology , are starting to materialize thanks mainly to the impressive progresses in the experimental control of light and matter at the quantum level @xcite . on the other hand , some very central issues are still to be addressed at the theoretical level , which can be summarized into one straight question : which quantum features are ultimately needed to outperform the operation of classical devices ? metrology @xcite is one field where considerable debate around such a question has been spurned in recent years . while in some metrological setups entangled probes can lead to an extra gain in precision for the estimation of unobservable parameters compared to separable probes @xcite , such an enhancement can fade away under the most common sources of noise @xcite . at the same time , other means to achieve supraclassical performances even without using entanglement have been devised @xcite . somehow disappointingly , one might then conclude that quantum correlations in the form of entanglement are neither necessary nor sufficient for quantum - enhanced metrology in general . here we focus on a specific , highly relevant metrological setting , namely optical interferometry @xcite . the most pressing mission of optical interferometry is arguably the revelation of weak phase shifts induced by gravitational waves @xcite . optical interferometric setups traditionally involve a mach zender interferometer , in which a relative phase is acquired between the two arms and needs to be detected at the output @xcite . it is convenient to model theoretically such a setup as a dual - arm channel , where a phase shift is applied to one arm only , while the identity operation is applied to the other arm @xcite . note that , in practice , the implementation of the scheme requires an additional phase reference beam ; see e.g. the discussion in @xcite . if , as it is customary , the generator of the phase shift is known _ a priori _ , then tailored nonclassical resources such as single - mode squeezed states or two - mode entangled states can be exploited to improve the precision of phase estimation beyond the classical shot noise level . the mathematical techniques for assessing the ultimate precision limits allowed by quantum mechanics for parameter estimation are beautifully rooted in information geometry and find widespread applications @xcite . in this paper , we explore optical interferometry in a black box setting where the generator of the phase shift on one arm is not known _ a priori_. the aim of this analysis is to elucidate exactly which characteristics of continuous - variable quantum states are necessary and sufficient for them to act as sensitive probes to not just one , but a variety of possible local dynamics . we identify such essential characteristics with genuinely quantum correlations between the two modes entering the interferometer , of a general type commonly referred to as quantum discord @xcite . this finding resonates with the analogous one for finite - dimensional systems , where the novel paradigm of black box metrology has been very recently introduced @xcite . the worst - case precision enabled by a two - party probe state in a black box phase estimation setting defines the interferometric power of the state ( section [ sec2 ] ) . remarkably , we obtain a closed formula for this operational quantifier in the relevant instance of two - mode gaussian probes ( which include squeezed and thermal states ) , thus assessing their potential usefulness for practical sensing technologies ( section [ sec3 ] ) . we then identify gaussian states which offer , in principle , optimized performances in optical metrology with or without entanglement and even in the presence of high thermal noise at the probe preparation stage ( section [ sec4 ] ) . let us begin by formalizing the black box paradigm @xcite for optical interferometry @xcite . we consider a bosonic continuous - variable system of two modes @xmath0 and @xmath1 , respectively described by the annihilation operators @xmath2 and @xmath3 . we can define a vector of quadrature operators ( in natural units , @xmath4 ) as @xmath5 , where @xmath6 and @xmath7 ( and similarly for mode @xmath1 ) . the canonical commutation relations are then compactly expressed as @xmath8 = i \omega_{kl}$ ] , with @xmath9 being the two - mode symplectic form @xcite . a two - mode state @xmath10 is prepared as the input of an interferometer , see fig . [ gipfig ] . mode @xmath0 enters a black box where it undergoes a unitary transformation @xmath11 whose full specifics are unknown _ a priori_. in analogy with the finite - dimensional case @xcite , we need to restrict the generator @xmath12 to have a nondegenerate spectrum , in order to avoid trivial dynamics . in the present continuous - variable setting , the most natural and maximally informative choice for the spectrum of @xmath12 is a harmonic one . with this prescription , we can then decompose the black box transformation as follows without any loss of generality , @xmath13 where @xmath14 is a standard phase shift generated by the number operator @xmath15 , and @xmath16 is an arbitrary unitary transformation . the transformed state of the two modes after the black box is @xmath17 one can then perform a measurement on the output state , to construct an estimator @xmath18 for the parameter @xmath19 . one can iterate the probing trial a large number @xmath20 of times ( or equivalently , one can run @xmath20 parallel experiments if one has the availability of @xmath20 independent copies of the black box ) , to improve the statistical accuracy of the estimator . in mathematical terms , the variance of any estimator for the parameter @xmath19 , defined as @xmath21 , is constrained by the cramr - rao bound @xcite , @xmath22 where the quantity at the denominator is known as the quantum fisher information ( qfi ) @xcite and can be interpreted as the squared speed of evolution of the probe state @xmath23 under an infinitesimal transformation @xmath24 @xcite . under a smoothness hypothesis , the qfi can be defined as @xcite @xmath25 via the uhlmann fidelity @xcite @xmath26\right\}^2\,.\ ] ] for single - parameter estimation , the bound in ( [ cramerrao ] ) is asymptotically tight ( for @xmath27 ) . this means that the qfi directly quantifies the precision ( intended as the inverse of the variance of the estimator per trial ) that can be achieved with the input probe state @xmath10 , for the estimation of the parameter @xmath19 embedded in the local transformation @xmath28 , by means of a specific optimized measurement on the output state @xmath23 . for this reason , the qfi is conventionally adopted as the figure of merit in quantum metrology @xcite . with this in mind , the interferometric power ( ip ) of the state @xmath10 , with respect to the probing mode @xmath0 , is then defined as @xmath29 where the @xmath30 is a normalization factor adopted here for consistency with the finite - dimensional definition of ip @xcite . the quantity @xmath31 evaluates the worst - case precision guaranteed by using @xmath10 as a probe , where the minimization runs over all possible choices of local dynamics generated by a hamiltonian @xmath12 with harmonic spectrum . in practice , probe states @xmath10 with higher ip embody more reliable resources for metrology , as they ensure a smaller variance in the estimation of @xmath19 even if uncontrollable unitary fluctuations @xmath16 occur in conjunction with the designed phase shift @xmath32 ; in general , this can happen even in absence of entanglement @xcite . notice that , by definition , the ip is invariant under local unitary operations , @xmath33 = { \cal p}^a[\rho_{ab}]$ ] @xcite . this follows by observing that unitaries on @xmath1 are irrelevant for the qfi , while unitaries on @xmath0 can be reabsorbed in the minimization of eq . ( [ ipcv ] ) . notice however that , in spite of the convexity of the qfi , the ip is _ not _ convex . one can namely show the following inconclusive chain of inequalities . given two states @xmath10 and @xmath34 and a probability @xmath35 , one has @xmath36 \\ & \leq & { \cal f}[p \rho^{\phi,\hat{v}_a}_{ab } + ( 1-p ) \tau^{\phi,\hat{v}_a}_{ab } ] \\ & \leq & p { \cal f}[\rho^{\phi,\hat{v}_a}_{ab } ] + ( 1-p ) { \cal f}[\tau^{\phi,\hat{v}_a}_{ab } ] \\ & \geq & 4 p { \cal p}^a[\rho_{ab } ] + 4(1-p ) { \cal p}^a[\tau_{ab}]\,,\end{aligned}\ ] ] where we used the definition of ip in the first and last inequalities , and the convexity of the qfi in the middle one . in particular , one can construct straightforward examples where a state with nonzero ip is obtained by mixing two states @xmath10 and @xmath34 with zero ip ; this happens when the minimum @xmath16 is different for @xmath10 and @xmath34 . in the following , we restrict our attention to a fully gaussian scenario . namely , the probe states @xmath10 are assumed to be two - mode gaussian states , and the local dynamics @xmath37 is assumed to be gaussian ( also known as linear ) , i.e. , preserving the gaussian character of the states it acts upon . it is in order to recall that a gaussian state @xmath10 is represented by a gaussian characteristic function in phase space , and is completely specified by the first and second moments of the quadrature operators @xcite , collected respectively in the vector @xmath38 and in the covariance matrix @xmath39 , where @xmath40 $ ] and @xmath41 $ ] ( with @xmath42 ) . as the first moments can be adjusted by local displacements , and since the ip is invariant under local unitary operations , in what follows we can consider without any loss of generality states with zero first moments @xmath43 , described entirely by their covariance matrices . the latter will correspond to physical states in the hilbert space provided the _ bona fide _ condition @xmath44 which incarnates the robertson - schrdinger uncertainty relation , is satisfied . concerning the local dynamics , the gaussianity restriction amounts to requiring that the generator @xmath12 be at most quadratic in the canonical operators @xmath45 . given the decomposition in ( [ blackbox ] ) , and noting that @xmath32 is already a gaussian unitary , this requirement is passed on @xmath16 . in general , up to irrelevant displacements , a gaussian unitary @xmath16 is associated via the metaplectic representation to a symplectic transformation ( i.e. a real matrix which preserves the symplectic form ) acting by congruence on covariance matrices @xcite . by virtue of this correspondence , together with well established results of symplectic algebra and gaussian quantum information @xcite , we can now translate the scheme of fig . [ gipfig ] and the above equations at the phase space level , as follows . the probe state @xmath10 will be described by its covariance matrix @xmath46 . the black box unitary @xmath37 corresponds to a symplectic transformation @xmath47 , with @xmath48 being a phase - space rotation of an angle @xmath19 in phase space . furthermore , @xmath49 can be written in general according to the euler decomposition @xcite , @xmath50 , where @xmath51 , with @xmath52 , is a squeezing transformation . in this way , eq . ( [ blackbox ] ) translates into @xmath53 . from eq . ( [ rhophi ] ) , the transformed state after the black box has a covariance matrix given by @xmath54 the gaussian ip is of a two - mode gaussian probe with covariance matrix @xmath46 is then defined as @xmath55 the fidelity @xmath56 between two ( undisplaced ) two - mode gaussian states , eq . ( [ ulmafid ] ) , which enters in the definition ( [ qfifid ] ) of the qfi , can be computed from the respective covariance matrices @xmath57 and @xmath58 as @xcite @xmath59^\frac12\big\}^{-1}\,,\ ] ] where @xmath60,\\ \lambda & = & 16 \det[({\boldsymbol{\sigma}}_1+i { { \boldsymbol{\omega}}})/2 ] \det[({\boldsymbol{\sigma}}_2+i { { \boldsymbol{\omega}}})/2],\\ \upsilon & = & \det[({\boldsymbol{\sigma}}_1+{\boldsymbol{\sigma}}_2)/2].\end{aligned}\ ] ] notice that alternative yet related studies of qfi for gaussian states can be found e.g. in refs . @xcite . we now recall that , by local symplectic operations , every two - mode covariance matrix @xmath61 can be transformed in a standard form with all diagonal @xmath62 subblocks , @xmath63 , @xmath64 , @xmath65 , where @xmath66 . exploiting once more the invariance of the ( gaussian ) ip under local unitaries , we now proceed to evaluate eq . ( [ ipg ] ) on probe states with covariance matrix in standard form . in such case , the minimization over @xmath67 in ( [ ipg ] ) turns out to be solved simply by @xmath68 . the value of @xmath69 which yields the minimum in ( [ ipg ] ) is instead less trivial , and can be written as an analytical yet too cumbersome function of @xmath70 to be reported here @xcite . after some tedious algebra , we arrive at one of the main results of this paper : a closed formula for the gaussian ip of all two - mode gaussian states . this is independent of the standard form used for the explicit evaluation , and can be recast in terms of the four local symplectic invariants of an arbitrary covariance matrix , defined as @xmath71 , @xmath72 , @xmath73 , and @xmath74 . the formula reads @xmath75 where @xmath76 we can now analyze the properties of the gaussian ip for two - mode gaussian states . in @xcite , the ip has been proven to capture a peculiar nonclassical feature of bipartite states of a finite - dimensional system : their amount of quantum correlations beyond entanglement , of the so - called discord type @xcite . we will now show that the same interpretation holds in the infinite - dimensional gaussian case . first of all , the gaussian ip vanishes if and only the state is a zero - discord state ( also known as classical - quantum state ) @xcite . in the gaussian case , under a natural constraint of bounded mean energy per mode , the only classical - quantum states are product states @xcite . from eq . ( [ ipgg ] ) , we find indeed that the only two - mode gaussian states with vanishing gaussian ip are product states , characterized by the invariants @xmath77 . all correlated two - mode gaussian states are therefore useful for black box optical interferometry , returning a nonzero qfi for any possible local gaussian dynamics . furthermore , the gaussian ip is invariant under local unitary operations as already mentioned , and it can be shown to be monotonically nonincreasing under arbitrary gaussian quantum operations on subsystem @xmath1 . the proof follows from the definition of qfi and can be adapted from the finite - dimensional case @xcite . namely , suppose a gaussian probe state with covariance matrix @xmath78 is obtained from the state with covariance matrix @xmath46 by the action of a completely positive trace - preserving and gaussianity - preserving map ( a gaussian quantum channel ) on @xmath1 . any such a map commutes with the unitary phase shift applied on @xmath0 , so it can be moved after the black box and considered as part of the output measurement . since @xmath79 defines the optimal precision achieved by the best possible output measurement , the fisher information associated to @xmath80 can only be smaller or equal , which proves the claim . altogether , these properties allow us to conclude that the gaussian ip is a faithful measure of bipartite discord - type correlations @xcite for gaussian states . with the result of eq . ( [ ipgg ] ) , the ip becomes the only currently known faithful measure of discord - type correlations which is computable both for two - qubit states @xcite and for two - mode gaussian states , which are respectively the pillars of discrete - variable and continuous - variable bipartite quantum information processing . ( [ ipgg ] ) acquires a very simple form for states characterized by @xmath81 in standard form ( in which case the optimal @xmath69 in ( [ ipg ] ) reduces to @xmath82 ) , which include the relevant classes of squeezed thermal states ( @xmath83 ) and mixed thermal states ( @xmath84 ) : @xmath85 notice that in this simple case the gaussian ip is symmetric under swapping of the two modes @xmath0 and @xmath1 , but this is not true for general two - mode gaussian states , as clear from eq . ( [ ipgg ] ) . for pure states , specified by @xmath86 , one has in particular @xmath87 , which is a monotonic function of the marginal mixedness of each subsystem . this means that the gaussian ip reduces to a gaussian entanglement monotone @xcite on pure states . this is , once more , a desired property for a discord - type quantifier , and holds analogously in the finite - dimensional case @xcite . for @xmath88 entangled ( lighter ) and separable ( darker ) gaussian states . the standard quantum limit @xmath89 ( solid line ) and the heisenberg limit @xmath90 ( dashed line ) are indicated . , height=200 ] it is particularly interesting to study the scaling of the worst - case qfi , namely the gaussian ip , with the mean photon number of the probing subsystem @xmath0 , @xmath91=\frac{{\text{tr}}\ , { { \boldsymbol{\alpha}}}-2}{4}\,,\ ] ] which conventionally defines the resource count in optical interferometry @xcite . a numerical exploration of random two - mode gaussian states @xmath46 , as shown in fig . [ gipvsn ] , reveals two distinct regimes . as expected , separable probe states can never surpass the standard quantum limit ( or shot noise limit ) , given by a linear scaling of the ip with @xmath92 ; entangled states , on the other hand , can have ip scaling at most quadratically with @xmath92 , reaching up to the so - called heisenberg limit . a class of states with the maximum possible gaussian ip in absence of entanglement , for instance , is given in standard form by @xmath93 , in the limit @xmath94 ; for these states , @xmath95 spanning the solid line in fig . [ gipvsn ] . entangled states with maximum gaussian ip at fixed @xmath92 are instead pure two - mode squeezed states , sitting on the dashed line in fig . [ gipvsn ] , for which ( as mentioned before ) @xmath96 however , there are a considerable number of entangled states which perform worse than shot noise , which means that their entanglement does not translate into a practical quantum enhancement for black box metrology . motivated by the above observation , we perform a thorough analysis of the interplay between the gaussian ip @xmath97 , rescaled by the local mean photon number @xmath92 , and the entanglement of two - mode gaussian states . the latter can be conveniently measured by the logarithmic negativity @xcite , which is a decreasing function of the smallest symplectic eigenvalue @xmath98 of the partial transpose of the covariance matrix , @xmath99 where @xmath100 with @xmath101 @xcite . [ gipvse ] shows a comparison between the two quantities , which reveals that @xmath102 is bounded from above and from below at fixed @xmath103 . to derive the expression of the bounds analytically , we start from the ansatz that the extremal states are to be found within the class of entangled squeezed thermal states . we can reparametrize their standard form covariances as @xmath104 , with @xmath105 , and perform a constrained optimization of @xmath106/[(a-1 ) ( a \tilde\nu + b \tilde\nu -\tilde\nu ^2 + 1)]\ ] ] at fixed @xmath107 , subject to the _ bona fide _ condition ( [ bonafide ] ) . we then find that the upper boundary ( solid line ) in fig . [ gipvse ] is given by states with @xmath108 in the limit @xmath109 , for which @xmath110 the lower boundary has two branches , see fig . [ gipvse ] . for @xmath111 ( where @xmath112 is the real root of the polynomial @xmath113 ) , i.e. @xmath114 , the extremal states ( dotted line ) have @xmath115/(1-\tilde\nu ) , b=\sqrt{2(\tilde\nu+1)}+\tilde\nu+2 $ ] , for which @xmath116 for @xmath117 , i.e. @xmath118 , the extremal states ( dashed line ) are pure two - mode squeezed states , described by @xmath119 , for which @xmath120 , plotted versus the logaritmic negativity @xmath103 for @xmath88 entangled gaussian states . the dashed line accommodates pure states . see text for details of the other boundary curves.,height=200 ] this analysis reveals several interesting facts which can be relevant for applications . first , there is a minimum threshold in entanglement to beat necessarily the shot noise limit in black box metrology : all two - mode gaussian states with @xmath121 achieve @xmath122 , while some less entangled states can be outperformed by separable , more discordant states . second , pure states eventually offer the _ worst _ possible metrological performance in black box optical interferometry for a given ( sufficiently high ) degree of entanglement . conversely , highly thermalized states such as the ones on the upper boundary of fig . [ gipvse ] can attain significantly higher gaussian ip per local mean photon number , at equal degree of entanglement . this is a very practical situation where the combined effect of entanglement and state mixedness surprisingly results in an enhancement of discord - type correlations useful for an operative task ( namely interferometry in this case ) , somehow giving shape to the abstract statistical predictions of ref . @xcite . finally , we like to point out that fig . [ gipvse ] is comparable to fig . 1 ( right panel ) of @xcite , which features the gaussian entropic discord versus the gaussian entanglement of formation , although the extremal states are different . in particular , for separable states both the gaussian discord and the gaussian ip divided by @xmath92 can reach at most one @xcite , while they are unbounded for entangled states . overall this confirms the intimate connection between ip and discord . in conclusion , we extended the paradigm of black box parameter estimation to the technologically important setting of optical interferometry . we defined the operative notion of interferometric power for a two - mode probe system , and specialized its definition to the case of gaussian states and local gaussian phase dynamics . we derived a closed formula for the gaussian interferometric power of all two - mode gaussian states . by studying it against the mean photon number and the entanglement of the probes , we singled out classes of extremal gaussian states which guarantee the best possible metrological precision in a worst - case scenario . these states can be highly thermalized , which eases the demands for their implementation in laboratory . this work develops a conceptual and practical advance for the characterization and exploitation of general nonclassical correlations in continuous - variable systems , and complementing ref . @xcite it shows that their role in metrology transcends specific schemes and hilbert space dimensions . the formalism applied here can be immediately useful to calculate other discord - type quantities for gaussian states , which capture geometrically their sensitivity under local unitary transformations , e.g. the local quantum uncertainty @xcite , the discriminating strength @xcite , and the discord of response @xcite ( see also @xcite ) . _ note added._upon completion of the present manuscript , a preprint appeared @xcite where a similar measure is independently defined , and explicitly computed only for the subclass of symmetric two - mode squeezed thermal states . a. farace , a. de pasquale , l. rigovacca and v. giovannetti , new j. phys . * 16 * , 073010 ( 2014 ) ; l. rigovacca , _ non - classical correlations in quantum states _ , msc thesis ( scuola normale superiore , pisa , 2014 ) . g. adesso , s. m. giampaolo , and f. illuminati , phys . a * 76 * , 042334 ( 2007 ) ; a. monras , g. adesso , s. m. giampaolo , g. gualdi , g. b. davies , and f. illuminati , phys . rev . a * 84 * , 012301 ( 2011 ) ; s. gharibian , phys . a * 86 * , 042106 ( 2012 ) ; s. m. giampaolo , a. streltsov , w. roga , d. bruss , and f. illuminati , phys . a * 87 * , 012313 ( 2013 ) ; w. roga , d. buono , and f. illuminati , arxiv:1407.7063 ( 2014 ) .

the interferometric power of a bipartite quantum state quantifies the precision , measured by quantum fisher information , that such a state enables for the estimation of a parameter embedded in a unitary dynamics applied to one subsystem only , in the worst - case scenario where a full knowledge of the generator of the dynamics is not available _ a priori_. for finite - dimensional systems , this quantity was proven to be a faithful measure of quantum correlations beyond entanglement . here we extend the notion of interferometric power to the technologically relevant setting of optical interferometry with continuous - variable probes . by restricting to gaussian local dynamics , we obtain a closed formula for the interferometric power of all two - mode gaussian states . we identify separable and entangled gaussian states which maximize the interferometric power at fixed mean photon number of the probes , and discuss the associated metrological scaling . at fixed entanglement of the probes , highly thermalized states can guarantee considerably larger precision than pure two - mode squeezed states .

@xcite derived the equivalence of mass and energy ( @xmath0 ) by considering an object of mass @xmath3 that simultaneously emits two electromagnetic packets , each with energy @xmath4 in opposite ( @xmath5 and @xmath6 ) directions . by momentum conservation in the rest frame of the object , it does not change its velocity after emission . seen from a frame moving at velocity @xmath7 ( i.e. , along the axis defined by the emissions ) , the two packets are each doppler shifted ( in opposite directions ) , so that the total energy of these packets is higher in the moving frame than the rest frame by @xmath8 , where @xmath9 . @xcite argued that by energy conservation , the object must lose energy in the moving frame by an amount that is greater than what it loses in the rest frame by exactly this difference . @xcite had already shown in his earlier paper introducing special relativity , that the kinetic energy of a mass @xmath3 moving at velocity @xmath10 is @xmath11 . appealing to this result , @xcite concluded that the mass of the emitting object must decline from @xmath3 to @xmath12 , where @xmath13 . here i show that the same result can be derived from conservation of momentum , without invoking any results from special relativity . that is , the derivation uses only effects that are first order in @xmath1 , and does not employ the second - order effects that characterize special relativity . consider as above an object emitting the two electromagnetic packets that , viewed in its rest frame are equal and opposite . by momentum conservation , the dual ejection leaves the object at rest in this frame . see figure [ fig : frames ] . now consider the same event from a frame that is moving _ perpendicular _ ( with velocity @xmath15 ) relative to the emission directions . by symmetry , the wave packets are still equal , but they are no longer opposite : because of the aberration of starlight ( first discovered by james bradley in 1729 ) , the packets will both appear to be moving slightly upward , at an angle @xmath16 . denote the emitted energies of the packets _ in the moving frame _ by @xmath4 , and denote the mass of object in this frame before and after emission by @xmath3 and @xmath17 . by a variety of arguments elaborated below , the magnitude of the momenta of the two packets in this frame are @xmath18 hence , because of aberration of starlight , the vertical components of these momenta will be ( to first order in @xmath1 ) @xmath19 equating the total @xmath20-momentum in the moving frame before and after emission yields , @xmath21 which can be solved to obtain , @xmath22 note that in carrying out this derivation , i explicitly ignored terms higher than first order in @xmath23 , in particular when i adopted @xmath24 . hence , the result strictly applies only in the limit @xmath25 , i.e. , in the rest frame . this can be expressed as an equivalence between energy and rest - mass , @xmath26 i address the question of how this result can be generalized to moving bodies in [ sec : moving ] . in the derivation , i used the relation between the energy @xmath27 and momentum @xmath28 for ( monodirectional ) electromagnetic fields , @xmath29 of course , this can be derived from special relativity , but the orientation here is to derive equation ( [ eqn : emc2 ] ) with no recourse to relativity , nor to concepts of a similar vintage , such as photons . @xcite recapitulates @xcite s manipulations of maxwell s equations to derive the electromagnetic energy flux density @xmath30 , where @xmath31 and @xmath32 are the electric and magnetic fields . he then develops a similar manipulation of maxwell s equations ( together with the lorentz force law ) to derive the momentum density @xmath33 , where @xmath34 is the magnetic induction . combining these two equations for monodirectional electromagnetic waves in free space yields equation ( [ eqn : pegamma ] ) . this shows that this relation rests directly on the maxwell / lorentz equations , although whether anyone actually derived the expression for @xmath35 prior to the simplification of vector notation is not clear . however , @xcite already uses @xmath36 for isotropic electromagnetic radiation in his thermodynamic derivation of stefan s law . here @xmath37 is the pressure and @xmath38 is the energy density . this expression already implies @xmath39 for monodirectional electromagnetic waves . as emphasized in [ sec : derivation ] , by carrying out the derivation only to first order in @xmath23 , i ultimately restricted its validity to bodies at rest . put differently , if the true relation between mass and energy had the form , @xmath40 , the derivation would have proceeded exactly the same way . there are two paths to generalizing the result to moving bodies . the first is to adopt the results of special relativity . this is the approach of @xcite , who derived @xmath0 using momentum conservation when light is emitted in an arbitrary direction . in special relativity , equation ( [ eqn : pgammamov ] ) is exact , so the derived relation between mass and energy is exact to all orders in @xmath23 . this approach is pedagogically useful : like einstein s derivation , it makes use of special relativity , but it is simpler and more direct . however , as a historical and logical exercise , one may also ask how equation ( [ eqn : em0c2 ] ) could have been generalized if it had been discovered prior to special relativity . such a generalization follows from a simple thought experiment . imagine a box filled with warm gas , whose thermal energy ultimately resides in the kinetic energy of the atoms . at the time , this picture was controversial but at least some physicists ( e.g. , boltzmann ) held to it . light is emitted from two holes in the box , similarly to the situation in [ sec : derivation ] . the energy of the light packets is drawn from the kinetic energy of the atoms in the box , some of which now move more slowly . by equation ( [ eqn : emc2 ] ) , the box has lost not only energy , but also mass . however , since the box contains no inter - atom potential energy , the mass ( i.e. , inertia ) of the box must be the sum of the mass ( inertia ) of the atoms in it . as the number of these has not changed , the mass of some of the atoms must have been reduced by exactly the amount of reduced mass of the box , which is exactly the same as the kinetic energy lost from these atoms divided by @xmath41 . that is , kinetic energy also contributes to inertia . up to this point , i have derived @xmath0 without ever making use of @xcite s postulate that @xmath42 is the same in all frames of reference , nor of any of the results that he derived from this postulate . i now show that special relativity , including the universality of @xmath42 , can be derived from this equation . first , @xcite shows that @xmath0 leads to the growth of inertia with velocity , @xmath43 . to permit clarification of a subtle point , i repeat that derivation here , beginning with the newtonian equation relating force to the increase of kinetic energy , @xmath44 . using the definitions , @xmath45 , @xmath46 , @xmath47 , this can be written @xmath48 , or @xmath49 substituting in the just derived @xmath0 yields @xmath50 at this point , there may be some question as to whether the one may pull `` @xmath42 '' out of the derivative , since it has not yet been shown to be `` constant '' . but @xmath42 is a _ constant _ in any one frame : the point that has not yet been addressed is whether it is _ invariant _ under frame changes . in the present case , the observer is not changing frames : it is the mass that is accelerating . the quantities @xmath51 , @xmath3 , @xmath10 , and @xmath42 are all as measured in the observer frame , which is inertial . we then obtain , @xmath52 whose solution is @xmath53 where @xmath54 is an integration constant , which we identify with the rest mass . from this point , it is straightforward to derive the other relations of special relativity by well - known arguments . for example , as a fast train passes by , a passenger and a bystander each throw tennis balls transverse to the motion of the train ( with equal strength ) in such a way that they hit and each bounces back directly to its respective thrower . the balls must each return at the speed they were launched or the train passenger could detect her own motion . thus , they must have equal and opposite momenta . the bystander reckons that the passenger s ball is more massive and therefore concludes it s transverse velocity is smaller , which can only be true if time passes more closely . by similar traditional arguments , one can go on to derive length contraction , etc . in this way , one can prove that the speed of light is the same in all frames of reference rather than assuming it . while the result obtained here is obviously not new , there are three reasons for establishing this result using a new derivation . first , the expression @xmath0 is zeroth order in @xmath1 , in sharp contrast to the majority of results from special relativity , which are second order . it seems more elegant , therefore , to derive this expression using first - order arguments , rather than relying on second - order expressions . second , because the derivation is more elegant , it has pedagogical value , i.e. , it is easier to transmit to students . third , because the derivation is independent of special relativity , it raises the question of why @xmath0 was not derived earlier than 1905 . in particular , the elements needed to derive it ( momentum conservation , aberration of starlight , and the proportionality between electromagnetic energy and momentum ) were all in place by 1884 . indeed , once one realizes that electromagnetic waves have momentum ( even if one does not yet know the exact expression for this quantity ) , it follows immediately from momentum conservation and aberration of starlight that a light - emitting object must lose mass . as reviewed by @xcite , during the 25 years before special relativity there were many efforts to express the mass of particles in terms of their energy divided by @xmath41 . but these differed from the arguments given here ( and that i have argued could have been given at least as early as 1884 ) by two important features . first , they generally centered around evaluations of the ultimately rather nebulous electromagnetic self - energy of charged particles rather than the kinetic properties of all matter ( charged or neutral ) . second , these evaluations did not recognize ( at least explicitly ) that when an object emitted energy , it also lost mass . indeed , the very complexity of the arguments developed in this era compared to the absolute simplicity of the derivation in [ sec : derivation ] makes it even more puzzling why no one hit on the latter .

the equivalence of mass and energy is indelibly linked with relativity , both by scientists and in the popular mind . here i prove that @xmath0 by demanding momentum conservation of an object that emits two equal electromagnetic wave packets in opposite directions in its own frame . in contrast to einstein s derivation of this equation , which applies energy conservation to a similar thought experiment , the new derivation employs no effects that are greater than first order in @xmath1 and therefore does not rely on results from special relativity . in addition to momentum conservation , it uses only aberration of starlight and the electromagnetic - wave momentum - energy relation @xmath2 , both of which were established by 1884 . in particular , no assumption is made about the constancy of the speed of light , and the derivation proceeds equally well if one assumes that light is governed by a galilean transformation . in view of this , it is somewhat puzzling that the equivalence of mass and energy was not derived well before the advent of special relativity . the new derivation is simpler and more transparent than einstein s and is therefore pedagogically useful .

in this paper i assume the gravitational system of units with @xmath17 , and the signature of the metric tensor @xmath18 . the schwarzschild anti - de sitter metric in standard polar coordinates @xmath19 has the form @xmath20 where the cosmological constant @xmath8 is negative , and @xmath21 denotes the mass of the black hole . alternatively , one can work in suitably chosen eddington finkelstein - type coordinates , that are regular at the horizon . an interested reader is referred to @xcite . the horizon of the black hole is located at @xmath22\ ] ] ( a real positive root of the equation @xmath23 ) . the motion of the fluid is described by the conservation laws @xmath24 where @xmath25 is the energy momentum tensor . here @xmath2 , @xmath1 , and @xmath5 denote the energy density , the pressure , and the baryonic density , respectively . in what follows we introduce the specific enthalpy @xmath26 ( note that the specific enthalpy is sometimes defined as @xmath27 , which is different form the convention used in this paper ) . the symbol @xmath28 denotes the components of the metric tensor ; @xmath29 are the components of the four - velocity of the fluid . for spherically symmetric and stationary flows @xmath30 , and all quantities appearing in eqs . ( [ wbc ] ) are functions of @xmath9 only . assuming that the solution is smooth , one can integrate eqs . ( [ wbc ] ) . this yields @xmath31 the above equations constitute a starting point for the discussion of the bondi - type accretion solutions . in this work i deal with two classes of equations of state . the isothermal equations of state @xmath0 , where @xmath32 is a constant @xcite , and polytropic equations of state @xmath4 , where @xmath33 and @xmath34 are constant . in both cases @xmath5 , @xmath2 , @xmath35 , and @xmath1 are related by simple formulas , provided that the flow of the gas is smooth , i.e. , there are no shock waves or contact discontinuities . for the isothermal equations of state one has @xmath36 where @xmath37 is a constant . in the case of the polytropic equation of state , it is easy to show that @xmath38 the choice of the constant @xmath39 is subject to the physical interpretation . in the following i assume standard normalization with @xmath40 . by setting @xmath41 one can recover eqs . ( [ agg ] ) . it is also important to stress that the conservation law @xmath42 in eqs . ( [ wbc ] ) is , de facto , the definition of @xmath5 . this quantity can have different physical interpretation , depending on the context . for the gas of a conserved number of massive particles , the density @xmath5 can be expressed as the number density of the particles times the mean rest - mass of the particle . on the other hand , the dynamics of the photon gas with the equation of state @xmath43 is described by equations @xmath44 only . in this case it is also possible to introduce a function @xmath5 satisfying the conservation law @xmath42 , provided that the flow of the gas is smooth . such a function has a natural interpretation of the specific entropy ( note that it is not conserved across possible discontinuities in the flow ) . by differentiating eqs . ( [ www ] ) with respect to @xmath9 one can obtain the following expression for @xmath45 : @xmath46 - \frac{m}{2r } + \frac{\lambda}{6 } r^2}{(u^r)^2 - c_s^2 \left [ 1 - \frac{2m}{r } - \frac{\lambda}{3}r^2 + ( u^r)^2 \right]},\ ] ] where @xmath47 , and @xmath48 is the speed of sound . one has @xmath49 for isothermal equations of state , and @xmath50 for polytropes . formally , this constitutes a `` step backwards '' in the process of finding of solutions , but it gives an insight into their structure . let @xmath51 be a function of @xmath9 such that @xmath52 \right\ } \nonumber \\ & \equiv & f_1(r , u^r ) . \label{dyn_a}\end{aligned}\ ] ] then @xmath53 - \frac{m}{2r } + \frac{\lambda}{6 } r^2 \right\ } \nonumber \\ & \equiv & f_2 ( r , u^r ) . \label{dyn_b}\end{aligned}\ ] ] equations ( [ dyn_a ] ) and ( [ dyn_b ] ) can be treated as a dynamical system , whose phase portrait consists of the graphs of @xmath11 versus @xmath9 , or more precisely , the graphs of @xmath12 belong to the orbits of eqs . ( [ dyn_a ] ) and ( [ dyn_b ] ) . the dynamical system defined by eqs . ( [ dyn_a ] ) and ( [ dyn_b ] ) has critical ( fixed ) points @xmath54 where @xmath55 , that is @xmath56 & = & 0 , \\ c_{s\ast}^2 \left [ 1 - \frac{2m}{r_\ast } - \frac{\lambda}{3}r_\ast^2 + ( u^r_\ast)^2 \right ] - \frac{m}{2r_\ast } + \frac{\lambda}{6 } r_\ast^2 & = & 0 . \label{critb}\end{aligned}\ ] ] here , and in what follows , the quantities referring to the critical point will be denoted with an asterisk . in order to find critical points one has to specify the equation of state . it is , however , clear that @xmath57 and also @xmath58 thus , one can easily show that at the critical point @xmath59 where @xmath60 is the radial component of the three - velocity . thence it is common to use the term `` sonic point '' instead of `` critical point '' . for @xmath61 , i.e. , in the schwarzschild case , and for standard equations of state , @xmath54 is a saddle point . thus , there is an accretion solution passing through the sonic point . in the following i will reserve the term `` sonic point '' for a critical saddle point , as opposed to other types of critical points . the existence and number of critical points will be discussed in sec . [ criticalpoints ] for the isothermal equations of state . it is quite illuminating to start with an analysis of the asymptotic behavior of isothermal solutions . for isothermal equations of state eqs . ( [ www ] ) yield @xmath62 where @xmath63 is a positive constant . let us assume an asymptotic expansion of the form @xmath64 . one gets @xmath65 there are two ways in which this equation can be satisfied in the leading order for @xmath66 . in both cases the term on the right - hand side has to cancel with the leading term at the left - hand side . if the leading order term on the left - hand side is @xmath67 , then @xmath68 , and @xmath69 ( the condition that the term @xmath70 is not the leading one ) . if the leading order term is @xmath70 , we have @xmath71 , and @xmath72 . these two possibilities lead to @xmath73 and @xmath74 , respectively . in either case @xmath75 . for @xmath76 the two asymptotics coincide ; one has @xmath77 . it can be seen from the examples given in @xcite that the two exponents @xmath78 correspond to the branches of the solution that are asymptotically subsonic and supersonic respectively ( note that for @xmath79 we have @xmath80 ) . if @xmath76 , the two branches still exist , but with the same asymptotic behavior . it is clear that for @xmath14 asymptotic behavior of the form @xmath64 is not admitted . this suggests that for @xmath14 global solutions do not exist . the following example confirms this expectation . and sample metric parameters @xmath82 , @xmath83 . the solid line depicts transonic solutions . the vertical line denotes the position of the horizon of the black hole . the graph shows solutions for @xmath84 , 1 , and 1.02 . ] an explicit homoclinic solution can be found for the equation of state @xmath81 . for this equation of state eq . ( [ yyy ] ) can be written as @xmath85 here i will be mainly interested in transonic solutions . in this case the value of @xmath63 should be determined by requiring that the solution passes through the sonic point ( a saddle critical point of the dynamical system defined by eqs . ( [ dyn_a ] ) and ( [ dyn_b ] ) ) . the location of the sonic point @xmath86 is given by @xmath87 or @xmath88 ( cf . [ criticalpoints ] ) . the above polynomial equation has real positive solutions only for @xmath89 , i.e. , when its discriminant is not positive . this follows from a very simple reasoning ; it is presented in sec . [ criticalpoints ] without restricting to @xmath15 only . for @xmath89 the location of the sonic point is given by @xmath90.\ ] ] the square of the radial component of the velocity at the sonic point is given by eq . ( [ trala ] ) ; the value of the constant @xmath63 appearing in eq . ( [ wwa ] ) that corresponds to transonic solutions can be expressed as @xmath91 equation ( [ wwa ] ) is a quartic polynomial equation in @xmath92 , and it can be solved exactly . let @xmath93 where @xmath94 is given by eq . ( [ wwb ] ) . the domain of the transonic solution is given by the condition @xmath95 . let @xmath96 denote the largest root of the equation @xmath97 , and let us set @xmath98 @xmath99 { \left ( \frac{a r}{4 } \right)^2 + \sqrt{\delta } } + \sqrt[3 ] { \left ( \frac{a r}{4 } \right)^2 - \sqrt{\delta}},\ ] ] where in the expression for @xmath100 we choose real roots . then the two branches of the transonic solution can be written as @xmath101 ( this branch is subsonic outside @xmath86 ) and @xmath102 for the branch that is supersonic outside @xmath86 . for values of the constant @xmath63 other than that given by eq . ( [ wwb ] ) the expressions @xmath103 give remaining accretion solutions , that are not transonic . once the expressions for @xmath11 are found , all other quantities can be trivially computed . the density @xmath5 is given by the second equation in eqs . ( [ www ] ) . the expressions for @xmath2 and @xmath35 are obtained from eqs . ( [ agg ] ) . examples of solutions for the equation of state @xmath81 are depicted in fig . 1 . here @xmath82 , @xmath83 , and the solutions are plotted for @xmath104 , and 1.02 , respectively . the discussion of the asymptotic behavior of isothermal solutions given in sec . [ sec_asymptotic ] and the solutions shown in fig . 1 suggest that for isothermal equations of state @xmath0 with @xmath105 one should expect the existence of two critical points on the phase diagram of @xmath106 , at least for a certain range of parameters . these should be a saddle point belonging to the homoclinic orbit , and a center enclosed by this orbit . i will now argue that this is indeed the case . in addition , a simple reasoning gives a restriction on the mass of the black hole allowing for transonic accretion . for isothermal equations of state @xmath0 eqs . ( [ crita ] ) and ( [ critb ] ) yield @xmath107 which is equivalent to the cubic equation @xmath108 dividing by @xmath109 one can write the above equation as @xmath110 where @xmath111 for @xmath112 and @xmath105 the above equation has a real negative root . this is a trivial consequence of the fact that @xmath113 and @xmath114 . thus , a real and positive root can exist only if the discriminant @xmath115 is nonpositive . a straightforward calculation shows that this condition is equivalent to @xmath116 which can be interpreted as an upper bound on the black hole mass allowing for the transonic accretion . in this case ( [ eqf ] ) has three real roots , two of them being positive . this last statement follows directly from the analysis of the complex roots in the cardano formula @xmath117{-q + i \sqrt{|w| } } + \sqrt[3]{-q - i \sqrt{|w|}}.\ ] ] conversely , for @xmath75 and @xmath112 the discriminant @xmath118 is always positive , and there is just one real root . this root is positive : in this case @xmath119 and , of course , @xmath120 . in principle , the analysis of the critical points can be pursued further by computing the jacobians of the right - hand side of eqs . ( [ dyn_a ] ) and ( [ dyn_b ] ) , i.e. , @xmath121 at critical points , and analyzing their eigenvalues . this can be easily done , say with _ wolfram mathematica_. one can show that the critical points are either a saddle ( jacobian ( [ jac ] ) has real eigenvalues with different signs ) or a center ( eigenvalues of ( [ jac ] ) are imaginary ) . more precisely , it can be shown that the eigenvalues are of the form @xmath122 , but the resulting expression for @xmath123 is lengthy , and the sign of @xmath123 is not immediately clear . a case - by - case study shows that the critical point with smaller @xmath86 is a saddle point ( @xmath124 ) , and the critical point with larger @xmath86 is a center ( @xmath125 ) , as expected . for the data depicted in fig . 1 , i.e. , @xmath15 , @xmath82 , and @xmath83 , the critical points are located at @xmath126 and @xmath127 . in the first case the eigenvalues of ( [ jac ] ) are @xmath128 . in the second they are @xmath129 . the mass limit ( [ masslimit ] ) is illustrated in fig . 2 . it shows a sequence of transonic solutions obtained for increasing values of the black hole mass and the equation of state @xmath81 . the solid line in this graph depicts the limiting case with the mass @xmath130 . note that the `` homoclinic loop '' gets smaller and smaller with the increasing mass , and it disappears for @xmath131 . sample numerical solutions with polytropic equations of state were given in @xcite , exhibiting characteristic homoclinic behavior . here i note that no global polytropic solutions exist ( i.e. , eqs . ( [ www ] ) can not be satisfied asymptotically ) , irrespective of the assumed parameters of the polytropic equation of state . this can be seen immediately , if one attempts to repeat the reasoning of sec . [ sec_asymptotic ] . from eqs . ( [ www ] ) and ( [ wwz ] ) one has @xmath132 the square root in the above formula behaves asymptotically at least as @xmath133 , due to the @xmath8 term . the only way in which this divergent behavior could be cancelled ( to yield the constant appearing on the right - hand side ) is to have @xmath35 vanishing sufficiently fast . this is clearly impossible since @xmath134 . of course , the unity in the expression for @xmath35 comes from the @xmath5 term in the formula for the energy density @xmath2 , i.e. , a contribution due to the rest - mass of the particles of the gas . and @xmath83 . the graphs correspond to four values of the black hole mass : @xmath135 , @xmath136 , @xmath137 , and @xmath131 , where @xmath138 is the maximal mass for which a sonic point exists . ] a very natural interpretation of the above results is to say that the existence of the global ( asymptotic ) solutions of the bondi - type accretion in the schwarzschild anti - de sitter spacetime is not directly related to the algebraic form of the equation of state . global solutions exist for relativistic matter models that can be associated with the gas of massless particles . in this context it is important to note that the equation of state @xmath43 , a limiting case among isothermal equations of state , is a simple well - known model of the photon gas . this interpretation is also confirmed by the observation that the term that forbids asymptotic solutions in the case of polytropic equations of state is directly connected with the non - vanishing rest - mass of the gas particles . of course , the simplest ( and perhaps slightly naive ) explanation would be to refer to the behavior of radial null and time - like geodesics in the schwarzschild anti - de sitter spacetimes . they are analyzed in detail in @xcite . time - like radial geodesics are described by @xmath139 where @xmath140 denotes the proper time , and @xmath141 is constant ( the energy ) . here @xmath142 plays a role of the effective potential . since it diverges asymptotically as @xmath16 , the range of the motion of the particle with finite energy @xmath141 is limited . in terms of the coordinate time the above equation can be written as @xmath143.\end{aligned}\ ] ] the equation describing null geodesics has the form @xmath144 meaning that a photon can travel to infinity . it is also interesting to note that a similar homoclinic behavior was observed in the spherically symmetric models of bondi - type accretion on reissner nordstrm black holes @xcite . in this case it is caused by the electric charge term , and the homoclinic loop is located inside the horizon of the black hole . as a by - product of the analysis presented in this note i obtained a restriction on the maximum mass of the black hole that allows for the transonic accretion of isothermal fluids with @xmath105 . this bound is given by a remarkably simple formula , and it is discussed in sec . [ criticalpoints ] . i am grateful to jerzy knopik and filip ficek for many fruitful discussions . 99 j. karkowski , e. malec , _ bondi accretion onto cosmological black holes _ d87 , 044007 ( 2013 ) p. mach , e. malec , j. karkowski , _ spherical steady accretion flows : dependence on the cosmological constant , exact isothermal solutions , and applications to cosmology _ , phys . d88 , 084056 ( 2013 ) p. mach , e. malec , phys . rev . d88 , _ stability of relativistic bondi accretion in schwarzschild-(anti-)de sitter spacetimes _ , 084055 ( 2013 ) h. bondi , _ on spherically symmetrical accretion _ , mon . not . r. astron . 112 , 195 ( 1952 ) f.c . michel , _ accretion of matter by condensed objects _ , astrophys . space sci . 15 , 153 ( 1972 ) t.k . das , b. czerny , _ hysteresis effects and diagnostics of the shock formation in low angular momentum axisymmetric accretion in the kerr metric _ , new astronomy 17 , 254 ( 2012 ) n. cruz , m. olivares , j.r . villanueva , _ the geodesic structure of the schwarzschild anti - de sitter black hole _ , class . . 22 1167 ( 2005 ) j. karkowski , b. kinasiewicz , p. mach , e. malec , z. wierczyski , _ universality and backreaction in a general - relativistic accretion of steady fluids _ , phys . d73 021503(r ) ( 2006 ) p. mach , e. malec , _ on the stability of self - gravitating accreting flows _ , phys . d78 , 124016 ( 2008 ) v.i . dokuchaev , yu.n . eroshenko , _ accretion with back reaction _ , d84 , 124022 ( 2011 ) j. karkowski , e. malec , k. roszkowski , z. wierczyski , _ transonic and subsonic flows in general relativistic radiation hydrodynamics _ , acta phys . b40 , 273 ( 2009 ) e. malec , t. rembiasz , _ general relativistic versus newtonian : a universality in spherically symmetric radiation hydrodynamics for quasistatic transonic accretion flows _ , phys . d 82 , 124005 ( 2010 ) d.b . ananda , s. bhattacharya , t.k . das , _ acoustic geometry through perturbation of mass accretion rate : i radial flow in general static spacetime _ , arxiv:1406.4262 ( 2014 ) a.r . amani , h. farahani , _ phantom accretion onto the schwarzschild anti de - sitter black hole _ , international journal of theoretical physics 51 , 1498 ( 2011 ) e.o . babichev , v.i . dokuchaev , yu.n . eroshenko , _ black hole mass decreasing due to phantom energy accretion _ , phys . 93 , 021102 ( 2004 ) g.c . colvero , m. ujevic , _ potential flows in the reissner - nordstrm(anti ) de sitter metric : numerical results _ d80 , 084010 ( 2009 ) s. chandrasekhar , _ a limiting case of relativistic equilibrium _ , in general relativity : papers in honour of j.l . synge , ed . l. oraifeartaigh , clarendon press , oxford 1972 e.o . babichev , v.i . dokuchaev , yu.n . eroshenko , _ perfect fluid and scalar field in the reissner - nordstrm metric _ , journal of experimental and theoretical physics 112 , 784 ( 2011 )

the aim of this paper is to clarify the distinction between homoclinic and standard ( global ) bondi - type accretion solutions in the schwarzschild anti - de sitter spacetime . the homoclinic solutions have recently been discovered numerically for polytropic equations of state . here i show that they exist also for certain isothermal ( linear ) equations of state , and an analytic solution of this type is obtained . it is argued that the existence of such solutions is generic , although for sufficiently relativistic matter models ( photon gas , ultra - hard equation of state ) there exist global solutions that can be continued to infinity , similarly to standard michel s solutions in the schwarzschild spacetime . in contrast to that global solutions should not exist for matter models with a non - vanishing rest - mass component , and this is demonstrated for polytropes . for homoclinic isothermal solutions i derive an upper bound on the mass of the black hole for which stationary transonic accretion is allowed . in a series of 3 papers @xcite karkowski , malec , and i have discussed stationary , bondi - type accretion flows in schwarzschild de sitter and schwarzschild anti - de sitter spacetimes . we derived analytic solutions for three `` isothermal '' equations of state of the form @xmath0 , where @xmath1 is the pressure , and @xmath2 denotes the energy density , with @xmath3 , and 1 . for polytropic equations of state of the form @xmath4 , where @xmath5 is the baryonic density and @xmath6 and @xmath7 are constant , we have computed numerical solutions . while in the sector of the positive cosmological constant @xmath8 all these solutions behave in a similar fashion , their behavior differ significantly for negative @xmath8 . in particular , isothermal transonic solutions with @xmath3 , and 1 cover the entire space outside the black hole horizon , i.e. , they can be continued to arbitrarily large radii ( similarly to standard bondi - type solutions in the schwarzschild case @xcite ) . the situation is different for polytropic solutions , and this behavior is illustrated in figs . 2 and 3 of @xcite . when a polytropic solution is continued outwards , one encounters a finite radius @xmath9 , where the derivative of the radial velocity component @xmath10 diverges . this behavior corresponds to the existence of homoclinic orbits in the phase diagram of the radius @xmath9 versus the radial component of the velocity @xmath11 . given the equations describing stationary bondi - type accretion one can construct a dynamical system with the phase portrait consisting of the graphs of different solutions @xmath12 describing the accretion flow @xcite . the homoclinic orbit of this dynamical system consists of two transonic solutions the solutions that pass through a saddle - type critical ( fixed ) point , at which the local value of the speed of sound is equal to the modulus of the three velocity of the fluid ( the so - called sonic point ) . given the above examples , one could have an impression that those homoclinic solutions are somehow essentially related to the assumed polytropic equation of state ( as opposed to the isothermal form ) . this is not quite true , and the aim of this paper is to clarify these issues . it is possible to show , and i do that in sec . [ sec_asymptotic ] of this paper , that for isothermal equations of state a power - law asymptotic behavior is admitted only if @xmath13 . this suggests the existence of homoclinic solutions also for isothermal equations of state with @xmath14 , and indeed an analytic example of such a solution can be given for @xmath15 ( sec . [ solk14 ] ) . on the other hand , one can show ( sec . [ sec_poly ] ) that global ( extending to infinity ) solutions do not exist for polytropic equations of state , irrespective of the assumed values of the speed of sound . interestingly , such solutions are not permitted due to the term in the expression for the specific enthalpy that represents the contribution from the rest - mass of gas particles . the above observations suggest an analogy with the properties of radial time - like and null geodesics in the schwarzschild anti - de sitter spacetime @xcite . for time - like geodesics the term proportional to @xmath16 in the effective potential forbids the motion of a particle with a finite energy to infinity . in contrast to that , a massless particle can travel to arbitrary large radii . it seems quite plausible that the occurrence of homoclinic solutions is generic , and it is characteristic for nonrelativistic matter , where the energy density has a nonvanishing contribution from the rest - mass of the gas particles . somewhat on the margin of the above considerations i show that for isothermal equations of state with @xmath14 transonic accretion solutions exist only for sufficiently small black holes . the appropriate limit on the mass of the black hole is derived in sec . [ criticalpoints ] . relativistic bondi - type accretion flows were investigated in many recent papers . for instance , self - gravitating flows were analyzed in @xcite , and also in @xcite . radiation transfer was included in @xcite . among very recent developments one should also notice @xcite . there are also papers dealing with schwarzschild anti - de sitter spacetimes . in @xcite amani and farahani discussed phantom accretion onto schwarzschild anti - de sitter black holes , basing on the idea presented in @xcite . accretion onto schwarzschild anti - de sitter black holes is also considered in @xcite .

in our fdtd simulations maxwell s equations are time - integrated on a computational grid , with a drude - lorentz model assumed for the dielectric function@xcite : @xmath63 where the first term is the drude free - electron contribution and the second contains lorentz oscillators corresponding to interband transitions . @xmath64 and @xmath65 are the free electron plasma frequency and relaxation time , @xmath66 , @xmath67 , @xmath68 are transition frequency , oscillator strength and decay rate for the lorentz terms . to accurately reproduce the experimental dielectric functions ( au from ref.@xcite and si from ref.@xcite ) we treat these as fit parameters . for au we use @xmath69=4 , and @xmath70=4.054 , @xmath67=(0.43 , 0.634 , 0.755 , 1.059 ) , @xmath71=8.76ev , @xmath72=0.068ev , @xmath73=(2.67 , 3.03 , 3.54 , 4.23)ev , and @xmath74=(0.458 , 0.641 , 0.892 , 0.959)ev . for si we use @xmath69=7 without any drude term : @xmath70=1.89 , @xmath67=(1.198 , 0.963 , 1.021 , 1.164 , 1.407 , 2.259 , 1.869 ) , @xmath73=(3.39 , 3.51 , 3.68 , 3.86 , 4.06 , 4.25 , 4.61)ev , and @xmath74=(0.188 , 0.203 , 0.239 , 0.269 , 0.283 , 0.265 , 0.0)ev . fig.[fig : theorys1 ] plots our model dielectric functions along with the experimental ones , showing an excellent agreement . the computational cell comprises a 2 nm cubic grid . perfectly matched layer ( pml ) boundary conditions@xcite are applied at the edges of the cell in the vertical direction . in the lateral direction , we use periodic boundary conditions for the uniform illumination case , and pml for the focused . the index of refraction of slg was measured by ellipsometry for wavelengths up to 750 nm in ref.@xcite . to extrapolate to longer wavelengths , we assume undoped slg and use the universal optical conductance @xmath75 , to get the dielectric function : @xmath76 where @xmath77=0.335 nm is slg s thickness and @xmath78 is the fine structure constant . in the thin film limit , eq.[eq : slg_index ] yields : @xmath79 a value of @xmath80 ensures that eq.[eq : slg_index ] matches the ellipsometric experimental data of ref.@xcite at smaller wavelengths . we fit a drude - lorentz model to both the ellipsometric data at small wavelengths and eq.[eq : slg_index ] at longer wavelengths , fig.[fig : theorys2]a . in fig.[fig : theorys2]b the resulting slg absorption is plotted . the drude - lorentz model for slg uses @xmath69=4 with @xmath70=2.148 , @xmath67=(64.8 , 2.92 , 1.69 ) , @xmath71=1.34ev , @xmath72=0.7ev , @xmath73=(1.0 , 4.0 , 4.56)ev , and @xmath74=(5.41 , 2.77 , 1.0)ev . within fdtd s 2 nm grid , slg is treated as an effective 2 nm thick slab , thus its dielectric function is scaled to reproduce the correct absorption and reflection properties according to @xmath81 , i.e. @xmath821.192 and @xmath83(10.85 , 0.488 , 0.283 ) . in a three - layer system , the normal incidence fresnel equations for reflection and transmission amplitudes are@xcite : @xmath84 where @xmath85 , @xmath86 and @xmath87 , with @xmath60 the film thickness . here we assume the incoming medium @xmath88 , the film @xmath89 and the semi - infinite substrate @xmath90 . reflection and transmission coefficients are @xmath91 and @xmath92 . in the thin film limit , using the slg dielectric function given by eq.[eq : slg_index ] , we get : @xmath93 where @xmath94 , @xmath95 is the film dielectric constant , @xmath78 is the fine structure constant , and @xmath96 is the n - layer graphene ( nlg ) thickness . this results into the absorption : @xmath97 while for a nlg suspended inside a uniform medium ( i.e. @xmath98 ) we find @xmath99 . these equations are valid for n<10@xcite . for @xmath100 the thin film limit breaks down and the optical paths inside the film must be taken into account . regarding reflection , this is typically dominated by the substrate since the difference in refractive index is usually largest between air and the substrate . however , in the @xmath98 case the third term in the right hand side of eq.[eq : refl ] is the only nonzero term , resulting into nlg reflection : @xmath101 . this is very small for slg ( @xmath102 at 600 nm for suspended slg@xcite ) but it increases quadratically with n. m spot illumination at normal incidence ( tm ) , for 500 , 600 , 700 nm . the color - coding is in logarithmic scale . ( b ) slg absorption distribution ( vertical axis ) as a function of illumination wavelength ( horizontal axis ) . the three wavelengths studied in ( a ) correspond to the three characteristic cases : absorption under the illumination spot , extended absorption under the contact , and absorption in the exposed slg far from the illumination spot . ( c ) the same as in ( b ) but for te polarization . ( d ) same as above for unpatterned contacts ( both polarizations yield identical results ) . absorptions are normalized to the incident light flux.,width=340 ] fig.[fig : theorys3 ] plots the response of our system considering the absorption in both exposed and covered slg for both polarizations and patterned and unpatterned contacts . for simplicity we only consider the @xmath21=410 nm case with 13 ridge periods . we find that in all cases , other than tm - excitation on the patterned contact , the absorption in the exposed slg is similar to that of a slg on top of sio@xmath4 in the absence of the contact ( green dotted line ) . interestingly , there is some absorption in the covered slg as well . in the tm - excitation of the patterned contact we obtain a similar modulation with wavelength as that in the exposed slg , albeit of significantly smaller magnitude . we note that , in order to directly compare with the exposed slg absorption , we still normalize to the power illuminating the exposed slg area . had we normalized to the flux illuminating the whole contact , then the covered slg absorption would appear with a much smaller magnitude than in fig.[fig : theorys3 ] . we also obtain absorption in the other cases as well , peaked at@xmath0500 nm . this , however , is irrelevant to the grating and spps , as it is there that au becomes most transparent . au is strongly absorptive at high energies because of the onset of interband transitions from its d - electrons@xcite while it is strongly reflective at small energies because of its conduction electrons@xcite . the inset in fig.[fig : theorys3]d plots the reflection , transmission and absorption coefficients through a 50 nm thick au film in air , displaying au s transparency window at@xmath0500 nm . we explore the case of focused illumination on the grating , which more closely resembles the experiments . to better facilitate the simulation and avoid having laterally scattered light re - enter the computational cell , we remove the lateral periodic boundary conditions and adopt pml boundary conditions@xcite , so that any light scattered towards the sides of the computational cell permanently exits the calculations . we also consider a much larger exposed slg area for better visualization . we adopt a tm polarized 1@xmath48m - wide plane source illuminating the 13-period @xmath21=410 nm grating at a non - symmetric position , as depicted in fig.[fig : theorys4 ] . the frequency domain electric - field intensity profiles for @xmath103=500 , 600 and 700 nm are shown in logarithmic scale in fig.[fig : theorys4]a . no scattering occurs at 500 nm , while the most intense is seen at 700 nm . the system s full response is shown in fig.[fig : theorys4]b , where we plot the slg absorption throughout the length of the structure ( vertical axis ) for different wavelengths ( horizontal axis ) . the illumination source is again a tm - polarized 1@xmath48 m wide spot as shown in the inset schematic . absorption is normalized to the peak incoming flux per unit area . three distinct regions emerge : up to 550 nm , absorption only occurs in the covered slg directly underneath the illumination source . for 550nm@xmath104650 nm , increased absorption is found in an extended area ( several microns away from the illumination spot ) in the covered slg , pointing towards light diffraction into spps in the au - sio@xmath4 interface . above 650 nm there is strong absorption in the exposed slg beyond the grating . this points towards light diffraction into spps in the au - air interface , which propagates and reaches the exposed slg at the grating s edge . in fig.[fig : theorys4]c we plot the same absorption map for te illumination of a patterned contact , while in fig.[fig : theorys4]d for illumination of an un - patterned contact . in both these cases no absorption is found , except directly underneath the illumination spot . it is thus clear that spp - mediated effects are dominating the response for tm - polarized illumination . the sio@xmath4 ( 300nm)/si substrate has two effects . first , it provides some interference - based enhancement in the slg absorption@xcite . fig.[fig : theorys5]a plots the exposed slg absorption for the system described in fig.1 on top of a 300 nm sio@xmath4/si substrate . the response is similar to the semi - infinite sio@xmath4 case , except for an overall modulation due to the interference effects in the sio@xmath4 dielectric spacer . the net interference - enhancement ( i - e ) effect on absorption ( i.e. without the patterned contact ) is plotted in fig.[fig : theorys5]b . the dielectric spacer can thus be used as an additional degree of freedom ( i.e. the spacer s index and thickness ) in optimizing the system s response . we also simulated the asymmetric contact layouts studied in fig.1d on top of 300 nm sio@xmath4/si and found that they produce a very similar response irrespective of the finite dielectric spacer . this is expected , since i - e partially cancels out when considering asymmetric absorption . the second effect of the 300 nm sio@xmath4/si substrate is that the leaked spp of the au - air interface will be reflected back from the si substrate . fig.[fig : theorys6 ] plots the field intensities along the two au interfaces as well as the corresponding fourier transform amplitudes ( see fig.3b - e for the 300 nm sio@xmath4/si case ) . we note a deeper `` beat '' modulation of the fields on the au - sio@xmath4 interface , due to the back reflected fields of the `` leaked '' au - air spp . this is also apparent in the fourier amplitude . however , no frequency shifts are observed in the latter . we thus conclude that , other than some small amplitude modulations , the spp structure is largely unaffected by the sio@xmath4 being 300 nm thick or semi - infinite .

the combination of plasmonic nanoparticles and graphene enhances the responsivity and spectral selectivity of graphene - based photodetectors . however , the small area of the metal - graphene junction , where the induced electron - hole pairs separate , limits the photoactive region to sub - micron length scales . here , we couple graphene with a plasmonic grating and exploit the resulting surface plasmon polaritons to deliver the collected photons to the junction region of a metal - graphene - metal photodetector . this results into a 400% enhancement of responsivity and a 1000% increase in photoactive length , combined with tunable spectral selectivity . the interference between surface plasmon polaritons and the incident wave introduces new functionalities , such as light flux attraction or repulsion from the contact edges , enabling the tailored design of the photodetector s spectral response . this architecture can also be used for surface plasmon bio - sensing with direct - electric - readout , eliminating the need of complicated optics . graphene - based photodetectors ( pds)@xcite have been reported with ultra - fast operating speeds ( up to 262ghz from the measured intrinsic response time of graphene carriers@xcite ) and broadband operation from the visible and infrared@xcite up to the thz@xcite . the simplest graphene - based photodetection scheme relies on the metal - graphene - metal ( mgm ) architecture@xcite , where the photoresponse is due to a combination of photo - thermoelectric and photovoltaic effects@xcite . for both mechanisms , the presence of a junction is required to spatially separate excited electron - hole ( e - h ) pairs@xcite . at the metal - graphene junction , a work - function difference causes charge transfer and a shift of the graphene fermi level underneath the contact@xcite , compared to that of graphene away from the contact@xcite , resulting into a build - up of an internal electric field ( photovoltaic mechanism)@xcite and into a difference of seebeck coefficients ( photo - thermoelectric mechanism)@xcite . an alternative way to create a junction is to use a set of gate electrodes to electrostatically dope graphene@xcite . for both photovoltaic and photo - thermoelectric mechanisms , however , the spatial extend of the junction is@xmath0100 - 200nm@xcite , which reduces the photoactive area to a fraction of the diffraction limited laser spot size in a typical scanning current microscopy experiment . furthermore , suspended undoped graphene only absorbs 2.3% of light@xcite which , while remarkably high for a one atom thick material , is low in absolute terms for practical applications . this is further reduced by a factor of @xmath1 for graphene on a dielectric substrate of refractive index @xmath2 ( see methods ) . additionally , in highly doped graphene the absorption decreases even further@xcite . one approach is to extend the junction region in order to capture more light . in a vertical ( i.e. with doping profile perpendicular to the device s surface ) p - i - n semiconductor pd this is achieved by ion - implantation with tailored dose and energy@xcite . in the mgm configuration , however , the lateral nature ( i.e. doping profile parallel to the device s surface ) of the junction does not allow a straightforward doping profile engineering and , thus far , to the best of our knowledge , no techniques have been reported to reliably do that . as an alternative , several graphene - based vertical architectures have been proposed , including all - graphene@xcite , or graphene integrated with semiconductor layers , such as other two dimensional ( 2d ) materials@xcite or si@xcite . in the latter cases , however , graphene is not the absorbing material and the spectral response is thus far limited to above the band gap of the semiconductor layer . furthermore , while these approaches have demonstrated high responsivities ( up to@xmath3a / w in ref.@xcite by employing mos@xmath4 as light capturing material ) they do come with the cost of smaller operation speed ( up to 100khz in ref.@xcite ) as compared to the graphene - based pds operating at speeds up to 50 gbit / s in the optical link reported in ref.@xcite . improving graphene absorption in the ultra - fast mgm configuration is thus critical . various solutions have been proposed , such as the integration of graphene into an optical microcavity@xcite ( @xmath520-fold enhancement ) or onto a planar photonic crystal cavity@xcite ( 8-fold enhancement ) , to take advantage of the multiple passes of the trapped light through graphene , or its coplanar integration with a si integrated photonic waveguide@xcite ( @xmath510-fold enhancement ) . another solution is the integration of plasmonic nanostructures on graphene@xcite to exploit the strongly enhanced electromagnetic near - fields@xcite associated with the localized surface plasmon resonances ( lspr)@xcite . lsprs originate from the resonant coherent oscillation of the metal s conduction electrons in response to the incident radiation . the resulting enhanced near - fields surrounding the nanostructures promote light absorption in the materials around them@xcite . we previously reported a @xmath6 enhancement in photoresponse@xcite when radiation is focused close to the nanostructures . in this approach , however , light absorbed around nanostructures far from the junction ( where efficient charge separation occurs ) does not fully contribute to the photoresponse@xcite . an ideal alternative would be to enable light collection in one part of a device and then guide it into the junction region at the contact edge . this can be achieved by exciting surface plasmon polaritons ( spps ) on the metal contacts . spps are surface - bound waves propagating on a metal - dielectric interface and originate from the coupling of light with the metal s free electrons@xcite . their excitation can be achieved by means of an integrated diffraction grating@xcite , and their delivery to the active region ( junction at contact edge ) will enhance the overall absorption . thus , the contact now becomes a light collector . such an approach was demonstrated with semiconductor - based near infra red ( nir ) pds@xcite in order to reduce the semiconductor active area without compromising light absorption . a smaller active area results into reduced carrier transit time@xcite and reduced capacitance@xcite , thus increased operation speed . in particular , a circular ( bull s eye ) grating@xcite was used to deliver spps into a subwavelength circular aperture on top of vertical si@xcite or ge@xcite schottky photodiodes , while a linear grating was used to deliver spps into a subwavelength linear slit in a lateral metal - gaas - metal photodiode configuration@xcite . operation speeds beyond 100ghz where estimated@xcite , while responsivity enhancements ( compared to a device without the grating ) were up to @xmath7 for linear gratings@xcite and over @xmath8 for circular gratings@xcite . no compromise in operation speed was reported due to the presence of the grating@xcite . this approach was also applied in mid - ir detection@xcite , by delivering light into a quantum cascade detector , and in thz detection@xcite , by delivering light into a gaas / algaas 2d - electron - gas bolometer . here we apply the spp grating coupler concept to a mgm pd and demonstrate a@xmath0400% increase in responsivity and a@xmath01000% increase in photoactive length . furthermore , we show that this offers a solution to another problem : in order to have a net response under uniform illumination of the whole mgm pd area , one must break the mirror symmetry between the two contacts@xcite . in contrast to the metal - semiconductor - metal case , applying a bias is not practical because it would result into a large dark current , due to the semimetallic nature of graphene@xcite . using different metallizations for the two contacts is an option@xcite , but increases the fabrication steps . in our approach this problem is addressed by using different contact grating structures . one can utilize the interference between spps and incident waves and create novel asymmetric contact designs that produce complex spectral responses , such as switching the light flux between the two contacts edges , enabling new functionalities , such as label- and optics - free direct - electrical - readout plasmonic biosensing . to explore the design opportunities offered by a spp grating coupler on the metal contacts , we first carry out numerical simulations using the finite - difference time - domain ( fdtd ) method@xcite . the spp wavevector on a metal - dielectric interface is@xcite : @xmath9 where @xmath10 , @xmath11 are the dielectric functions for the metal and dielectric medium respectively , and the spp existence condition is @xmath12@xcite . @xmath13 is larger than any propagating wave in the dielectric , whose wavevector is @xmath14 . this momentum mismatch between spps and propagating waves implies that spps can not decay into free propagating waves , but also that one can not directly excite spps from free waves on a smooth metal surface@xcite . one way to overcome this is by diffraction , whereby the continuity of the component of momentum parallel to the surface is broken and incident light can scatter into spps@xcite . this can be achieved by a nano - slit@xcite , a diffracting element@xcite , or a grating coupler@xcite . in the latter case , in particular , a periodic array of metal ridges and grooves delivers the additional momentum according to@xcite : @xmath15 where @xmath16 is the parallel component of incident wavevector , @xmath17 the incident angle , @xmath18 the grating s reciprocal lattice vector , @xmath19 the grating pitch , and @xmath20 the diffraction order . we select a grating of 50 nm au bars periodically placed at a pitch of 620 nm on top of a 50 nm au contact film , as depicted in fig.[fig : theory1]a , at a 1:1 ratio of ridge and groove widths . these grating parameters are chosen since they were shown to be optimal for yielding a high percentage of incident light scattered into spps ( @xmath020% in refs.@xcite ) . a termination `` step '' of width @xmath21 extends beyond the last ridge . for simplicity , we assume the graphene / contact structure to be on top of a semi - infinite sio@xmath4 substrate . the dielectric functions of au@xcite and graphene@xcite are treated through a drude - lorentz model , as explained in ref.@xcite and in methods . inserting au s dielectric function in eqs.[eq : spp],[eq : spp2 ] , with @xmath22=1 for air , yields @xmath23=645 nm for the vacuum wavelength of the spps on the au / air interface . it is also possible to have spps in the au / sio@xmath4 interface . given that @xmath22=2.13 for sio@xmath4@xcite , they are excited at @xmath23=930 nm . in the first set of calculations the device is illuminated by a normally incident plane wave polarized perpendicular to the grating ( transverse magnetic - tm ) . this polarization is required because the spps are themselves tm waves@xcite , having both longitudinal ( parallel to the propagation direction ) and transverse ( perpendicular to the surface ) electric field components . the exposed part of the single layer graphene ( slg ) in - between the contacts is kept fixed at 1000 nm , while the width of the au contact , thus of the slg unexposed ( buried ) part , is varied depending on the number of ridges and the size @xmath21 of the termination step . periodic and absorbing boundaries are assumed in the lateral and vertical directions respectively ( see methods ) . fig.[fig : theory1]b plots the absorption in the exposed slg normalized to the flux incident on the exposed slg area , for @xmath21=0 and 410 nm . strong enhancement peaks , reaching up to 5% absorption , are found between 630 - 700 nm . there are also secondary small peaks at@xmath0900 - 950 nm , assigned to the au / sio@xmath4 interface spps . the green dashed line indicates the absorption within the 1000 nm wide slg on top of the semi - infinite sio@xmath4 substrate in the absence of the grating , as derived in the thin film limit [ see methods ] : @xmath24 where @xmath25 is the absorption coefficient for suspended slg in air@xcite and @xmath26=1.46 is the sio@xmath4 refractive index in the above wavelength range@xcite . a three - fold wavelength - selective increase in absorption is observed , due to coupling with spps scattered from the grating . we now consider the parameter @xmath21 . fig.[fig : theory1]b indicates that for certain wavelengths ( e.g.@xmath0630 nm ) , there are opposite extremes of absorption for the two different steps . such an asymmetry can be instrumental in designing contact layouts that exhibit a net photovoltage even under uniform illumination . for the two cases pointed by the two arrows of fig.[fig : theory1]b , we report in graph fig.[fig : theory1]c the cumulative absorption in the exposed slg as we move away from the grating edge . in the @xmath21=410 nm case , absorption is strongly enhanced at the edge of the grating , and starts leveling off@xmath0300 nm away from it . in contrast , for @xmath21=0 absorption is minimal close to the grating and starts picking up only@xmath5300 nm from the grating edge . spp interference causes one contact to `` pull '' light close to its edge and the other to `` push '' it away from it . the situation reverses at@xmath0700 nm . to confirm that these light `` attraction '' and `` expulsion '' effects are not related to interferences within the contact , i.e. that they are independent of contact width , we perform the same calculations for 9 , 11 and 13 ridges . while we observe some small interference effects at longer wavelengths ( @xmath5730 nm , especially for @xmath21=0 ) , within the primary wavelength range of interest ( 630 - 700 nm ) the absorption is independent of grating size . furthermore , the response is well saturated above 10 grating periods , consistent with numerical studies on the influence of the number of grating periods@xcite . . the line is a fit of eq.[eq : spp_interf ] , which assumes spp+incident wave interference . the top inset color - codes the absorption distribution in the exposed slg as a function of @xmath21 . ( c ) asymmetric contact layout with @xmath21=0 and 410 nm , as shown in the insets , with 600 nm exposed slg width . the asymmetric absorption ( i.e. @xmath27 , where @xmath28 , @xmath29 are the slg absorptions in the right and left halves of the slg channel ) , is proportional to the net photovoltage under coherent uniform illumination . the insets color - code the absorption profiles for the two peaks indicated by the arrows . ( d ) asymmetric absorption when the 410 nm contact is replaced by a 770 nm one.,width=302 ] the mechanism responsible for these sharp contrasts in absorption is interference : the excited spps travel down the termination step towards the exposed slg and interfere with the incident waves there , as depicted in fig.[fig : theory1b]a . interference between spp waves was reported in pure metal systems@xcite . to confirm that what shown in fig.[fig : theory1]b is due to interference of spps with the incident wave , we examine a series of @xmath21 values , and plot the absorption in the exposed slg within the first 100 nm from the contact edge . these calculations are performed at 680 nm , i.e. in - between the two peaks in fig.[fig : theory1]b . an oscillatory response is observed in fig.[fig : theory1b]b ( points ) , which is characteristic of interference between two waves with a variable phase between them . the total field at the slg will be @xmath30 , where @xmath31 for an incident wave normalized to unit amplitude , and @xmath32 is the spp amplitude , with @xmath33 the relative spp electric field strength compared to the input field , @xmath13 the spp wave vector calculated from eq.[eq : spp ] for the au / air interface , and @xmath34 a constant phase . we thus fit the slg absorption of fig.[fig : theory1b]b to : @xmath35 where @xmath36 is evaluated from eq.[eq : theory1 ] . as discussed later , spps on the au / air interface can leak into the dielectric substrate@xcite providing an extra loss mechanism . to account for these losses we scale the imaginary part of the wave vector @xmath37 according to @xmath38 . we treat @xmath33 , @xmath34 and @xmath39 as adjustable parameters and fit eq.[eq : spp_interf ] to the simulation of fig.[fig : theory1b]a ( note that the oscillation period is dependent only on @xmath40 ) . an excellent fit ( line ) is obtained with @xmath33=1.87 , @xmath41/5 and @xmath39=9 , confirming spp excitation , propagation and interference . fig.[fig : theory1b]b ( top ) color codes the absorption within the first 500 nm of exposed slg as a function of @xmath21 . the oscillation of light `` attraction '' and `` expulsion '' from the contact edge is apparent . the implication of eq.[eq : spp_interf ] is is that the asymmetry in photovoltage is larger than what seen in fig.[fig : theory1]b . e.g. , if the two contacts shown in fig.[fig : theory1]b are placed across each other with a 600 nm gap , then all the light accumulated from both contacts will be funneled close to only one of them . such a scheme offers great flexibility in designing asymmetric contacts suitable for uniform illumination , potentially eliminating the need for different metallizations@xcite . we explore this asymmetric contact design in fig.[fig : theory1b]c ( see insets ) . the asymmetric absorption is defined as @xmath42 , where @xmath43 is the slg absorption in the 300 nm area close to the right ( @xmath21=0 ) contact and @xmath44 is the slg absorption in the 300 nm area close to the left ( @xmath21=410 nm ) contact . @xmath45 is then proportional to the expected net photovoltage . an antisymmetric response is obtained , as shown in fig.[fig : theory1b]c . in particular , at 650 nm ( point ( i ) in fig.[fig : theory1b]c ) all the flux is `` pulled '' to the left contact , while at 700 nm ( point ( ii ) in fig.[fig : theory1b]c ) all the flux is `` pushed '' to the right contact . i.e. , a spectrally - selective region less than 50 nm wide is created , within which the photovoltage abruptly changes sign . outside this region ( @xmath46580 nm and @xmath47800 nm ) both contacts have similar responses , thus the net photovoltage is zero . yet , this is not the only response function available . plotted in fig.[fig : theory1b]d is the case of a left contact with @xmath21=770 nm . the response is found to be symmetric around 660 nm . in both cases , the peak absorption is 2 - 3 times higher than what a 600 nm slg on sio@xmath4 would absorb in the absence of the contacts . in addition , optimizing the contacts just for the highest absorption ( i.e. without creating `` clear '' symmetric or antisymmetric response functions ) , the peak absorption exceeds 6% , i.e. a 4-fold increase compared to slg on sio@xmath4 . if we further limit ourselves to the first 100 nm from the contact edge ( i.e. within the junction ) , then fig.[fig : theory1b]a gives an 8-fold enhancement . further exploration of the system s response , including absorption in the unexposed ( buried ) slg under the contact , different light polarizations , unpatterned contacts and sio@xmath4 ( 300nm)/si substrate , for both uniform and focused illumination , are consistent with what discussed above ( see methods ) . a detailed analysis of all the spps that can be launched in our system is performed by considering a patterned contact with a large @xmath21=20@xmath48 m , illuminated by a narrow 1@xmath48 m width tm - polarized source positioned on top of the grating , as shown by the arrow in fig.[fig : theory2]a . in contrast to plane wave incidence , for which @xmath49 and where only spps compatible with @xmath50 are excited ( see eq.[eq : spp2 ] ) , the focused beam has @xmath51 incident wave vector components . it thus allows the full spectrum of spps to be excited according to eq.[eq : spp2 ] . absorbing boundaries are employed in all directions to avoid scattered light from re - entering the computational cell . in fig.[fig : theory2]a the electric - field intensity distribution is plotted at 650 nm . strong scattering and fields extending many microns away from the grating are observed , both in the au / air and au / sio@xmath4 interfaces . an intensity oscillation is also observed in the latter , with a period@xmath01.4@xmath48 m . spps have both longitudinal and transverse field components , the latter being perpendicular to the metal surface@xcite . fig.[fig : theory2]b , c plots the longitudinal @xmath52 and transverse @xmath53 electric field amplitudes at the au / air and au / sio@xmath4 interfaces , as a function of distance @xmath54 from the grating . a simple decay curve is obtained at the top interface , but an oscillating decay curve is obtained for the bottom interface . fig.[fig : theory2]d , e report the spatial fourier transform on these fields . a single peak is found in the top interface , and two in the bottom one : one at exactly the same wavevector as in the top interface , and the other at a larger wavevector . by repeating this procedure at different illumination frequencies we get the spp dispersion shown in fig.[fig : theory2]f . lines denote the theoretical dispersion curves from eq.[eq : spp ] using au s dielectric function and assuming either an au / air or au / sio@xmath4 interface . squares indicate the peaks obtained from the simulations by the spatial fourier transforms . in the au / air interface only the au / air spp dispersion curve emerges , while both au / air and au / sio@xmath4 spp dispersions emerge in the au / sio@xmath4 interface . spps are surface waves bound on a metal / dielectric interface because they exist below the light - cone of the dielectric ( i.e. the phase - space of free propagating modes defined by @xmath55)@xcite . the spp at the au / sio@xmath4 interface is below both the air and sio@xmath4 light - cones ( i.e. @xmath56 , @xmath57 ) , thus can not couple to any free radiation states . on the other hand , the spps at the au / air interface are below the air light - cone and within the sio@xmath4 light - cone ( i.e. @xmath58 ) , thus they can leak ( tunnel ) into the free radiation states of substrate , explaining why we obtain two spp signals in the au / sio@xmath4 interface . such leaky waves have been used for spp characterization within the context of leakage radiation microscopy@xcite ( a far - field optical method analyzing the leaked spp waves in glass substrates to characterize spp propagation on the top interface of a flat or nanostructured metal film@xcite ) . in a semi - infinite substrate they will just propagate away . in a finite one , on the other hand , part of the leaked waves will be reflected back to the interface and contribute more to the slg absorption . the dominant effect in photoresponse , however , remains in the au / air spp , as inferred by the fourier amplitudes in figs.[fig : theory2]b - e , and confirmed by the absorption in fig.[fig : theory1]b , with minimal contribution from au / sio@xmath4 spps at 930 nm . = 600 nm is assumed for the exposed slg . an analyte ( purple ) is deposited on both sensor arms . ( b ) asymmetric absorption ( proportional to the net photovoltage ) as a function of analyte thickness for four sensor arm pairs . the operating wavelength is 678 nm . ( c ) response curve slope as a function of small sensor arm length.,width=340 ] besides photodetection , spp+incident wave interference in a mgm architecture also lends itself to label - free surface plasmon biosensing@xcite , whereby slg assumes the role of an integrated transducer providing direct electrical readout , thus eliminating the need for optical measurements . the use of slg as an integrated transducer was reported in a dielectric waveguide sensor geometry@xcite , but not in surface plasmon sensing . fig.[fig : biosensor ] assumes that one termination step ( sensor arm hereafter ) has length @xmath59 and the other @xmath60 , so that they are at the highest slopes of fig.[fig : theory1b]a , i.e. in the midplane of the interference oscillation , with one of them at a positive slope and the other at a negative one . in this setup , the two contacts are at an accidental degeneracy , producing the same interference between spp and incident wave , thus zero net photovoltage under uniform illumination . if now the dielectric environment around the sensor arms changes by the presence of an analyte , it will cause an increase in @xmath13 , thus an additional phase to both spps . having the two arms on a different slope in the response curve of fig.[fig : theory1b]a introduces an asymmetry , thus a net photovoltage . the larger the dielectric change , the larger the photovoltage . also , the longer the sensor arms , the higher the sensitivity , as the spp will travel a longer distance , therefore sampling more analyte . fig.[fig : biosensor ] numerically tests this idea simulating four pairs of sensor arms , tuned to operate at 678 nm according to fig.[fig : theory1]c : ( i ) @xmath59=410 nm , @xmath60=750 nm , ( ii ) @xmath59=775 nm , @xmath60=1075 nm , ( iii ) @xmath59=1070 nm , @xmath60=1400 nm and ( iv ) @xmath59=1435 nm , @xmath60=1730 nm . in cases ( i)-(iv ) the left arm alternates from being on a positive to a negative slope of the interference diagram of fig.[fig : theory1]c . thus , the photovoltage in the presence of the analyte is also expected to alternate . for simplicity , the analyte is assumed to be a thin film deposited on the sensor arms and to have an @xmath2=1.55 , an average value for dry protein films@xcite . fig.[fig : biosensor]b plots the asymmetric absorption as a function of analyte thickness . we obtain a linear response , with slope increasing the longer the sensor arms . in fig.[fig : biosensor]c the absolute value of the slope is plotted as a function of the small sensor arm length @xmath59 , and a good linear fit is obtained . tuning the arm dimensions thus provides an additional tool for controlling and tuning the device s performance and sensitivity . at long arm lengths , spp losses will limit the sensitivity , but they could be overcome , e.g. by increasing the metal thickness and/or reducing the substrate index to limit spp coupling to the substrate . having demonstrated the design versatility of grating - coupled gpds , we now turn to the experimental validation of our predicted spp enhanced slg absorption and photodetection . the general architecture of our experimental devices with plasmonic grating coupler is shown in fig.[graph : graphene_grating]a . slg is contacted with metallic source and drain electrodes . in order to quantify the enhancement relative to the no - grating case , we fabricate the grating coupler on top of one of the contacts and leave the other contact flat . this is the simplest way to break the contacts symmetry and allows the generation of net non - zero photoresponse , even with both contacts illuminated . our devices are fabricated as follows . graphene is produced by mechanical exfoliation of graphite onto an si+300 nm sio@xmath4@xcite and characterized by optical microscopy@xcite and raman spectroscopy@xcite . subsequently , the source and drain contacts are prepared by e - beam lithography and a base metallization layer is deposited by thermal evaporation of 4 nm cr and 50 nm au , employing a lift - off step . the 620 nm period grating is then defined in a further e - beam lithography step by performing a second thermal evaporation of 50 nm au followed by lift - off . fig.[graph : graphene_grating]b shows a scanning electron micrograph ( sem ) of the device . a slight asymmetry between the ridges and grooves is detected due to overexposure during e - beam lithography , but this does not change the spectral characteristics of the grating , solely determined by its period . afterwards , the samples are bonded into a chip carrier for electrical and optical characterization . we perform wavelength dependent photovoltage mapping to determine the spatial pattern of the devices photoresponse . fig.[graph : grating_pv ] plots the photovoltage maps at different incident wavelengths for polarization perpendicular to the grating ( tm - polarization , 100x ultralong working distance objective , numerical aperture na=0.6 ) . fig.[graph : grating_pv ] also shows the structured grating contact and the flat contact without perturbations . at 514 nm the photoresponse occurs predominantly at the contacts edges and is of similar magnitude , but opposite polarity , to that of a standard mgm pd@xcite . at 633 nm , fig.[graph : grating_pv]b indicates that the influence of the grating starts emerging . 2 - 3@xmath48 m away from the edge of the patterned contact , a photoresponse is visible , even though no junction is present . the effect is much more pronounced at 785 nm , where the entire structured contact becomes photosensitive , fig.[graph : grating_pv]c , and the photoresponse is enhanced@xmath0400% compared to the flat contact edge . furthermore , the responsivity of the device is polarization dependent , as shown in fig.[graph : grating_pv]d . the strongest photoresponse occurs for perpendicular polarization ( 0@xmath61,tm - polarized light ) . m spot size illuminates the grating 4@xmath48 m away from the metal / slg junction . ( a ) schematic of the simulation system . 300 nm sio@xmath4 on si is assumed for closer matching to the experiments . ( b ) absorption in slg . two polarizations and a flat contact case are considered.,width=340 ] to have a direct comparison with the experiments , we perform calculations with a focused beam illumination ( 1@xmath48 m width ) on top of the grating,@xmath04@xmath48 m away from the contact edge , as depicted in fig.[graph : theory_focused]a . the termination step length is taken as 1250 nm ( estimated from the sem image in fig.[graph : graphene_grating ] ) and we also include the si substrate with 300 nm sio@xmath4 . fig.[graph : theory_focused]b plots the absorption in the exposed slg 4@xmath48 m away from the illumination spot for tm polarization , te polarization and a flat contact . both the theoretical spectral and polarization responses are in excellent agreement with experiments , and verify the strong responsivity above 700 nm . this contrasts with the normally incident plane wave illumination case , below 700 nm ( see fig.[fig : theory1]b ) . this is understood by considering eq.[eq : spp2 ] and fig.[fig : theory2]f . for a normally incident plane wave and focused illumination , @xmath62 is zero and nonzero , respectively . since the latter case is less restrictive for spp excitation , it results into a wider spp spectrum at both au / air and au / sio@xmath4 interfaces , thus into a wider responsivity compared to the plane wave case ( see methods for details ) in conclusion , we demonstrated the coupling to graphene of surface plasmon polaritons excited in a metallic plasmonic grating and its exploitation in graphene - based photodetection with enhanced responsivity and polarization selectivity . depending on its dimensions , highly tunable spectral selectivity below 50 nm bandwidth can be achieved . further , the symmetry of the photodetector can be broken making it operable under full illumination , despite identical metal source and drain contacts . the underlying mechanism involves the coupling of light into spps on the patterned contact , and their propagation to the exposed slg area . for uniform coherent illumination , these spps can further interfere with the waves directly incident on the exposed slg , offering a novel tuning capability where the light flux can be attracted or repelled from the contact edge by design . the whole contact thus becomes a highly tunable polarization- and spectral - selective photosensitive area . spps and incident wave interference can potentially be employed for ( bio- ) sensing by tailoring the grating dimensions . this may allow a novel plasmonic sensing architecture with high sensitivity and small footprint with direct electrical readout and without complicated optics . + we acknowledge funding from eu graphene flagship ( no . 604391 ) , erc grant hetero2d , epsrc grants ep / k01711x/1 , ep / k017144/1 , eu grant genius , a royal society wolfson research merit award .

this supplemental material presents the derivation of the noise - induced line properties of the @xmath38 limit cycles . let us rewrite the order parameters from ( [ orderparameter ] ) as @xmath122 from the equations of motion ( [ eq : motion ] ) it is straightforward to find the functions @xmath123 of the differential equations ( we omit these details for brevity ) @xmath124 adding noise back to the equations of motion ( [ eq : motion ] ) yields @xmath125 the random process @xmath126 is characterized by @xmath127 in a @xmath38 limit cycle the noise - free evolution of @xmath128 is determined by the evolution of the total phase given by @xmath129 . the presence of noise gives @xmath130 since @xmath131 we obtain @xmath132 . \label{eq : commonfcorrelation}\ ] ] the emission line is finally given by @xmath133 which is a lorentzian centred at @xmath134 and of width @xmath135 .

we predict the spontaneous modulated emission from a pair of exciton - polariton condensates due to coherent ( josephson ) and dissipative coupling . we show that strong polariton - polariton interaction generates complex dynamics in the weak - lasing domain way beyond hopf bifurcations . as a result , the exciton - polariton condensates exhibit self - induced oscillations and emit an equidistant frequency comb light spectrum . a plethora of possible emission spectra with asymmetric peak distributions appears due to spontaneously broken time - reversal symmetry . the lasing dynamics is affected by the shot noise arising from the influx of polaritons . that results in a complex inhomogeneous line broadening . condensation of exciton - polaritons ( ep s ) in semiconductor microcavities formed by two distributed bragg mirrors with quantum wells between them has been experimentally observed @xcite . being incoherently excited in the microcavity , ep condensates are in general out of thermodynamic equilibrium . ep condensates refuel their particle depot through absorption of cavity photons and emit coherent light due to tunneling of the composite ep states through distributed bragg mirrors . sample inhomogeneity , either accidental or intentional , can induce several condensation centers ( cc s ) @xcite . at low enough pumping , one expects a system of disconnected bec droplets emitting light at different uncorrelated frequencies . as the pumping increases the condensates tend to establish mutual coherence and emit in a laser mode @xcite . already two ccs can synchronize and emit at a single joint frequency @xcite . this is possible because the condensates exchange particles due to josephson coupling and adjust their emission frequencies , which in turn depend on the number of condensed particles due to the polariton - polariton repulsion . in addition to the coherent josephson coupling , there can be a dissipative ( radiative ) coupling between cc s , which reflects the dependence of the losses in the system on the symmetry of singe - particle states . that new stationary regime called weak lasing takes place when pumping rates reside between some minimal and maximal rates of losses @xcite . in the weak lasing regime , the system is stabilized by the formation of specific many - particle states which adjust the balance between gain and loss in the system . in this letter we show that in the weak lasing regime two ccs can emit not only at a _ single _ frequency , but also at a whole _ frequency comb _ which in principle contains an infinite number of equidistant lines of coherent lase - like radiation . this emission reflects the fact of formation of spontaneous selfsustained anharmonic oscillations of both the occupation numbers and the relative phase between the condensates , in sharp contrast to previously reported damped josephson oscillations @xcite . we study possible emission spectra and the way they are affected by noise . while the emission frequency of single - line ep lasers resides in the ev range @xcite , the modulation frequency of comb emission can be adjusted to be in the terahertz and sub - terahertz range . filtering out of the high - frequency component through optical demodulation yields the low - frequency coherent signal as a new promising type of coherent terahertz emitters . the ep self - induced oscillation is also a novel mechanism of optical frequency comb generation as compared to mode - locked lasers @xcite and optical microresonators @xcite . consider two coupled ep condensates with order parameters @xmath0 where @xmath1 are the occupations of the two condensates , @xmath2 is the total phase and @xmath3 is the phase difference . the time evolution of @xmath4 is governed by the langevin equations ( @xmath5 ) @xcite @xmath6 where @xmath7 label the condensates . the parameter @xmath8 describes the difference between the rates of losses @xmath9 and pumping @xmath10 , @xmath11 denote the singe - particle energies of the condensates , the parameters @xmath12 and @xmath13 define dissipative and coherent coupling between the condensates , respectively , and @xmath14 is the polariton - polariton interaction constant . the last term in eq . is the gaussian white noise satisfying @xmath15 and @xmath16 . due to gauge invariance , only the frequency detuning @xmath17 is relevant and in what follows we will count the frequency from @xmath18 . rescaling time we can fix @xmath19 and , since rescaling the condensate amplitudes is equivalent to a change of @xmath14 , we can set @xmath20 without loss of generality . the dissipative coupling induces a relative phase @xmath21 dependent dissipation in the system . this can be observed from the eigenvalues @xmath22 which control the condensate evolution @xmath23 in the absence of interaction @xcite . for sufficiently large @xmath24 one of the eigenmodes turns unstable . therefore , the dissipative coupling acts as a phase - selective pump which depends on the relative phase @xmath21 : it pumps one eigenmode while keeping the other one lossy . in this regime nontrivial weak lasing states are formed ( see ref.@xcite for a complete account ) . first we consider the noise - free case ( @xmath25 ) in order to determine the attractors of the system . two nontrivial solutions f@xmath26 to eqs . were identified in ref.@xcite . they are characterized by nonzero time - independent triplets @xmath27 with the total occupation @xmath28/gjr^{\pm}$ ] , @xmath29 , @xmath30 , @xmath31 , where @xmath32 . the total phase @xmath2 of the condensates satisfies @xmath33 and the rhs in ( [ eq : totalphase ] ) gives a time - independent frequency @xmath34 for these solutions . the two centers evolve in a coherent fashion @xmath35 and @xmath36 defines the blue - shift of the emission line with respect to the average single - particle frequency . in the subspace @xmath37 these states correspond to fixed points @xcite . in parts of the control parameter space these states are stable , and should manifest themselves as weak lasing states . the @xmath38 states loose stability at @xmath39 @xcite @xmath40\ ] ] we plot the two instability curves in the @xmath41 space at fixed @xmath42 in fig . [ fig1 ] ( solid lines ) . the @xmath38 states are unstable in the shaded areas lc@xmath26 . in particular they are both unstable in the joint area lc@xmath43 and lc@xmath44 , where the trivial solution @xmath45 is unstable as well . what are then the stable stationary states of the system , if any ? the answer is obtained by linearizing the phase space flow around @xmath38 in the subspace @xmath46 . at @xmath47 two corresponding eigenvalues are purely imaginary @xmath48 , with their real parts changing sign . as a result , a supercritical hopf bifurcation occurs , where stable limit cycles lc@xmath49 with frequency @xmath50 are born around the respective unstable fixed points f@xmath49 @xcite . appear in the shaded areas in the @xmath51-parameter space . stable lc@xmath49 are born through hopf bifurcations at the solid @xmath52 line and turn unstable at the dashed @xmath53 lines where they undergo period doubling bifurcations . they are the only stable attractors to coexist in the central ( yellow ) region . here @xmath54 . ] away from the bifurcation line the lcs increase the oscillation amplitudes , deform , and change their frequency . the coexistence region of lc@xmath43 and lc@xmath44 grows in size as the josephson tunneling is reduced . at the hopf bifurcation , where a lc emerges , @xmath55 and also @xmath56 become periodic functions of time with period @xmath57 . then they may be expanded in a fourier series with frequency harmonics @xmath58 and @xmath59 the integration of the constant term @xmath60 in the fourier series of @xmath56 results in a linear time dependence , @xmath61 , similar to @xmath38 . therefore @xmath62 where the functions @xmath63 are periodic in time . the fourier spectrum of @xmath64 is equidistant with frequency harmonics positioned at @xmath65 . approaching the dashed lines @xmath66 in fig.[fig1 ] , the corresponding lc turns unstable and undergoes a period doubling bifurcation . this gives rise to a new stable period - doubled lc , which however again quickly undergoes a period doubling bifurcation . a period doubling route to chaos along a feigenbaum scenario leads to chaotic attractors @xcite . therefore just two coupled exciton - polariton condensates suffice to produce an extremely rich and complex synchronized dynamics . experimentally the polariton order parameter is detected by analyzing the emitted light from the microcavity . in near - field measurements , only small parts of the sample , like one condensation center , can be probed . our aim is to calculate the spectral density @xmath67 of the radiation corresponding to different nontrivial attractors . applying a fourier transformation ( ft ) we have @xmath68 in the fixed points @xmath38 @xmath69 , and the time dependence comes from the evolution of the total phase @xmath70 , @xmath71 with constants @xmath72 . thus the condensates emit light at frequency @xmath36 fully synchronized . the emission spectrum consists of only one peak , in contrast to the case of noninteracting polaritons , where two separated peaks are expected . ( a),(b ) and a period doubled lc@xmath43 ( c ) . here @xmath73 , @xmath42 . the small arrows indicate the position of the respective n=0 peak . ( a ) @xmath74 with @xmath75 ( upper panel ) , @xmath76 ( lower panel ) . ( b ) the same as ( a ) but for @xmath77 . ( c ) @xmath74 for @xmath78 ( upper panel ) and @xmath79 ( lower panel , period doubled lc spectrum ) . for visualization , a small artificial lorentzian line width was added to all emission lines . , title="fig : " ] ( a),(b ) and a period doubled lc@xmath43 ( c ) . here @xmath73 , @xmath42 . the small arrows indicate the position of the respective n=0 peak . ( a ) @xmath74 with @xmath75 ( upper panel ) , @xmath76 ( lower panel ) . ( b ) the same as ( a ) but for @xmath77 . ( c ) @xmath74 for @xmath78 ( upper panel ) and @xmath79 ( lower panel , period doubled lc spectrum ) . for visualization , a small artificial lorentzian line width was added to all emission lines . , title="fig : " ] since the limit cycles lc@xmath26 are characterized by an equidistant spectrum , we first numerically compute the corresponding frequency positions , and then calculate the intensity of each frequency harmonics using a fourier series expansion . the resulting spectra are shown in fig.[fig2](a , b ) . close to the hopf bifurcation , there is only one considerable emission peak originating from the @xmath80 spectral line ( fig.[fig2](a , b ) upper panels ) . further away from the hopf bifurcation , the satellite peaks grow to form a frequency comb with asymmetric tails ( fig.[fig2](a , b ) lower panels ) . the comb also acquires several peak maxima , with the highest peak originating from a satellite with nonzero @xmath81 ( fig.[fig2](a , b ) lower panels ) . when the lc undergoes a period doubling bifurcation , the comb becomes twice as dense ( fig.[fig2](c ) ) . the typical modulation frequency is independent of the polariton - polariton interaction constant @xmath14 and is of the order of the coupling constant @xmath82 . for typical dissipative rates in semiconductor microcavities @xmath83 one expects condensate pairs with dissipative coupling @xmath84 and below . this would permit to generate frequency combs with terahertz separation between the individual peaks . additional reduction of this separation by period doubling can shift the modulation frequency into the millimeter range . finally , we consider the influence of noise in eq .. in general , it will broaden the peaks discussed so far , and can lead to a merging of peaks with small enough spacing . the emission spectrum can be obtained using the wiener - khinchin theorem , @xmath85 where @xmath86 is the auto - correlation of the , now , random process @xmath64 . the @xmath38 states are periodic orbits in the full four - dimensional phase space , and the dynamics along these periodic orbits is parameterized by the total phase @xmath2 . while fluctuations off the periodic orbit will relax back , fluctuations along the orbit do not , and will enforce diffusion of @xmath2 on the orbit . the latter fluctuations can be shown to form a lorentzian line with the full width at half maximum ( fwhm ) given by @xmath87 @xcite . note that the fwhm is inversely proportional to the number of particles in the condensate , as it should be for a laser . in contrast to the @xmath38 states , the lc@xmath49 states are formed by the motion on a torus in the full phase space . the stability of the attractor demands that fluctuations off the torus relax back . fluctuations along the torus surface enforce a diffusion on it . the two nontrivial phases which diffuse , are the total phase @xmath2 and the second phase angle , which characterizes the position on the limit cycle . close to the hopf bifurcation , and in the presence of only a few satellite peaks , we can obtain a closed formula for the line width . to parameterize the lc we introduce two time arguments : one originating from the total phase and the other from the lc phase . noise in these time arguments , according to eq . , leads to @xmath88 where the periodic function @xmath89 has been expanded in a fourier series with coefficients @xmath90 and @xmath91 is the velocity of the noise - free trajectory along the lc in the three - dimensional space @xmath92 . the noise term @xmath93 is the projected noise along the lc , while @xmath94 $ ] is the noise added to the rhs of eq . for @xmath95 note that perturbing the time argument of @xmath89 already accounts for a part of the noise @xmath96 , because @xmath97 is time periodic . however close to the hopf bifurcation this periodic part is negligible compared to the dc part of @xmath97 , so that the above separation of noise in the time arguments is justified . the choice of the two time arguments is convenient when characterizing the dynamics of the noise - free lc . in the presence of noise however it leads to nonzero correlations @xmath98 denoting @xmath99 , @xmath100 , @xmath101 , we have @xmath102 with an asymmetric dependence of @xmath103 on @xmath104 . the intensity of the noise @xmath105 is periodic in time , which originates from the oscillation of occupation numbers and relative phase for evolution along the lc . experimentally , the measurement time spans many lc periods and one can use the average value @xmath106 . then @xmath107 we obtain a lorentzian for every emission line , with an @xmath104-dependent width @xmath108 according to eq . . the @xmath104-dependence of the line broadening @xmath108 follows from eq.([asymmetry ] ) and shows two remarkable features . first , there is an @xmath104-symmetric line broadening with increasing @xmath104 which is very strong and proportional to @xmath109 . second , there is an asymmetric contribution @xmath110 which originates form nonzero correlations @xmath111 . it may lead to a satellite peak becoming more narrow than the main peak @xmath112 , and can further enhance the asymmetry of the spectrum , as compared to the noise - free case . at @xmath113 , for ( a ) condensation center 1 and ( b ) condensation center 2 . the arrows indicate the central peak . here @xmath114 . ] to calculate the line width @xmath115 numerically , we denote by @xmath116 the normalized tangent vector along the lc in the coordinates @xmath117 . then @xmath118 is the noise in @xmath119 projected onto this tangent and we can evaluate from eq . @xmath120 we show the spectrum with inhomogeneous line broadening compared to the noise - free case in fig.[fig3 ] . due to noise , the asymmetry of the spectrum is enhanced and the strict equidistance of emission lines is relaxed for strong enough line broadening . we note that this shape together with the equidistance of emission peaks is very reminiscent of experimentally obtained spectra in @xcite . dissipative coupling between coexisting exciton - polariton condensates in semiconductor microcavities together with strong polariton - polariton repulsion leads to a rich dissipative nonlinear dynamics already for two coupled condensates . we showed that , in addition to full synchronization @xcite , formation of limit cycles gives rise to frequency combs of equidistant asymmetric spectral lines . the frequency offset and line spacing of the combs are tunable through the control parameters . through period doubling , the line spacing can be additionally reduced by an order of magnitude . this modulated emission can be useful for terahertz and sub - 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dimensional space @xmath121 , which becomes essential in the presence of noise . these new solutions are limiting two - dimensional tori in the full four - dimensional space @xmath121 , which will become particularly important in the presence of noise . n. v. alexeeva , i. v. barashenkov , k. rayanov and s. flach , phys . rev . a * 89 * , 013848 ( 2014 ) . please see supplemental material at [ url will be in- serted by publisher ] for the derivation of the lorentzian line width .

it is generally believed that cosmic rays with energies up to the `` ankle '' at around @xmath7 are predominantly of galactic origin @xcite and that energies up to around @xmath8 can be achieved by first order fermi acceleration in shocks produced by supernovae exploding into the interstellar medium @xcite . recently the fly s eye detector revealed a change in the cosmic ray composition which is correlated with a dip in the total energy spectrum @xcite located at the ankle . around this dip the spectrum first steepens and then flattens again to a spectral index of around @xmath9 which is even smaller than the index of @xmath10 corresponding to the spectrum below the steepening . the data are consistent with a superposition of a steep power law spectrum of heavy nuclei and a flatter spectrum of protons which overtakes the former component at energies above the ankle . it is expected that this latter high energy proton component is of extragalactic origin . furthermore , on 15 october 1991 , the fly s eye observed an event at @xmath11 ( @xmath12 errors ) @xcite , which is the event with the highest energy ever recorded . interesting enough , the world s second highest energy air shower of @xmath13 was recorded at yakutsk @xcite located within 7.8 degrees from the fly s eye event ( see fig . 2 ) . in this paper we will assume that these events were caused by relativistic particles . there is plenty observational evidence that agns and radiogalaxies contain relativistic termination shocks which are likely to produce high energy cosmic rays . it therefore seems natural to extend the standard diffusive shock acceleration scenario which works well for supernova shocks at lower cosmic ray energies to larger extragalactic shock scales in order to explain the origin of this higher energetic extragalactic component . however , it turns out to be hard to explain the highest energy events by this mechanism . an interesting option is provided by decaying or annihilating topological defects which could be left over from phase transitions in the early universe at temperatures corresponding to some grand unification scale . the rest of the paper is organized as follows : we first reconsider in section 2 the source spectrum cutoff energy @xmath14 for shock accelerated cosmic rays as a function of shock size and average magnetic field strenght on this lenght scale and evaluate it for some typical observational numbers . in section 3 we discuss propagation effects on protons and show that at least the @xmath15 event is difficult to reconcile with the observational knowledge of typical extragalactic shock parameters in this acceleration scenario . we therefore discuss in section 4 some other options for these highest energy events . in section 5 we suggest that such events could alternatively be produced by topological defects . finally we summarize our findings in section 6 . in relativistic shocks the cutoff energy @xmath14 for the source spectrum of accelerated cosmic rays is in the test particle approximation always given by @xmath16 , the product of the charge @xmath17 of the cosmic ray particle , the magnetic field @xmath18 and the size @xmath19 of the shock , multiplied by some factor of order unity @xcite ( we use natural units , _ i.e. _ @xmath20 , throughout this paper ) . however , it turns out that for the highest energies the mean free path of the particle becomes comparable to the shock size @xmath19 , which sometimes is not properly accounted for . we therefore calculate here our own approximation for @xmath14 . the acceleration of a cosmic ray particle of energy @xmath21 in an astrophysical shock is governed by the equation @xmath22 where @xmath23 is the energy dependent acceleration time . in the statistical point of view the slope @xmath24 of the energy spectrum @xmath25 of the particle flux is related to @xmath23 and @xmath26 , the mean ( in general also energy dependent ) escape time by @xcite @xmath27 for first order fermi acceleration at nonrelativistic shocks caused by supernovae , @xmath23 is usually given by @xmath28 where @xmath29 , @xmath30 are the up- and downstream velocities of the shock and @xmath31 and @xmath32 are the corresponding diffusion coefficients , respectively . diffusion is dominated by magnetic pitch angle scattering caused by inhomogeneities in the magnetic field @xcite . therefore , the mean free path @xmath33 is bounded from below by some multiple @xmath34 of the gyroradius @xmath35 and @xmath31 and @xmath32 can for ultrarelativistic particles be estimated by @xmath36 for nonrelativistic shocks , @xmath34 is usually set equal to 1 @xcite . however , as we deal with the highest energetic extragalactic cosmic ray component , we have to consider relativistic shocks because they provide the most powerful accelerators . monte - carlo simulations of such relativistic shocks yield @xmath37 @xcite . furthermore , the acceleration turns out to be enhanced compared to eq . ( [ tacc ] ) by about a factor of 10 in highly inclined @xcite and by about a factor of 13.5 in parallel @xcite relativistic shocks , respectively . putting everything together and maximizing @xmath23 from eq . ( [ tacc ] ) we arrive at @xmath38 on the other hand , as long as the diffusion approximation is valid , _ i.e. _ as long as @xmath39 corresponding to @xmath40 , the escape time is given by @xmath41 , whereas for @xmath42 the particles are freely streaming out of the shock region to a good approximation and @xmath43 . using eqs . ( [ q ] ) and ( [ taccmin ] ) , we thus get @xmath44 defining the cutoff energy @xmath14 as the energy where the source spectral index becomes 3 ( remember that the slope of the energy spectrum observed at the earth was around 2.7 in the region of highest energies ) yields @xmath45 this is compatible or even higher as compared to similar estimates @xcite . we have assumed here that the magnetic field is parallel to the shock normal . if that is not the case there will be an electric field @xmath46 in the shock rest frame ( @xmath47 is the shock velocity in the lab frame ) . this causes drift acceleration of charged particles to a maximal energy given by @xmath48 which is around one order of magnitude larger than eq . ( [ ec ] ) if @xmath49 approaches the speed of light . however , the electric field @xmath50 is expected to be much smaller in general due to plasma effects so that rather special conditions have to be fulfilled in order that such high energies can be approached . throughout the rest of this section and the next section we will restrict our discussion to protons ( @xmath51 ) . we will comment on nuclei as possible candidates for events with energies above @xmath52 in section 4.3 . let us now look at some observational numbers for @xmath19 and @xmath18 and evaluate the corresponding cutoff energy for a proton . cesarsky @xcite cited the example of a galaxy encounter ( ngc 4038/39 @xcite ) as a location of a strong relativistic shock with a magnetic field of about @xmath53 on a scale of about @xmath54 , leading to @xmath55 , significantly too small to explain the origin of cosmic rays with energies as high as the fly s eye event of @xmath56 . potentially more interesting candidates for acceleration beyond @xmath52 are revealed by the `` hot spots '' and radio lobes of cyga with @xmath57 , @xmath58 and @xmath59 , @xmath60 , respectively @xcite , cited by quenby @xcite which lead to a cutoff energy @xmath61 . there are also some indirect indications from gamma ray astronomy that in some quasars protons could be shock accelerated to energies of about @xmath0 @xcite . there is one nearby object where pure application of formula ( [ ec ] ) on observational indications for @xmath19 and @xmath18 leads to an @xmath14 larger than @xmath56 . this is if one takes the whole virgo cluster with an extension of @xmath62 and a intracluster magnetic field @xmath63 which is compatible with observations @xcite and leads to @xmath64 . however , it is highly improbable that the whole virgo cluster moving through intercluster space forms a relativistic shock of such an enormous extension , effective in coherent cosmic ray particle acceleration . based on examples of the sort presented above there is some common belief that for protons the highest source energy achievable by diffusive shock acceleration in quasars and radiogalaxies is around @xmath65 @xcite . we will nevertheless show in the next section that even if this is true , it is still difficult to explain the observed fly s eye event because of the information we have on its arrival direction . up to now we were only talking about the source energy spectrum . however , a proton traveling through space is in general subject to interactions , mainly with photons and magnetic fields . the latter effect leads to a curved path with a radius given by the gyroradius @xmath66 the former effect leads to scattering and an effective energy loss as long as the proton energy lies above a kinematical threshold energy @xmath67 depending on the angle @xmath68 between the incoming photon momentum and the negative nucleon momentum . here @xmath69 is the nucleon mass , @xmath70 is the typical energy of an incoming photon and @xmath71 is the center of mass energy threshold for the particular reaction under consideration . the most important ones are electron pair production @xmath72 with @xmath73 , and pion production , @xmath74 , with @xmath75 , with @xmath76 and @xmath77 the electron and pion mass , respectively . for a fraction of its propagation time depending on the effective neutron lifetime the proton will actually transform into a neutron which does , however , not have much influence on this treatment as protons and neutrons have similar interactions with the mbr . thus , nucleons interact with photons of the microwave background radiation ( mbr ) producing @xmath78 pairs and pions above an energy of about @xmath79 and @xmath80 , respectively . the latter effect is known as the greisen - zatsepin - kuzmin ( gzk ) effect @xcite . there are also plenty of infrared and optical photons around luminous agns and galaxy clusters and especially in their central regions leading to correspondingly smaller nucleon threshold energies . however , their number density is in general not much bigger than that of the mbr photons and the cross section at the correspondingly higher center of mass energy is even smaller . this implies that possible interactions with these higher energy photons does not substantially reduce the mean free path below @xmath81 , the typical mean free path in the mbr @xcite . as this is much larger than typical galaxy sizes , we will restrict our considerations to interactions with the mbr . that means that we are on the safe side and get lower limits on energy losses and upper limits on the corresponding possible travel distances @xcite . we now ask the question in what distance range a source causing nucleon induced events of energy @xmath82 on earth could be if the maximal source energy is @xmath83 . to this end we introduce the `` longitudinal nucleon energy '' @xmath84 besides the nucleon energy @xmath85 in the comoving frame . the last inequality holds because of eq . ( [ eth ] ) as long as reactions are kinematically allowed . by performing a lorentz transformation from the comoving frame to the center of mass frame corresponding to a gamma factor @xmath86 one can see that after scattering the nucleon energy in the comoving frame is given by @xmath87 where @xmath88 and @xmath89 are momentum and energy of the nucleon after scattering and @xmath90 is the scattering angle , evaluated in the center of mass frame , respectively . above @xmath52 the energy loss is dominated by pion production @xcite for which these quantities are related to @xmath91 by @xmath92 . as we are only interested in an estimate we set @xmath93 and neglect the energy dependence of the mean free path ( which in the energy range we are interested in is a good approximation @xcite ) @xmath94 in the following calculation . because of eq . ( [ eprime ] ) the energy change in a scattering event @xmath95 relative to the lower energy is given by @xmath96 furthermore , we neglect energy loss due to cosmological redshift as we are dealing with non - cosmological distances at these high energies . by integrating from lower to higher energies one can show that the mean @xmath97 and the variance @xmath98 of the distance as a function of @xmath82 and @xmath83 can be estimated by @xmath99 the first term in the variance is due to the fluctuation of the number of scatterings and the second one is due to the fluctuation of the energy transfer @xmath100 of eq . ( [ xi ] ) around its mean @xmath101 averaged over the center of mass scattering angle @xmath90 . we have numerically integrated eq . ( [ esti ] ) . the results are shown in fig . 1a and 1b . we see that for @xmath102 , the lowest possible energy for the fly s eye event the distance must be smaller than @xmath103 and @xmath104 on the @xmath105 level for a source energy @xmath106 and @xmath107 , respectively . for the best fit energy of @xmath15 we get @xmath108 and @xmath109 for the corresponding @xmath105 upper limits of distance for the same source energies . thus from the energy point of view an agn or a galaxy cluster constituting a large scale shock with the intercluster medium could be marginally able to cause events like the @xmath56 fly s eye event by shock accelerated protons , if it is not much further away than @xmath1 . note that this number actually means the path length for which the distance is a lower limit which can be overtaken if the path is curved . however , we now show that the arrival direction of such a nucleon would then have to lie within about 10 degrees of the direction of its source if conventional wisdom about magnetic fields is used . let us first discuss deflection caused by magnetic fields . unfortunately , not much is known about extragalactic magnetic fields . faraday rotation measurements of extragalactic radio sources seem to suggest fields of the order of @xmath110 which could be homogeneous on large scales @xcite . most estimates are of this order or below @xcite , a more recent one being as low as @xmath111 @xcite . the bending angle @xmath112 in radian for a proton traveling in a magnetic field satisfies @xmath113 where @xmath114 is the differential path length and @xmath115 is given by eq . ( [ gyro ] ) . a @xmath110 field leads to a maximal bending angle of around 10 degrees for a proton with arrival energy of @xmath56 . this maximum can only be reached if the magnetic field is perpendicular to the proton path and does not change its polarization considerably on a scale of @xmath103 . otherwise the bending angle is reduced at least by a factor @xmath116 where @xmath117 is the magnetic field coherence length scale . therefore , even if the typical intercluster field would be as high as @xmath118 @xcite the bending angle would still not be larger than @xmath119 if the coherence length scale is @xmath120 . a proton can also be deflected by our own galactic magnetic field which is of the order of @xmath121 @xcite . based on radio telescope observations of faraday rotations @xcite its coherent component is supposed to have a cylindrical structure of diameter @xmath122 and height of order @xmath123 and being polarized in the direction of decreasing galactic longitude in the outer region . the random component is supposed to be of the same order of magnitude . given the arrival direction of the fly s eye and the yakutsk events shown in fig . 2 we see that the corresponding path length through the field is less than @xmath124 resulting in a maximum bending angle of @xmath125 . we finally remark that even though magnetic fields in galaxy clusters as high as @xmath126 as already mentioned @xcite could cause a significant deflection this does not influence our argumentation . it only means that if our proton encountered such `` magnetic lenses '' it should point approximately to the last encountered . there still has to be a nearby galaxy cluster in its arrival direction . furthermore , the above mentioned fact that a proton partly transforms into a neutron during propagation even tends to decrease the bending further . the propagation direction could in principle also be changed by scattering with photons , but the scattering angles involved are much too small . to see that we use our notation from above and note that the scattering angle @xmath127 in the comoving frame obeys @xmath128 maximizing with respect to @xmath90 and using @xmath129 leads to an estimate independent from the final state @xmath130 in the reaction @xmath131 , @xmath132^{1/2 } \la10^{-11}\,,\label{sc2}\ ] ] where we have used @xmath133 and @xmath134 . because the mean free path at these energies is of order @xmath81 @xcite we expect only a few scattering events during traveling over a distance of @xmath1 which thus never can lead to a significant change in the propagation direction . the arrival direction of the fly s eye event is given by @xmath135 and @xmath136 . no potentially interesting object with a sufficiently powerful shock acceleration engine of the scale discussed in the previous section is located within @xmath1 in that direction . this can be seen from fig . 2 where we show the directions to important nearby galactic objects , galaxy clusters and agns . one of the three prominent fr - ii radio galaxies listed in @xcite and thought to contribute to the proton spectrum between @xmath137 and @xmath52 ( _ i.e. _ below the gzk threshold ) , namely 3c111 , has the coordinates @xmath138 , @xmath139 and lies thus @xmath140 away from that direction . but it is at least @xmath141 away . the only quasars within around @xmath119 of the arrival direction are 3c147 and 3c159 @xcite , both of them at a distance of at least @xmath142 . the only seyfert galaxy within @xmath119 is mcg 08.11 at an angular distance of @xmath143 and a distance of at least @xmath144 @xcite . although this is much nearer than the before mentioned quasars it produces a much lower radio flux at the earth and therefore seems also not likely to produce high cosmic ray fluxes . under the quite improbable assumptions discussed at the end of the previous section the virgo cluster seems to be the nearest possible candidate however , it is located around @xmath145 away from that direction . we saw in the previous section that especially the fly s eye event is difficult to explain as a proton within the standard shock acceleration scenario . we therefore now like to discuss some other options beginning with secondary particles produced by shock accelerated protons . is the shower development of the highest energy fly s eye event consistent with what would be expected if it was caused by a photon ? at these high energies the photon begins to interact with the earth s magnetic field already _ above _ the atmosphere . fitting the shower shape with a three parameter gaisser - hillas shower development function @xcite gives a depth of first interaction of @xmath146 and thus seems to indicate a first interaction above the atmosphere . however , it could also indicate that the fitting function is simply inappropriate at these high energies . indeed , it is now believed that fitting heavy nuclei induced showers can lead to similar negative values @xcite . taking the lpm effect into account the average shower maximum is expected to be somewhat larger than for a proton induced shower with an average width about twice as large as the corresponding proton profile width @xcite . as the fluctuations are expected to be large this could still be compatible with the reported maximum at @xmath147 . photons of such a high energy have a secondary origin in the standard scenario as they are produced by decay of pions or @xmath78 interactions which in turn are produced by the interactions of the cosmic ray protons with the mbr . the photon mean free path becomes comparable with or larger than that of the protons above a few @xmath148 @xcite . the exact value of the photon to proton ratio depends on the universal radio background and the intergalactic magnetic field strength . the former leads to additional losses due to electron pair production . the latter leads to an inefficient electromagnetic cascade development @xmath149 , @xmath150 ( @xmath151 is the background photon ) due to synchrotron cooling of the electrons even for fields as low as @xmath152 . typical estimates for the photon to proton ratios are considerably smaller than 1 above @xmath52 @xcite . that makes the problem even harder as there have to be ( even more abundant and of higher energy ) primary protons acting as the source for such photons . wolfendale @xcite claims that the photon to proton ratio could exceed unity above @xmath153 if the source energy cutoff @xmath14 is much above @xmath65 . this possibility thus runs into trouble with our discussion of @xmath14 in section 2 . could the highest energy fly s eye event have been an extragalactic neutrino produced as a secondary of a shock accelerated proton ? because neutrinos essentially lose no energy apart from redshift in going over cosmological distances it could have been produced by a proton interacting near its acceleration site thus avoiding excessive subsequent energy losses due to downscattering . however , it turns out that the neutrino yield above @xmath137 is considerably smaller than one for all reasonable injection spectra . furthermore , at the highest energies the spectral index observed at the earth is predicted to be 0.5 larger than the corresponding proton spectral index @xcite . as even at these energies the neutrino nucleon cross section is still by a factor of at least @xmath154 smaller than the nucleon nucleon ( and also the gamma nucleon ) cross section the average interaction depth is much larger than the atmospheric depth . all that leads to the conclusion that the event rate due to neutrinos should be much smaller than that due to protons at the same energy . one would also expect the neutrino induced showers to start predominantly at high depths _ i.e. _ near the horizon @xcite . therefore , if a neutrino caused the event then it was a very atypical one and we would expect much more proton and even photon events at the same energy . in the shock acceleration scenario neutrons are also produced as secondaries of protons or heavy nuclei as they are neutral and can not be accelerated directly . furthermore , due to instability they have only a finite range which for @xmath56 is about @xmath155 . the puzzle is therefore not solved if our events are caused by neutrons . looking at eqs . ( [ ec ] ) and ( [ emax ] ) one realizes that heavy nuclei can reach maximal energies which are higher by a factor @xmath2 compared to the protons . however , heavy nuclei lose energy not only due to the processes which dominate the energy loss of protons but also due to the giant dipole resonance which leads to photodisintegration . above @xmath52 the corresponding energy loss rates are about a factor 10 higher than those for protons @xcite and are typically due to proton stripping reactions . for example , a @xmath156fe nucleus being launched with an energy of @xmath65 will be below @xmath52 after traveling @xmath157 @xcite . thus , for our purposes nuclei are only interesting when they are of galactic origin . there are two galactic sites which could provide acceleration to interesting energies for heavy nuclei . the first one is the termination shock of the galactic wind caused by the milky way @xcite . this leads to maximal energies of @xmath158 @xcite where @xmath159 is the galactic wind speed and @xmath160 is the shock lifetime . even for @xmath161 this is significantly too low and one is forced to use quite extreme parameters to reach beyond @xmath52 . the second site would be even more natural to produce predominantly high energy heavy nuclei , namely young supernova remnants which form a pulsar wind shock @xcite . this is because pulsars can have quite large surface magnetic fields of order @xmath162 leading to fields of order @xmath163 on scales of @xmath164 cm @xcite . the pulsar wind can be relativistic so that application of eq . ( [ emax ] ) leads to @xmath165 which for @xmath166 is about one order of magnitude smaller than the highest observed energies . adopting the more conservative estimate eq . ( [ ec ] ) leads to a short fall of about two orders of magnitude . indeed , as can be seen from fig . 2 , the crab nebula lies near the arrival direction of our events and is expected to have magnetic fields of the order of @xmath167 on a scale of @xmath168 @xcite leading to estimates for the energy cutoff quite similar to the above mentioned . however , a calculation including the uncertainty of the latter one gives a relative angle of @xmath169 which is more than @xmath105 away . furthermore , taking into account deflection effects due to the coherent galactic magnetic field component mentioned in section 3.3 increases the angular distance as the path should be bent towards the galactic north pole ( see fig . 2 ) . for example , for @xmath170 the angular difference from the arrival direction would be @xmath171 . it thus seems that only large bending by almost @xmath172 could explain a possible origin from the crab . for @xmath173 the larmor radius is @xmath174 at these energies which indeed comes near the required amount of bending . in that case , however , the arrival direction is not expected to be correlated with the source location in a simple way and the propagation of these particles should better be considered as diffusion in the magnetic field . the source could therefore be any galactic site being able to produce the required source energy and this possibility can not be completely excluded although , as mentioned above , current models fall short in energy . it should be noted that the shower maximum at @xmath147 allows no definite distinction between a proton and a heavy nucleus induced shower as the expected numbers for these options are @xmath175 and @xmath176 for iron , respectively @xcite . there are still problems left in interpreting the @xmath56 event as caused by a heavy nucleus . as so it is possible that heavy nuclei could be disintegrated already at the source of acceleration @xcite . furthermore , the fly s eye data between @xmath137 and @xmath52 suggest the transition to a lighter component as we already mentioned . there were some other suggestions how one could get to higher source energies . for example , colgate @xcite claimed that in the relativistic plasma of agn jets energies as high as @xmath177 could be reached due to a plasma pinch effect similar than that used in tokamaks . note however that due to fig . 1a and 1b the source of the highest energy events could still not be much further away than @xmath178 . because the larmor radius grows with energy the possible bending angle caused by magnetic fields could also not be enhanced significantly beyond 15 degrees so the problem remains . it was suggested @xcite that high energy events could be caused by relativistic dust grains . the lateral shower profile caused by a dust grain entering the atmosphere is expected not to show a broad maximum but instead to have a more or less constant lateral spread as long as the grain remains large and energetic enough to produce secondary particles . however , the highest energy fly s eye event showed a quite `` normal '' shower development typical for a primary proton or possibly a photon which leaves the dust grain hypothesis to seem not very likely . furthermore there is a tendency for these grains to break up by interactions with photons and gas atoms in the interstellar medium @xcite . topological defects ( tds ) @xcite could have been formed in the early universe during phase transitions associated with spontaneous breaking of symmetries implemented in unified models of high energy interactions . such tds are magnetic monopoles , cosmic strings , domain walls , superconducting cosmic strings , textures , etc . tds are topologically stable but can nevertheless be destroyed due to physical processes like collapse or annihilation @xcite . in that case the energy stored in the defects is released in the form of massive quanta of the fields like gauge fields and higgs fields associated with the broken symmetry . these `` x '' particles released from the tds would typically decay into quarks and leptons . hadronization of the quarks would produce jets of hadrons containing mainly light mesons ( pions ) together with a small fraction ( @xmath179 ) of nucleons . the gamma rays and neutrinos from the decay of the pions would thus be the dominant particles in the final decay products of the x particles . the mass @xmath180 of the x particles is typically of the order of the symmetry - breaking scale which in grand unified theories ( guts ) can be @xmath181 , or even the planck scale @xmath182 . the decay of the x particles released from tds can thus give rise to nucleons , gamma rays and neutrinos with energies up to @xmath183 , very much higher than what can be achieved by astrophysical shock acceleration mechanism . the cosmic ray particles can thus be produced directly in this scenario , and no acceleration mechanism is needed . the production spectra of the nucleons , gamma rays and neutrinos in the td scenario are determined by the physics of fragmentation of quarks into hadrons . extrapolation @xcite of qcd based hadronization models ( which describe well the gev scale collider data ) to the extremely high energies gives a power - law approximation @xcite to the differential production spectra with a power - law index @xmath1841.32 for nucleons as well as pions . the decay of the neutral pions thus gives a differential gamma ray production spectrum also with @xmath24=1.32 . it is to be emphasized , however , that there is a great deal of uncertainty in extrapolating the low energy qcd models of hadronization to the extremely high energies involved in the present situation . moreover , the gamma ray production spectrum can be somewhat different from the proton production spectrum if one considers the gamma rays generated by the charged leptons ( electrons and positrons ) in the primary decay products of the x particles . the electrons and positrons coming from the decay of the charged pions in the hadronic jets also contribute to the overall primary gamma rays . the main point , however , is that the production spectra of cosmic ray particles in the td scenario can in principle be considerably flatter than in the standard shock acceleration scenario . the latter , to recall , by and large produces differential production spectra with @xmath185 . one consequence of a relatively flat production spectrum in the td scenario would be the `` recovery '' @xcite of the evolved proton spectrum after the gzk `` cutoff '' @xcite . while this is heartening from the point of view of prospects for detecting protons above the gzk `` cutoff '' , too flat a proton spectrum may cause problems in that it may give rise to excessive gamma ray flux at much lower energies , as discussed below . in any case , as first discussed in ref . @xcite , the photon - to - proton ratio in the evolved spectra can be considerably larger than 1 above @xmath52 in the td scenario @xcite ( because of the _ primary _ gamma rays which outnumber the protons by a factor of at least 10 at production , and also because of higher transparency of gamma rays relative to the protons at these energies ) , and so the cosmic rays above @xmath52 are predicted to be mainly _ primary _ gamma rays rather than protons . gamma rays as well as protons of ultrahigh energies generate lower energy gamma rays by @xmath186 and @xmath187 collisions with the photons ( @xmath151 ) of the background radiation fields . the electromagnetic component of the energy lost by the photons and protons in these collisions cascades down to lower energies by electromagnetic cascading in the universal radio background ( urb ) , the microwave background ( cmbr ) , and in the infrared background ( irb ) ( in order of decreasing energy of the propagating photon ) . recently it has been realised @xcite that the measured flux of extragalactic gamma rays in the @xmath188 region @xcite provides an upper limit on the total energy density of the cascade - initiating electromagnetic radiation that can possibly be released in the universe due to @xmath187 and @xmath189 interactions . this , in turn , restricts the shapes of the proton as well as the primary photon spectra in the highest energy region . the authors of ref . @xcite claim that a proton spectrum with @xmath190 at injection and extending to @xmath177 would by itself give rise to a @xmath188 gamma ray flux exceeding the measured flux by a factor of 2 if the evolved proton spectrum is normalized @xcite to observed particle flux at @xmath191 . this question has recently been studied in detail @xcite by a careful numerical calculation of the cascading process including the gamma rays generated by both @xmath187 as well as the @xmath189 processes . it is found that whether or not the predicted @xmath188 gamma ray flux exceeds the measured value depends strongly on the level of the irb as well as on its cosmological evolutionary history both of which are rather uncertain , and so a firm conclusion in this regard can not be drawn at this stage . nevertheless , the authors of ref . @xcite have suggested that the possible problem arising from requirement of consistency with the measured @xmath188 gamma rays can be avoided if the cosmic rays above @xmath148 are mainly gamma rays and not protons . the preliminary analysis of ref . @xcite shows that this is possible provided the injection spectrum of the primary gamma rays above @xmath52 in the td model is made somewhat steeper ( @xmath192 ) compared to the protons ( @xmath193 ) and @xmath194 is demanded to be @xmath195 at injection so that the proton component is made negligible compared to the photons . the average multiplicity in the hadronic jets arising out of the decay of the x particles is also required to be somewhat higher than what naive extrapolation of the low energy qcd based models of jet fragmentation indicates . while all these phenomenological requirements need to be substantiated on more theoretical grounds , the general conclusion that seems to arise from the above discussion is that the highest energy cosmic ray particles in the td scenario _ should be _ mainly gamma rays and not protons . and , of course , primary neutrinos @xcite should be at least as abundant as the gamma rays , perhaps even more . could the @xmath56 fly s eye event be a primary gamma ray due to td collapse or annihilation ? as already mentioned in section 4.1 above , the shower development is not in contradiction with what is expected for a primary gamma . the resulting electromagnetic shower can in fact be very similar @xcite to a proton - induced shower , although some differential parameters , e.g. , muon / electron ratio at large distances from the core of the shower can in principle be used for effective separation @xcite of these photon - induced showers from the proton - induced ones . how could one distinguish between the td option and the galactic heavy nuclei hypothesis which seems to be the least problematic option within the standard picture ? heavy nuclei are expected to produce substantially more muons compared to gammas of the same energy @xcite . it should therefore also be possible to draw a decision between these options as more statistics is available at these highest energies . the lack of any obviously identifiable astrophysical source for the event is not a problem for the td scenario because tds are not necessarily expected to be associated with any astrophysical sources such as galaxy clusters or agns . the td model thus seems to offer an attractive option in this regard . it is , however , expected that the same td annihilation event would also produce _ lower energy _ gamma rays which would arrive at earth at roughly the same time and with same arrival direction as the @xmath56 fly s eye event . unfortunately , the casa array @xcite capable of detecting such gamma rays was not operating at the time when the above fly s eye event was recorded . however , the cygnus array @xcite capable of detecting gamma rays above about @xmath196 was operating and it detected no event @xcite that can be associated with the fly s eye event . if the _ integral _ primary gamma ray spectrum between @xmath8 and @xmath52 due to td annihilation is taken to be approximately proportional to @xmath197 , ( @xmath198 ) , and if one ( optimistically ) takes the flux at @xmath52 as @xmath199 1 per @xmath200 , then above @xmath8 and in an area of @xmath201 ( roughly the area of the cygnus array ) one would expect an integral flux @xmath202 events per @xmath201 . the non - detection by the cygnus array of any gamma ray event in the @xmath196 region coincident with the fly s eye event can then be interpreted in terms @xmath112 being @xmath203 ( i.e. , a relatively flat spectrum ) in the td model at energies below @xmath52 . ( note that in conventional scenarios @xmath112 is usually taken to be @xmath199 1 ) . for example , if in the td model one takes @xmath204 0.32 @xcite and neglects the attenuation due to interaction with the cmbr ( thus overestimating the expected flux ) , then @xmath205 events in @xmath201 , and so cygnus may have missed the event . this point , however , needs further investigation , and will be discussed elsewhere . we are lead to the conclusion that protons arriving at the earth with energies of @xmath56 or above are very likely to have come from an agn or a galaxy cluster not further away than @xmath1 if they were produced there via the standard diffusive shock acceleration mechanism . even then the necessary conditions to be fulfilled in such relativistic strong shocks seem highly improbable as long as the shock parameters have to be compatible with observational data . furthermore , the arrival direction of such protons have to be within around 15 degrees in the direction of their source . the @xmath56 fly s eye event and the highest energy yakutsk event were therefore very likely not protons produced within the standard diffusive shock acceleration scenario as they do not point to some possible source being nearer than @xmath1 . some other explanations for such events like that being produced by secondaries of shock accelerated protons were discussed . within the astrophysical shock scenario the most promising , although also problematic option seem to be heavy nuclei of galactic origin which could be accelerated in pulsar wind shocks driven by young supernova remnants . we therefore conclude that at least some improvements in the understanding of the current acceleration picture have to be made in order to explain the highest energy cosmic rays observed . it seems possible that a completely new production mechanism for such particles is necessary . we suggested that the td model could be a promising option . it is curious that such an exotic option seems to have less difficulties in explaining these ultra - high energy particles . all other current options appear to require suspension of belief in seemingly well substantiated observational numbers or indicate incomplete understanding of the underlying physical process . we list all the options discussed here together with their problems in table 1 . we would like to thank james cronin , eugene parker , hongyue dai , stirling colgate , gene loh , cy hoffman and corbin covault for highly valuable discussions . in addition , we are grateful to paul sommers and rene ong for their suggestions concerning the manuscript . we would also like to thank chris hill for his collaboration in earlier papers dealing with topological defects and his enthusiastic contributions to this topic . this work has been supported , in part , by nsf , nasa and the doe at the university of chicago , by the doe and by nasa through grant nagw 2381 at fermilab and by the alexander - von - humboldt foundation . _ note added : _ during preparation of this manuscript we became aware of a paper by p. sommers @xcite which essentially deals with the same topic . the conclusions reached concerning the shock acceleration scenario are quite similar . in difference , however , the aim of our paper was a more detailed discussion of all possible explanations related to relativistic particles and especially to note the emergence of the td scenario as an interesting option . std d. j. bird et . al . , _ phys . lett . _ * 71 * , 3401 ( 1993 ) . t. k. gaisser : _ cosmic rays and particle physics _ , cambridge 1990 . we are grateful to paul sommers for providing us with these numbers and a discussion of it . : m. nagano and f. takahara , world scientific , singapore ( 1991 ) . , published by the institute for cosmic ray research , university of tokyo , ed . : m. nagano ( september 1993 ) . n. n. efimov et . al . in ref . @xcite , p. 20 . t. a. egorov in ref . @xcite , p. 35 . 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j. w. cronin et . al . , efi 94 - 05 , submitted to nucl . meth .. d. e. alexandreas et . al . , _ nucl . meth . _ * a311 * , 350 ( 1992 ) . c. hoffmann , private communications . p. sommers in ref . @xcite , p. 23 . we are grateful to corbin covault for helping us with this figure . & & additional + sources & shock acc . & drift acc . & problems + agns & @xmath206 & @xmath207 & distance+direction + pulsars & @xmath208 & @xmath209 & direction + galactic wind & & + + & & additional + sources & shock acc . & drift acc . & problems + pulsars & @xmath210 & @xmath211 & direction + galactic wind & & + + & & problem + td s & & exotic + * figure 1a and 1b : * distance of source versus source energy for protons of arrival energies of @xmath212 and @xmath15 , respectively . plotted from bottom to top are the average distances and the maximal distances at the @xmath12 , @xmath213 and @xmath105 level , respectively . * figure 2 : * arrival directions of the highest energy events seen by fly s eye and the yakutsk experiment in galactic coordinates @xcite . this plot is centered around the galactic anticenter with the middle horizontal line being the projection of the galactic plane . also shown are nearby galaxy clusters ( big circles ) , agn s ( small circles ) and galactic supernova remnants ( light circles ) . as discussed in section 3.3 the galactic magnetic field is supposed to have a coherent component near the galactic plane which in the outer region is polarized in the direction of decreasing galactic longitude . therefore , the apparent arrival direction of cosmic rays coming from one of the objects shown should be shifted by @xmath214 to lower latitude .

in this paper we show that the conventional diffusive shock acceleration mechanism for cosmic rays associated with relativistic astrophysical shocks in active galactic nuclei ( agns ) has severe difficulties to explain the highest energy cosmic ray events . we show that protons above around @xmath0 could have marginally been produced by this mechanism in an agn or a rich galaxy cluster not further away than around @xmath1 . however , for the highest energy fly s eye and yakutsk events this is inconsistent with the observed arrival directions . galactic and intergalactic magnetic fields appear unable to alter the direction of such energetic particles by more than a few degrees . we also discuss some other options for these events associated with relativistic particles including pulsar acceleration of high @xmath2 nuclei . at the present stage of knowledge the concept of topological defects left over from the early universe as the source for such events appears to be a promising option . such sources are discussed and possible tests of this hypothesis are proposed . 2gcm^-2 # 1#23.6pt 6.5 in 8.5 in -0.25 in + g. sigl@xmath3 , d. n. schramm@xmath3 and p. bhattacharjee@xmath4 + _ @xmath5department of astronomy & astrophysics + enrico fermi institute , the university of chicago , chicago , il 60637 - 1433 _ + _ @xmath6nasa / fermilab astrophysics center + fermi national accelerator laboratory , batavia , il 60510 - 0500 _ + _ @xmath4indian institute of astrophysics + sarjapur road , koramangala , bangalore 560 034 , india _

one of the most impressive demonstrations of the specifically non - classical features of quantum mechanics is the violation of bell s inequalities by entangled photon pairs @xcite . the violation of bell s inequalities shows that it is impossible to explain the statistical predictions of quantum theory by assigning a complete set of polarization components to each photon before the measurement . this implies that the measurement results for a specific polarization direction should not be interpreted as a general property of the photon . in particular , photons can not be classified as either x or y polarized , even though only these two outcomes are observed in precise polarization measurements along these axes . in the following , the non - classical correlations of entangled photons are analyzed by applying finite resolution measurements @xcite . a measurement setup for the simultaneous measurement of non - commuting polarization components is presented in section [ sec : onephoton ] . in section [ sec : twophotons ] , this measurement concept is applied to entangled photon pairs . it is shown how information on all four polarization components responsible for the violation of bell s inequalities can be obtained from a single measurement setup . the analysis of the measurement statistics shows that the violation of bell s inequalities arises from negative joint probabilities similar to the ones obtained in the single photon measurement setup . the statistics derived from the finite resolution measurement thus allows an identification of the local non - classical properties responsible for the violation of bell s inequalities . the polarization of light can be characterized by the stokes parameters , defined as the intensity difference between orthogonally polarized modes . a complete description of polarization requires three stokes parameters . all stokes parameters can then be written as components of this three dimensional vector . in terms of the annihilation operators for right and left circular polarization , @xmath0 and @xmath1 , the stokes parameters read @xmath2 for a single photon , these operators have eigenvalues of @xmath3 , as observed in measurements using polarization filters . however , a polarization filter is only sensitive to one component of the stokes vector at a time , while completely randomizing the information potentially carried by the other two components . this limitation can be overcome by applying finite resolution measurements to obtain information on one polarization component while limiting the noise introduced in the other components . it is then possible to study correlations between the non - commuting polarization components of a single photon . ( 420,270 ) ( 0,200)(1,0)360 ( 25,230)(55,15 ) beam ( 25,215)(55,15 ) displacer ( 30,175)(45,35 ) ( 30,200)(3,-1)45 ( 75,185)(1,0)285 ( 120,230)(60,15 ) polarization ( 120,215)(60,15 ) rotation ( 150,192.5 ) ( 220,220)(55,15 ) polarizer ( 225,170)(45,45 ) ( 225,215)(1,-1)45 ( 240,200)(0,-1)120 ( 255,185)(0,-1)105 ( 335,245)(60,15 ) detector ( 335,230)(60,15 ) array ( 360,160)(10,65 ) ( 335,140)(60,15 ) @xmath4 ( 370,170)(370,175)(390,180 ) ( 390,180)(402,183)(402,185 ) ( 370,200)(370,195)(390,190 ) ( 390,190)(402,187)(402,185 ) ( 370,185)(370,190)(390,195 ) ( 390,195)(402,198)(402,200 ) ( 370,215)(370,210)(390,205 ) ( 390,205)(402,202)(402,200 ) ( 140,75)(60,15 ) detector ( 140,60)(60,15 ) array ( 215,70)(65,10 ) ( 285,67)(60,15 ) @xmath5 ( 225,70)(230,70)(235,50 ) ( 235,50)(238,38)(240,38 ) ( 255,70)(250,70)(245,50 ) ( 245,50)(242,38)(240,38 ) ( 240,70)(245,70)(250,50 ) ( 250,50)(253,38)(255,38 ) ( 270,70)(265,70)(260,50 ) ( 260,50)(257,38)(255,38 ) figure [ branch ] illustrates the experimental setup for a finite resolution measurement of two orthogonal components of the stokes vector , @xmath6 and @xmath7 . a beam displacer is used to couple the transversal position of the photon with the polarization component @xmath6 . the resolution of this measurement is given by the ratio of the displacement and the width of the beam . after the measurement of @xmath6 , the polarization component @xmath7 is measured by a @xmath8 rotation of the polarization axes and a polarizer . however , the resolution of the @xmath7 measurement is limited by the polarization noise induced in the beam displacer . the detector arrays record the continuous measurement values @xmath9 obtained in the measurement of @xmath6 for the two final measurement values of @xmath7 . the finite resolution measurement of @xmath6 is described by the measurement operator @xcite , @xmath10 the probability of a measurement of @xmath9 followed by a measurement of @xmath11 for an input state @xmath12 is then given by @xmath13 ( 420,470 ) ( 195,210)(60,20)@xmath14 ( 0,15)(420,200 ) single photon polarization statistics for photons polarized along the @xmath15 direction ( @xmath4 eigenstates ) . the resolution is @xmath16 for the initial @xmath17 measurement in the beam displacer.,title="fig:",width=226 ] ( 205,0)(40,20)@xmath9 ( 195,450)(60,20)@xmath18 ( 0,255)(420,200 ) single photon polarization statistics for photons polarized along the @xmath15 direction ( @xmath4 eigenstates ) . the resolution is @xmath16 for the initial @xmath17 measurement in the beam displacer.,title="fig:",width=226 ] ( 205,240)(40,20)@xmath9 if the input state is in the @xmath19 eigenstate of @xmath7 , the measurement statistics are @xmath20 \lefteqn{p(s_{1m};s_2\!=\!-1 ) = } \nonumber \\ & & \frac{1}{\sqrt{2\pi\delta\!s^2 } } \exp\left(-\frac{s_{1m}^2 + 1}{2\delta\!s^2}\right ) \sinh^2\left(\frac{s_{1m}}{2 \delta\!s^2}\right),\end{aligned}\ ] ] as shown in figure [ onephoton ] for a resolution of @xmath21 . the results show that the intuitive assumption that @xmath17 should be statistically independent of @xmath11 is wrong even for an eigenstate of @xmath7 . instead , the high values of @xmath9 are clearly correlated with `` quantum jumps '' to @xmath22 . as discussed in @xcite , this implies non - vanishing probability contributions from @xmath5 in the statistics of the @xmath4 eigenstate . @xmath23 + 1 & \frac{1}{4 } & \frac{1}{2 } & \frac{1}{4 } \\[0.3 cm ] -1 & \frac{1}{4 } & - \frac{1}{2 } & \frac{1}{4 } \\[0.3 cm ] \hline \end{array}\ ] ] if the measurement statistics is interpreted in terms of gaussian contributions with a variance of @xmath24 centered around the actual values of @xmath17 , it appears that , in addition to the quantized eigenvalue results of @xmath25 , results of @xmath26 must also be taken into account . moreover , the total probability for @xmath26 remains zero because the negative joint probability of @xmath27 for @xmath26 and @xmath5 cancels the positive joint probability of @xmath28 for @xmath26 and @xmath4 . this negative joint probability also explains the coexistence of correlations between @xmath17 and @xmath11 with a total probability of zero for @xmath5 . the full set of joint probabilities obtained for @xmath29 is shown in table [ smallstat ] . ( 420,420 ) ( 20,340)(60,60 ) ( 30,380)(40,15)photon ( 30,365)(40,15)pair ( 30,350)(40,15)source ( 85,375)(15,15)@xmath30 ( 30,320)(15,15)@xmath31 ( 80,370)(1,0)300 ( 137,355)(27,21 ) ( 137,370)(3,-1)27 ( 164,361)(1,0)216 ( 220,365.5 ) ( 287,352)(27,27 ) ( 287,379)(1,-1)27 ( 296,370)(0,-1)84 ( 305,361)(0,-1)75 ( 380,346)(6,39 ) ( 281,280)(39,6 ) ( 383,306)(0,-1)110 ( 383,196)(1,1)15 ( 383,196)(-1,1)15 ( 337,246)(40,15)@xmath32 ( 300.5,250)(0,-1)54 ( 300.5,196)(1,1)15 ( 300.5,196)(-1,1)15 ( 254.5,218)(40,15)@xmath32 ( 50,340)(0,-1)300 ( 44,256)(21,27 ) ( 50,283)(1,-3)9 ( 59,256)(0,-1)216 ( 54.5,200 ) ( 41,106)(27,27 ) ( 41,133)(1,-1)27 ( 50,124)(1,0)84 ( 59,115)(1,0)75 ( 35,34)(39,6 ) ( 134,100)(6,39 ) ( 114,37)(1,0)110 ( 224,37)(-1,1)15 ( 224,37)(-1,-1)15 ( 146,40)(40,15)@xmath33 ( 170,119.5)(1,0)54 ( 224,119.5)(-1,1)15 ( 224,119.5)(-1,-1)15 ( 170,122.5)(40,15)@xmath33 ( 267,165)(150,15)coincidence counts ( 348,2)(70,70 ) ( 358,40)(50,15)@xmath34 ( 358,19)(50,15)@xmath35 ( 348,84.5)(70,70 ) ( 358,122.5)(50,15)@xmath34 ( 358,101.5)(50,15)@xmath36 ( 265.5,2)(70,70 ) ( 275.5,40)(50,15)@xmath37 ( 275.5,19)(50,15)@xmath35 ( 265.5,84.5)(70,70 ) ( 275.5,122.5)(50,15)@xmath37 ( 275.5,101.5)(50,15)@xmath36 finite resolution measurements thus reveal that negative joint probabilities are an integral part of local quantum statistics . since this property of quantum statistics contradicts the assumptions made about elements of reality in the formulation of bell s inequalities @xcite , it is possible to explain the violation of bell s inequalities by applying the same analysis to the polarization statistics of entangled photon pairs . ( 400,440 ) ( 40,420)(150,20)@xmath38 ( 40,260)(150,150 ) ( 0,335)(30,10)@xmath33 ( 105,235)(40,10)@xmath32 ( 250,420)(150,20)@xmath39 ( 250,260)(150,150 ) ( 210,335)(30,10)@xmath33 ( 315,235)(40,10)@xmath32 ( 40,190)(150,20)@xmath40 ( 40,30)(150,150 ) ( 0,105)(30,10)@xmath33 ( 105,5)(40,10)@xmath32 ( 250,190)(150,20)@xmath41 ( 250,30)(150,150 ) ( 210,105)(30,10)@xmath33 ( 315,5)(40,10)@xmath32 figure [ setup ] shows the setup for a coincidence measurement for entangled photons . the two branches @xmath30 and @xmath31 are set up as illustrated in figure [ branch ] . a maximal violation of bell s inequalities is obtained for an input state of @xmath42 where the letters @xmath43 and @xmath44 denote eigenstates of right and left polarization for photon @xmath30 and photon @xmath31 , respectively . this input state is an eigenstate of the correlation @xmath45 with an eigenvalue of @xmath46 . the maximal value obtained by assigning eigenvalues of @xmath3 to the operators @xmath47 in equation ( [ eq : k ] ) is @xmath48 . therefore , @xmath49 maximally violates the bell s inequality @xmath50 . figure [ stats ] shows the measurement statistics obtained for a resolution of @xmath51 . at this resolution , the quantum noise introduced in the measurement of @xmath6 is still very low , so that the original properties of @xmath7 are preserved . therefore , the statistics clearly reveal the non - classical features of correlations between @xmath6 and @xmath7 . in particular , the peaks of the results obtained for @xmath52 are at values of @xmath53 , far beyond the eigenvalue limits of @xmath3 . moreover , the peaks are actually sharper than the resolution of @xmath51 would allow in a classical context . @xmath54 \left(s_1(b),s_2(b)\right ) & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } \\[-0.3 cm ] & ( -1,-1 ) & ( 0,-1 ) & ( 1,-1 ) & ( -1,1 ) & ( 0,1 ) & ( 1,1 ) \\[0.5 cm ] \hline & & & & & & \\ ( 1,1 ) & \frac{\sqrt{2}-1}{16\sqrt{2 } } & - \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}+1}{16\sqrt{2 } } & \frac{\sqrt{2}-1}{16\sqrt{2 } } & \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}+1}{16\sqrt{2 } } \\[0.5 cm ] ( 0,1 ) & \frac{1}{8\sqrt{2 } } & - \frac{1}{4\sqrt{2 } } & - \frac{1}{8\sqrt{2 } } & \frac{1}{8\sqrt{2 } } & \frac{1}{4\sqrt{2 } } & -\frac{1}{8\sqrt{2 } } \\[0.5 cm ] ( -1,1 ) & \frac{\sqrt{2}+1}{16\sqrt{2 } } & \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}-1}{16\sqrt{2 } } & \frac{\sqrt{2}+1}{16\sqrt{2 } } & - \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}-1}{16\sqrt{2 } } \\[0.5 cm ] \hline & & & & & & \\ ( 1,-1 ) & \frac{\sqrt{2}-1}{16\sqrt{2 } } & - \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}+1}{16\sqrt{2 } } & \frac{\sqrt{2}-1}{16\sqrt{2 } } & \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}+1}{16\sqrt{2 } } \\[0.5 cm ] ( 0,-1 ) & -\frac{1}{8\sqrt{2 } } & \frac{1}{4\sqrt{2 } } & \frac{1}{8\sqrt{2 } } & - \frac{1}{8\sqrt{2 } } & -\frac{1}{4\sqrt{2 } } & \frac{1}{8\sqrt{2 } } \\[0.5 cm ] ( -1,-1 ) & \frac{\sqrt{2}+1}{16\sqrt{2 } } & \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}-1}{16\sqrt{2 } } & \frac{\sqrt{2}+1}{16\sqrt{2 } } & - \frac{1}{8\sqrt{2 } } & \frac{\sqrt{2}-1}{16\sqrt{2 } } \\[0.5 cm ] \hline \end{array}\ ] ] @xmath54 \left(s_1(b)\!,s_2(b)\right ) & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } & \hspace{1.8 cm } \\[-0.3 cm ] & ( -1,-1 ) & ( 0,-1 ) & ( 1,-1 ) & ( -1,1 ) & ( 0,1 ) & ( 1,1 ) \\[0.5 cm ] \hline & & & & & & \\ ( 1,1 ) & -2 & -2 & -2 & + 2 & + 2 & + 2 \\[0.5 cm ] ( 0,1 ) & 0 & -1 & -2 & + 2 & + 1 & 0 \\[0.5 cm ] ( -1,1 ) & + 2 & 0 & -2 & + 2 & 0 & -2 \\[0.5 cm ] \hline & & & & & & \\ ( 1,-1 ) & -2 & 0 & + 2 & -2 & 0 & + 2 \\[0.5 cm ] ( 0,-1 ) & 0 & + 1 & + 2 & -2 & -1 & 0 \\[0.5 cm ] ( -1,-1 ) & + 2 & + 2 & + 2 & -2 & -2 & -2 \\[0.5 cm ] \hline \end{array}\ ] ] as in the case of a single photon , the statistics may be interpreted as a sum of gaussian contributions with a variance of @xmath24 centered around the actual values of @xmath17 . the non - classical features then arise from the negative joint probabilities at @xmath55 and/or @xmath56 . the sharpness and the shift of the peaks at @xmath52 are explained by the negative probability at @xmath57 . table [ negprop ] shows the full set of joint probabilities obtained from the measurement results for @xmath29 . as shown in table [ kvalue ] , each of the 36 measurement results in table [ negprop ] corresponds to a well defined value of @xmath58 . in accordance with the probability maxima in figure [ stats ] , the values of @xmath59 are found at @xmath60 or @xmath61 for @xmath62 , at @xmath63 or @xmath64 for @xmath65 , at @xmath63 and @xmath61 for @xmath37 and @xmath35 , and at @xmath60 and @xmath64 for @xmath34 and @xmath36 . the broadness of the peaks in the measurement statistics observed for @xmath66 in figure [ stats ] is explained by the positive probability contribution for @xmath67 at @xmath57 . the steep slopes of the peaks for @xmath52 is likewise explained by the negative probability contribution for @xmath68 at @xmath57 . the regions of low probability in figure [ stats ] are explained by the near cancellation of negative and positive probabilities for values of @xmath69 at @xmath60 or @xmath64 for @xmath37 and @xmath35 , at @xmath63 or @xmath61 for @xmath34 and @xmath36 , at @xmath63 and @xmath64 for @xmath62 , and at @xmath60 and @xmath61 for @xmath65 . the total probability distribution of k values then reads @xmath70 the violation of bell s inequalities is therefore the result of negative joint probabilities for the non - commuting polarization components of the entangled photon pair . the formulation of bell s inequalities is based on the assumption that the operator variables can be represented by their eigenvalues . this assumption reflects the definition of elements of reality given in the famous paper by einstein , podolsky and rosen ( epr ) @xcite : `` _ 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 . _ '' however , this argument breaks down if the statistics include negative probabilities . if one value of a physical quantity has a probability equal unity , it is still possible that another value of the same property has positive and negative joint probabilities . in particular , the discussion of single photon polarization in section [ sec : onephoton ] revealed the presence of contributions from @xmath26 , even though no eigenvalue of @xmath6 corresponds to this result . while it is possible to predict with certainty that no precise measurement of @xmath6 can produce this result , this certainty does not apply to the result of finite resolution measurements . while the total probability for @xmath26 is always zero , the joint probabilities shown in tables [ smallstat ] and [ negprop ] are not . negative probabilities thus introduce a measurement dependent ambiguity into the selection of elements of reality that contradicts the assumptions of bell s inequalities . note that negative probabilities cause no conceptual problems as long as the uncertainty principle applies to all measurements . indeed , the uncertainty principle can be interpreted as a consequence of negative joint probabilities since it must be impossible to isolate an event associated with a negative probability . uncertainty guarantees that negative probabilities are always `` covered up '' by quantum noise in the measurement process . effectively , actual measurement results can only be associated with a region of phase space sufficiently large to include more positive than negative probability contributions . finite resolution measurements of single photon polarization allow simultaneous measurements of non - commuting stokes parameter components . by applying this type of measurement to entangled photon pairs , details of the violation of bell s inequalities can be obtained in a single measurement setup . it is possible to represent the statistics of the photon pair polarization in a table of 36 joint probabilities for the non - commuting polarization components . non - classical features arise from the negative probabilities at values of @xmath26 . these features not only explain the violation of bell s inequalities , but also establish a connection between entanglement and the non - classical properties of individual quantum systems . i would like to acknowledge support from the japanese society for the promotion of science , jsps .

by using finite resolution measurements it is possible to simultaneously obtain noisy information on two non - commuting polarization components of a single photon . this method can be applied to a pair of entangled photons with polarization statistics that violate bell s inequalities . the theoretically predicted results show that the non - classical nature of entanglement arises from negative joint probabilities for the non - commuting polarization components . these negative probabilities allow a `` disentanglement '' of the statistics , providing new insights into the non - classical properties of quantum information . + keywords : bell s inequalities , photon statistics , entanglement

microblogging platforms such as twitter provide rapid access to situation - sensitive information that people post during mass convergence events such as natural disasters . rapid crisis response can be aided by processing these tweets @xcite in real - time . different stake - holders ( e.g. different humanitarian organizations ) have different informational needs . for example , to better understand the severity and status of an event , most of these organizations rely on situational awareness information . some others look for information on a specific concern like reports of damage to key infrastructure in the area such as airports , bridges , buildings , communication infrastructure , etc . typically , the first step in extracting situational awareness information from these tweets involves classifying them into different informational categories such as infrastructure damage , shelter needs or offers , relief supplies , etc . for instance , one such application , aidr @xcite , classifies twitter messages into different categories in real time . however , even after the automatic classification step , each category still contains thousands of messages many of which are important . additional in - depth analysis is required to create a coherent situational awareness summary for disaster managers to understand the situation , which can be rapidly changing . in this paper , we seek to extract important topical information from microblogging platforms and generate summaries for the identified topics . for example , within the tweets categorized as infrastructure damage related , the reader can examine the status of airports , buildings , bridges , etc . provided this information has been reported . drilling down into sub - topics and examine the set of tweets from which the information was extracted , we must group tweets dealing with similar information into sets . these sets should be labeled with a concept name . + * summarizing messages in disaster - related categories : * summarizing tweets is significantly more challenging than summarizing news articles . the difficulty arises because tweets are often written in informal and non - standard language as opposed to the formal language used in news articles . to address the real - time nature of our application and the need for a more readable , more informative , and more easily understandable summary , we propose a novel _ two - step summarization process _ that uses a fast extractive summarization technique @xcite followed by an abstractive summarization step that improves the information coverage and readability of the final summary . rudra et al . used extractive summarization to summarize a set of tweets @xcite . for example , consider the following tweets collected during the nepal earthquake in 2015 : 1 . tribhuvan international airport closed after the quake 2 . airport closed after 7.9 earthquake in kathmandu an abstractive summary of these tweets would be as follows : + tribhuvan international airport closed + after 7.9 earthquake in kathmandu . note that the latter is more compact freeing up words that can be used for additional information coverage . information coverage can be improved in a summary by including as many content words as possible . for example , rudra et al . have showed that maximizing the coverage of content words produces effective summaries of disasters @xcite . however , we observed that many such content words are semantically similar and capturing one of those in final summary will suffice to provide adequate information coverage . hence , in this work , we collate similar nouns and verbs to develop concept and event clusters . we propose a word - graph based abstractive summarization technique that combines information from semantically similar tweets ( extracted in first step ) and applies an ilp - based content word ( numeral , location , concept , event ) coverage method to generate the summary . although abstractive summarization @xcite produces more compact and informative sentences , the algorithms in general are time - consuming . hence , if the abstractive approach is run over the entire incoming set of tweets , it may not be possible to produce the results in real - time ( which is one of the important requirements during disasters ) . in order to circumvent this problem , first we extract a set of important tweets from the whole set using a fast but effective extractive summarization approach . in the second step , we use abstractive summarization to choose and rewrite the most important tweets among them , remove redundancy and improve the readability of the tweets . + * identification and summarization of micro - topics : * to provide information about events at a finer granularity such as when an airport has been shut , or re - opened , school suspended , communication cut or restored etc . that happen during a crisis situation , our method first identifies micro - topics ( i.e. , small - scale events ) and then generates summaries for each of these micro - topics at a finer granularity level . in this work , we use the nepal earthquake dataset @xcite comprising of several million tweets collected and initially classified by the aidr platform @xcite . our contribution lies in the two - step extractive - abstractive summarization approach ( section [ sec : summarize ] ) that is efficient and yet generates better summaries with respect to information coverage , diversity , coherence , and readability . experimental results in section [ sec : topic_summarization ] also confirm that our extracted topics and summaries related to those topics outperform traditional lda based methodologies . finally , we conclude our paper in section [ sec : conclu ] . real - time information posted by affected people on twitter helps improve disaster relief operations @xcite . however , relief organizations can plan more effectively if they have access to crucial information from the tweets @xcite . kedzie et al . @xcite proposed an extractive summarization method to summarize disaster event - specific information from news articles . in contrast , several researchers have attempted to utilize information from twitter to retrieve important situational updates from millions of posts on disaster - specific events @xcite . more recently , sophisticated methods for automatically generating summaries by extracting the most important tweets on the event @xcite have been proposed . to generate summaries in real - time , a few approaches for online summarization of tweet streams have recently been proposed @xcite . the methods mentioned above generate extractive summaries that are merely a collection of tweets . ideally , we prefer an abstractive summary composed of important content from tweets instead of the whole tweets . such a summary should also be more readable than a collection of tweets . furthermore , the summaries should not contain redundant information . to this end , olariu @xcite proposed a bigram word - graph - based summarization technique , which is capable of handling online streams of tweets in real - time and also generates abstractive summaries . each bigram represents a node in the graph and new words are added in real - time from incoming new tweets . however , the method does not consider pos - tag information of nodes and thus can spuriously fuse tweets having the same bigram but are otherwise unrelated . furthermore , it is a general method that does not consider the typicality of disaster related tweets , for example earlier rudra et al @xcite showed that during disasters content words ( nouns , verbs , numerals ) vary quite slowly compared to any other general events like sports , movies etc . in our proposed abstractive summarization framework , we have incorporated such domain dependent features to make the summary more coherent , informative , and useful . banerjee , mitra , and sugiyama proposed a graph - based abstractive summarization method on news articles @xcite . several new sentences are generated using a graph where words are nodes , edges are added between two consecutive words present in a sentence and an optimization problem is formulated that selects the best sentences from the new sentences to optimize the overall quality of the summary . the optimization problem ensures that redundant information is not conveyed in the final generated summary . however , the graph construction and path generation is computationally expensive and can not be used in real - time . we combine the positive aspects of the above studies - ( a ) we employ extractive summarization to reduce the number of tweets , and on the reduced set run an algorithm adapated from the technique proposed by banerjee et al . @xcite for tweet fusion ( b ) we use pos tags along with the words in each bigram to avoid spurious tweet fusions and ( c ) we employ disaster - specific content words to determine the importance of a disaster - related tweet @xcite . further , we also focus on template - based topic extraction and summarizing information over those topics . we use the nepal earthquake 2015 twitter data from crisisnlp @xcite . the dataset consists of 27 million messages from april 25th to april 27th obtained using different keywords ( e.g. nepal earthquake , nepalquake , nepalquakerelief , nepalearthquake , kathmanduquake , quakenepal , earthquakenepal , @xmath0 , etc . ) . in this work , we selected aidr @xcite classified messages from three categories for which the machine confidence was @xmath1 0.80 . the selected classes and messages in each of the three classes are as follows : \1 . * missing , trapped , or found people : * 10,751 2 . * infrastructure and utilities : * 16,842 3 . * shelter and supplies : * 19,006 messages . given the machine - categorized messages by aidr , in this section we present our two step automatic summarization approach to generate summaries from each class . we consider the following key characteristics / objectives while developing an automatic summarization approach : 1 . a summary should be able to capture most situational updates from the underlying data . that is , the summary should be rich in terms of information coverage . 2 . as most of the messages on twitter contain duplicate information , we aim to produce summaries with less redundancy while keeping important updates of a story . 3 . twitter messages are often noisy , informal , and full of grammatical mistakes . we aim to produce more readable summaries as compared to the raw tweets . 4 . the system should be able to generate the summary in real - time , i.e. , the system should not be heavily overloaded with computations such that by the time the summary is produced , the utility of that information is marginal . the first three objectives can be achieved through abstractive summarization and near - duplicate detection , however , it is very difficult to achieve that in real - time ( hence violating the fourth constraint ) . in order to fulfill these objectives , we follow an extractive - abstractive framework to generate summaries . we define our overall summarization framework as concept based abstractive summarization ( * conabs * ) . in the first phase ( extractive phase ) , we use the approach proposed by rudra et al . @xcite and select a subset of tweets that cover most of the information produced and then run abstractive summarization over that . we generate the paths using the extractive - abstractive framework proposed by rudra et al @xcite . our goal is to select the best paths from these generated tweet paths with the objective of generating a readable and informative summary . to this end , we formulate an ilp based technique that selects final paths and generates the summary . + * concept and event extraction : * [ sec : concept_extraction ] given that aidr classified messages into the categories mentioned above , we extract important concepts and events associated with them . for example , the ` infrastructure ' class contains information about building collapse , temples and whether the airport is open or closed . we observed that such micro - level information mainly consists of two core nuggets , a noun part which we call as a concept ( e.g. , airport ) and a verb part , which we call as an event ( e.g. , closed ) . in our summarization process , we capture information about these concepts by using nouns , because concepts are in general denoted by the nouns @xcite . for this purpose , ( i ) extract all the nouns from the dataset , ( ii ) develop a complete undirected weighted graph where nouns are nodes and weight of the edge between two nouns is their semantic similarity score ( we have used lin similarity measure ) , and ( iii ) run affinity clustering method to cluster semantically similar nouns ( e.g. , airport , flight ) . each of the identified clusters represents a particular concept . + ritter et al @xcite proposed a method to extract events from tweets but this method takes significant amount of time to tag large stream of tweets . this creates a bottleneck in real - time summarization process . hence , we can not use their method directly in our proposed summarization approach . we observed that main verbs generally represent such events like ` collapsed ' , ` killed ' , ` injured ' , ` blocked ' . we construct a complete undirected weighted graph by taking the verbs and apply clustering technique over the graph ( similar to concept extraction ) . each cluster of verbs represents one event . for example , verbs like ` injured ' , ` wounded ' are clubbed into one cluster and represent one event . + * ilp formulation * + for abstractive summarization phase , we redefine the content words . content words consist of numerals , places ( this is similar to that adopted during the extractive phase ) , concepts , and events . the ilp - based technique optimizes based upon three factors - ( i ) presence of content words : the formulation tries to maximize the number of these parameters in the final summary which in turn takes care of diversity by reducing the probability of choosing the same content word multiple times . ( ii ) informativeness of a path , i.e. , finding importance of a path based on centroid - based ranking @xcite , and ( iii ) _ linguistic quality score _ that captures the readability of a path using a trigram confidence score @xcite . [ table : ilp - parameters ] the summarization of @xmath2 words is achieved by optimizing the following ilp objective function , whereby the highest scoring _ tweet - paths _ are returned as output of summarization , the equations are as follow : @xmath3 subject to the constraints @xmath4 @xmath5 \\\ ] ] @xmath6\ ] ] where the symbols are as explained in table [ table : ilp - parameters ] . the objective function considers both the number of _ tweet - paths _ included in the summary ( through the @xmath7 variables ) as well as the number of important content - words ( through the @xmath8 variables ) included . the constraint in eqn . [ eqn : length_constraint ] ensures that the total number of words contained in the _ tweet - paths _ that get included in the summary is at most the desired length @xmath2 ( user - specified ) while the constraint in eqn . [ eqn : content - word - constraint ] ensures that if the content word @xmath9 is selected to be included in the summary , i.e. , if @xmath10 , then at least one _ tweet - path _ in which this content word is present is selected . similarly , the constraint in eqn . [ eqn : tweet_constraint ] ensures that if a particular _ tweet - path _ is selected to be included in the summary , then the content words in that _ tweet - path _ are also selected . we use the gurobi optimizer @xcite to solve the ilp . after solving this ilp , the set of _ tweet - paths _ @xmath11 such that @xmath12 , represent the summary at the current time . in this section , we compare the performance of our proposed framework with state - of - the - art abstractive and disaster - specific summarization techniques . given the aidr classified messages from three classes , we perform date - wise split starting from 25th april to 27th april . * baseline approaches : * we use three state - of - the - art summarization approaches described below : 1 . * cowts : * an extractive summarization approach specifically designed for generating summaries from disaster - related tweets @xcite . * apsal : * an affinity clustering based summarization technique proposed by kedzie et al . it mainly considers news articles and focuses on human - generated information nuggets to assign salience score to those news articles while generating summaries . * towgs : * an online abstractive summarization approach proposed by olariu @xcite . it is designed for informal texts like tweets . they consider bigrams as nodes and build word graph using these nodes . to generate a summary , they start from most frequent bigrams to explore different paths . in our case , we modified it to generate event - specific summaries as it was originally not proposed to do so . * evaluation using expert generated data * we took summaries generated by experts from the disaster management domain . during nepal earthquake , un ocha ( united nations office for the coordination of humanitarian affairs ) among other humanitarian organizations used aidr s output ( i.e. , machine classified messages ) for their disaster response efforts . in this case , the experts were given the machine classified messages that they analyzed to generate a situational awareness report for each informational category . we consider these reports as our gold standard summaries . figure [ fig : rouge_score ] shows the improvement by our method over baseline techniques in terms of rouge-1 recall score which basically indicates in percentage the amount of more ( important ) information covered in the generated summaries . we can see that conabs performs significantly better compared to other three baselines - the improvement ranges from 10% to 40% . + * evaluation using crowdsourcing : * we perform crowdsourced evaluation using the crowdflower crowdsourcing platform . we take summaries generated from each class using our proposed method and all three baselines for each day in total we use 9 summaries . a crowdsourcing task , in this case , consists of four summaries ( i.e. , one proposed and three from baseline methods ) and the four criteria with their description ( as described below ) along with a scale from 1 ( very bad ) to 5 ( very good ) for each criterion . for each task , we asked five different annotators to read each summary carefully and provide scores for each criterion . the exact description of the crowdsourcing task is as follows : _ `` the purpose of this task is to evaluate machine - generate summaries using tweets collected during the nepal earthquake happened in 2015 . each task given below has 4 summaries of length 200 words generated by 4 different algorithms on the same set of tweets ( thousands in this case ) belonging to a particular topic . given the summaries and their topic , we are interested in comparing them based on the following criteria : information coverage , diversity and readability''_. the definitions of various criteria we used in the task and discussion of the results are as follows : + * information coverage * corresponds to the richness of information a summary contains . for instance , a summary with more informative sentences ( i.e. , crisis - related information ) is considered better in terms of information coverage . our proposed method is able to capture very good situational information / updates in case of infrastructure and missing classes for both of the days chosen while it performs fairly in the shelter class . in 4 cases , it performs better than the three competing techniques , and it performs equally well with cowts , towgs , and apsal in 3 , 1 , 1 cases respectively . figure [ fig : crowd_info_cover ] shows the detailed ratings of users for 25th and 26th april . * diversity * corresponds to the novelty of sentences in a summary . a good summary should contain diverse informative sentences . while we do not apply any direct parameter in our ilp framework to control diversity , in our abstractive ilp method , we not only rely on the importance score of paths but also coverage of different content words , which helps in capturing information from various dimensions . this is also quite clear from figure [ fig : crowd_diversity ] . in seven out of nine cases , conabs generated summaries that are comparable to other baseline techniques . * readability * measures how easy it is to read the summary . a good summary should be easily readable , well formed , coherent , and have fewer grammatical errors . we used a linguistic quality score in our final ilp framework to generate coherent summaries . our system chooses paths with higher linguistic scores . summaries generated by conabs were rated to be equal or better than the other baselines in 8 ( of nine ) cases . figure [ fig : crowd_readability ] shows that our summaries lowest readability score was 3 . its performance is particularly good on 26th april where it is marked 4 ( good ) for all cases . conabs performs as well or better than other baseline techniques in most of the cases . [ table : summary_sample ] table [ table : summary_sample ] shows summaries generated by conabs and cowts ( both disaster - specific methodologies ) from the same set of messages ( i.e tweets form infrastructure class posted on 26th april ) . the two summaries are quite distinct . we find that summary returned by cowabs is more informative and diverse in nature compared to cowts . for instance , we can see the cowabs summary contains information about flights , damages of buildings , and information sources . in this approach , we have measured semantic lin similarity based on wordnet for nouns and verbs . however , we observe that in case of verbs this similarity metric does not perform well . as a result , some unrelated verbs may be clustered and some important information may be missed in final summary . in future , we try to use better semantic similarity measures to resolve this problem . + * time taken for summarization : * as stated earlier , one of our primary objectives is to generate the summaries in real time . hence , we analyze the execution times of the various techniques . for infrastructure , missing , and shelter classes , our proposed method conabs takes 25.947 , 17.915 , and 26.663 seconds on average ( over three days ) respectively , to generate the summaries . the time taken by conabs is comparable with other real time summarization methods like cowts , and towgs . however , apsal requires more time due to large nonnegative matrix factorization and computation of large similarity matrices . following a major crisis , a number of small - scale sub - events such as ` power outage ' , ` bridge closure ' etc . normal lda based topic detection techniques do not capture micro - level sub - events . moreover , according to un ocha , such lda topics are too general to act upon @xcite . in this section , we capture sub - events / topics from messages classified in a particular category ( e.g. infrastructure damage ) . we define a sub - topic as a combination of a noun and a verb where noun represents a concept and verb represents an event ( as described in the previous section ) . however , in this section we seek dependency relations between nouns and verbs , which is important to declare some information as an event / topic . table [ table : topic_phrase ] provides examples of some sub - topic phrases from various aidr classes . these sub - topics show important yet very specific events after the major earthquake crisis . for example , these include ` shut down of airports ' , ` resume of flight operation ' , emergency declared etc . most of the existing topic detection methodologies represent topics as a bag of words . in our case we try to capture the semantics between nouns and events using dependency relationships . for instance , in table [ table : topic_phrase ] , ` flight ' is related to ` shut ' , and ` road ' is related to ` crack ' . to the best of our knowledge , no prior work on processing tweet streams during disasters has attempted to combine nouns and events to generate such micro - level topic phrases . [ table : topic_phrase ] * assigning nouns to events : * in the sub - topic extraction methodology , we have found that it is often non - trivial to associate nouns to the context of an event in a tweet . for example , the words ` says ' and ` toppled ' in the sentence ` # china media says buildings toppled in # tibet http://t.co/o7vsywtgsk ' were identified as events @xcite . the noun ` building ' is related to the term ` toppled ' but it is not related to the verb ` says ' . hence , ( ` building',`toppled ' ) forms a valid topic phrase whereas ( ` building',`says ' ) is not a topic phrase . it is observed that sometimes such nouns may not always appear prior or adjacent to the events in a tweet . for example , in ` india sent 4 ton relief material , team of doctors to nepal ' , ( ` relief',`sent ' ) forms a valid topic phrase but the noun ` relief ' appears after the event ` sent ' . if a noun is directly associated / connected with an event ( edge exists between noun and event in dependency tree ) , we associate that noun with the event . we use pos tagger @xcite , event detector @xcite , and dependency parser @xcite for tweets to extract the association information . + * ranking topic phrases : * in this part , we rank the identified topic phrases . we only keep those topic phrases for which its constituent noun and event occur more than a certain threshold value in this case we set it as 10 . next , we compute szymkiewicz - simpson overlap score between noun ( n ) and event ( e ) as follows : @xmath13 where x indicates the set of tweets containing n and y indicates the set of tweets containing e. finally , we rank the topic phrases based on the similarity scores computed as per equation [ eqn : topic_rank_eqn ] . + * summarizing topic phrases : * after identifying topic phrases , we try to summarize the tweets corresponding to each of these topic phrases . basically , we search the words present in topic phrases and retrieve those tweets that match . finally , content words ( nouns , numerals , verbs ) based extractive summarization @xcite technique is applied over the retrieved tweets to generate a summary for each of the identified topics . table [ table : topic_summary ] provides examples of identified topic phrases and their summaries . [ table : topic_summary ] the micro - level topics and summaries can be useful for various stakeholders in a disaster scenario . for instance , _ communication cut _ can help government to plan , _ airline held , flight cancel _ can facilitate stranded foreigners to make proper departure planning while _ medicine send _ may enable the relief agency to connect supply to demand center . + * evaluating topic phrases : * to measure the accuracy of our proposed method for topic phrases identification , we check what fraction of nouns are correctly associated with the corresponding events . for this purpose , we compared the accuracy of our algorithm with a simple baseline algorithm in which nouns occurring within a window of 3 words on either side of the event were selected as being related to the event . averaging over all the different classes ( infrastructure , missing , shelter ) , the baseline algorithm obtains precision of 0.72 , whereas our method obtains precision of 0.95 . next , we evaluate the importance and utility of our identified topics . for this purpose , we performed user studies . for each day , we extract top ten topics based on our proposed methodology for each of the three classes . in a similar way , we identified ten topics using the lda based topic summarization approach proposed by arora et al @xcite . each topic is represented by two words having the highest probability of belonging to that topic . we use a crowdsource based evaluation methodology to judge the utility of our topic based summarization approach over the lda based technique . we asked five question to the workers on crowdflower as follows : * ( q1 ) relevance of the generated topics to the high - level category . ( on a scale : from 1 ( not related at all ) to 5 ( highly related ) ) ; * ( q2 ) which method provides more situational awareness ( m1 or m2 ) ; * ( q3 ) which method shows less redundant topics ( m1 or m2 ) ; * ( q4 ) which method generates more semantically meaningful topic ( m1 or m2 ) ; * ( q5 ) usefulness of topic keywords for situational awareness ( scale:1 - 5 ) . by showing the top ten topics from both methods , we asked 15 different workers to answer each question . with reference to the relevance to the high - level topics ( i.e. , q1 ) and usefulness of topic keywords ( i.e. , q5 ) , out of nine cases , our method performs better than the baseline in six cases and in rest of the three cases it is on par with the baseline . figure [ fig : topic_crowd_score ] shows detailed ranking of users for 25th april for q1 and q2 . for questions 2 , 3 , 4 , and 5 , our method performs better than the baseline in six cases which demonstrates utility of our proposed topic detection scheme during crisis scenario . a large number of tweets are posted during disaster events . for better situational awareness , a concise , categorical as well as multi - faceted representation of the tweets is necessary . we presented a novel framework to summarize information in crisis - related tweets in two different forms : ( a ) general situation update summary and ( b ) specific flash point activity reports thus producing pointed information about a place and/or an event . to present such a diverse yet coherent picture , a deep understanding of the tweets posted during such scenario is necessary - we believe the series of innovations that have been undertaken in this work has been an outcome of thorough analysis of such tweets . we also have performed extensive evaluation using experts to determine the useful of our approach . in future , we will deploy the system so that it can be of help for any future disaster event . k. rudra was supported by a fellowship from tata consultancy services .

the use of microblogging platforms such as twitter during crises has become widespread . more importantly , information disseminated by affected people contains useful information like reports of missing and found people , requests for urgent needs etc . for rapid crisis response , humanitarian organizations look for situational awareness information to understand and assess the severity of the crisis . in this paper , we present a novel framework ( i ) to generate abstractive summaries useful for situational awareness , and ( ii ) to capture sub - topics and present a short informative summary for each of these topics . a summary is generated using a two stage framework that first extracts a set of important tweets from the whole set of information through an integer - linear programming ( ilp ) based optimization technique and then follows a word graph and concept event based abstractive summarization technique to produce the final summary . high accuracies obtained for all the tasks show the effectiveness of the proposed framework . < ccs2012 > < concept > < concept_id>10002951.10003317.10003338.10003345</concept_id > < concept_desc > information systems information retrieval diversity</concept_desc > < concept_significance>500</concept_significance > < /concept > < concept > < concept_id>10002951.10003317.10003347.10003352</concept_id > < concept_desc > information systems information extraction</concept_desc > < concept_significance>500</concept_significance > < /concept > < concept > < concept_id>10002951.10003317.10003347.10003356</concept_id > < concept_desc > information systems clustering and classification</concept_desc > < concept_significance>500</concept_significance > < /concept > < concept > < concept_id>10002951.10003317.10003347.10003357</concept_id > < concept_desc > information systems summarization</concept_desc > < concept_significance>500</concept_significance > < /concept > < concept > < concept_id>10002951.10003317.10003371.10010852.10010853</concept_id > < concept_desc > information systems web and social media search</concept_desc > < concept_significance>300</concept_significance > < /concept > < /ccs2012 > * keywords : * disaster events ; twitter ; situational information ; classification ; summarization ; topic search .

let @xmath0 be a closed , connected , oriented three - manifold . the embedded contact homology ( ech ) and the heegaard floer homology of @xmath0 are invariants that have been studied and computed for many manifolds . ech was defined by hutchings using a contact form on @xmath0 , see @xcite , and heegaard floer homology was defined in @xcite by ozsvth - szab using a heegaard decomposition of @xmath0 . these two homology theories have very distinct flavors , but they have recently been shown to be isomorphic by colin - ghiggini - honda @xcite . more specifically , they construct an isomorphism @xmath1 . here @xmath2 is a version of heegaard floer homology of @xmath0 with the opposite orientation . one could instead consider the heegaard floer cohomology of @xmath0 with the right orientation . one can define a canonical absolute grading on ech and heegaard floer homology by homotopy classes of oriented 2-plane fields on @xmath0 . we now explain why this is a natural grading set to consider . first note that the orientation of @xmath0 induces a bijection between the homotopy classes of oriented 2-plane fields and the homotopy classes of nonvanishing vector fields . let @xmath3 denote the set of homotopy classes of nonvanishing vector fields on @xmath0 . one also defines @xmath4 to be the set of equivalence classes of vector fields on @xmath0 , where two vector fields are said to be equivalent if they are homotopic in the complement of an embedded three - dimensional ball . let @xmath5 denote the quotient map . we observe that @xmath6 acts on @xmath3 as follows . let @xmath7\in{\text{vect}}(y)$ ] and let @xmath8 be a small embedded ball in @xmath0 . fix a trivialization of @xmath9 . so we can see @xmath10 as a map @xmath11 . without loss of generality , we can assume that @xmath10 is constant in @xmath8 . let @xmath12 be the vector field given by the composition of the quotient map @xmath13 with the hopf map @xmath14 . one can check that the homotopy class @xmath15 $ ] does not depend on the choice of @xmath8 or @xmath10 . so @xmath7 + 1 $ ] is defined to be @xmath15 $ ] . by induction , one defines an action of @xmath6 on @xmath3 . it follows from the definition that @xmath16 is invariant under this action . moreover , one can check that for @xmath17 , any two elements in @xmath18 differ by @xmath19 , for some @xmath20 . in fact , the set @xmath18 is an affine space over @xmath21 , where @xmath22 is the divisibility of @xmath23 . one can prove all these facts using the pontryagin - thom construction , for example . now , since all versions of ech and heegaard floer homology split acording to spin@xmath24 structures and they are relatively graded on @xmath21 for the appropriate @xmath22 for each spin@xmath24 structure , the set @xmath3 is a possible choice for an absolute grading set . in fact , absolute gradings with values on @xmath3 were defined without any extra choices on ech and heegaard floer homology @xcite . for @xmath25 , let @xmath26 and @xmath27 denote the subgroups of @xmath2 and @xmath28 , respectively , consisting of all elements of grading @xmath29 . [ mainthm ] let @xmath1 be the isomorphism constructed by colin - ghiggini - honda . then @xmath30 maps @xmath26 to @xmath27 for all @xmath29 . we recall that both @xmath2 and @xmath28 admit a map @xmath31 whose mapping cone is denoted by @xmath32 and @xmath33 , respectively . the latter groups are simpler than the former and they are also isomorphic . in fact , in order to show that @xmath30 is an isomorphism , colin , ghiggini and honda first construct an isomorphism @xmath34 . they also show that the following diagram commutes . @xmath35^{\widehat{\phi } } \ar[r]^{\iota _ * } & hf(-y ) \ar[d]^{\phi}\\ \widehat{ech}(y ) \ar[r]^{\iota _ * } & ech(y ) } \label{eq : comm}\ ] ] here the horizontal maps @xmath36 denote the natural maps given by the mapping cone construction . in order to show that @xmath30 preserves the absolute grading , it is enough to prove that both maps @xmath36 and @xmath37 do . the map @xmath37 is defined as a composition @xmath38 as follows . @xmath39 here @xmath40 is the homology of a chain group computed from the page of an open book decomposition @xmath41 of @xmath0 and @xmath42 is the homology of a subcomplex of the ech chain complex @xmath43 generated by orbits sets that intersect @xmath44 exactly @xmath45 times , where @xmath46 is a certain contact form on @xmath47 . the maps @xmath48 , @xmath49 and @xmath50 are all isomorphisms and we will show that all of them preserve the grading . this paper is organized as follows . in section [ sec : hf ] , we review the definition heegaard floer homology , using lipshitz s cylindrical reformulation , and the absolute grading on it . we explain how to compute its `` hat''-version from a page of an open book decomposition , following @xcite . we also prove that @xmath48 preserves the grading . in section [ sec : ech ] , we recall the definition of ech and its absolute grading . we explain its `` hat''-version and the relationship between ech and open book decompositions . we show how to construct the map @xmath50 and we prove that it preserves the grading . in section [ sec : thm ] , we recall some of the steps to construct the isomorphism @xmath49 and we finish the proof of theorem [ mainthm ] . in this section , we will recall the definition of heegaard floer homology using the cylindrical reformulation from @xcite and the absolute grading on it from @xcite . a heegaard diagram is a triple @xmath51 , where @xmath52 is a closed oriented surface of genus @xmath53 and the tuples @xmath54 and @xmath55 are @xmath53-tuples of disjoint circles on @xmath52 which are linearly independent in @xmath56 . given such a heegaard diagram , one can construct a closed oriented three - manifold by considering @xmath57 $ ] , attaching disks to it along the circles @xmath58 and @xmath59 and filling the rest with two three - dimensional balls . any closed oriented three - manifold @xmath0 can be obtained this way from a heegaard diagram . a pointed heegaard diagram is a quadruple @xmath60 where @xmath51 is a heegaard diagram and @xmath61 is a point on @xmath52 in the complement of all of the circles @xmath62 and @xmath63 . given a pointed heegaard diagram @xmath60 , an intersection point is a @xmath53-tuple @xmath64 , where @xmath65 and @xmath66 is a permutation of @xmath67 . the chain complex @xmath68 is the @xmath6-module generated by the intersection points . the differential @xmath69 is defined as follows . consider the manifold @xmath70\times\sigma$ ] with the symplectic form @xmath71 , where @xmath72 and @xmath73 are the coordinates on the @xmath74 and @xmath75$]factors , respectively , and @xmath76 is an area form on @xmath52 . if @xmath64 and @xmath77 , one defines a moduli space @xmath78 , which is basically the moduli space of holomorphic curves in @xmath79 with boundary on the union of the cylinders @xmath80 and @xmath81 , and that converge to the union of chords @xmath75\times x_i$ ] as @xmath82 and to the union of @xmath75\times y_j$ ] as @xmath83 . for details , see @xcite . now @xmath84 is defined to be the signed count of certain embedded holomorphic curves in @xmath78 that do not intersect @xmath85\times\{z\}$ ] . it turns out that @xmath86 , so one defines @xmath87 to be the homology of this chain complex . we now recall the definition of the other versions of heegaard floer homology . the complex @xmath88 is defined to be the @xmath6-module generated by @xmath89 $ ] , where @xmath90 is an intersection point and @xmath20 . the differential is defined by @xmath91=\sum_{{\mathbf{y}},a } c({\mathbf{x}},{\mathbf{y}},a ) [ { \mathbf{y}},n - n_z(a)],\ ] ] where @xmath92 is a relative homology class in @xmath79 , the number @xmath93 is the signed count of certain holomorphic curves in @xmath94 in @xmath92 and @xmath95 is the intersection number of @xmath92 and @xmath85\times\{z\}$ ] . one can show that @xmath86 . one can now define @xmath96 to be the subgroup of @xmath88 generated by @xmath89 $ ] , for @xmath97 . this is a subcomplex because @xmath98 , whenever @xmath92 can be represented by a holomorphic curve . one also defines @xmath99 to be the quotient of @xmath88 by @xmath96 . the homologies of these various complexes are denoted , respectively , by @xmath100 the @xmath31 map is defined by @xmath101=[{\mathbf{x}},m-1]$ ] . so there is a short exact sequence @xmath102 this sequence induces an exact triangle . @xmath103^{u } & & hf^+(\sigma,{\bm{\alpha}},{\bm{\beta}},z ) \ar[dl ] \\ & \ar[lu ] \widehat{hf}(\sigma,{\bm{\alpha}},{\bm{\beta}},z ) & } \label{tri : hf}\end{aligned}\ ] ] we will now recall the absolute grading on all these homology groups . let @xmath104 be a pair consisting of a self - indexing morse function @xmath105 on @xmath0 and a gradient - like vector field @xmath106 , i.e. @xmath107 , whenever @xmath108 . we assume that @xmath105 has exactly one index 0 and one index 3 critical points . we also assume that all stable and unstable manifolds intersect transversely . for each index 1 critical point @xmath109 , let @xmath110 denote the unstable manifold containing @xmath109 and , for each index 2 critical point @xmath111 , let @xmath112 denote the stable manifold containing @xmath111 . the pair @xmath104 is said to be compatible with the heegaard diagram @xmath113 if * @xmath114 , * @xmath115 and @xmath116 , for all @xmath117 . an intersection point @xmath90 determines @xmath53 flow lines @xmath118 connecting the points @xmath109 to the points @xmath111 . the basepoint @xmath61 determines a flow line @xmath119 from the index 0 critical point to the index 3 critical point . outside the union of small neighborhoods of @xmath120 , the vector field @xmath106 does not vanish . the absolute grading @xmath121 is the homotopy class of an appropriate extension of this nonvanishing vector field to the union of these small neighborhoods , as we briefly explain . figure [ nbhd](a ) illustrates two transverse vertical sections of the vector field @xmath106 in a small neighborhood of @xmath122 , for some @xmath123 and figure [ nbhd](b ) illustrates a vertical section of @xmath106 in a small neighborhood of @xmath119 . now we substitute @xmath106 in these neighborhoods by the vector fields illustrated on figure [ defn ] . we note that in the neighborhood of @xmath119 , the vector field on figure [ defn](b ) has a circle of zeros . we modify the vector field in neighborhood of this circle so that it rotates clockwise on the @xmath124-plane . then we define @xmath121 to be the homotopy class of this vector field . for more details on this construction , see @xcite . nbhd.eps ( 10,-3)@xmath125-plane ( 40,-3)@xmath126-plane ( 27.5,-7)(a ) ( 86,-7)(b ) defn.eps ( 10,-3)@xmath125-plane ( 40,-3)@xmath126-plane ( 27.5,-7)(a ) ( 86,-7)(b ) now , for an intersection point @xmath90 and @xmath20 , we define @xmath127)={\text{\textnormal{gr}}}({\mathbf{x}})+2n$ ] . the following was the main theorem of @xcite . [ hf : absgrad ] for a pointed heegaard diagram @xmath60 , @xmath128 defines an absolute grading on any version of heegaard floer homology @xmath129 , satisfying the following properties : 1 . if @xmath90 and @xmath130 are in the same spin@xmath24 structure , then @xmath131 , where @xmath22 is the divisibity of the first chern class of the given the spin@xmath24 structure . 2 . if @xmath132 is a contact structure on @xmath0 and @xmath133 is the contact class in @xmath32 , then @xmath134 is the homotopy class of the reeb vector field @xmath135 for any contact form @xmath46 for @xmath132 . the grading @xmath128 is invariant under heegaard moves , so it is does not depend on the pointed heegaard diagram . the grading @xmath128 respects the cobordism maps . it follows from the definition that the map @xmath136 in the exact triangle has degree 0 . in this subsection , we will recall how to compute @xmath32 from the page of an open book decomposition and we will show how to adapt the absolute grading to it . an _ open book _ is a pair @xmath41 , where @xmath44 is a compact oriented surface with boundary and @xmath66 is a diffeomorphism of @xmath44 which is the identity on @xmath137 . we will always assume that @xmath137 is connected . given such a pair , one constructs a three - manifold as follows . let @xmath138 be the mapping cylinder of @xmath66 , i.e. the quotient of @xmath139 $ ] by the equivalence relation given by @xmath140 . we obtain a closed three - manifold by further quotienting @xmath138 by the equivalence relation given by @xmath141 for all @xmath142 and @xmath143 . for every closed oriented three - manifold @xmath0 , there exists an open book @xmath41 giving rise to @xmath0 by the above construction . so @xmath41 is called an open book decomposition of @xmath0 . let @xmath41 be an open book decomposition of @xmath0 . by ( * ? ? ? * lemma 2.1.1 ) , we can assume that there exists a diffeomorphism of a neighborhood of @xmath137 in @xmath44 to @xmath144\times \partial s$ ] such that the monodromy @xmath66 is given by @xmath145 in this neighborhood . then @xmath41 gives rise to a heegaard decomposition as follows . the heegaard surface is @xmath146 , where @xmath147 denotes @xmath148 . if we denote the genus of @xmath44 by @xmath53 , then @xmath52 has genus @xmath45 . we choose a set of properly embedded arcs @xmath149 of @xmath44 such that @xmath150 is a disk . one can then let @xmath151 , where @xmath152 is seen as an arc in @xmath153 and @xmath154 is its copy in @xmath155 . one also lets @xmath156 , where @xmath157 is the simplest arc in @xmath155 which is isotopic to @xmath154 and extends @xmath158 to a smooth curve in @xmath52 , see figure [ fighf ] ( ) . hence @xmath51 is a heegaard diagram for @xmath0 . so @xmath159 is a heegaard diagram for @xmath160 . for each @xmath161 , the circle @xmath154 intersects @xmath157 in @xmath155 at three points . we label them @xmath162 , as in figure [ fighf ] ( ) . we fix a basepoint @xmath163 . one defines @xmath164 to be the subcomplex of @xmath165 generated by @xmath45-tuples of intersection points contained in @xmath153 . one also defines @xmath166 to be the quotient @xmath167 , where two @xmath45-tuples of intersection points in @xmath153 are equivalent if they differ by substituting @xmath168 by @xmath169 . there is an induced differential on @xmath166 . it is shown in ( * ? ? ? * theorem 4.9.4 ) that the homology of this chain complex is isomorphic to @xmath170 . the absolute grading on the complex @xmath166 is well - defined as the following simple lemma shows . if @xmath90 is a @xmath45-tuple of intersection points in @xmath153 containing @xmath168 and @xmath171 , then @xmath172 . therefore @xmath128 is well - defined on the quotient @xmath166 . we observe that @xmath90 and @xmath173 are in the same spin@xmath24 structure and that @xmath174 . therefore , by theorem [ hf : absgrad](a ) , @xmath175 moreover , by definition , the map @xmath176 preserves the absolute grading . therefore the isomorphism @xmath177 preserves the absolute grading . [ hfobd1 ] [ hfobd2 ] we observe that @xmath0 can also be decomposed as @xmath178 , where @xmath179 is glued to @xmath180 . this decomposition can also be obtained by considering the open book @xmath181 , where @xmath182 is an annulus and @xmath183 is an extension of @xmath66 to @xmath184 which is the identity in a small neighborhood of @xmath185 . let @xmath46 be a contact form on @xmath0 which is _ adapted _ to @xmath41 , i.e. the reeb vector field @xmath135 is a positively transverse to @xmath186 for all @xmath73 and positively tangent to the binding @xmath187 . for each generator @xmath188 , we will now construct a vector field @xmath189 in the homotopy class of @xmath121 which differs from @xmath135 only in a small set . the inclusion @xmath190 can be extended to a diffeomorphism @xmath191 , as in figure [ fighf ] ( ) . we will use this description of @xmath52 for the rest of this section . let @xmath192 and @xmath193 be the sets of @xmath194 and @xmath195 curves under this diffeomorphism . note that @xmath196 and that @xmath197 . we can also assume that @xmath198 and that @xmath199 and @xmath157 coincide outside a small neighborhood of @xmath200 , see figure [ fighf ] ( ) . we isotope @xmath61 from @xmath201 to @xmath202 in the complement of the @xmath194 and @xmath195 circles . we now choose a morse function @xmath105 and a gradient - like vector field @xmath106 for @xmath105 such that @xmath104 is compatible with @xmath159 and such that the index two and one critical points are contained in @xmath203 and @xmath204 , respectively . fix @xmath205 and let @xmath206\cup[1-{\varepsilon},1])$ ] . since @xmath106 and @xmath135 are positively transverse to @xmath44 on @xmath207 , we can assume that @xmath208 in @xmath207 . for each @xmath209 , let @xmath210 be a small neighborhood of @xmath211 in @xmath212 containing @xmath213 and let @xmath214 be a small thickening of it . we can see @xmath215 and @xmath216 as subsets of @xmath184 . we can assume that @xmath104 is chosen so that @xmath217 $ ] contains @xmath218 , where @xmath110 and @xmath219 denote the unstable and stable manifolds of the index one and two critical points corresponding to @xmath220 and @xmath62 , respectively . for @xmath221 , we define @xmath222\big)\setminus m_0 \subset y. \label{def : m}\ ] ] let @xmath223 be a small neighborhood of the flow line passing through @xmath61 and let @xmath224 be the vector field obtained by modifying @xmath106 in @xmath223 as in figure [ defn](b ) . we note that @xmath224 does not vanish in @xmath225 . for a generator @xmath188 , its grading @xmath121 is the homotopy class of a vector field obtained by modifying @xmath224 in @xmath226 . we will now show how to homotope @xmath224 in the complement @xmath226 to obtain a vector field which mostly coincides with @xmath135 . [ lem : iso ] the vector field @xmath224 is homotopic to a nonvanishing vector field @xmath227 in @xmath228 relative to its boundary such that @xmath227 coincides with @xmath135 in @xmath229 . the proof of this lemma is essentially the last paragraph of the proof of ( * ? ? ? * theorem 1.1(b ) ) . we will rewrite it here for the reader s sake . let @xmath230 . it follows from the definition of the arcs @xmath231 that @xmath232 is topologically a disk with @xmath233 boundary punctures . so , @xmath8 is diffeomorphic to a three - ball with @xmath233 interior punctures . we can choose this diffeomorphism so that @xmath234 is contained on the @xmath124-plane in @xmath235 , as shown in figure [ hfmod ] ( ) . under this diffeomorphism , @xmath236 is a union of very small punctured balls centered at the @xmath233 punctures from above . the vector field @xmath224 is depicted in figure [ hfmod ] ( ) . a horizontal section of @xmath106 can be seen in figure [ hfmod ] ( ) , where the green punctured disks represent @xmath236 . we can isotope @xmath224 so the closed orbit contained on this horizontal section coincides with the binding @xmath237 outside @xmath238 , see figure [ hfmod ] ( ) . the vector field obtained by this isotopy can be chosen to coincide with @xmath135 in @xmath239 . so we let @xmath227 be this vector field . [ hf1 ] [ hf2 ] [ hf3 ] [ lj ] [ hf4 ] now we choose disjoint neighborhoods of the flow lines corresponding to each @xmath240 which are contained in @xmath226 . we define @xmath189 to be the vector field obtained by modifying @xmath227 in these neighborhoods as in [ sec : absgrad ] . it follows from lemma [ lem : iso ] that @xmath241={\text{\textnormal{gr}}}({\mathbf{x}})\in{\text{vect}}(y)$ ] . in this section , we will recall the definition of the ech chain complex and its absolute grading . for more details , see @xcite . let @xmath0 be a closed , oriented three - manifold , let @xmath46 be a contact form on @xmath0 and let @xmath242 . the ech chain complex @xmath43 is generated by finite orbit sets @xmath243 , where @xmath122 are distinct embedded reeb orbits , @xmath244 are positive integers , and @xmath245 whenever @xmath122 is hyperbolic . the chain complex @xmath43 splits by homology classes as @xmath246 where @xmath247 is the subcomplex generated by the orbit sets @xmath248 whose total homology class @xmath249=\gamma\in h_1(y)$ ] . now consider two orbit sets @xmath248 and @xmath250 whose total homology classes equal @xmath251 and let @xmath252 be a 2-chain in @xmath253 $ ] such that @xmath254 let @xmath255 denote a trivialization of @xmath132 over all of the reeb orbits @xmath122 and @xmath256 and let @xmath257 be the relative first chern class of @xmath258 with respect to @xmath255 . let @xmath259 be the relative self - intersection number of @xmath252 with respect to @xmath255 , as explained in @xcite , and let @xmath260 where @xmath261 denotes the conley - zehnder index of the reeb orbit @xmath262 with respect to the trivialization @xmath255 . in @xcite , hutchings defined the _ ech index _ to be @xmath263 one can show that @xmath264 does not depend on @xmath255 and only depends on the relative homology class of @xmath252 . if we change the relative homology class of @xmath252 , the ech index @xmath265 changes by a multiple of @xmath22 , where @xmath22 is the divisibility of @xmath266 . so , @xmath267 is well defined in @xmath21 . hence @xmath247 is relatively graded by @xmath21 . the differential @xmath268 is defined as follows . let @xmath269 be the symplectization of @xmath0 , where @xmath72 denotes the @xmath74-coordinate , and let @xmath270 be a cylindrical almost - complex structure on @xmath271 . the coefficient of @xmath272 in @xmath268 is the signed count of @xmath270-holomorphic curves @xmath273 in @xmath271 that have ech index @xmath274 , and that converge to @xmath275 as @xmath82 and @xmath272 as @xmath83 , modulo translation . it is shown in @xcite that for a generic @xmath270 the differential @xmath69 is well - defined and @xmath86 . so one defines @xmath276 to be the homology of @xmath247 for a given @xmath270 . it follows from @xcite that @xmath276 does not depend on the choice of @xmath270 , so we will omit @xmath270 from the notation . we will also denote by @xmath277 the homology of @xmath43 . it follows from the definition of @xmath69 that @xmath278 moreover , by taubes @xcite , for two contact forms @xmath279 , there exists an isomorphism @xmath280 . therefore one can write @xmath28 . we note that there are additional structures on @xmath277 that do depend on the contact form , e.g. ech capacities . we now recall the definition of the absolute grading on ech from @xcite . this grading takes values on the set of homotopy classes of oriented 2-plane fields on @xmath0 . the orientation of @xmath0 induces an isomorphism from this set to the set of homotopy classes of nonvanishing vector fields @xmath3 . let @xmath248 be an orbit set . the absolute grading @xmath281 is the homotopy class of the vector field obtained by modifying the reeb vector field in disjoint neighborhoods of the reeb orbits @xmath122 , as follows . for each @xmath161 , fix a small tubular neighborhood of @xmath122 and choose a braid @xmath282 with @xmath244 strands in that neighborhood . let @xmath283 be the union of the braids @xmath282 . a trivialization @xmath284 of @xmath132 over each @xmath122 , induces a framing @xmath284 on each @xmath282 . let @xmath255 denote this framing on @xmath283 . now , for each component @xmath285 of @xmath283 , let @xmath286 denote a small neighborhood of @xmath285 in @xmath0 . we can choose these neighborhoods so that @xmath286 and @xmath287 do not intersect for different components @xmath285 and @xmath288 . the framing on @xmath285 induces a diffeomorphism @xmath289 and a trivialization of @xmath290 , identifying @xmath291 and @xmath292 . using the previous identifications , one can define a vector field @xmath293 on @xmath286 as @xmath294 one now constructs a vector field on @xmath0 by defining it to be given by @xmath293 in each neighborhood @xmath286 and to equal the reeb vector field in the complement of the union of the neighborhoods @xmath286 . let @xmath295 be the homotopy class of this vector field . now define @xmath296 here @xmath297 denotes the writhe of @xmath282 with respect to @xmath284 and @xmath298 one can check that @xmath281 does not depend on the choice of @xmath255 or @xmath283 . in @xcite , hutchings proved the following proposition . let @xmath275 and @xmath272 be orbit sets with @xmath299=[\sigma]\in h_1(y)$ ] and let @xmath22 denote the divisibity of @xmath300)\in h^2(y;{{\mathbb z}})$ ] . then @xmath301 for @xmath29 , we denote by @xmath302 the submodule of @xmath277 consisting of the elements of grading @xmath262 . it follows from @xcite that for two contact forms @xmath279 , the isomorphism @xmath280 restricts to an isomorphism @xmath303 for every @xmath29 . so we can write @xmath27 for @xmath29 . the @xmath31 map is defined similarly to the differential . for an orbit set @xmath275 , one defines @xmath304 to be the signed count of @xmath270-holomorphic curves @xmath273 in @xmath271 with @xmath305 that go through a fixed basepoint in @xmath271 . the @xmath31 map is a degree @xmath306 chain map @xmath307 . the chain complex @xmath308 is defined to be the mapping cone of @xmath31 . the homology of @xmath308 is denoted by @xmath309 . again , it follows from @xcite that the @xmath31 map in homology does not depend on @xmath270 or @xmath46 so one can write @xmath33 . we obtain an exact triangle , as follows . @xmath310^{u } & & ech(y ) \ar[dl ] \\ \label{tri : ech } & \ar[lu ] \widehat{ech}(y ) & } \end{aligned}\ ] ] we define the absolute grading on @xmath308 so that the map @xmath311 has degree 0 . hence for @xmath29 , we can write @xmath312 . we note that the map @xmath313 has degree 1 . in this subsection , we will show that the cobordisms maps in ech defined by hutchings and taubes in @xcite preserve the absolute grading . this fact will be used in the next subsection . let @xmath46 be a contact form on @xmath0 . the symplectic action of an orbit set @xmath314 is defined to be @xmath315 . for @xmath316 , the filtered ech chain complex @xmath317 is defined to be the subcomplex of @xmath43 generated by all orbit sets @xmath275 with @xmath318 . since the differential decreases the action , the subgroup @xmath317 is indeed a subcomplex . its homology is denoted by @xmath319 and it is independent of the almost - complex structure by ( * ? ? ? * theorem 1.3(a ) ) . for @xmath320 , let @xmath321 be a 3-manifold with contact form @xmath322 . an exact symplectic cobordism from @xmath323 to @xmath324 is a pair @xmath325 , where @xmath79 is a compact 4-manifold , @xmath326 is a symplectic form , @xmath327 and @xmath328 for @xmath320 . according to ( * ? ? ? * theorem 1.9 ) , such corbordisms induce maps @xmath329 the maps @xmath330 are constructed by taking the composition of the corresponding map in seiberg - witten floer homology and the isomorphism from ech to seiberg - witten floer homology . [ lem : cob ] let @xmath331,d\lambda)$ ] be an exact cobordism from @xmath332 to @xmath333 . then , for every @xmath316 , the map @xmath334,\lambda)$ ] preserves the absolute grading , i.e. @xmath334,\lambda)$ ] maps @xmath335 to @xmath336 for every @xmath29 . the maps @xmath334,\lambda)$ ] are defined as a composition of maps @xmath337 here @xmath338 and @xmath339 are appropriate filtered seiberg - witten floer cohomology groups , as explained in @xcite . the second map in is a filtered version of the cobordism maps defined in @xcite . now it follows from the definition of these maps that if an element of @xmath338 has grading @xmath340\in{\text{vect}}(y)$ ] , then its image in @xmath341 is the sum of elements of ( possibly different ) gradings @xmath342 $ ] such that for each such @xmath343 there exists an almost - complex structure @xmath270 on @xmath253 $ ] satisfying @xmath344 now by considering @xmath345 , we conclude that @xmath346 and @xmath347 are homotopic . hence @xmath348 . so the second map in preserves the absolute grading . now , the first and third maps in preserve the grading by @xcite . therefore @xmath334,\lambda)$ ] preserves the grading . we now recall from @xcite how to compute @xmath349 and @xmath277 from an open book decomposition . we fix an open book decomposition @xmath41 of @xmath0 and write @xmath350 , as in [ sec : hfobd ] . let @xmath351 and let @xmath73 denote the @xmath75$]-coordinate in @xmath47 as in [ sec : hfobd ] . we again assume that @xmath66 satisfies @xmath145 in a neighborhood of @xmath137 . let @xmath46 be a contact form on @xmath47 such that @xmath135 is parallel to @xmath352 in @xmath47 . hence the torus @xmath353 is foliated by reeb orbits . up to a small isotopy of @xmath66 , we can assume that all the reeb orbits in the interior of @xmath47 are nondegenerate and that @xmath353 is a negative morse - bott torus . after a morse - bott perturbation , we obtain a pair of reeb orbits @xmath354 on @xmath353 . let @xmath355 be the chain complex generated by orbit sets contructed from reeb orbits in the interior of @xmath47 and @xmath356 and let @xmath357 be the chain complex generated by orbit sets contructed from reeb orbits in the interior of @xmath47 and @xmath354 . the differential in both cases counts morse - bott buildings of ech index 1 . let @xmath358 and @xmath359 denote the homology of these chain complexes . now let @xmath360 and @xmath361 denote the quotients of @xmath358 and @xmath359 by the equivalence relations generated by @xmath299\simeq [ e\gamma]$ ] , respectively . in @xcite , colin , ghiggini and honda proved that @xmath360 and @xmath361 are well - defined and they constructed isomorphisms @xmath362 we will now show that @xmath360 and @xmath361 have well - defined gradings and that and preserve the gradings . we start by recalling the construction of the isomorphism @xmath363 . we write @xmath364)$ ] where @xmath106 is a tubular neighborhood of the binding @xmath365 , which is again a solid torus . let @xmath366 be a contact form on @xmath106 which is nondegenerate in the interior of @xmath106 such that the reeb vector field of @xmath366 is positively transverse to the interior of the pages and positively tangent to the binding and such that @xmath367 is a _ positive _ morse - bott torus . the precise construction of @xmath366 will not be necessary here and we refer the reader to @xcite . we denote by @xmath368 and @xmath369 the elliptic and hyperbolic orbits obtained after a morse - bott pertubation . let @xmath370 be an increasing sequence such that @xmath371 . following @xcite , we can choose a family of contact forms @xmath372 on @xmath0 which equal @xmath46 in a neighborhood of @xmath47 and a positive multiple of @xmath366 in a neighborhood of @xmath106 such that @xmath373 is a morse - bott contact form and all reeb orbits in @xmath374 $ ] have action larger than @xmath375 . so as in @xcite , we can perturb @xmath372 to a sequence of contact forms @xmath376 satisfying , in particular , the following conditions : * @xmath377 coincides with @xmath373 outside neighborhoods of the morse - bott tori . * the reeb orbits of @xmath373 of action less than @xmath375 are nondegenerate and are either the reeb orbits of @xmath46 and @xmath366 in the interior of @xmath47 and @xmath106 , respectively , or one of the orbits @xmath378 , @xmath379 , @xmath368 or @xmath369 . hence @xmath380 is generated by elements of the form @xmath381 , where @xmath382 is an orbit set contructed from reeb orbits in the interior of @xmath106 and @xmath383 , and @xmath384 is a generator of @xmath357 . for @xmath316 , let @xmath385 be the subcomplex of @xmath386 generated by orbit sets @xmath275 with action @xmath387 and whose total homology class intersects a page up to @xmath388 times . following @xcite , we can define another increasing sequence @xmath389 with @xmath390 such that the maps @xmath391 below are well - defined . @xmath392 here @xmath393 is defined by the equation @xmath394 , where @xmath395 and @xmath396 are the differentials in @xmath357 and @xmath355 , respectively . it follows from ( * ? ? ? * lemma 9.7.2 ) that the maps @xmath391 are chain maps so they induce maps @xmath397 . following ( * ? ? ? 3.2.3 ) , there are chain maps @xmath398 which are given by cobordism maps as in [ sec : cob ] . so we obtain a directed system @xmath399^{\sigma_k}\ar[d]^{\iota_k } & ecc^{l_k}(y,\lambda_k')\ar[d]^{\phi_k}\\ ecc^{\flat , l_{k+1}'}_{\le k+1}(n,\lambda)\ar[r]^{\sigma_k } & ecc^{l_{k+1}}(y,\lambda_{k+1 } ' ) } \label{eq : iota}\ ] ] where @xmath400 denotes the inclusion . the maps @xmath401 induce maps in homology with respect to which one can take the direct limit @xmath402 . there is also a nondegenerate contact form @xmath403 and cobordism maps @xmath404 . it is shown in ( * ? ? ? 3.2.3 ) that the direct limit of these maps is an isomorphism @xmath405 now we note that @xmath406 . therefore the maps @xmath391 give rise to a map @xmath407 the calculations in @xcite imply that the image of the map @xmath408 given by @xmath299\mapsto[\gamma]-[e\gamma]$ ] is contained in the kernel of @xmath409 . hence we obtain a map @xmath410 it is shown in ( * ? ? ? * theorem 9.8.3 ) that @xmath363 is an isomorphism . we will now prove a lemma that will be useful to show that the absolute grading is well - defined in @xmath360 and that @xmath363 preserves the grading . [ lem : greh ] let @xmath275 be an orbit set obtained from the reeb orbits of @xmath46 in the interior of @xmath47 , respectively , and the orbits @xmath378 , @xmath379 , @xmath368 or @xmath369 . then @xmath411 is well - defined . moreover , @xmath412 to see that @xmath275 has a well - defined grading , first note that there exists @xmath413 such that @xmath414 for every @xmath415 . so we define @xmath281 using the contact form @xmath377 for some @xmath415 . it follows from lemma [ lem : cob ] that the maps @xmath401 preserve the grading . so @xmath416 is well - defined . to prove , we can restrict to the case when @xmath275 does not contain @xmath378 , @xmath379 , @xmath368 or @xmath369 . the general case is a straightforward consequence of this case . let @xmath255 be a trivialization of @xmath132 over @xmath275 and let @xmath283 be a link as in [ abs : ech ] so that @xmath417 , where @xmath418 denotes the sum of the writhes of all components of @xmath283 . let @xmath419 . we can choose a trivialization @xmath420 of @xmath421 that is trivial with respect to the morse - bott torus containing @xmath422 . let @xmath423 be a knot obtained by pushing @xmath422 in a direction which is transverse to the morse - bott torus containing @xmath422 such that @xmath423 is in the interior of @xmath424 . then @xmath425 . now let @xmath426 a the disk in @xmath424 bounding @xmath422 . it follows from ( * ? ? ? * lemma 3.4(d ) ) that @xmath427 moreover , @xmath428 therefore it follows from that holds . the module @xmath360 has a well - defined absolute grading and the isomorphism @xmath429 preserves the grading . it follows from lemma [ lem : greh ] that the grading on @xmath358 is well - defined . we recall that @xmath360 is the quotient of @xmath358 by the equivalence relation given by @xmath299\sim[e\gamma]$ ] . by lemma [ lem : greh ] that @xmath430 . so the grading on @xmath360 is well - defined . let @xmath275 be an orbit set in @xmath431 for some @xmath388 . since @xmath395 decreases the grading by @xmath432 , it follows that @xmath433 . now , by lemma [ lem : greh ] , @xmath434 . hence for all @xmath435 , @xmath436 so @xmath391 preserves the grading . now , it is tautological that the inclusion @xmath400 in preserves the grading . moreover , by lemma [ lem : cob ] , the maps @xmath401 and the isomorphism preserve the grading . hence after passing to homology and taking the direct limit we conclude that @xmath409 , and hence @xmath363 , preserve the grading . it also follows from lemma [ lem : greh ] that the gradings on @xmath359 and @xmath361 are well - defined . we now define two chain maps as follows . @xmath437 here @xmath438 and @xmath439 do not contain @xmath379 . these maps descend to homology and to the quotients @xmath361 and @xmath360 . it follows from @xcite that these maps fit into an exact triangle @xmath440 & & ech(n,\partial n,\lambda ) \ar[dl]^{\iota _ * } \\ \label{tri : echrel } & \ar[lu]^{\pi _ * } \widehat{ech}(n,\partial n,\lambda ) & } \end{aligned}\ ] ] where the map @xmath441 is a version of the @xmath31 map . moreover there exists an isomorphism @xmath442 such that @xmath363 and @xmath443 give an isomorphism from to . it follows from that @xmath36 increases the grading by @xmath432 and that @xmath444 preserves the grading . hence we obtain the following long exact sequences . @xmath445 & ech_{\rho-1}(n,\partial n)\ar[r]\ar[d]^{\psi_1 } & \widehat{ech}_\rho(n,\partial n ) \ar[r]\ar[d]^{\widehat{\psi}_1 } & ech_{\rho}(n,\partial n)\ar[r]\ar[d]^{\psi_1}&\dots \\ \dots\ar[r]^-u & ech_{\rho-1}(y)\ar[r ] & \widehat{ech}_\rho(y ) \ar[r ] & ech_{\rho}(y)\ar[r]^-u&\dots } \label{eq : commdiag}\ ] ] here @xmath446 and we dropped the dependence on @xmath46 . in fact , because of the isomorphisms @xmath363 and @xmath443 , the modules on the upper row of do not depend on @xmath46 . it follows from that @xmath443 preserves the grading . we end this section with the a discussion of @xmath447 and the grading on it . the chain complex @xmath447 is a subcomplex of @xmath357 generated by orbits sets which intersect @xmath44 exactly @xmath45 times where @xmath53 is the genus of @xmath44 . this inclusion induces an absolute grading on @xmath447 and a map in homology . by composing this map with the quotient map @xmath448 we obtain a map @xmath449 which preserves the grading . it is shown in @xcite that @xmath450 is an isomorphism . let @xmath451 . therefore @xmath452 is an isomorphism and it preserves the grading . in this section , we will prove that the absolute grading is preserved under the isomorphism from heegaard floer homology to ech defined by colin - ghiggini - honda in @xcite . we now recall the construction of the map @xmath49 on the chain level @xmath454 this map is defined by counting rigid holomorphic curves with an ech - type index equal to 0 . we now review the relevant moduli spaces and this ech - type index . throughout this section we fix an open book @xmath41 for @xmath0 satisfying the conditions given in [ sec : hfobd ] and we let @xmath455 be the mapping torus of @xmath66 . we denote by @xmath53 the genus of @xmath44 and we let @xmath46 be a contact form on @xmath0 which is adapted to @xmath41 . in order to prove that @xmath49 is an isomorphism , it is necessary to make a more specific choice of @xmath46 as it is done in @xcite , but this particular choice does not affect the absolute grading . let @xmath456 be the map @xmath457 and let @xmath458 , where @xmath459 . we also round the corners of @xmath8 . now define @xmath460 and @xmath461 , where @xmath76 is a certain area form on @xmath44 . then @xmath462 is a symplectic manifold with boundary . it has a positive end , which is diffeomorphic to @xmath463 $ ] and a negative end , which is diffeomorphic to @xmath47 . the map @xmath16 restricts to a symplectic fibration @xmath464 which admits a symplectic connection whose horizontal space is spanned by @xmath465 . now if we take a copy of @xmath466 on the fiber @xmath467 and take its symplectic parallel transport along @xmath468 , we obtain a lagrangian submanifold of @xmath462 , which is denoted by @xmath469 . for each @xmath470 we denote by @xmath471 the corresponding component of @xmath469 . we will consider @xmath270-holomorphic maps @xmath472 where @xmath473 is a riemann surface with boundary and punctures , both in the interior and on the boundary . a puncture @xmath474 is said to be positive or negative if the @xmath72-coordinate of @xmath475 converges to @xmath476 or @xmath477 , respectively , as @xmath478 . now to each generator @xmath90 of @xmath166 we can associate a subset of @xmath463 $ ] given by the union of @xmath479 $ ] , for all @xmath480 . we will still denote the union of these chords by @xmath90 . given @xmath90 , an orbit set @xmath248 in @xmath447 and an admissible almost - complex structure @xmath270 , one defines @xmath481 to be the moduli space of @xmath270-holomorphic maps @xmath472 satisfying the following conditions : 1 . @xmath482 and each component of @xmath483 is mapped to a different @xmath471 . the boundary punctures are positive and the interior punctures are negative . 3 . at each boundary puncture , @xmath484 converges to a different chord @xmath479 $ ] and every chord @xmath479 $ ] is such an end of @xmath484 . 4 . at an interior puncture , @xmath484 converges to an orbit @xmath122 with some multiplicity . for each @xmath161 , the total multiplicity of all ends converging to @xmath122 is @xmath244 . the energy of @xmath484 is bounded . let @xmath485 denote the compactification of @xmath486 obtained by compactifying @xmath74 to @xmath487 . a continuous map @xmath488 satifying ( a)(d ) above can be compatified to a map @xmath489 mapping @xmath490 to @xmath491 . two such maps @xmath484 , @xmath10 are said to be homologous if the images of @xmath492 and @xmath493 are homologous in @xmath494 . let @xmath495 denote the set of homology classes of such maps @xmath488 . for a homology class @xmath496 , one defines its ech - index @xmath497 as follows . let @xmath488 be a continuous map satifying ( a)(d ) above such that @xmath498=a$ ] and let @xmath499 be its compactification . now note that one can view @xmath500 as a sub - bundle of @xmath501 . we choose an orientation of the arcs @xmath152 , which gives rise to a nonvanishing vector field along each @xmath152 . this vector field induces a trivialization @xmath255 of @xmath500 along @xmath502 . we extend this trivialization arbitrarily along @xmath503 $ ] and along @xmath504 . let @xmath505 denote the first chern class of @xmath506 relative to @xmath255 . now let @xmath507 and @xmath508 be distinct embedded surfaces in @xmath485 given by pushing @xmath509 off along vectors field which are transverse to it and trivial with respect to @xmath255 along the boundary . for more details see @xcite . then @xmath510 is defined to be the signed count of intersections of @xmath507 and @xmath508 . now let @xmath511 be a real , rank one subbundle of @xmath500 along @xmath512 $ ] defined as follows . at @xmath513 , let @xmath514 and at @xmath515 , let @xmath516 in @xmath500 . then @xmath511 is defined by rotating counterclockwise by the minimum possible amount as we travel along @xmath512 $ ] . one defines @xmath517 to be the sum of the maslov indices of @xmath511 along each @xmath479 $ ] with respect to @xmath255 . the ech - index is defined as @xmath518 now @xmath519 is defined as follows . the coefficient of an orbit set @xmath275 in @xmath519 is the ( signed ) count of maps @xmath484 in @xmath481 with @xmath520)=0 $ ] . as explained in @xcite , for a generic @xmath270 this count is finite and all the maps that are counted are embeddings . showing that the map @xmath49 induces an isomorphism on homology is far from trivial and it is done in @xcite . we remark that @xcite , they only construct @xmath49 over @xmath521 coefficients . our construction and , in particular proposition [ prop : indiso ] , do not depend on an eventual choice of signs . we first recall a relative version of the pontryagin - thom construction . let @xmath10 and @xmath523 be nonvanishing vector fields on a closed and oriented three - manifold @xmath0 . assume that @xmath10 and @xmath523 coincide in @xmath524 , where @xmath31 is an open set in @xmath0 . let @xmath255 be a trivialization of @xmath525 and let @xmath526 be a regular value of both @xmath10 and @xmath523 seen as maps @xmath527 . the one - manifolds @xmath528 and @xmath529 inherit framings by considering the isomorphisms of the normal bundles with @xmath530 given by @xmath531 and @xmath532 along @xmath533 and @xmath534 , respectively . now if @xmath533 and @xmath534 are contained in the interior of @xmath31 and are homologous in @xmath31 , there is a link cobordism @xmath535 $ ] from @xmath533 to @xmath534 . that is , @xmath273 is a surface such that @xmath536 . the framings on @xmath533 and @xmath534 induce a framing on @xmath273 along @xmath537 which we denote by @xmath538 . [ lem : pt ] let @xmath10 and @xmath523 be nonvanishing vector fields and @xmath533 and @xmath534 the links as above . let @xmath273 be an immersed cobordism from @xmath533 to @xmath534 and let @xmath539 denote the number of self - intersections of @xmath273 . let @xmath538 denote the framing on @xmath273 along @xmath537 which is induced by the framings on @xmath533 and @xmath534 . then @xmath540-[w]=c_1(nc,\nu)+2\delta(c).\ ] ] first assume that @xmath273 is an embedded surface . we can find a framing @xmath541 of @xmath273 which coincides with @xmath538 along @xmath542 . it follows from the pontryagin - thom contruction that @xmath7-[w]$ ] equals the difference of the framings @xmath541 and @xmath538 along @xmath543 . but this difference is given by @xmath544 . now the general case follows from ( * ? ? ? * lemma 2.3 ) . we write @xmath248 . let @xmath488 be an immersion such that @xmath498=a$ ] and let @xmath545 denote its continuous compactification . we note that by rounding the corners of @xmath485 , we obtain a trivial cobordism from @xmath47 to itself which we denote by @xmath546 . here we identify @xmath547 $ ] . let @xmath283 be the union of disjoint braids @xmath282 around @xmath122 with total multiplicity @xmath244 . then by moving @xmath548 in the direction of some transverse vector field near @xmath549 , we obtain an immersed cobordism @xmath273 from @xmath550 to @xmath551 , where @xmath173 is the union of @xmath90 with segments on the arcs @xmath152 . up to an isotopy , we can assume that @xmath550 is transverse to @xmath552 . we also note that , under our identification , @xmath553)\times\{1\}$ ] . let @xmath554 be the subset defined by . let @xmath189 be the vector field defined in [ sec : hfobd ] and let @xmath555 be the vector field defined in [ sec : absgrad ] whose homotopy class is @xmath556 . let @xmath557 be a small open neighborhood of @xmath558 . then , by lemma [ lem : iso ] , we can assume that @xmath189 and @xmath555 coincide in @xmath559 . we will now use lemma [ lem : pt ] . we first choose a nonvanishing tangent vector field along the arcs @xmath152 . that induces a trivialization of @xmath560 . since @xmath561 , we obtain a trivialization of @xmath562 . we extend it arbitrarily to @xmath563 . we can now extend this trivialization to a trivialization @xmath255 of @xmath242 on @xmath564 . so @xmath565 is a trivialization of @xmath566 where the reeb vector field @xmath135 in @xmath557 is mapped to @xmath567 . we observe that @xmath568 . the framing can be calculated by considering the preimage of a vector near @xmath569 . the corresponding link gives a framing of the normal bundle @xmath570 which coincides with @xmath255 . now we compute @xmath571 . first note that this link is contained in @xmath238 . we observe that @xmath571 is a link @xmath572 which is a slight pertubation of the union of the flow lines corresponding to the points @xmath480 and the points @xmath573 from figure [ fighf ] ( ) . the framing on @xmath572 is a trivialization of @xmath574 which we denote by @xmath538 . the link @xmath572 is isotopic to @xmath173 so we can assume that @xmath575 . so we can see @xmath538 as a trivialization of @xmath576 along @xmath577 . as in lemma [ lem : pt ] , the framing on @xmath551 is also denoted by @xmath538 . by lemma [ lem : pt ] , @xmath578 the trivialization @xmath255 also gives rise to a trivialization of @xmath576 along @xmath577 and @xmath551 . we claim that @xmath579 to prove the claim , we first compute the difference @xmath580 . this difference is given by @xmath581 as framings of @xmath582 , since @xmath583 along @xmath551 . we will now identify @xmath584 , where @xmath585 is the union of the flow lines corresponding to @xmath90 and @xmath586 is the union of the flow lines corresponding to all @xmath587 . let @xmath588 be a vector field along @xmath572 which is trivial with respect to @xmath255 and which coincides with the vector field tangent to each @xmath152 as chosen above along @xmath586 . it follows from the isotopy @xmath589 and from the definition of @xmath255 that at the index two or one critical points , @xmath588 is tangent to the stable or unstable submanifolds , respectively . moreover , the vector field @xmath588 rotates a quarter of a turn counterclockwise about each component of @xmath590 as we go from an index two to an index one critical point . so along each component of @xmath586 , the trivializations @xmath538 and @xmath255 differ by a half - turn clockwise . now we compute the difference @xmath581 along each component of @xmath585 as we go from an index one to an index two critical point . if @xmath588 rotates rotates a quarter of a turn counterclockwise about a certain component , we again obtain a contribution of @xmath591 to @xmath581 . in that case , this component will contribute by @xmath592 to @xmath593 . now say that @xmath588 differs from a quarter of a turn counterclockwise rotatation by @xmath19 counterclockwise half - turns , for some @xmath20 . then we obtain a contribution of @xmath594 to @xmath581 and @xmath595 to @xmath593 . therefore the total difference @xmath581 along @xmath572 is @xmath596 and we have proven . it remains to compute @xmath597 . for that , we will use a classical construction in topology . consider a generic section of @xmath598 which is trivial with respect to @xmath255 along @xmath537 . we move @xmath273 in the direction of this section and we obtain a surface @xmath599 which intersects @xmath273 tranversely . then @xmath600 where @xmath601 denotes the signed count of intersections of @xmath273 and @xmath599 . now @xmath273 and @xmath599 can be completed to surfaces in the homology class of @xmath92 . but these surfaces are not necessarily @xmath255-trivial . in fact , the linking number of @xmath537 and @xmath602 in @xmath549 is @xmath603 . following a standard calculation in ech , see e.g. @xcite , we obtain @xmath604 recall that @xmath38 . by proposition [ prop : indiso ] , @xmath49 preserves the grading . moreover , it follows from our constructions in [ sec : hfobd ] and [ sec : echobd ] that @xmath48 and @xmath50 preserve the grading . therefore @xmath37 preserves the grading . to3em , _ the embedded contact homology index revisited _ , new perspectives and challenges in symplectic field theory , crm proc . lecture notes , vol . 49 , amer . soc . , providence , ri , 2009 , pp .

in joint work with yang huang , we defined a canonical absolute grading on heegaard floer homology by homotopy classes of oriented 2-plane fields . a similar grading was defined on embedded contact homology by michael hutchings . in this paper we show that the isomorphism between these homology theories defined by colin - ghiggini - honda preserves this grading .

an ultracold fermi gas provides us the unique opportunity that we can systematically study physical properties of a many fermion system at various interaction strengths , by adjusting the threshold energy of a feshbach resonance@xcite . indeed , by using this advantage , the so - called bcs ( bardeen - cooper - shrieffer)-bec ( bose - einstein condensation ) crossover@xcite has experimentally been realized in @xmath3k@xcite and @xmath4li@xcite fermi gases , where a bcs - type fermi superfluid continuously changes into the bec of tightly bound molecules , with increasing the strength of a pairing interaction . in this sense , we can now deal with a fermi superfluid and a bose superfluid in a unified manner . recently , the spin susceptibility has become accessible in the bcs - bec crossover regime of an ultracold fermi gas@xcite . here , `` spin '' is actually pseudospin describing one of the two atomic hyperfine states contributing to the pair formation . using this thermodynamic quantity , we can examine to what extent the spin degrees of freedom are active in the bcs - bec crossover region . theoretically , the possibility of the so - called spin - gap phenomenon has been discussed in the crossover region near the superfluid phase transition temperature @xmath2@xcite , where the spin susceptibility is anomalously suppressed by preformed spin - singlet cooper pairs . since preformed cooper pairs also cause the pseudogap phenomenon@xcite ( where the single - particle density of states exhibits a gap - like structure even in the normal state ) , the spin - gap phenomenon and pseudogap phenomenon are deeply related to each other in the cold fermi gas system . so far , the spin susceptibility has theoretically been discussed in a uniform fermi gas@xcite , although a real ultracold fermi gas is always prepared in a trap potential . in this paper , thus , taking this realistic situation into account , we study how spatially inhomogeneous pairing fluctuations affect the spin - gap phenomenon in a trapped unitary fermi gas . for this purpose , we employ the extended @xmath0-matrix approximation ( etma ) developed in the uniform system@xcite , to include effects of a harmonic trap within the local density approximation ( lda)@xcite . in a uniform fermi gas , it has been shown that etma correctly describes the bcs - bec crossover behavior of the spin susceptibility@xcite , which makes us expect that this strong - coupling theory is also valid for the trapped case . we briefly note that the ordinary @xmath0-matrix approximation@xcite , as well as the strong - coupling theory developed by nozires and schmitt - rink@xcite , are known to unphysically give negative spin susceptibility in the bcs - bec crossover region , although these theories have successfully explained various many - body phenomena in the bcs - bec crossover region . using the combined etma with lda , we calculate the local spin susceptibility @xmath1 in the normal state near @xmath2 . throughout this paper , we take @xmath5 , for simplicity . we consider a two - component fermi gas , described by the bcs hamiltonian , @xmath6 where @xmath7 is a creation operator of a fermi atom with pseudospin @xmath8 . @xmath9 is the kinetic energy in the @xmath10-spin component , which is measured from the fermi chemical potential @xmath11 , where @xmath12 is an atomic mass , and @xmath13 is an infinitesimally small effective magnetic field to calculate the spin susceptibility . the pairing interaction @xmath14 is assumed to be tunable . the unitarity limit ( which we are dealing with in this paper ) is characterized by the vanishing inverse @xmath15-wave scattering length ( @xmath16 ) , which is related to the interaction strength @xmath14 as @xmath17 where @xmath18 is a cut - off momentum . in lda , effects of a harmonic trap potential @xmath19 can be conveniently incorporated into the theory by simply replacing the chemical potential @xmath20 with the position - dependent one @xmath21@xcite , where @xmath22 is a trap frequency . the lda single - particle thermal green s function then has the form , @xmath23 where @xmath24 is the fermion matsubara frequency , and @xmath25 . the lda self - energy @xmath26 describes fluctuation corrections to single - particle fermi excitations . in etma , it is diagrammatically described as fig . [ fig1 ] , which gives , @xmath27 here , @xmath28 is the boson matsubara frequency , and @xmath29 means the opposite component to @xmath10-spin . we briefly note that the ordinary @xmath0-matrix approximation@xcite is immediately reproduced by simply replacing the etma green s function @xmath30 in eq . ( [ eq2 - 4 ] ) with the bare one , @xmath31 in eq . ( [ eq2 - 4 ] ) , @xmath32 is the particle - particle scattering matrix , given by @xmath33 where @xmath34 is the lowest - order pair - correlation function , describing fluctuations in the cooper channel . in the extended @xmath0-matrix approximation ( etma ) . ( b ) particle - particle scattering matrix @xmath32 . the solid line and double - solid line represent the bare green s function @xmath35 and the etma green s function @xmath36 , respectively . the wavy line describes the pairing interaction @xmath14.,width=226 ] in the present formalism , the superfluid phase transition temperature @xmath2 is determined from the condition that the thouless criterion is satisfied at the trap center as @xmath37@xcite . as usual , we solve this equation , together with the equation for the total number @xmath38 of fermi atoms , given by @xmath39 to self - consistently determine @xmath2 and @xmath11 . here , @xmath40 is the local number density of fermi atoms with @xmath10 spin . above @xmath2 , we only solve the lda number equation ( [ eq2 - 6b ] ) , to determine the chemical potential @xmath11 . the local spin susceptibility @xmath1 is calculated from , @xmath41}{\partial h}=\lim_{h\rightarrow 0}\frac{n_{\up}(r)-n_{\down}(r)}{h}. \label{eq2 - 6d}\ ] ] in this paper , we numerically evaluate eq . ( [ eq2 - 6d ] ) , by taking a small but finite value of @xmath13 . of a trapped unitary fermi gas , as a function of the spatial position @xmath42 measured from the trap center . at each temperature , the spatial position @xmath43 at which @xmath1 takes a maximal value is shown as the filled circle . the horizontal dotted line shows the maximal value ( @xmath44 ) of the scaled spin susceptibility in the case of a uniform unitary fermi gas at @xmath2 shown in panel ( b)@xcite . @xmath45 is the thomas - fermi radius , where @xmath46 is the lda fermi energy at the trap center . @xmath47 is given in eq . ( [ eq3 - 1 ] ) . ( b ) spin susceptibility @xmath48 in a _ uniform _ unitary fermi gas@xcite . the filled circle shows the spin - gap temperature @xmath49 at which @xmath48 takes the maximal value , @xmath50 , where @xmath51 is the spin susceptibility of a free fermi gas at @xmath52 . @xmath53 is the fermi temperature in a uniform fermi gas.,width=453 ] figure [ fig2](a ) shows the local spin susceptibility @xmath1 in a trapped unitary fermi gas above @xmath2 . here , @xmath1 is normalized by the zero - temperature spin susceptibility @xmath54 in an assumed uniform free fermi gas with the particle density being equal to the density at @xmath42 in the trapped case , given by @xmath55 where @xmath56 . since the density profile monotonically decreases as one goes away from the trap center ( see fig . [ fig3](a ) . ) , pairing fluctuations become weak around the edge of the gas cloud even at @xmath2 . on the other hand , atoms feel a high scaled - temperature @xmath57 around the edge of the gas cloud , because the lda local fermi temperature , @xmath58^{2/3}/2 m , \label{eq.app}\ ] ] is low in the low - density region . ( see fig . [ fig3](b ) . ) as a result , the local spin susceptibility @xmath1 is suppressed thermally around the edge of the gas cloud , as in the case of a simple free fermi gas at high temperatures . thus , one has @xmath59 in this spatial region , as seen in fig . [ fig2](a ) . this ordinary thermal effect becomes weak , as one approaches the trap center , because of the decrease of the scaled temperature @xmath57 , as shown in fig . [ fig3](b ) . as a result , @xmath60 increases , as one approaches the trap center from the outer region of the gas cloud . however , fig . [ fig2](a ) shows that the scaled spin susceptibility @xmath61 is suppressed in the vicinity of the trap center , @xmath62 ( where @xmath63 is the thomas fermi radius ) , in spite of the fact that the scaled temperature @xmath57 still decreases with decreasing @xmath42 in this spatial region ( because of the monotonic spatial variation of the density profile shown in fig . [ fig3](a ) ) . thus , this suppression is not due to the simple thermal effect , but is considered as the spin - gap phenomenon originating from strong pairing fluctuations enhanced in the trap center near @xmath2 . indeed , in the spatial region @xmath64 , where @xmath43 is the position at which @xmath60 takes a maximal value , @xmath60 is found to _ increase _ with increasing the temperature . while this temperature dependence is opposite to the case of a uniform free fermi gas ( where the spin susceptibility monotonically _ decreases _ with an increase of the temperature ) , it is consistent with the temperature dependence of the spin susceptibility in the spin - gap regime ( @xmath65 ) of a uniform fermi gas@xcite . ( see fig . [ fig2](b ) . ) as shown in fig . [ fig2](a ) , the spatial region , @xmath64 , becomes narrower at higher temperatures , to eventually vanish at @xmath66 , reflecting the weakening of pairing fluctuations . in a trapped ultracold fermi gas at various temperatures . ( b ) scaled temperature @xmath57 , as a function of @xmath42 . the inset shows @xmath57 magnified around the trap center , where @xmath43 is the peak position of @xmath60 in fig . [ fig2](a ) . the horizontal dotted line in the inset shows the spin - gap temperature @xmath67 in a uniform fermi gas at the unitarity.,width=453 ] a uniform fermi gas at the unitarity is known to exhibit the so - called universal thermodynamics@xcite , where the fermi energy @xmath46 ( or equivalently the fermi temperature @xmath53 ) is the unique energy scale , because of the vanishing inverse scattering length @xmath16 . in the present trapped case , the scaled local spin susceptibility in lda is expected to behave as , @xmath68 the same universal function @xmath69 in eq . ( [ eq3 - 1b ] ) is also expected in the uniform case as @xmath70 where @xmath71 is the spin susceptibility in a uniform unitary fermi gas , and @xmath72 is the zero - temperature susceptibility in a uniform free fermi gas . @xmath53 is the fermi temperature in a uniform free fermi gas . using the relation between eqs . ( [ eq3 - 1b ] ) and ( [ eq3 - 1c ] ) , together with the fact that the scaled temperature @xmath57 is related to the spatial position through eq . ( [ eq.app ] ) , we can relate the spatial variation of @xmath60 in fig . [ fig2](a ) to the temperature dependence of @xmath73 in fig . [ fig2](b ) . indeed , the maximal value @xmath74 at the spin gap temperature @xmath67 in a uniform unitary fermi gas ( fig . [ fig2](b ) ) just equals the peak value of @xmath75 at @xmath76 in the trapped case ( fig . [ fig2](b ) ) , and the latter result is independent of the value of @xmath0 . in addition , the inset in fig . [ fig3](b ) shows that the local scaled temperature @xmath77 in the trapped case always equals the spin gap temperature @xmath78 obtained in the uniform case . these universal results indicate that the observations of the spatial variation of the spin susceptibility @xmath1 , as well as the density profile @xmath79 , in a trapped fermi gas at the unitarity enable us to evaluate the spin - gap temperature @xmath80 in a _ uniform _ unitary fermi gas . in this regard , we briefly note that the relation between a uniform fermi gas and a trapped one become complicated when @xmath81 . in this case , the lda spin susceptibility @xmath1 in a trap also depends on @xmath82 in addition to @xmath57 , where @xmath83^{1/3}$ ] is the lda local fermi momentum . as a result , @xmath1 is related to the spin susceptibility in a uniform fermi gas , not only at various scaled temperatures @xmath84 , but also at various interaction strengths @xmath85 , where @xmath86 is the fermi momentum in a uniform fermi gas . to summarize , we have discussed magnetic properties of a unitary fermi gas in a harmonic potential above @xmath2 . including strong pairing fluctuations within the framework of the extended @xmath0-matrix approximation ( etma ) , as well as effects of a harmonic trap within the local density approximation ( lda ) , we showed that , near @xmath2 , the local spin susceptibility is anomalously suppressed in the trap center due to the formation of preformed singlet cooper pairs . the spatial region where this spin - gap phenomenon occurs becomes wide with decreasing the temperature . we also confirmed that the so - called universal thermodynamics hold for the spin susceptibility . we pointed out that , using this , we can determine the spin - gap temperature @xmath80 in a uniform unitary fermi from the observation of the spatial variation of the local spin susceptibility in the trapped case . in this paper , we have treated effects of a harmonic trap within lda , where spatial correlations are completely ignored . in addition , the present analyses is restricted to the unitarity limit . improving these issues remains as our future problems . since a real ultracold fermi gas is always trapped in a harmonic potential , our results would be useful for the study of how the spatial inhomogeneity affects thermodynamic properties of this system in the bcs - bec crossover region , as well as how to observe the spin gap temperature @xmath80 in a unitary fermi gas . we would like to thank t. kashimura , r. watanabe , d. inotani and p. van wyk for useful discussions . this work was supported by the kipas project in keio university . h.t . and r.h were supported by the japan society for the promotion of science . y.o was supported by grant - in - aid for scientific research from mext and jsps in japan ( no.25400418 , no.15h00840 ) . 99 v. gurarie , and l. radzihovsky , ann . phys . * 322 * , 2 ( 2007 ) . s. giorgini , s. pitaevskii , s. stringari , rev . * 80 * , 1215 ( 2008 ) . i. bloch , j. dalibard , w. zwerger , rev . * 80 * , 885 ( 2008 ) . c. chin , r. grimm , p. julienne , e. tiesinga , rev . * 82 * , 1225 ( 2010 ) . d. m. eagles , phys . rev . * 186 * , 456 ( 1969 ) . p. nozires and s. schmitt - rink , j. low temp . phys . * 59 * , 195 ( 1985 ) . c. a. r. sa de melo , m. randeria , and j. r. engelbrecht , phys . lett . * 71 * , 3202 ( 1993 ) . r. haussmann , phys . b * 49 * , 12 975 ( 1994 ) . y. ohashi , and a. griffin , phys . lett . * 89 * , 130402 , ( 2002 ) . c. a. regal , m. greiner , and d. s. jin , phys . * 92 * , 040403 ( 2004 ) . m. w. zwierlein , c. a. stan , c. h. schunk , s. m. f. raupach , a. j. kerman , and w. ketterle , phys . 92 * , 120403 ( 2004 ) . j. kinast , s. l. hemmer , m. e. gehm , a. turlapov , and j. e. thomas , phys . rev . lett . * 92 * , 150402 ( 2004 ) . m. bartenstein , a. altmeyer , s. riedl , s. jochim , c. chin , j. h. denschlag , and r. grimm , phys . lett . * 92 * , 203201 ( 2004 ) . c. sanner , e. j. su , a. keshet , w. huang , j. gillen , r. gommers , and w. ketterle , phys . lett . * 106 * , 010402 ( 2011 ) . a. sommer , m. ku , g. roati , and m. w. zwierlein , nature ( london ) * 472 * , 201 ( 2011 ) . lee , t. t. wang , t. m. rvachov , j .- h . choi , w. ketterle , m .- s . heo , phys . rev . a * 87 * , 043629 ( 2013 ) . f. palestini , p. pieri , and g. c. strinati , phys . lett . * 108 * , 080401 ( 2012 ) . t. kashimura , r. watanabe , y. ohashi , phys . a * 86 * , 043622 ( 2012 ) . t. enss , and r. haussmann , phys . lett . * 109 * , 195303 ( 2012 ) . g. wlazlowski , p. magierski , j. e. drut , a. bulgac , and k. j. roche , phys . * 110 * , 090401 ( 2013 ) . h. tajima , t. kashimura , r. hanai , r. watanabe , and y. ohashi , phys . rev . a * 89 * , 033617 ( 2014 ) . s. tsuchiya , r. watanabe , y. ohashi , phys . a * 80 * , 033613 ( 2009 ) ; 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we theoretically investigate magnetic properties of a unitary fermi gas in a harmonic trap . including strong pairing fluctuations within the framework of an extended @xmath0-matrix approximation ( etma ) , as well as effects of a trap potential within the local density approximation ( lda ) , we calculate the local spin susceptibility @xmath1 above the superfluid phase transition temperature @xmath2 . we show that the formation of preformed singlet cooper pairs anomalously suppresses @xmath1 in the trap center near @xmath2 . we also point out that , in the unitarity limit , the spin - gap temperature in a uniform fermi gas can be evaluated from the observation of the spatial variation of @xmath1 . since a real ultracold fermi gas is always in a trap potential , our results would be useful for the study of how this spatial inhomogeneity affects thermodynamic properties of an ultracold fermi gas in the bcs - bec crossover region . pacs numbers : 03.75.hh , 05.30.fk , 67.85.lm .

radiation - pressure driven wind models for main sequence and evolved ob stars developed over the past two decades have proven enormously successful in accounting for the gross properties of such outflows . the essential concept underpinning these models is that momentum is extracted most efficiently from the radiation field via line opacity ( lucy & solomon 1970 ; castor , abbott & klein 1975 , hereafter cak ) . with the inclusion of lines , cak showed that the effective radiation force can be increased by several orders of magnitude above that due to electron - scattering alone , thus facilitating mass loss even from stars radiating at around 0.1% of their eddington limit . a decade later it had become clear that the model was well able to predict time - averaged mass loss rates and terminal velocities in agreement with empirical estimates ( friend & abbott 1986 ) . more recently still , the goal has been to obtain a deeper understanding of instabilities inherent in the line - driving mechanism ( e.g. owocki , castor & rybicki 1988 ; puls , owocki & fullerton 1993 ) . these impressive achievements have all been set in the context of one - dimensional ( spherical ) geometry . at the same time , accretion disks have come to be accepted as important components in a wide variety of astrophysical settings . these too can produce intense radiation fields at effective radiation temperatures comparable with those of ob stars . as an example , such disks are inferred from observations to be present in high state cataclysmic variables ( cvs , see warner 1995 ) . moreover , observations also show these same systems give rise to very high velocity winds that most probably emanate from deep within the gravitational potential in the vicinity of the accreting star ( see drew 1997 ) . another example of outflow associated with a luminous disk may be provided by higher mass ysos ( the bn - type objects and herbig be stars , see mundt & ray 1994 and references therein ) . in fact , in these objects it would not even be necessary to sustain a high mass accretion rate to achieve an intense radiation field as one is already produced by the young ob star . finally , shifting up the luminosity scale many orders of magnitude , the accretion disk model is now ` standard ' for active galactic nuclei , and in this context as well , there is direct evidence of fast outflow in the broad absorption line ( or bal ) qsos ( e.g. weymann et al . 1991 ) . given the success of one - dimensional models of line - driven winds from hot stars , it is natural to ask : what is the nature of line - driven winds from a star plus luminous accretion disk ? in practice , calculating models for winds in disk systems is complex because of the intrinsically two - dimensional , axisymmetric character of the problem . to render this problem amenable to analytic solution , previous studies have generally found it necessary to introduce simplifying or ad hoc assumptions . for example , in seeking a steady state accretion disk wind solution , vitello & shlosman ( 1988 ) found it necessary to enforce a radiation force term that increased with height above the disk and required a very particular variation in the ionization state of the gas a matter that gave them cause for concern . more recently an analytic disk wind model has been designed by murray _ et al._@xmath2(1995 ) specifically for agn . in order to simplify the problem they introduced heuristic assumptions which allowed the equations of motion in the radial and polar angle directions to be solved separately . unfortunately , the outcome of their calculations depends in a basic way on these assumptions . in this study , we face the multi - dimensional character of the disk wind problem directly , by adopting numerical methods to solve the dynamical equations from first principles . the first numerical treatment of the problem was formulated as long ago as 1980 ( icke 1980 , see also icke 1981 ) . icke ( 1980 ) set up what , in terms of today s computing power , would now be regarded as a very modest calculation aimed at determining the character of an outflow from a disk driven by continuum ( electron - scattering ) radiation pressure only . in 1981 he incorporated rotation into his treatment and obtained results that we have found to be broadly comparable with our own as a partial test of our independent formulation . like icke , we choose a two - dimensional computational domain in which the central accreting object and inner disk are well - resolved . in our view this is important because of the evidence that disk winds in cataclysmic variables , a natural first test - bed for our results , originate close to the white dwarf . recently pereyra , kallman & blondin ( 1997 , see also pereyra 1997 ) have also presented numerical calculations of the two - dimensional structure of cv disk winds , albeit at a resolution too coarse to capture the inner disk structure or the subsonic portion of the outflow . here , we use non - uniform meshes at high resolution to capture the structure of the wind in both the subsonic and supersonic regimes . moreover , we employ a carefully - tested adaptive numerical integration technique to compute the line force directly within the sobolov approximation . we make no restrictive geometric assumptions with regard to the flux integrals involved and show , on holding the force multiplier constant , that the line - driving force should be constant near the disk . the original motivation for these calculations was to obtain self - consistent dynamical models for cv winds that remove the need to apply ad hoc kinematic structures in modelling observed ultraviolet spectral line profiles . recent high quality observations obtained with the hubble space telescope ( e.g. mason et al . 1995 ) have shown that the kinematic models designed for iue data are already inadequate ( knigge & drew 1997 ) , and serve to emphasise the need for realistic rather than simplistic models . our models incorporate a star , for which we adopt the mass and radius appropriate to a white dwarf , and a geometrically thin accretion disk that is a source both of radiation and mass . in our formulation of the problem , the velocity field has three components that are functions of two spatial coordinates ( sometimes referred to as a 2.5d formulation ) . for this reason , exact implementation of a force multiplier into the radiation force on spectral lines is formidable and some level of approximation required . we describe the formal solution of the problem and our approximations in section 2 , with most of the analytic details given in appendices . we have developed numerical methods to compute both the radiation field from , and the radiation driving force , associated with the disk . these are coupled to the hydrodynamical code zeus-2d ( stone & norman 1992 ) . we describe our numerical methods and tests in section 3 . our particular interests are to explore the impact upon the mass loss rate and outflow geometry of ( i ) varying the system luminosity and ( ii ) varying the radiation field geometry by changing the relative contributions of the central star and disk radiation fields . crudely speaking , we find that the mass loss rate is an extremely strong function of luminosity , while the outflow geometry and its temporal behaviour are controlled by the radiation field s geometry . some of our results have already been presented in a short communication ( proga , drew & stone 1997 ) . a full description of our results is given in section 4 . these are discussed together with their likely relevance and perceived limitations in section 5 . the paper ends , in section 6 , with our conclusions . to calculate the radiation force from a disk we need first to specify the disk s geometry and its radiation field . consistent with existing conventions , we assume a flat , keplerian , geometrically - thin and optically - thick disk . we calculate the disk radiation field from the surface brightness of the so - called @xmath0-disk ( shakura & sunyaev 1973 ) . for models including a radiant central star , we take into account its irradiation of the disk , and assume that the disk re - emits all absorbed energy locally as a black body . the irradiation increases the disk temperature primarily in the inner part of the disk . in the presence of a luminous central star ( cs ) , the disk temperature profile is altered such that the temperature decreases monotonically with radius , whereas a pure @xmath0-disk is characterised by a temperature maximum at 1.36 stellar radii . see appendix a for further details . we approximate the radiative line force by means of the formalism introduced by cak . a key assumption in their method is that the sobolev approximation is valid , i.e. the radiation force exerted as a result of absorption of radiation coming from the central radiant object along a direction @xmath3 depends mainly on the velocity gradient along @xmath3 , at the point of absorption . cak showed that for a spherically - expanding flow the radiation force due to a large ensemble of spectral lines can be expressed by @xmath4 where @xmath5 , is the radiation force due to electron scattering , and @xmath6 is the frequency integrated flux from the star . the quantity @xmath7 , called the force multiplier , represents the increase in radiation force over the pure electron scattering case when lines are included . in the sobolev approximation , it is a function of the optical depth parameter @xmath8 where @xmath9 is the density , @xmath10 is the thermal velocity , and @xmath11 is the velocity gradient along the radial direction ( the only non - zero component in spherical symmetry ) . cak found a fit to their numerical results for the force multiplier , such that @xmath12 where @xmath13 and @xmath0 are parameters of the fit . they also showed that this expression for the force multiplier could be reproduced by assuming a simple statistical model of the distribution of lines in strength and frequency . this statistical model indicates that @xmath13 is proportional to the total number of lines involved and @xmath0 is the ratio of optically - thick to optically - thin lines . subsequently the cak formalism to approximate the radiation force due to lines has undergone a number of refinements ( e.g. , abbott 1982 ; pauldrach , puls & kudritzki 1986 ; stevens & kallman 1990 , gayley 1995 ) . many of the modifications are specific to 1d spherically - symmetric stellar winds . in this work we find the refinement introduced by owocki , castor and rybicki ( 1988 , ocr hereafter ) is important . the formal specification of the force multiplier used by cak allows an unlimited increase of the radiation line force with decreasing @xmath14 , which is clearly unphysical ( a point remarked upon by cak ) . instead one expects the force multiplier to saturate at some maximum value for very low @xmath14 as all lines , including the most optically thick , contribute to the radiation force : a further decrease of @xmath14 does not activate any more lines . for example , this saturation can be seen in the radiation force calculations of abbott ( 1982 ) in which he accounted for lines of the first to sixth stages of ionization of the elements h - zn ( see his figure 3 ) . generally he confirmed the cak results . however his results showed that for @xmath15 , m(t ) falls away from the cak approximation ( equation 3 ) as the force multiplier becomes less sensitive to @xmath14 , the optical depth parameter . ocr considered this problem in terms of a line strength distribution . they modified the simple cak statistical model by cutting off the maximum line strength and thereby limiting the effect of very strong lines : @xmath16\ ] ] where @xmath17 and @xmath18 is a parameter determining the maximum value , @xmath19 achieved for the force multiplier . equation ( 4 ) shows the following limiting behaviour : @xmath20 where @xmath21 . the maximum value of the force multiplier is in reality a function of physical parameters of the wind and radiation field . in a number of studies ( cak , abbott 1982 , stevens & kallman 1990 , gayley 1995 ) it has been shown that @xmath19 is of the order of a few thousand . to adapt the cak formalism designed for ob stars to the disk wind case we need to accommodate two essential differences : ( 1 ) a stellar wind can be well approximated by a 1d radial flow while a disk wind is in general a 3d flow ; ( 2 ) the stellar radiation field is spherically symmetric while the disk radiation is axially symmetric , as a consequence of the disk geometry and the non - uniform disk intensity . first we consider the 3d nature of the flow . rybicki & hummer ( 1978 , 1983 ) generalized the sobolev method to the 3d case in which the flow velocity along a line of sight is not necessarily monotonic . in such a case , radiative coupling between distant parts of the flow must be taken into account . at this stage , where the flow properties are not known , we do not consider this effect . however , a further consequence of generalising the sobolev method to 3 dimensions is that the sobolev optical depth s dependence on the photon line - of - flight velocity gradient , @xmath22 , may in practice become a complicated function of the velocity , velocity derivatives and position . in the generalized sobolev method @xmath23 where the velocity gradient along the line of sight may be written @xmath24 and @xmath25 is the symmetric rate - of - strain tensor . expressions for the components of @xmath25 in the spherical polar coordinate system are given in batchelor ( 1967 ) . the complexity of the disk radiation field , together with the generalized optical depth parameter @xmath14 , mean that the radiation line force at a given location due to the total radiating surface becomes a complicated integral in which the dependences on geometry , the radiation field and local optical depth are no longer separable . for a 2.5d wind , we evaluate the disk radiation force in three steps . first , we calculate the radiation flux due to a surface element of the disk at a point above the disk , @xmath26 ( see appendix a ) . then we calculate the radiation force exerted by this flux via electron scattering , @xmath27 , and via an ensemble of lines @xmath28 ( see appendix b ) . finally we integrate @xmath29 and @xmath30 over the total disk surface visible at the point in question ( i.e. , we exclude the disk region shadowed by the cs , see appendix b ) . our calculations of the radiation force contributed by the cs assume that it radiates as a blackbody at a fixed temperature and without any limb - darkening . we express the cs luminosity in alpha disk luminosity units @xmath31 . the method of calculation of the radiation force from the cs is the same as for the disk , and takes into account disk occultation of the cs . in practice , evaluating the generalised optical depth ( equation 7 ) using all terms in @xmath32 is computationally prohibitive . instead , we keep only the dominant terms ( see section 3.2 and appendix c ) . we use the 2d eulerian finite difference code zeus-2d ( stone & norman 1992 ) to calculate the wind structure . we have extended the code to include radiation forces due to electron scattering and line driving , i.e. we solve : @xmath33 @xmath34 where @xmath35 is the gravitational acceleration of the cs , and @xmath36 is the total radiation force per unit mass . we describe how @xmath36 is evaluated numerically in section 3.2 . the gas in the wind is taken to be isothermal with a sound speed @xmath37 . our calculations are performed in spherical polar coordinates @xmath38 with @xmath39 at the centre of the accreting cs . we assume axial symmetry about the rotation axis of the accretion disk ( @xmath40 ) . thus we assume that all quantities are invariant in @xmath41 . our standard computational domain is defined to occupy the radial range @xmath42 , where @xmath43 is the cs radius , and angular range @xmath44 ( in section 4.1 we compare the results to a solution computed on a grid ten times larger in radial extent , i.e. @xmath45 ) . the @xmath46 domain is discretized into zones . the gridding needs to be such as to ensure that the subsonic portion of the model outflow is sampled by at least a few grid points in both @xmath47 and @xmath48 . this requirement and the nature of the problem combine to demand an increasingly fine mesh toward the disk plane : here the density declines dramatically with height , and , moreover , the velocity in the wind increases rapidly . our standard numerical resolution consists of 100 zones in each of the @xmath47 and @xmath48 directions , with fixed zone size ratios , @xmath49 . the smallest radial grid zone has dimension @xmath50 at @xmath51 , while the smallest angular grid zone is @xmath52 near @xmath53 . gridding in this manner ensures good spatial resolution close to the radiating surfaces of the disk plane and the cs . in addition , to check that our solutions are resolved , we have computed the evolution of two of our models ( our fiducial unsteady and steady wind cases see sections 4.1 and 4.2 respectively ) at twice this resolution , i.e. using 200 zones in each of the @xmath47 and @xmath48 directions , with @xmath54 . we find that the global properties of the solutions ( such as the terminal velocity and mass loss rate ) differ by no more than 10% between the high and standard resolution models . the boundary conditions are specified as follows . at @xmath55 , we apply an axis - of - symmetry boundary condition . for the outer radial boundary , we apply an outflow boundary condition . for the inner radial boundary @xmath56 and for @xmath57 , we apply reflecting boundary conditions . the initial density profile is given by the condition of hydrostatic equilibrium in the latitudinal direction @xmath58 where @xmath59 is the density in the first grid zone above the equatorial plane . physically , @xmath60 is analogous to the density in the photosphere of the disk at the base of the wind . the interior of the disk itself is treated as negligibly thin and is excluded from the models ( for a disk temperature of @xmath61 k at @xmath62 , the disk scale height @xmath63 is @xmath64 ) . the value chosen for @xmath60 is arbitrary . typically we choose @xmath65 . as discussed in section 4.4 , we find the gross properties of the winds are unaffected by the value of @xmath59 provided it is large enough that the acceleration of the wind up to the sonic point is resolved with at least a few grid points . we set a lower limit to the density on the grid of @xmath66 and enforce it at all times in all models . for the initial velocity field , we adopt @xmath67 , and @xmath68 . we find only the initial , transient evolution of the wind is affected by this choice of initial velocity conditions . at late times , we find all solutions for the same model parameter values have the same time - averaged properties regardless of the initial conditions . in order to represent steady conditions in the photosphere at the base of the wind , during the evolution of each model we continue to apply the constraints that in the first zone above the equatorial plane the radial velocity @xmath69 , the rotational velocity @xmath70 remains keplerian , and the density is fixed at @xmath71 at all times . we have found that this technique when applied to calculations of spherically symmetric line - driven winds from stars , produces a solution that relaxes to the appropriate cak solution within a few dynamical crossing times . the integrals ( appendix b ) that express the radiation force have to be evaluated numerically . to include the contribution from all radiating surface elements properly , the size of each surface element seen from a point w in the disk wind has to be in proportion to the distance from w to this surface element . in other words , the sampling should be uniform in apparent solid angle at w. a more subtle issue is that the vector character of the radiation force means that cancellation of opposing contributions to its net value can occur ( e.g. , in the radial component for points close to the disk plane ) . care must therefore be taken in the numerical scheme that inappropriate gridding does not misrepresent such cancellations and thereby introduces spurious , fluctuating force terms . to satisfy both these important requirements we calculate the radiative force using grids of radiating surface elements that are adapted to every point w in the wind . the need to use adaptive integration methods is demonstrated by the tests discussed in section 3.3 . the integration scheme applied to the disk component of the radiation force uses a 2d version of gaussian quadrature such that the number of quadrature points increases with increasing @xmath48 . because of foreshortening , the integrands ( equations b3 and c7 ) are strong functions of the position on the disk plane for @xmath48 close to @xmath72 they reach a maximum close to the point @xmath73 , ( @xmath74 ) , wherein @xmath75 is defined relative to @xmath47 by @xmath76 for @xmath57 and @xmath77 ( see appendix a ) . note that for @xmath78 , the point @xmath73 is very close to the point on the disk for which the size of a surface element has to be the smallest ( i.e. @xmath73 lies close to the projection of w onto the disk plane ) . these properties of the integrands allow us to evaluate the radiation force accurately using a manageable number of grid points because the finest resolution on the disk plane is necessary mainly close to the point @xmath73 . for the region far from @xmath73 , we can afford less dense coverage of the disk surface . in evaluating the integrals involved in the disk component of the radiation force ( equations b3 and c7 ) , we generally break the inner radial integral over ( @xmath79 ) into two sub - integrals , spanning ( @xmath80 ) and ( @xmath81 ) . for cases where the point @xmath73 falls within the inner , or beyond the outer , edge of the disk just the one integral over ( @xmath79 ) is performed . the discretization of the disk surface uses 128 , 256 , 512 , 1024 radial quadrature points for @xmath82 , @xmath83 , @xmath84 , and @xmath85 , respectively . we calculate the outer azimuthal integral over the angular range ( @xmath86 ) with 128 and 512 quadrature points for @xmath87 and @xmath88 , respectively . for the cs radiation force ( equations b9 and c3 ) , we use a 2d version of the trapezium method with the number of quadrature points increasing with decreasing @xmath47 . in the polar direction , the stellar surface is divided up into 5000 and 500 grid points for @xmath89 and @xmath90 respectively . in the azimuthal direction , the resolution is 101 grid points for all @xmath47 . the evaluation of the line radiation force integrals is a major element in the computational cost of these simulations . the situation is exacerbated by the need to sample the subsonic part of the wind , close to the disk plane , reasonably well both in @xmath48 and @xmath47 . indeed we find we can not afford a full recalculation of the radiation line force for all locations , at every time step . therefore we have to seek a working compromise between regular updating of the radiation line force and maintaining its accuracy in space . this suggests two contrasting ways of approximating the line force calculations : ( 1 ) the wind streamlines can be assumed to favour particular approximations to the velocity gradient that allow the radiation line force to be updated every time step ; ( 2 ) the wind velocity can be assumed not to change rapidly with time at a given location , with the implication the line force need only be updated after some time interval taking into account the exact velocity vector . the first approach amounts to a simplification of @xmath32 ( equation 8) that enables the time - varying velocity gradient to be taken out of the integration over the radiating surfaces . for present purposes , this is our preferred option and our particular implementation of it is presented in appendix c. in physical terms , our treatment is equivalent to assuming that the transmission of disk light to points w in the wind will be controlled by the velocity shear in the vertical direction , and that the cs light has mainly to propagate through radial velocity shear . in specifying @xmath32 , this allows us to consider just @xmath91 ( where @xmath92 is height above the disk plane ) when computing the disk s contribution to the force and just @xmath93 when calculating the cs component . this treatment should work well for the cs component throughout most of the computational domain and it should also be adequate for the disk component except , perhaps , close to the disk plane where neglect of the radial and azimuthal derivatives of the rotational component of motion ( @xmath94 ) will result in underestimation of the line force ( i.e. inclusion of these terms would enhance @xmath32 for @xmath95 ) . this neglect has less impact on the cs force component because foreshortening of the spherical stellar surface acts to reduce the weight of contributions from the larger angle , higher shear lines of sight with respect to the more nearly radial , low shear directions . within the present context in which unsteady flow has to be described , the second numerical approach of only updating the line force after a number of time steps requires more computational time . it does become both viable and accurate where the outflow achieves a steady state rather than a configuration that is steady only in a time - averaged sense . we are able to assess the impact of approximating @xmath32 upon our simulations in such cases . we return to this issue in section 4.4 . the hydrodynamical algorithms implemented in the zeus-2d code have been extensively tested already ( stone & norman 1992 ) , thus here the primarily concern is with tests of the integration of the radiative force term in the code . as a first check , we have tested that our numerical methods are able to reproduce the appropriate cak wind solutions for spherically symmetric radiation fields . on the other hand , tests of fully multidimensional radiation fields are more difficult to construct . thus our tests are restricted to the asymptotic behaviour of the electron - scattering component of the radiation force in a few cases where we know the analytical solution . as @xmath48 approaches @xmath72 , we can treat the disk as an isothermal infinite plane . therefore as @xmath96 we expect the radial component of @xmath97 to approach zero , while the latitudinal component of @xmath97 becomes @xmath48-independent and equal to that exerted by the local disk flux . for high @xmath47 and low @xmath48 , the cs can be approximated by a point source . thus the radial component of @xmath98 should decrease there like @xmath99 and the latitudinal component of the radiation force should vanish . the radiation force calculated according to the numerical scheme described in 3.2 is a smooth function of @xmath47 and @xmath48 , and agrees with the above asymptotic solutions to within 5% . the discrepancy is attributable to fact that our disk is not isothermal even for points close to the disk , regions of different temperature contribute to the force . ( 80,170 ) ( 0,0 ) in the case that the disk is isothermal and without any cs irradiation ( @xmath100 ) , the @xmath48-component of equation b3 can be written as @xmath101 for points very close to the disk plane and far from the disk edges the solution of equation 12 should be equal to @xmath102 . figure 1 illustrates how the numerical estimate of equation ( 12 ) depends on the number of grid points for @xmath62 . as few as 128 quadrature points produces a satisfactory integral for points w at @xmath103 . for higher @xmath48 however , only solutions based on 1024 quadrature points give an accurate estimate . recall that in order to resolve the subsonic portion of the acceleration zone , we adopt a nonuniform grid with the smallest zones having an angular extent of only @xmath104 . hence , there are many grid points in the region @xmath105 where the use of densely - sampled adaptive quadratures is critical . before settling on the discretization described in 3.2 we tried using the trapezium method to integrate the radiation force due to the disk . to achieve as good agreement with the asymptotic solutions as obtained using the gaussian scheme of 3.2 , the trapezium method requires at least 2 orders of magnitude more grid points . we have also tried a modified version of the trapezium method using an exponential distribution of grid points around the point @xmath73 although this method gave better results than uniformly distributed grid points , it remained inferior to gaussian quadrature . we have also experimented with the number of grid points used in calculating the radiation force due to the cs . this calculation is less demanding than that for the disk component . a reasonable number of grid points ( as specified in 3.2 ) gives satisfactory agreement with the analytical solution for the asymptotic case . our numerical models are specified by a number of parameters . the cs is specified by its mass @xmath106 and radius @xmath43 . in all our calculations we assume @xmath107 m@xmath108which yields the cs radius @xmath109 cm using the mass - radius relation for co white dwarfs due to hamada & salpeter ( 1961 ) . the accretion disk is characterized by the mass accretion rate through it , @xmath110 ( which we treat as a free parameter ) , and by the sound speed @xmath37 ( fixed at 14 @xmath111 ) . finally , the line - driving force is determined by the force multiplier parameters , @xmath13 , @xmath0 , and @xmath19 , the maximum value allowed for the force multiplier , and by the thermal speed @xmath10 which sets the line widths . as a starting point , we adopt typical ob star values for the force multiplier parameters , @xmath13 and @xmath0 ( i.e. @xmath112 , @xmath113 , see gayley 1995 ) , and subsequently vary @xmath0 . tables 1 and 2 specify the parameter values of all the models discussed in sections 4.1 , 4.2 , and 4.3 below . . full list of model parameters . the second column shows the parameters for model 2 , while the third column indicates the parameter range explored for those parameters which we have varied . [ cols= " < , < , < " , ] our simulations suggest that there are two kinds of flow that might arise from luminous accretion disks ( proga , drew , & stone 1997 ) . we describe a representative example of each of these two types of outflow in some detail first ( sections 4.1 and 4.2 ) . these are followed by a limited parameter survey in which we focus on the effects of varying three key parameters the disk luminosity , the relative luminosity of the cs ( @xmath114 ) and the force multiplier index @xmath0 . finally we draw attention to the role of some of the model assumptions in section 4.4 . ( 180,590 ) ( 0,0 ) ( 0,205 ) ( 0,410 ) ( 90,0 ) ( 90,205 ) ( 90,410 ) we begin the presentation of our results by describing the properties and behaviour of our model in which @xmath115 @xmath116and the central star is assumed to be dark ( @xmath117 ) . this is our model 2 ( see table 2 ) . figure 2 presents a sequence of density maps and velocity fields ( left and right hand panels ) from model 2 plotted in the @xmath118 plane . the figure displays the results of our high resolution ( @xmath119 grid points ) run . the length of the arrows in the right - hand panels is proportional to @xmath120 . the pattern of the direction of the arrows is an indication of the shape of the instantaneous streamlines . after @xmath121 time units ( we define the time unit as the orbital period at the surface of the cs , @xmath122 ) the disk material fills the grid for @xmath123 and it remains in that region for the rest of the run . figure 2 shows that the high density region usually corresponds to regions of low velocity . the variation in the orientation of the velocity arrows in the righthand panels indicates the flow is time - dependent and , moreover , it is clear that in some cases negative radial velocities ( i.e. infall ) are possible . the time dependence persists even after 80 @xmath124 . however it is important to note as discussed below that the gross properties of the flow ( such as the mass loss rate ) , settle down to steady time - averages when averaged over timescales on the order of 100 sec ( @xmath125 ) or more . in model 2 , the flow is complex with a few filaments sweeping outwards , typically , and various knots and clumps of gas moving both upwards and downwards . the direction and speed of motion at any one position is apt to change unpredictably with time ; the velocity magnitudes at @xmath126 are typically less than @xmath127 . in the flow there is also a region where the material moves in a quite organized fashion . for @xmath128 beyond @xmath129 the material moves along nearly straight trajectories and leaves the outer boundary of the grid with velocities ranging from @xmath130 up to @xmath131 . however , at no location in even this part of the flow does the velocity ever exceed the local escape velocity ( which decreases from @xmath132 at @xmath51 to @xmath133 at @xmath134 ) . this does not mean that the mass loss will necessarily stall at a larger radius . to investigate the nature of the outflow on larger scales , we have calculated this model on a computational domain that is ten times larger in the radial direction , i.e. it extends from @xmath43 to @xmath135 . we use a grid of 100 angular zones and 150 radial zones for this model , so that the numerical resolution in the inner region of the grid ( i.e. @xmath136 ) is identical to our standard case . the results of this calculation show that the integrated mass loss rate at @xmath135 is the same as that at @xmath137 . moreover , the fast stream does continue to accelerate beyond @xmath138 , so that @xmath139 rises from @xmath11100 km s@xmath140 at @xmath138 up to 1300 km s@xmath140 or so at @xmath141 ( which is well above the escape velocity at this point ) . that the fast outflow eventually exceeds escape velocity is not suprising since at large radii the density of the flow is so low that the radiation force reaches its maximum value set by @xmath19 : thus both gravity and the radiation force will scale with radius as @xmath99 . consequently , not only can the total force never change sign ( because the ratio of the radiation force to gravity is constant ) , but also the total force decreases in magnitude rapidly , so that the flow velocity no longer changes significantly . more importantly , the model on the larger grid indicates the mass loss is completely dominated by material arising from the inner ( @xmath142 ) region of the disk . ( 100,480 ) ( 0,0 ) ( 0,160 ) ( 0,320 ) next we consider the time and angle dependence of the gross properties of the flow at large radii . figure 3 is a plot of the angular dependence of density , radial velocity , mass flux density , and accumulated mass loss rate at @xmath143 at the same times as figure 2 . the accumulated mass loss rate is given by : @xmath144 the gas density is a very strong function of angle for @xmath48 between @xmath145 and @xmath146 . between the disk mid - plane at @xmath53 and @xmath147 , @xmath9 drops by @xmath148 orders of magnitude , as might be expected of a density profile determined by hydrostatic equilibrium ( see equation 11 ) . for @xmath149 , the wind domain , @xmath9 varies between @xmath150 and @xmath151 . for @xmath152 , density again decreases exponentially , but this time to so low a value that it becomes necessary to replace it by the numerical lower limit @xmath153 . the region with @xmath154 is not relevant to our analysis as it has no effect on the disk flow . the radial velocity at @xmath137 varies around zero with an amplitude @xmath155 for @xmath156 . over the angular range @xmath157 , @xmath139 increases from @xmath158 up to @xmath159 . the cumulative mass loss rate is negligible for @xmath152 because of the very low prevailing gas density . beginning at @xmath160 , @xmath161 increases to @xmath162 at @xmath163 . then , in the region close to the disk plane , where the gas density starts to rise very sharply and where the motion is typically more complex , the cumulative mass loss rate is subject to enormous fluctuations ( some of which may even be negative ! ) . in the example shown as figure 3 , the total mass loss rate through the outer boundary , @xmath164 @xmath165 reaches @xmath166 . this figure is most certainly dominated by the contribution from the slow - moving region very close to the disk mid - plane a contribution that is very markedly time - dependent . ( 100,170 ) ( 0,0 ) to provide some insight into the time dependence , figure 4 shows the time evolution of volume - averaged quantities for run 2 . the total mass on the grid , @xmath167 , is subject only to small changes . it increases by @xmath168 during the first 60 @xmath124 , and then decreases again , dropping back to 1.006 of the initial value , @xmath169 , by 170 @xmath124 . after this time @xmath167 starts to fluctuate between @xmath11.006 @xmath169 and @xmath170 . by contrast , the total mass loss rate is seemingly a strong function of time . initially , it rises steeply , peaking at @xmath171 at 50 @xmath124 . then it plummets to @xmath172 at @xmath173 and starts oscillating with decreasing amplitude . these large swings are entirely a consequence of the complex character of the flow close to the disk mid - plane . a much steadier , and consistently positive , cumulative mass loss rate is achieved if the integration over polar angle is stopped at @xmath174 ( see figure 3 ) . a further justification for stopping the integration at this angle , is that @xmath161 is then the mass loss rate associated with just the hypersonic outflow that easily escapes the system . we find that unsteady outflow such as that described above in section 4.1 persists as long as the disk radiation field is dominant ( small @xmath114 in our parameterisation ) . however , as the radial component of the radiation field is increased with respect to the latitudinal ( @xmath48 ) component , by adding in light from a central star ( cs ) , we find that the volume occupied by unsteady outflow diminishes . indeed , in the presence of a strong radiation force due to the cs , a disk wind can even settle into a steady state . ( 180,220 ) ( 0,0 ) ( 90,0 ) ( 100,170 ) ( 0,0 ) ( 100,170 ) ( 0,0 ) model 8 is a contrasting example of a strong outflow in a steady state . in it , we set the cs luminosity equal to the intrinsic disk luminosity ( i.e. @xmath175 ) , and chose @xmath176 @xmath116 . the remaining model parameters and the initial conditions are as specified in tables 1 and 2 . figure 5 is the density map and velocity field for this model after 240@xmath124 . the flow is almost in a steady state by then , with the gas density a smooth function of position . the flow may be described as organized and regular . small changes with time still occur , but only very close to the disk plane . figure 6 presents the wind properties as function of @xmath48 at @xmath177 after 240@xmath124 . on this surface , the flow density varies between @xmath178 and @xmath179 for @xmath180 . within the same @xmath48 range , @xmath139 increases from low values on the order of 100 @xmath111 , up to @xmath181 as @xmath48 decreases . the accumulated mass loss rate is @xmath182 at @xmath183 . this is a factor of @xmath1400 increase with respect to the mass loss rate obtained in model 2 , for just a factor of @xmath184 increase in total luminosity . figure 7 shows the time evolution of volume - averaged quantities in model 8 . this should be contrasted with the equivalent figure for model 2 ( figure 4 ) . as in model 2 all the quantities plotted are subject to fluctuations of the wind properties near the disk plane , where the flow does not quite settle into a steady state . however the magnitude of these fluctuations of the total mass loss rate has collapsed from a factor of 10 ( model 2 , figure 4 ) to around 1.5 ( model 8 , figure 7 ) . as is fitting for a first exploration of radiation - driven wind models from disks , we aim to examine only the parameter space of our models that will define the major trends in disk wind behaviour . table 2 lists the models considered . we emphasise a survey of how the mass loss rate , outflow velocity and geometry change with disk luminosity and relative cs luminosity . in view of the important formal role that the force multiplier index @xmath0 is known to play in determining one - dimensional stellar wind solutions ( i.e. @xmath185 ) , we have also calculated a few models in which the index @xmath0 been set equal to the relatively extreme values of 0.4 and 0.8 ( see gayley 1995 ) . so that we might focus on this dependence , we arbitrarily hold @xmath13 and @xmath19 constant . ( 100,310 ) ( 0,0 ) ( 0,155 ) ( 100,310 ) ( 0,0 ) ( 0,155 ) in figure 8 we show ( a ) the derived ratio , @xmath186 , as a function of @xmath187 and ( b ) the wind mass loss rate , @xmath188 , as a function of the total effective luminosity , @xmath189 l_d m_{max}$ ] , for various @xmath114 and @xmath0 . we define @xmath188 as the cumulative mass loss rate for the region well above the disk plane in which the highly supersonic , organized flow is located ( i.e. the angular integral is stopped early enough to avoid the exponential density profile of the disk and any lower - velocity complex flow component at @xmath48 near @xmath72 ) . the total effective luminosity is the total luminosity of the system , @xmath189 l_d$ ] , multiplied by the maximum value of the force multiplier @xmath19 ( see the discussion leading to equation 4 ) ; it is measured in figure 8 in units of the classical eddington value . in figure 8a it can be seen that @xmath186 is a very strong function of @xmath187 for @xmath100 . at low @xmath187 there is virtually no disk wind at all . the outflow turns on sharply for @xmath190 @xmath116 , and then there is a flattening out of @xmath186 to follow a power law of index @xmath191 for @xmath192 @xmath116 . a similar trend is apparent from the @xmath193 and @xmath194 models , with the difference that increased @xmath114 at a fixed mass accretion rate @xmath187 results in a higher mass loss rate . but when , instead , the absolute values of the calculated mass loss rates are considered just as a function of the total effective luminosity , this family of curves for different @xmath114 collapses , most impressively , into a single curve . this is shown in figure 8b . that this occurs shows that @xmath188 is not itself sensitive to the geometry of the driving radiation field ( provided that @xmath195 is higher than the eddington limit ) . a further point to note from figure 8b is that a disk together with cs will produce a fast wind for @xmath196 if the effective luminosity is higher than @xmath12 times the eddington limit . ( 180,430 ) ( 0,0 ) ( 0,210 ) ( 90,0 ) ( 90,210 ) our models for @xmath197 and @xmath198 indicate that the wind mass loss rate is very sensitive to the rapidity with which the force multiplier saturates at its maximum value ( i.e. the limit in which all lines have become optically - thin ) . for @xmath199 , @xmath188 is @xmath200 orders of magnitude higher than for @xmath201 . higher @xmath0 yielding higher mass loss simply reflects the fact that the force multiplier is higher for a given @xmath14 ( where @xmath14 is the optical depth parameter defined in equation 7 ) . the extremity of the effect shows that it does not require large shifts in the relative magnitudes of the radiation force and effective gravity to make the difference between negligible and efficient mass loss . interestingly , there seems to be no change in the power law dependence of @xmath188 on luminosity between @xmath113 and @xmath202 , in contrast to the behaviour of one - dimensional stellar wind solutions . figure 9 presents how the wind radial velocity at @xmath203 changes with @xmath187 and @xmath204 l_d m_{max}$ ] for various @xmath0 . the figure we quote is @xmath139 of the gas at a representative angle in the supersonic part of the outflow this is usually @xmath139 at the polar angle where @xmath205 peaks in the fast stream ( see figure 3 ) . generally , the wind velocity is a weaker function of the disk luminosity than @xmath188 . figure 9b suggests that @xmath139 increases with @xmath204 l_d m_{max}$ ] along one universal curve at fixed @xmath0 , mirroring the single relation found also for the mass loss rate . this curve is not as smooth as that for @xmath188 because of inherent imprecision in our method of determining @xmath139 . the radial velocity is a strong function of @xmath48 and also may change with time . to obtain a smoother curve we would have had to calculate many of our models for longer and then derive a consistent set of time averages rather than make ` by eye ' measurements as here . despite this , the trend is clear enough that @xmath139 scales with @xmath204 l_d m_{max}$ ] in our wind models . while it is true of our models that the integrated mass loss rate and typical radial outflow velocity are not sensitive to the particular geometry of the radiation field , this does not mean that the radiation geometry has no role to play . panels b , c and d of figure 10 compare the flow pattern from three models in which the mass accretion rate ( and therefore disk luminosity ) is held fixed at @xmath206 m@xmath108 yr@xmath140 and the luminosity of the central star is varied using @xmath207 and 3 . note this implies that the total luminosity ( disk plus star ) is increasing . it can be seen that , as the contribution of the central star to the radiation field grows , the flow becomes more equatorial . this is unsurprising given that the increasing contribution from the central star boosts the radial component of the radiation force , while contributing little to the ( negative ) @xmath48 component . however , it is important to realise that the flow geometry also responds , for a fixed radiation geometry , to a change in the driving luminosity . panels a and b of figure 10 compare the flow pattern from two models in which the mass accretion rate ( and therefore disk luminosity ) is changed from @xmath208 m@xmath108 yr@xmath140 to @xmath176 m@xmath108 yr@xmath140 , while the central star is assumed to be dark , i.e. @xmath100 . the contrast between these two panels shows how a more luminous disk will power a stronger , more vertically - directed wind . specifically , an increase of a factor @xmath102 in the mass accretion rate is sufficient to divert the fast boundary stream from @xmath209 to @xmath210 . there are two parameters in our models listed in table 1 whose values seem arbitrary , yet which may appear important in determining our solutions . these are the density at the base of the wind @xmath59 and the maximum value of the force multiplier @xmath19 . in this section , we discuss the effect on our models of varying these parameters . we also discuss the tests we have performed of the sensitivity of our solutions to the approximations made in calculating the line - driving . first , we consider the effect of varying @xmath19 . in principle , @xmath19 is a function of @xmath13 , @xmath0 and @xmath18 ( cf . equation 6 ) . in this paper , we have treated @xmath13 and @xmath0 as free parameters and studied the effect of varying @xmath0 in section 4.3 . in this case , @xmath18 was also varied to keep @xmath19 fixed . instead , here we investigate the effect of varying @xmath18 ( and therefore @xmath19 ) for fixed @xmath13 and @xmath0 . as pointed out in section 2 , @xmath19 is not an arbitrary quantity , but we expect it only to take effect in regions where the optical depth @xmath14 is small . since @xmath211 is likely to be fixed near the wind base in the higher @xmath14 domain , we may anticipate that moderate changes in @xmath18 ( @xmath19 ) are more likely to have a bearing on the strongly supersonic flow and alter , for example , the wind terminal velocity . our tests indicate this is the case . of course very low @xmath18 ( @xmath19 ) can adversely affect the driving near the base of the flow and , if low enough , quench the wind altogether . if we treat @xmath19 as the parameter identifying the scale of the line driving force , we can use it to set the following rough lower limit on the total luminosity needed to produce a fast disk wind : @xmath212 where @xmath213 is the eddington luminosity . next , we consider the effect of varying the density @xmath59 at the base of the wind . the radiation force due to lines per unit mass is a function of gas density ( equations 24 ) such that the higher the density , the lower the force . for @xmath48 near @xmath72 , the total force acting on the gas is nearly equal to the radiation force since the effective gravity near the disk mid - plane is small . in this region , therefore , the boundary density @xmath60 controls not only the radiation line force but also the total force acting on the gas . we have tested the sensitivity of our model winds to changes in our assumed value for @xmath60 . we find that for @xmath214 , the flow is transsonic ( i.e. the subsonic portion of the flow is resolved on our numerical mesh ) , and the properties of the outflow do not depend on the value of @xmath60 . on the other hand , for @xmath60 @xmath215 the line radiation force per unit mass is high at the outset , giving rise to a steady supersonic outflow even at the wind base ( i.e. the flow becomes supersonic in less than one grid point ) . this is clearly unphysical : if the radiation force per unit mass were this high in a real disk , subsonic outflow would have begun at much higher densities deeper in the disk . the highest allowed value for the density at the base of the wind must be less than the density at the midplane of the disk . in principle , the density along the disk plane can be determined self - consistently from disk structure models . at the same time , the density at the base of the wind must be large enough to produce a transsonic wind . disk structure models have shown that the density in the disk midplane is @xmath216 ( e.g. pringle 1981 , carroll _ et al._@xmath21985 ) , thus the value @xmath217 we adopt in our models can be seen to be entirely satisfactory . finally , we have also examined the sensitivity of our solutions to the assumptions we adopt to compute the line driving force ( see appendix c ) . while it is infeasible to evaluate the generalized cak force in our models at every timestep , it is feasible to evaluate it at a particular time for all locations in the flow , in order to compare the exact calculation with the force computed approximately . typically , we find the biggest discrepancies very close to the disk surface : the full treatment of the generalized force yields an acceleration up to an order of magnitude higher than that given by our approximation . this is because of the extra contributions from terms depending on @xmath94 in the rate of strain tensor which we drop . however , as the flow is accelerated , these terms are quickly overwhelmed by terms which depend on @xmath218 and @xmath219 that are included in our approximations . thus , a few degrees above the surface of the disk , our approximate form for the radiation force is in good agreement with the full expression . moreover , as the optical depth in the lines decreases , the force reaches its maximum value @xmath19 and becomes independent of the approximations we adopt . typically the force saturates at just a few stellar radii in response to the declining wind density . still , it is possible that the increased radiation force close to the disk plane in a more exact treatment may affect the solutions by , e.g. , increasing the mass loss rate in the wind for a given disk luminosity . thus , the development of an efficient computational scheme that can relax to a hydrodynamical solution consistent with the full form of the generalized line - driving force is important : we will present such results in a future communication . however , the current tests of our approximations give no indication that the key features of our results ( the unsteady nature of radiation driven disk winds , or the overall two - dimensional geometry of such winds ) are sensitive to an improved representation of the radiation force . the most dramatic result of these models is the discovery of unsteady outflow in many of the cases that we have considered . this component when present occurs in the base of the outflow near the disk . it is characterised by large amplitude density and velocity variations . it is important to ask what is the origin of this behaviour . we identify several factors which contribute to it . the first , and fundamental driver of the behaviour , is the difference in the height dependence of the vertical components of gravity and the radiation force . the former increases linearly whilst the flux integral central to the latter is nearly constant in the brightest parts of the disk . the consequence is that mass lifted off the disk plane by radiation pressure is susceptible to stalling as the increasing gravity takes effect . in this circumstance , mass loss can only be established if a segregation can occur in which denser concentrations of mass fall back toward the disk plane , while the interspersed lower density gas ( in which the line - driving force per unit mass is larger ) continues to be accelerated outward by the radiation force . if this separation were only required to occur in the subcritical part of the flow , gas pressure effects might then act to smooth the density profile , thereby preventing the development of unstable behaviour . in practice , the radiation force term continues to be at a disadvantage with respect to gravity out to greater heights in all our models where the disk is the only source of radiation . a critical aspect that facilitates the unsteady behaviour is the multi - dimensional character of the flow . in one dimension , it is likely that the increase in gravity with height would prevent an outflow being established at all in the case of a sub - eddington disk . however , in two dimensions , streamlines can merge laterally , with the result that higher density regions , in which the radiation force per unit mass and acceleration is reduced , are created alongside lower density gas that can be more readily accelerated to form the outflow . the contrast between a nearly planar flow from a disk and a spherical flow from a star is relevant here ; the effects of streamline convergence would be reduced by geometric dilution in the latter . we are certainly not the first to appreciate the significance of the increase in effective gravity with height for disk winds . for example , in their essentially one - dimensional treatment , vitello & shlosman ( 1988 ) dealt with the problem by deriving an ionization structure for the wind which ensured the radiation force tracked the rising gravity term . in contrast , we take the view that unsteady behaviour is likely to be a natural characteristic of disk winds and therefore see no need to condition our calculations to eliminate it . given the inherent instability present in the outflow , it is not surprising that our models show complex behaviour . all that is required to excite such behaviour are modest perturbations . these will arise in our models for several reasons related to the physics of the problem . for example , our initial conditions are not a perfect equilibrium state rather , small radial pressure gradients excite both radial and vertical oscillations of the disk that can seed perturbations in the outflow . vertical oscillations of the disk continue to be driven as dense material falls back onto the disk from the flow . in fact , the tendency of pressure - supported disks to undergo vertical oscillations ( e.g. , cox , & everson 1980 ; lin , papaloizou , & savonije 1990 ) may ensure the flow will never reach a steady state . in addition to these small amplitude perturbations associated with the lack of perfect hydrostatic equilibrium in the initial state , there are large amplitude velocity perturbations associated with the transients generated during the establishment of the outflow . finally , there is considerable velocity shear between the dense disk wind and the lower density fast stream defining the upper envelope of the flow . there is evidence in our simulations that this shear gives rise to kelvin - helmholtz instabilities . it is well - known that even 1d radiation - driven stellar winds are subject to powerful instabilities ( ocr ) . it is plausible this instability will be present in radiation - driven disk winds also , although the instability tends to produce strong shocks perpendicular to the outflow which we do not observe in our simulations . even without these ( as described above ) there are other physical effects that will in any case lead to complex , unsteady flow . a related and important feature of our calculations is that the addition of a strong radial component to the radiation field associated with a bright central star ` organizes ' the flow into a steady state . the effect is almost certainly caused by the fact that the streamlines near the surface of the disk will be directed outwards of the purely vertical by the added stellar radiation . thus , the effective gravity along the streamlines no longer increases , and the mechanism of the unsteady behavior ( that gravity exceeds the radiation force at some distance from the disk ) no longer operates . empirically , we have seen that , in the luminosity domain where the disk wind is robust , a cs half as luminous as the disk ( @xmath220 ) is sufficient to make this difference . our present calculations have been motivated by and designed for the case of winds from cv . we now consider whether the dynamical structures and mass loss rates predicted by them are likely to be appropriate . the primary evidence for the existence of winds in cvs is contained within ultraviolet observations of high - state non - magnetic systems ( dwarf novae in outburst and nova - like variables ) . in low inclination non - eclipsing systems , the profiles of the stronger resonance lines include broad blueshifted absorption indicating outflow . the maximum expansion velocities inferred are on the order of a few thousand km s@xmath140 and are thus comparable with the typical white dwarf escape velocity . a point of contrast between the line profile shapes seen in ob stars and cvs , is that deepest absorption is achieved near terminal velocity in the former , but near line centre in the latter ( e.g. see data presented by prinja & rosen 1995 ) . in high inclination eclipsing systems , the p cygni absorption is replaced by broad high contrast line emission . the order of magnitude decrease in expansion speed with respect to a spherically - symmetric ms star wind ( @xmath221 km s@xmath140 ; howarth & prinja 1989 ) is particularly significant . associated with the changed mass flux and the restricted opening angle of the outflow is a density that can be up to @xmath1100 times higher than would be expected of a spherically - symmetric stellar wind the efficiency of hi line emission would presumably rise by a still larger factor . all of these effects are substantive changes in the right direction , suggesting that a radiation - driven disk wind model for massive ysos is worth further investigation . adopting a mass accretion rate consistent with values inferred from observations of high - state cv ( i.e. @xmath222 m@xmath108 yr@xmath140 , e.g. warner 1987 ) leads us to consider a qualitative comparison of model 3 with observation . this model is characterised by complex dense flow near the equatorial plane bounded by a fast stream ( see figure 10a ) . typical velocities in these two components are @xmath223 km s@xmath140 and @xmath224 km s@xmath140 respectively . on viewing such an object at low inclination , we would expect to see high - velocity blueshifted absorption due to the fast stream combined with a substantial low - velocity absorption component originating in the more slowly churning equatorial gas . at high inclinations the low - velocity component should still be apparent in absorption , while the high velocity gas will appear in emission if it is no longer seen in projection against the bright inner disk . hence both at low inclination and at high inclination , the kinematic structure of the model outflow appears to be capable of matching the characteristics indicated by observation . clearly , it will be appropriate to confirm qualitative impression of agreement by carrying out detailed line profile synthesis based on these models . when this is undertaken , it may well be appropriate to think again about the boundary condition currently imposed at the surface of the white dwarf . in the interests of simplicity we have thus far ignored the possibility that there may yet be a significant component of boundary layer emission between the disk and star , and have not allowed any mass loss from the star itself . in a crude way , the @xmath175 models give some idea as to what impact the presence of a hot white dwarf and non - planar boundary layer might have . mass loss from the star could very well add significantly to the total column contained within the fast stream particularly if the star is allowed to rotate at a significant fraction of break - up . these are , however , issues that amount to the introduction of further free parameters that should be faced in the future , rather than taken on board now , at the outset . if , as our models suggest , there is an equatorial zone of complex time - dependent flow , there are consequences of this that may be directly observable . we find the flow varies on timescales of order of the local orbital period , i.e. a few tens of seconds in the vicinity of a white dwarf . if this behaviour is present in real systems and gives rise to a granularity on a spatial scale not too small compared with the total extent of the effective resonance line - forming region , we can expect the low velocity absorption component to vary on this timescale . this prediction is just within the realms of testability using highly time resolved hst spectra . the effect may be looked for both in high and low inclination systems we now come to the question of the comparison between model and observed mass loss rates . we find for @xmath225 m@xmath108 yr@xmath140 that the mass loss rate in the wind is @xmath226 m@xmath108 yr@xmath140 for @xmath113 , rising to almost @xmath227 m@xmath108 yr@xmath140 for @xmath202 ( see table 2 and figure 8) . the reason for this sensitivity is that the higher value of @xmath0 causes the force multiplier to achieve its maximum value earlier in the flow . observational lower limits based on profile fitting uncorrected for unknown ion abundances span much the same range ( e.g. drew 1997 , prinja & rosen 1995 ) . estimates based on ionization models require @xmath228 in the region of a few percent ( hoare & drew 1993 ) . in view of our expectation that the present calculations are liable to underestimate the wind mass loss , this initial comparison is very encouraging indeed . however , it is also true that there is yet much work to be done to determine internally consistent choices for the parameters @xmath13 and @xmath0 that control the radiation force multiplier . thus far we have just used values typical for single hot stars . lastly , we note that the mass loss is modelled as showing a sharp cut - off as the product of the total luminosity and maximum force multiplier decreases below a critical value . presently this is twice the eddington limit , and translates at small @xmath114 and @xmath113 into @xmath229 @xmath116 . there is a parallel to this behaviour in ultraviolet observations of dwarf novae undergoing outburst , where it has been noted that p - cygni absorption features are apt to disappear very suddenly as the decline from maximum light begins . a good example of this was seen early in a decline of su uma ( woods , drew & verbunt 1990 ) , when a factor of 2 decrease in the uv continuum erased what had been prominent blueshifted absorption in civ 1549 and other lines at maximum . another aspect of this is that different systems apparently present very different levels of mass loss , despite the expectation that the high state viscosity , and hence mass accretion rates , can not vary by more than a factor of a few ( e.g. compare and contrast the weak blueshifted absorption features in ss cyg , during outburst , with the extremely strong features in rw sex , a nova - like variable , prinja & rosen 1995 ) . ultimately this effect will provide a useful quantitative calibration of radiation driven disk wind models against observation . for the timebeing , it is again encouraging that the cut - off occurs at a mass accretion rate comparable with those believed to be attained during outburst . the models presented in this paper have all been calculated for white dwarf accretion disks . we show below how our models might be scaled to produce guideline mass loss rates and expansion velocities for other applications . we introduce a set of primed dimensionless variables . first , it is natural to scale lengths to the stellar radius , @xmath43 : @xmath230 and define the unit time @xmath231 , as earlier , such that @xmath232 the unit velocity is accordingly @xmath233 . for the white dwarf case this is @xmath234 . translational velocity and the sound speed then become : @xmath235 and @xmath236 in our models , @xmath237 . the eddington factor expressed in terms of just the disk luminosity is @xmath238 where @xmath239 is the thompson scattering cross - section divided by the mass of the hydrogen atom . using these new variables , the equation of motion can be rewritten in the dimensionless form : @xmath240 where the scaling to a dimensionless density via @xmath241 is trivial . this equation has three parameters : @xmath242 , @xmath243 , @xmath114 , and depends on one dimensionless function the locally - determined force multiplier , @xmath244 . we can approximate and hence simplify this somewhat . first , for many cases of interest , the electron scattering terms will be of minor importance compared to the line acceleration terms and so may be neglected . second , we may conclude from the empirical absence of a dependence upon @xmath114 in the relations between total luminosity and either mass loss rate or outflow velocity ( figs . 8b & 9b ) that the disk and cs driving terms can be combined to yield : @xmath245 wherein @xmath246 is a factor encompassing all the geometric aspects of the radiation force calculation . a priori it was not possible to assume that the dynamics might be reducible to such a form . it only remains to provide a scaling to allow mass loss rates to be estimated for other applications . this can be extracted from the definition of the dimensionless eddington factor , @xmath243 , in that we can define a fiducial mass time derivative such that @xmath247 . the mass loss rate will then scale as @xmath248 in figs . 8b and 9b we provide as alternate ordinates the quantities @xmath249 and @xmath250 in order to facilitate rescaling of our results to other contexts for which @xmath251 and @xmath252 can be estimated . although the models presented in this paper have been motivated primarily by observations of cvs , there are clearly other astrophysical systems to which our results may be relevant . here we discuss just two such cases : accretion disks associated with active galactic nuclei ( agn ) , and massive young stellar objects . the presence of broad , blueshifted absorption lines in quasar spectra ( osterbrock 1989 ) is often interpreted as evidence for a line - driven disk wind . recently , murray et al ( 1995 ) have constructed dynamical models for such winds based on the solution of the one - dimensional ( radial ) equation of motion subject to certain assumptions about how the gas is loaded onto radial streamlines via vertical motions . it is not clear whether these assumptions lead to a good representation of the streamlines in a fully two - dimensional solution such as presented here . the bulk of the radiative flux in quasars comes from or near the central source , implying our models with very large values for the parameter @xmath114 should be most appropriate to these systems . the most extreme value of @xmath114 we have considered is 10 ( model 14 ) . in it , we find strongly radial flow confined to angles of less than 30 degrees from the disk midplane with little time - dependence . however , the radiation from the central source in the agn case is very much harder than that produced locally in the disk and , as the former increasingly dominates over the latter with increasing height above the disk photosphere , it is plausible that the force multiplier would become a function of position to reflect this ( see vitello & shlosman 1988 , murray et al . this is not an effect that our present models include . at least a simplified treatment of the photoionization and recombination of the wind material is required before the two - dimensional structure of quasar winds can be examined self - consistently . in the case of high mass young stellar objects , e.g. the bn - type objects and herbig be stars , photoionization effects are not an overriding concern in that the literature already contains force multiplier parameters designed for the appropriate effective temperature range . a more fundamental issue is the nature and extent of their circumstellar disks , as this can not yet be said to have been defined compellingly . that disks of some kind are present has been entertained by many ( simon et al . 1985 ; hamann & persson 1989 ; chandler , carlstrom & scoville 1995 to mention a few ) . a major phenomenological challenge of these objects is the dynamical origin of their often extremely bright , yet modestly velocity - broadened ( @xmath253 km s@xmath140 ) hydrogen line emission . if , like classical t tau stars , these objects are in an active accretion phase , the ratio of stellar to disk luminosity may not be too extreme . for instance , an early b star accreting at a rate of @xmath254 m@xmath108 yr@xmath140 would be described by @xmath255 . since , for @xmath256 , the disk s light is dominated by the reprocessed component , there is no susbtantive difference between @xmath257 and any higher value of @xmath114 . thus , our model 14 with @xmath257 may again be crudely indicative of the outflow geometry we might expect for such systems . the expectation is therefore that the outflow would be equatorial and steady . a more interesting point , however , is that the flow is very likely to be very much more dense and significantly less rapidly expanding than a conventional early - type stellar wind . specifically , the effective eddington number ( @xmath258 ) for an early b star is likely to be in the region of 20 or so , while the scaling variables , @xmath252 and @xmath251 ( section 5.3 ) , are respectively @xmath259 km s@xmath140 and @xmath260 @xmath116 . these numbers combine with the results in figs . 8b and 9b to yield mass loss rates estimates in excess of 10@xmath261 @xmath116and maximum expansion velocities of @xmath259 km s@xmath140 ( i.e. @xmath262 ) . this amounts to an order of magnitude increase in @xmath188 and a factor of a few decrease in expansion speeds with respect to a spherically symmetric ms star wind ( @xmath263 m@xmath108 yr@xmath140 , @xmath264 km s@xmath140 ; howarth & prinja 1989 ) . the net impact of both these differences and the restricted opening angle of the outflow could be to raise the density , with respect to a normal ms stellar wind , by a factor of a few tens and the efficiency of h i line emission by a factor of 1001000 perhaps . all of these effects are substantive changes in the right direction , suggesting that a radiation - driven disk wind model for massive ysos is worthy of further investigation . there are a number of limitations of the present analysis which are worthy of mention and further investigation . perhaps the most important relate to the approximations adopted here to represent the radiation force . we have already discussed , in section 4.4 , the tests we have performed to check the sensitivity of our models to an improved representation of the general line - driving force in a multidimensional wind . based on these tests , we conclude it is unlikely that the major results of this paper ( for example , the two - dimensional geometry of line - driven winds from disks , or the existence of unsteady behavior in low luminosity systems ) will change with a formalism which includes all terms in the radiation force on lines . however , quantities such as the mass loss rate and terminal velocity reported here should only be considered accurate to factors of a few . here , we also wish to point out that in a rotating wind there are azimuthal forces even in axisymmetry ( because not all terms in the velocity gradient projected along the line of sight @xmath265 are symmetric in @xmath41 , see equations 8 and a2 ) ; these forces may change the angular momentum of the gas and effect the dynamics of the wind . while we expect such effects to be small , we have yet to study them in detail . we shall report the results of our calculations using a more general treatment of the radiation force on lines in a two - dimensional , rotating wind in a future communication . of course , a more fundamental concern is whether the sobolev approximation should even apply in principle to the multidimensional and time - dependent flows considered here . in adopting the sobolev approximation , we have ignored non - local radiative transfer effects . because the velocity field in some of the models reported here is neither monotonic nor steady , non - local effects such as shadowing can be expected to affect the solutions . a proper study of these effect requires the use of algorithms for multidimensional transport of line radiation in a rotating wind , which is beyond the scope of the present work . however , it may be anticipated that the inclusion of shadowing would have a similar effect to increased @xmath114 for the reason that shadowing should mostly reduce the driving of the slow equatorial component and have little impact on the relatively well - organized fast boundary stream . other effects which bear further investigation are the inclusion of mass loss from the central star . this however requires a realistic prescription for the properties of the radiation field and gas flow in the interaction region ( boundary layer ) between the cs and accretion disks . since magnetic fields are likely to be central to the production of angular momentum transport in accretion disks ( balbus & hawley 1997 ) , it would also be fruitful to consider the effect of a global magnetic field anchored in the disk on the properties of the wind . finally , we have considered the two - dimensional structure of winds assuming an isothermal equation of state . thermal pressure effects can be expected to be important only in the subsonic acceleration zone , which we find is generally small in spatial extent . nevertheless , we have not modeled the transition between the optically - thick ( and therefore adiabatic ) gas inside the disk , and the optically thin wind above . in principle , the radial variation in the internal structure of the disk caused by the radial variation in temperature might affect conditions at the base of the wind . this inadequacy is not so serious given that the main seat of the outflow is the relatively small area of the innermost disk ( @xmath266 ) . clearly dynamical models which consider the internal magnetohydrodynamics of an optically thick , turbulent accretion disk ( brandenburg et al 1995 ; stone et al 1996 ) and radiation pressure on spectral lines in the wind region above the disk are the most appropriate description of real disks ; such models await future studies . using numerical methods to solve the two - dimensional , time - dependent equations of hydrodynamics , we have studied radiation driven winds from luminous accretion disks . in so doing we have accounted for the radiation force mediated by spectral lines using a generalized multidimensional formulation of the sobolev approximation . our primary conclusions are the following . \(1 ) we find radiation driven winds from luminous accretion disks are intrinsically unsteady : the outflow consists of large amplitude density and velocity fluctuations , with some regions of dense material undergoing infall . this behavior is rooted in the difference in the variation with height of the vertical component of gravity and the radiation force . since the former increases , it grows until it overwhelms the radiation force , causing high density material ( in which the radiation force per unit mass is low ) to stall . despite the fact that instantaneous values in the wind are variable , time - averaged values are constant . \(2 ) the contribution of a strong radial component to the driving radiation field from a bright central star serves to ` organize ' the outflow into a steady state . very bright central stars produce steady transsonic disk winds . moreover , the region producing unsteady outflow is reduced as the luminosity of the disk is increased . \(3 ) regardless of whether the flow is steady or unsteady , we find the time - averaged geometry of the flow typically consists of a dense , nearly equatorial , low velocity flow confined to angles within @xmath267 to @xmath146 of the equatorial plane , bounded by a lower density , high velocity flow in a channel at larger angles . in the absence of a wind directly from the central , star the gas density in the polar regions is so low as to be of no dynamical or observable significance . most of the mass loss occurs within a few stellar radii of the central star . \(4 ) the geometry of the radiation field is a major factor in controlling the geometry of the outflow . increasing the luminosity of the star at a fixed disk luminosity produces a radial wind confined to smaller regions near the equatorial plane . conversely , increasing the disk luminosity at a fixed stellar luminosity produces a more polar outflow . \(5 ) the total mass loss rate and terminal velocity in the wind depends on the total luminosity of the star plus disk system , but is insensitive to the outflow geometry or whether the wind is steady or unsteady . no outflow is produced if the effective luminosity of the disk ( that is , the luminosity of the disk times the maximum value of the force multiplier associated with the line - driving force ) is less than the eddington limit . above the eddington limit , the mass loss rate in the wind scales with the effective luminosity as a power law with index of about 1.5 . the effective luminosity can be increased either by increasing the accretion rate in the disk , or by increasing the brightness of the central star . the ratio of the mass loss rate in the wind to the accretion rate increases sharply , reaching a few percent for the most luminous disks considered . this study has been motivated primarily by high resolution spectroscopic observations of winds from disks in cv systems . the overall structure of disk winds revealed by our calculations , i.e. a dense equatorial wind with a fast polar outflow , appear to be in agreement with the kinematics inferred for real systems . furthermore , the magnitude of the mass loss rates obtained on adopting a force multiplier parameterisation known to be applicable to ob stars overlaps the range that has been deduced from observation . we plan more detailed comparison of line profiles computed from our models with observational data in the future . future applications also include high mass ysos with circumstellar disks ( in which case outflow from the central star must also be allowed ) , and agn ( in which case photoionization of the wind by the central source must be taken into account ) . * acknowledgments : * this research has been supported by a research grant from pparc , and by nasa through hst grant go-6494 . computations were performed at the pittsburgh supercomputing center . we use a spherical polar coordinate system with an origin at point c , the center of the cs . colatitude ( @xmath48 ) is measured from the rotation axis of the disk , and azimuth ( @xmath41 ) is measured from a plane perpendicular to the disk plane , containing the point c and a point w above the disk ( see figure a.1 ) . we define the location of a wind point , w , and a disk point , d , by the co - ordinates @xmath268 and @xmath269 respectively . the distance between d and w then is @xmath270 where @xmath271 is the angle wcd and @xmath272 . the direction d toward w can be defined by the unit vector @xmath273 . using the coordinates of points d and w , @xmath274 the intensity of an @xmath0-disk at point d is ( e.g. , pringle 1981 ) @xmath275 where @xmath276 and @xmath43 are the mass and radius of the central star , @xmath187 is the accretion rate through the disk ( shakura & sunayev 1973 ) . the total luminosity of an @xmath0-disk is @xmath277 in the presence of a luminous cs , the intensity radiated by an optically - thick @xmath0-disk changes due to heating of the disk by the cs radiation . to calculate this illumination effect , it is convenient to describe the location of the cs surface point s in a spherical polar coordinate system @xmath278 in which the origin is at the point d. the colatitude @xmath279 , is now measured from the dc axis and the azimuth @xmath280 , is measured from the plane perpendicular to the disk surface that contains both the points d and c. expressing the central star luminosity in @xmath281 units @xmath282 and assuming that the cs surface is isothermal , the cs intensity then is @xmath283 the stellar energy absorbed per unit time by a surface element of the disk is @xmath284 where , @xmath285 . assuming that the disk reemits the absorbed energy locally as a black body , the disk intensity due to irradiation can be written as @xmath286 thus , using equations a3 , a6 and a8 , we can express the intensity of the steady state disk illuminated by the cs as @xmath287 finally , at a point w , the radiation flux from a disk surface element between ( @xmath288 ) and ( @xmath289 ) is @xmath290 where @xmath291 is the projection of w on the disk plane ( @xmath292 ) . ( 80,300 ) ( 0,0 ) to calculate the radiation flux from the cs at a point w , we describe the location of a point s in a spherical polar coordinate system whose origin is at the centre of the star ( point c ) . the angle of colatitude @xmath293 is measured from the cw axis , while the azimuth @xmath294 is measured from the plane perpendicular to the disk surface containing the points w and c. the distance between s and w is then @xmath295 . the unit vector , @xmath296 specifying the direction sw has the following components : @xmath297 at the point w , the radiation flux due to the stellar surface element between ( @xmath298 , @xmath299 ) and ( @xmath300 , @xmath301 ) is @xmath302 the radiation force due to electron scattering , per unit mass , contributed by the disk surface element along @xmath3 is @xmath303 where @xmath239 is the mass scattering coefficient of free electrons . we assume that the mean mass of the particle is equal to the proton mass , @xmath304 . thus @xmath305 , where @xmath306 is the thomson cross section . using equations a9 and a10 we obtain from equation b1 the radiation force per unit mass from the total disk surface acting a point w : namely , @xmath307 where @xmath308 is the vector - valued integral @xmath309 in which the primed quantities are expressed in @xmath43 units . the integration limits , @xmath310 , @xmath311 , and @xmath312 are functions of position because of the need to account for the shadowing of the disk by the cs . the upper limit on the radial integration is always the outer radius of the disk , @xmath313 . the integral with respect to @xmath41 is calculated assuming symmetry about the @xmath314 plane . using the cak formalism ( see 2 ) the radiation force per unit mass , due to spectral lines along @xmath3 , contributed by a disk surface element is @xmath315 where @xmath244 is the force multiplier . following the analogy with @xmath316 , the total radiation force due to lines from the disk is @xmath317 where @xmath318 is the vector - valued integral @xmath319 the radiation force due to electron scattering from a stellar surface element is @xmath320 and using equations a6 and a12 , the angle - integrated force becomes @xmath321 where @xmath322 is the vector - valued integral @xmath323 by analogy with equation b2 , the stellar component of the radiation force due to lines is @xmath324 where @xmath325 finally , the total radiation force per unit mass acting on a particle at point w is @xmath326 or using the vector - valued integrals b3 , b6 , b9 and b11 : @xmath327 to calculate the radiation force from the cs we consider the simple case where we assume that @xmath32 is dominated by terms associated with the radial component of the velocity @xmath328 where @xmath329 ( see rybicki & hummer 1983 ) . if we further assume @xmath330 , i.e. that the star is a point source , then @xmath331 this is exactly the case considered by cak . as we described in 2 , an advantage of this approximation is that , in the calculation of the radiation line force due to the whole star , we can move a time dependent , velocity factor outside the integral in equation b11 : @xmath332 \int_0^{\theta_u } \int_{\phi_i}^{\phi_o } ~\hat{m } \frac{\sin \theta_s ( r \cos \theta_s - r_\ast ) r_\ast^2 } { 3d_\ast^3 } d\theta_s d\phi_s.\ ] ] thus we need to calculate the integral only once at the beginning of the hydrodynamic calculations and update the radiation line force every time step only through @xmath333 . however even in a purely radial wind , time - dependent calculations become very costly if we take into account a star with a finite disk ( for example , see the appendix of cak , pauldrach , puls , kudritzki 1986 , friend & abbott 1986 ) . we also make use of an analogous major simplification of the disk radiation force . assuming that the gradient of the velocity along the vertical direction is the dominant term , equation ( 8) reduces to @xmath334 where @xmath335 . numerically , this form of @xmath32 is constructed from its equivalent form in spherical coordinates : @xmath336 in this case , equation b6 can be expressed as @xmath337 \nonumber \\ & & \left(1- \left(\frac{1}{{r'}_d}\right)^{1/2 } + \frac{x { r'_d}^2}{3\pi}\left(\arcsin \frac{1}{r'_d } - \frac{1}{r'_d } \left(1 - \left(\frac{1}{r'_d}\right)^2\right)^{1/2}\right)\right ) d\phi_d dr'_d .\end{aligned}\ ] ] in the present form we still find the time - dependent @xmath338 term within the integrand . strictly , we can not move the factor within square brackets in front of the integral because @xmath338 is dependent on line - of - sight and hence position on the disk plane . therefore we make an approximation that @xmath339 as far as @xmath338 is concerned . introducing a new variable , @xmath340 , equation c6 can be rewritten @xmath341 \int_0^{\phi_u } \int_{r_i}^{r_o } { \hat{n}}~ \frac{r'~\cos~\theta } { { r'_d}^2 { d'_d}^3 } \left(\frac{r ' \cos\theta}{d'_d}\right)^{2\alpha}\nonumber \\ & & \left(1- \left(\frac{1}{{r'}_d}\right)^{1/2 } + \frac{x { r'_d}^2}{3\pi}\left(\arcsin \frac{1}{r'_d } - \frac{1}{r'_d } \left(1 - \left(\frac{1}{r'_d}\right)^2\right)^{1/2}\right)\right ) d\phi_d dr'_d . \end{aligned}\ ] ] now both the time - dependent factors appear outside the integral . note that our approximation gives exactly the same result as equation c7 for @xmath342 because then the factors in square brackets are unity in both equations c6 and c7 ( see equation 5 ) . for @xmath343 , equation c7 gives values lower than c6 because the integrand in equation c7 is smaller than the integrand in equation c6 by a factor @xmath344 . however for a given @xmath18 , @xmath343 occurs when @xmath14 is very small . small @xmath14 will mainly be associated with regions high above the disk where the gas density is low . bearing in mind that a foreshortened disk element at low @xmath345 contributes at low weight compared to an element with @xmath346 , we can see that equation c7 is quite a reasonable approximation of equation c6 even for @xmath343 . abbott d.c . 1982 , apj , 259 , 282 batchelor g.k . 1967 , an introduction to fluid mechanics ( cambridge : cambridge university press ) balbus , s.a . , & hawley , j.f . , 1997 , rev . , in press . brandenburg a. , nordlund a. , stein r.f . , torkelsson u. 1995 , apj , 446 , 741 carroll b.w . , cabot w. , mcdermott p.n . , savedoff m.p . , van horn h.m . 1985 , apj , 296 , 529 castor j.i . , abbott d.c . , & klein r.i . 1975 , apj , 195 , 157 ( cak ) chandler c. j. , carlstrom j. e. , scoville n. z. 1995 , apj , 446 , 793 cox j.p . , & everson b.l . 1980 , apjss , 52 , 451 drew j.e . 1997 , in _ accretion phenomena and related outflows _ , eds . d. wickramasinghe , l. ferrario , g. bicknell , asp conf . sers , in press friend d.b . , abbott d.c . 1986 , apj , 311 , 701 gayley k. g. 1995 , apj , 454 , 410 hamada t. , salpeter e.e . 1961 apj , 134 , 683 hamann f. , persson s. e. 1989 , apj , 339 , 1078 hoare m.g . , & drew j.e . 1993 , mnras , 260 , 647 howarth i.d . , prinja r.k . 1989 apjs , 69 , 527 icke v. 1980 , aj , 85 , 329 icke v. 1981 , apj , 247 , 152 knigge c. , drew j.e . 1997 , apj , in press lin , d.n.c . , papaloizou , j.c.b . , & savonije , g.j . 1990 , apj , 364 , 326 lucy , l.b . , & solomon , p. 1970 , apj , 159 , 879 mason , k.o . , drew j.e . , cordova f.a . , horne k. , hilditch r. , knigge c. , lanz l. , & meylan t. 1995 , mnras , 274 , 271 mundt r. , ray t.p . 1994 , in _ the nature and evolutionary status of herbig ae / be stars _ , eds . th , m. r. prez , e.p.j . van den heuvel , asp conf.sers . 62 , 237 murray , n. , chiang , j. , grossman , s.a . , & voit , g.m . 1995 , apj , 451 , 498 owocki s.p . , castor j.i . , rybicki , g.b . 1988 , apj , 335 , 914 osterbrock , d.e . , 1989 , astrophysics of gaseous nebulae and active galactic nuclei ( mill valley : university science books ) pauldrach a. , puls j. , & kudritzki r.p . 1986 , a&a , 164 , 86 pereyra n.a . , kallman t.r . blondin j.m . 1997 , apj , 477 , 368 pereyra , n.a . , 1997 . thesis , university of maryland . prinja , r.k . , & rosen , r. 1995 , mnras , 273 , 461 pringle j.e . 1981 , araa , 19 , 137 proga , d. , drew , j.e . , & stone , j.m . 1997 , in preparation . puls , j. , owocki , s.p . , & fullerton , a. 1993 , a&a , 279 , 457 rybicki g.b . , & hummer d.g . 1978 , apj , 219 , 654 rybicki g.b . , & hummer d.g . 1983 , apj , 274 , 380 shakura n.i . , sunyaev r.a . 1973 a&a , 24 , 337 simon m. , peterson d. m. , longmore a. j. , storey j. w. v. , tokunaga a. t. 1985 , apj , 300 , 32 stevens i.r . , & kallman t.r . 1990 , apj , 365 , 321 stone j.m . , & norman m.l . 1992 , apjs , 80 , 753 stone j.m . , hawley j.f . , gammie c.f . , & balbus s.a . 1996 , apj , 463 , 656 vitello p.a.j . , & shlosman i. 1988 , apj , 327 , 680 warner b. 1987 mnras , 227 , 23 warner b. 1995 , cataclysmic variable stars , ( cambridge : cambridge university press ) weymann r.j . , morris s.l . , foltz c.b . , hewett p.c . 1991 , apj , 373 , 23 woods j.a . , drew j.e . , & verbunt f. 1990 , mnras , 245 , 323

we study the two - dimensional , time - dependent hydrodynamics of radiation - driven winds from luminous accretion disks in which the radiation force is mediated primarily by spectral lines . we assume the disk is flat , keplerian , geometrically thin , and optically thick , radiating as an ensemble of blackbodies according to the @xmath0-disk prescription . the effect of a radiant central star is included both in modifying the radial temperature profile of the disk , and in providing a contribution to the driving radiation field . angle - adaptive integration techniques are needed to achieve an accurate representation of the driving force near the surface of the disk . our hydrodynamic calculations use non - uniform grids to resolve both the subsonic acceleration zone near the disk , and the large - scale global structure of the supersonic wind . we find that line - driven disk winds are produced only when the effective luminosity of the disk ( i.e. the luminosity of the disk times the maximum value of the force multiplier associated with the line - driving force ) exceeds the eddington limit . if the dominant contribution to the total radiation field comes from the disk , then we find the outflow is intrinsically unsteady and characterised by large amplitude velocity and density fluctuations . both infall and outflow can occur in different regions of the wind at the same time . the cause of this behaviour is the difference in the variation with height of the vertical components of gravity and radiation force : the former increases while the latter is nearly constant . on the other hand , if the total luminosity of the system is dominated by the central star , then the outflow is steady . in either case , we find the two - dimensional structure of the wind consists of a dense , slow outflow , typically confined to angles within @xmath145 degrees of the equatorial plane , that is bounded on the polar side by a high - velocity , lower density stream . the flow geometry is controlled largely by the geometry of the radiation field a brighter disk / star produces a more polar / equatorial wind . global properties such as the total mass loss rate and terminal velocity depend more on the system luminosity and are insensitive to geometry . the mass loss rate is a strong function of the effective eddington luminosity ; less than one there is virtually no wind at all , whereas above one the mass loss rate in the wind scales with the effective eddington luminosity as a power law with index 1.5 . matter is fed into the fast wind from within a few stellar radii of the central star . our solutions agree qualitatively with the kinematics of outflows in cv systems inferred from spectroscopic observations . we predict that low luminosity systems may display unsteady behavior in wind - formed spectral lines . our study also has application to winds from active galactic nuclei and from high mass ysos . epsf hydrodynamics instabilities methods : numerical accretion discs stars : mass - loss cataclysmic variables

over the last few decades uo@xmath0 has been one of most widely studied actinide oxides due to its technological importance as standard fuel material used in nuclear reactors . there exists currently considerable interest in understanding the behavior of nuclear fuel in reactors which is a complex phenomenon , influenced by a large number of materials properties , such as thermomechanical strength , chemical stability , microstructure , and defects . especially , knowledge of the fuel s thermodynamic properties , such as specific heat , thermal expansion , and thermal conductivity , is essential to evaluate the fuel s performance in nuclear reactors.@xcite these thermodynamic quantities are directly related to the lattice dynamics of the fuel material.@xcite dolling _ et al . _ @xcite were the first to measure phonon dispersion curves of uo@xmath0 , using the inelastic neutron scattering technique in 1965 ; their seminal article has become the standard reference for uranium dioxide s phonon spectrum . later the vibrational properties of uo@xmath0 were investigated in detail by schoenes,@xcite using infrared and raman spectroscopic techniques . a good agreement with phonon frequencies obtained from inelastic neutron scattering was observed . @xcite more recently , livneh and sterer @xcite studied the influence of pressure on the raman scattering in uo@xmath0 and livneh @xcite demonstrated the resonant coupling between longitudinal optical ( lo ) phonons and u@xmath1 crystal field excitations in a raman spectroscopic investigation . a theoretical investigation of the phonon spectra of uo@xmath0 was reported recently by yin and savrasov @xcite who employed a combination of a density - functional - theory ( dft ) based technique and a many - body approach . according to their results , the low thermal conductivity of uo@xmath0 stems from the large anharmonicity of the lo modes resulting in no contribution from these modes in the heat transfer . @xcite investigated the phonon properties of uo@xmath0 using an empirical interatomic potential based on the shell model and observed that the calculated thermodynamic properties including the specific heat are in good agreement with available experimental data . devey @xcite employed recently the generalized gradient approximation with additional coulomb @xmath2 ( gga+@xmath2 ) to compute the main phonon mode frequencies at the brillouin zone center which were in reasonable agreement with experimental data . very recently , sanati _ et al._@xcite used the gga and gga+@xmath2 approaches to investigate phonon density of states and elastic and thermal constants , which were found to be in reasonably good agreement with experimental data . in spite of the already performed studies , further investigations are needed . especially , the full dispersions of the phonons in reciprocal space have not yet been considered . also , important quantities such as the thermal expansion coefficient and heat capacity are directly related to the lattice vibrations but these quantities have not yet been studied _ ab initio _ from the calculated phonon spectrum . the objective of this study is to contribute to a detailed understanding of the lattice vibrations of uo@xmath0 . using the first - principles approach , based on the dft we have calculated phonon dispersion curves and phonon density of states of uo@xmath0 . the calculated phonon properties are compared with the available experimental data from inelastic neutron scattering and raman spectroscopy along with a detailed discussion . furthermore , several thermodynamic properties have been computed taking the influence of lattice vibrations into account . here , we report the lattice contribution to the heat capacity as function of temperature as well as temperature and volume ( in the quasiharmonic approximation ) . the dependence of the total free energy on the lattice constant of uo@xmath0 as a function of temperature has calculated , from which we derive the thermal expansion coefficient . the thermal expansion coefficient as well as lattice heat capacity compare favorably to available experimental data up to 500 k , which is the temperature range in which the influence of anharmonicity can be neglected . the electronic structure of uo@xmath0 has been discussed in the past years . @xcite dft calculations within the generalized gradient approximation ( gga ) underestimate the influence of the strong on - site coulomb repulsion between the @xmath3 electrons . an improved @xmath3 electronic structure description can be obtained with the gga+@xmath2 approach , in which a supplementary on - site coulomb repulsion term is added ; this approach correctly gives the electronic band gap of uo@xmath0.@xcite while the gga+@xmath2 approach would appear preferable for description of uo@xmath0 s electronic structure , we encountered specific problems when using this method . some of the phonon branches became negative away from the zone center . this artifact might be related to the occupation matrix of @xmath3 states that would require an additional stabilizing constraint in the gga+@xmath2 method.@xcite using conversely the spin - polarized gga approach , we found that such difficulties did not occur . the phonon dispersion spectrum presented below is hence computed with the gga exchange - correlation for antiferromagnetically ordered uo@xmath0 and is found to be in good agreement with experiment.@xcite here , we have determined the phonon dispersion curves and density of states ( dos ) in the quasiharmonic approximation using the direct method . @xcite by displacing one atom in a supercell ( of 96 atoms ) from its equilibrium position , non - vanishing hellmann - feynman forces were generated . due to the high symmetry of the face - centered cubic ( fcc ) lattice of uo@xmath0 , only one atom for uranium ( u ) and for oxygen ( o ) was needed to be displaced . the actual shift of the atoms in the supercell had an amplitude of 0.03 and was taken along the [ 001 ] direction only , on account of the cubic symmetry of uo@xmath0 . in the calculation of the resulting forces we employed the projector augmented wave ( paw ) pseudopotential approach within the vienna ab - initio simulation package ( vasp).@xcite the phonon code @xcite has been used to extract the force constant matrix from the hellmann - feynman forces and to subsequently calculate the phonon dispersion curves and dos . for the thermodynamic quantities we consider the total free energy of uo@xmath0 , including the phonon contribution , @xmath4 where @xmath5 is the helmholtz free energy at a given strain @xmath6 . the phonon free energy contribution @xmath7 is expressed as @xmath8 , \label{eqn : phononenergy}\end{aligned}\ ] ] where @xmath9 is the phonon dos , computed as mentioned above . we note that the free electronic energy , @xmath10 , is not considered in the present study , because the thermal electronic contribution is known to be negligible in the temperature range up to 1000k , which is the range of interest in this work.@xcite the static lattice energy @xmath11 ) appearing in eq . ( [ eqn : helmholtz ] ) can be expressed as @xmath12 where @xmath13 is the static lattice energy at zero strain , @xmath14 are the elastic constants , and @xmath15 is the equilibrium volume at @xmath16 k. the static lattice energies have also been calculated using the vasp code.@xcite in our calculations we have used a 2@xmath172@xmath172 supercell containing 96 atoms with a 4@xmath174@xmath174 @xmath18-point mesh in the brillouin zone ( bz ) . the perdew - wang parametrization@xcite of the gga functional was used . the kinetic energy cut - off for the plane waves was set at 600 ev and the energy criterion used for convergence was 10@xmath19 ev . the force acting on each ion was converged until less than 0.01 ev / . once the phonon dos has been calculated , the thermal expansion of uo@xmath0 can be evaluated straightforwardly . first , the phonon dos with static lattice energy is calculated for several volumes around the @xmath20 k equilibrium volume . subsequently , the total free energies are calculated for these different volumes at constant temperature using eqs . ( [ eqn : helmholtz])-([eqn : staticenergy ] ) . after the free energy has been calculated its minimum gives the corresponding equilibrium volume at the considered temperature . by repeating the process for different temperatures , the thermal expansion coefficient @xmath21 defined by @xmath22 is obtained ; here @xmath23 is the lattice constant . a further thermodynamic quantity , the lattice contribution to the specific heat can be derived from @xmath24 at a fixed temperature in the quasiharmonic approximation . uo@xmath0 crystallizes in the cubic fluorite structure ( caf@xmath0 ) belonging to the @xmath25 space group ( no . 225 ) and there are three atoms per primitive unit cell with one u atom ( wyckoff position @xmath26 ) and two inequivalent o atoms ( wyckoff positions @xmath27 ) . therefore , there are generally nine phonon branches . before turning to the description and analysis of the calculated lattice dynamics let us briefly consider the ground - state properties of uo@xmath0 . as mentioned above the employed dft framework is that of the spin - polarized gga . the calculated equilibrium lattice constant _ a _ and bulk modulus _ b _ , which we have obtained by a birch - murnaghan @xmath28 order fit,@xcite are presented in table [ bulk ] , where these are compared to experimental lattice properties.@xcite our calculated equilibrium volume as well as the bulk modulus compare reasonably well with the experimental data and with results from molecular dynamics simulations.@xcite compared to the experimental lattice constant the lattice constant computed here is 1.2% smaller . this can be attributed to a too strong binding of the @xmath3 orbitals which become too much delocalized in the spin - polarized - gga approach . in the gga+@xmath2 approach the @xmath3 orbitals are more localized and their contribution to the bonding reduced , @xcite which leads to a theoretical lattice parameter which is larger than the experimental one.@xcite .calculated equilibrium lattice constant _ a _ ( in ) and bulk modulus _ ( in gpa ) of uo@xmath0 . theoretical values obtained in this work are compared to values from molecular dynamics simulations@xcite ( md ) , as well as to experimental data@xcite ( exp . ) . [ cols="<,<,^",options="header " , ] computed for three lattice constants @xmath29 , 5.406 , and 5.406 , compared to the measured@xcite dos ( for @xmath30 , at @xmath31=296 k ) . [ fig2 ] ] figure [ fig2 ] shows the calculated and measured phonon dos . the blue , red , and green lines with square , triangle , and diamond symbols indicate the phonon dos computed for @xmath23=5.330 , 5.406 , and @xmath32 , respectively . the experimental data@xcite ( @xmath33 , @xmath31=296k ) are plotted with the black line . the u contribution to the calculated phonon doss gives rise to a higher intensity and narrower peak widths in the lower frequency region . the more broadened dos with lower intensity that occurs in the higher frequency region is mainly derived from the oxygen atoms . a notable difference between the phonon dos at the three lattice parameters is the size and position of the phonon gap occurring for frequencies of about @xmath34 thz . for the experimental lattice parameter @xmath35 the computed gap practically closes . the experimental phonon dos spectrum at this lattice parameter shows a minimum at about 6 thz , in reasonable agreement , considering some experimental broadening . we note that the trend of decreasing gap with larger lattice constant continues , leading to a closing of the gap computed for larger lattice constants ( not shown ) . overall , the computed phonons dos of both the theoretical equilibrium ( @xmath36 ) and the experimental lattice parameter are in good accordance with the measured spectrum . the phonon dos of the theoretical lattice parameter agrees best with the experimental data at higher frequencies ( 7 to 13 thz ) , where the peaks coincide with the measured ones . as mentioned earlier , the lo2 mode lies both lower in the computed spectra and is more dispersive than in the measurements . we note that the recent gga+@xmath2 calculations@xcite provide a sharper dos peak at 17 thz , due to a flatter lo2 dispersion near the zone boundaries . at the zone center the lo2 branch lies however much deeper than in the experiment , at 10 thz ( _ vs. _ 17 thz in experiment ) . the calculated phonon dos enables us to evaluate some thermodynamic quantities which depend on the lattice vibrations . we start with the thermal expansion . the phonon contribution to the total free energy of uo@xmath0 increases with increasing temperature and hence becomes progressively responsible for changes of the lattice parameters . to compute the thermal expansion of uo@xmath0 we have first computed the total free energy , including the phonon contribution , for various lattice parameters , from which we computed the temperature - dependent lattice constant . figure [ fig3 ] ( bottom ) shows the calculated variation of the lattice constant with temperature . the red curve gives the spline interpolation of the calculated lattice constants , shown by the symbols . the thermal expansion coefficient @xmath38 was subsequently evaluated by differentiating the spline fit . the upper panel of fig . [ fig3 ] shows the calculated thermal expansion coefficient , which is in good agreement with experimental data@xcite up to 500 k. the deviation between the calculated and measured data slightly increases above 500 k and becomes significant at around 1000 k. this might be due to an increased electronic contribution to the thermal expansion . at very low temperatures , in the region between 0 and 50 k , a deviation is also observed between the calculated and measured data . the origin of this deviation is not unambiguously clear . we note however that uo@xmath0 undergoes a magnetic phase transition at 31 k ( see ref . ) , which may add an additional influence on the lattice parameter . of uo@xmath0 . the experimental data are those of taylor.@xcite bottom : computed temperature - dependent lattice parameter of uo@xmath0 ( open squares ) and spline fit function . [ fig3 ] ] next , we employ the helmholtz free energy to compute the lattice heat capacity at constant volume ( @xmath39 ) and at constant pressure ( @xmath40 ) . [ hc ] shows theoretical results for @xmath39 , computed with the harmonic approximation , and @xmath40 , computed with the quasiharmonic approximation , as well as experimental results@xcite for @xmath40 up to 1000 k. first the lattice contribution to the specific heat @xmath39 was computed as a function of temperature for the equilibrium lattice constant , @xmath36 . subsequently , the specific heat at constant pressure was derived according to @xmath41 where @xmath42 , @xmath23 , and @xmath43 are the calculated linear thermal expansion coefficient , the equilibrium lattice constant , and the bulk modulus ( see , e.g. , ref . ) . [ hc ] illustrates that there is a very good agreement between the computed heat capacity @xmath40 and the experimental data.@xcite note that the sharp anomaly in the experimental data at @xmath44 k is due to the aforementioned magnetic phase transition,@xcite which effect is not included in the calculations . clearly , the specific heat at constant pressure is in much better agreement with the experimental data at higher temperatures than @xmath39 , which is mainly due to the thermal expansion of uo@xmath0 . conversely , evaluating @xmath40 for the theoretical equilibrium lattice constant or for the experimental lattice constant @xmath35 only gives very minor differences . the computed @xmath40 curves fall somewhat below the experimental @xmath40 data for temperatures in the range of 400 to 1000 k. the uranium atoms in uo@xmath0 are in the paramagnetic state at higher temperatures.@xcite therefore , the remaining difference between experimental and computed data above @xmath45 k could be due to the magnetic entropy contribution to the specific heat or , alternatively , it could be due to the anharmonic effects . the magnetic entropy contribution to the specific heat was investigated for @xmath46 to 300 k in refs . and . at higher temperatures it saturates to approximately @xmath47 , corroborating a magnetic @xmath48 state on the uranium atoms . it provides to a relatively small magnetic entropy contribution that would lead to a small increase of the computed @xmath40 data ( by about 3 jmol@xmath49k@xmath49 ) . at constant volume , @xmath39 , computed within the harmonic approximation ( full curve ) , and a constant pressure , @xmath40 , computed with the quasiharmonic approximation ( dashed lines ; for @xmath36 and the experimental @xmath35 ) . the experimental data for lower and higher temperatures , full circles and full squares , are taken from huntzicker and westrum ( ref . ) and grnvold _ et al . _ ( ref . ) , respectively . ] we have performed first - principles calculations to investigate the lattice vibrations and their contribution to thermal properties of uo@xmath0 . we find that the calculated phonon dispersions are in good agreement with experimental dispersions@xcite measured using inelastic neutron scattering . computing the phonon dispersions for various lattice constants , we observed a softening of the phonon frequencies with decreasing lattice constant . furthermore , the band gap between ta and lo modes at high - symmetry zone - boundary points are found to depend significantly on the volume . this gap almost closes at @xmath35 , consistent with a pseudogap detected in the inelastic neutron experiment . also , raman and infrared active modes have been determined as a function of volume . the agreement with experimental data and with results obtained with molecular dynamics simulations is overall very good , with an exception of the infrared lo mode that appears underestimated in our first - principles calculations . including the phonon contribution to the free energy , the heat capacity and the thermal expansion coefficient of uo@xmath0 have been computed . both thermal quantities are found to agree well with experimental data for temperatures up to 500 k. the good correspondence of the computed and measured thermal data exemplifies the feasibility of performing first - principles modeling of the thermal properties of the important nuclear fuel material uo@xmath0 . this work has been supported by svensk krnbrnsle - hantering ab ( skb ) , the swedish research council ( vr ) , the swedish national infrastructure for computing ( snic ) , and by research project no . cz.1.07/2.3.00/20.0074 of the ministry of education of the czech republic . d. l. thanks p. pavone , university of leoben , leoben , austria , and u. d. wdowik , pedagogical university , cracow , poland for fruitful discussions .

we report first - principles calculations of the phonon dispersion spectrum , thermal expansion , and heat capacity of uranium dioxide . the so - called direct method , based on the quasiharmonic approximation , is used to calculate the phonon frequencies within a density functional framework for the electronic structure . the phonon dispersions calculated at the theoretical equilibrium volume agree well with experimental dispersions . the computed phonon density of states ( dos ) compare reasonably well with measurement data , as do also the calculated frequencies of the raman and infrared active modes including the lo / to splitting . to study the pressure dependence of the phonon frequencies we calculate phonon dispersions for several lattice constants . our computed phonon spectra demonstrate the opening of a gap between the optical and acoustic modes induced by pressure . taking into account the phonon contribution to the total free energy of uo@xmath0 its thermal expansion coefficient and heat capacity have been _ ab initio _ computed . both quantities are in good agreement with available experimental data for temperatures up to about 500k .

many - body systems have an intriguing property : under the right circumstances , local interactions can conspire to produce long - range or global effects . this behavior leads to phase transitions in statistical mechanics , and it also appears in combinatorial problems such as 3-sat . if we consider quantum systems , the situation is more complicated , due to non - commuting measurements and the possibility of entanglement . this leads to new kinds of quantum phase transitions @xcite , and new examples such as the local hamiltonian problem @xcite . a basic question in all of these examples is : if we know local information about various parts of a system , what can we say about the system as a whole ? this paper gives one answer to this question , for quantum systems . suppose we have an @xmath0-qubit system , and we are given a collection of local density matrices @xmath1 , where each @xmath2 describes a subset @xmath11 of the qubits . we say that @xmath1 are `` consistent '' if there exists a global state @xmath3 ( on all @xmath0 qubits ) whose reduced density matrices match @xmath1 ; in other words , for all @xmath12 , @xmath13 . clearly , if @xmath1 are consistent , then whenever two density matrices @xmath2 and @xmath14 describe overlapping subsets of qubits ( @xmath15 ) , they must agree on the intersection @xmath16 ; that is , @xmath17 . this gives a necessary condition for consistency . however , the above condition is not sufficient to guarantee consistency . to see this , consider the following example : we have three qubits , and we are told that qubits 1 and 2 are in the bell state @xmath18 , and qubits 2 and 3 are also in the same state @xmath19 . more formally , let @xmath20 , @xmath21 , and let @xmath22 , @xmath23 . in this case , @xmath24 and @xmath25 both agree on qubit 2 , since @xmath26 . but there is no state @xmath3 on all three qubits such that @xmath27 and @xmath28 ; one way to see this is to apply the strong subadditivity inequality , @xmath29 . thus the consistency of @xmath1 would seem to be a more subtle question . we prove the following result : [ consistent - rhos ] if @xmath1 are consistent with some state @xmath4 , then they are also consistent with a state @xmath5 of the form @xmath6 , where each @xmath7 is a hermitian matrix acting on the qubits in @xmath30 , and @xmath31 . here , @xmath4 means that @xmath3 is a positive definite matrix . the state @xmath5 is known as a gibbs state . essentially , this result says that a gibbs state @xmath5 can simulate an arbitrary state @xmath4 , with respect to an observer who can only access subsets @xmath32 of the qubits . for example , consider a physical system with local interactions , described by a hamiltonian @xmath33 . it is easy to see that the ground state of @xmath33 can be approximated by @xmath34 , for @xmath35 large ; and since @xmath33 is a sum of local terms , @xmath36 is a gibbs state . our result extends this simple observation to a much more general setting . actually , we prove the following more general result : consider a finite quantum system , and let @xmath10 be observables ( hermitian matrices ) . without loss of generality , assume that the collection of matrices @xmath37 is linearly independent ( over @xmath38 ) . we say that a state @xmath39 has expectation values @xmath40 if @xmath41 for all @xmath42 . [ consistent - ts ] if there exists some state @xmath43 which has expectation values @xmath40 , then there exists a state @xmath44 which has the same expectation values @xmath40 , and is of the form @xmath45 , where @xmath46 . this statement holds even when the observables @xmath10 do not commute . this result was previously proved by jaynes , as part of the maximum entropy principle in statistical mechanics @xcite . jaynes showed that the gibbs state @xmath44 is the state which maximizes the entropy @xmath47 subject to the constraints @xmath48 ; implicitly , he also showed that the gibbs state @xmath44 is always feasible , in the sense that it can produce the same expectation values @xmath49 as an arbitrary state @xmath43 . however , jaynes motivation was somewhat different from ours . jaynes was interested in statistical mechanics , which deals with large systems with many degrees of freedom and only a few constraints . feasibility is not usually a concern in such cases , while the maximum - entropy property is crucial in making plausible inferences about the `` true '' state of the system . in this paper , we focus on finite quantum systems , with many non - commuting constraints ; we are interested in the relationship between local constraints and the global state of the system . for us , feasibility of the gibbs state is important , since it is possible for the system to become overdetermined . statistical inference is less important , because the systems we study are small enough that their state can be completely determined ( at least in principle ) . rather than viewing this as an inference problem , we can speak directly about what states are allowed under a given set of constraints . finally , we prove our result using a technique which is different from jaynes . jaynes used the lagrange dual of the entropy - maximization problem , while we use some analytic properties of the partition function . our analysis bears some resemblance to classical results on exponential families in statistics @xcite although the technical details are quite different . our proof also contains some geometric intuition which may be of interest . first , we will review some useful facts about the partition function for a gibbs state . then we will prove theorem [ consistent - ts ] , and obtain theorem [ consistent - rhos ] as a special case . recall the situation described in theorem [ consistent - ts ] : we have a finite quantum system , and observables @xmath10 , such that @xmath37 are linearly independent ( over @xmath38 ) . we are interested in states of the form @xmath50 where @xmath51 . @xmath52 is called the partition function , and we also define the log partition function @xmath53 . note that , in the above definition , we can translate each observable @xmath54 by a multiple of the identity , without changing the state @xmath55 . more precisely , if we define new observables @xmath56 , with @xmath57 , we have that : @xmath58 using subscripts @xmath59 and @xmath60 to denote the two sets of observables , we arrive at the same state , @xmath61 , although the partition functions are different , @xmath62 . the log partition function @xmath63 has some nice analytic properties : it is convex , and its derivatives encode the expectation values of the observables @xmath54 . we briefly sketch these results , which can be found in quantum statistical mechanics @xcite , as well as quantum information geometry @xcite . proof sketch : this follows from some facts in matrix analysis @xcite . first , the golden - thompson inequality : if @xmath65 and @xmath66 are hermitian matrices , then @xmath67 next , a matrix version of hlder s inequality : for any matrix @xmath65 , define the frobenius or hilbert - schmidt norm to be @xmath68 . also , let @xmath69 denote the unique positive semidefinite square root of @xmath70 . then we have that , for all square matrices @xmath65 and @xmath66 , @xmath71 for @xmath72 , @xmath73 . @xmath74 proof sketch : use `` parameter differentiation '' @xcite : if @xmath33 is a hermitian matrix which depends on a parameter @xmath76 , and @xmath77 and @xmath78 exist and are continuous , then @xmath79 exists and is equal to @xmath80 proof : we are given expectation values @xmath40 , and we want to find a state @xmath81 that has these expectation values . ( here , @xmath82 is the partition function , and @xmath83 is the log partition function . ) by translating the observables @xmath54 , we can assume that @xmath84 , for all @xmath42 . we can now restate the problem in terms of the log partition function : we are looking for some @xmath85 such that @xmath86 . we know there exists a state @xmath43 which has the desired expectation values @xmath40 . now choose some observables @xmath87 , such that the set @xmath88 is complete and linearly independent ( in other words , any @xmath89-dimensional hermitian matrix can be written uniquely as a real linear combination of the matrices in this set ) . let @xmath90 be the expectation values of @xmath39 for the observables @xmath87 ; that is , @xmath91 . by translating the @xmath92 , we can assume that @xmath93 , for all @xmath94 . we will consider states of the form @xmath95 ( here , @xmath96 is the partition function , and @xmath97 is the log partition function . ) completeness of the @xmath54 and the @xmath92 implies that we can write @xmath39 in the form @xmath98 for some @xmath99 . this implies that @xmath100 for some @xmath99 . furthermore , we claim that there is a unique point @xmath101 such that @xmath102 has the expectation values @xmath103 and @xmath104 . this is because the expectation values @xmath103 and @xmath104 uniquely determine the state @xmath39 , and setting @xmath98 uniquely determines the values of @xmath105 and @xmath106 . this in turn follows from the completeness and linear independence of the @xmath54 and the @xmath92 . so we conclude that @xmath100 at exactly one point @xmath101 . to complete the proof , we will carry out the following plan : we will show that @xmath107 as @xmath108 , where @xmath109 denotes the norm of the vector @xmath101 . this implies that @xmath110 as @xmath111 ; and hence @xmath86 for some @xmath85 . ( see figure [ fig - geometry ] for a simple example that shows the geometric intuition for the proof . ) let @xmath112 be the unique point where @xmath113 vanishes . we claim that @xmath112 is the unique global minimum of @xmath63 . [ since @xmath63 is convex ( proposition [ convexity ] ) , it follows that @xmath63 is bounded below , and @xmath112 is a global minimum . also , @xmath63 is differentiable everywhere on the domain @xmath114 , which has no boundaries ( proposition [ derivatives ] ) ; so any extremum @xmath101 must satisfy @xmath100 . but this happens only at @xmath112 , and so @xmath112 is the unique global minimum . ] let @xmath115 be the set of all unit vectors in @xmath114 . define the function @xmath116 , for @xmath117 , and @xmath118 . say we fix @xmath119 . we claim that there exists some @xmath120 such that , for all @xmath121 , @xmath122 . [ since @xmath112 is the unique global minimum , we have that @xmath123 , for all @xmath121 . moreover , @xmath124 is a continuous function of @xmath121 , and @xmath115 is a compact set , hence its image @xmath125 is compact . hence @xmath124 must be bounded away from @xmath126 , for all @xmath121 . ] next we claim that , for all @xmath121 , and for all @xmath127 , @xmath128 . [ fix any @xmath121 . @xmath129 is a differentiable function of @xmath130 , so by the mean value theorem , there exists some @xmath131 such that @xmath132 . in addition , since @xmath63 is convex , @xmath133 is nondecreasing in @xmath130 . this proves the claim . ] now , say we are given some @xmath101 , and assume that @xmath134 . we can write @xmath101 in the form @xmath135 for some unit vector @xmath117 . then we have : @xmath136 from this , we conclude that @xmath107 as @xmath108 . we will use the following fact : if @xmath139 is continuous , and @xmath140 as @xmath141 , then @xmath142 is bounded below , and @xmath142 attains its minimum at some point @xmath143 . [ to see this , let @xmath144 , choosing @xmath145 large enough that @xmath146 . note that @xmath115 is bounded ; otherwise , there would exist a sequence @xmath147 such that @xmath148 and @xmath149 , a contradiction . also , note that @xmath115 is closed ; this is because the interval @xmath150 $ ] is closed , and @xmath142 is continuous . so we have that @xmath115 is compact . this implies that @xmath151 is compact . hence @xmath151 is closed and bounded ; also note that @xmath152 . this implies that @xmath142 is bounded below , and attains its minimum . ] from this , we conclude that @xmath153 attains its minimum at some point @xmath154 . @xmath64 has no boundaries , and @xmath153 is differentiable everywhere on @xmath64 , so it follows that @xmath155 . this completes the proof . @xmath74 proof : we will obtain theorem [ consistent - rhos ] as a special case of theorem [ consistent - ts ] . the basic idea is that specifying the local density matrices @xmath1 is equivalent to specifying the expectation values of all pauli matrices on the subsets @xmath32 . let @xmath156 , @xmath157 and @xmath8 denote the pauli matrices for a single qubit , and define @xmath158 . we can construct @xmath0-qubit pauli matrices by taking tensor products @xmath159 . any @xmath89-dimensional hermitian matrix can be written as a real linear combination of @xmath0-qubit pauli matrices . furthermore , the @xmath0-qubit pauli matrices are orthogonal with respect to the hilbert - schmidt inner product : @xmath160 if @xmath161 , and 0 otherwise . we make the following claim : let @xmath3 be a density matrix on @xmath0 qubits , and let @xmath39 be a density matrix on a subset of the qubits @xmath162 , with @xmath163 . we claim that @xmath164 , if and only if , for all pauli matrices @xmath60 on the subset @xmath165 , @xmath166 . ( notation : we write @xmath0-qubit pauli matrices in the form @xmath167 , where @xmath60 acts on the subset @xmath165 , and @xmath168 acts on the rest of the qubits . ) the ( @xmath169 ) direction is obvious , but we need to show ( @xmath170 ) . we write @xmath3 and @xmath39 as linear combinations of pauli matrices , with real coefficients @xmath171 and @xmath172 : @xmath173 we know that , for all pauli matrices @xmath60 on the subset @xmath165 , @xmath174 . but this implies : @xmath175 which proves the claim . 99 s. sachdev , _ quantum phase transitions _ , cambridge university press , 2000 . d. aharonov and t. naveh , `` quantum np - a survey , '' arxiv : quant - ph/0210077 . jaynes , `` information theory and statistical mechanics ii , '' phys . ( 2 ) 108 ( 1957 ) , pp.171 - 190 . jaynes , `` information theory and statistical mechanics , '' lectures at brandeis in 1962 . reprinted in _ e.t . jaynes : papers on probability , statistics and statistical physics _ , r.d . rosenkrantz ( ed . ) , d. reidel publishing company , 1983 . brown , _ fundamentals of statistical exponential families _ , institute of mathematical statistics , 1986 . ingarden , h. janyszek , a. kossakowski and t. kawaguchi , `` information geometry of quantum statistical systems , '' tensor ( n.s . ) 37 ( 1982 ) , pp.105 - 111 . r. bhatia , _ matrix analysis _ , graduate texts in mathematics , springer - verlag , 1997 . wilcox , `` exponential operators and parameter differentiation in quantum physics , '' j. mathematical physics 8 ( 1967 ) , pp.962 - 982 . that satisfies @xmath176 ; we have one observable @xmath177 . we know there exists some state @xmath43 that satisfies @xmath176 ; in this case , @xmath39 also satisfies @xmath178 , and we let @xmath179 play the role of the `` extra '' observables . as the graph shows , @xmath180 vanishes at exactly one point ; @xmath181 ; and @xmath182 vanishes for some @xmath105 . ]

suppose we have an @xmath0-qubit system , and we are given a collection of local density matrices @xmath1 , where each @xmath2 describes some subset of the qubits . we say that @xmath1 are `` consistent '' if there exists a global state @xmath3 ( on all @xmath0 qubits ) whose reduced density matrices match @xmath1 . we prove the following result : if @xmath1 are consistent with some state @xmath4 , then they are also consistent with a state @xmath5 of the form @xmath6 , where each @xmath7 is a hermitian matrix acting on the same qubits as @xmath2 , and @xmath8 is a normalizing factor . ( this is known as a gibbs state . ) actually , we show a more general result , on the consistency of a set of expectation values @xmath9 , where the observables @xmath10 need not commute . this result was previously proved by jaynes ( 1957 ) in the context of the maximum - entropy principle ; here we provide a somewhat different proof , using properties of the partition function .

* * let us consider an unbalanced michelson interferometer with long path @xmath50 and short path @xmath51 . the path difference for normal incidence is @xmath52 . simple ray tracing shows for an incident angle @xmath53 the paths through the interferometer change with @xmath5 according to @xmath54 where @xmath55 is the lateral offset between the two rays coming from the short and long arms at the output port of the beam splitter . the angle - dependent path difference affects the photon counts in the phase - dependent basis , as the inset in fig . the lateral offset reduces the interference visibility , as shown in fig . 1 . we can see this if the incoming ray is replaced by a single - mode gaussian beam with an amplitude of @xmath56 and variance @xmath4 . the intensity at the output is then given by @xmath57 from which we obtain the visibility in eq . . here , @xmath58 denotes the relative phase between the paths . + * * the relay - lens system , consisting of free space ( @xmath59 ) and lens ( @xmath60 ) transmission , @xmath61 can be inserted into the long arm of an unbalanced michelson interferometer ( see fig . 1(a ) ) in order to remove the spatial evolution caused by the path - length difference . here , @xmath62 denotes the focal length of the lenses . note that image inversion at mirrors is excluded as this happens in both paths with no effect on the path indistinguishability . + * * light from a 404 nm cw laser with an average power of 6mw pumps a periodically poled potassium titanyl phosphate ( ppktp ) crystal inside a sagnac interferometer . this generates polarization entangled photon pairs at 776 nm and 842 nm via type - ii spdc : @xmath63 , where @xmath64 denotes _ horizontal _ ( _ vertical _ ) polarization . the downconverted photons are spectrally filtered with 12 nm band - pass filters ( bpf ) and pump photons are removed with long - pass filters ( lpfs ) . the ratio of coincidence - to - single counts is measured to be 0.07 . + * * the 776 nm photon from the polarization - entangled photon pair is sent through a single - mode fiber ( smf ) to the time - bin qubit converter ( tqc ) . a polarization controller ( fpc ) ensures the faithful transmission of the polarization state . at the pbs of the tqc , the photon is either transmitted or reflected into the short or long path , respectively . to ensure both photons leave the interferometer at the desired output port , a quarter - wave plate is inserted in each path . directly after the interferometer the photon passes through a polarizer ( pol ) set to an equal superposition between the polarizations . this allows , at the cost of 50% transmission loss , to erase polarization information for each time - bin state , which completes the map @xmath65 and @xmath66 from polarization to time - bin state . the time - bin qubit is then coupled into a smf with a collection efficiency of 0.67 , which gives a total transmission through the tqc of 0.24 . the authors thank jacob koenig , rolf horn , evan meyer - scott , and patrick coles for useful discussions , and martin laforest for lending us the hamamatsu em - ccd camera . we gratefully acknowledge supports through the office of naval research ( onr ) , the canada foundation for innovation ( cfi ) , the ontario research fund ( orf ) , the canadian institute for advanced research ( cifar ) , the natural sciences and engineering research council of canada ( nserc ) , and industry canada . 99 j. brendel , n. gisin , w. tittel , and h. zbinden , pulsed energy - 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classicality of time - polarization entanglement is bounded with an estimate of the bell - chsh inequality violation @xcite . despite the absence of an active phase control , required to set measurement bases for the time - bin qubit deterministically , we search for the maximally obtainable violation by varying the measurement basis . more specifically , we first set the measurement basis for the polarization qubit to @xmath68 @xmath69 @xmath70 using wave plates , and slowly and continuously change the path - length difference of the multimode time - bin qubit analyzer ( mm - tqa ) by externally heating it . this allows us to scan projection measurements for the time - bin qubit in the superposition bases . 6(a ) shows coincidences between a polarization qubit ( two detectors , i.e. d1 and d2 ) and a time - bin qubit ( three temporal modes ) . detections in the _ early / late _ bin ( _ middle _ bin ) corresponds to a projection of the time - bin qubit onto @xmath71 @xmath69 @xmath72 ( @xmath73 @xmath69 @xmath74 ) . owing to the absence of the second output of the tqa , we consider all possible expectation values @xmath75 between any two points in time , i.e. @xmath76 and @xmath77 ( see fig . 6(b ) ) , which is defined as here , @xmath79 are the coincidence counts for the projections @xmath80 , where i , j @xmath28\{1,2 } and superscript(@xmath81 ) denotes two outcomes of the projection measurement . among all the computed expectation values , we find the absolute maximum expectation value . we then change the measurement basis for the polarization qubit to @xmath82 @xmath69 @xmath83 and repeat the procedure . finally , we compute the chsh - bell inequality parameter and find the value of @xmath44=2.42 @xmath1 0.05 , which is clearly above classical bound @xmath85=2 . to see whether this value agrees with the measured visibilities , we model the two - qubit state with noise , described by an asymmetric depolarization channel , on a time - bin qubit . the output state is then described by where @xmath87 and @xmath88 are the depolarization probability and single - qubit pauli operator along the channel j , respectively . here , @xmath89 denotes the input state , described in eq . of the main text . assuming unbiased depolarizations in the superposition bases , i.e. @xmath90 , we calculate expectation values @xmath75 = tr @xmath91 for given measurement bases . using the definition of visibility , we further represent the chsh - bell parameter @xmath45 = @xmath92 as a function of entanglement visibilities . we find @xmath45 = 2.47 @xmath1 0.02 is in accordance with our measured value @xmath93 . our multimode time - bin qubit analyzer is passively stabilized by enclosing it with black hardboard . in order to assess the phase stability of the mm - tqa , we perform joint - projection measurement onto the superposition bases over a half an hour . the time - bin qubit is projected onto a @xmath94 and the polarization qubit alternatively between @xmath95 or @xmath96 . this allows us to calculate the expectation values , defined as @xmath97 , where @xmath79 denotes joint - detection counts when a 842 nm photon is projected onto @xmath26 and a 776 nm photon onto @xmath27 . as shown in fig . 7 , the average expectation value @xmath98 remains always higher than 0.65 , which is well above the required value @xmath99 for verifying entanglement , given entanglement visibility @xmath100 = 0.952 ( see fig . 5 in the main text ) . as entanglement is fundamentally important and an effective distribution of entanglement is a necessary condition for secure qkd @xcite , we need to verify whether or not entanglement is present in our experiment . this is especially important in the absence of a complete security analysis of a qkd implementation . we assume that the spontaneous parametric down - conversion process generates a pair of photons with negligible multiple - photon - pairs event . each photon is a polarization qubit and the pair of photons is potentially entangled . by detecting one photon in the pair , we can herald the generation of the second photon . after the conversion from polarization qubit to time - bin qubit , the second one is transmitted to the mm - tqa . suppose that alice holds the polarization qubit while bob holds the time - bin qubit . to include conversion and transmission losses of the time - bin qubit , we enlarge the dimension of bob s system from @xmath101 to @xmath102 by adding a dimension corresponding to no photon arriving at bob . hence , the final state @xmath103 shared by alice and bob is a @xmath104-dimensional state . we need to verify whether or not the state @xmath103 is entangled using only the measurement results @xmath105 and @xmath106 without further assumptions on the state . since the measurements of alice and bob are block - diagonal with respect to the subspaces of total photon number , as we will show below in eq . and , we can also assume without loss of generality that the state @xmath103 shows the same structure . this follows from the fact that the measurement structure allows us to assume that a quantum non - demolition measurement of the total photon number is executed before the actual measurement itself . in order to verify entanglement , we need to know how to accurately describe the measurements on the polarization qubit and on the time - bin qubit . for the polarization qubit , we measure it in the _ horizontal_/_vertical _ or _ diagonal_/_anti - diagonal _ basis , i.e. along the z- or x - axis in the bloch sphere . in the _ horizontal_/_vertical _ basis , these measurements are represented as @xmath107 0 & 0 \\[0.3em ] \end{bmatrix } , m_\text{v}=\begin{bmatrix } 0 & 0 \\[0.3em ] 0 & 1 \\[0.3em ] \end{bmatrix } , m_\text{d}=\frac{1}{2}\begin{bmatrix } 1 & 1 \\[0.3em ] 1 & 1 \\[0.3em ] \end{bmatrix},\text { and } m_\text{a}=\frac{1}{2}\begin{bmatrix } 1 & -1 \\[0.3em ] -1 & 1 \\[0.3em ] \end{bmatrix } , \label{eq : meas_alice } \end{aligned}\ ] ] where the subscript indicates measurement outcome , and h , v , d , or a denotes the _ horizontal _ , _ vertical _ , _ diagonal _ , or _ anti - diagonal _ polarization . on the other side , for the time - bin qubit , the photon loss in the long path or the short path of the mm - tqa could be different from each other . hence , the operators corresponding to measurement of the time - bin qubit in the _ early_/_late_-time basis or in the superposition bases could deviate from the ideal case . without loss of generality , we can choose the relative phase between the _ early_-time and _ late_-time basis states in the superposition basis to be zero . therefore , in the basis where the basis states are no photon , one photon in the _ early _ bin and one photon in the _ late _ bin , these measurements can be written as @xmath108 0 & \eta_s & 0 \\[0.3em ] 0 & 0 & 0 \end{bmatrix } , m_\text{l } = \frac{1}{4 } \begin{bmatrix } 0 & 0 & 0\\[0.3em ] 0 & 0 & 0 \\[0.3em ] 0 & 0 & \eta_l \end{bmatrix } , m_\text{x } = \frac{1}{4 } \begin{bmatrix } 0 & 0 & 0\\[0.3em ] 0 & \eta_l & \sqrt{\eta_l\eta_s } \\[0.3em ] 0 & \sqrt{\eta_l\eta_s } & \eta_s \end{bmatrix},\text { and } \nonumber \\ & m_{\emptyset}=i_{3\times 3}-m_\text{e}-m_\text{l}-m_\text{x } , \label{eq : meas_bob } \end{aligned}\ ] ] where the subscript e , l , x , or @xmath109 means that the measurement outcome is _ early _ time , _ late _ time , the superposition of the _ early _ time and _ late _ time , or no detection , respectively . no - detection events are due to detection inefficiency and the absence of the second output in the mm - tqa . in eq . , @xmath110 or @xmath111 is the respective transmission efficiency in the long path or the short path of the mm - tqa . note that in our experiment @xmath110 and @xmath111 are very close to each other . after knowing the description of alice s and bob s joint state @xmath103 and also that of their measurements , we can verify entanglement by the negative partial - transpose ( npt ) criterion @xcite . the npt criterion is used because this criterion is satisfied if and only if a state is entangled , given the state is @xmath112- or @xmath104-dimensional @xcite . the npt criterion has been applied to verify entanglement in qkd systems , such as in @xcite . explicitly , we verify entanglement by solving the following semi - definite program ( sdp ) : @xmath113=\mathcal{v}_{+\text{z}}\operatorname{tr}[\rho(m_\text{h}\otimes m_\text{e}+m_\text{v}\otimes m_\text{e } ) ] \\ & & \operatorname{tr}[\rho(m_\text{v}\otimes m_\text{l}-m_\text{h}\otimes m_\text{l})]=\mathcal{v}_{-\text{z}}\operatorname{tr}[\rho(m_\text{v}\otimes m_\text{l}+m_\text{h}\otimes m_\text{l } ) ] \\ & & \operatorname{tr}[\rho(m_\text{d}\otimes m_\text{x}-m_\text{a}\otimes m_\text{x})]=\mathcal{v}_\text{xy}\operatorname{tr}[\rho(m_\text{d}\otimes m_\text{x}+m_a\otimes m_\text{x } ) ] , \end{array}\ ] ] where @xmath114 is the partial - transpose operation on a subsystem , such as on the polarization - qubit subsystem , and @xmath115 denotes the tensor product . note that , we formulate the last three constraints according to the measured visibilities . the first two of them are based on entanglement visibilities @xmath116 and @xmath117 conditioned on measurement outcomes of the time - bin qubit being early time and late time , respectively . the last constraint is based on entanglement visibility @xmath118 , where the time - bin qubit comes out in the _ middle _ bin . since the mm - tqa has only one output , we can not differentiate the case when the photon comes out from the second output if this output exists from the case when the photon is lost over the transmission . hence , we can not formulate two constraints based on @xmath118 . in our experiment , we verified that within experimental errors the visibilities @xmath119 . so , for solving the sdp program in eq . we set @xmath120 . using the measured results @xmath121 and @xmath122 , the sdp program in eq . is not feasible , signifying that the state @xmath103 must be entangled . furthermore , by numerically checking over which values of @xmath123 and @xmath118 the sdp program in eq . is not feasible , we are able to upper bound the required visibilities @xmath123 and @xmath118 in order to certify the presence of entanglement in the system . the numerical results are shown in fig . 5 of the main text . from this figure , one can see that our visibility result at any observed incident angle witnesses entanglement with high confidence . finally , we would like to note two points . first , the constrains considered in eq . are independent of the transmission or conversion loss of the photon arriving at the mm - tqa , and even independent of the common photon loss in the two different paths of the mm - tqa . therefore , the upper bounds on the visibilities @xmath123 and @xmath118 obtained for verifying entanglement are independent of all of these different losses . second , we numerically verified that the classical boundary , as shown in fig . 5 of the main text , is even independent of the relative loss between the two paths of the mm - tqa .

time - bin encoded photonic qubits are robust against decoherence in birefringent media such as optical fibers @xcite , which has led to many successful quantum communication demonstrations , including a mach - zehnder interferometer - based system @xcite , plug - and - play quantum key distribution ( qkd ) @xcite , differential phase shift ( dps ) @xcite and coherent one - way ( cow ) @xcite protocols , quantum teleportation @xcite , and elements of quantum repeaters @xcite . despite these successes , time - bin encoding is considered impractical for turbulent free - space or multimode fiber channels , because spatial and temporal mode distortions hinder the interference required for analysis of time - bin superposition states . here we present an efficient time - bin qubit analyzer , assisted by passive relay optics , that matches the spatial mode of the interfering paths . our system demonstrates interference visibility of 89% , even for strongly distorted optical input modes emerging from multimode fiber under variable angles of incidence . we measure a high level of entanglement between a time - bin qubit and a separate polarization qubit , thereby demonstrating the feasibility of time - bin based quantum communication over optical multimodal links . polarization - encoded photons have been preferred for most quantum communication experiments in free space @xcite due to their robustness against atmospheric turbulence @xcite . however , over long distances or when the communicating parties are in motion , it can be challenging to maintain alignment of polarization reference frames . in addition , building polarization preserving optics and telescopes is often difficult and expensive . in particular , active or adaptive optics , utilized to compensate angular fluctuations of the link in real time , may incur unavoidable polarization fluctuations @xcite , leading to practical challenges for polarization - based quantum communications @xcite . time - bin encoding is an alternative which has many advantages over polarization . despite this , it has not been implemented for quantum communications over a long - distance free - space channels . this is mainly due to turbulence - induced effects on the wavefront @xcite and path @xcite , which make it difficult to perform interferometric measurements necessary for time - bin state projection onto superposition bases . for instance , ursin _ et al . _ @xcite measured turbulence - induced angle of incidence ( aoi ) errors of up to @xmath0rad , in addition to scintillation effects . an even larger contribution may come from telescope pointing errors between communicating parties , which can be as high as @xmath1 1.04mrad ( 0.06 @xmath2 ) for a moving platform @xcite . most importantly , these angle variances lead to distinguishable paths in unbalanced michelson and mach - zehnder interferometers , which form the basis for typical time - bin qubit analyzer ( tqa ) implementations . we analyze the performance reduction of a tqa with variable aoi , assuming a michelson interferometer with path difference @xmath3 , for an incident gaussian beam with width @xmath4 incident with an angle @xmath5 . the angle - dependent interference visibility follows @xmath6 where @xmath7 denotes the system visibility at zero angle ( see appendix ) . for @xmath8 mm and @xmath9 m , chosen to achieve a clear separation of the time bins given our detector timing jitter and channel - induced dispersion , we expect the visibility to drop to 0.70 for @xmath10mrad and @xmath7 = 0.91 . furthermore , the relative phase between the two paths is very sensitive to the aoi , with a predicted @xmath11shift per 349 nanorad input angle variation ( see appendix ) . the relationship eq . is verified experimentally with a single - mode beam ( see fig . 1(b ) ) , generated by a continuous - wave ( cw ) laser at 776 nm . as shown in fig . 1(c ) , the initial interference visibility of @xmath12 = 0.91 @xmath1 0.01 decreases rapidly with aoi when no correction optics are implemented , as expected from eq . . next , the same laser beam is sent through a 1m - long step - index multimode fiber , thereby distorting it into a multimodal beam @xcite which mimicks the effect of turbulent atmosphere ( fig . 1(d ) , see @xcite for comparison ) . despite lengthy and careful alignment we are only able to obtain a maximum visibility of @xmath13 = 0.16 @xmath1 0.01 , which , as shown in fig . 1(e ) , drops with aoi . current solutions to this behavior include spatial - mode filtering using single - mode optical fibers , which , however , discard most of the impinging photons @xcite . these observations clearly show that , given the expected angular deviations reported for free - space quantum channels , it would be technically very challenging to achieve a reliable , stable and efficient operation of time - bin qubit receiver using standard interferometers . these interference challenges are overcome by utilizing relay optics in the long arm of the unbalanced michelson interferometer , as depicted in fig . 1(a ) . effectively , the relay optics reverse differences in the evolution of the spatial mode over length @xmath3 in the long arm ( see appendix ) . with our new tqa design , for a single - mode beam , an interference visibility of @xmath14= 0.91 @xmath1 0.01 is obtained , which remains constant as the aoi is varied ( see fig . 1(c ) ) . the improvement is further confirmed by measurements with a multimode beam ( fig . 1(d ) ) where the high visibility of @xmath15=0.89 @xmath1 0.01 ( fig . 1(e ) ) demonstrates that the interferometer design is robust against highly distorted beams . the measured optical throughput of the interferometer is 0.74 including multimode fiber - coupling efficiency of 0.87 , showing that the relay lenses faithfully symmetrize the two paths of the interferometer . the utility of our interferometer design , as a working tqa for multimodal quantum signals , is demonstrated with the experimental setup depicted in fig . 2 . a source generates polarization - entangled photon pairs centered at 776 nm and 842 nm ( see appendix ) with initial entanglement visibilities of @xmath16 = 0.979 @xmath1 0.013 and @xmath17 = 0.901 @xmath1 0.012 . the 776 nm polarization qubit is converted into a time - bin qubit , resulting in the hybrid state @xmath18 where @xmath19h@xmath20 ( @xmath19v@xmath20 ) and @xmath19e@xmath20 ( @xmath19l@xmath20 ) denotes the quantum state in which the 842 nm photon is in _ horizontal _ ( _ vertical _ ) polarization mode and the 776 nm photon in _ early _ ( _ late _ ) temporal mode ( see appendix ) . the 776 nm photon is sent through a 1m - long step - index multimode fiber to artificially distort the spatial mode ( see fig . 3(a ) ) and the temporal mode , with a measured dispersion of about 50ps , drastically exceeding the photon s coherence time of 3.2ps @xcite , prior to entering our multimode time - bin qubit analyzer ( mm - tqa ) . the output of the interferometer is coupled into a multimode fiber for delivery of the photons to the detector ( si - avalanche photodiode ) . verification of the mm - tqa performance is done by measuring entanglement visibilities between the polarization and time - bin qubits . each qubit is first projected onto basis states , i.e. @xmath21 , where @xmath22 denotes @xmath19h(v)@xmath20 and @xmath19e(l)@xmath20 for a 842 and a 776 nm photon , respectively . the coincidence counts are used to calculate a visibilty , defined as @xmath23 , and the average value of @xmath24 = 0.952 @xmath1 0.011 is observed . here , @xmath25 denotes the joint - detection counts between a 842 nm photon projected onto @xmath26 and a 776 nm photon onto @xmath27 , where i , j @xmath28 \{+z , -z } ( see fig . 3(b ) ) . more important for the mm - tqa operation is the projection of the time - bin qubits onto superposition states , i.e. @xmath29 , where @xmath30 . to measure the visibility , we vary the relative phase @xmath31 between basis states using motorized wave - plates acting on the polarization qubit . a complete scan of the phase along the @xmath32-plane of the bloch sphere is performed within 10 seconds , yielding the average visibility @xmath33 = 0.804 @xmath1 0.006 ( see fig . 3(c ) ) . to + 0.2@xmath2 over 20 seconds . the inset shows the expectation value without relay optics as a function of aoi . due to aoi - induced phase fluctuations , the value rapidly changes with aoi and thus yields an average value of zero ( red circle ) . measured visibility @xmath34 bounds the value @xmath35 . these phase fluctuations are corrected by the relay optics , allowing a near constant expectation value . ] the entanglement verification measurements are also carried out for different aoi , demonstrating that the measured visibilities ( in fig . 3(d ) ) are constant within experimental errors . due to the coupling geometry of the photons into the multimode detector fiber , the photon - collection efficiency of 0.87 decreases with aoi and drops to zero at @xmath36 degrees . however , the fact that the interference visibility is essentially invariant to variation of aoi confirms the robustness of our mm - tqa . without relay optics , a variance of aoi also introduces phase fluctuations of the time - bin qubit analyzer . for instance , dixon _ et al . _ @xcite measured an angular deflection of 400 femtorad using interferometric phase measurements . from our theoretical model , we anticipate 5@xmath11-shift with an aoi of only 1.75 @xmath37rad ( 0.0001@xmath2 ) ( see appendix ) . in order to assess the phase stability of our mm - tqa with aoi , we measure expectation values , defined as @xmath38 , for aois changing from -0.20@xmath2 to + 0.20@xmath2 continuously over 20 seconds . as shown in fig . 4 , the measured @xmath39-values remain almost constant within experimental errors , showing that our mm - tqa prevents the aoi - caused phase fluctuations . moreover , by performing continuous projection measurements over a half an hour , we also test the long - term stability of our passively stabilized mm - tqa and observed the measured @xmath39-values change very slowly during the measurement period ( see supplementary information ( si ) ) . hence , this mm - tqa could be used for reference - frame - independent ( rfi ) qkd in which secret keys are extracted under slow drift of measurement bases @xcite . the suitability of the multimode tqa for quantum communications is further substantiated by examining a chsh - bell inequality @xcite from measurements on the hybrid entangled state . we prepare two bases on xz - plane of a bloch sphere for the polarization qubit , i.e. , @xmath40 . for the time - bin qubit , one basis ( @xmath41 ) is set deterministically , while the second one ( @xmath42 ) is made to drift using a heat gun , owing to the absence of active phase control in our current tqa . a search for the maximally achievable chsh - bell parameter is performed by scanning the second basis ( @xmath43 ) along the xy - plane of the time - bin states . a maximum value of @xmath44 = 2.42 @xmath1 0.05 is found . this is in good agreement with the predicted parameter @xmath45 = 2.47 @xmath1 0.02 calculated assuming the state described in eq . is exposed to noise modeled by an asymmetric depolarization channel ( see si ) . a more rigorous verification of entanglement takes practical assumptions into account , including the photon loss in the channel and path - dependent transmission in the mm - tqa . we numerically find minimum visibilities required to detect the presence of entanglement for arbitrary 2 @xmath46 3-dimensional quantum system ( see si ) . measured visibilities @xmath47 = 0.95 @xmath1 0.02 and @xmath48 = 0.81 @xmath1 0.02 averaged over various aois are well above the obtained classical bound , revealing clear evidence of entanglement ( see fig . 5 ) . note that the obtained numerical results are valid regardless of the system efficiency and mismatched transmission between the two paths of tqa . 3-dimensional quantum state . our experimental results for various angles ( green circles ) are well above the classical bound ( black line ) . ] while theoretically @xmath7 @xmath49 1 , the performance of the current mm - tqa shows about @xmath7 = 0.9 , which is expected to be improved considering several approaches . first , the overlap of the spatial modes in the interferometer can be improved with careful custom design and selection of optical elements and optimization of beam diameters . second , appropriate mode matching optics in both paths will improve the interference quality because of dispersion symmetrization at the expense of increasing complexity of the system . in conclusion , we demonstrated a novel time - bin qubit analyzer , robust against multimode distortions on the quantum link as expected from turbulence - induced effects , i.e. wavefront distortions and path - delay fluctuations , as well as telescope pointing errors . the main benefit of our analyzer is that it is entirely based on passive optics without any active feedback systems for beam correction . despite strong spatial and temporal distortions of the incident photon modes , we observe a robust entanglement visibility of 0.80 in the superposition bases which remains constant with variable angles of incidence . at the same time , our mm - tqa has a very high throughput efficiency of 0.74 from input to output , mainly limited by optical surface losses which can be improved . our results open the door for implementing time - bin based quantum communication experiments over turbulent free - space and multimodal channels such as with moving platforms , including aircraft and satellites , or through non - polarization maintaining windows . furthermore , as a quantum state receiver , the multimode time - bin qubit analyzer should also be suitable for cow , dps , and reference - frame - independent qkd protocols .

we are interested in the eigenvalue statistics for the operator @xmath9 on the hilbert space @xmath0 , where @xmath3 and @xmath10 are iid random variables following real symmetric absolutely continuous distribution @xmath5 ( that is @xmath11 for any @xmath12 ) satisfying @xmath13 for some @xmath14 . we will be working under the condition @xmath7 . + for anderson tight binding model , eigenvalue statistics was first studied by molchanov@xcite for one - dimension and minami@xcite for higher dimensions . other similar works includes germinet - klopp @xcite , geisinger @xcite , dolai - krishna @xcite . in above mentioned works poisson statistics was shown in pure point regime . the class of models described by are not ergodic , but recently many results are obtained in non - ergodic cases . in @xcite killip - stoiciu studied eigenvalue statistics for cmv matrices and showed that i ) for @xmath15 the process is clock , ii ) for @xmath16 the process is @xmath17-ensemble , and iii ) for @xmath18 it is poisson . in case of @xmath19 , dolai - mallick @xcite showed that the statistics is poisson . dolai - krishna @xcite showed level repulsion for higher dimensional model with decaying randomness . + cmv matrices are representation of unitary operators and so it is expected that this trend would also hold even for one - dimensional anderson like models . one such work is of avila - last - simon @xcite where they showed quasi - clock behaviour for ergodic jacobi operator in region of absolutely continuous spectrum . in @xcite krichevski - valk - viag showed the sine-@xmath17 process for @xmath16 . similar work in continuous analogue are done by kotani @xcite where gap between eigenvalue was studied and limit laws is obtained for @xmath15 . kotani - nakano@xcite and nakano@xcite showed that the statistics for @xmath15 is clock and for @xmath16 it is circular @xmath17-ensemble . in these works , the randomness is via brownian motion on a compact manifold and the decay is implemented as multiplication to the randomness . in @xcite kotani - quoc studied the distribution of individual eigenvalue in continuous case when the potential is compound poisson process . + in this work we look at the eigenvalue statistics inside the continuous spectrum for @xmath7 . in delyon - simon - souillard@xcite showed that for @xmath20 , the essential spectrum of @xmath1 is @xmath21 $ ] , they also showed that in the case of @xmath15 and @xmath22 the spectrum is continuous in that region . + here we study the eigenvalue statistics in the region @xmath8 . for @xmath23 the sequence of measures @xmath24 is defined by @xmath25 where @xmath26 is the restriction of @xmath1 onto the subspace @xmath27 . + the main result in this work is the following : [ mainthm ] consider @xmath1 as in then for @xmath7 and @xmath28 there exists a subsequence @xmath29 such that @xmath30 converges to a clock process in distribution . to study eigenvalue statistics , we work with prfer phase . for coherence , we provide the derivation to obtain the phase function in this setup . + for any @xmath31 , there exists a real sequence @xmath32 such that @xmath33 by looking at the transfer matrix ( with @xmath34 ) @xmath35 in case @xmath36 , there exists @xmath37 such that @xmath38 . using the transformation @xmath39 and setting @xmath40 , we get @xmath41 here we take the principal branch of logarithm with the branch cut @xmath42 , so @xmath43 . as a result of above transformations , we have @xmath44 so defining @xmath45 , and using the fact that @xmath34 we have the recursion formula @xmath46 and also define @xmath47 ( in the case of no randomness , we have @xmath48 ) . it is clear that @xmath49 for @xmath50 , because @xmath51 . therefore to identify the eigenvalues of @xmath26 inside @xmath8 one has to only look at those @xmath52 for which @xmath53 . let @xmath54 denote the eigenvalues of @xmath26 ( arrange in decreasing order ) . then for any @xmath55 fixed , there exists @xmath12 such that @xmath56 $ ] , so one only needs to consider the eigenvalues satisfying @xmath57 since @xmath28 , we can assume that all the eigenvalues under consideration are of the form @xmath58 where @xmath59 . set @xmath37 such that @xmath60 , then @xmath61 hence define @xmath62 ( which are arranged in increasing order ) . most of the work done here to show that @xmath63 converges to @xmath64 in probability . + + here we consider @xmath5 with unbounded support ( one can also take @xmath65 to be less than @xmath66 ) , we are only able to show convergence in probability . to do this , since @xmath7 , there exists a sequence of measurable sets @xmath67 ( described in ) where after a point all the potential is decaying fast enough and @xmath68\xrightarrow{n\rightarrow\infty}1 $ ] . so limit of @xmath69 can be computed as element of @xmath70 ( @xmath71 is important as seen in lemma [ lem6 ] and can not be proven for any other exponent ) . the construction of sequence @xmath67 is done after lemma [ lem5 ] and in lemma [ lem6 ] the convergence rates in @xmath71 are computed . lemma [ lem3 ] and [ lem5 ] are necessary to get one - to - one correspondence between @xmath72 and @xmath73 ( see for correspondence ) and they are completely general ( does not depend upon randomness ) . in lemma [ lem4 ] and [ lem2 ] we get the limit . finally we combine the results for the proof of the main theorem . if the measure @xmath5 is compactly supported or @xmath22 , the convergence in lemma [ lem6 ] can be shown to be @xmath71 on @xmath74 itself and the theorem can be proved for @xmath15 . following lemma gives the continuity of @xmath75 . this is done by using the fact that even though @xmath76 is discontinuous , the singularity of the @xmath77 is never reached . final expression is basically lipschitz continuity statement . it can be noted that the neighbourhood where continuity is obtained depends only on @xmath52 . [ lem3 ] given @xmath52 , for @xmath78 the function @xmath79 is continuous in @xmath80 in a neighbourhood of @xmath81 for a.e @xmath82 . using @xmath83 for @xmath84 , and writing @xmath85 as @xmath86 for @xmath87 where @xmath88 is chosen such that that @xmath89 , some simple estimation gives @xmath90 using this recursively till @xmath91 and then using @xmath92 , we get @xmath93|p - q|.\ ] ] this completes the proof . the above lemma and properties of green s function give the next lemma which show that the functions @xmath94 are increasing . [ lem5 ] the @xmath95 are increasing functions . we have the recurrence relation ( setting @xmath96 and @xmath97 ) @xmath98 which modifies to @xmath99 where @xmath100 . using , we get @xmath101 and using definition of green s function @xmath102 where @xmath103 , @xmath104 are eigenfunctions corresponding to eigenvalues @xmath105 of @xmath26 . by definition of @xmath106 we have @xmath107 let us denote the function @xmath108 defined by @xmath109 , this is an increasing continuous function ( using the properties of last expression in and the fact that @xmath110 is decreasing in the concerned region ) with @xmath111 hence for @xmath112 we get @xmath113 using @xmath114 and @xmath115 is positive on @xmath116 . finally using continuity of @xmath117 and the fact that @xmath118 is decreasing we get that @xmath117 is increasing function . set @xmath119 , and observe ( using @xmath120 ) @xmath121 in particular @xmath122 as a consequence of equation and simplicity of spectrum , there exists an enumeration of @xmath123 such that @xmath124 where @xmath125 is defined such that @xmath126 ( both of them have different sign and none of them are zero ) . from now on we will use this enumeration whenever it is required . + from here we assume that @xmath7 , so there exists @xmath17 such that @xmath127 . we drop dependence of @xmath52 in notation now onward . define the sequence of sets @xmath128 observe that ( choose @xmath78 to be large ) @xmath129\leq n+c|\sin\theta|^\delta\sum_{n\geq n}\frac{1}{n^{\delta(\alpha-\beta)}}<\infty.\ ] ] so using borel - canterlli lemma we have @xmath130=0 $ ] . so the random variable @xmath131 is almost everywhere finite , since @xmath132 define the set @xmath133 and observe that @xmath68\xrightarrow{n\rightarrow\infty}1 $ ] . [ lem6 ] for @xmath134 large enough : @xmath135\leq c n^{1 - 2\beta}\ ] ] for any @xmath136 . for @xmath137 and @xmath138 define @xmath139 using @xmath140 for @xmath141 , we have @xmath142&={\mathop{\mathbb{e}}}_{\omega\in b_n}\left[\left|\sum_{n = n}^{m-1}\left(\im\ln\left(1-\frac{a_n\omega_n \sin y^\omega_n(\theta)}{\sin\theta}e^{-i y^\omega_n(\theta)}\right)-\frac{a_n\omega_n\sin^2 y^\omega_n(\theta)}{\sin\theta}\right)\right|^2\right]\nonumber\\ & \leq { \mathop{\mathbb{e}}}_{\omega\in b_n}\left[\left(\sum_{n = n}^{m-1}\left|\im\ln\left(1-\frac{a_n\omega_n \sin y^\omega_n(\theta)}{\sin\theta}e^{-i y^\omega_n(\theta)}\right)-\frac{a_n\omega_n\sin^2 y^\omega_n(\theta)}{\sin\theta}\right| \right)^2\right]\nonumber\\ & \leq 4{\mathop{\mathbb{e}}}_{\omega\in b_n}\left[\left(\sum_{n = n}^{m-1}\left|\frac{a_n\omega_n}{\sin\theta}\right|^2\right)^2\right]\leq 4\left(\sum_{n = n}^{m-1}\frac{1}{n^{2\beta}}\right)^2\leq c_1 n^{2(1 - 2\beta)}.\end{aligned}\ ] ] using @xmath143 for reals , we have @xmath144&= { \mathop{\mathbb{e}}}_{\omega\in b_n}\left[\left|\sum_{n = n}^{m-1}\frac{a_n\omega_n\sin^2 y^\omega_n(\theta)}{\sin\theta}\right|^2\right]\nonumber\\ & = \sum_{m , n = n}^{m-1}{\mathop{\mathbb{e}}}_{\omega\in b_n}\left[\frac{a_n\omega_n\sin^2 y^\omega_n(\theta)}{\sin\theta}\frac{a_m\omega_m\sin^2 y^\omega_m(\theta ) } { \sin\theta}\right]\nonumber\\ & = \sum_{n = n}^{m-1}{\mathop{\mathbb{e}}}_{\omega\in b_n}\left[\left(\frac{a_n\omega_n\sin^2 y^\omega_n(\theta)}{\sin\theta}\right)^2\right]\nonumber\\ & \qquad+\sum_{n\leq m < n < m}2{\mathop{\mathbb{e}}}_{\omega_n\in a_n}[\omega_n]{\mathop{\mathbb{e}}}_{\omega\in b_n}\left[\frac{a_na_m \omega_m \sin^2 y^\omega_n(\theta)\sin^2 y^\omega_m(\theta)}{\sin^2\theta}\right]\nonumber\\ & \leq c_2 n^{1 - 2\beta}.\end{aligned}\ ] ] since @xmath94 is independent of @xmath145 and the fact that @xmath11 , we have @xmath146={\mathop{\mathbb{e}}}_{\omega_n\in a_n}[\omega_n]{\mathop{\mathbb{e}}}_{\omega\in b_n}[\omega_m f(y^\omega_n , y^\omega_m)]=0~~\text{for } n > m\ ] ] combining and we obtain . it should if noted that above proof still works if we assume that @xmath10 satisfies @xmath147=0\ ] ] for any @xmath148 which is a bounded function of @xmath149 . in case @xmath22 , we can take @xmath15 because ( define @xmath150 ) @xmath151\leq c\sum_{n\geq n}\frac{1}{n^{\alpha\delta}}\xrightarrow{n\rightarrow\infty}0.\ ] ] so @xmath152 is defined as @xmath153 along with @xmath154 . now we can prove the convergence of @xmath155 . this is divided in two parts . in following lemma , convergence of @xmath156 is shown . then in next lemma , convergence of @xmath157 is shown . [ lem2 ] for given @xmath52 , there exists @xmath158 $ ] measurable , such that for all @xmath159 @xmath160\xrightarrow{l\rightarrow\infty}0\ ] ] let @xmath161 , and observe that @xmath162\\ & \qquad\leq \mathbb{p}\left[\omega\in b_{n}:|\tilde{y}^\omega_m(\theta)-\tilde{y}^\omega_n(\theta)|>\epsilon\right]+\mathbb{p}[b^c_{n}].\end{aligned}\ ] ] using chebyshev s inequality in we get @xmath163\leq \frac{cn^{1 - 2\beta}}{\epsilon^2}.\ ] ] hence @xmath164\leq \mathbb{p}[b^c_{n}]+ \frac{cn^{1 - 2\beta}}{\epsilon^2 } \xrightarrow{n\rightarrow \infty}0\ ] ] so there exists a measurable function @xmath165 for which hold . + [ lem4 ] given @xmath52 and @xmath12 , for @xmath159 we have @xmath166\xrightarrow{l\rightarrow\infty}0\ ] ] set @xmath167 for some @xmath168 , and we have @xmath169\nonumber\\ & \leq \mathbb{p}\left[\omega\in b_{n_l}:\sup_{|x|<k}\left|\tilde{y}^\omega_{l+1}\left(\theta+\frac{x}{l}\right)-\tilde{y}^\omega_{l+1}(\theta)\right| > \epsilon\right]+\mathbb{p}[b_{n_l}^c].\end{aligned}\ ] ] second part goes to zero by definition . for the first part , we observe that @xmath170 using lemma [ lem6 ] and chebyshev s inequality for first set on rhs , we have @xmath171\leq c_1\frac{n_l^{1 - 2\beta}}{\epsilon^2}.\ ] ] next we focus on second set in rhs of ( [ set ] ) . let @xmath172 and observe that @xmath173\leq \sum_{n=1}^{n_l}\mathbb{p}[(a_n^l)^c]\leq \frac{c_m}{(\ln l)^\delta |\sin\theta|^\delta } \sum_{n=1}^{n_l } \frac{1}{n^{\alpha\delta}}<\frac{c_2}{(\ln l)^\delta}\xrightarrow{l\rightarrow\infty}0.\ ] ] setting @xmath174 and we have @xmath175\xrightarrow{l\rightarrow\infty}1 $ ] . for @xmath176 , using we have @xmath177\frac{k}{l}\\ & \leq c_3k \frac{(\ln l)^{n_l}}{l}=o\left(e^{(\ln l)^\eta \ln\ln l -\ln l}\right)\xrightarrow{l\rightarrow\infty}0.\end{aligned}\ ] ] using above @xmath178\\ & \leq \mathbb{p}[c_l^c]+\mathbb{p}\left[\omega\in c_l:\sup_{|x|<k}\left|y^\omega_{n_l}\left(\theta+\frac{x}{l}\right)-y^\omega_{n_l}(\theta)+\frac{n_l x}{l}\right|>\frac{\epsilon}{3}\right]=\mathbb{p}[c_l^c]\end{aligned}\ ] ] combining , and we have the result . + it should be noted that in case of lemma [ lem2 ] and [ lem4 ] , the convergence is shown for the entire sequence . the problem that arises in proving clock process for entire sequence is the fact that the limit points of the sequence @xmath179 ( @xmath180 is the fractional part of @xmath80 ) is entire @xmath181 $ ] for almost every @xmath182 $ ] . before going into proof of main theorem , we choose the subsequence @xmath29 such that @xmath183 for some @xmath184 $ ] . let @xmath55 real valued with @xmath56 $ ] . given @xmath159 , uniform continuity of @xmath185 implies the existence of @xmath186 such that @xmath187 combining lemma [ lem4 ] and [ lem2 ] gives @xmath188\xrightarrow{l\rightarrow\infty}0.\ ] ] since @xmath189 is fixed , above statement can be modified to @xmath190\xrightarrow{l\rightarrow\infty}0.\ ] ] set @xmath191 and , using the enumeration resulting from lemma [ lem5 ] , for @xmath192 we get @xmath193 using the minimality of definition for @xmath125 , we have @xmath194 , so set @xmath195 , which provides @xmath196 for @xmath192 , using and above @xmath197 so @xmath198\nonumber\\ & \qquad\leq \mathbb{p}[\omega_l^c]+{\mathop{\mathbb{e}}}_{\omega\in\omega_l}\left[\left|e^{i \sum_k f(l(e^{\omega , l}_k - e_0))}-e^{i\sum_k f(-2(k\pi+\phi^{\omega , l}_\theta)\sin\theta)}\right|\right]\nonumber\\ & \qquad\leq \mathbb{p}[\omega_l^c]+{\mathop{\mathbb{e}}}_{\omega\in\omega_l}\left[\left|e^{i \sum_k \left[f(2l(\cos\left(\theta+\frac{x^{\omega , l}_k}{l}\right)-\cos\theta))-f(-2(k\pi+\phi^{\omega , l}_\theta)\sin\theta)\right]}-1\right|\right]\nonumber\\ & \qquad\leq \mathbb{p}[\omega_l^c]+{\mathop{\mathbb{e}}}_{\omega\in\omega_l}\left[\sum_k\left|f\left(2l\left(\cos\left(\theta+\frac{x^{\omega , l}_k}{l}\right)-\cos\theta\right)\right)-f(-2(k\pi+\phi^{\omega , l}_\theta)\sin\theta)\right|\right]\nonumber\\ & \qquad\leq \mathbb{p}[\omega_l^c]+\epsilon\sup_{\omega\in\omega_l}\#\{i:|x^{\omega , l}_i|<k \}\leq \mathbb{p}[\omega_l^c]+\frac{2\epsilon k}{\pi}.\end{aligned}\ ] ] last line follows using @xmath199 now using the subsequence @xmath29 , it is clear that @xmath200 converges , and set @xmath201 combining and gives @xmath202= { \mathop{\mathbb{e}}}_\omega\left[e^{i \sum_k f(-2(k\pi+\tilde{g}^\omega_\theta)\sin\theta)}\right]\ ] ] completing the proof . + the authors would like to thank fumihiko nakano for helpful comments and suggestions . the author dhriti r. dolai is supported by ncleo milenio de fsica mathemtica , icm grant rc120002 . the author anish mallick is partially supported by imsc project 12-r&d - ims-5.01 - 0106 .

in this work we study the spectral statistics for anderson model on @xmath0 with decaying randomness whose single site distribution has unbounded support . here we consider the operator @xmath1 given by @xmath2 , @xmath3 and @xmath4 are real iid random variables with absolutely continuous symmetric distribution @xmath5 such that @xmath6 . in case of @xmath7 , we are able to show that the eigenvalue process in @xmath8 is the clock process . * keywords : * local eigenvalue statistics , clock process , anderson model

as follows from the time inversion symmetry and the kramers theorem , the electronic states in centrosymmetric systems are at least doubly - degenerate . on the other hand , in the three- , two- and one - dimensional systems lacking a center of space inversion , the degeneracy of free bloch single - electron states is removed with exception of particular points and directions of the brillouin zone . the removal of spin degeneracy occurs due to spatially antisymmetric part of the single - particle periodic hamiltonian @xmath4 with allowance for the spin - orbit interaction . in terms of the effective hamiltonian @xmath5 the interaction appears as a spin - dependent contribution odd in the electron wave vector @xmath6 . this contribution is responsible for a number of fascinating and important effects being actively studied nowadays , see , e.g. , refs . [ ] . in nano- and heterostructures , the quantum confinement strongly modifies free - carrier dispersion . particularly in quantum wells ( qws ) , the parabolic conduction band turns into series of two - dimensional subbands shifted , in parallel , along the energy axis . in the absence of inversion center , the spin - orbit interaction splits each subband with the splitting described by linear and , sometimes , cubic @xmath6 terms in the 2@xmath22 effective hamiltonian.@xcite because of the more complex valence band structure , the dispersion of holes is also much more complicated than that in the conduction band . rashba and sherman @xcite were the first to calculate the spin splitting of the topmost heavy- and light - hole subbands in qws grown along @xmath7 $ ] from zinc - blende lattice semiconductors by using the bulk effective hamiltonian ( four - by - four matrix ) consisting of the conventional luttinger hamiltonian and spin - dependent terms of the order of @xmath8 . they imposed the simplest conditions for the four - component hole envelope wave function , @xmath9 , on the boundaries of the qw and obtained @xmath10-linear terms in the effective hamiltonians of two - dimensional hole subbands . direct extension of the procedure developed in ref . for the realistic models of quantum confinement , particularly , including the effects of finite barrier , deems impossible owing to the presence of @xmath11 spin - dependent term in the bulk @xmath12 hamiltonian . this term makes consistent matching of the valence - band wave functions really challenging , since it leads to a @xmath13 contribution to the velocity operator @xmath14 and , therefore , to a singularity of the flux @xmath15 at the interface . here we develop the @xmath16-band @xmath17 model to calculate spin splittings of hole subbands in qws , which allows us to avoid the @xmath11-term problem . moreover , we propose additional terms in the boundary conditions for the 14-component envelope which naturally describe the interface heavy - light hole mixing arising due to anisotropy of chemical bonds at the interfaces . @xcite the developed @xmath17 approach presents an independent alternative to atomistic calculations of the spin - orbit splittings in qws . @xcite the paper is organized as follows : sec . [ sec : sym ] presents the symmetry analysis of the spin - dependent terms in the valence band hamiltonian , sec . [ sec:14 ] outlines the 14-band @xmath18 model and the spin - orbit splittings in the bulk semiconductor as well as boundary conditions for the qw structures ; numerical results and analytical approximations are presented in sec . [ sec : res ] , and sec . [ sec : concl ] contains brief conclusions . we begin the symmetry analysis by reminding that , in a bulk zinc - blende - lattice semiconductor , the expansion of spin - dependent part @xmath19 of the electron effective hamiltonian in the conduction band @xmath20 starts from the nonzero cubic term @xmath21 where @xmath22 is the band parameter , @xmath23 and @xmath24 are pseudovectors composed of the pauli matrices in the coordinate system @xmath25 $ ] , @xmath26 $ ] , @xmath7 $ ] and the cubic combinations @xmath27 etc . the expansion of the 4@xmath24 effective hamiltonian in the @xmath28 valence band starts from the first order term @xmath29/@xmath30 , where @xmath31 , @xmath32 , @xmath33 are the angular momentum matrices in the basis of spherical harmonics @xmath34 . @xcite although the t@xmath35 point symmetry allows this term , the coefficient is nonzero only due to the @xmath17 mixing between the valence states and the remote @xmath36 states , it is quite small and may be neglected for the gaas - based systems.@xcite in the @xmath28 band , the cubic-@xmath6 term contains three linearly independent contributions , as follows , @xmath37\ : , \end{gathered}\ ] ] where , for further convenience , two of the three coefficients are presented as products of the parameter @xmath38 which has the dimension of @xmath39 and dimensionless factors @xmath40 and @xmath41.@xcite it is noteworthy that the first term is non - relativistic in its origin , it is symmetry - allowed for the @xmath1 band . two terms in the second line of eq . are relativistic and one can see that the last summand contains the `` dangerous '' contribution proportional to @xmath11 . in the following symmetry analysis we consider only symmetrical qw structures grown along the crystallographic axes [ 001 ] , [ 110 ] or [ 111 ] and having the point symmetries ( i ) d@xmath42 , ( ii ) c@xmath43 and ( iii ) c@xmath44 , respectively . the cartesian coordinates are conveniently chosen along the axes ( i ) @xmath45,\ : y \parallel [ 010],\ : z \parallel [ 001]$ ] or @xmath46,\ : y_1 \parallel [ 110],\ : z_1 \parallel [ 001]$ ] , ( ii ) @xmath47,\ : y_2 \parallel [ 001],\ : z_2 \parallel [ 110]$ ] , and ( iii ) @xmath48,\ : y_3 \parallel [ \bar{1}10],\ : z_3 \parallel [ 111]$ ] . in a qw structure the @xmath28 valence band is split into the heavy- and light - hole - like states . in the following we choose the basic states at the @xmath49-point ( @xmath50 ) transforming under the symmetry operations as the bloch functions @xmath51 or @xmath52/\sqrt{6 } , \\ & & \psi_{lh}^{(2 ) } \equiv |\gamma_8 , -1/2 \rangle = [ 2 \downarrow \mathcal z_j + \uparrow ( \mathcal x_j - { \rm i } \mathcal y_j)]/\sqrt{6}\ : . \nonumber\end{aligned}\ ] ] here @xmath53 are the spin - up and spin - down two - component columns , @xmath54 and @xmath55 are the periodic orbital bloch functions in the chosen coordinate system @xmath56 ( @xmath57 ) , and for simplicity we omit the index @xmath58 in the spin columns . the states in the heavy - hole subbands @xmath59 and light - hole subbands @xmath60 transform according to the @xmath20 spinor representation of the point group d@xmath42 whereas the eigenstates @xmath61 and @xmath62 form the bases for the @xmath63 representation . in the method of invariants@xcite , the 2@xmath22 matrix effective hamiltonian in each hole subband is decomposed into a linear combination of four basis matrices . since both direct products @xmath64 and @xmath65 reduce to the same direct sum of irreducible representations @xmath66 , the basis matrices can be chosen common for the subbands of @xmath20 and @xmath63 symmetries . if the basic functions of the spinor representations are chosen in the form , , then the set of basic matrices includes the identity matrix ( @xmath67 representation ) , pseudospin matrix @xmath68 ( @xmath69 representation ) and two pseudospin matrices @xmath70 transforming as the pair of wave vector components @xmath71 ( @xmath72 representation ) . this allows one to write the linear-@xmath6 term in the effective hamiltonian as @xmath73}_n = \beta^{(n)}_1 ( \sigma_{x_1 } k_{y_1 } + \sigma_{y_1 } k_{x_1 } ) = \beta^{(n)}_1 ( \sigma_{x } k_{x } - \sigma_{y } k_{y } ) \:,\ ] ] where @xmath74 or @xmath75 , @xmath76 , and @xmath77 are the subband parameter . in eq . , the effective hamiltonian is presented in the two coordinate systems @xmath78 and @xmath79 relevant for the @xmath80 structures . we stress that the same form of the effective hamiltonian for the heavy - hole and light - hole subbands results from the special order of the bloch functions in eqs . and . both heavy- and light - hole states transform according to the same spinor representation @xmath72 of the group c@xmath43 . among components @xmath81 , @xmath82 only @xmath83 and @xmath84 transform according to equivalent representations . as a result , the linear-@xmath6 term has the form @xmath85}_n = \beta^{(n)}_2 \sigma_{z_2 } k_{x_2}\:,\ ] ] with @xmath86 being the subband parameters . the pair of functions ( [ psihh ] ) and the states @xmath87 transform according to the reducible representation @xmath88 of the c@xmath44 point group . the direct product @xmath89 does not contain the @xmath36 representation which means that the @xmath10-linear splitting of the heavy - hole states is symmetry - forbidden . the first non - vanishing spin - dependent contribution to the heavy - hole effective hamiltonian is cubic in @xmath6 and has the form @xcite @xmath90 } = \gamma_1^{(\nu ) } \sigma_{x_3 } k_{y_3 } \left ( k_{y_3}^2 - 3k_{x_3}^2 \right ) \hspace{1 cm } \\ + \gamma_2^{(\nu ) } \sigma_{y_3 } k_{x_3 } \left ( k_{x_3}^2 - 3k_{y_3}^2 \right ) + \gamma_3^{(\nu ) } \sigma_{z_3 } k_{y_3 } \left ( k_{y_3}^2 - 3k_{x_3}^2 \right ) \nonumber\end{aligned}\ ] ] with three independent parameters @xmath91 , @xmath92 and @xmath93 . by contrast , @xmath10-linear terms are allowed in the dispersion of light - hole subbands . indeed , the functions ( [ psilh ] ) and the light - hole states @xmath75 transform according to the two - dimensional representation @xmath94 . the product @xmath95 contains a @xmath36 representation meaning that the @xmath10-linear light - hole splitting is described by @xmath96 } = \beta_3^{(\nu ) } \left ( \sigma_{x_3 } k_{y_3 } - \sigma_{y_3 } k_{x_3 } \right)\:,\ ] ] with a single parameter @xmath97 . the 14-band @xmath98 model , called sometimes 5-level @xmath98 model or the 14@xmath214 extended kane model , displays the full symmetry of a zinc - blende - lattice crystal and describes in detail the electron dispersion in the vicinity of the @xmath49 point in materials.@xcite the model includes the @xmath99 and @xmath100 valence bands formed from the orbital bloch functions @xmath101 , @xmath102 , @xmath103 ( @xmath1 representation in the coordinate system @xmath78 ) , the lowest conduction band @xmath104 formed from the invariant orbital function @xmath105 ( @xmath67 symmetry ) and the higher conduction bands @xmath106 and @xmath107 originating from the @xmath1-symmetry orbital functions @xmath108 , @xmath109 , @xmath110 . for the spinor @xmath49-point bloch functions @xmath111 @xmath112 , we use the notations @xmath113 and @xmath114 , the @xmath20 basis is taken in the form @xmath115@xmath105 , @xmath116@xmath105 , and the @xmath117 basis is given by eqs . ( [ psihh ] ) and ( [ psilh ] ) , the @xmath63 basic functions are also taken in the canonical form.@xcite the model contains eight parameters of the 14-band model , namely , the band gap @xmath118 , the energy distance @xmath119 between the @xmath107 and @xmath120 states , spin - orbit splittings @xmath121 and @xmath122 of the valence and higher conduction bands , interband matrix elements of the momentum operator @xmath123 and , finally , the interband matrix element of the spin - orbit interaction between the valence and higher conduction bands defined by @xmath124 where @xmath125\bm p)/4m_0 ^ 2c^2 $ ] is the spin - orbit hamiltonian , @xmath126 is the spin - independent single - electron periodic potential , @xmath127 is the speed of light , and @xmath128 is the free - electron mass . note that hereafter we ignore the difference between the generalized momentum operator @xmath129 and the operator @xmath130 because the @xmath131 correction is usually negligibly small.@xcite the 14@xmath214 hamiltonian matrix @xmath132 is a sum of the diagonal matrix elements @xmath133 and off - diagonal matrix elements linear either in @xmath134 or in @xmath6 . as frequently used in the simplified multiband @xmath98 models , @xcite we ignore the free - electron term @xmath135 , which in the case of qw structures reduces the number of boundary conditions at an interface from 28 to 14 and simplifies numerical calculations . .[tab : mass ] analytic expressions for the effective mass of an electron in the @xmath20 conduction band and the luttinger parameters @xmath136 , @xmath137 and @xmath138 for the @xmath99 band.@xcite [ cols="^,^,^ " , ] to conclude , we have presented here the 14-band @xmath18 model extended to allow for the reduced microscopic symmetry of qw interfaces which makes it possible to calculate the spin - orbit splitting of hole subbands in qws . we proposed a simple boundary condition , eq . , which takes into account heavy - light hole mixing at the interface due to anisotropic orientation of interface chemical bonds . main contributions to the hole spin splitting are identified . the developed model has been applied to calculate the valence - band spin splittings in ( 001 ) qws , but it can be used as well for qws of any crystallographic orientation including the ( 110 ) and ( 111 ) orientations . the results of numerical calculations are well described by the developed analytical theory . moreover , we have demonstrated that the large values of the spin splitting for the topmost heavy - hole subband predicted in ref . on the basis of atomistic calculations can be ascribed to the relatively strong interface - induced mixing of heavy- and light - hole states . 99 f. meier and b. zakharchenya , eds . , optical orientation ( 1984 ) . i. uti , j. fabian , and s. sarma , rev . 76 , 323 ( 2004 ) . j. fabian , a. matos - abiague , c. ertler , p. stano , and i. uti , acta phys . slovaca * 57 * , 565 ( 2007 ) . m. i. dyakonov , ed . , spin physics in semiconductors ( springer - verlag : berlin , heidelberg , 2008 ) . kusraev and g. landwehr , eds . , semicond . * 23 * , n11 ( 2008 ) , special issue on optical orientation . semiconductors * 42 * , n8 ( 2008 ) , special issue in memory of v.i . l. w. molenkamp and j. nitta , eds . , semicond . * 24 * , n6 ( 2009 ) , special issue on effects of spin - orbit interaction on charge transport . jiang , and m.q . weng , phys . rep . * 493 * , 61 ( 2010 ) . r. winkler , _ spin - orbit coupling effects in two - dimensional electron and hole systems _ ( springer , 2003 ) . ivchenko , _ optical spectroscopy of semiconductor nanostructures _ ( alpha science international , harrow , uk , 2005 ) . e. i. rashba and e. y. sherman , phys . a * 129 * , 175 ( 1988 ) . i. l. aleiner and e. l. ivchenko , jetp letters * 55 * , 692 ( 1992 ) . ivchenko , a.yu . kaminski , and u. roessler , phys . b * 54 * , 5852 ( 1996 ) . o. krebs and p. voisin , phys . lett . * 77 * , 1829 ( 1996 ) . l. vervoort , r. ferreira , and p. voisin , phys . b. * 56 * , 12744 ( 1997 ) ; 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we propose the 14-band @xmath0 model to calculate spin - orbit splittings of the valence subbands in semiconductor quantum wells . the reduced symmetry of quantum well interfaces is incorporated by means of additional terms in the boundary conditions which mix the @xmath1 conduction and valence bloch functions at the interfaces . it is demonstrated that the interface - induced effect makes the dominating contribution to the heavy - hole spin splitting . a simple analytical expression for the interface contribution is derived . in contrast to the 4@xmath24 effective hamiltonian model , where the problem of treating the @xmath3 term seems to be unsolvable , the 14-band model naturally avoids and overcomes this problem . our results are in agreement with the recent atomistic calculations [ j .- w . luo et al . , phys . rev . lett . * 104 * , 066405 ( 2010 ) ] .

supermassive black holes ( bhs ) are ubiquitous in the centre of massive galaxies . for most of the time they are in a quiescent state , but sometimes they can accrete matter from the surroundings and power an active galactic nucleus ( agn ; ho 2008 ) . stars orbiting around the central bh of a galaxy interact with each other , increasing the probability for one of them to be scattered on a low angular momentum orbit ( alexander 2012 ) . a tidal disruption event ( tde ) could thus occur contributing to the bh flaring on time - scales of months or years ( e.g. rees 1988 ; phinney 1989 ) . for solar mass stars , this occurs when the ( non - spinning ) bh mass is less than about @xmath2 . for heavier bhs , these stars cross the horizon and are fully swallowed before being tidally disrupted ( macleod , ramirez - ruiz & guillochon 2012 ) . as a consequence , tdes contribute to the detection of otherwise quiescent bhs in inactive ( or weakly active ) galaxies in a mass interval somewhat complementary to that probed in surveys of bright agns and qsos ( vestergaard & osmer 2009 ) . the total tidal disruption of a single star of mass @xmath3 and radius @xmath4 moving on a parabolic orbit around the central bh of a galaxy of mass @xmath5 occurs when its pericentre radius @xmath6 is less than about the so - called bh tidal radius @xmath7 corresponding to the distance where the bh tidal force overcomes the star self - gravity at its surface ( hills 1975 ; frank & rees 1976 ) . on the contrary , if @xmath8 , the star undergoes less distortion and suffers only partial disruption . the value of the impact parameter @xmath9 defines how deep the disruption is ( guillochon & ramirez - ruiz 2013 , 2015a ) . roughly , only about half of the produced stellar debris remains bound to the bh and accretes on to it , powering the emission of a characteristic flare ( e.g. rees 1988 ; phinney 1989 ) . in the regime of partial tdes , the star , if on a bound orbit , could transfer a fraction of its mass to the bh every time it passes through the pericentre of its orbit , thus powering one flare for every orbital period and maybe ` spoon - feeding ' the quiescent luminosity of weakly active galaxies ( macleod et al . 2013 ) . tdes are quite rare events , with estimated rates of @xmath10 @xmath11 @xmath12 ( e.g. donley et al . 2002 ) . despite this and sparse observations , a few tdes have been observed mainly in the optical - uv ( renzini et al . 1995 ; gezari et al . 2006 , 2008 , 2009 , 2012 ; komossa et al . 2008 ; van velzen et al . 2011 ; wang et al . 2011 , 2012 ; cenko et al . 2012a ; gezari 2012 ; arcavi et al . 2014 ; chornock et al . 2014 ; holoien et al . 2014 ; vinko et al . 2015 ) and soft x - ray bands ( bade , komossa & dahlem 1996 ; komossa & bade 1999 ; komossa & greiner 1999 ; grupe , thomas & leighly 1999 ; greiner et al . 2000 ; li , ramesh & kristen 2002 ; halpern , gezari & komossa 2004 ; komossa 2004 , 2012 , 2015 ; komossa et al . 2004 ; esquej et al . 2007 , 2008 ; cappelluti et al . 2009 ; maksym , ulmer & eracleous 2010 ; lin et al . 2011 , 2015 ; saxton et al . 2012 , 2015 ; maksym et al . 2013 ; donato et al . 2014 ; khabibullin & sazonov 2014 ; maksym , lin & irwin 2014 ) , but also in the radio and hard x - ray bands ( bloom et al . 2011 ; burrows et al . 2011 ; levan et al . 2011 ; zauderer et al . 2011 ; cenko et al . 2012b ; hryniewicz & walter 2016 ; lei et al . many theoretical studies have been carried out to understand the physics of tdes and model their accretion luminosity light curves ( hereafter just light curves or flares ) , considering stars approaching the bh on a variety of orbits , from parabolic to bound ( nolthenius & katz 1982 ; bicknell & gingold 1983 ; carter & luminet 1985 ; luminet & marck 1985 ; luminet & carter 1986 ; rees 1988 ; evans & kochanek 1989 ; phinney 1989 ; khokhlov , novikov & pethick 1993a , b ; laguna et al . 1993 ; diener et al . 1995 , 1997 ; ivanov & novikov 2001 ; kobayashi et al . 2004 ; rosswog , ramirez - ruiz & his 2008 , 2009 ; guillochon et al . 2009 ; lodato , king & pringle 2009 ; ramirez - ruiz & rosswog 2009 ; strubbe & quataert 2009 ; kasen & ramirez - ruiz 2010 ; lodato & rossi 2010 ; amaro - seoane , miller & kennedy 2012 ; macleod et al . 2012 , 2013 ; guillochon & ramirez - ruiz 2013 , 2015a ; hayasaki , stone & loeb 2013 ) . so far , only single - star tdes have been taken into account . however , most of the stars in the field are in binaries ( duquennoy & mayor 1991b ; fischer & marcy 1992 ) ; hence , it is worth also studying close encounters between binaries and galactic central bhs which can lead to the disruption of both the binary members . the topic was first discussed by mandel & levin ( 2015 ) , suggesting that in a binary - bh encounter under certain conditions both binary components may undergo tidal disruption in sequence immediately after the tidal binary break - up . a double - peaked flare is expected to occur , signature of such a peculiar event . in this paper , we present for the first time the results of a series of smoothed particle hydrodynamics ( sph ) simulations performed using the gadget2 code ( springel 2005 ; the code can be freely downloaded from http://wwwmpa.mpa-garching.mpg.de/gadget/ ) in the aim at studying the physics of double tidal disruptions and at characterizing the expected light curves . as a first exploratory study , we consider parabolic encounters of binaries with galactic central bhs in the newtonian regime , in order to explore which are the most favourable conditions for the occurrence of double - peaked flares . in particular , we address the following questions . are all simulated encounters leading to double - peaked light curves or are there cases of single - peaked light curves ? how can we disentangle the different outcomes ? how prominent are the double peaks ? the paper is organized as follows . in section 2 , we resume the conditions required for double tdes and the associated space of binary parameters ( mandel & levin 2015 ) . in section 3 , we initialize low - resolution sph simulations of binary - bh encounters with different @xmath6 values of the centre of mass ( cm ) of the binaries around the bh . not all encounters can lead to double tdes , and in section 4 we introduce a classification of the obtained outcomes . in section 5 , we show the results of a selected sample of high - resolution simulations and the light curves directly inferred from them . section 6 sums up results and conclusions . we are here interested in identifying the set conditions for the sequential tidal disruption of binary stars around galactic central bhs , following mandel & levin ( 2015 ) . tidal break - up of a binary on a parabolic orbit around a bh occurs if the binary cm around the bh enters a sphere of radius @xmath13 where @xmath14 and @xmath15 are the binary semimajor axis and total mass ( miller et al . 2005 ; sesana , madau & haardt 2009 ) . we notice that binary break - up comes before single - star tidal disruptions , given that @xmath16 ( see equations [ eq1 ] and [ eq2 ] ) . tidal break - up occurs when the specific angular momentum ( in modulus ) of the binary cm at pericentre becomes less than @xmath17 orbits which allow tidal binary break - up are called loss cone orbits ( merritt 2013 ) . a binary on a loss cone orbit is broken up after one pericentre passage , over a time - scale @xmath18 , corresponding to the orbital period of a binary on a circular orbit at the same distance from the bh . both stars of a binary can undergo a sequential tidal disruption immediately after the tidal binary break - up only if the specific angular momentum of the binary cm around the bh at the closest approach , defined as @xmath19 instantly changes from being greater than @xmath20 to becoming less than @xmath21 , where @xmath22 in this way , the binary enters intact the region of single - star tdes . this occurs if the binary experiences a large enough change @xmath23 , at least of the order of @xmath24 , in the specific circular angular momentum @xmath25 , over a time - scale @xmath26 . interactions with surrounding stars and/or massive perturbers can promote such a change ( perets , hopman & alexander 2007 ; alexander 2012 ) . we consider empty the portion of the loss cone , in phase space , corresponding to binaries that break up before entering the region of single - star tdes , and full the portion of the loss cone corresponding to binaries which can enter intact the region of single - star tdes ( merritt 2013 ) . in order to evaluate the distribution of the binary parameters associated with double disruptions , it is useful to determine @xmath27 , defined as the distance of the binary from the bh before experiencing the change @xmath23 in @xmath28 , separating the two regimes . considering two - body relaxation over a time - scale @xmath29 $ ] ( spitzer & hart 1971 ) as the main mechanism which drives changes in angular momentum , it is known that the change in specific circular angular momentum @xmath30 over a period @xmath26 is of the order of @xmath31 ( merritt 2013 ) . thus , the critical condition @xmath32 enables us to infer @xmath27 . if the binary is orbiting inside a bahcall - wolf density profile @xmath33 ( bahcall & wolf 1976 ) , @xmath27 reads @xmath34^{4/9 } \bigl(\frac{m_{\rm bh}}{m_{\rm * } } \bigr)^{28/27 } a_{\rm bin}^{4/9}\\ \sim10^{7 } \rm r_{\rm \odot } \bigl(\frac{1.3\times 10 ^ 6 \rm pc^{-3}}{\it n_{\rm 0}}\bigr)^{4/9 } \bigl(\frac{0.3 \rm pc}{\it r_{\rm 0}}\bigr)^{7/9 } \bigl(\frac{10}{\ln \lambda}\bigr)^{4/9 } \\ \times \bigl(\frac{\it m_{\rm bh}}{10^{6 } \rm m_{\rm \odot}}\frac{1 \rm m_{\rm \odot}}{\it m_{\rm * } } \bigr)^{28/27 } \bigl(\frac{\it a_{\rm bin}}{10 \rm r_{\rm \odot}}\bigr)^{4/9 } , \label{eq9}\end{gathered}\ ] ] taking @xmath35 and @xmath36 as for the milky way ( merritt 2010 ) . we note that @xmath27 is comparable to the radius of gravitational influence of a bh @xmath37 ( peebles 1972 ; merritt 2000 ) . a binary carries internal degrees of freedom , and in particular the relative velocity of the two binary components , @xmath38 , is clearly smaller than the orbital velocity of the binary cm relative to the bh , @xmath39 . the velocity of the two stars relative to the centre of mass of the stellar binary gives then a small contribution to the specific angular momentum of each binary star relative to the bh at @xmath40 that approximately is @xmath41 sequential disruptions are expected to be favoured when @xmath42 is small . indeed , the smaller @xmath42 is , the more each binary component has an orbit around the bh similar to the one of the binary cm , i.e. a similar pericentre passage . thus , we require @xmath43 where we approximated @xmath44 . for @xmath45 , @xmath46 , @xmath47 we need @xmath48 . hence , the second condition for double tdes , which joins the condition on @xmath23 , is the involvement of close binaries . furthermore , very close binaries are required in order to avoid their evaporation due to interactions with field stars before tidal binary break - up ( merritt 2013 ) . in the full loss cone regime , the parameter space of binaries that can undergo double tdes can be inferred from the rate of binary entrance in the region of stellar tdes per unit of @xmath49 and @xmath14 as found in mandel & levin ( 2015 ) : @xmath50 where @xmath51 is the probability for a binary to enter directly the single tde region ( merritt 2013 ) and @xmath52^{-1 } a_{\rm bin}^{-1}$ ] is the distribution function for @xmath14 given in @xmath53 ( 1924 ) , with @xmath54 and @xmath55 being the maximum and the minimum semimajor axes of stellar binaries in a generic galactic field . integration of equation [ eq10 ] over @xmath49 , between @xmath27 and @xmath56 , enables us to evaluate the number of binaries that may undergo sequential tidal disruption of their components per unit of time and unit of @xmath14 . the resulting integral scales as @xmath57^{-1 } r_{\rm * } a_{\rm bin}^{-19/9}. \label{eqrate}\ ] ] from kepler s law , we can connect @xmath14 with the internal orbital period of the stellar binaries @xmath58 to infer the number of events per unit of time and unit of @xmath58 . the resulting rate is @xmath59^{-1 } r_{\rm * } p_{\rm bin}^{-47/27}.\ ] ] we use this scaling to extract the initial conditions of our sph simulations . thus , in the case of solar mass stars ( i.e. @xmath60 ) , the contribution of double tdes to all tdes could be approximately estimated by integrating equation [ eqrate ] over @xmath14 between @xmath61 and @xmath62 and dividing it by the corresponding integral obtained after integration over @xmath49 of equation [ eq10 ] , with @xmath63 in place of @xmath14 ( also in equation [ eq9 ] ) . to @xmath14 in equation [ eq10 ] , @xmath64 . ] this ratio scales as @xmath65^{-1}$ ] , which gives a maximum of @xmath66 per cent assuming @xmath67 and @xmath68 and considering that the multiplicity of stars is single : double @xmath69 50:50 for 100 solar - type stars ( duquennoy & mayor 1991b ) , disregarding uncertainties in the number of very close binaries . the definition of the parameter space of binaries that may be double tidally disrupted is fundamental to guide us to sensibly define the initial conditions of a small number of representative low - resolution simulations aimed at checking different outcomes from different initial parameters , and particularly from different pericentre radii of the binary cm . the simulations in this paper are performed using the treesph code gadget2 ( springel 2005 ) . in sph codes , a star is represented by a set of gas particles . each particle is characterized by a spatial distance , the smoothing length , over which its properties are ` smoothed ' by its kernel function , i.e. evaluated by summing the properties of particles in the range of the kernel according to the kernel itself ( price 2005 ) . in particular , in gadget2 the smoothing length of each particle is defined so that its kernel volume contains a constant mass , and is allowed to vary with time , thus adapting to the local conditions . the kernel adopted here is the one used most commonly and is based on cubic splines ( monaghan & lattanzio 1985 ) . on the other hand , gravitational interactions between particles are computed through a hierarchical oct - tree algorithm , which significantly reduces the number of pair interactions needed to be computed . the definition of a gravitational softening length @xmath70 , where @xmath71 is the total number of particles , prevents particle overlapping . gadget2 enables us to follow the temporal evolution of single particle properties and to infer from them tde light curves ( see section [ subsection ] ) . we run 14 low - resolution simulations of parabolic encounters between equal - mass binaries and bhs ( le runs ) to test the nature of the outcomes for different initial conditions , varying binary parameters , @xmath5 and @xmath6 . the stellar binaries are first evolved in isolation for several dynamical times to ensure their stability . the bh force is implemented in the code analytically , as a newtonian potential , and particles which fall below the innermost stable circular orbit radius @xmath72 are excised from simulations . we consider equal solar mass stars modelled as polytropes of index 5/3 and we sample each of them with @xmath73 particles . some correspondent high - resolution simulations are presented in section [ subsection ] . the initial binary internal orbital periods @xmath58 and semimajor axes @xmath14 are extracted according to the distributions described in section [ basics ] . based on the work of duquennoy & mayor ( 1991a ) , we consider binaries with @xmath74d ( @xmath75 ) @xmath76d ( @xmath77 ) to be circular , binaries with @xmath78d @xmath79d ( @xmath80 ) to have internal eccentricities distributed according to a gaussian of mean 0.3 and standard deviation 0.15 and binaries with @xmath81d @xmath82yr ( @xmath83 ) to have internal eccentricities which follow a thermal distribution @xmath84 . in order to avoid immediate collisions between the binary components , the initial pericentre radius of the internal binaries ( i.e. @xmath85 ) is set greater than twice the sum of the stellar radii , which are @xmath86 with @xmath87 for @xmath88 and @xmath89 for @xmath90 , according to kippenhahn & weigert ( 1994 ) , @xmath60 for @xmath91 . binaries are then placed on parabolic orbits around the bh at an initial distance 10 times greater than the tidal binary break - up radius @xmath40 , thus preventing initial tidal distortions from the bh . bhs of masses @xmath92 and @xmath93 are considered . the nominal pericentre distances @xmath6 are generated between @xmath61 and @xmath94 ( mandel & levin 2015 ) . stars are placed on keplerian orbits , and their positions and velocities relative to their binary centre of mass and to the bh are assigned accordingly . the initial internal binary plane is set , arbitrarily , perpendicular to the orbital plane around the bh . the results of these simulations are shown in section [ following ] . tables [ 1 ] and [ 2 ] summarize the results of our low - resolution simulations as a function of @xmath5 , @xmath14 and @xmath6 . .outcomes of our low - resolution sph simulations of parabolic binary - bh encounters ( @xmath91 , @xmath60 ) as a function of @xmath14 and @xmath6 . here @xmath95 , @xmath96 , @xmath97 , @xmath98 . td - tde stands for total double tde , atd - tde for almost total double tde ( i.e. more than @xmath99 of stellar mass lost ) , pd - tde for partial double tde , mg for merger , bbk for binary break - up without stellar disruptions.[1 ] [ cols="^,^,^,^,^,^,^ " , ]

in a galactic nucleus , a star on a low angular momentum orbit around the central massive black hole can be fully or partially disrupted by the black hole tidal field , lighting up the compact object via gas accretion . this phenomenon can repeat if the star , not fully disrupted , is on a closed orbit . because of the multiplicity of stars in binary systems , also binary stars may experience in pairs such a fate , immediately after being tidally separated . the consumption of both the binary components by the black hole is expected to power a double - peaked flare . in this paper , we perform for the first time , with gadget2 , a suite of smoothed particle hydrodynamics simulations of binary stars around a galactic central black hole in the newtonian regime . we show that accretion luminosity light curves from double tidal disruptions reveal a more prominent knee , rather than a double peak , when decreasing the impact parameter of the encounter and when elevating the difference between the mass of the star which leaves the system after binary separation and the mass of the companion . the detection of a knee can anticipate the onset of periodic accretion luminosity flares if one of the stars , only partially disrupted , remains bound to the black hole after binary separation . thus knees could be precursors of periodic flares , which can then be predicted , followed up and better modelled . analytical estimates in the black hole mass range @xmath0 show that the knee signature is enhanced in the case of black holes of mass @xmath1 . [ firstpage ] hydrodynamics - methods : numerical - binaries : close - galaxies : kinematics and dynamics - galaxies : nuclei

the investigation of non - classical correlations in mixed states of composite quantum systems has attracted strong attention in recent years . while in pure states such correlations can be identified with entanglement @xcite , in the case of mixed states , separable ( unentangled ) states , defined in general as convex mixtures of product states @xcite , i.e. , as states which can be generated by local operations and classical communication ( locc ) , may still exhibit non - classical features . the latter emerge from the possible non - commutativity of the different products and lead , for instance , to a finite value of the quantum discord @xcite and other recently introduced related quantifiers of quantum correlations @xcite . these quantifiers include the one - way information deficit @xcite , the geometric discord @xcite , generalized entropic measures @xcite and more recently the local quantum uncertainty @xcite and the trace distance discord @xcite . while entanglement is certainly necessary for quantum teleportation @xcite and for an exponential speed - up in pure state based quantum computation @xcite , interest on these new measures has been triggered by the existence of mixed state based quantum algorithms like that of @xcite , able to achieve an exponential speedup over the best classical algorithms for a certain task , with vanishing entanglement @xcite but finite quantum discord @xcite . and various operational interpretations of the quantum discord and other related measures have been provided @xcite . in this article we will concentrate on the quantum discord @xcite and the generalized entropic measures of @xcite , which include as particular cases the von neumann based one - way information deficit @xcite and the geometric discord @xcite , and which represent a generalized information deficit . the quantum discord as well as all other related measures require a rather complex minimization over a local measurement or operation which has limited their applicability to small systems or special states . the optimization problem for the quantum discord was in fact recently shown to be np complete @xcite . the advantage of the generalized entropic formalism is , first , the possibility of using simpler entropic forms like the linear entropy , which , as will be discussed in section 2 , enables an easier evaluation ( it does not require the diagonalization of the density matrix ) and a more direct experimental access ( it can be determined without a full state tomography ) . this entails that an explicit solution of the associated optimization problem for certain states can be achieved . the generalized formalism also allows to identify some universal properties , i.e. valid for any entropic form ( and not just for a particular choice of entropy ) satisfied by the post - measurement state . we first provide in section 2 an overview of the main concepts and properties associated with these measures . we then apply these measures to examine the quantum correlations of spin pairs in the exact ground state of finite spin @xmath1 chains with @xmath0-type couplings in a transverse magnetic field , through their entanglement , quantum discord and information deficit . all separations between the pairs are considered . several important studies of the quantum discord in spins chains have been made @xcite , but the relation with the generalized information deficit and the differences between their optimizing measurements in these spin pairs have not yet been analyzed in detail . we have recently investigated these aspects for an @xmath2 spin chain in @xcite , and will here extend this analysis to the anisotropic @xmath0 case . it is first shown that in contrast with the pair entanglement , the quantum discord and the information deficit exhibit , for the exact ground state of these chains , common features such as an appreciable finite value below the critical field , for all separations . moreover , they approach a finite _ common non - zero value _ @xcite at the remarkable factorizing field @xcite that these chains can exhibit in the anisotropic case . on the other hand , we will also show that important differences between the quantum discord on the one side , and the standard and generalized information deficit on the other side , do arise in the minimizing local spin measurement that defines them . while in the quantum discord the direction of the latter is always orthogonal to the transverse field , in the other measures it exhibits a perpendicular to parallel transition as the field increases , which is present for all separations and which reflects significant qualitative changes in the reduced state of the pair . this difference indicates a distinct response of the minimizing measurement of these quantities to the onset of quantum correlations . we start by providing a brief overview of the basic notions . a pure state @xmath3 of a bipartite system @xmath4 is separable iff ( if and only if ) it is a product state @xmath5 . otherwise it is entangled . the schmidt decomposition @xcite @xmath6 where @xmath7 denote orthonormal states for subsystem @xmath8 and @xmath9 , @xmath10 , allows to easily distinguish separable pure states ( @xmath11 ) from entangled states ( @xmath12 ) . here @xmath13 is the schmidt rank of @xmath3 ( @xmath14 $ ] , with @xmath15 the hilbert space dimensions of @xmath8 ) . pure sate entanglement can be measured by the entanglement entropy @xcite @xmath16 where @xmath17 , are the reduced states of @xmath8 and @xmath18 is the von neumann entropy . we will set in what follows @xmath19 , such that @xmath20 for a maximally entangled two - qubit state ( @xmath21 , @xmath22 ) . on the other hand , a general mixed state @xmath23 ( @xmath24 , @xmath25 ) of a bipartite system @xmath4 is separable iff it can be expressed as a _ convex mixture _ of product states @xcite : @xmath26 where @xmath27 and @xmath28 denote mixed states for subsystem @xmath29 . otherwise , it is entangled . the meaning is that a separable state can be created by locc , i.e. , alice prepares a state @xmath30 with probability @xmath31 and tells bob to prepare a partner state @xmath32 . for pure states @xmath33 , eq.([3s ] ) is equivalent to the previous definition ( @xmath34 ) , but in the case of mixed states , product states @xmath35 are just a very particular case of separable states . the latter also include : a ) _ classically correlated states _ , i.e. states diagonal in a standard product basis @xmath36 , @xmath37 where @xmath38 and @xmath39 are orthonormal states of @xmath8 , b ) _ classically correlated states from one of the subsystems _ , say @xmath40 , which are of the form @xmath41 where @xmath42 and @xmath43 are states of @xmath44 , and which are then diagonal in a _ conditional _ product basis @xmath45 with @xmath46 the eigenstates of @xmath43 ( the case ( [ 4cc ] ) recovered when all @xmath43 commute ) , and c ) convex mixtures of product states which are not of the previous forms a ) or b ) . the latter typically possess entangled eigenstates . for this reason , it is much more difficult to determine whether a mixed state is separable or entangled . the well known positive partial transpose criterion @xcite ( @xmath47 , with @xmath48 for @xmath49 ) provides a necessary criterion for separability , which is sufficient for two - qubit or qubit - qutrit states . for mixed states , the marginal entropies @xmath50 , @xmath51 no longer provide a measure of entanglement . instead , it is possible to use the entanglement of formation @xcite , defined through the convex roof extension of the pure state definition : @xmath52 where the minimization is over all decompositions of @xmath23 as convex mixtures of pure states ( @xmath53 , @xmath27 ) and @xmath54 is the entanglement entropy of the pure state @xmath55 . ( [ 6ef ] ) vanishes iff @xmath23 is separable , and reduces to the entanglement entropy ( [ 2ee ] ) for pure states . it is an entanglement monotone @xcite , i.e. , it does not increase by locc , staying unaltered under local unitary operations @xmath56 . its evaluation is , however , difficult in general . a general analytic expression has been derived just for the two - qubit case @xcite , which will be specified in sec . while the marginal entropies are no longer entanglement indicators , it can still be shown @xcite that if @xmath57 or @xmath58 , @xmath23 is entangled , i.e. , @xmath59 eq . ( [ s ] ) provides an _ entropic criterion for separability _ @xcite ( necessary but not sufficient in general ) , which can be also extended to more general entropic forms @xcite and which will be invoked in sec . 2.3 . for the classically correlated states ( [ 4cc ] ) or in general ( [ 5cc ] ) , there is a complete local measurement on @xmath40 which leaves the state unaltered . this is not the case for entangled states nor for separable states not of the form ( [ 4cc ] ) or ( [ 5cc ] ) . let us recall here that a general positive operator valued measurement ( povm ) @xcite ) on system @xmath4 is defined by a set of operators @xmath60 satisfying @xmath61 , such that the probability of outcome @xmath62 and the joint state after such outcome are @xmath63 the post - measurement state if the outcome is unknown is then @xmath64 standard projective measurements correspond to the case where the @xmath65 are orthogonal projectors ( @xmath66 ) , while a local measurement on @xmath40 corresponds to @xmath67 . by a complete local measurement on @xmath40 we will mean one based on rank one orthogonal projectors @xmath68 . it is then apparent that the states ( [ 4cc ] ) and ( [ 5cc ] ) remain unchanged ( @xmath69 ) after a local measurement on @xmath40 based on the projectors @xmath70 . for the states ( [ 4cc ] ) ( but not necessarily ( [ 5cc ] ) ) there is also a local measurement on @xmath44 ( that based on the projectors @xmath71 ) which leaves them unchanged . the quantum discord @xcite is a measure of quantum correlations which , unlike the entanglement of formation , can distinguish the classically correlated states ( [ 5cc ] ) from the rest of separable states : it vanishes iff @xmath23 is of the form ( [ 4cc ] ) or ( [ 5cc ] ) , being positive in the other separable states c ) , and reduces to the entanglement entropy ( [ 2ee ] ) in the case of pure states . it can be defined as the minimum difference between two distinct quantum versions of the mutual information , or equivalently , of the conditional entropy : @xmath72\nonumber\\&= & \mathop{\rm min}_{m_b } s(a|b_{m_b})-s(a|b)\,,\label{d2}\end{aligned}\ ] ] where the minimization is over all local measurements @xmath73 on @xmath40 and @xmath74 are , respectively , the standard quantum mutual information and conditional entropy while @xmath75 are the mutual information and conditional entropy after the local measurement @xmath73 , with @xmath76 the reduced state of @xmath44 after outcome @xmath62 . ( [ d2 ] ) is always non - negative @xcite , a property which arises from the concavity of the _ conditional _ von neumann entropy @xcite . in the case of complete local projective measurements @xmath73 we have @xmath77 where @xmath78 and @xmath79 is the post - measurement state ( [ rhop ] ) . it is then apparent that if the state is of the form ( [ 4cc ] ) or ( [ 5cc ] ) , a measurement @xmath73 based on the projectors @xmath80 leads to @xmath81 and hence @xmath82 . for all other states ( i.e. , entangled states or separable states not of the form ( [ 4cc ] ) or ( [ 5cc ] ) ) , @xmath83 . in the case of pure states , @xmath84 while @xmath85 if @xmath73 is any complete local measurement , entailing @xmath86 . for mixed states , the quantum discord can be related to the entanglement of formation @xmath87 with a third system @xmath88 purifying the whole system @xcite . the mutual information @xmath89 is a measure of all correlations between @xmath44 and @xmath40 , being non - negative and vanishing just for product states @xmath90 . the bracket in ( [ d2 ] ) can then be interpreted as the difference between all correlations ( classical+quantum ) present in the original state minus the classical correlations left after the local measurement on @xmath40 , which leaves then the quantum correlations . the evaluation of eq . ( [ d2 ] ) is , nevertheless , difficult in the general case , being in fact an np complete problem @xcite due to the minimization over all possible local measurements @xmath73 . nonetheless , the minimum is always attained for measurements based on rank one projectors @xmath91 , not necessarily orthogonal @xcite . the one - way information deficit can be considered as an alternative measure of quantum correlations , with basic properties similar to those of the quantum discord . it can be defined as @xcite @xmath92 where @xmath79 is the post - measurement state ( [ rhop ] ) and @xmath73 is here restricted to complete local projective measurements on @xmath40 , such that @xmath79 is of the form ( [ 5cc ] ) . like the quantum discord , eq.([id ] ) is a non - negative quantity which also vanishes just for the states ( [ 4cc ] ) or ( [ 5cc ] ) , and which also reduces to the entanglement entropy ( [ 2ee ] ) in the case of pure states . these properties will be shown below in a more general context , although they are also apparent from the alternative expression @xmath93 where @xmath94 is the relative entropy @xcite , a quantity satisfying @xmath95 , with @xmath96 iff @xmath97 . ( [ idr ] ) can be shown by noting that @xmath79 is the diagonal part of @xmath23 in the basis defined by the projective measurement ( the minimization in ( [ idr ] ) can in fact be extended to all @xmath79 of the form ( [ 5cc ] ) @xcite ) . nevertheless , differences with the quantum discord may arise in the minimizing measurement , as discussed in the next section . we also note that if the minimizing measurement of @xmath98 is projective and in the basis of eigenstates of @xmath99 , then @xmath100 and eqs.([d2])([sacb3 ] ) lead to @xmath101 . otherwise @xmath102 , since for projective measurements eqs . ( [ d2])([sacb3 ] ) imply @xmath103\leq s(\rho'_{ab})-s(\rho_{ab})$ ] . ( [ i d ] ) admits a simple interpretation in terms of the entanglement generated between the system and a measuring apparatus @xmath104 performing the complete local measurement @xcite . the measurement on the local basis @xmath105 can be represented through a unitary operator @xmath106 satisfying @xmath107 , where @xmath108 is the initial state of the apparatus and @xmath109 an orthogonal basis of @xmath104 , such that @xmath110 since @xmath111 , it is seen that eq . ( [ i d ] ) is the difference between the entropy of the subsystem @xmath112 and that of the total system @xmath113 after the measurement , and according to eq . ( [ s ] ) , such difference can be positive only if there is entanglement between @xmath112 and @xmath104 . thus , a positive @xmath114 indicates that entanglement between @xmath112 and @xmath104 is generated by _ any _ complete local measurement @xmath73 . on the other hand , if @xmath115 , then @xmath23 is of the form ( [ 5cc ] ) and for a measurement in the basis @xmath116 , @xmath117 is clearly separable , so that no entanglement is generated by this measurement . it can be shown @xcite that eq . ( [ i d ] ) coincides in fact with the minimum distillable entanglement between @xmath112 and @xmath104 generated by the complete local measurement on @xmath40 . a similar interpretation for the quantum discord in terms of the minimum partial distillable entanglement can also be obtained @xcite . other operational interpretations can be found in @xcite . it is possible in principle to extend eq . ( [ i d ] ) to more general entropic forms , since in contrast with the quantum discord ( [ d2 ] ) , its positivity is not related to specific properties of the von neumann entropy @xmath118 , as shown below . we consider here generalized entropies of the form @xcite @xmath119 where @xmath120 , with @xmath121 the eigenvalues of @xmath122 and @xmath123 a smooth strictly concave real function defined for @xmath124 $ ] and satisfying @xmath125 . these entropies fulfill the same basic properties as the von neumann entropy , with the exception of additivity : we have @xmath126 , with @xmath127 iff @xmath122 is a pure state ( @xmath128 ) , while all @xmath129 are maximum for the maximally mixed state @xmath130 , where @xmath131 is the hilbert space dimension of the system . moreover , they are strictly concave , i.e. , @xmath132 , for @xmath133 , @xmath27 , with equality iff all @xmath134 are coincident . the von neumann entropy is obviously recovered for @xmath135 . concavity of @xmath129 implies the fundamental majorization property @xmath136 where @xmath137 indicates that @xmath138 is _ majorized _ by @xmath122 @xcite ( also denoted as @xmath138 _ more mixed _ than @xmath122 ) : @xmath139 where @xmath140 , @xmath141 denote the eigenvalues of @xmath122 and @xmath138 sorted in _ decreasing _ order ( equality in ( [ prec0 ] ) obviously holds for @xmath142 ) . if the dimensions of @xmath122 and @xmath138 differ , eq . ( [ prec ] ) still holds ( for @xmath143 ) after completing with zeros the smallest set of eigenvalues . conversely , while the reverse of eq . ( [ prec ] ) does not necessarily hold , indicating that majorization provides a more strict concept of mixedness or disorder than that defined by a single choice of entropy , it does hold if @xmath144 @xmath145 @xmath146 of the previous form @xcite : @xmath147 eq . ( [ prec ] ) remains actually valid for more general entropic forms ( like increasing functions @xmath148 of @xmath149 or in general , schur concave functions @xcite ) , but eq . ( [ prec2 ] ) indicates that the forms ( [ sf ] ) are already sufficient to capture majorization . among the various properties implied by majorization , we mention that for states with the same dimension , @xmath137 iff @xmath122 is a convex mixture of unitary transformations of @xmath122 @xcite , i.e. , iff @xmath150 , with @xmath151 unitary and @xmath53 . now , for any projective measurement ( local or non - local ) performed on the system @xmath4 , it can be easily shown that @xmath152 @xmath145 @xmath149 , i.e. , @xmath153 the reason is that the post measurement state @xmath79 conserves just the diagonal elements @xmath154 of @xmath23 in a certain orthonormal basis @xmath155 determined by the projectors and hence , @xmath156 , where @xmath157 and @xmath158 denote here the eigenvalues and eigenvectors of @xmath23 . this relation is not restricted to rank one projectors ( just choose an orthonormal basis @xmath155 where @xmath79 is diagonal ) , so that it holds for local projective measurements . ( [ prec4 ] ) remains actually valid for any measurement satisfying @xmath159 , i.e. , which leaves the maximally mixed state @xmath160 unchanged @xcite . note also that strict concavity of @xmath149 implies @xmath161 iff @xmath69 , as is apparent from the previous demonstration . in fact , if the off diagonal elements of @xmath23 in the measured basis are sufficiently small , a second order expansion of @xmath162 leads to @xcite @xmath163 where the fraction is always positive due to the strict concavity of @xmath146 ( and should be replaced by its limit @xmath164 if @xmath165 ) . ( [ norm ] ) is essentially the square of a weighted norm of the off - diagonal elements of @xmath23 in the measured basis ( i.e. , of those lost in the measurement ) , and is therefore non - negative , vanishing ( if @xmath166 @xmath145 @xmath167 ) only if all off - diagonal elements are zero . we may then define the quantity @xcite @xmath168 where the minimization is again over all complete local measurements on @xmath40 . ( [ if ] ) is non - negative , due to eq . ( [ prec4 ] ) , and vanishes iff @xmath69 , i.e. , iff @xmath23 is already of the classically correlated form ( [ 4cc ] ) or ( [ 5cc ] ) . it therefore vanishes only for the states with zero quantum discord . it obviously also remains invariant under local unitary operations . in the case of pure states , it can be shown @xcite that the minimum of eq . ( [ if ] ) is always attained for a measurement in the basis @xmath169 determined by the schmidt decomposition ( [ 1sd ] ) , i.e. , in the basis formed by the eigenstates of @xmath99 , which leads to @xmath170 it therefore reduces to the _ generalized entanglement entropy _ @xmath171 of the pure state . the entanglement entropy can then be identified with the minimum information loss due to a local measurement @xcite . it is apparent that for pure states , @xmath172 , a property which does not hold in the general case . in the case of the von neumann entropy , @xmath173 becomes the standard information deficit ( [ i d ] ) and eq . ( [ ifpure ] ) implies that for pure states , it will coincide with the standard ( von neumann ) entanglement entropy , like the quantum discord . nevertheless , an important difference arises in the minimizing measurement , since that for the latter becomes undetermined in the case of pure states ( it can be any measurement based on rank one projectors @xcite ) , whereas all @xmath173 , including @xmath114 , require a measurement in the basis @xmath169 , which is fully undetermined only in the case of maximally mixed marginals . like the standard information deficit , @xmath173 is also an indicator of the minimum entanglement between the system and the measurement apparatus @xmath104 generated by a complete local measurement . the von neumann entropic criterion for separability ( [ s ] ) can actually be extended to any @xmath149 @xcite : @xmath174 the validity of eq . ( [ sfg ] ) for all @xmath149 is stronger than the von neumann based criterion ( [ if ] ) @xcite , and equivalent to the disorder criterion of separability @xcite ( @xmath175 @xmath176 @xmath177 ) . by the same arguments given below eq . ( [ oper ] ) , it follows that a positive @xmath173 , i.e. , @xmath178 , is indicating the existence of entanglement between @xmath112 and @xmath104 after _ any _ complete local projective measurement on @xmath40 . ( [ ifpure ] ) reflects an universal property exhibited by the local measurement minimizing @xmath173 for pure states : it is the same for all @xmath149 . such measurement , i.e. , a measurement in the basis @xmath169 determined by the schmidt decomposition of the pure state , is also optimum , for _ all _ @xmath149 , for the mixture of the pure state with the maximally mixed state @xcite , @xmath179\,.\label{mix}\ ] ] these states exhibit then an unambiguous _ least disturbing local measurement _ , in the sense that it minimizes all @xmath173 and leads to a `` least mixed '' post - measurement state @xmath180 which _ majorizes _ any other post - measurement state emerging after a local measurement . this property does not hold for an arbitrary initial state @xmath23 . in the general case , the projective measurement @xmath181 minimizing @xmath173 may depend on the choice of entropy @xmath149 . it can be shown that it must satisfy the necessary stationary condition @xcite @xmath182=0\,,\label{stat}\ ] ] where @xmath183 denotes the derivative of @xmath146 and @xmath79 is the post - measurement state ( [ rhop ] ) . ( [ stat ] ) implies , explicitly , @xmath184=0 $ ] , where @xmath185 and @xmath186 , with @xmath187 the eigenstates of @xmath43 . the minimizing measurement basis will not coincide in general with the eigenstates of @xmath99 , even though this holds for certain states , like pure states and the mixtures ( [ mix ] ) . ( [ stat ] ) shows that the eigenstates of @xmath99 will be stationary for any state @xmath23 where the non - zero off - diagonal elements are of the form @xmath188 with @xmath189 _ and _ @xmath190 , where @xmath191 and @xmath192 , @xmath193 are the eigenstates of @xmath194 and @xmath99 respectively @xcite . in the case of the quantum discord , and for @xmath73 restricted to complete local projective measurements , eq . ( [ stat ] ) is to be replaced by ( here @xmath195 ) @xcite @xmath182-[f'(\rho'_{ab}),\rho_b]=0\ , . \label{statqd}\ ] ] more explicit expressions can be obtained for a two - qubit system , where we may write a general state as @xmath196 where @xmath197 , @xmath198 , with @xmath199 the pauli operators , and @xmath200 the identity . since @xmath201 and @xmath202 for @xmath203 , we have ( @xmath204 @xmath205 a complete projective measurement on @xmath40 corresponds to a spin measurement along the direction of a unit vector @xmath206 , represented by projectors @xmath207 . after this measurement , eq . ( [ stat2q ] ) becomes @xmath208\ , . \label{rhopk}\ ] ] eq . ( [ stat ] ) leads then to the explicit equation @xcite @xmath209 where @xmath210 , @xmath211 are the eigenvalues of @xmath79 , with @xmath212 , and @xmath213 is a proportionality factor . in the case of the quantum discord , eq . ( [ statqd ] ) leads to a similar equation , with @xmath214 and @xmath215 , where @xmath216 are the eigenvalues of @xmath217 @xcite . one of the advantages of the generalized information deficit ( [ if ] ) is the possibility of using simple entropic forms which can be more easily evaluated ( and measured ) than the von neumann entropy . for instance , if @xmath218 , eq . ( [ sf ] ) becomes the so called linear entropy @xmath219 which follows from the linear approximation @xmath220 in the von neumann entropy , but is actually a quadratic function of @xmath122 , i.e. , a linear function of the purity @xmath221 . it is the simplest entropic form and its evaluation does not require the knowledge of the eigenvalues of @xmath122 ( see eq . ( [ s2tq ] ) below ) . moreover , purity , and hence @xmath222 , can be experimentally determined without a full state tomography @xcite . ( [ s2 ] ) is actually the @xmath223 case of the tsallis entropies @xcite , obtained for @xmath224 : @xmath225 eq . ( [ ts ] ) approaches the von neumann entropy @xmath118 for @xmath226 , being strictly concave for @xmath227 . we have normalized ( [ s2 ] ) and ( [ ts ] ) such that @xmath228 for a maximally mixed two - qubit state . in the case ( [ s2 ] ) , it is first seen that for post - measurements states @xmath79 , @xmath229 where @xmath230 . hence , the local projective measurement minimizing @xmath231 , which is that maximizing the post - measurement purity @xmath232 , leads to the post - measurement state with the minimum hilbert - schmidt distance to the original state . the associated deficit @xmath233 coincides , apart from a constant factor , with the geometric discord @xcite . for pure states , @xmath234 will then coincide with the linear marginal entropies : @xmath235 in two qubit systems , eq . ( [ i2p ] ) is just the squared _ concurrence _ @xcite of the pure state @xmath23 . while as a measure the geometric discord fails to satisfy some additional properties fulfilled by the quantum discord or the information deficit @xcite , it offers the enormous advantage of a simple analytic evaluation in qudit - qubit systems @xcite , as discussed below , also admitting through the purity a more direct experimental access . moreover , eq . ( [ norm ] ) shows that if @xmath23 is close to the maximally mixed state @xmath160 , all @xmath173 will become proportional to @xmath234 @xcite , as in this case @xmath236 is nearly constant . in fact , all @xmath129 are linearly related to @xmath222 in this limit @xcite . any state of a general system @xmath4 can be written in the form ( [ stat2q ] ) , replacing the pauli operators by a complete set of orthogonal operators @xmath237 in @xmath44 and @xmath40 satisfying @xmath238 , @xmath239 : @xmath240 where @xmath241 and @xmath242 ( now a @xmath243 matrix ) are again given by eq . ( [ vm ] ) . the @xmath244 entropy can then be readily evaluated as @xmath245\ , , \label{s2tq}\ ] ] where @xmath246 . if @xmath40 is now a qubit , the state after a spin measurement along direction @xmath206 on @xmath40 , will have the form ( [ rhopk ] ) with @xmath247 . we then obtain , using eq.([s2tq ] ) , @xmath248 where @xmath249 is a @xmath250 positive semidefinite symmetric matrix . hence , @xmath251 . its minimum @xmath234 can then be evaluated analytically as @xcite @xmath252 where @xmath253 is the largest eigenvalue of @xmath254 , the minimizing spin measurement being along the direction of the corresponding eigenvector . eq.([i2 m ] ) is valid for an arbitrary qudit - qubit state @xmath23 . let us notice that the stationary condition ( [ stat ] ) or ( [ stk ] ) reduces , for the linear entropy , precisely to the eigenvalue equation @xmath255 , as in this case @xmath256 and hence , @xmath257 , @xmath258 and @xmath259 @xcite . this indicates that the stationary measurements are those along the direction of the eigenvectors of @xmath254 . for arbitrary @xmath227 , we may similarly define the quantities ( in what follows @xmath260 ) @xmath261\,,\label{irq2}\end{aligned}\ ] ] where @xmath262\,,\;\;q>0\,,\ ] ] are the _ renyi _ entropies @xcite , which are just increasing functions of the tsallis entropies ( [ ts ] ) ( and also approach the von neumann entropy for @xmath263 ) . eqs.([iq])([irq ] ) are again non - negative , vanishing iff @xmath23 is of the form ( [ 4cc ] ) or ( [ 5cc ] ) , and approach the von neumann information deficit ( [ i d ] ) for @xmath263 . ( [ irq2 ] ) is again just an increasing function of @xmath264 ( for fixed @xmath23 ) and does not depend on the addition of an uncorrelated ancilla @xmath88 to @xmath44 ( @xmath265 ) , as @xmath266 cancels out . an analytic expression for @xmath267 valid for any two - qubit state can also be obtained @xcite . we consider a spin @xmath1 system with xyz couplings of arbitrary range , immersed in a transverse magnetic field @xmath40 along the @xmath268 axis . the hamiltonian reads @xmath269 where @xmath270 are the ( dimensionless ) components of the local spin at site @xmath271 , and @xmath272 the coupling strengths . the hamiltonian ( [ h ] ) commutes with the @xmath273 spin parity operator @xmath274 , irrespective of the coupling range , anisotropy , dimension , or geometry of the system @xcite , @xmath275=0,\;\;p_z=\exp[i\pi\sum_i ( s_{iz}+1/2)]=\prod_i(-\sigma_{iz})\ , , \label{pz}\ ] ] where @xmath276 . the non - degenerate eigenstates of @xmath277 will then have a definite @xmath273 parity @xmath278 . consequently , the reduced density matrix of an arbitrary spin pair @xmath279 in any non - degenerate eigenstate @xmath280 , @xmath281 will then commute with the @xmath273 parity operator of the pair @xmath282 : @xmath283=0 $ ] . in the standard basis @xmath284 , @xmath285 will therefore be an @xmath286-type state of form @xmath287 where the coefficients are all real ( since @xmath277 is real in the full standard basis ) and given by ( @xmath288 ) @xmath289 with @xmath290 . it corresponds to @xmath291 and @xmath292 along @xmath268 in ( [ stat2q ] ) ( @xmath293 ) , with @xmath242 _ diagonal _ , i.e. , @xmath294 , with @xmath295 , @xmath296 . positivity of @xmath285 implies @xmath297 , @xmath298 , with @xmath299 , @xmath300 non - negative . the single spin density matrix is @xmath301 both @xmath285 and @xmath302 will obviously be typically mixed due to the entanglement with the rest of the chain . in what follows we will consider translational invariant systems such that @xmath303 is site independent , i.e. , @xmath304 @xmath145 @xmath279 , implying @xmath305 . in this situation , @xmath306 and @xmath307 , @xmath308 @xmath145 @xmath149 . the entanglement of the pair can be measured by the entanglement of formation ( [ 6ef ] ) , which for two qubit states can be evaluated as @xcite @xmath309 where @xmath88 is the concurrence @xcite . for the states ( [ rij ] ) with @xmath310 , the concurrence of the pair is given by @xmath311\ , . \label{cij}\ ] ] the pair entanglement is of parallel type ( as in the bell states @xmath312 ) if the first entry in ( [ cij ] ) is positive and antiparallel ( as in @xmath313 ) if the second entry is positive @xcite ( just one of them can be positive ) . on the other hand , the quantum discord of the pair can be readily evaluated with the expressions ( [ rij ] ) and ( [ rhopk ] ) ( see @xcite for details ) . the ensuing minimization over the spin measurement direction @xmath206 ( we will consider here just projective measurements ) will normally lead to the direction corresponding to maximum correlation , according to general arguments of @xcite . in the @xmath0 chains which will be considered , i.e. , @xmath314 , with @xmath315 and @xmath316 , the quantum discord for the states ( [ rij ] ) will always prefer a measurement along the @xmath317 axis , irrespective of the field intensity @xcite . the information deficit ( [ i d ] ) can be evaluated in a similar way . in contrast with the quantum discord , the optimizing measurement direction will be affected by the field intensity , exhibiting a smooth transition from the @xmath317 to the @xmath268 direction as the field increases for the systems considered , as discussed below . the angle @xmath318 between @xmath206 and the @xmath268 axis can be determined from eq . ( [ stk ] ) , which leads explicitly to @xmath319 when @xmath320 @xcite , which is a transcendental equation ( as the @xmath321 depend on @xmath318 ) . the quadratic information deficit ( [ i2 ] ) can , however , be analytically evaluated with eq . ( [ i2 m ] ) . here @xmath254 is already diagonal , @xmath322 . assuming @xmath323 , as will occur in the cases considered , we obtain @xmath324}\nonumber\\ & = & { \textstyle 4{\rm min}[\alpha^2+\beta^2,\frac{a_+^2+a_-^2}{4}+\frac{c^2-(a_+-a_-)c+(\alpha-\beta)^2}{2}]}\,,\;\;\ ; \label{i2a}\end{aligned}\ ] ] with the minimizing measurement direction @xmath206 along the @xmath268 ( @xmath317 ) axis if the first ( second ) entry is minimum : @xmath325 this entails that as the field @xmath40 increases from @xmath326 , a sharp @xmath327 transition in the minimizing measurement direction will take place for @xmath328 , reflecting the change in the largest eigenvalue of the matrix @xmath254 . this transition becomes softened in the von neumann information deficit ( [ i d ] ) , where @xmath206 will evolve smoothly from the @xmath317 to the @xmath268 axis within a narrow field interval located in the vicinity of the @xmath328 transition . a measurement transition also occurs for other values of @xmath329 in the quantities ( [ iq])([irq ] ) ( see @xcite for an example ) . in figs . [ f1][f2 ] we show results for the exact ground state of a finite chain with @xmath330 spins coupled through cyclic ( @xmath331 ) first neighbor anisotropic @xmath0 couplings ( @xmath314 , @xmath332 for @xmath333 ) , for which the reduced pair states ( [ rij ] ) will depend just on the separation @xmath334 between the spins of the pair . the exact values of the elements of the density matrix ( [ rij ] ) can be obtained , for any size @xmath330 or separation @xmath335 , through the jordan - wigner fermionization of the model @xcite and its analytic parity dependent diagonalization @xcite ( see appendix ) . we will set @xmath336 , with @xmath337 . this involves no loss of generality as the sign of @xmath338 can be changed by a local rotation of angle @xmath339 around the @xmath268 axis at even sites ( assuming @xmath330 even in cyclic chains ) , which will not affect the value of the correlation measures , and the @xmath317 axis can be chosen along the direction of maximum coupling . [ f1 ] depicts the behavior with increasing field @xmath40 of the one way information deficits @xmath340 ( eq . ( [ i d ] ) ) and @xmath328 ( eqs.([i2])-([i2a ] ) ) of spin pairs in the exact definite parity ground state for the anisotropic case @xmath341 , together with that of the quantum discord ( [ i d ] ) and the concurrence ( [ cij ] ) . it is first seen that @xmath342 , @xmath328 and @xmath343 exhibit a similar qualitative behavior , acquiring appreciable finite values for _ any _ separation @xmath335 in the interval @xmath344 , in marked contrast with the concurrence , which is appreciable just for first and second neighbors ( except for the immediate vicinity of the factorizing field , see below ) . the @xmath273 parity symmetry is essential for this result . in fact , all measures converge to a _ finite common value _ , independent of the separation @xmath335 , at the factorizing field @xcite @xmath345 existing for @xmath346 , where the system possesses a pair of degenerate completely separable exact ground states @xcite given by @xmath347 and @xmath348 , where @xmath349 is the single spin state forming an angle @xmath350 with the @xmath351 direction and @xmath352 . actually , in the finite case this field coincides with _ the last parity transition _ of the exact ( and hence of definite parity ) ground state @xcite , such that the latter approaches , as side limits at @xmath353 , the definite parity combinations @xcite @xmath354 here @xmath355 ( @xmath356 ) is the ground state limit for @xmath357 ( @xmath358 ) . discarding the overlap @xmath359 , which is negligible if @xmath330 and @xmath350 are not too small ( @xmath360 for small @xmath350 ) , eq . ( [ stt ] ) leads to a _ common _ reduced state for _ any _ pair @xmath279 , given by @xcite @xmath361 this is a separable mixed state and therefore , it leads to a zero concurrence for any pair , as seen in fig . [ f1 ] ( where results at @xmath362 correspond to the side limits ( [ stt ] ) ) . however , it is not of the classically correlated form ( [ 4cc ] ) or ( [ 5cc ] ) if @xmath363 or @xmath364 , i.e. if @xmath365 are non - orthogonal and distinct , leading then to a common appreciable value of @xmath343 , @xmath342 , @xmath328 and in fact all @xmath366 . we also notice that the same reduced state ( [ sttr ] ) is obtained from the mixture @xmath367 , which represents the low temperature limit of the thermal state @xmath368 $ ] at @xmath353 . it is then possible to obtain straightforward analytic expressions for the side limits of @xmath343 @xcite , @xmath328 and @xmath342 at the factorizing field through the state ( [ sttr ] ) , which leads to @xmath369 and @xmath370 in ( [ rij ] ) , with @xmath371 . that for @xmath328 is particularly clean and given by @xmath372 with the minimizing measurement at @xmath362 being along @xmath268 if @xmath373 and along @xmath317 if @xmath374 . ( [ i2bs ] ) applies for all separations @xmath335 . for small chains , the results are similar but the effects of the parity transitions of the ground state ( it undergoes @xmath375 parity transitions as the field increases from @xmath326 , the last one at @xmath353 @xcite ) are now appreciable trough the finite discontinuities exhibited by @xmath328 , @xmath342 and @xmath343 , as seen in fig . [ f2 ] . at the factorizing field , these discontinuities arise from the overlap @xmath376 , which now can not be strictly neglected . it leads to an additional term @xmath377 in eq . ( [ sttr ] ) , which originates slightly distinct side limits of @xmath343 @xcite and also @xmath328 and @xmath342 at @xmath362 . moreover , it also leads to small but finite and distinct common side limits of the concurrence at @xmath353 @xcite , which was known to reach full range in its vicinity @xcite . all these side limits are , nevertheless , still independent of the pair separation @xmath335 . in the case of @xmath328 , they are given , for @xmath378 , by @xmath379 which corrects the upper line in eq . ( [ i2bs ] ) for finite @xmath330 ( or @xmath380 ) and @xmath381 ( @xmath382 ) corresponds to the right ( left ) side limit . the side limits of the concurrence are @xmath383 , as obtained from ( [ cij ] ) @xcite . the behavior of the quantum discord for longer range ferromagnetic - type couplings is qualitatively similar @xcite . moreover , a factorizing field still exists for longer range couplings with a constant anisotropy @xmath384 @xcite , in which case the reduced pair state at @xmath362 is again given by eq . ( [ sttr ] ) with @xmath385 , and eqs . ( [ i2bs])([i2bsn ] ) remain then valid . in fig . [ f3 ] we compare the behavior of @xmath328 , @xmath342 and @xmath343 for first neighbors in the chains of figs . [ f1 ] and [ f2 ] , with that of the associated entanglement monotone , i.e. , the squared concurrence @xmath386 for @xmath328 and the entanglement of formation @xmath387 for @xmath342 and @xmath343 , such that both quantities coincide for pure states . it is seen that for strong fields , differences are very small , in agreement with the weak entanglement of the pair with the rest of the chain in this regime ( @xmath388 is almost pure ) . the strong differences arise for @xmath389 , and especially in the vicinity of the factorizing field , due to the arguments exposed above . for @xmath390 the reduced pair state becomes appreciably mixed in the definite parity ground states , including the states ( [ stt ] ) at the factorizing field , due to the entanglement with the rest of the chain . significant differences between @xmath366 ( and @xmath343 ) with the corresponding entanglement monotone become then feasible . it is also seen that @xmath328 is in this case an upper bound of @xmath386 for all fields , whereas @xmath342 is not a upper bound of @xmath387 for low fields while @xmath343 is not a upper bound even for strong fields , indicating the lack of an order relationship between @xmath343 and @xmath387 even in this regime . in the case of @xmath328 , it is easy to show from eqs . ( [ cij ] ) and ( [ i2a ] ) that for @xmath286 states , it is always an upper bound of @xmath386 when the minimizing measurement is along @xmath268 @xcite . in , fact , for strong fields @xmath391 , a perturbative expansion @xcite for the present chain leads to @xmath392 , @xmath393 , @xmath394 and @xmath395 , where @xmath396 hence , in this limit @xmath397 and @xmath398 , both positive , whereas @xmath399 becomes negative . although @xmath342 , @xmath328 and @xmath343 show a similar qualitative behavior , both measures @xmath342 , and @xmath328 exhibit a more pronounced maximum , in comparison to that of the quantum discord , as appreciated in figs . [ f1]-[f3 ] . this reflects the transition in the orientation of their local minimizing spin measurements as the field increases , which , as mentioned above , is not present in the quantum discord . the latter prefers in the present system a measurement along the @xmath317 axis , even for large fields and for any separation between the spins , following the strongest correlation @xcite . as seen in fig.[f4 ] and as previously stated , @xmath328 exhibits instead a sharp transition from a direction _ parallel to the @xmath317 axis _ ( @xmath400 ) to a direction _ parallel to the @xmath268 axis _ ( @xmath401 ) i.e. , parallel to the field . this transition takes place , in the case shown in fig . [ f1 ] , for all separations @xmath335 at @xmath402 . in the case of the information deficit @xmath342 , the transition becomes smooth , as the angle @xmath318 takes all the intermediate values between @xmath403 and @xmath404 ( as determined by eq . ( [ cg ] ) ) for all separations in a narrow field interval centered at the @xmath328 critical field , as also seen in fig . [ f4 ] . the value of the field where the transition in the optimizing local measurement for @xmath328 occurs , depends on the anisotropy but only slightly on the separation @xmath335 , except in the @xmath2 limit ( @xmath405 ) , as can be seen in the top panel of fig . [ f5 ] . the same holds for the field interval where the `` transition '' ( actually the evolution from @xmath404 to @xmath326 of the measurement angle @xmath318 ) in @xmath342 takes place ( bottom panel of fig.[f5 ] ) . in the case of @xmath328 , if @xmath406 the measurement transition for _ all _ separations @xmath335 occurs _ exactly _ at the factorizing field @xmath407 , as follows from eq . ( [ i2bs ] ) . the measurement transition reflects essentially the qualitative change experienced by the reduced state of the pair for increasing fields . away from the @xmath2 limit , the dominant eigenstate of @xmath285 ( that with the largest eigenvalue ) for not too low fields is the entangled state @xmath408 with @xmath409 and @xmath410 . above the measurement transition field ( i.e. , when the optimum measurement is parallel to the field ) , @xmath411 becomes small ( @xmath412 ) , indicating that the pair is approximately aligned with the field . instead , below the transition field @xmath411 increases , approaching @xmath364 for @xmath413 ( where @xmath414 becomes a parallel bell state ) and the least disturbing measurement is along @xmath317 . for very low fields the dominant eigenstate may shift to the antiparallel bell state @xmath415 arising from the central block of ( [ rij ] ) , and in this case the measurement along @xmath317 is still preferred . on the other hand , in the @xmath2 limit , @xmath416 in ( [ rij ] ) and the dominant eigenstate is either @xmath417 at low fields , or @xmath418 for strong fields , and the measurement transition of @xmath328 indicates essentially the field where the sharp transition in the dominant eigenstate ( from maximally entangled to separable ) takes place @xcite . such measurement transition for increasing fields persists even at finite temperatures @xcite . we have examined the behavior of the quantum discord and the standard and quadratic one - way information deficit of spin pairs in the exact definite parity ground state of a finite anisotropic cyclic @xmath0 spin @xmath1 chain in a transverse field . we have first provided a brief overview of the quantum discord , the standard von neumann based one - way information deficit and the generalized information deficit , which contains the standard as well the quadratic deficit as particular cases , and which can be interpreted as a measure of the minimum entanglement generated between the system and the measurement apparatus after a complete local projective measurement . the first important result is that the behavior of all these measures is quite distinct from that of the pair entanglement for fields below the critical field , acquiring finite appreciable values for _ all separations _ of the spins of the pair . moreover , they reach ( as side limits ) a common ( independent of the separation ) finite value at the factorizing field , which in a finite chain is the field where the last ground state parity transition takes place . these finite limits can be evaluated analytically . the entanglement of pairs also reaches full range in its vicinity , although its value is much smaller and vanishes at this field except for very small samples . parity effects are of crucial importance for the proper description of these measures in finite systems below the critical field . the second important result is that the behavior of the optimizing local spin measurement of both the standard and generalized information deficit is quite distinct from that optimizing the quantum discord , exhibiting a transition in the direction of the spin measurement , from that of maximum correlation to that parallel to the field . the details of this transition depend on the choice of entropy ( it is sharp for @xmath328 , and smooth for @xmath342 ) . the quantum discord prefers instead that of maximum correlation even for strong fields . hence , the quantum discord , which is based on the minimization of a conditional entropy , `` detects '' in this way this direction @xcite , while the information deficits , based on the minimization of a total entropy , are more sensible to changes in the structure of the reduced state of the pair . a final comment is that the generalized formalism permits the use of simple entropic forms involving just low powers of the density matrix , leading to measures of the form ( [ iq ] ) or ( [ irq ] ) which can be more easily evaluated and optimized , and which are also more easily accessible from the experimental side . we briefly discuss here the exact solution of the _ finite _ cyclic @xmath0 chain with first neighbor couplings , which requires to take into account exactly the parity effects @xcite . the jordan wigner transformation @xcite allows to rewrite the hamiltonian ( [ h ] ) in the @xmath0 case ( @xmath314 ) for @xmath419 , @xmath333 , and for each value @xmath420 of the @xmath273 parity @xmath274 , as a quadratic form in fermion creation and annihilation operators @xmath421 , @xmath422 defined by @xmath423 $ ] , with the reverse transformation given by @xmath424 $ ] . this leads to @xmath425 where @xmath426 and @xmath331 , @xmath427 , @xmath428 @xcite . in ( [ qd ] ) , @xmath429 , @xmath430 and @xmath431 the last form ( [ qd ] ) is obtained through a parity dependent discrete fourier transform @xmath432 , followed by a bcs - 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we examine the behavior of quantum correlations of spin pairs in a finite anisotropic @xmath0 spin chain immersed in a transverse magnetic field , through the analysis of the quantum discord and the conventional and quadratic one way - information deficits . we first provide a brief review of these measures , showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms . all these measures coincide with an entanglement entropy in the case of pure states , but can be non - zero in separable mixed states , vanishing just for classically correlated states . it is then shown that their behavior in the exact ground state of the chain exhibits similar features , deviating significantly from that of the pair entanglement below the critical field . in contrast with entanglement , they reach full range in this region , becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field . it is also shown , however , that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them . both analytical and numerical results are provided .

the first stars in the @xmath7cdm universe are believed to have formed inside dark - matter - dominated minihalos filled with mostly neutral , metal - free gas of virial temperature @xmath8 k , when @xmath1 molecules formed in sufficient abundance to cool the gas radiatively to @xmath9 k. if , as currently thought , these stars were massive , hot , and luminous , they may have contributed significantly to the reionization of the universe , which cmb polarization observations by wmap indicate was highly ionized by @xmath10 @xcite . the release of ionizing uv radiation by minihalos and other sources ( e.g. stars in more massive halos , with @xmath11 k , or miniquasars ) , required to explain reionization , must have been accompanied by radiation release at energies below the h lyman limit , as well , however . this may , in turn , have limited the @xmath1 abundance inside minihalos and their ability to form stars , thereby limiting their contribution to cosmic reionization . in the absence of dust and at densities below the three - body formation regime ( @xmath12 @xmath13 ) , the most important reaction for the production of @xmath1 is @xmath14 ( e.g. , shapiro & kang 1987 and refs . therein ) with reaction rate @xmath15 @xcite . once formed , @xmath1 can be destroyed by collisions with other species @xmath16 or by photodissociation via lyman - werner band photon absorption @xmath17 the latter process becomes dominant once a substantial uv background is built up between 912 and 1110 @xmath18 , providing a feedback mechanism against the formation of new radiation sources @xcite . in this paper we explore the impact of another feedback mechanism , the photodissociation of @xmath0 , @xmath19 the cross - section for photodissociation of @xmath0 is well fitted by @xcite @xmath20 where @xmath21 is the photon energy in ev . the cross section is zero below a threshold of @xmath22ev , the binding energy of the second electron . in the absence of the uv background , the primary mode of @xmath0 destruction is the formation of @xmath1 ( eq . [ [ fh2 ] ] ) , ) mutual neutralization with @xmath23 can provide another efficient channel for @xmath0 destruction . however , typically the fractional ionization of minihalos is much lower . ] so introducing the @xmath0 photodissociating flux reduces the @xmath1 formation rate by a factor @xmath24 where @xmath25 is the photodissociation rate per @xmath0 ion , @xmath26 is the hydrogen atom number density and @xmath27 is the number density of photons with energy @xmath21 . breaks down when its value exceeds @xmath28 , since for such uv intensities @xmath29 reaction becomes a dominant channel of @xmath1 production ( assuming reaction rates given by @xcite ) . note also that @xmath30 is still uncertain to within a factor of a few ( see glover et al . 2006 ) , and this uncertainty carries over to @xmath31 when @xmath32 . ] hence the importance of this mechanism depends primarily on the local density ratio of @xmath0 photodissociating photons and hydrogen atoms . the impact of @xmath0 photodissociation differs from that of @xmath33 by two fundamental characteristics . first , the time required for @xmath0 abundance to approach equilibrium is very short ( typically less than 10000 years ) , while for @xmath1 the equilibration time can exceed the hubble time . therefore , when gas is exposed to a _ transient _ uv flux , produced by nearby pop iii stars , for example , @xmath0 photodissociation can generally be ignored , as it does not affect the subsequent thermal and chemical evolution . secondly , photons that make up the @xmath1 photodissociating background are destroyed after a few percent of the hubble time , as they redshift into one of the lyman series resonances , and must be replenished continuously . by contrast , photons that constitute the @xmath0 photodissociating background are very rarely destroyed , which allows them to accumulate over time . consequently the importance of @xmath0 photodissociation increases over time , and as we show in this paper , by the time a significant ( @xmath34 % ) fraction of the universe is ionized , @xmath35 photodissociation may result in a drastic reduction of the molecular hydrogen abundance . this in turn may lead to a reduced star formation rate and delay the progress of reionization . recently , @xcite considered the suppression of @xmath1 formation due to the photodissociation of @xmath0 and @xmath36 . whereas @xcite focused on the local feedback around and inside hii regions created by pop iii stars , we treat the problem globally and also consider long range effects due to the much lower optical depth of the universe below the lyman limit . the paper is organized as following . in 2 and 3 , we estimate the intensity of @xmath0 photodissociating flux produced by uv and x - ray sources , respectively . in 4 , we discuss the implication of our results for gas cooling in minihalos . since the first radiation sources are expected to form within overdense gas clouds , only the escaping fraction of their ionizing photons , @xmath4 , was available for ionization of the diffuse igm . the rest was absorbed within the host halos and , via the process of radiative recombination , converted into lower energy uv photons . since the universe during that epoch is transparent to most non - ionizing uv photons , ) lyman resonances , which , following their absorption by hydrogen atoms , are further split into two or more lower energy photons . for ly@xmath37 photons , the optical depth is also very high , but in their case the absorption in almost all cases is followed by reemission , with the destruction probability being extremely low @xcite . also , at the very early stage of reionization ( @xmath38 ) the presence of @xmath39 molecules makes the universe opaque in the lyman - werner range . however , since their initial abundance ( @xmath40 ) is already very low , the number of photons they destroy is negligible . ] almost all of them add to the @xmath0 photodissociation background . neglecting recombinations in the diffuse igm , the mean ionization is @xmath41 , where @xmath42 is the total number of ionizing photons per baryon produced up to this point . inside halos , the recombination time is quite short , and so the number of ionizations taking place there , @xmath43 , is almost equal to the number of electron recombinations to @xmath44 states ( i.e. , recombinations which do not result in emission of additional ionizing photons ) , @xmath45 . therefore the average @xmath0 photodissociating rate is given by @xmath46 where @xmath47 is the mean baryon density and @xmath48 is the average cross - section per recombination photon times the average number of photons per recombination , @xmath49 . note that since emissivity , @xmath50 , is proportional to @xmath51 , @xmath48 is in fact independent of @xmath52 and @xmath53 . using osterbrock s ( 1989 , sec . 4.3 ) calculation of the recombination spectrum , @xmath54 , and assuming that the temperature of the recombining gas is close to @xmath55 k , we find @xmath56 . by combining equations ( [ sup ] ) and ( [ zet ] ) , we can estimate the importance of the @xmath0 photodissociation due to recombination radiation . assuming that most of the recombinations occurred recently , we find that , the recombination radiation alone will suppress the @xmath1 formation rate by @xmath57 where @xmath58 is the local overdensity . here we have neglected recombinations in the diffuse intergalactic medium ( igm ) and the associated @xmath0 dissociating photons from these recombinations , but these would only further increase @xmath59 . cosmological redshift can affect the photodissociation rate by shifting the spectrum to longer wavelengths . initially this leads to an increase in @xmath60 due to the @xmath61 dependence of the cross - section for @xmath62 ev . eventually , as more and more of the spectrum is shifted below the threshold , the cosmological redshift begins to decrease the dissociation rate . for recombination photons this redshift effect is small , and the transition to @xmath60-depression occurs at a redshift factor of @xmath63 , see figure [ fig : zdep ] . unlike ionizing photons , whose intensity is heavily attenuated both in stellar atmospheres and in their host galaxies , most of the photons with frequencies below the lyman limit escape freely into the igm . from then on , photons with frequency below ly@xmath64 undergo no evolution apart from cosmological redshift . by contrast , within a small fraction of the hubble time , most photons with frequency between ly@xmath64 and the lyman limit are split by cascade into two or more photons after being redshifted into one of the hydrogen resonances . most of the cascade products , which include lines such as ly@xmath37 , h@xmath37 , and h@xmath64 , as well as a continuum spectrum produced by the two photon transition @xmath65 , are above the @xmath66 ev threshold for @xmath0 photodissociation . the relative importance of these directly emitted @xmath0 dissociating photons depends on the nature of the uv sources . figure [ fig : zetaratio ] shows the increase of the @xmath0 dissociation rate due to inclusion of direct emission from metal - poor pop iii stars , which we calculated using the stellar atmosphere models of @xcite . predictably , for very massive pop iii stars , with surface temperatures @xmath67 k , adding the stellar continuum below the lyman limit to the recombination spectrum increases the photodissociation rate by only @xmath68 . if , on the other hand , most of the early ionizing flux was produced by stars with masses below @xmath69 , whose continuum emission is stronger at lower frequencies , then the total @xmath0 dissociation rate would be tripled at least . likewise , direct emission may be important if most of the uv photons were produced by miniquasars . for example , assuming that their spectrum can be approximated by a power law , @xmath70 , with a cutoff below @xmath71 ev , adding the directly emitted photons to the recombination products increases the total photodissociation rate by a factor of @xmath72 . it has been suggested that x - ray photons could contribute a large fraction of the energy emitted by the first radiation sources ( e.g. * ? ? ? * ) . by increasing the number of free electrons , x - rays can boost the production of @xmath0 , and thus of @xmath1 , providing a positive feedback to the formation of new sources @xcite . this effect , however , would be at least partially offset by an increase of the @xmath0 photodissociating background , caused by conversion of x - rays into uv photons . the absorption of an x - ray photon is followed by release of a non - thermal electron , which then loses some of its energy by inelastic collisions with atoms before it can thermalize its energy by elastic scattering with ions and other electrons . when the gas ionization fraction is low ( @xmath73 ) , the photoelectron splits most of its energy evenly between collisional ionizations and excitations of hydrogen atoms @xcite . using electron - hydrogen excitation cross - sections @xcite , we find that around @xmath74 of the excitations are to the 2p level , which are followed by emission of a ly@xmath37 photon . most of the remaining excitations are to the 3p level , which decays via emission of one h@xmath37 photon and a subsequent two - photon decay from the 2s level . the ly@xmath37 , h@xmath37 and two - photon continuum each produce roughly equal contributions to @xmath0 photodissociation . per ionization , the average intensity - weighted cross - section for these photons is @xmath75 . due to the low number of uv photons produced during this phase , the formation of @xmath1 is not strongly affected @xmath76 after the ionized fraction climbs above @xmath77 , most of the energy of the non - thermal electrons is converted to heat . however , simultaneously with the growth of the ionized fraction , the temperature of the gas rises , and as it crosses @xmath55 k , the collisions between thermal electrons and atoms begin to dissipate the energy added by x - rays , mainly via emission of ly@xmath37 photons . neglecting gas clumping , we find that the number of emitted ly@xmath37 photons per hydrogen atom is @xmath78 assuming for simplicity that @xmath3 and @xmath79 are constants , we can rewrite the equation ( [ na ] ) as @xmath80 where @xmath81 is the thompson optical depth from the epoch of partial ionization by x - rays . if x - ray preionization contributes at least half of the @xmath82 measured by wmap ( i.e. @xmath83 ) , hydrogen atomic de - excitations in the diffuse igm may produce @xmath84 ly@xmath37 photons per baryon . the suppression of @xmath1 formation due to @xmath0 photodissociation by ly@xmath37 photons is @xmath85 since the energy of ly@xmath37 photons ( 10.2 ev ) is far above the @xmath86 photodissociation threshold ( 0.75 ev ) , the photodissociation rate grows roughly as @xmath87 , where @xmath88 is the redshift at which the photon was emitted . in the case of an extended period of partial ionization , @xmath59 may be increased by a factor of a few , possibly exceeding @xmath89 . since , when the igm temperature rises above @xmath55 k , the formation of new minihalos is suppressed , the impact of @xmath86 photodissociating flux produced by x - ray conversion is relevant only for minihalos which have formed some time ago or for halos with @xmath11 k , which also rely on @xmath1 cooling to form stars . as shown by our calculations , @xmath0 photodissociation reduces the formation of @xmath1 molecules by a factor of @xmath90 where @xmath6 is a constant of order a few , whose value depends on the type of radiation source and the growth history of the radiation background . thus , by the time a significant fraction ( @xmath91 ) of the universe becomes ionized , @xmath0 photodissociation can significantly reduce the @xmath1 formation rate in regions with overdensities of up to a few thousands , i.e. in the interior regions of minihalos . the equilibrium abundance of molecular hydrogen during this stage would be determined by the balance between its formation and destruction rates ( eqs . [ [ fh2 ] ] and [ [ fdes ] ] ) @xmath92 where @xmath93 is the @xmath1 destruction rate by the lyman - werner photons . thus a reduction of @xmath0 abundance by a factor @xmath59 translates into the same reduction of the @xmath1 abundance and , in minihalos , a comparable increase of the cooling time . indirectly , @xmath0 photodissociation may affect the cooling in the central regions of minihalos even during the early stages of reionization . the maximum density that gas can reach in the core region of a minihalo is limited by the amount of entropy it is able to radiate away during collapse . the lower density gas prevalent during the early collapse phase would be susceptible to @xmath0 dissociation from even a relatively low intensity @xmath0 dissociating flux , and the resulting lowered @xmath1 abundance would limit its ability to radiate away entropy via @xmath1 cooling . furthermore , the density and @xmath1 abundance at the center depend on the conditions in the low density outer regions , through their contributions to both the total pressure and the self - shielding ability of the halo . we plan to investigate these effects further with numerical radiation - hydrodynamic simulations in the future . lc thanks the mcdonald observatory for the w.j . mcdonald fellowship . mk gratefully acknowledges support from the institute for advanced study . this work was partially supported by nasa astrophysical theory program grants nag5 - 10825 and nng04g177 g to p. r. s.

during the epoch of reionization , the formation of radiation sources is accompanied by the growth of a @xmath0 photodissociating flux . we estimate the impact of this flux on the formation of molecular hydrogen and cooling in the first galaxies , assuming different types of radiation sources ( e.g. pop ii and pop iii stars , miniquasars ) . we find that @xmath0 photodissociation reduces the formation of @xmath1 molecules by a factor of @xmath2 , where @xmath3 is the mean ionized fraction in the igm , @xmath4 is the fraction of ionizing photons that escape from their progenitor halos , @xmath5 is the local gas overdensity and @xmath6 is an order unity constant which depends on the type of radiation source . by the time a significant fraction of the universe becomes ionized , @xmath0 photodissociation may significantly reduce the @xmath1 abundance and , with it , the primordial star formation rate , delaying the progress of reionization .

protostellar disks are a ubiquitous outcome of the rotating collapse of dense molecular cloud cores in the standard paradigm of low - mass star formation ( e.g. , * ? ? ? * ; * ? ? ? their existence has been confirmed around young stellar objects across a broad range in mass from objects in the brown dwarf regime , to those with masses of up to @xmath6 ( e.g. , * ? ? ? * ; * ? ? ? * ) as well as in a wide variety of star forming environments ( e.g. , * ? ? * ; * ? ? ? * ; * ? ? ? numerical simulations of collapsing cloud cores reveal that disks can form within @xmath7 from the onset of core collapse @xcite . these early so - called class 0 systems are difficult to study observationally as they are still embedded within their progenitor cloud cores @xcite . numerical simulations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) suggest that the earliest periods ( @xmath8 ) of disk formation are rather tumultuous , as infall from the parent cloud core induces gravitational instability driven mass accretion . depletion of the gas reservoir by this mechanism then gives way to a much more quiescent period of accretion in which gravitational torques act to transport mass inward while transporting angular momentum outward @xcite . indeed , the subsequent class i and ii phases are respectively marked by a decline in the rate of accretion from the surrounding natal environment , and its eventual cessation @xcite . hence , it is during the class ii phase , once the central star is optically visible , that the disk properties are most easily amenable to observational investigation . one result to emerge from observational studies of young stellar objects and their disks is the correlation between protostellar mass @xmath1 and the inferred accretion rate @xmath0 from the disk , for which the power law exponent is typically estimated to be @xmath9 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? although this correlation appears to hold across multiple orders of magnitude in both @xmath1 and @xmath0 , fitting the accretion rates of brown dwarfs and t tauri stars together may be misleading . in the brown dwarf regime , as well as for low mass t tauri stars ( i.e. , those objects with mass @xmath10 ) , a least squares fit yields @xmath11 . for intermediate and upper mass t tauri stars ( @xmath12 ) , the equivalent fit yields a value for @xmath13 of @xmath14 ; suggestive that different physical mechanisms may be responsible for accretion across the sequence of protostellar masses @xcite . studies by @xcite and @xcite have sought to explain the @xmath15 scaling in the context of viscous models for the disk evolution , wherein the turbulent viscosity has ad hoc spatial dependence of the form @xmath16 . @xcite link the disk evolution to the properties of the parent cloud core , providing a self - consistent basis for the results of their study . however , their models require that the ratio of rotational to gravitational energy be uniform across all cloud core masses . @xcite have even attempted to ( weakly ) incorporate the additional effects of magnetic fields ( in high temperature regions of the disk ) in quasi steady state models , but were also unable to fully account for the observed correlation . in this paper we present a study of the quasi steady state evolution of viscous circumstellar disks surrounding young stellar objects , following the cessation of mass accretion onto the protostar - disk system ( definitively class ii objects ) . these disks inherit initial conditions roughly consistent with the results of numerical simulations of the earlier burst phase ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and undergo diffusive evolution wherein angular momentum redistribution is driven by self - gravity , which we parameterize in terms of an effective kinematic viscosity ( following * ? ? ? we add to this a simplified argument for angular momentum conservation that correlates disk size with protostellar mass at the start of our simulations . with these assumptions , we are able to reproduce many features of the observed correlation between @xmath0 and @xmath1 for young protostellar systems . recent observations of disks using near - infrared polarization imaging @xcite have found that disks around four recently outbursting ( fu ori ) sources have large - scale ( hundreds of au ) spiral arms and arcs that are consistent with models of gravitational instability . added to previous near - infrared detections of spiral structure in smaller disks ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , there is a growing realization that meaningful spiral structure , arcs , and gaps exist in myr - old disks ( see the review by @xcite on the seeds survey by the subaru telescope ) . new efforts are being made to use gravitational instability driven disk evolution models to predict the near - infrared scattered light patterns as may be seen by the subaru or gemini telescopes , or the millimeter dust emission patterns that may be seen with the alma telescope @xcite . furthermore , numerical simulations are also being extended to include long - term residual infall from the molecular cloud to the disk ( even after the parent cloud core may have dissipated ) , which may be needed to keep gravitational instability active after several myr @xcite . our model in this paper studies gravitational torque driven evolution in a simplified manner . it does not however include residual mass infall from the cloud , which may be a subject of future work . in this paper we seek to characterize the bulk of transport within the disk through the action of gravitational torques , in the same spirit as the models of e.g. , @xcite and @xcite . our aim is to explain the global behavior of disks in which the mass accretion rate is predominantly set by the action of gravitational torques acting through most of the disk . other accretion mechanisms may be necessary in the innermost sub - au regions of the disk , possibly introducing short - term time variability . the above studies typicaly invoke the magnetorotational instability @xcite as the transport mechanism in the hot inner disk , however it is worthwhile to keep in mind that the region @xmath17 from the star is generally thought to be the outflow driving zone ( e.g. , * ? ? ? * ; * ? ? ? * ) from which significant amounts of angular momentum and mass are carried away from the disk . we construct a model for the temporal evolution of self - gravitating , axisymmetric thin disks on a radial grid with logarithmic spacing , and consisting of @xmath18 annular elements . discretization of the radial grid allows us to write the relevant partial differential equations as sets of ordinary differential equations , with one equation for each coordinate position in @xmath19 . the spatial derivatives are approximated using second - order accurate central differencing . integration of the system through time is handled using a variable order adams - bashforth - moulton solver ( e.g. , * ? ? ? combining together the fluid equations for mass and momentum conservation yields a diffusion - like equation that governs the temporal evolution of the disk surface mass density @xmath20 ( e.g. , * ? ? ? * ; * ? ? ? * ) : @xmath21,\ ] ] where @xmath22 is the disk angular frequency ( obtained assuming centrifugal balance ) , and @xmath23 is the effective kinematic viscosity ( detailed in section [ subsec : viscosity ] ) . a precise determination of @xmath24 requires a thorough accounting of the contribution to the gravitational potential made by the disk itself , which can be calculated explicitly using the elliptic integral of the first kind ( e.g. , * ? ? ? * ) . however , the central point - mass dominates the system s gravitational potential , with the contribution from the disk increasing @xmath24 only slightly . for the sake of computational convenience we thus adopt a simplified procedure by approximating the total gravitating mass at a radius @xmath19 to be @xmath25 in which @xmath26 denotes the innermost radius of the simulation domain ( and the assumed disk inner edge ) . the action of ( [ eqn : diskevolutioneqn ] ) is to transport material within the disk to ever smaller radii , while a small fraction of disk material is simultaneously transported to larger radii , thereby preserving the system s total angular momentum . for these simulations , the disk edge @xmath27 is always @xmath28 , the computational domain s outer boundary . thus , material that exits the simulation can only do so through @xmath26 . we impose a free * outflow * boundary condition there , and any material crossing @xmath26 is assumed to be accreted onto the central protostar , which we model as a point mass . temporal evolution of the disk is governed by the viscous stresses acting on the disk material . these stresses are typically subsumed into a dimensionless parameter @xmath29 that characterizes the efficiency of angular momentum transport . @xcite developed the most commonly invoked prescription of this kind , proposing a turbulent kinematic viscosity of the form @xmath30 which is the product of the turbulent velocity @xmath31 and the size @xmath32 of the largest eddies in the turbulent pattern . as turbulence is quickly dissipated by shocks in a highly supersonic flow , the turbulent velocity is often taken to be roughly equal to the local sound speed of the disk medium , @xmath33 . an upper limit to the size of the largest eddies that form can similarly be argued to be roughly equal to the disk half - thickness @xmath34 ; hence @xmath35 where we have used @xmath36 , and imposed that the disk be everywhere in vertical hydrostatic equilibrium . taken together , these arguments imply that @xmath29 should be less than unity . determinations based on measurements of the accretion rates from disks surrounding young stellar objects in the taurus complex suggest that @xmath37 @xcite . however , if and why @xmath29 should be spatially uniform and constant in time remains unclear . one mechanism that may act as an effective @xmath29-viscosity is the magnetorotational instability ( mri ) . @xcite have demonstrated that a weak magnetic field can make an otherwise hydrodynamically stable disk become unstable to the mri . simulations in ideal magnetohydrodynamics ( mhd ) find that the nonlinear outcome of mri can amplify and sustain turbulence that can then lead to angular momentum transport @xcite . however , in a largely neutral medium ( such as in a protostellar disk ) the ionization fraction must be large enough for the neutral - ion collision frequency to be greater than the local epicyclic frequency @xcite . many studies have been performed to determine in which regions of protostellar disks the mri may be active @xcite . for example , @xcite find that the mri can be active at distances greater than 5 au from the central star , while @xcite finds that the disk midplane is largely inactive to the mri after accounting for the effect of stellar winds and magnetic mirroring of cosmic rays , in addition to using a different critical electron fraction for the magnetic coupling . simulations of the mri with non - ideal mhd are very sensitive to ionization structure ( e.g. , * ? ? ? hence the determination of an effective @xmath29 for the mri in local simulations , let alone for a global disk model or for different evolutionary stages , is hardly established . * and references therein ) suggest a value for @xmath29 anywhere in the range of @xmath38 . when using @xmath29 as a comparator in this study we use @xmath39 , based on observational constraints ( hartmann et al . , 1998 ) . gravitational torques may represent an alternative source for angular momentum transport in cold and/or massive disks . using self - consistent cooling , @xcite showed that an @xmath29 prescription based on the gravitational instability agrees with the @xcite description very well , which assumes that the viscous heating is locally balanced by the cooling . @xcite have also used smoothed particle hydrodynamics to show that these disks often possess tightly wound spiral arms that can be approximated with a local treatment . here we consider a perturbation in an otherwise axisymmetric thin disk , which has the form of an annulus of width @xmath40 and increased local mass @xmath41 ( e.g. , the formation of a spiral arm ) . the growth condition for a perturbation depends on whether its self - gravity is greater than the tidal acceleration acting on it . that is , @xmath42 a natural length scale thus emerges , in excess of which perturbations of this nature are stabilized by their rotation , @xmath43 furthermore , with the assumption of vertical hydrostatic equilibrium , the disk self - gravity is supported by gas pressure in the vertical direction . this additional constraint implies @xmath40 must be at least larger than the disk half - thickness @xmath34 , @xmath44 @xcite originally formulated these arguments , summarizing the instability criterion as : @xmath45 this condition can additionally be rephrased in terms of the disk mass . multiplying equation ( [ eqn : toomresqcriterion ] ) by the square of the disk outer radius , @xmath46 , and then approximating the disk mass as @xmath47 , toomre s @xmath3 criterion implies @xmath48 from this statement one can conclude that provided the disk is thin , even a relatively low - mass disk will exhibit the effects of self - gravity . disks of this variety may not be uncommon , possibly forming during the earliest stages of star formation ( e.g. , * ? ? ? the importance of self - gravity in providing an effective way of redistributing angular momentum at the earliest stages of star formation has been recently recognized @xcite . @xcite noted that gravitational torques could be parameterized as an effective kinematic viscosity , constructed dimensionally using the length scales arising from toomre s analysis : the maximum size of the region over which angular momentum is transferred being roughly @xmath49 , together with a time - scale of approximately @xmath50 , produces an effective kinematic viscosity of the form @xmath51 this is clearly analogous to the standard @xmath29 prescription of @xcite with @xmath52 ( see equation ( [ eqn : alphaviscosity ] ) ) . additionally , by using equation ( [ eqn : toomresqcriterion ] ) we can also write @xmath53 hence , a convenient feature of parameterizing the gravitational torques via equation ( [ eqn : convenientnu ] ) is that we need not explicitly evaluate the energy equation during the disk evolution . this allows us to study the evolution of the disk while circumventing the complex issues related to the disk thermodynamics . nevertheless , to validate the consistency of our results with existing 1d simulation work ( e.g. , * ? ? ? * ; * ? ? ? * ) as well as higher dimensional models , such as the 2 + 1d models of @xcite , we do follow the temperature evolution of the disk implicitly . in calculating the temperature of the disk we assume it evolves isothermally up to some critical density @xmath54 , and in a polytropic manner thereafter . the critical surface mass density at which this transition occurs is @xmath55 , and corresponds to a critical volume density of @xmath56 for a gas disk in vertical hydrostatic equilibrium at @xmath57 @xcite . matching the isothermal and non - isothermal regimes , the effective vertically integrated gas pressure as a function of surface mass density can therefore be expressed as @xcite @xmath58 here , @xmath59 is the sound speed corresponding to a medium of predominantly molecular hydrogen ( with an admixture of helium ) that is isothermal with @xmath57 . for the ratio of the specific heats we used @xmath60 . the gas temperature , via the ideal gas equation of state @xmath61 , is then simply @xmath62,\ ] ] where @xmath63 is the mean molecular mass ( which we take to be 2.36 ) , @xmath64 is the proton mass , and @xmath65 is boltzmann s constant . angular momentum transport by gravitational instability ( or more appropriately , gravitational torques , as being described herein ) has been shown to remain effective in the regime of @xmath66 @xcite . in their 2006 study specifically , it was found that radial profiles of the toomre @xmath3 parameter were both near - uniform and noticeably larger during periods of quiescence in the mass accretion rate , and never falling below @xmath67 during this phase . although the persistence of gravitational instabilities is not usually expected for these values of @xmath3 , their 2 + 1d simulations revealed that weak spiral structures formed early on in the formation of the disk were then sustained by a swing amplification mechanism@xcite . in fact , strong observational evidence for spiral structure has actually been found in several - million year old disks such as hd 100546 @xcite , ab aurigae @xcite , and hd 135344b @xcite , all in support of this conjecture . we investigate the temporal evolution of more than 200 initial protostar - disk configurations . the parameter space of our models include initial protostellar masses of @xmath68 , corresponding to the range of intermediate- to upper - mass t tauri stars . this range is divided into intervals of @xmath69 up to @xmath70 , and intervals of @xmath71 thereafter . each protostar harbors a disk that extends from @xmath72 to an initial size @xmath27 that is @xmath73 , ensuring that the simulation s outer boundary has no influence on the disk evolution . we note that since we model the late - time quiescent evolution of the disk , the time @xmath74 in our simulations represents the state of a disk that is already @xmath75 old ( e.g. , see figure 1 in * ? ? ? our simulations end after an additional @xmath76 , congruent with observational estimates of disk lifetimes in the post - embedded phase ( see * ? ? ? * and references therein ) . ( solid black lines ) , angular velocity @xmath24 ( dashed black lines ) , and temperature @xmath77 ( solid orange lines ) , at the beginning of the simulation ( @xmath74 ; top ) , and at its end ( @xmath78 ; bottom ) . the protostar s initial mass is @xmath79 ; its associated disk , @xmath69 . the correlation between surface mass denisty with radius as @xmath80 is also shown to guide the eye ( dotted black line ) . , title="fig : " ] ( solid black lines ) , angular velocity @xmath24 ( dashed black lines ) , and temperature @xmath77 ( solid orange lines ) , at the beginning of the simulation ( @xmath74 ; top ) , and at its end ( @xmath78 ; bottom ) . the protostar s initial mass is @xmath79 ; its associated disk , @xmath69 . the correlation between surface mass denisty with radius as @xmath80 is also shown to guide the eye ( dotted black line ) . , title="fig : " ] we presume that each protostar - disk system is formed as a result of the collapse of a rotating prestellar core . numerical simulations of this process ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) show that a disk formed in this manner undergoes an initial burst mode of accretion driven by gravitational instabilities . at the end of this phase , the disk settles into a more quiescent phase , which we model in this paper , and is characterized by a disk roughly @xmath81 in radius . we follow the numerical results of @xcite and adopt a prescription weakly correlating the disk mass ( at @xmath74 ) with the host protostar s mass , @xmath82 . this produces disk mass fractions in the range of @xmath83 at the low mass end , to @xmath84 for the highest mass stars in our study . mass fractions of this order are greater than what is typically reported for disks with similar ages to those in our study ( e.g. , * ? ? ? however , at face value , up to 10% of the disk mass values reported by @xcite would still fall in this regime ( see their figure 10 ) . moreover , there is significant uncertainty in the disk masses estimated from dust emission . dust grain growth to sizes @xmath85 , for example , decrease the millimeter wavelength opacity and allow for higher disk mass estimates ( e.g. , * ? ? ? * ; * ? ? ? @xcite have computed synthetic observations from which they argue that a combination of partial optical thickness and emission further from the rayleigh - jeans limit can lead to underestimates of the actual disk mass when assuming optically thin rayleigh - jeans emission from submillimeter observations ( see their figure 7 ) . between @xmath26 and @xmath27 the local surface mass density within the disk scales as @xmath80 , and is exponentially tapered thereafter . this is consistent with theoretical estimates for the minimum - mass solar nebula @xcite , and the extrapolated radial profiles of observationally resolved disks ( e.g. , * ? ? ? * ) . for completeness we have also investigated the evolution of disks with two alternative initial profiles : @xmath86 and @xmath87 . however , the action of ( [ eqn : diskevolutioneqn ] ) and ( [ eqn : lp1987viscosity ] ) readjusts the surface mass density distribution of the disk to be @xmath88 within @xmath89 , rendering the initial distinction inconsequential for the subsequent evolution . evolution toward @xmath90 in an environment with near keplerian rotation and weak temperature variation will invariably lead to a profile @xmath80 , as can be seen from equation ( [ eqn : toomresqcriterion ] ) . we estimate the scaling of the initial disk radial extent @xmath27 with central object mass @xmath1 as follows . a parcel of material initially located at a cylindrical radius @xmath19 that possesses a specific angular momentum @xmath91 and that falls in from the outermost mass shell of the cloud core , can be expected to land in the plane of the disk at @xmath92 . here , we have used a rough equality of core mass @xmath93 and final central object mass @xmath1 . for a rotating and collapsing cloud core we expect surface mass density and rotation profiles @xmath86 and @xmath94 , respectively @xcite . in this case , @xmath95 and @xmath96 , so that @xmath97 . we therefore expect disk sizes to directly correlate with protostellar mass approximately as @xmath98 where we use the empirically motivated scale that @xmath99 corresponds to @xmath100 . we use this relation as a proxy for determining disk sizes in our models at @xmath74 . in figure [ fig : diskevolutionexample ] we present snapshots of the radial profiles of the disk surface mass density @xmath101 ( solid black lines ) , angular velocity @xmath24 ( dashed black lines ) , and temperature @xmath77 ( solid orange lines ) at times @xmath74 ( top ) and @xmath102 ( bottom ) . as the overall surface density declines with time , the disk edge moves steadily outward . most of the mass redistribution occurs within this first @xmath103 , during which time the disk size increases to @xmath104 . over a subsequent @xmath103 the disk size increases by only another @xmath105 this late - time quiescent evolutionary phase is also characterized by values of @xmath3 of order unity throughout the disk . such behavior is consistent with that seen in more robust 2d simulations of disk evolution such as those performed by @xcite . the instantaneous mass accretion rate onto the protostar is calculated from the change in the total disk mass between time steps , as mass loss from the disk occurs only through the inner boundary of the simulation domain ( being that @xmath27 is always @xmath28 ) . figure [ fig : mdot - t_toomreq ] illustrates the time evolution of mass accretion rates between @xmath106 and @xmath76 for protostars with initial masses of @xmath68 . . the accretion rate for each protostar - disk system is initially relatively constant . once the system settles into a quasi steady state , the accretion rate declines as @xmath107 @xcite . the length of time any one system requires to reach the quasi steady state scales roughly as @xmath108 . as a result , the difference in @xmath0 between the least and most massive systems decreases with time . ] . for comparison , the disk evolution for these systems is governed by a standard @xmath29 parameterization ( e.g. , * ? ? ? * ) , with @xmath109 . there is an approximately one order of magnitude difference in @xmath0 between the least and most massive protostar - disk systems . the constant value of this range of @xmath0 on a logarithmic scale is in sharp contrast to the decrease in range seen in disks whose viscosity is described by gravitational torques . ] the mass accretion rates onto the star are relatively constant at all masses during the first @xmath89 as the disks approach a quasi steady state , characterized by the radial mass accretion rate throughout the disk being uniform . the range in @xmath0 between the least and most massive systems spans more than three orders of magnitude during this time : the @xmath71 protostar accretes material from its disk at a rate of several times @xmath110 ; the @xmath111 protostar , at a rate of a few times @xmath112 . once a disk has settled into a quasi steady state however , @xmath0 begins to decline as @xmath107 ( as found by * ? ? ? transition into this regime is a direct consequence of the parameterization of the effective kinematic viscosity in terms of toomre s @xmath3 criterion specifically the strong dependence of @xmath23 on @xmath101 in equation ( [ eqn : convenientnu ] ) . we find that the length of time preceding the transition scales approximately as @xmath108 . this causes the range of accretion rates spanned by systems with different initial masses to decrease with time . for contrast , in figure [ fig : mdot - t_alpha ] we provide an example of the analogous evolution of @xmath0 for disks in which the effective kinematic viscosity is described by the spatio - temporally constant classical @xmath29 parameterization of equation ( [ eqn : alphaviscosity ] ) @xcite . @xmath29 to have a fiduciary value of @xmath113 in these models , consistent with estimates inferred from fitting disk similarity solutions to statistically significant samples of disk observations spanning different ages ( e.g. , * ? ? ? the disks modeled in this fashion exhibit a steady decline in their mass accretion rates with time , at the same rate irrespective of the mass of the system . it is clear that for these objects there is no change in the range of @xmath0 spanned by the least and most massive protostar - disk systems . the narrower ( and constant in logarithmic space ) range of accretion rates in these models makes it less likely that they can fit the observed @xmath15 correlation . for the regime of intermediate to upper mass t tauri stars ( @xmath12 ) , the exponent of the power law correlation @xmath13 , between mass accretion rate @xmath0 and protostellar mass @xmath1 , can be taken to be approximately @xmath114 ( e.g. , * ? ? ? * ; * ? ? ? * ) . in figure [ fig : mdot - m_composite ] we present mass accretion rates from more than 200 individual simulations , which reflect the evolution of protostars with masses of @xmath4 , and their disks , over @xmath76 . although material is being accreted from the disk and onto the protostar in these simulations , the change in @xmath1 with time is negligible compared to the order of magnitude changes in @xmath0 over the same period . a single simulation thus produces a seemingly vertical evolutionary track within the figure . for clarity , we plot only those values of @xmath0 at every 1,000th time step from the individual simulations . variations in the initial disk size@xmath27 , as determined through equation ( [ eqn : diskradiusrelation])cause protostar - disk systems with the same initial mass to follow slightly different evolutionary trajectories . the open circles are observational measurements of @xmath0 for protostars in the same mass range as those of our simulation , from the compilation of @xcite . a least squares fit to this data produces @xmath115 . if we were to consider isochrones connecting together systems of different mass at the same age in figure [ fig : mdot - m_composite ] , we would see that @xmath13 decreases as a function of protostellar age . * this is due to the faster convergence to the self - similar solution for models of higher mass . the effect is * most apparent in the difference in the slope of the lines that form the upper and lower envelope bounding the simulation data at @xmath74 and @xmath76 , respectively . note that only a handful of the observational measurements fall outside of this envelope . the most significant outliers are the particularly high accreters found at the lowest masses . however , we suspect these objects are likely younger than those represented in our simulations , and still enshrouded by the material of their natal environment . infall from their surroundings can induce disk instability and fragmentation . the resulting burst mode of accretion ( e.g. , * ? ? ? * ; * ? ? ? * ) is known to dominate the quiescent mode of accretion ( modeled here ) at early times . another challenge to our models comes from observations of the orion nebula cluster by @xcite , who find that the highest mass objects have a longer time scale for decrease of @xmath0 than do lower mass objects , so that the @xmath15 relation becomes steeper at later times . this is the opposite trend of our models . it could be that the accretion of mass on to the disk ( not a part of our model ) is significant in the more massive systems , and that their evolution to the self - similar solution is then delayed in comparison to low mass disks . this effect is outside the scope of our current model , but is a subject for future work . versus protostellar mass @xmath1 . open circles are observational measurements of objects with mass @xmath116 from ( * ? ? ? * and references therein ) . we evolve more than two hundred different initial protostar - disk configurations over @xmath76 . filled circles are the mass accretion rates for these systems at every 1,000th time step . the orange line is the least squares fit to the observed mass accretion rates , and produces a @xmath115 . the blue line is the equivalent fit to the values of @xmath0 from our simulations , and produces a @xmath117 . ] in order to make a fit of the @xmath15 from our model data , we need to consider the relative weighting of models of different mass . to explore this possibility we randomly sample the spectrum of @xmath15 results from our simulations using an initial mass function ( imf ) of the form proposed by @xcite . initial protostellar masses are acquired from the chabrier imf . from these we determine a disk mass @xmath82 ( as in section [ subsec : initialconditions ] ) and size @xmath27 ( equation [ eqn : diskradiusrelation ] ) at time @xmath74 for each model . we then uniformly sample each model s temporal history in order to determine a specific value of @xmath0 . after 100 such `` measurements '' we are then able to estimate a value of @xmath13 . figure [ fig : slopehistogram ] presents a histogram summarizing the value of @xmath13 from 10,000 such samplings . the average value of the exponent of the power law correlation in @xmath15 is @xmath118 , with a standard deviation of @xmath119 . this places the value of @xmath13 as determined from our simulations well within the error bounds of the observationally determined value * @xmath120*. for the correlation between mass accretion rate and protostellar mass , @xmath121 , for 10,000 samples generated by method described in the text . the distribution of slopes is reasonably well fit by a gaussian curve with mean @xmath122 and variance @xmath123 ( dashed black line ) . ] in figure [ fig : syntheticmdot - m ] we present one example of this sampling procedure for which the simulation data produces a typical value for the power law exponent of @xmath124 . in figure [ fig : numberdistributions ] we also present the number distributions of protostellar masses , and of mass accretion rates , for the observational measurements and the randomly selected simulation points . by visual inspection , the observational measurements and the sample drawn from the simulations are in general agreement . however , we can quantify this agreement statistically , as well as across repeated samples , to evaluate the likelihood with which our model is capable of reproducing the observed correlation . for each of the 10,000 samplings represented in figure [ fig : slopehistogram ] , we construct the corresponding number distributions by mass and mass accretion rate , as in figure [ fig : numberdistributions ] . we then perform a two - sample kolmogorov - smirnov test to evaluate the null hypothesis that the number distributions by mass and mass accretion rate between the observed and our randomly generated samples have been drawn from the same underlying distributions . figure [ fig : pvaluedistributions ] presents two histograms that summarize the @xmath125 values resulting from this test , where the @xmath125 value is an estimate of the probability that the two distributions are representative of a singular underlying sample ; considered unlikely for @xmath126 . in general , we find @xmath125 to be sufficiently large such that the null hypothesis can not be strictly ruled out . however , in roughly half of our randomly generated samples , we are unable to reproduce the magnitude spread in mass accretion rates that exists among observational measurements . as we discussed earlier , this is likely the result of younger objects than those represented in our simulations , in which infall from the surrounding environment is likely to dominate the accretion mode . generated by the selection criterion discussed in the text . for reference , the open circles are the observational measurements of t tauri stars in the mass range of @xmath127 , from ( * ? ? ? * and references therein ) . the orange line is the least squares fit to the observational data , and yields a value for the power law exponent of @xmath128 . the filled circles are the data obtained from our simulations , and the blue line is the least squares fit to these points , yielding @xmath124 . ] ( top ) , and by mass accretion rate @xmath0 ( bottom ) , of both the observational measurements ( orange ) and simulation data ( blue ) , gray regions indicating overlap , as they appear in figure [ fig : syntheticmdot - m ] . a kolmogorov - smirnov two - sample test , comparing the number distributions of the observational measurements to their corresponding simulation counterparts , produces a @xmath125 value of roughly @xmath129 in each case . the randomly selected simulation data is thus statistically indistinguishable from the observational measurements.,title="fig : " ] ( top ) , and by mass accretion rate @xmath0 ( bottom ) , of both the observational measurements ( orange ) and simulation data ( blue ) , gray regions indicating overlap , as they appear in figure [ fig : syntheticmdot - m ] . a kolmogorov - smirnov two - sample test , comparing the number distributions of the observational measurements to their corresponding simulation counterparts , produces a @xmath125 value of roughly @xmath129 in each case . the randomly selected simulation data is thus statistically indistinguishable from the observational measurements.,title="fig : " ] . for each sample we determine the number distribution by protostellar mass @xmath1 and mass accretion rate @xmath0 and compare those distributions to their observational counterparts using a kolmogorov - smirnov two - sample test . the @xmath125 value resulting from each of these tests is depicted above , with @xmath130 denoting the fraction of the 10,000 simulations in the indicated bucket . comparisons between these number distributions in @xmath1 appear at left , and in @xmath0 at right . ] in this paper , we have shown that the observed power law correlation between mass accretion rate @xmath0 and protostellar mass @xmath1 can be explained within the framework of gravitational torque driven transport . we parameterize the effects of the gravitational torques as an effective kinematic viscosity using toomre s @xmath3 criterion @xcite , noting that this prescription resembles but also differs from the classical @xmath29 model of @xcite . we carry out more than 200 individual simulations of protostellar disks in order to examine the time evolution of their mass accretion rates in the @xmath131 plane . the rates associated with a particular protostellar mass agree with those inferred from observational studies of t tauri disks across a broad spectrum of protostellar masses . the observed scatter in @xmath0 arises naturally as a result of the temporal evolution of the protostar - disk system through this plane . we are able to use a simple statistical argument , resampling our simulations onto the initial mass function of @xcite , to show that even with limited sampling , our simulation results are sufficiently robust to be able to reproduce the observed correlation . the initial disk masses presented in this paper are somewhat greater than is often reported in the literature ( by a factor of @xmath132 , e.g. , * ? ? ? however , current estimates for disk masses based on dust emission may have been systematically underestimated @xcite . nevertheless , as the efficacy of our transport mechanism is dependent on disk mass , it is possible that that additional physics may be required at late times to remove the remaining disk material within observed disk lifetimes @xcite . conversely , there is growing evidence that some disks may persist for several myr and have noticeable spiral structure , arcs , and gaps that may be indicative of gravitational instability @xcite . one way to understand this is by invoking gas accretion on to the disks for an extended period of several myr @xcite . accretion on to the disk is not a part of our current model and is a subject of future work . in a recent paper , @xcite offer a physically different model explanation . in their view , @xmath0 is initially mass independent , and declines in a self - similar manner with a somewhat different power law index than in our model , due to the use of an @xmath29-viscosity . mass accretion is then quenched when a model wind mass loss rate ( that depends on the x - ray luminosity of the protostar ) equals the mass accretion rate . the physical view here is that the observed mass - dependent x - ray luminosity sets the @xmath15 relation by reducing the accretion rate when it drops to the level of the wind mass loss rate . mathematically however , both models depend partially upon a bimodal mass accretion rate history . it is only the physical explanation of the two phases , and the specific mathematical shape of their curves , that differ . the observed correlation in @xmath15 may therefore be fit by a variety of models that have common mathematical elements , but differ substantially enough in their physics that they will hopefully lead to interesting observational comparator tests in the future . we thank the anonymous referee for comments that have significantly improved the discussion in the paper . sb acknowledges support from a discovery grant from the natural sciences and engineering research council ( nserc ) of canada . this research has made use of nasa s astrophysics data system .

we model the mass accretion rate @xmath0 to stellar mass @xmath1 correlation that has been inferred from observations of intermediate to upper mass t tauri stars that is @xmath2 . we explain this correlation within the framework of quiescent disk evolution , in which accretion is driven largely by gravitational torques acting in the bulk of the mass and volume of the disk . stresses within the disk arise from the action of gravitationally driven torques parameterized in our 1d model in terms of toomre s @xmath3 criterion . we do not model the hot inner sub - au scale region of the disk that is likely stable according to this criterion , and appeal to other mechanisms to remove or redistribute angular momentum and allow accretion onto the star . our model has the advantage of agreeing with large - scale angle - averaged values from more complex nonaxisymmetric calculations . the model disk transitions from an early phase ( dominated by initial conditions inherited from the burst mode of accretion ) into a later self - similar mode characterized by a steeper temporal decline in @xmath0 . the models effectively reproduce the spread in mass accretion rates that have been observed for protostellar objects of @xmath4 , such as those found in the @xmath5 ophiuchus and taurus star forming regions . we then compare realistically sampled populations of young stellar objects produced by our model to their observational counterparts . we find these populations to be statistically coincident , which we argue is evidence for the role of gravitational torques in the late time evolution of quiescent protostellar disks . & accretion ; accretion disks ; hydrodynamics ; stars : formation ; stars : protostellar disks

we are interested in the following semilinear heat equation : @xmath17 where @xmath18 , @xmath19 denotes the laplacian in @xmath20 , and @xmath4 or @xmath21 if @xmath22 . it is well known that for each initial data @xmath23 the cauchy problem has a unique solution @xmath24 for some @xmath25 , and that either @xmath26 or @xmath27 in the latter case we say that the solution blows up in finite time , and @xmath28 is called the blow - up time . in such a blow - up case , a point @xmath29 is called a blow - up point if @xmath0 is not locally bounded in some neighborhood of @xmath30 , this means that there exists @xmath31 such that @xmath32 when @xmath33 . we denote by @xmath8 the blow - up set , that is the set of all blow - up points of @xmath7 . given @xmath34 , we know from velzquez @xcite ( see also filippas and kohn @xcite , filippas and liu @xcite , herrero and velzquez @xcite , merle and zaag @xcite ) that up to replacing @xmath7 by @xmath35 , one of following two cases occurs : - case 1 ( non degenerate rate of blow - up ) : for all @xmath36 , there is an orthonormal @xmath37 matrix @xmath38 and @xmath39 such that @xmath40 where @xmath41 - case 2 ( degenerate rate of blow - up ) : for all @xmath42 , there exists an even integer @xmath43 such that @xmath44 where @xmath45 , @xmath46 if @xmath47 and @xmath48 for all @xmath49 . + according to velzquez @xcite , if case 1 occurs with @xmath50 or case 2 occurs with @xmath51 for all @xmath52 , then @xmath53 is an isolated blow - up point . herrero and velzquez @xcite and @xcite proved that the profile with @xmath50 is generic in the case @xmath54 , and they announced the same for @xmath55 , but they never published it . bricmont and kupiainen @xcite , merle and zaag @xcite show the existence of initial data for such that the corresponding solutions blow up in finite time @xmath28 at only one blow - up point @xmath53 and verify the behavior with @xmath50 . the method of @xcite also gives the stability of the profile ( @xmath50 ) with respect to perturbations in the initial data ( see also fermanian , merle and zaag @xcite , @xcite for other proofs of the stability ) . in @xcite and @xcite , the authors prove the stability of the profile ( @xmath50 ) with respect to perturbations in the initial data and also in the nonlinearity , in some class allowing lower order terms in the solution and also in the gradient . all the other asymptotic behaviors are suspected to be unstable . when @xmath56 in , we do not know whether @xmath53 is isolated or not , or whether @xmath8 is continuous near @xmath53 . in this paper , we assume that @xmath53 is a non - isolated blow - up point and that @xmath8 is continuous locally near @xmath53 , in a sense that we will precisely define later . our main concern is the regularity of @xmath8 near @xmath53 . the first relevant result is due to velzquez @xcite where the author showed that the hausdorff measure of @xmath8 is less or equal to @xmath57 . no further results on the description of @xmath8 were known until the contributions of zaag @xcite , @xcite and @xcite ( see also @xcite for a summarized note ) . in @xcite , the author proves that if @xmath8 is locally continuous , then @xmath8 is a @xmath58 manifold . he also obtains the first singularity description near @xmath53 . more precisely , he shows that ( see theorems 3 and 4 in @xcite ) for some @xmath59 and @xmath60 , for all @xmath36 , @xmath61 and @xmath62 such that @xmath63 , @xmath64 where @xmath65 is defined in ( @xmath66 ) . moreover , for all @xmath67 , @xmath68 as @xmath69 with @xmath70 if @xmath71 zaag in @xcite further refines the asymptotic behavior and gets to error terms of order @xmath72 for some @xmath14 . this way , he obtains more regularity on the blow - up set @xmath8 . the key idea is to replace the explicit profile @xmath65 in by a non - explicit function , say @xmath73 , then to go beyond all logarithmic scales through scaling and matching . in fact , for @xmath73 , zaag takes a symmetric , one dimensional solution of that blows up at the same time @xmath28 only at the origin , and behaves like with @xmath66 . more precisely , he abandons the explicit profile function @xmath65 in and chooses a non explicit function @xmath74 as a first order description of the singular behavior , where @xmath75 is defined by @xmath76 he shows that for each blow - up point @xmath77 near @xmath53 , there is an optimal scaling parameter @xmath78 so that the difference @xmath79 along the normal direction to @xmath8 at @xmath77 is minimum . hence , if the function @xmath80 is chosen as a first order description for @xmath0 near @xmath81 , we escape logarithmic scales . more precisely , for all @xmath61 and @xmath62 such that @xmath63 , @xmath82 for some @xmath14 . note that any other value of @xmath83 in gives an error of logarithmic order of the variable @xmath12 ( the same as in ) . exploiting estimate yields geometric constraints on @xmath8 which imply the @xmath84-regularity of @xmath8 for all @xmath85 . a further refinement of given in @xcite yields better estimates in the expansion of @xmath0 near @xmath81 . moreover , some following terms in the expansion of @xmath0 near @xmath81 contain geometrical descriptions of @xmath8 , resulting in more regularity of @xmath8 , namely the @xmath11-regularity . + in this work , we want to know whether the @xmath11-regularity near @xmath53 proven in @xcite for @xmath66 would hold in the case where @xmath7 behaves like near @xmath30 with @xmath86 since the author in @xcite and @xcite obtains the result only when @xmath66 , this corresponds to @xmath87-dimensional blow - up set ( the codimension of the blow - up set is one , according to @xcite ) . in our opinion , in those papers the major obstacle towards the case lays in the fact that the author could not refine the asymptotic behavior with @xmath88 to go beyond all logarithmic scales and get a smaller error term in polynomial orders of the variable @xmath12 . it happens that a similar difficulty was already encountered by fermanian and zaag in @xcite , when they wanted to find a sharp profile in the case with @xmath50 , which corresponds to an isolated blow - up point , as we have pointed out right after estimate . such a sharp profile could be obtained in @xcite only when @xmath89 ( which corresponds also to @xmath90 ) : no surprise it was @xmath91 , the dilated version of @xmath92 , the one - dimensional blow - up solution mentioned between estimates and . as a matter of fact , the use of @xmath92 was first used in @xcite for the isolated blow - up point in one space dimension ( @xmath89 and @xmath90 ) , then later in higher dimensions with a @xmath87 dimensional blow - up surface ( @xmath93 and still @xmath90 ) in @xcite . the interest of @xmath92 is that it provides a one - parameter family of blow - up solutions , thanks to the scaling parameter in , which enables to get the sharp profile by suitably choosing the parameter . handling the case @xmath94 remained open , both for the case of an isolated point ( @xmath95 ) and a non - isolated blow - up point ( @xmath96 ) . from the refinement of the expansion around the explicit profile in @xmath97 in , it appeared than one needs a @xmath98-parameter family of blow - up solutions obeying . such a family was constructed by nguyen and zaag in @xcite , and successfully used to derive a sharp profile in the case of an isolated blow - up point ( @xmath99 ) , by fine - tuning the @xmath100 parameters . in this paper , we aim at using that family to handle the case of a non - isolated blow - up point ( @xmath93 and @xmath101 ) , in order to generalize the results of zaag in @xcite , @xcite and @xcite , proving in particular the @xmath102 regularity of the blow - up set , under the mere hypothesis that it is continuous . the main result in this paper is the following . [ theo:1 ] take @xmath55 and @xmath103 . consider @xmath7 a solution of that blows up in finite time @xmath28 on a set @xmath8 and take @xmath34 where @xmath7 behaves locally as stated in with @xmath104 . if @xmath8 is locally a @xmath58 manifold of dimension @xmath105 , then it is locally @xmath11 . theorem [ theo:1 ] was already proved by zaag @xcite only when @xmath106 . thus , the novelty of our contribution lays in the case @xmath107 and @xmath22 . under the hypotheses of theorem [ theo:1 ] , zaag @xcite already proved that @xmath8 is a @xmath58 manifold near @xmath53 , assuming that @xmath8 is continuous . therefore , theorem [ theo:1 ] can be restated under a weaker assumption . before stating this stronger version , let us first clearly describe our hypotheses and introduce some terminologies borrowed from @xcite ( see also @xcite and @xcite ) . according to velzquez @xcite ( see theorem 2 , page 1571 ) , we know that for all @xmath108 , there is @xmath109 such that @xmath110 where @xmath111 is the orthogonal projection over @xmath112 , where @xmath113 is the so - called `` weak '' tangent plane to @xmath8 at @xmath53 . roughly speaking , @xmath114 is a cone with vertex @xmath53 and shrinks to @xmath112 as @xmath115 . in some `` weak '' sense , @xmath8 is @xmath116-dimensional . in fact , here comes our second hypothesis : we assume there is @xmath117 such that @xmath118 and @xmath119 , where @xmath120 is at least @xmath121-dimensional , in the sense that @xmath122,s)$ such that $ \gamma_i(0 ) = b$ and $ \gamma'_i(0 ) = v_i$.}&\quad \end{array}\ ] ] the hypothesis means that @xmath123 is actually non - isolated in @xmath121 independent directions . we assume in addition that @xmath53 is not an endpoint in @xmath124 in the sense that @xmath125 this is the stronger version of our result : * theorem [ theo:1]. * _ take @xmath55 and @xmath103 . consider @xmath7 a solution of that blows up in finite time @xmath28 on a set @xmath8 and take @xmath34 where @xmath7 behaves locally as stated in with @xmath104 . consider @xmath126 such that @xmath127 and @xmath120 is at least @xmath128-dimensional ( in the sense ) . if @xmath53 is not an endpoint ( in the sense ) , then there are @xmath60 , @xmath129 and @xmath130 such that @xmath131 and the blow - up set @xmath8 is a @xmath11-hypersurface locally near @xmath53 . _ + let us now briefly give the main ideas of the proof of theorem [ theo:1 ] . the proof is based on techniques developed by zaag in @xcite and @xcite for the case when the solution of equation behaves like with @xmath132 . as in @xcite and @xcite , the proof relies on two arguments : * the derivation of a sharp blow - up profile of @xmath0 near the singularity , in the sense that the difference between the solution @xmath0 and this sharp profile goes beyond all logarithmic scales of the variables @xmath12 . this is possible thanks to the recent result in @xcite . * the derivation of a refined asymptotic profile of @xmath0 near the singularity linked to geometric constraints on the blow - up set . in fact , we derive an asymptotic profile for @xmath0 in every ball @xmath133 for some @xmath36 and @xmath77 a blow - up point close to @xmath53 . moreover , this profile is continuous in @xmath77 and the speed of convergence of @xmath7 to each one in the ball @xmath133 is uniform with respect to @xmath77 . if @xmath77 and @xmath123 are in @xmath8 and @xmath134 , then the balls @xmath133 and @xmath135 intersect each other , leading to different profiles for @xmath0 in the intersection . however , these profiles have to coincide , up to the error terms . this makes a geometric constraint which gives more regularity for the blow - up set near @xmath53 . let us explain the difficulty raised in @xcite and @xcite for the case @xmath136 . consider @xmath137 for some @xmath60 and introduce the following self - similar variables : @xmath138 then , we see from that for all @xmath139 , @xmath140 under the hypotheses stated in theorem [ theo:1 ] , zaag @xcite proved in proposition 3.1 , page 513 and in section 6.1 , pages 530 - 533 that for all @xmath141 for some @xmath142 and @xmath143 , there exists an @xmath144 orthogonal matrix @xmath145 such that @xmath146 where @xmath147 , @xmath148 , @xmath145 is continuous in terms of @xmath77 such that @xmath149 spans the tangent plane @xmath150 to @xmath8 at @xmath77 and @xmath151 are the normal directions to @xmath8 at @xmath77 , @xmath152 is the weighted @xmath153 space associated with the weight @xmath154 . note that the estimate implies ( see appendix c in @xcite ) . when @xmath106 , in order to refine estimate , the author in @xcite subtracts from @xmath155 a 1-dimensional solution with the same profile . let us do the same when @xmath156 , and explain how the author succeeds in handing the case @xmath132 and gets stuck when @xmath136 . to this end , we consider @xmath157 with @xmath158 a radially symmetric solution of in @xmath159 which blows up at time @xmath28 only at the origin with the profile with @xmath104 ( see appendix a.1 in @xcite for the existence of such a solution ) . if the @xmath160-dimensional solution @xmath161 is considered in @xmath162 , then it blows up on the @xmath9 vector space @xmath163 in @xmath162 . in particular , if we introduce @xmath164 then , @xmath165 is a radially symmetric solution of which satisfies @xmath166 noting that @xmath161 and @xmath165 maybe considered as solutions defined for all @xmath167 ( and independent of @xmath168 ) , and given that @xmath169 and @xmath170 have the same behavior up to the first order ( see and ) , we may try to use @xmath165 as a sharper ( though non - explicit ) profile for @xmath170 . in fact , we have the following classification ( see corollary [ coro : class ] below ) : + _ - case 1 : there is a symmetric , real @xmath171 matrix @xmath172 such that @xmath173 - case 2 : there is a positive constant @xmath174 such that @xmath175 _ if @xmath132 ( @xmath176 ) , the author in @xcite noted the following property @xmath177 therefore , choosing @xmath178 such that @xmath179 , we see from and that @xmath180 from the classification given in and , only holds and @xmath181 if we return to the original variables @xmath0 and @xmath182 through and , then follows from the transformation together with estimate ( see appendix c in @xcite ) . in other words , @xmath183 serves as a sharp ( though non - explicit ) profile for @xmath170 in the sense of . using estimate together with some geometrical arguments , we are able to prove the @xmath184-regularity of the blow - up set , for any @xmath85 . then , a further refinement of up to order of @xmath185 together with a geometrical constraint on the blow - up set @xmath8 results in more regularity for @xmath8 , which yields the @xmath11-regularity . + if @xmath136 , the matrix @xmath186 in has @xmath187 real parameters . therefore , applying the trick of @xcite ( see above ) only allows to manage one parameter ; there remain @xmath188 real parameters to be handled . this is the major reason which prevents the author in @xcite and @xcite from deriving a similar estimate to , hence , the refined regularity of the blow - up set . fortunately , we could overcome this obstacle thanks to a recent result by nguyen and zaag @xcite ( see proposition [ prop : cons ] below ) where the authors show that for all symmetric , real @xmath189 matrix @xmath190 , there is a solution @xmath191 of equation in @xmath192 such that @xmath193 hence , choosing @xmath194 , we see from , and that @xmath195 exploiting estimate and adapting the arguments given in @xcite and @xcite , we are able to prove the @xmath11-regularity of the blow - up set . + the next result shows how the @xmath11-regularity is linked to the refined asymptotic behavior of @xmath155 . more precisely , we link in the following theorem the refinement of the asymptotic behavior of @xmath155 to the second fundamental form of the blow - up set at @xmath77 . [ theo:2 ] under the hypotheses of theorem [ theo:1 ] , there exists @xmath196 and @xmath60 such that for all @xmath197 , there exists a continuous @xmath198 symmetric matrix @xmath186 such that for all @xmath199 , @xmath200 for some @xmath201 , where @xmath202 is a continuous symmetric matrix representing the second fundamental form of the blow - up set at the blow - up point @xmath77 along the unitary normal vector @xmath203 . moreover , @xmath204 in section [ sec:2 ] , we give the main steps of the proof of theorems [ theo:1 ] and [ theo:2 ] . we leave all long and technical proofs to section [ sec : proofall ] . in this section we give the main steps of the proof of theorems [ theo:1 ] and [ theo:2 ] . all long and technical proofs will be left to the next section . we proceed in 3 parts corresponding to 3 separate subsections . for the reader s convenience , we briefly describe these parts as follows : * part 1 : we derive a sharp blow - up behavior for solutions of equation having the profile with @xmath205 such that the difference between the solution and this sharp blow - up behavior goes beyond all logarithmic scales of the variable @xmath206 . the main result in this step is stated in proposition [ prop : expodecay ] . * part 2 : through the introduction of a local chart , we give a geometrical constraint on the expansion of the solution linked to the asymptotic behavior ( see proposition [ prop : geocon ] below ) . this geometrical constraint is a crucial point which is the bridge between the asymptotic behavior and the regularity of the blow - up set . * part 3 : using the sharp blow - up behavior derived in part 1 , we first get the @xmath84- regularity of the blow - up set @xmath8 ( see proposition [ prop : c1alph ] below ) , then together with the geometrical constraint , we achieve the @xmath207-regularity of @xmath8 ( see proposition [ prop : c11eta ] below ) . with this better regularity and the geometric constraint , we further refine the asymptotic behavior ( see proposition [ prop : furre ] below ) and use again the geometric constraint to get @xmath11 -regularity of @xmath8 , which yields the conclusion of theorems [ theo:1 ] and [ theo:2 ] . the reader should be noticed that parts 1 and 2 are independent , whereas part 3 is a combination of the first two parts . throughout this paper , we work under the hypotheses of theorem [ theo:1 ] . since @xmath8 is locally near @xmath208 a manifold of dimension @xmath209 , we may assume that there is a @xmath58 function @xmath210 such that @xmath211 for some @xmath142 and @xmath212 with @xmath129 . in what follows , @xmath103 is fixed , and for all @xmath213 , we denote by @xmath214 the first @xmath160 coordinates of @xmath215 , namely @xmath216 , and by @xmath217 the last @xmath9 coordinates of @xmath215 , namely @xmath218 . we usually use indices @xmath219 , @xmath220 for the range @xmath221 and indices @xmath222 , @xmath223 , @xmath224 for the range @xmath225 . in this subsection , we use the ideas given by fermanian and zaag @xcite together with a recent result by nguyen and zaag in @xcite in order to derive a sharp ( though non - explicit ) profile for blow - up solutions of in the sense that the first order in the expansion of the solution around this sharp profile goes beyond all logarithmic scales of @xmath12 and reaches to polynomial scales of @xmath12 . in fact , we replace the 1-scaling parameter @xmath226 in by a @xmath227-parameters family , which generates a substitution for @xmath75 and serves as a sharp profile for solutions having the behavior with @xmath228 . the main result in this part is proposition [ prop : expodecay ] below . consider @xmath229 . if @xmath230 and @xmath169 are defined as in and , then we know from @xcite that @xmath231 and @xmath232 the first step is to classify all possible asymptotic behaviors of @xmath233 as @xmath234 goes to infinity . to do so , we shall use the following result which is inspired by fermanian and zaag @xcite : [ prop : class ] assume that @xmath235 and @xmath236 are two solutions of verifying @xmath237 where @xmath238 for some @xmath239 . then , one of the two following cases occurs : + - case 1 : there is a symmetric , real @xmath198 matrix @xmath240 such that @xmath241 - case 2 : there is @xmath242 such that @xmath243 the proof follows from the strategy given in @xcite for the difference of two solutions with the radial profile @xmath244 . note that the case when @xmath132 was treated in @xcite . since some technical details are straightforward , we briefly give the main steps of the proof in section [ sec:3 ] and just emphasize on the novelties . an application of proposition [ prop : class ] with @xmath245 and @xmath246 yields the following corollary directly : [ coro : class ] as @xmath234 goes to infinity , one of the two following cases occurs : + - case 1 : there is a symmetric , real @xmath198 matrix @xmath172 continuous as a function of @xmath77 such that @xmath247 - case 2 : there is @xmath242 such that @xmath248 note that the continuity of @xmath249 comes from the continuity of @xmath155 with respect to @xmath77 , where @xmath155 behaves as in . in particular , zaag @xcite showed the stability of the blow - up behavior with respect to blow - up points ( see proposition 3.1 and section 6.1 in @xcite ) . in the next step , we recall the recent result by nguyen and zaag @xcite , which gives the construction of solutions for equation with some prescribed behavior . [ prop : cons ] consider @xmath103 . for all @xmath250 , where @xmath251 is the set of all symmetric , real @xmath198-matrix , there exists a solution @xmath252 of defined on @xmath253 such that @xmath254 where @xmath165 is the radially symmetric , @xmath160-dimensional solution of satisfying . see theorem 3 in @xcite . although that result is stated for the case @xmath255 , we can extend it to the case when @xmath256 by considering solutions of as @xmath160-dimensional solutions , those artificially generated by adding irrelevant space variables @xmath257 to the domain of definition of the solutions . the following result is a direct consequence of corollary [ coro : class ] and proposition [ prop : cons ] : [ prop : expodecay ] there exist @xmath258 and a continuous matrix @xmath259 , such that for all @xmath229 and @xmath260 , @xmath261 where @xmath262 is the solution constructed as in proposition [ prop : cons ] , @xmath263 is given in proposition [ prop : class ] . moreover , we have + @xmath264 for all @xmath265 , @xmath266 where @xmath267 . + @xmath268 for all @xmath269 , @xmath270 where @xmath271 . from and , we have for any @xmath189 symmetric matrix @xmath190 , @xmath272 choosing @xmath194 , we get @xmath273 note that an alternative application of proposition [ prop : class ] with @xmath274 and @xmath275 yields either or . however , the case is excluded by . hence , follows . since we showed in corollary [ coro : class ] that @xmath276 is continuous , the same holds for @xmath277 . as for , it is a direct consequence of the following lemma which allows us to carry estimate from compact sets @xmath278 to sets @xmath279 : [ lemm : extvel ] assume that @xmath280 satisfies @xmath281 for some @xmath282 . then for all @xmath283 and @xmath284 such that @xmath285 , we have @xmath286 this lemma is a corollary of proposition 2.1 in velzquez @xcite and it is proved in the course of the proof of proposition 2.13 in @xcite ( in particular , pp . 1203 - 1205 ) . let us derive from lemma [ lemm : extvel ] . if we define @xmath287 , straightforward calculations based on yield @xmath288 where @xmath289 for some @xmath290 . from merle and zaag @xcite ( theorem 1 ) , we know that for @xmath234 large enough , @xmath291 which follows @xmath292 if @xmath293 , then we use the kato s inequality @xmath294 to derive equation from and . applying lemma [ lemm : extvel ] together with estimate yields for all @xmath295 and @xmath284 for some @xmath296 large such that @xmath285 , @xmath297 which yields . the estimate directly follows from by the transformation . this ends the proof of proposition [ prop : expodecay ] . in this subsection , we follow the idea of @xcite to introduce local @xmath298-charts of the blow - up set , and get a geometric constraint mechanism on the blow - up set ( see proposition [ prop : geocon ] below ) which is a crucial step in linking refined asymptotic behaviors of the solution to geometric descriptions of the blow - up set . consider @xmath229 and @xmath103 , we introduce the local @xmath298-chart of the blow - up set at the point @xmath77 as follows : @xmath299 where @xmath300 and @xmath301 for some @xmath302 and @xmath303 , then the set @xmath304 is locally near @xmath77 defined by @xmath305 where @xmath306 and @xmath307 are of norm 1 , and respectively , normal and tangent to @xmath304 at @xmath77 . by definition , we have @xmath308 let @xmath145 be the orthogonal matrix whose columns are @xmath309 and @xmath310 , namely that @xmath311 and define @xmath312 then we see from that @xmath313 satisfies and @xmath314 note from that the point @xmath315 in the domain of @xmath313 becomes the point @xmath316 in the domain of @xmath7 , where @xmath317 now , fix @xmath229 and consider an arbitrary @xmath318 . from , we have @xmath319 if we differentiate with respect to @xmath320 with @xmath321 , we get @xmath322 if we fix @xmath123 as the projection of @xmath323 on the blow - up set in the orthogonal direction to the tangent space to the blow - up set at @xmath77 , then @xmath123 has the same components on the tangent space spanned by @xmath324 as @xmath325 . in particular , @xmath326 the following proposition gives a geometric constraint on the expansion of @xmath313 , which is the bridge linking the refined asymptotic behavior to the refined regularity of the blow - up set . [ prop : geocon ] assume that @xmath327 then , there exists @xmath328 ( @xmath329 is introduced in proposition [ prop : expodecay ] ) such that for all @xmath229 , @xmath330 , @xmath331 and @xmath332 , it holds that @xmath333 + c e^{-\frac{(1 + \alpha^*)s}{2}}s^{c_0},\label{est : geoconst}\end{aligned}\ ] ] where @xmath238 , @xmath334 and @xmath123 is defined by . note that the proof of proposition [ prop : geocon ] was given in @xcite only when @xmath132 . of course , that proof naturally extends to the case when @xmath335 . since our paper is relevant only when @xmath136 and proposition [ prop : geocon ] presents an essential link between the asymptotic behavior of the solution and a geometric constraint of the blow up set , we felt we should give the proof of this proposition for the completeness and for the reader s convenience . as said earlier , this section just gives the main steps of the proof of theorem [ theo:1 ] , and because the proof is long and technical , we leave it to section [ sec : ap1 ] . in this subsection , we give the proof of the @xmath11-regularity of the blow - up set ( theorems [ theo:1 ] and [ theo:2 ] ) . we proceed in 2 steps : * step 1 : we derive from proposition [ prop : expodecay ] that @xmath336 is @xmath184 for all @xmath85 . then we apply proposition [ prop : geocon ] with @xmath337 to improve the regularity of @xmath336 which reaches to @xmath207 for all @xmath85 . * step 2 : using the @xmath338-regularity and the geometric constraint in proposition [ prop : geocon ] , we refine the asymptotic behavior given in proposition [ prop : expodecay ] , which involves terms of order @xmath339 . exploiting this refined asymptotic behavior together with the geometric constraint , we derive that @xmath336 is of class @xmath11 , which is the conclusion of theorem [ theo:1 ] . from the information obtained on the @xmath11-regularity , we calculate the second fundamental form of the blow - up set , which concludes the proof of theorem [ theo:2 ] . we first derive the @xmath341-regularity of the blow - up set for all @xmath85 from proposition [ prop : expodecay ] . then we apply proposition [ prop : geocon ] with @xmath342 to get @xmath207-regularity for all @xmath85 . in particular , we claim the following : [ prop : c1alph ] under the hypotheses of theorem [ theo:1 ] , @xmath8 is the graph of a vector function @xmath343 for any @xmath85 , locally near @xmath53 . more precisely , there is an @xmath344 such that for all @xmath345 and @xmath346 such that @xmath347 , we have for all @xmath348 , @xmath349 . in fact , we exploit the estimate to find out a geometric constraint on the blow - up set @xmath8 , which implies some more regularity on @xmath8 . since the argument follows the same lines as in section 4 , @xcite for the case @xmath132 , and no new ideas are needed for the case @xmath136 , we will just sketch the proof by underlying the most relevant aspects in section [ sec : c1alpha ] for the reader s sake . the next proposition shows the @xmath350-regularity of the blow - up set . [ prop : c11eta ] there exists @xmath351 such that for each @xmath229 , the local chart defined in satisfies for all @xmath332 and @xmath352 , @xmath353 note that the case @xmath132 was already proven in @xcite ( see lemma 3.4 , page 516 ) . here we use again the argument of @xcite for the case @xmath136 . using the estimate given in proposition [ prop : expodecay ] and parabolic regularity , we see that for all @xmath354 and @xmath355 , @xmath356 consider @xmath229 and @xmath357 , where @xmath238 is such that @xmath358 for some @xmath359 , @xmath360 for @xmath361 , and @xmath362 is arbitrary in @xmath363 . for @xmath364 , we consider @xmath365 defined as in . since @xmath366 is @xmath184 for any @xmath85 , we use with @xmath342 to write for @xmath367 , @xmath368 since @xmath369 is arbitrary in @xmath370 , we get @xmath371 which gives @xmath372 if @xmath373 , then @xmath374 and @xmath375 since @xmath376 . therefore , @xmath377 since @xmath378 is arbitrary in @xmath363 , @xmath373 covers a whole neighborhood of @xmath379 , namely @xmath380 where @xmath381 , this concludes the proof of proposition [ prop : c11eta ] . in this part , we shall use the @xmath350-regularity of the blow - up set together with the geometric constraint in order to refine further the asymptotic behavior . in particular , we claim the following : [ prop : furre ] there exist @xmath382 , @xmath383 and continuous functions @xmath384 for all @xmath385 with @xmath386 and @xmath387 , where @xmath388 , @xmath389 , such that for all @xmath229 and @xmath390 , @xmath391 where @xmath392 is defined in . the proof of this proposition is based on ideas of @xcite where the case @xmath132 was treated . as in @xcite , the geometric constraint given in proposition [ prop : geocon ] plays an important role in deriving . since the proof is long and technical , we leave it to section [ sec : futre ] . let us derive theorem [ theo:1 ] from propositions [ prop : furre ] and [ prop : geocon ] . in particular , theorem [ theo:1 ] is a direct consequence of the following : [ prop : idenc2 ] for all @xmath229 , we have for all @xmath348 , @xmath393 , @xmath394 where @xmath395 is introduced in proposition [ prop : furre ] , @xmath396 is the @xmath219-th vector of canonical base of @xmath162 , and @xmath397 is the kronecker symbol . from , and the fact that estimate also holds in @xmath398 by parabolic regularity , we derive for all @xmath354 and @xmath399 , @xmath400 for some @xmath401 . + note that if @xmath387 , then there is a unique index @xmath402 such that @xmath403 and @xmath404 for @xmath405 , @xmath406 note also from the definition of @xmath392 ( see below)that @xmath407 and that @xmath408 . therefore , yields @xmath409 take @xmath402 arbitrarily and @xmath410 where @xmath411 and @xmath412 , and note that @xmath413 if @xmath220 is odd , and that if @xmath414 , then either @xmath415 or @xmath416 for some @xmath417 , the above identity yields @xmath418 similarly , we have @xmath419 now using proposition [ prop : geocon ] , we write for @xmath410 and @xmath420 , @xmath421 using this estimate together with and , we obtain @xmath422 from proposition [ prop : furre ] , we see that @xmath423 using this estimate and noticing that the same proof of proposition [ prop : c11eta ] holds with @xmath424 , we derive @xmath425 putting this estimate into and note that @xmath426 , we find that @xmath427 since @xmath428 is taken arbitrarily belonging to @xmath370 , identity holds for all @xmath402 . this concludes the proof of proposition [ prop : idenc2 ] . from the definition of the local chart , we have for all @xmath348 , @xmath429 . hence , we deduce from the expression of the second fundamental form of the blow - up set at the point @xmath77 along the unitary basic vector @xmath203 : for all @xmath430 , @xmath431 in addition , since @xmath384 is continuous , we conclude that the blow - up set is of class @xmath11 . this completes the proof of theorem [ theo:1 ] . the estimate directly follows from propositions [ prop : furre ] and [ prop : idenc2 ] . indeed , the sum in estimate can be indexed as @xmath432 where @xmath433 is the @xmath223-th canonical basis vector of @xmath162 . from and the definition of @xmath392 ( see below ) , we write @xmath434 which yields . as for , we note from that for all @xmath386 with @xmath387 that ( recall that @xmath435 ) , @xmath436 hence , we write from , @xmath437 using again the definition of @xmath392 ( see below ) , we see that @xmath438 and @xmath439 . recall that @xmath191 does not depend on @xmath440 for @xmath412 . hence , for all @xmath441 , @xmath442 which is . this concludes the proof of theorem [ theo:2 ] . in this subsection , we give the proof of proposition [ prop : class ] . the formulation is the same as given in @xcite for the difference of two solutions with the radial profile ( @xmath255 ) . therefore , we sketch the proof and emphasize only on the novelties . note also that the case @xmath132 was treated in @xcite . the operator @xmath449 is self - adjoint on @xmath450 . its spectrum consists of eigenvalues @xmath451 the eigenfunctions corresponding to @xmath452 are @xmath453 where @xmath454 } \frac{m!}{i!(m - 2i)!}(-1)^i\xi^{m - 2i } \quad \text{for}\;\ ; m \in { \mathbb{n}},\ ] ] satisfy @xmath455 the component of @xmath456 on @xmath392 is given by @xmath457 if we denote by @xmath458 the orthogonal projector of @xmath152 over the eigenspace of @xmath449 corresponding to the eigenvalue @xmath452 , then @xmath459 since the eigeinfunctions of @xmath449 span the whole space @xmath152 , we can write @xmath460 where @xmath461 . we also denote @xmath462 where @xmath463 for @xmath479 and @xmath480 , see lemma 2.7 , page 1197 in @xcite . for @xmath481 , see appendix b.1 , page 545 in @xcite for a similar calculations . for @xmath482 , see page 523 in @xcite , where the calculation is mainly based on the following regularizing property of equation by herrero and velzquez @xcite ( control of the @xmath483-norm by the @xmath152-norm up to some delay in time , see lemma 2.3 in @xcite ) : @xmath484 this ends the proof of lemma [ lemm : evoilr ] . see proposition 2.6 , page 1196 in @xcite for the existence of a dominating component , where the proof relies on @xmath479 and @xmath480 of lemma [ lemm : evoilr ] . if case @xmath480 occurs with @xmath495 , we write from @xmath481 of lemma [ lemm : evoilr ] : for all @xmath385 with @xmath501 , @xmath502 where we used and from which @xmath503 and @xmath504 . since @xmath505 is only equal to @xmath506 or @xmath507 if @xmath501 , estimate follows after integration . estimate immediately follows from @xmath479 of lemma [ lemm : evoilr ] . this ends the proof of proposition [ prop : dom ] . let us now derive proposition [ prop : class ] from proposition [ prop : dom ] . indeed , we see from proposition [ prop : dom ] that if case @xmath508 occurs , we already have exponential decay for @xmath467 . if case @xmath480 occurs with @xmath509 , we write from part @xmath479 of lemma [ lemm : evoilr ] , @xmath510 since @xmath511 in a neighbourhood of infinity , this gives @xmath512 which yields . if case @xmath480 occurs with @xmath495 , by definition of @xmath513 , we derive from that there is a symmetric , real @xmath198-matrix @xmath249 such that @xmath514 which is . this concludes the proof of proposition [ prop : class].@xmath515 we give the proof of proposition [ prop : c1alph ] in this section . the proof uses the argument given in @xcite treated for the case @xmath132 . here we shall exploit the refined estimate to obtain a geometric constraint on the blow - up set . without loss of generality , we assume @xmath516 and @xmath517 . under the hypotheses of proposition [ prop : c1alph ] , we know that @xmath518 with @xmath103 . if we introduce @xmath519 then @xmath520 consider @xmath521 and @xmath522 in @xmath523 such that @xmath521 as well as @xmath524 are in @xmath525 and @xmath526 as well as @xmath527 are in @xmath304 . for all @xmath528 such that @xmath529 , we use with @xmath530 , then with @xmath531 and @xmath532 ) to find that @xmath533 where @xmath534 is defined as @xmath535 since @xmath536 is @xmath58 , we have @xmath537 let us fix @xmath538 such that @xmath539 for some @xmath540 , then we have @xmath541 . hence , if @xmath542 , we have by , @xmath543 similarly , by changing the roles of @xmath521 and @xmath524 , we get @xmath544 where @xmath545 is defined as in . + from a taylor expansion for @xmath546 near @xmath547 , we write @xmath548 for some @xmath215 between @xmath379 and @xmath549 . since and also hold in @xmath550 by parabolic regularity , we deduce that @xmath551 from @xcite ( see theorem 1 ) , we know that @xmath552 . substituting all these above estimates into yields @xmath553 therefore , we have @xmath554 we claim from , and the following : @xmath555 indeed , if @xmath556 , then we have by and , @xmath557 if @xmath558 , then we do as above and use instead of to obtain . + from , and , we get @xmath559 hence , we obtain @xmath560 is the tangent plan of @xmath8 at @xmath527 . in the other hand , we claim that @xmath561 where @xmath562 is the surface of equation @xmath563 , @xmath564 is the tangent plan of @xmath562 at @xmath527 . indeed , we note that @xmath565 and @xmath566 , hence , follows from @xmath567 . combining , , together with the relation @xmath568 yields @xmath569 if we denote @xmath570 , then we have by the relation , @xmath571 hence , @xmath572 which yields . this concludes the proof of proposition [ prop : c1alph].@xmath515 [ [ sec : ap1 ] ] a geometric constraint linking the blow - up behavior of the solution to the regularity of the blow - up set . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ this section is devoted to the proof of proposition [ prop : geocon ] . the proof follows ideas given in @xcite . recall from the hypothesis that @xmath573 for some @xmath574 and @xmath303 , and that @xmath429 , we have for all @xmath575 , latexmath:[\[\label{est : varphic1alp } @xmath592 the point @xmath593 . using , and , we write @xmath594 \right\}\cdot q_be_m\\ & = \left\{\sum_{i = 1}^\ell \big[y_i - e^{\frac s2}\gamma_{a , i}(e^{-\frac s2}\tilde{y})\big]\eta_i(a)\right\}\cdot q_be_m.\end{aligned}\ ] ] from , we write for @xmath595 , @xmath596 and for @xmath597 , @xmath598 using yields @xmath599 hence , if we write @xmath600 then @xmath601 term @xmath602 . from proposition [ prop : expodecay ] and the parabolic regularity , we have that @xmath603 this implies @xmath604 similarly , from and , @xmath605 from and , we deduce that @xmath606 using , we have for @xmath607 , @xmath608 note that @xmath609 , we then take the taylor expansion of @xmath610 near @xmath611 up to the first order to get @xmath612 using and yields @xmath613 we prove proposition [ prop : furre ] in this subsection . we first refine estimate and find following terms in the expansion which is of order @xmath617 . using the geometric constraint , we show that all terms of order @xmath617 must be identically zero , which gives a better estimate for @xmath618 . we then repeat the process and use again proposition [ prop : geocon ] in order to get the term of order @xmath619 and conclude the proof of proposition [ prop : furre ] . from @xmath479 and @xmath482 of lemma [ lemm : evoilr ] , we write for all @xmath472 , @xmath631 and @xmath632 the estimate then follows after integration of the above inequalities . as for , we just use part @xmath481 of lemma [ lemm : evoilr ] and ( note that @xmath633 by definition ) . this ends the proof of lemma [ lemm : equbeta3 ] . [ lemm : re_es12 ] there exist @xmath634 and continuous functions @xmath384 for all @xmath385 with @xmath386 and @xmath635 such that for all @xmath229 and @xmath636 , @xmath637 for some @xmath638 , where @xmath392 is defined by . we first show that there is @xmath634 such that @xmath639 from , if @xmath640 , we are done . if @xmath641 , we apply lemma [ lemm : equbeta3 ] with @xmath642 to get @xmath643 and @xmath644 hence , @xmath645 estimate then follows by repeating this process a finite number of steps . now using and lemma [ lemm : equbeta3 ] with @xmath646 , + - if @xmath386 and @xmath647 , we integrate on @xmath648 to derive @xmath649 - if @xmath386 and @xmath650 , by integrating on @xmath651 $ ] , we deduce that there exists continuous functions @xmath384 such that @xmath652 this concludes the proof of lemma [ lemm : re_es12 ] . now we shall use the geometric constraint on the asymptotic behavior of the solution given in proposition [ prop : geocon ] to show that all the coefficients @xmath653 with @xmath386 and @xmath654 in lemma [ lemm : re_es12 ] have to be identically zero . in particular , we claim the following : from , and the fact that the estimate given in lemma [ lemm : re_es12 ] also holds in @xmath660 by parabolic regularity , we write for all @xmath354 and @xmath661 , @xmath662 take @xmath357 , where @xmath663 and @xmath664 , then use proposition [ prop : c11eta ] and , we obtain @xmath665 for some @xmath574 . from and , we get @xmath666 from and proposition [ prop : c11eta ] , we see that @xmath667 as @xmath485 . since @xmath384 is continuous , @xmath401 , @xmath668 from definition , and @xmath407 we derive by passing to the limit in , @xmath669 by the orthogonality of the polynomials @xmath670 , this yields @xmath671 take @xmath672 arbitrary with @xmath386 and @xmath650 , then there exists @xmath354 such that @xmath673 , which implies that @xmath674 . this ends the proof of lemma [ lemm : exps12sd ] . from lemmas [ lemm : exps12sd ] and [ lemm : equbeta3 ] , we see that for all @xmath675 , @xmath676 and @xmath677 for some @xmath401 . integrating this equation between @xmath234 and @xmath678 if @xmath650 and between @xmath679 and @xmath234 if @xmath647 , we get @xmath680 hence , @xmath681 with this new estimate , we use again lemma [ lemm : equbeta3 ] with @xmath682 to show that there exists @xmath683 such that for all @xmath684 , @xmath685 and @xmath686 this new equation implies that for all @xmath386 and @xmath684 , + - if @xmath650 or @xmath687 , we have @xmath688 , + - if @xmath387 , we obtain the existence of continuous functions @xmath384 such that @xmath689 this concludes the proof of proposition [ prop : furre ] . h. zaag . regularity of the blow - up set and singular behavior for semilinear heat equations . in _ mathematics & mathematics education ( bethlehem , 2000 ) _ , pages 337347 . world sci . , river edge , nj , 2002 .

we consider @xmath0 , a solution of @xmath1 which blows up at some time @xmath2 , where @xmath3 , @xmath4 and @xmath5 . define @xmath6 to be the blow - up set of @xmath7 , that is the set of all blow - up points . under suitable nondegeneracy conditions , we show that if @xmath8 contains a @xmath9-dimensional continuum for some @xmath10 , then @xmath8 is in fact a @xmath11 manifold . the crucial step is to derive a refined asymptotic behavior of @xmath7 near blow - up . in order to obtain such a refined behavior , we have to abandon the explicit profile function as a first order approximation and take a non - explicit function as a first order description of the singular behavior . this way we escape logarithmic scales of the variable @xmath12 and reach significant small terms in the polynomial order @xmath13 for some @xmath14 . the refined asymptotic behavior yields geometric constraints of the blow - up set , leading to more regularity on @xmath8 . tej - eddine ghoul@xmath15 , van tien nguyen@xmath15 and hatem zaag@xmath16

recent experimental breakthroughs in spinor boson - einstein condensate , such as the sub - poissonian spin correlations generated by atomic four - wave spin mixing @xcite , the atomic squeezed states realized in the spin-1 ultracold atomic ensembles @xcite , and the antiferromagnetic spatial ordering observed in a quenched one - dimensional spin-1 gas @xcite , are all in connection with the vacuum fluctuations and recall attentions to the finite particle number effect beyond the mean - field treatment . the vacuum fluctuations become a significant subject in more and more experimental facts , e.g. , atomic quantum matter - wave optics , atomic spin squeezing and quantum information . as one of the active frontiers , the spin-1 ultracold atomic ensemble is often adopted . with the basic interaction form @xmath1 , the properties of such a three - component spinor condensate @xcite have been demonstrated experimentally @xcite and two different phases reflecting fundamental properties of spin correlation are identified : the so - called polar and ferromagnetic states for @xmath2 ( @xmath3na ) and @xmath4 ( @xmath5rb ) atomic condensates respectively . the mixture of two spinor condensate with the ferromagnetic and polar atoms , respectively , show more attractive quantum effects luo , xu09,xu2,zhang1,zhang2 , yushi , wangdajun , wangdajun2 . with the help of sympathetic cooling , the bec mixtures of na and rb have been realized and it is interesting to observe the interspecies interaction induced immiscibility between the two condensates @xcite . the ground state of the condensate with @xmath2 has been predicted to be either polar ( @xmath6 ) or antiferromagnetic ( @xmath7 within the mean - field treatment , where the condensate is usually described by a coherent state . however , the many - body theory by law , pu and bigelow @xcite pointed out that the ground state of @xmath2 atoms is a spin singlet with properties ( @xmath8 ) drastically different with the results predicted by the mean field theory . soon , ho and yip @xcite show that this spin singlet state is a fragmented condensate with anomalously large number fluctuations and thus has fragile stability . the remarkable nature of this super - fragmentation is that the single particle reduced density matrix gives three macroscopic eigenvalues ( @xmath9 ) with large number fluctuations @xmath10 . similar considerations were also addressed by koashi and ueda @xcite . the signature of fragmentation is then refer to the anomalously large fluctuations of the populations in the zeeman levels . this is a super - poissonian correlation character , and the large number fluctuations shrink rapidly as the experimentally adventitious perturbations exist , such as magnetic field or field gradient . in this paper we will report the influence of external magnetic field on the spinor condensate with @xmath2 , but on the premise of doping many ferromagnetic atoms in it . the interspecies spin coupling interaction arises and we propose a valid procedure to observe and control the fragmented states . if the ferromagnetic atoms in the mixture are condensed , the ground state favors all atoms aligned along the same direction and provides a uniform and stable background which can delay the rapidly shrinking of the number fluctuations when the inter - species coupling interaction is adjusted . the back action from polar atoms on to the more stable ferromagnetic atoms is negligible . doping ferromagnetic atoms into spin-1 polar condensate can effectively influence the vacuum fluctuations and will have potential applications in quantum information and quantum - enhanced magnetometry . we consider the mixture of two spinor condensates of @xmath11 ferromagnetic and @xmath12 polar atoms , respectively . the intra - condensate atomic spin-1 interaction takes the standard interaction form @xmath13 with @xmath14 . the inter - condensate interaction between the ferromagnetic and polar atoms is @xmath15 , which is more complicated because collision can occur in the total spin @xmath16 channel between different atoms @xcite . the parameters @xmath17 , and @xmath18 are related to the @xmath19-wave scattering lengths in the three total spin channels and the reduced mass @xmath20 for atoms in different species , and @xmath21 projects an inter - species pair into spin singlet state . within the single spatial - mode approximation ( sma ) @xcite for each of the two spinor condensates , the spin - dependent hamiltonian for the mixture finally reads as @xmath22where @xmath23 ( @xmath24 ) are defined in terms of the @xmath25 spin-1 matrices with @xmath26 , and @xmath27 creates a ferromagnetic ( polar ) atom in the hyperfine state @xmath28 . the operator @xmath29creates a singlet pair with one atom each from the two species , similar to @xmath30for intra - species spin - singlet pair @xcite when @xmath31 . the interaction parameters are @xmath32 @xmath33 and @xmath34 which can be tuned through the control of the frequency of the trapping potential xu09 . , at fixed values of @xmath35 , @xmath36 , and @xmath37 . the total numbers of the two species are @xmath38 , and we consider the full - space with total magnetization @xmath39 a variable . black dashed and red solid lines denote the number distributions in the ferromagnetic and polar condensate respectively . all interaction parameters are in the units of @xmath40,width=288 ] when the interspecies scattering parameters are calculated in the degenerate internal - state approximation ( dia ) @xcite , the low - energy atomic interactions can be mostly attributed to the ground - state configurations of the two valence electrons , and the non - commutative term @xmath41 can be neglected @xcite . the ground states are classified into four distinct phases : ff , mm@xmath42 , mm@xmath43 , and aa by three critical values of @xmath44 , @xmath45 , and @xmath46 @xcite . in this paper we discuss the atom number distribution and fluctuation in an external magnetic field . the spin - dependent hamiltonian in the magnetic field reads , @xmath47where only the linear zeeman terms are considered . as the su(2 ) symmetry is broken in a spinor mixture , one can not eliminate the linear zeeman effect through a spin rotation @xcite . meanwhile the quadratic zeeman energy , typically 2 orders of magnitude weaker than the linear terms , is negligible in the calculation of number distributions . furthermore , we take @xmath48 in the following discussion , which can easily realized through adjusting the trapping frequency . we consider the direct product of the fock states of the two species @xmath49 , and do not restrict the model in the subspace with zero total magnetization @xcite . instead , we consider the full space including all possible system magnetization @xmath50 . using the full quantum approach of exact diagonalization , we study the response of the two species to the external magnetic field @xmath51 for @xmath52 . the three critical points for the phase boundaries are approximately @xmath53 . the field dependence of the population is shown in fig . 1 for the mm@xmath43 phase at @xmath54 , where polar atoms are partly polarized in the oppsite direction as the ferromagnetic atoms @xcite . we notice that the ferromagnetic atoms ( black dashed lines ) are very sensitive to the magnetic field , i.e. atoms quickly redistribute in the @xmath55 component and the magnetization of ferromagnetic condensate @xmath56 saturates immediately . the ferromagnetic atoms actually form a stable condensate and provide a uniform magnetic background in the mixture . the polar atoms present a stepwise increase ( decrease ) in the atom number distribution @xmath57 when the field increases . for small @xmath51 and positive @xmath58 , the system favors a negative magnetization ( @xmath59 ) of polar condensate , and @xmath60 will reverse and tend to saturate for large magnetic field . we notice that the super - fragmented state featured with @xmath61 remarkably arises around a special value of @xmath62 . the situation becomes more simple if the parameter @xmath63 is negative , that is , in the ff phase ( or mm@xmath42 phase ) , where polar atoms are fully ( partly ) polarized in the same direction as the ferromagnetic atoms . the enhanced ferromagnetic effect and the external magnetic field jointly suppress the atom number distribution @xmath64 and @xmath65 of the polar condensate to zero , and at the same time saturate @xmath55 and the magnetization @xmath60 without magnetization reversal . and @xmath51 at fixed values of @xmath35 and @xmath36 . this column only shows the results of polar atoms with @xmath64 and @xmath66 . when the extra magnetic field parameter @xmath51 ( in the units of @xmath67 ) increases , there are serval critical points associated with @xmath68 . all interaction parameters are in the units of @xmath40,width=259 ] at fixed values of @xmath35 , @xmath69 and @xmath36 . the total numbers of the two species are @xmath70 , @xmath71 . black solid , red dash - dot , and blue dashed lines denote atom numbers and the fluctuations on the @xmath72 and @xmath73 sub - levels respectively . all interaction parameters are in the units of @xmath40,width=240 ] the super - fragmented state , with equal population @xmath74 and zero magnetization @xmath75 , has been predicted in the pure spin-1 polar condensate and described by a spin singlet state in the form @xmath76 @xcite . @xmath77 is invariant under spin rotations and commutes with @xmath78 and @xmath79 . in the ground state of the system subject to an external magnetic field @xmath80 , one can see a rapid shrink of the spin-0 component distribution @xmath64 and an unbalanced population @xmath81 . the super - fragmented state then reduces to a much more generic fragmented state : a two component number state with essentially zero fluctuations . such state with polar interaction was not likely realized in typical experiments due to its fragility towards any perturbation breaking spin rotational symmetry . for the spin-1 polar condensate doped with many ferromagnetic atoms , we can retrieve this super - fragmented state in the presence of an external field . for some special values of the magnetic field , both the spin-0 component population and number fluctuations would not shrink but revive to macroscopic orders of the super - fragmented state . in fig . 2 , we illustrate the revival points for three inter - species coupling parameters @xmath82 . these revival points are found to move towards larger value of @xmath51 as @xmath82 increases . as learned from previous studies @xcite , the mean - field treatment is efficient for atomic interaction of the ferromagnetic type . the much more stable ferromagnetic condensate in the mixture can be formulated in the mean field treatment as a boson - enhanced effective magnetic field . this simplifies the hamiltonian ( [ hhh ] ) as @xmath83where @xmath84 @xmath85 @xmath86 the criterion for the emergence of super - fragmented state is @xmath87 , where the magnetic field ( @xmath51 ) , the optical trapping frequency ( @xmath88 ) , and the number of the doped ferromagnetic atoms ( @xmath89 ) are all adjustable . when the magnetic field matches the condition that @xmath90 and @xmath91 cancel each other , we may achieve the super - fragmented state in a magnetic field . the three critical points in fig . 2 are found to agree with the numerical results exactly . next , we turn to the situation with population imbalance in the two species . 3 illustrates the location of the critical point when the inter - species coupling parameter @xmath82 is fixed to be 2.5 and the atom numbers for the two species are @xmath92 and @xmath71 . as the mean - field picture works well for the ferromagnetic atoms , we still get the crucial point @xmath93 in fig . 3 . when equal population @xmath94 occurs for the polar condensate , the number fluctuations also instantaneously reach to the macroscopic levels . our numerical results for the fluctuation relation ( @xmath95 ) agree exactly with the algebraic results in @xcite for pure polar condensate . with the emergence of equal population @xmath9 regarded as a sign of anti - ferromagnetic spin interaction , the inter - species spin coupling interaction @xmath63 can be estimated by the location of the critical magnetic field . of the polar condensate on both @xmath68 and magnetic field @xmath51 at fixed values of @xmath35 and @xmath96 . the total numbers of the two species are @xmath97 . black solid , red dot , and blue dash - dot lines denote the numbers on the 1 , 0 , and @xmath73 sub - levels respectively . all interaction parameters are in the units of @xmath40,width=240 ] aa phase is another super - fragmented state which have been predicted in the absence of magnetic fields @xcite . it is a many - body spin singlet , which requires exactly the same atoms number of the two species ( @xmath98 ) , and total spins from different species polarized in opposite directions . in the notation of the angular momentum representation @xmath99 aa phase is denoted as @xmath100 with @xmath101 and @xmath102 the total spin quantum numbers of the ferromagnetic atoms , polar atoms , and the mixture and @xmath103 and @xmath39 the corresponding @xmath104-components . the intra - species angular momentum states involved in the aa phase , @xmath105 and @xmath106 , should obey the constraint @xmath107 . the interesting feature of aa phase is the equal distribution of atoms in the six components ( @xmath9 ) and the large number fluctuations . using the full quantum approach of exact diagonalization , and considering the full space including all possible system magnetization @xmath108 , we study the responses of the aa phase to the external magnetic field with @xmath109 , and compare the results with the super - fragmented state in the pure condensate @xcite . in fig . 4 , we compare these two typical fragmented ground states : the inter - species entangled singlet @xmath110 and the pure polar singlet @xmath111 , which belong to two special phases characterized by typical values of the interaction parameter : @xmath112 , and @xmath113 . we find that the responses of the @xmath64 component to the magnetic field are quite different . 4a shows the influence of a magnetic field on a pure polar condensate . as @xmath51 increases , the 0-component distribution @xmath64 ( red dashed line ) shrink rapidly , which agree with the algebraic results in @xcite @xmath114 . 4b shows the influence of a magnetic field on the polar atoms in the aa phase . we find that @xmath64 does not shrink rapidly in the beginning , instead , it increases first and remains in a high value for a certain range of @xmath51 . compared with the pure polar spin - singlet state , the inter - species entangled singlet seems to be more difficult to be magnetized as @xmath51 increases . when all the polar atoms are forced to arrange in the opposite direction of the ferromagnetic ones , the bose - enhanced magnetic background formed by ferromagnetic condensate will be canceled . the system spontaneously breaks into a high symmetry state when the interaction parameter @xmath63 goes across the phase transition point @xmath115 . the magnetic behaviors of the two species are then identical ( except for the atom mass ) , and both ferromagnetic atoms and polar atoms are equally to be magnetized when the field intensity is increased . compared with the pure singlet in fig . 4a , the involved total atoms are doubled , and at least a field strength @xmath116 is needed to saturate the magnetization . the external magnetic field can be used to characterize these two spin - singlets through tracing the atoms numbers of @xmath64 component of the polar atoms . if we refer to more general case beyond the dia approximation , the @xmath18 term of the hamiltonian ( [ ham ] ) should be considered . we notice that@xmath117 \neq 0,\left [ \mathbf{\hat{f}}^{2},\hat{\theta}_{12}^{\dag } \hat{% \theta}_{12}\right ] = 0,\]]which means in general they do not belong to a set of commutative operators . however , we can numerically study the phase transition through the order parameter @xmath118 zhang2 . to see more clearly the role played by the parameter @xmath119 on the fragmentation , we numerically diagonalize the hamiltonian ( [ ham ] ) with @xmath52 . and @xmath64 in the polar condensate on magnetic coefficient @xmath51 and @xmath120 at fixed values of @xmath35 , @xmath121 and @xmath36 . the total numbers of the two species are @xmath52 . black solid line and red dashed line denote the value of @xmath55 and @xmath64 respectively . all interaction parameters are in the units of @xmath40,width=259 ] in fig.5 , we illustrate the influence of a small @xmath122 to the population @xmath123 and @xmath55 of super - fragmented state which has been retrieved in mm@xmath124 phase in the presence of the magnetic field . we find that the crucial point is still located at @xmath93 , but a tiny @xmath125 will elevate the @xmath64 component to a dominated value , meanwhile suppress the @xmath55 and @xmath65 components to lower level . the high occupation on @xmath126 component is an evidence of the nematic order @xcite , but the signature of fragmentation is still obvious . away from for the critical point where @xmath127 are suppressed to be nearly zero , the @xmath55 ( @xmath65 ) component is linearly increased ( decreased ) to meet the conditions that the system magnetization should be increased . for @xmath128 , both @xmath126 and @xmath66 shrink , the system reduced to a much more generic fragmented state with the system magnetization @xmath129 which increases linearly with @xmath51 . this situation is more like the pure polar condensate in a weak magnetic field . , @xmath130 at fixed values of @xmath131 , @xmath132 and @xmath36 . this graph only shows the results of @xmath64 and @xmath133 when the interaction parameter @xmath130 equals to @xmath134 . the total numbers of the two species are @xmath52 and we restrict the problem in full - space without the external magnetic field . black solid lines and red dash lines denote the ferromagnetic and polar condensate respectively . all interaction parameters are in the units of @xmath40,width=249 ] the negative @xmath18 term encourages pairing two different types of atoms into singlets @xcite . in fig . 6 , the influences of @xmath135 singlet pairing coefficient @xmath136 on the numbers and quantum fluctuations of the two species are illustrated . we notice that the typical @xmath9 number distributions arise both in the @xmath137 and @xmath138 regions when @xmath139 reaches to @xmath134 . the number fluctuation @xmath140 gives two steady values , which represent two typical fragmented ground state : the inter - species entangled fragmented state for @xmath141 , and the inter - species independent fragmented state for @xmath142 . the fluctuations for these two states @xmath143are found to match the numerical results in fig . to conclude , we studied the ground state properties of a binary mixture of ferromagnetic and polar spinor condensates in a magnetic field . using the full quantum approach of exact diagonalization , we can study the competition between magnetic linear zeeman effect and the inter - species spin coupling interaction @xmath63 . the large vacuum fluctuation of number distributions on the three zeeman levels inside the polar condensate is worthy of investigation . we point out that the fragmentation properties of polar condensate can be adjusted through the magnetic field ( p ) , trapping frequency @xmath144 , and the number of doped ferromagnetic atoms ( @xmath89 ) . the ferromagnetic condensate is involved to provide a uniform and stable background which can delay the rapidly shrinking of the large number fluctuations . we illustrated the influences of the magnetic parameter @xmath51 , and identified two typical fragmented state with total spin @xmath145 . the positive inter - species spin coupling interaction ( @xmath146 ) can effectively entangle the different species , while for @xmath147 the different species on their @xmath148 manifold are essentially independent . we propose a possible mechanism to effectively measure the inter - species spin coupling interactions through applying a magnetic field , as well as discriminate the two types of many - body spin singlets . our work highlights the significant promises for experimental work on sodium and rubidium atomic condensate mixtures and provide some useful information for the study of photo - association of heteronuclear molecules . this work is supported by the nsf of china under grant nos . 11204204 , 11347181 , the nsf of shanxi province under grant nos . 2012021003 - 2 , 2014021011 - 1 and the fund of taiyuan university of technology for young teachers . yz is also supported by the national basic research program of china ( 973 program ) under grant no . 2011cb921601 , program for changjiang scholars and innovative research team in university ( pcsirt)(no . irt13076 ) . c.d . hamley , c.s . gerving , t.m . hoang , e.m . bookjans and m.s . chapman , nature physics , vol . 4 . 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we study the ground state quantum fragmentation in a mixture of a polar condensate and a ferromagnetic condensate when subject to an external magnetic field . we pay more attentions to the polar condensate and find that it will be less fragile in the mixture when perturbed by the magnetic field . both atom numbers and the number fluctuations in the spin-0 component will keep in a high magnitude of order of @xmath0 when the magnetization of the system is increased . the role of the ferromagnetic condensate is to provide a uniform and stable background which can delay the rapid shrink of the 0-component population and make it possible to capture the super - fragmentation . our method has potential applications in measuring the inter - species spin - coupling interaction through adjusting the magnetic field .

given a simple undirected graph @xmath4 with vertex set @xmath5 , let @xmath6 be the set of all real symmetric @xmath3 matrices @xmath7 $ ] such that for @xmath8 , @xmath9 if and only if @xmath10 . there is no condition on the diagonal entries of @xmath11 . the set @xmath12 is defined in the same way over hermitian @xmath13 matrices , and every problem we consider comes in two flavors : the real version , involving @xmath14 , and the complex version , involving @xmath12 . there are known examples where a question of the sort we examine here has a different answer when considered over hermitian matrices rather than over real symmetric matrices @xcite , @xcite , but for each question that is completely resolved in the present paper , the answer over @xmath12 proves to be the same as that obtained over @xmath14 . the inverse eigenvalue problem for graphs asks : given a graph @xmath0 on @xmath1 vertices and prescribed real numbers @xmath15 , is there some @xmath16 ( or @xmath17 , alternatively ) such that the eigenvalues of @xmath18 are exactly the numbers prescribed ? in general , this is a very difficult problem . some contributions to its solution appear in @xcite , @xcite , @xcite , @xcite . a more modest goal is to determine the maximum multiplicity @xmath19 of an eigenvalue of a matrix in @xmath6 . this is easily seen to be equivalent to determining the minimum rank @xmath20 of a matrix in @xmath6 since @xmath21 . this problem has been intensively studied . some of the major contributions appear in the papers @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . a variant of this problem is the study of @xmath22 , the minimum rank of all positive semidefinite @xmath23 . the hermitian maximum multiplicity @xmath24 , hermitian minimum rank @xmath25 , and hermitian positive semidefinite rank @xmath26 are defined analagously . a problem whose level of difficulty lies between the inverse eigenvalue problem and the minimum rank problem for graphs is the inverse inertia problem , which we now explain . [ d1 ] given a hermitian @xmath3 matrix @xmath11 , the _ inertia _ of @xmath11 is the triple @xmath27 where @xmath28 is the number of positive eigenvalues of @xmath11 , @xmath29 is the number of negative eigenvalues of @xmath11 , and @xmath30 is the multiplicity of the eigenvalue @xmath31 of @xmath11 . then @xmath32 and @xmath33 . if the order of @xmath18 is also known then the third number of the triple is superfluous . the following definition discards @xmath34 . [ d2 ] given a hermitian matrix @xmath11 , the _ partial inertia _ of @xmath11 is the ordered pair @xmath35 we denote the partial inertia of @xmath11 by @xmath36 . we are interested in the following problem : [ inverse inertia problem ] given a graph @xmath37 on @xmath38 vertices , for which ordered pairs @xmath39 of nonnegative integers with @xmath40 is there a matrix @xmath41 such that @xmath42 ? the hermitian inverse inertia problem is the same question with @xmath12 in the place of @xmath14 . it is well known @xcite that in the case of a tree @xmath43 most questions over @xmath44 are equivalent to questions over @xmath45 , and in particular if @xmath46 is a forest and @xmath47 , then there exists a diagonal matrix @xmath48 with diagonal entries from the unit circle such that @xmath49 . in those sections concerned with the inverse inertia problem for trees and forests we thus assume without loss of generality that every matrix in @xmath50 is in fact in @xmath51 . in this paper we give a complete solution to the inverse inertia problem for trees and forests . the statement of our solution is a converse to an easier pair of lemmas that apply not just to forests but to any graph . [ northeast lemma ] [ l5 ] let @xmath37 be a graph and suppose that @xmath17 with @xmath52 . then for every pair of integers @xmath53 and @xmath54 satisfying @xmath55 , there exists a matrix @xmath56 with @xmath57 . if in addition @xmath18 is real , then @xmath58 can be taken to be real . in other words , thinking of partial inertias or hermitian partial inertias as points in the cartesian plane , the existence of a partial inertia @xmath59 implies the existence of every partial inertia @xmath60 anywhere `` northeast '' of @xmath59 , as long as @xmath61 does not exceed @xmath38 . we prove this lemma in section [ s2 ] by perturbing the diagonal entries of @xmath18 . to state the second lemma we need to introduce an indexed family of graph parameters . [ dmd ] let @xmath37 be a graph with @xmath38 vertices . for any @xmath62 we define @xmath63 , the _ maximal disconnection _ of @xmath37 by @xmath64 vertices , as the maximum , over all induced subgraphs @xmath46 of @xmath37 having @xmath65 vertices , of the number of components of @xmath46 . for example , @xmath66 is the number of components of @xmath37 , and if @xmath43 is a tree then @xmath67 is the maximum vertex degree of @xmath43 . since an induced subgraph can not have more components than vertices , we always have @xmath68 . as far as we can determine , @xmath63 is not a known family of graph parameters . it is , however , related to the toughness @xmath69 of a graph , which can be defined @xcite as @xmath70 for a recent survey of results related to toughness of graphs , see @xcite . there is also some relation between @xmath63 and vertex connectivity : a graph @xmath37 on @xmath38 vertices is @xmath64-connected , @xmath71 , if and only if @xmath72 whenever @xmath73 . [ stars and stripes lemma ] [ lmd ] let @xmath37 be a graph with @xmath38 vertices , let @xmath62 be such that @xmath74 , and choose any pair of integers @xmath75 and @xmath76 such that @xmath77 , @xmath78 , and @xmath79 . then there exists a matrix @xmath41 such that @xmath42 . this lemma is proved in section [ s2 ] ; the idea of the proof is that each partial inertia in the diagonal `` stripe '' from @xmath80 northwest to @xmath81 can be obtained by combining the adjacency matrices of `` stars '' at each of the @xmath64 disconnection vertices together with , for each of the remaining components , a matrix of co - rank @xmath82 and otherwise arbitrary inertia . these two lemmas provide a partial solution to the inverse inertia problem for any graph . our main result for trees and forests is that for such graphs , and exactly such graphs , the partial solution is complete . let @xmath37 be a graph on @xmath38 vertices . then @xmath39 is an _ elementary inertia _ of @xmath37 if for some integer @xmath64 in the range @xmath83 we have @xmath84 , @xmath85 , and @xmath86 . the elementary inertias of a graph @xmath37 are exactly those partial inertias that can be obtained from @xmath37 by first applying the stars and stripes lemma and then applying the northeast lemma . the partial solution given by these lemmas is the following : if @xmath39 is an elementary inertia of a graph @xmath37 , then there exists a matrix @xmath87 with @xmath42 . this is proved as observation [ ob_elini ] in section [ s2 ] . [ th_treemain ] the stars and stripes lemma and the northeast lemma characterize the partial inertias of exactly forests , as follows : 1 . let @xmath46 be a forest , and let @xmath88 with @xmath42 . then @xmath39 is an elementary inertia of @xmath46 . 2 . conversely , let @xmath37 be a graph and suppose that for every @xmath87 , @xmath89 is an elementary inertia of @xmath37 . then @xmath37 is a forest . of course claim 1 also applies for @xmath47 , since for @xmath46 a forest any matrix in @xmath50 is diagonally congruent to a matrix in @xmath51 having the same partial inertia . claim 2 of theorem [ th_treemain ] is a corollary to known results , here called theorem [ th1 ] . we prove claim 1 of theorem [ th_treemain ] at the end of section [ s5 ] . in section [ s4 ] we show that determining the set of possible inertias of any graph with a cut vertex can be reduced to the problem of determining the possible inertias of graphs on a smaller number of vertices . the formula we obtain is a generalization of the known formula for the minimum rank of a graph with a cut vertex . in section [ s5 ] we describe elementary inertias in terms of certain edge - colorings of subgraphs , and we show that the same cut - vertex formula proven in section [ s4 ] for inertias also holds when applied to the ( usually smaller ) set of elementary inertias . applying these parallel formulas inductively to trees and forests then gives us a proof of claim 1 of theorem [ th_treemain ] . in section [ s6 ] we outline an effective procedure for calculating the set of partial inertias of any tree , using the results of section [ s3 ] to justify some simplications , and we calculate a few examples . in section [ s7 ] we again consider more general graphs , and demonstrate both an infinite family of forbidden inertia patterns , and the first example of a graph that is not inertia - balanced . the concept of an inertia - balanced graph was introduced in @xcite , and determining whether a graph is inertia - balanced is a special case of the inverse inertia problem . [ d3 ] a hermitian matrix @xmath11 is _ inertia - balanced _ if @xmath90 a graph @xmath91 is _ inertia - balanced _ if there is an inertia - balanced @xmath23 with @xmath92 . a graph @xmath91 is _ hermitian inertia - balanced _ if there is an inertia - balanced @xmath93 with @xmath94 . our formulation , unlike the definition in @xcite , is symmetric in allowing @xmath95 . this doubles the set of inertia - balanced matrices of odd rank , but the two definitions are equivalent when applied to graphs since @xmath41 if and only if @xmath96 . barioli and fallat @xcite proved that every tree is inertia - balanced . theorem [ th_treemain ] , once proved , will imply a slightly stronger result . the intuition for expecting a graph to be inertia - balanced comes from many small examples in which achieving an eigenvalue of high multiplicity appears to become increasingly difficult as the imbalance increases between the number of eigenvalues that are higher and the number that are lower than the target multiple eigenvalue . the behavior observed in these small examples can be stated formally in terms of the following definitions . [ dstripe ] a set @xmath97 of ordered pairs of integers is called _ symmetric _ if whenever @xmath98 , then @xmath99 . a symmetric nonempty set @xmath97 of ordered pairs of nonnegative integers is called a _ stripe _ if there is some integer @xmath100 such that @xmath101 for every @xmath98 , and we specify the particular constant sum by saying that @xmath97 is a _ stripe of rank @xmath100 . _ a stripe @xmath97 is _ convex _ if the projection @xmath102 is a set of consecutive integers . the set @xmath103 is symmetric , the set @xmath104 is a stripe , and the stripe @xmath105 is convex . [ ob_symmetry ] given a graph @xmath37 of order @xmath38 and an integer @xmath100 in the range @xmath106 , the set @xmath107 is a stripe of rank @xmath100 . the same is true for @xmath108 with @xmath100 in the range @xmath109 . symmetry comes from the fact that @xmath96 if and only if @xmath41 , and similarly for @xmath12 . the sets are nonempty by the definitions of @xmath110 and @xmath25 and the northeast lemma . a graph @xmath37 is _ inertia - convex on stripes _ or _ hermitian inertia - convex on stripes _ if each of the stripes defined in observation [ ob_symmetry ] ( with @xmath41 or @xmath111 , respectively ) is convex . in other words , a graph is inertia - convex on stripes if each stripe of possible partial inertias does not contain a gap . [ corollary to theorem [ th_treemain ] ] [ cor_stripes ] every forest is inertia - convex on stripes . let @xmath46 be a forest . by theorem [ th_treemain ] , each of the stripes defined in observation [ ob_symmetry ] is the set of elementary inertias of some fixed rank @xmath100 . for each fixed @xmath64 with @xmath112 we obtain a set of elementary inertias which is a union of convex stripes . it follows that for any fixed @xmath100 , the set of elementary inertias of rank @xmath100 is the union of convex stripes of rank @xmath100 as @xmath64 varies over all allowed integers . since a union of convex stripes of the same rank is a single convex stripe , each of the stripes defined in observation [ ob_symmetry ] is convex . it has been an outstanding question if there is any graph that is not inertia - balanced . at the aim workshop in palo alto in october 2006 , the prevailing opinion was that such a graph does not exist @xcite . in section [ s7 ] we give an example of a graph that is not inertia - balanced . first we show that every graph satisfies a condition that is much weaker than inertia - balanced ( except in the case of minimum semidefinite rank @xmath113 ) . the counterexample graph and new condition together allow us to completely determine which sets can occur as the complement of the set of possible partial inertias of a graph @xmath37 with @xmath114 . the possible excluded partial inertia sets giving minimum semidefinite rank @xmath115 or greater remain unclassified . for the most part our notation for graphs follows diestel @xcite . we make use specifically of the following notation throughout : * all graphs are simple , and a graph is formally an ordered pair @xmath116 where @xmath117 is a finite set and @xmath118 consists of pairs from @xmath117 . when referring to an individual edge , we abbreviate @xmath119 to @xmath120 or @xmath121 . the vertex set of a graph @xmath37 is also referred to as @xmath122 , and the edge set as @xmath123 . * for @xmath124 , @xmath125 $ ] is the subgraph of @xmath37 induced by @xmath97 and @xmath126 is the induced subgraph on @xmath127 . we write @xmath128 rather than @xmath129 and @xmath130 rather than @xmath131 . * the number of vertices of a graph @xmath37 is denoted @xmath132 . * @xmath133 is the complete graph on @xmath1 vertices . * @xmath134 is called the _ star graph _ on @xmath1 vertices . this is the same as the complete bipartite graph @xmath135 . * @xmath136 is the path on @xmath1 vertices . paths are described explicitly by concatenating the names of the vertices in order ; for example , @xmath137 denotes the graph @xmath138 . * if @xmath139 is a vertex of @xmath0 , @xmath140 is the degree of @xmath139 . * @xmath141 . we conclude the introduction with some elementary facts about inertia , and include short proofs to keep the paper self - contained . [ p1 ] let @xmath11 be a hermitian @xmath3 matrix and let @xmath142 be a principal submatrix of @xmath18 of size @xmath143 . then @xmath144 by the interlacing inequalities @xcite @xmath145 where @xmath146 are the eigenvalues of @xmath11 and @xmath147 are the eigenvalues of @xmath142 , arranged in decreasing order . if @xmath148 . otherwise , let @xmath149 be the largest integer with @xmath150 . then @xmath151 and @xmath152 . if @xmath153 . otherwise , let @xmath154 be the largest integer with @xmath155 . then @xmath156 and @xmath157 . similarly , @xmath158 . [ subadditivity ] [ p2 ] let @xmath11 , @xmath142 be hermitian @xmath3 matrices and let @xmath159 . then @xmath160 if @xmath161 , the first inequality is true , so assume that @xmath162 . let @xmath163 and @xmath164 . then @xmath165 and @xmath166 . by the weyl inequalities @xcite , @xmath167 therefore @xmath168 . similarly , @xmath169 . [ p3 ] let @xmath11 be a hermitian @xmath3 matrix and let @xmath170 be a hermitian rank 1 matrix ( so @xmath171 is real - valued ) . then @xmath172 and @xmath173 let @xmath174 . then @xmath175 , @xmath176 . by proposition [ p2 ] , @xmath177 the argument is similar if @xmath178 . [ d4 ] let @xmath179 be the set of nonnegative integers , and let @xmath180 . we define the following sets : @xmath181}= { \mathbb{n}}^2_{\ge i}\cap { \mathbb{n}}^2_{\le j } , \\ & & { \mathbb{n}}^2_i= { \mathbb{n}}^2_{[i , i ] } \ \mbox{(the \textit{complete stripe } of rank $ i$)}.\end{aligned}\ ] ] we note that a stripe of rank @xmath182 is a nonempty symmetric subset of @xmath183 . [ d5 ] given a graph @xmath0 , we define @xmath184 and @xmath185 we call @xmath186 the _ inertia set _ of @xmath0 and @xmath187 the _ hermitian inertia set _ of @xmath0 . now suppose @xmath188 and let @xmath23 with @xmath189 . since @xmath190 , we have @xmath191 . we record this as [ ob4 ] given a graph @xmath0 on @xmath1 vertices , @xmath192 } $ ] and @xmath193 } $ ] . the fact that every real symmetric matrix is also hermitian immediately gives us : [ obvious ] for any graph @xmath37 , @xmath194 and @xmath195 . the northeast lemma , as stated in the introduction , substantially shortens the calculation of the inertia set of a graph . [ proof of northeast lemma ] let @xmath0 be a graph and suppose that @xmath196 , and let @xmath197 be given with @xmath198 and @xmath199 . we wish to show that @xmath200 . if in addition @xmath201 , we must show that @xmath188 . let @xmath93 with @xmath202 . if @xmath203 there is nothing to prove , so assume @xmath204 . it suffices to prove that there exists a @xmath205 with @xmath206 , because then an analogous argument can be given to prove that there is a @xmath207 with @xmath208 and these two facts may be applied successively to reach @xmath209 . we also need to ensure that when @xmath18 is real symmetric @xmath58 is also real symmetric . choose @xmath210 such that @xmath211 is invertible and @xmath212 . then @xmath213 . let @xmath214 and then perturb the diagonal entries in order : for any @xmath215 let @xmath216 , so that @xmath217 . then @xmath218 for @xmath219 and by propositions [ p2 ] and [ p3 ] , @xmath220 for @xmath221 . it follows that every integer in @xmath222 is equal to @xmath223 for some @xmath224 . since @xmath225 by proposition [ p3 ] , @xmath226 for @xmath227 . then for some @xmath228 we have @xmath229 , and we can take @xmath230 . as desired , @xmath58 is real symmetric if @xmath18 is real symmetric , which completes the @xmath14 version of the northeast lemma as well as the @xmath12 version : within either one of the two inertia sets @xmath231 or @xmath232 , the existence of a partial inertia @xmath233 implies the existence of every partial inertia @xmath39 within the triangle @xmath234 or in other words every partial inertia to the `` northeast '' of @xmath233 . [ d6 ] if a graph @xmath0 on @xmath1 vertices satisfies @xmath235 } $ ] we say that @xmath0 is _ inertially arbitrary_. if a graph @xmath0 on @xmath1 vertices satisfies @xmath236 } $ ] we say that @xmath0 is _ hermitian inertially arbitrary_. [ ex1 ] the complete graph @xmath237 , @xmath238 . since @xmath239 ( the all ones matrix ) @xmath240 , @xmath241 . by the northeast lemma @xmath242}\subseteq{{\mathcal i}}(k_n ) $ ] . since @xmath243}= { \mathbb{n}}^2_{[1,n ] } $ ] by observation [ ob4 ] , @xmath133 is inertially arbitrary . [ ex2 ] the path @xmath244 , @xmath238 . a consequence of a well - known result of fiedler @xcite is that for a graph @xmath0 on @xmath1 vertices , @xmath245 if and only if @xmath246 . it follows from a theorem in @xcite that there is an @xmath247 with eigenvalues @xmath248 . then for @xmath249 , @xmath250 . by the northeast lemma , @xmath251}= { \mathbb{n}}^2_{[\operatorname{mr}(p_n),n ] } $ ] , so @xmath136 is also inertially arbitrary . the partial inertia set for a graph on @xmath1 vertices can never be smaller than the partial inertia set for @xmath136 . [ p6 ] if @xmath0 is any graph on @xmath1 vertices , @xmath252 } \subseteq { { \mathcal i}}(g ) $ ] . let @xmath253 with @xmath254 . let @xmath255 and let @xmath256 be the adjacency matrix of @xmath0 . by gershgorin s theorem , @xmath257 has eigenvalues @xmath258 , so @xmath259 . furthermore for @xmath260 , @xmath261 has partial inertia @xmath262 . it follows that @xmath252 } \subseteq { { \mathcal i}}(g ) $ ] . the fact that inertia sets are additive on disconnected unions of graphs ( observation [ ob21 ] ) gives us an immediate corollary . [ cor_ellcomp ] if @xmath37 is any graph on @xmath38 vertices and @xmath37 has @xmath263 components , @xmath264 } \subseteq { { \mathcal i}}(g)$ ] . the existence of a complete stripe of partial inertias of rank @xmath265 plays a role in the proof of our second lemma from the introduction . [ proof of the stars and stripes lemma ] let @xmath37 be a graph with @xmath38 vertices , and let @xmath266 be such that @xmath267 and @xmath268 has @xmath63 components , with @xmath74 . also , let @xmath39 be any pair of integers such that @xmath84 , @xmath85 , and @xmath79 . without loss of generality label the vertices of @xmath37 so that @xmath269 , and for each vertex @xmath270 let @xmath271 be the @xmath13 adjacency matrix of the subgraph of @xmath37 that retains all vertices of @xmath37 , but only those edges that include the vertex @xmath272 . if @xmath272 is isolated in @xmath37 then @xmath273 ; otherwise the subgraph is a star plus isolated vertices and @xmath274 . now @xmath268 is a graph with @xmath65 vertices and @xmath63 components , so by corollary [ cor_ellcomp ] there exists a matrix @xmath275 with @xmath276 . let @xmath277 be the direct sum of the @xmath278 zero matrix with @xmath58 , so that the rows and columns of @xmath277 are indexed by the full set @xmath122 , as is the case with the matrices @xmath279 . let @xmath280 . then @xmath281 , and by subadditivity of partial inertias ( proposition [ p2 ] ) we also have @xmath282 and @xmath283 . since @xmath284 we can apply the northeast lemma to conclude that @xmath285 . as mentioned in the introduction , the partial inertias which can be deduced from lemmas [ l5 ] and [ lmd ] are precisely the elementary inertias . [ d_elset ] let @xmath37 be a graph on @xmath38 vertices . then the _ set of elementary inertias of @xmath37 _ , @xmath286 , is given by @xmath287 we may also think of @xmath286 as follows : for each integer @xmath64 , @xmath288 , let @xmath289 and let @xmath290 . each nonempty @xmath291 is a possibly degenerate trapezoid , and @xmath292 [ ob_elini ] for any graph @xmath37 , we have @xmath293 . let @xmath37 be a graph on @xmath38 vertices , and suppose @xmath294 . then for some integer @xmath64 we have @xmath295 ( note that this implies @xmath74 . ) recall that @xmath68 , so @xmath296 it follows that there is an ordered pair of integers @xmath297 satisfying @xmath298 the stars and stripes lemma gives us @xmath299 , after which the northeast lemma gives us @xmath300 since @xmath40 . given a graph @xmath46 on @xmath100 vertices there is a smallest integer @xmath301 such that @xmath302 . if @xmath46 is inertia - convex on stripes then @xmath301 is the same as @xmath303 , and if @xmath46 is inertially arbitrary then @xmath301 is the same as @xmath304 . suppose that @xmath46 is @xmath268 as in the definition of @xmath63 , with @xmath267 and @xmath305 . then some trapezoid of elementary inertias of @xmath37 comes from the easy estimate that the maximum co - rank of arbitrary inertia for @xmath46 , i.e. @xmath306 , is at least the number of components @xmath263 of @xmath46 ( corollary [ cor_ellcomp ] ) . suppose we had an improved lower bound @xmath307 for this co - rank , a graph parameter that always satisfies @xmath308 . ( the improvement @xmath309 will be guaranteed , for example , if @xmath310 is additive on the components of @xmath46 and is at least @xmath82 on each component . ) we could then define a family of graph parameters analogous to @xmath63 by defining @xmath311 to be the maximum , over all subsets @xmath312 of size @xmath267 , of @xmath313 . replacing @xmath63 by @xmath311 would then give a stronger version of the stars and stripes lemma , and an expanded set of not - as - elementary inertias . for any graph @xmath37 , the stars and stripes lemma gives us a bound on the maximum eigenvalue multiplicity @xmath314 . [ cor_mbound ] let @xmath37 be a graph on @xmath38 vertices . then for any @xmath83 , @xmath315 . when this bound is attained , it is attained in particular on a set that includes the center of the stripe @xmath316 . [ cor_inertiabalanced ] let @xmath37 be a graph . if @xmath317 for some @xmath64 , then @xmath37 is inertia - balanced . [ exhn ] the _ @xmath38-sun _ @xmath318 is defined as the graph on @xmath319 vertices obtained by attaching a pendant vertex to each vertex of an @xmath38-cycle @xcite . we have @xmath320 and @xmath321 for @xmath322 . it follows that , in addition to @xmath323 and @xmath324 , @xmath325 contains every integer point @xmath39 within the trapezoid @xmath326 since for @xmath327 it is known that @xmath328 @xcite , this shows that the @xmath38-sun is inertia - balanced for @xmath327 . it is useful to note the following connection between the inverse inertia problem and the minimum semidefinite rank problem . [ ob7 ] the inertia set of a graph restricted to an axis gives @xmath329 and similarly for @xmath232 and @xmath26 . in other words , solving the inverse inertia problem for a graph @xmath330 on the @xmath331-axis ( or @xmath332-axis ) is equivalent to solving the minimum semidefinite rank problem for @xmath0 . one well - known result about minimum semidefinite rank is : [ @xmath333 @xcite , @xmath334 @xcite ] [ th1 ] given a graph @xmath0 on @xmath1 vertices , @xmath335 if and only if @xmath0 is a tree , and @xmath336 if and only if @xmath0 is a tree . as noted in example [ ex2 ] , if @xmath37 is not @xmath244 then @xmath337 , and therefore @xmath338 . it follows that @xmath339 are the only inertially arbitrary trees . if @xmath37 is not connected then any matrix in @xmath14 is a direct sum of smaller matrices , which shows that @xmath334 is additive on the components of a graph . [ obmainconverse ] let @xmath37 be a graph on @xmath38 vertices and let @xmath263 be the number of components of @xmath37 . then @xmath340 if and only if @xmath37 is a forest . this gives us a statement that implies the second claim of theorem [ th_treemain ] . [ comainconverse ] let @xmath37 be a graph . then @xmath341 is an elementary inertia of @xmath37 if and only if @xmath37 is a forest . let @xmath37 be a graph on @xmath38 vertices and let @xmath263 be the number of components of @xmath37 . since @xmath342 , @xmath343 is an elementary inertia of @xmath37 exactly for those integers @xmath182 in the range @xmath344 . in particular , @xmath341 is an elementary inertia if and only if @xmath340 . by observation [ obmainconverse ] , this is true if and only if @xmath37 is a forest . although the stars and stripes lemma only gives the correct value of @xmath345 when @xmath37 is a forest , we have already seen that it can give the correct values of @xmath110 and @xmath314 for some graphs containing a cycle . what is the class of graphs for which @xmath346 theorem [ th_treemain ] implies that this class includes all forests , and example [ exhn ] shows that the class includes the @xmath38-sun graphs @xmath318 for @xmath327 . [ ex3 ] @xmath347 , @xmath348 . let @xmath11 be the adjacency matrix of @xmath349 . then @xmath350 . since @xmath351 and @xmath352 , by the northeast lemma we have @xmath353 it follows that @xmath349 is not inertially arbitrary . as has already been noted , if @xmath93 with @xmath189 , then @xmath354 with @xmath355 , and if @xmath18 is real then @xmath356 is real . a consequence of observation [ ob_symmetry ] is [ symmetry property ] [ ob8 ] the sets @xmath186 and @xmath232 are symmetric about the line @xmath357 . the purpose of this section is to define some basic parameters associated with a tree , establish their fundamental properties , and relate them to the maximal disconnection numbers @xmath63 . in section [ s6 ] we will use these results to simplify the application of theorem [ th_treemain ] . in @xcite , johnson and duarte computed the minimum rank of all matrices in @xmath358 , where @xmath359 is an arbitrary tree . one of the graph parameters used by them , the _ path cover number _ of @xmath359 , is also needed in our work . it is defined as follows . [ d7 ] let @xmath359 be a tree . 1 . a _ path cover _ of @xmath359 is a collection of vertex disjoint paths , occurring as induced subgraphs of @xmath359 , that covers all the vertices of @xmath359 . the _ path cover number _ of @xmath359 , @xmath360 , is the minimum number of paths occurring in a path cover of @xmath359 . 3 . a _ path tree _ @xmath361 is a path cover of @xmath359 consisting of @xmath360 disjoint paths , say @xmath362 . extra edge _ is an edge of @xmath359 that is incident to vertices on two distinct @xmath363 s . clearly there are exactly @xmath364 extra edges . the theorem of duarte and johnson is [ th2 ] for any tree @xmath359 on @xmath1 vertices , @xmath365 as indicated , @xmath360 will also be used in our work . this is not surprising , because inertia is a refinement of rank . our use of @xmath360 will be made precise now . first , we need another definition . [ d8 ] let @xmath4 be a graph , and let @xmath366 . let @xmath367 that is , @xmath368 consists of all edges of @xmath0 that are incident to at least one vertex in @xmath369 . we define now an integer - valued mapping @xmath370 on the set of all subsets of @xmath371 by : @xmath372 [ ob9 ] for any graph @xmath37 , @xmath373 . [ ob9b ] let @xmath43 be a tree on @xmath38 vertices , and choose an integer @xmath64 in the range @xmath83 . then * for every @xmath374 with @xmath267 , @xmath375 and * for some @xmath374 with @xmath267 , @xmath376 in any forest , the number of components plus the number of edges equals the number of vertices . let @xmath124 with @xmath377 . the forest @xmath378 has @xmath379 edges and @xmath65 vertices , so it has @xmath380 components . by definition of @xmath381 , @xmath382 , or equivalently , @xmath383 . since @xmath384 for some @xmath97 with @xmath385 , the second statement follows . our first theorem in this section is the following : [ th3 ] let @xmath359 be a tree with @xmath386 and let @xmath360 denote its path cover number . then @xmath387 the second equality is a direct consequence of observation [ ob9b ] . the first equality will be proved by induction on @xmath388 , but first we prove it directly for several special cases . these special cases will also be used in the proof of theorem [ th3 ] . [ ob10 ] theorem [ th3 ] holds for any path . let @xmath136 denote the path on @xmath1 vertices , @xmath389 . the degree of any vertex in @xmath136 is at most two , so for any @xmath390 , @xmath391 . the result follows by observation [ ob9 ] . [ cor4 ] theorem [ th3 ] holds for any tree @xmath359 with @xmath392 . [ ob11 ] theorem [ th3 ] holds for @xmath349 , for any @xmath393 . label the pendant vertices of @xmath349 by @xmath394 and the vertex of degree @xmath395 by @xmath396 . it is known that @xmath397 . for @xmath398 , we have @xmath399 , while for any @xmath400 it is straightforward to see that @xmath401 . [ l12 ] let @xmath359 be a tree , and let @xmath402 . then @xmath403 . the proof is by induction on @xmath388 . corollary [ cor4 ] covers the base of the induction , so we proceed to the general induction step . let @xmath361 be any path tree of @xmath359 , consisting of paths @xmath404 . there are @xmath364 extra edges . we can assume without loss of generality that @xmath405 is a pendant path in @xmath361 ( so exactly one extra edge emanates from it ) , and we denote by @xmath139 the vertex of @xmath405 that is incident to an extra edge . we can also assume without loss of generality that no vertex of @xmath369 has degree @xmath396 or @xmath406 , since deleting such a vertex can not increase the value of the function @xmath407 that we are trying to bound from above . @xmath139 is an end vertex of @xmath405 . in this case @xmath408 also , @xmath409 . applying the induction hypothesis , we get @xmath410 @xmath139 is an internal vertex of @xmath405 . suppose first that one of the two end vertices of @xmath405 ( call it @xmath411 ) is at distance ( in @xmath405 ) of at least two from @xmath139 . then @xmath412 and @xmath413 . moreover , by the induction hypothesis , @xmath414 hence we may assume that @xmath405 has the form @xmath415 : @xmath416 we have @xmath409 . if @xmath417 then , by induction , @xmath410 if @xmath418 , then @xmath419 [ ob13 ] let @xmath359 be a tree that is not a star and for which @xmath420 . then there exists @xmath421 that has a unique non - pendant neighbor and at least one pendant neighbor . let @xmath422 be a vertex of degree @xmath423 . let @xmath363 be a path starting at @xmath422 , and of maximum length . denote by @xmath424 the terminal vertex of @xmath363 , by @xmath139 the predecessor of @xmath424 in @xmath363 ( note that @xmath425 ) , and by @xmath426 the predecessor of @xmath139 in @xmath363 ( it is possible that @xmath427 ) . then @xmath424 is a pendant neighbor of @xmath139 , and @xmath426 is the unique non - pendant neighbor of @xmath139 . a similar result appears as lemma 13 in @xcite . [ proof of theorem [ th3 ] ] as previously mentioned , the second equality comes from observation [ ob9b ] . the proof of the first equality is by induction on @xmath388 . the base of the induction is ensured by corollary [ cor4 ] . the theorem holds for any path and any star , by observations [ ob10 ] and [ ob11 ] . hence we may assume that @xmath359 is not a star and @xmath420 . let @xmath139 be as in observation [ ob13 ] , and let @xmath428 @xmath429 be its pendant neighbors . + in this case , @xmath431 . by the induction hypothesis , there exists @xmath432 such that @xmath433 . hence , @xmath434 so @xmath435 by lemma [ l12 ] . + in this case it is straightforward to see that @xmath437 . by the induction hypothesis , there exists @xmath438 such that @xmath439 . hence , for @xmath440 we have @xmath441 @xmath442 . + in this case it is straightforward to see that @xmath443 . by the induction hypothesis , there exists @xmath444 such that @xmath445 . if @xmath418 then @xmath446 , so we may assume that @xmath417 . we claim that @xmath447 . suppose otherwise that @xmath448 . then @xmath449 contradicting lemma [ l12 ] . let @xmath440 . then @xmath450 so @xmath451 by lemma [ l12 ] . this completes the proof that @xmath387 we pause to note a similar result to theorem [ th3 ] . given a tree @xmath43 , johnson and duarte @xcite ascertained that @xmath452 is the maximum of @xmath453 such that there exist @xmath454 vertices whose deletion leaves @xmath455 components each of which is a path ( possibly including singleton paths ) . it is obvious that this maximum is at most @xmath456 since any components are allowed in determining @xmath381 , and the converse is also true : if any component of the remaining forest is not a path , then deleting a vertex of degree greater than @xmath113 increases the value of @xmath457 . the observation of johnson and duarte can thus be seen as a corollary of theorem [ th3 ] . we will see the usefulness of allowing non - path components in section [ s6 ] , where we show that the lower values of @xmath381 provide an exact description of part of the boundary of @xmath458 . [ d9 ] let @xmath359 be a tree . 1 . a set @xmath459 is said to be _ optimal _ if @xmath435 . 2 . let @xmath460 . we say @xmath369 is _ minimal optimal _ if @xmath369 is optimal and @xmath461 . [ ob_cmd ] for a tree @xmath43 , @xmath462 theorem [ th2 ] , theorem [ th3 ] , definition [ d9 ] , and observation [ ob9b ] . [ ob_c_mr ] for a tree @xmath43 , @xmath463 let @xmath464 so @xmath465 . recall that @xmath466 for any integer @xmath64 , @xmath83 , so in particular @xmath467 and therefore @xmath468 . [ ob14 ] let @xmath359 be a tree and let @xmath459 be minimal optimal . then @xmath469 for every @xmath418 . [ ex4 ] we calculate @xmath470 for paths and stars . * @xmath471 : then @xmath472 so @xmath473 . * @xmath474 : let @xmath272 be the degree @xmath475 vertex of @xmath476 . then @xmath477 . so @xmath478 . [ p15 ] let @xmath359 be a tree and let @xmath479 be adjacent to @xmath480 pendant vertices @xmath428 , and at most one non - pendant vertex @xmath426 . then there is a path tree @xmath361 of @xmath359 in which @xmath481 . the claim is obvious if @xmath359 is a star , @xmath482 , so assume this is not the case . then @xmath139 is adjacent to exactly one non - pendant vertex . let @xmath361 be a path tree of @xmath359 . then at least @xmath483 of the vertices @xmath428 give single - vertex paths in @xmath361 . let @xmath363 be a path in @xmath361 containing @xmath139 . then @xmath363 contains at least one pendant neighbor of @xmath139 , say @xmath424 . then @xmath484 . note that @xmath485 , as @xmath361 is a path tree . if @xmath486 , then @xmath487 . otherwise , @xmath488 is a single - vertex path in @xmath361 . we can form a new path tree @xmath489 by replacing the path @xmath490 and the singleton path @xmath488 of @xmath361 by the pair of paths @xmath491 and @xmath492 . [ p16 ] let @xmath359 be a tree and let @xmath479 be adjacent to @xmath480 pendant vertices @xmath428 , and at most one non - pendant vertex @xmath426 . let @xmath493 . then @xmath494 and @xmath495 if @xmath442 , @xmath496 the proposition clearly holds if @xmath359 is a star , so we may assume that this is not the case . let @xmath489 be a path tree for @xmath497 . then @xmath498 is a path cover for @xmath359 , so @xmath499 . now let @xmath361 be a path tree for @xmath359 containing the path @xmath491 ( see proposition [ p15 ] ) . then @xmath500 , and @xmath501 is a path cover for @xmath497 . therefore , @xmath502 hence @xmath503 . now let @xmath369 be a minimal optimal set for @xmath497 , so @xmath504 . this implies that @xmath505 . let @xmath506 . since @xmath359 is not a star @xmath139 has a unique non - pendant neighbor @xmath426 . the vertices @xmath507 are adjacent to @xmath139 , so @xmath508 . then @xmath509 so @xmath510 is an optimal set for @xmath359 . it follows that @xmath511 now assume that @xmath442 and that @xmath369 is a minimal optimal set for @xmath359 . by observation [ ob14 ] , none of the vertices @xmath428 is in @xmath97 . if @xmath417 , lemma [ l12 ] implies @xmath512 a contradiction . therefore @xmath418 . let @xmath513 . then @xmath514 , so @xmath515 it follows from lemma [ l12 ] that @xmath516 is optimal for @xmath517 , implying @xmath518 hence @xmath519 . proposition [ p16 ] gives us a simple algorithm to calculate @xmath452 and thus the minimum rank of a tree . we will use the fact that if @xmath520 is a pendant vertex whose neighbor @xmath272 has degree @xmath113 , then any path in a minimal path cover that includes the vertex @xmath272 will also include the vertex @xmath520 , and @xmath452 = @xmath521 . [ ob_palgorithm ] let @xmath43 be a tree . then @xmath452 may be calculated as follows : 1 . set @xmath37 to @xmath43 and set @xmath455 to @xmath522 . 2 . if @xmath37 has a pendant vertex @xmath520 whose neighbor @xmath272 has degree @xmath113 , then replace @xmath37 by @xmath523 and repeat step 2 . 3 . if @xmath37 consists of a single edge or single vertex , then @xmath524 . if @xmath37 is a star on @xmath525 vertices , then @xmath526 . 4 . in all other cases ( by observation [ ob13 ] ) there will be some @xmath527 that is adjacent to @xmath528 pendant vertices @xmath529 and exactly one non - pendant vertex @xmath530 . replace @xmath37 by @xmath531 , replace @xmath455 by @xmath532 , and return to step 2 . the calculation of @xmath470 is not quite as straightforward as that of @xmath452 , although we can show one special case in which it is additive on subgraphs . for this we need the following definition . [ d14 ] let @xmath533 and @xmath0 be graphs on at least two vertices , each with a vertex labeled @xmath139 . then @xmath534 is the graph on @xmath535 vertices obtained by identifying the vertex @xmath139 in @xmath536 with the vertex @xmath139 in @xmath0 . the vertex @xmath139 in definition [ d14 ] is commonly referred to as a _ cut vertex _ of the graph @xmath534 . the next result determines @xmath537 when @xmath538 . [ th5 ] let @xmath497 and @xmath539 be trees each with a pendant vertex labeled @xmath139 . . then @xmath541 . let @xmath542 , @xmath543 be minimal optimal sets for @xmath497 , @xmath539 , respectively . then @xmath544 since @xmath545 , and @xmath546 by observation [ ob14 ] , by lemma [ l12 ] @xmath547 therefore , @xmath548 is an optimal set for @xmath359 by lemma [ l12 ] and @xmath549 . suppose now @xmath369 is a minimal optimal set for @xmath359 . by observation [ ob14 ] , @xmath417 . let @xmath550 , @xmath551 . since @xmath417 , @xmath552 . now @xmath553 then @xmath554 so we must have @xmath555 and @xmath556 [ cor6 ] let @xmath557 be a pendant vertex in a tree @xmath359 and suppose the neighbor of @xmath557 has degree @xmath406 . then @xmath558 . [ cor7 ] if a tree @xmath359 has exactly one vertex of degree @xmath559 , then . it is straightforward to see , by repeated application of corollary [ cor6 ] , that @xmath560 . [ d10 ] let @xmath43 be a tree and let @xmath561 be an integer such that @xmath562 . then @xmath563 [ obr_k ] for a tree @xmath43 , @xmath564 . the next theorem will play an important role in simplifying the computation of @xmath458 . [ th8 ] let @xmath359 be a tree with @xmath565 . then @xmath566 since @xmath565 , @xmath359 is not a path . therefore @xmath420 , implying @xmath567 . if @xmath568 , @xmath569 while @xmath570 then @xmath571 thus , the stronger conclusion in the special cases @xmath572 and @xmath568 has been established . we proceed by induction on @xmath388 . since @xmath359 can not be a path , the base of the induction is @xmath573 , and the only relevant tree @xmath359 with @xmath573 is @xmath574 . since @xmath575 the theorem holds in this case . consider now the general induction step . let @xmath359 be a tree on @xmath1 vertices , and let @xmath576 . note that if @xmath577 we are done . in particular , we can assume @xmath359 is not a star . we have to show that @xmath578 . by observation [ ob13 ] there exists @xmath421 that is adjacent to a unique non - pendant vertex @xmath426 , and to pendant vertices @xmath428 , where @xmath579 . + let @xmath580 . then @xmath581 by proposition [ p16 ] . this tells us both that we are allowed to assume the induction hypothesis on the tree @xmath582 ( which requires @xmath583 ) and that @xmath584 . now choose @xmath585 with @xmath586 and @xmath587 this choice is possible ( with equality ) by the definition of @xmath588 . we can assume without loss of generality that @xmath589 contains none of the vertices @xmath590 as follows : if @xmath591 we delete all @xmath592 s that belong to @xmath363 , possibly decreasing @xmath593 without changing @xmath594 . if at least one of @xmath428 , say @xmath424 , belongs to @xmath363 but @xmath595 , we replace @xmath424 by @xmath139 in @xmath363 and delete from @xmath363 all remaining @xmath592 , possibly decreasing @xmath593 and possibly increasing @xmath594 . we give the name @xmath263 to @xmath593 , so @xmath596 . suppose that @xmath595 . let @xmath597 . then @xmath598 , and @xmath599 . hence @xmath600 suppose that @xmath591 . let @xmath601 . then @xmath602 , and @xmath603 . by the induction hypothesis , @xmath604 choose @xmath605 with @xmath606 such that @xmath607 . let @xmath608 . then @xmath609 , and @xmath610 also , @xmath611 so @xmath612 then @xmath613 @xmath430 . + let @xmath614 . by corollary [ cor6 ] , we have @xmath615 , and clearly @xmath616 . since @xmath617 , @xmath618 . as in case 1 , we choose @xmath619 with @xmath620 and @xmath621 , and can assume without loss of generality that @xmath622 . suppose that @xmath595 . then @xmath623 , and applying the induction hypothesis , we have @xmath624 suppose that @xmath591 . . then @xmath625 . hence @xmath626 applying the induction hypothesis to @xmath497 , @xmath627 applying the induction hypothesis again to @xmath497 , @xmath628 [ cor_mdmonotone ] let @xmath43 be a tree . then for @xmath629 , @xmath630 it suffices to prove the case @xmath631 . here we have @xmath632 , and theorem [ th8 ] gives us @xmath633 . making the substitution @xmath634 from observation [ obr_k ] gives us the desired result . [ p17 ] let @xmath359 be a tree on @xmath393 vertices . then @xmath635 . we prove the proposition by induction on @xmath1 . the cases @xmath636 are obvious , so we consider the general induction step . the proposition holds if @xmath637 , as @xmath638 , so assume @xmath359 is not a star . by observation [ ob13 ] , there exists a vertex @xmath139 that is adjacent to exactly one non - pendant vertex @xmath426 and to pendant vertices @xmath428 , where @xmath579 . + it follows from corollary [ cor6 ] and the induction hypothesis that @xmath639 @xmath480 . + let @xmath580 . then @xmath640 , and by proposition [ p16 ] @xmath641 . then , by induction hypothesis , @xmath642 we conclude this section with a partial result toward the first claim of theorem [ th_treemain ] . [ dl_t ] for a tree @xmath43 we define @xmath643 , the _ minimum - rank stripe _ of @xmath43 , as the set @xmath644 for the moment the name `` minimum - rank stripe '' is not entirely justified , since it suggests that @xmath645 . in section [ s6 ] we will show that this is the case , but we can already show one direction of containment . [ th9 ] for any tree @xmath43 , @xmath646 . let @xmath647 . given any @xmath648 , we have @xmath77 , @xmath78 , and @xmath649 by observation [ ob_cmd ] . then by the stars and stripes lemma we have @xmath650 . [ cor_rightrank ] theorem [ th_treemain ] gives the correct value of @xmath651 for @xmath43 a tree . in this section we interrupt our discussion of inertia sets of trees in order to derive basic formulae about the inertia set of any graph with a cut vertex . we obtain formulae for inertia sets that are the analogue of theorem 16 in @xcite and theorem 2.3 in @xcite for minimum rank . [ d13 ] if @xmath363 , @xmath652 are subsets of @xmath653 , then @xmath654 addition of @xmath655 or more sets is defined similarly . [ d13b ] if @xmath363 is a subset of @xmath653 and @xmath1 is a positive integer , we let @xmath656_n = q\cap { \mathbb{n}}^2_{\le n}.\ ] ] we first consider the case of disconnected graphs . since the inertia of a direct sum of matrices is the sum of the inertias of the summands , we have : [ ob21 ] let @xmath657 . then @xmath658 and similarly for @xmath232 . we now determine the inertia set of a graph with a cut vertex see definition [ d14 ] . we first recall the following useful result @xcite , @xcite , which reduces the minimum rank problem for graphs to the case of @xmath406-connected graphs . [ hsieh ; barioli , fallat , hogben ] [ th11 ] with @xmath533 , @xmath0 and @xmath534 as in definition [ d14 ] , we have @xmath659 our next result generalizes this to inertia sets . [ th12 ] let @xmath533 and @xmath0 be graphs on at least two vertices with a common vertex @xmath139 and let @xmath660 . then @xmath661_n\cup \bigl [ { { \mathcal i}}(f - v)+{{\mathcal i}}(g - v)+\{(1,1)\}\bigr ] _ n\ ] ] and similarly for @xmath662 . we prove the complex hermitian version of the theorem ; the proof of the real symmetric version is the same but with the assumption that all matrices and vectors are real . let @xmath139 be the last vertex of @xmath663 and the first vertex of @xmath0 . _ reverse containment _ : + * i. * let @xmath664_n $ ] . then @xmath665 and there exist @xmath666 and @xmath667 such that @xmath668 , @xmath669 . let @xmath670 with @xmath671 and @xmath672 , and let @xmath673 be matrices of order @xmath1 . then @xmath674 and @xmath675 . by the subadditivity of partial inertias ( proposition [ p2 ] ) , @xmath676 and @xmath677 since @xmath678 by definition , and @xmath665 , @xmath679 by the northeast lemma ( lemma [ l5 ] ) . thus , we have @xmath680_n\subseteq{\mathrm{h}{{\mathcal i } } } { ( f { \underset{v}{\oplus}}g ) } $ ] . * ii . * now let @xmath681_n $ ] . then @xmath665 and there exist @xmath682 and @xmath683 with @xmath684 . let @xmath685 with @xmath686 and let @xmath687 with @xmath688 . choose @xmath689 , @xmath690 , @xmath691 such that @xmath692 by proposition [ p1 ] , @xmath693 and , similarly , @xmath694 since @xmath695 , and @xmath665 , by the northeast lemma , @xmath679 . so we have @xmath696_n \subseteq{\mathrm{h}{{\mathcal i } } } { ( f { \underset{v}{\oplus}}g ) } .\ ] ] _ forward containment _ : + now let @xmath697 . by observation [ ob4 ] , @xmath698 . let @xmath692 with @xmath671 . then @xmath699 if the first and third inequalities are strict , then @xmath700 the first equality implies that either @xmath701 or else @xmath702 , while the second equality implies that @xmath703 and @xmath704 . so this case does not occur and either @xmath705 or else @xmath706 * i. * @xmath707 . + then @xmath703 and @xmath704 . so @xmath708 , @xmath709 for some @xmath710 , @xmath711 . define @xmath712 then @xmath713 is congruent to @xmath714 } $ ] ; @xmath715 is congruent to @xmath716 } $ ] . hence @xmath717 also , @xmath718 by proposition [ p1 ] , @xmath719 such that @xmath720 it follows from and that @xmath721 hence , by the northeast lemma , @xmath722 and since these two vectors add up to @xmath723 , by and , we conclude that @xmath724 . since @xmath698 we get @xmath725_n $ ] . so in this case , @xmath726_n $ ] . * ii . * @xmath727 . + by proposition [ p1 ] , we have @xmath728 it follows that @xmath729 and @xmath730 and @xmath731 . by definition , @xmath732 , and since @xmath698 , @xmath733_n $ ] . so in this case , @xmath734_n.\ ] ] this completes the proof of the forward containment . it is straightforward to show that theorem [ th11 ] is a corollary of theorem [ th12 ] . the proof is not illuminating , so we do not include it . [ ex7 ] let @xmath735 and @xmath736 with @xmath139 a pendant vertex in @xmath574 and the degree @xmath406 vertex in @xmath737 . then @xmath738 is the graph below . @xmath739 from examples [ ex3 ] and [ ex2 ] we have ( 50,38 ) ( 5,20)@xmath740 : ( 22,4)(0,1)30 ( 20,6)(1,0)31 ( 20.75,28)(1,0)2.5 ( 22,28 ) ( 20.75,22.5)(1,0)2.5 ( 22,22.5 ) ( 27.5,22.5 ) ( 20.75,17)(1,0)2.5 ( 27.5,17 ) ( 33,17 ) ( 20.75,11.5)(1,0)2.5 ( 27.5,11.5 ) ( 27.5,4.75)(0,1)2.5 ( 33,11.5 ) ( 33,4.75)(0,1)2.5 ( 38.5,11.5 ) ( 38.5,4.75)(0,1)2.5 ( 43.5,4.75)(0,1)2.5 ( 38.5,6 ) ( 44,6 ) ( 65,20)@xmath741 : ( 82,4)(0,1)30 ( 80,6)(1,0)27 ( 80.75,22.5)(1,0)2.5 ( 82,22.5 ) ( 80.75,17)(1,0)2.5 ( 82,17 ) ( 87.5,17 ) ( 80.75,11.5)(1,0)2.5 ( 87.5,11.5 ) ( 87.5,4.75)(0,1)2.5 ( 93,11.5 ) ( 93,4.75)(0,1)2.5 ( 93,6 ) ( 98.5,4.75)(0,1)2.5 ( 98.5,6 ) ( 50,38 ) ( 0,10)@xmath742 : ( 40,4)(0,1)27 ( 38,6)(1,0)27 ( 38.75,22.5)(1,0)2.5 ( 40,22.5 ) ( 38.75,17)(1,0)2.5 ( 40,17 ) ( 45.5,17 ) ( 38.75,11.5)(1,0)2.5 ( 45.5,11.5 ) ( 45.5,4.75)(0,1)2.5 ( 51,11.5 ) ( 51,4.75)(0,1)2.5 ( 51,6 ) ( 56.5,4.75)(0,1)2.5 ( 56.5,6 ) ( 70,10)@xmath743 : ( 112,4)(0,1)21 ( 110,6)(1,0)20 ( 110.75,17)(1,0)2.5 ( 112,17 ) ( 110.75,11.5)(1,0)2.5 ( 112,11.5 ) ( 117.5,11.5 ) ( 117.5,4.75)(0,1)2.5 ( 112,6 ) ( 117.5,6 ) ( 123,4.75)(0,1)2.5 ( 123,6 ) it follows that ( 50,50 ) ( 5,20)@xmath744_6 $ ] is ( 42,4)(0,1)40 ( 40,6)(1,0)41 ( 40.75,39)(1,0)2.5 ( 42,39 ) ( 40.75,33.5)(1,0)2.5 ( 42,33.5 ) ( 47.5,33.5 ) ( 40.75,28)(1,0)2.5 ( 47.5,28 ) ( 53,28 ) ( 40.75,22.5)(1,0)2.5 ( 47.5,22.5 ) ( 53,22.5 ) ( 58.5,22.5 ) ( 40.75,17)(1,0)2.5 ( 53,17 ) ( 58.5,17 ) ( 64,17 ) ( 40.75,11.5)(1,0)2.5 ( 58.5,11.5 ) ( 64,11.5 ) ( 69.5,11.5 ) ( 47.5,4.75)(0,1)2.5 ( 53,4.75)(0,1)2.5 ( 58.5,4.75)(0,1)2.5 ( 64,4.75)(0,1)2.5 ( 69.5,4.75)(0,1)2.5 ( 69.5,6 ) ( 75,4.75)(0,1)2.5 ( 75,6 ) ( 100,20)and ( 50,50 ) ( 0,20)@xmath745_6 $ ] is ( 62,4)(0,1)40 ( 60,6)(1,0)41 ( 60.75,39)(1,0)2.5 ( 60.75,33.5)(1,0)2.5 ( 67.5,33.5 ) ( 60.75,28)(1,0)2.5 ( 67.5,28 ) ( 73,28 ) ( 60.75,22.5)(1,0)2.5 ( 67.5,22.5 ) ( 73,22.5 ) ( 78.5,22.5 ) ( 60.75,17)(1,0)2.5 ( 73,17 ) ( 78.5,17 ) ( 84,17 ) ( 60.75,11.5)(1,0)2.5 ( 78.5,11.5 ) ( 84,11.5 ) ( 89.5,11.5 ) ( 67.5,4.75)(0,1)2.5 ( 73,4.75)(0,1)2.5 ( 78.5,4.75)(0,1)2.5 ( 84,4.75)(0,1)2.5 ( 89.5,4.75)(0,1)2.5 ( 95,4.75)(0,1)2.5 then @xmath746_6 \cup [ { { \mathcal i}}(p_3 ) + { { \mathcal i}}(2k_1 ) + \{(1 , 1)\}]_6 $ ] is : ( 50,50 ) ( 42,4)(0,1)40 ( 40,6)(1,0)41 ( 40.75,39)(1,0)2.5 ( 42,39 ) ( 40.75,33.5)(1,0)2.5 ( 42,33.5 ) ( 47.5,33.5 ) ( 40.75,28)(1,0)2.5 ( 47.5,28 ) ( 53,28 ) ( 40.75,22.5)(1,0)2.5 ( 47.5,22.5 ) ( 53,22.5 ) ( 58.5,22.5 ) ( 40.75,17)(1,0)2.5 ( 53,17 ) ( 58.5,17 ) ( 64,17 ) ( 40.75,11.5)(1,0)2.5 ( 58.5,11.5 ) ( 64,11.5 ) ( 69.5,11.5 ) ( 47.5,4.75)(0,1)2.5 ( 53,4.75)(0,1)2.5 ( 58.5,4.75)(0,1)2.5 ( 64,4.75)(0,1)2.5 ( 69.5,4.75)(0,1)2.5 ( 69.5,6 ) ( 75,4.75)(0,1)2.5 ( 75,6 ) since @xmath747 and @xmath748 , @xmath749 by theorem [ th2 ] . since @xmath750 , @xmath751 , so @xmath752 is a minimal optimal set for @xmath359 . we observe that in this case @xmath645 . we pause to develop some additional fundamental properties of inertia sets before generalizing theorem [ th12 ] . the next result generalizes the fact @xcite that @xmath753 . [ p22 ] let @xmath0 be any graph on @xmath1 vertices and let @xmath139 be any vertex of @xmath0 . then we have : 1 . @xmath754_{n-1}\subseteq{{\mathcal i}}(g - v ) $ ] . 2 . @xmath755_{n-2}+\{(1,1)\ } $ ] . the same inclusions hold in the hermitian case . let @xmath756_{n-1 } $ ] . then @xmath757 . let @xmath23 with @xmath189 , and let @xmath58 be the principal submatrix of @xmath18 obtained by deleting the row and column @xmath272 . then @xmath758 and by the interlacing inequalities @xmath759 is one of @xmath209 , @xmath760 , @xmath262 , or @xmath761 . then one of these is in @xmath762 so by the northeast lemma , @xmath763 . this proves ( a ) . now let @xmath764_{n-2 } $ ] so @xmath765 . choose @xmath23 in such a way that the principal submatrix @xmath58 obtained by deleting row and column @xmath272 satisfies @xmath766 . then by the interlacing inequalities , @xmath36 is one of @xmath209 , @xmath767 , @xmath768 , or @xmath769 . since @xmath770 , @xmath771 by the northeast lemma applied to @xmath0 . this completes the proof of ( b ) . the proof of the hermitian case is the same , but with hermitian notation . [ p23 ] if @xmath139 is a pendant vertex of the graph @xmath0 and @xmath772 , then @xmath773 and @xmath774 , and similarly for @xmath775 and @xmath232 . as usual , the proofs of the real symmetric and hermitian versions do not differ materially . let @xmath139 be the first vertex of @xmath0 and let its neighbor @xmath776 be the second . let @xmath777 with @xmath686 . then @xmath778 and @xmath779 . by proposition [ p3 ] @xmath780 then @xmath781 and @xmath782 . similarly , @xmath774 . the following corollary of theorem [ th11 ] is very useful in simplifying the calculation of the minimum rank of a graph . [ ( * ? ? ? * lemma 38 ) ] [ p24 ] if the degree of @xmath139 is @xmath406 in @xmath534 , then @xmath783 the following result generalizes this fact to inertia sets . [ p25 ] if the degree of @xmath139 is @xmath406 in @xmath534 , and @xmath660 , then @xmath784_n,\ ] ] and similarly for @xmath662 . by theorem [ th12 ] it suffices to show that @xmath785_n\subseteq\bigl[{{\mathcal i}}(f)+{{\mathcal i}}(g)\bigr]_n.\ ] ] let @xmath786_n $ ] . then @xmath665 and @xmath787 with @xmath788 and @xmath789 . since @xmath139 is pendant in both @xmath533 and @xmath0 , by proposition [ p23 ] , @xmath790 and @xmath791 , so @xmath792 . since @xmath665 , @xmath793_n $ ] . replacing @xmath794 by @xmath795 uniformly proves the hermitian case . [ ex8 ] let @xmath796 and let @xmath139 be a pendant vertex in each of @xmath46 and @xmath37 so that @xmath738 is the graph below . @xmath797 by proposition [ p25 ] , @xmath798_7=\bigl[{{\mathcal i}}(s_4)+{{\mathcal i}}(s_4)\bigr]_7 $ ] . knowing that the inertia set @xmath799 is @xmath800 allows us to calculate @xmath801 , as depicted below . @xmath802 here , @xmath749 is attained only at the partial inertia @xmath803 , and one can easily check that @xmath804 , so that @xmath805 . we close this section with the generalization of theorem [ th12 ] . we first extend definition [ d14 ] . [ d15 ] let @xmath806 , @xmath807 , be graphs on at least two vertices with a common vertex @xmath139 and let @xmath808 be the graph on @xmath809 vertices obtained by identifying the vertex @xmath139 in each of the @xmath810 . we call @xmath0 the _ vertex sum _ of the graphs @xmath806 at @xmath811 . [ th13 ] let @xmath0 be a graph on @xmath393 vertices and let @xmath139 be a cut vertex of @xmath0 . write @xmath812 , @xmath813 , the vertex sum of @xmath806 at @xmath139 . then @xmath814_n \label{eq5 } \\ & & \cup\ \bigl[{{\mathcal i}}(g_1-v)+{{\mathcal i}}(g_2-v)+\cdots+{{\mathcal i}}(g_k - v)+\{(1,1)\}\bigr]_n , \nonumber\end{aligned}\ ] ] and similarly for @xmath232 . the idea of the proof is the same as in the proof of theorem [ th12 ] , which is showing that each side of equation is contained in the other . since each of the theorems cited applies equally well to @xmath795 as to @xmath794 , the same proof demonstrates both cases . _ forward containment _ : + we prove that @xmath815_n \label{eq6 } \\ & & \cup\ \bigl[{{\mathcal i}}(g_1-v)+{{\mathcal i}}(g_2-v)+\cdots+{{\mathcal i}}(g_k - v)+\{(1,1)\}\bigr]_n \nonumber\end{aligned}\ ] ] by induction on @xmath561 . for @xmath816 this follows from theorem [ th12 ] . assume holds for all integers @xmath817 with @xmath818 . let @xmath819 , the vertex sum of @xmath820 at @xmath139 and let @xmath821 . then by theorem [ th12 ] , @xmath822_n \\ & & \cup\ \bigl[{{\mathcal i}}(g^\prime - v)+{{\mathcal i}}(g_k - v)+\{(1,1)\}\bigr]_n.\end{aligned}\ ] ] but @xmath823 by observation [ ob21 ] . applying the induction hypothesis to @xmath824 we have @xmath825_{n^\prime } \\ & & \ \ \cup \ \bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k-1}-v)+\{(1,1)\}\bigr]_{n^\prime}\bigr\}+{{\mathcal i}}(g_k)\bigr]_n \\ & & \cup \ \bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k}-v)+\{(1,1)\}\bigr]_{n } \\ & = & \bigl[\bigl(\bigl[{{\mathcal i}}(g_1)+\cdots+{{\mathcal i}}(g_{k-1})\bigr]_{n^\prime}+{{\mathcal i}}(g_k)\bigr ) \\ & & \ \ \cup \ \bigl(\bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k-1}-v)+\{(1,1)\}\bigr]_{n^\prime } + { { \mathcal i}}(g_k)\bigr)\bigr]_n \\ & & \cup \ \bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k}-v)+\{(1,1)\}\bigr]_{n } \\ & = & \bigl[\bigl[{{\mathcal i}}(g_1)+\cdots+{{\mathcal i}}(g_{k-1})\bigr]_{n^\prime}+{{\mathcal i}}(g_k)\bigr]_n \\ & & \cup \ \bigr[\bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k-1}-v)+\{(1,1)\}\bigr]_{n^\prime } + { { \mathcal i}}(g_k)\bigr]_n \\ & & \cup \ \bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k}-v)+\{(1,1)\}\bigr]_{n } \\ & \subseteq & \bigl[{{\mathcal i}}(g_1)+\cdots+{{\mathcal i}}(g_k)\bigr]_n \\ & & \cup \ \bigl[\bigl(\bigl[{{\mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k-1}-v)\bigr]_{n^\prime-2}+\{(1,1)\}\bigr ) + { { \mathcal i}}(g_k)\bigr]_n \\ & & \cup \ \bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k}-v)+\{(1,1)\}\bigr]_{n}.\end{aligned}\ ] ] let @xmath826_n , \\ q_2 & = & \bigl [ { { \mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_k - v)+\{(1,1)\}\bigr]_{n } , \\ q_0 & = & \bigl[\bigl[{{\mathcal i}}(g_1-v)+\cdots+{{\mathcal i}}(g_{k-1}-v)\bigr]_{n^\prime-2}+\{(1,1)\ } + { { \mathcal i}}(g_k)\bigr]_n.\end{aligned}\ ] ] we show that @xmath827 . suppose that @xmath828 . then @xmath829 with @xmath830 , @xmath831 , @xmath832 , @xmath833 if @xmath834 , by proposition [ p22](a ) , @xmath835 and then @xmath836 . so suppose that @xmath837 . at least one of @xmath228 , @xmath817 is greater than @xmath31 . without loss of generality , assume @xmath838 . by proposition [ p6 ] , @xmath839 , and by proposition [ p22](a ) , @xmath840 . since @xmath841 , we have @xmath842 therefore , @xmath843 . without loss of generality , assume @xmath844 . by the northeast lemma @xmath845 . since @xmath665 , and @xmath846 , we again have @xmath836 . this completes the proof that @xmath847 . therefore @xmath848 , which is . _ reverse containment _ : + a proof by induction is not straightforward . however , one can show the two containments @xmath849_n\subseteq{{\mathcal i}}(g ) $ ] , and @xmath850_n\subseteq{{\mathcal i}}(g ) $ ] , by simply imitating each step in the proof of theorem [ th12 ] . as there are no new ideas in the proof , we omit it . the results of the previous section give us a way to inductively calculate the inertia set of any graph once we know the inertia sets of @xmath113-connected graphs . in this section we prove that the same inductive formula holds when calculating the set of elementary inertias . claim 1 of theorem [ th_treemain ] will then follow because a forest is a graph with no @xmath113-connected subgraph on @xmath851 or more vertices . it is convenient to describe the elementary inertias of a graph @xmath37 in terms of bicolored edge - colorings of certain subgraphs of @xmath37 . let @xmath37 be a graph on @xmath38 vertices , let @xmath97 be a subset of @xmath122 , and let @xmath852 and @xmath853 be disjoint subsets of @xmath854 . the ordered triple @xmath855 is called a _ bicolored span of @xmath37 _ if @xmath856 is a spanning forest of @xmath268 . ( a _ spanning forest _ of a graph consists of a spanning tree for each connected component . ) if @xmath855 is a bicolored span of @xmath37 , we say that the ordered pair @xmath857 is a _ color vector _ of @xmath37 . the set of color vectors of @xmath37 is denoted @xmath858 . the color vector counts how many edges of the spanning forest have been marked with either the first color or the second color , but it also counts the set @xmath97 of excluded vertices twice , as though each such vertex were marked simultaneously with both colors . because every spanning forest has the same number of edges , the quantity @xmath859 depends only on @xmath97 , and for a given size @xmath267 , @xmath860 is minimized if @xmath268 has @xmath63 components . if @xmath37 is a graph on @xmath38 vertices , @xmath861 is the number of components of @xmath37 , and @xmath862 is a bicolored span of @xmath37 , then @xmath863 . [ ob_cvec_stripe ] if @xmath37 is a graph on @xmath38 vertices and @xmath861 , then @xmath864 . if @xmath589 is a subset of @xmath865 , we define the _ northeast expansion of @xmath589 _ as @xmath866 for example , the northeast lemma is equivalent to the statement that , for @xmath37 a graph on @xmath38 vertices , @xmath867_{n } } \subseteq { { \mathcal i}}(g)$ ] . the prevalence of northeast expansions in this section leads us to define the following equivalence relation : given two sets @xmath868 , we say that @xmath869 is _ northeast equivalent to @xmath589 _ , written as @xmath870 , if @xmath871 . let @xmath37 be a graph on @xmath38 vertices , and let @xmath872 be an ordered pair of integers . we say that @xmath872 is a _ northeast color vector of @xmath37 _ if @xmath873 and if @xmath874 and @xmath875 for some color vector @xmath876 of @xmath37 . note that the set of all northeast color vectors of @xmath37 is @xmath877_{n}}$ ] . the term _ northeast color vector _ is actually a synonym for _ elementary inertia , _ as we now demonstrate . [ p_elementary ] let @xmath37 be a graph on @xmath38 vertices . then @xmath878_{n}}$ ] . we show both inclusions . _ forward inclusion . _ let @xmath39 be an elementary inertia of @xmath37 . then there exist a nonnegative integer @xmath64 and an ordered pair of integers @xmath879 such that @xmath880 let @xmath97 be chosen such that @xmath267 and @xmath268 has @xmath63 components , and let @xmath46 be a spanning forest of @xmath268 , so that @xmath46 has @xmath65 vertices and @xmath881 edges . we partition the edges of @xmath46 into two sets @xmath852 and @xmath853 with @xmath882 and @xmath883 edges respectively . it follows that @xmath884 is a color vector of @xmath37 . since @xmath40 , @xmath39 belongs to the set @xmath877_{n}}$ ] of northeast color vectors of @xmath37 . _ reverse inclusion . _ let @xmath885 be a northeast color vector of @xmath37 , and let @xmath855 be a bicolored span of @xmath37 such that @xmath886 and @xmath887 . letting @xmath888 , we can assume without loss of generality that @xmath97 is chosen among all sets of size @xmath64 in such a way as to minimize @xmath859 , or in other words that @xmath268 has @xmath63 components . under this assumption we have @xmath889 , so @xmath890 . we further have @xmath891 and @xmath892 , so @xmath872 is an elementary inertia of @xmath37 . we now state some set - theoretic results that allow us to simplify certain expressions involving @xmath893 and @xmath894_n$ ] . [ ob_settheoretic ] for @xmath895 and nonnegative integers @xmath896 , we have 1 . [ line1 ] @xmath897_n\bigr]_m = \bigl[\bigl[q\bigr]_m\bigr]_n = \bigl[q\bigr]_m$ ] . [ line1b ] @xmath898_{n } } \!\!}^\nearrow } \bigr]_m = { \bigl[{{q}^\nearrow}\bigr]_{m}}$ ] . [ line2 ] @xmath899_{m } } \sim \bigl[q\bigr]_m$ ] . [ line3 ] if @xmath869 is a stripe of rank @xmath100 , then @xmath900_n = \bigl [ q \bigr]_{n - m } + p$ ] . [ line4 ] @xmath901 implies @xmath902_m$ ] . [ line5 ] @xmath902_m$ ] implies @xmath902_n$ ] . these are all straightforward consequences of the definitions . [ p_settheoretic ] let @xmath263 , @xmath100 , and @xmath38 be nonnegative integers with @xmath903 and @xmath904 , suppose that @xmath895 satisfies @xmath902_{n-\ell}$ ] , and let @xmath905_{n}}$ ] . then 1 . [ setline1 ] @xmath906_{n-\ell}$ ] , 2 . [ setline2 ] @xmath907_{m } } = \bigl[p\bigr]_{m}$ ] , and 3 . [ setline3 ] @xmath907_{n } } = p$ ] . we have @xmath908_{n } } \sim \bigl[q\bigr]_n \sim q \sim \bigl[q\bigr]_{n-\ell } \sim { \bigl[{{q}^\nearrow}\bigr]_{n-\ell } } = \bigl [ { \bigl[{{q}^\nearrow}\bigr]_{n } } \bigr]_{n-\ell } = \bigl[p\bigr]_{n-\ell},\ ] ] @xmath909_{m } } = \bigl[{{{\bigl[{{q}^\nearrow}\bigr]_{n}}\!\!}^\nearrow}\bigr]_{m } = { \bigl[{{q}^\nearrow}\bigr]_{m } } = \bigl[{\bigl[{{q}^\nearrow}\bigr]_{n}}\bigr]_{m } = \bigl[p\bigr]_{m},\ ] ] and @xmath909_{n } } = \bigl[{{{\bigl[{{q}^\nearrow}\bigr]_{n}}\!\!}^\nearrow}\bigr]_{n } = { \bigl[{{q}^\nearrow}\bigr]_{n } } = p.\ ] ] we can apply this proposition immediately . first note that observations [ ob_cvec_stripe ] and [ ob_settheoretic ] ( [ line4 ] ) give us [ ob_c_equiv ] let @xmath37 be a graph on @xmath38 vertices with @xmath263 components . then @xmath910_{n-\ell}$ ] . observation [ ob_c_equiv ] and proposition [ p_elementary ] allow us to apply proposition [ p_settheoretic ] , by substituting @xmath858 for @xmath589 . [ ob_e_equiv ] let @xmath37 be a graph on @xmath38 vertices with @xmath263 components , and let @xmath100 be an integer in the range @xmath904 . then 1 . [ eqline1 ] @xmath911_{n-\ell}$ ] , 2 . [ eqline2 ] @xmath912_{m } } = \bigl[{{\mathcal e}}(g)\bigr]_{m}$ ] , and 3 . [ eqline3 ] @xmath912_{n } } = { { \mathcal e}}(g)$ ] . observation [ ob_e_equiv ] ( [ eqline3 ] ) can be viewed as a northeast lemma for elementary inertias . [ l_settheoretic ] let @xmath913 , , @xmath914 be subsets of @xmath865 , and suppose that for some collection @xmath915 of nonnegative integers we have @xmath916_{n_i}$ ] for @xmath917 . let @xmath918 and let @xmath919 . then @xmath920_{n } } = \bigl [ { { q_1}^\nearrow } + \cdots + { { q_k}^\nearrow } \bigr]_n = { \bigl[{{q_1}^\nearrow}\bigr]_{n_1 } } + \cdots + { \bigl[{{q_k}^\nearrow}\bigr]_{n_k}}.\ ] ] the first equality comes from the observation that @xmath921 . for the second equality , the reverse inclusion is easy to check . suppose then that we are given @xmath922_n,\ ] ] so there exist @xmath64 ordered pairs of integers @xmath923 with @xmath924 , @xmath925 , and @xmath926 . for any such collection @xmath927 , we can define two quantities , a _ surplus _ @xmath928 and a _ deficit _ @xmath929 so that @xmath930 and hence @xmath931 . if @xmath932 , then in every case we have @xmath933_{n_i}}$ ] , so @xmath934_{n_1 } } + { \bigl[{{q_2}^\nearrow}\bigr]_{n_2 } } + \cdots + { \bigl[{{q_k}^\nearrow}\bigr]_{n_k}}\ ] ] and we are done . but we can assume @xmath935 without loss of generality for the following reason : if @xmath936 , then @xmath937 also and for some integers @xmath182 and @xmath938 in the range @xmath939 we have @xmath940 and @xmath941 . since @xmath942_{n_i}\!\!}^\nearrow}$ ] , we can replace @xmath943 by either @xmath944 or @xmath945 , one of which must belong to @xmath946 , and simultaneously replace @xmath947 with respectively either @xmath948 or @xmath949 . this reduces both the value of @xmath76 and the value of @xmath950 , so we can assume without loss of generality that @xmath935 , giving the desired result . the following proposition is an immediate corollary . [ cor_settheoretic ] given @xmath895 and nonnegative integers @xmath896 , suppose that @xmath951_m$ ] . then @xmath920_{n } } = { \bigl[{{q}^\nearrow}\bigr]_{m } } + \sum\limits_{i=1}^{n - m } { { { \mathcal e}}(k_1)}.\ ] ] apply lemma [ l_settheoretic ] with @xmath952 , @xmath953 , @xmath954 , and for @xmath955 , @xmath956 and @xmath957 . we have abbreviated @xmath958_{1}}$ ] by the equivalent expression @xmath959 . with the necessary set - theoretic tools in place , we can proceed to demonstrate some properties of @xmath286 , starting with the fact that it is additive on the connected components of @xmath37 . [ p_21e ] let @xmath657 . then @xmath960 we first observe that for any bicolored span @xmath855 of @xmath37 , each entry of the triple is a disjoint union of corresponding entries from bicolored spans of the components @xmath961 , so @xmath962 now let @xmath963 and for each integer @xmath182 in the range @xmath964 , let @xmath965 . from observations [ ob_c_equiv ] and [ ob_settheoretic ] ( [ line5 ] ) we can conclude that @xmath966_{n_i}$ ] . since @xmath38 = @xmath967 , we can apply lemma [ l_settheoretic ] to obtain @xmath968_{n } } = { \bigl[{{{{\mathcal c}}(g_1)}^\nearrow}\bigr]_{n_1 } } + { \bigl[{{{{\mathcal c}}(g_2)}^\nearrow}\bigr]_{n_2 } } + \cdots + { \bigl[{{{{\mathcal c}}(g_k)}^\nearrow}\bigr]_{n_k } } , \ ] ] which by proposition [ p_elementary ] is equivalent to the desired conclusion . before stating and proving the cut vertex formula for elementary inertia sets , it will be useful to split the set @xmath286 into two specialized sets depending on a choice of vertex @xmath272 , and establish some of the properties of these sets . let @xmath37 be a graph and let @xmath272 be a vertex of @xmath37 . * if @xmath855 is a bicolored span of @xmath37 and @xmath969 , then we say that the ordered pair @xmath857 is a _ @xmath272-deleting color vector _ of @xmath37 . the set of @xmath272-deleting color vectors of @xmath37 is denoted @xmath970 . * if @xmath855 is a bicolored span of @xmath37 and @xmath971 , then we say that the ordered pair @xmath857 is a _ @xmath272-keeping color vector _ of @xmath37 . the set of @xmath272-keeping color vectors of @xmath37 is denoted @xmath972 . [ d_eplusminus ] let @xmath37 be a graph on @xmath38 vertices including @xmath272 . we define the set of _ @xmath272-deleting elementary inertias of @xmath37 _ as @xmath973_{n}}\ ] ] and the set of _ @xmath272-keeping elementary inertias of @xmath37 _ as @xmath974_{n}}.\ ] ] the first result we need is an immediate consequence of these definitions . [ p_l1 ] let @xmath37 be a graph with @xmath975 . then @xmath976 there are equivalent , simpler expressions for the set of @xmath272-deleting color vectors and @xmath272-deleting elementary inertias of @xmath37 . [ p_del_equiv ] let @xmath37 be a graph on @xmath238 vertices with @xmath975 . then @xmath977 and @xmath978_{n-2 } + \{(1,1)\ } = \bigl[{{\mathcal e}}(g - v ) + \{(1,1)\}\bigr]_n.\ ] ] the triple @xmath855 is a bicolored span of @xmath37 with @xmath969 if and only if the triple @xmath979 is a bicolored span of @xmath130 . it follows that the @xmath272-deleting color vectors @xmath39 in @xmath970 are exactly the vectors @xmath980 where @xmath872 is a color vector of @xmath130 . this gives us our first conclusion @xmath981 with the first conclusion as our starting point , we now have @xmath973_{n } } = \bigl [ { { { { \mathcal c}}(g - v)}^\nearrow } + \{(1,1)\ } \bigr]_{n}.\ ] ] since @xmath982 is a stripe of rank @xmath113 , by observation [ ob_settheoretic ] this simplifies to @xmath983_{n-2 } } + \{(1,1)\ } \\ & = & \bigl[{\bigl[{{{{\mathcal c}}(g - v)}^\nearrow}\bigr]_{n-1}}\bigr]_{n-2 } + \{(1,1)\ } \\ & = & \bigl[{{\mathcal e}}(g - v)\bigr]_{n-2 } + \{(1,1)\ } \\ & = & \bigl[{{\mathcal e}}(g - v ) + \{(1,1)\}\bigr]_n \end{aligned}\ ] ] which completes the proof . [ ob_c_minus_equiv ] let @xmath37 be a graph whose @xmath38 vertices include @xmath272 , and let @xmath263 be the number of components of @xmath130 . then @xmath984_{n+1-\ell}$ ] . by observation [ ob_c_equiv ] , @xmath985_{n-1-\ell}$ ] . proposition [ p_del_equiv ] and observation [ ob_settheoretic ] ( [ line3 ] ) then give us @xmath984_{n+1-\ell}$ ] . substituting @xmath986 into proposition [ p_settheoretic ] now gives us a result about @xmath272-deleting elementary inertias . [ ob_e_minus_equiv ] let @xmath37 be a graph whose @xmath38 vertices include @xmath272 , let @xmath263 be the number of components of @xmath130 , and let @xmath100 be an integer in the range @xmath904 . then 1 . [ eqminusline1 ] @xmath987_{n+1-\ell}$ ] , 2 . [ eqminusline2 ] @xmath988_{m } } = \bigl[{{{\mathcal e}}_v^{-}}(g)\bigl]_m$ ] , and 3 . [ eqminusline3 ] @xmath988_{n } } = { { { \mathcal e}}_v^{-}}(g)$ ] . similar results hold for the @xmath272-keeping color vectors and @xmath272-keeping elementary inertias : [ ob_c_plus_equiv ] let @xmath37 be a graph whose @xmath38 vertices include @xmath272 , and let @xmath989 . then @xmath990_{n-\ell}$ ] . it suffices to consider bicolored spans of the form @xmath862 , which of course satisfy @xmath991 . the set of @xmath272-keeping color vectors arising from such bicolored spans is exactly @xmath992 , from which the desired result follows by observation [ ob_settheoretic ] ( [ line4 ] ) . proposition [ p_settheoretic ] now gives us : [ ob_e_plus_equiv ] let @xmath37 be a graph whose @xmath38 vertices include @xmath272 , let @xmath989 , and let @xmath100 be an integer in the range @xmath904 . then 1 . [ eqplusline1 ] @xmath993_{n-\ell}$ ] , 2 . [ eqplusline2 ] @xmath994_{m } } = \bigl[{{{\mathcal e}}_v^{+}}(g)\bigr]_m$ ] , and 3 . [ eqplusline3 ] @xmath994_{n } } = { { { \mathcal e}}_v^{+}}(g)$ ] . it is possible to restrict the set of allowable bicolored spans that define @xmath972 and still obtain the full set of @xmath272-keeping color vectors of @xmath37 . [ p_con_equiv ] let @xmath37 be a graph with vertex @xmath272 , and let @xmath995 . suppose that @xmath872 belongs to @xmath972 . then there exists a bicolored span @xmath855 of @xmath37 with @xmath971 such that @xmath996 and such that @xmath997 is a bicolored span of @xmath130 . by the definition of @xmath972 , there exists a bicolored span @xmath855 of @xmath37 with @xmath971 such that @xmath996 . the vertex @xmath272 thus belongs to some component @xmath961 of @xmath268 , and those edges in @xmath852 and @xmath853 which are part of @xmath961 give a spanning tree @xmath998 of @xmath961 . there is no loss of generality if we assume that @xmath998 is constructed as follows : first , a spanning tree is obtained for each component of @xmath999 . each subtree is then connected to @xmath272 by way of a single edge , so that the degree of @xmath272 in @xmath998 is equal to @xmath1000 . with this assumption , @xmath1001 is a bicolored span of @xmath130 . the next key ingredient is a consequence of propositions [ p_del_equiv ] and [ p_con_equiv ] . [ p_l2 ] let @xmath37 be a graph on @xmath38 vertices , one of which is @xmath272 . then for @xmath1002 we have @xmath1003_{n-1 } \subseteq { { \mathcal e}}(g - v).\ ] ] given proposition [ p_l1 ] , proposition [ p_l2 ] is equivalent to an inclusion on elementary inertia sets which has already been proven for inertia sets as proposition [ p22 ] ( a ) : [ p22e ] for any graph @xmath37 and any vertex @xmath975 , @xmath1004_{n-1 } \subseteq { { \mathcal e}}(g - v).\ ] ] we need one more result before stating and proving the cut vertex formula for elementary inertias . [ p_l3 ] let @xmath1005 be a graph on @xmath38 vertices which is a vertex sum of graphs @xmath806 at @xmath139 , for @xmath813 . then @xmath1006_n.\ ] ] let @xmath37 , @xmath272 , @xmath38 , and @xmath1007 be as in the statement of the proposition . we first establish a related identity , @xmath1008 this holds because 1 . the sets @xmath1009 are disjoint , and their union is @xmath1010 , so subsets @xmath312 with @xmath971 are in bijective correspondence with collections of subsets @xmath1011 none of which contain @xmath272 . 2 . for any such set @xmath97 partitioned as a union of @xmath1012 , @xmath268 is a vertex sum at @xmath272 of the graphs @xmath1013 , and so the set @xmath854 is a disjoint union of @xmath1014 . 3 . a subgraph @xmath46 of the vertex sum @xmath268 is a spanning forest of @xmath268 if and only if @xmath46 is a vertex sum of graphs @xmath1015 each of which is a spanning forest of @xmath1013 . for each graph @xmath961 , let @xmath965 , so that @xmath1016 . since each graph @xmath961 contains the vertex @xmath272 , @xmath1017 . observations [ ob_c_plus_equiv ] and [ ob_settheoretic ] ( [ line5 ] ) then give us @xmath1018_{n_i-1}$ ] . thus by lemma [ l_settheoretic ] we have @xmath1019_{n-1 } } = { \bigl[{{{{{\mathcal c}}_v^{+}}(g_1)}^\nearrow}\bigr]_{n_1 - 1 } } + \cdots + { \bigl[{{{{{\mathcal c}}_v^{+}}(g_k)}^\nearrow}\bigr]_{n_k-1}}.\ ] ] we also have @xmath990_{n-1}$ ] , so by proposition [ cor_settheoretic ] we can add @xmath64 copies of @xmath959 to both sides to obtain @xmath1019_{n-1+k } } = { \bigl[{{{{{\mathcal c}}_v^{+}}(g_1)}^\nearrow}\bigr]_{n_1 } } + \cdots + { \bigl[{{{{{\mathcal c}}_v^{+}}(g_k)}^\nearrow}\bigr]_{n_k}}\ ] ] which gives the desired formula by observation [ ob_settheoretic ] ( [ line1 ] ) and definition [ d_eplusminus ] . the proof of the cut vertex formula depends on the following properties of @xmath286 , @xmath1020 , and @xmath1021 : * northeast equivalence i and iii ( observations [ ob_e_equiv ] and [ ob_e_plus_equiv ] ) , * additivity on components ( proposition [ p_21e ] ) , * splitting at @xmath272 ( proposition [ p_l1 ] ) , * the @xmath272-deleting formula ( proposition [ p_del_equiv ] ) , * domination by @xmath130 ( proposition [ p_l2 ] ) , and * the @xmath272-keeping cut vertex formula ( proposition [ p_l3 ] ) . [ th14e ] let @xmath37 be a graph on @xmath393 vertices and let @xmath272 be a cut vertex of @xmath37 . write @xmath1022 , @xmath813 , the vertex sum of @xmath806 at @xmath139 . then @xmath1023_n \label{eq5e } \\ & & \cup\ \bigl[{{\mathcal e}}(g_1-v)+{{\mathcal e}}(g_2-v)+\cdots+{{\mathcal e}}(g_k - v)+\{(1,1)\}\bigr]_n . \nonumber\end{aligned}\ ] ] we manipulate both sides to obtain the same set . define two sets @xmath1024_n\ ] ] and @xmath1025_n.\ ] ] by propositions [ p_del_equiv ] and [ p_21e ] , @xmath1026 and by proposition [ p_l3 ] , @xmath1027 , so by proposition [ p_l1 ] , @xmath1028 . the right hand side is @xmath1029_n \cup { q^{-}}.\ ] ] for each @xmath917 , let @xmath965 , so that @xmath1030_{n_i-1}$ ] ( observations [ ob_e_equiv ] ( [ eqline1 ] ) and [ ob_settheoretic ] ( [ line5 ] ) , since in each case @xmath1031 ) . starting with observation [ ob_e_equiv ] ( [ eqline3 ] ) and then applying lemma [ l_settheoretic ] both backwards and forwards , we have @xmath1032_n & = & \bigl [ { \bigl[{{{{\mathcal e}}(g_1)}^\nearrow}\bigr]_{n_1 } } + \cdots + { \bigl[{{{{\mathcal e}}(g_k)}^\nearrow}\bigr]_{n_k } } \bigr]_n \\ & = & \bigl [ \bigl[{{{{\mathcal e}}(g_1)}^\nearrow } + \cdots + { { { { \mathcal e}}(g_k)}^\nearrow } \bigr]_{n-1+k } \bigr]_n \\ & = & \bigl[{{\{(0,0)\}}^\nearrow } + { { { { \mathcal e}}(g_1)}^\nearrow } + \cdots + { { { { \mathcal e}}(g_k)}^\nearrow } \bigr]_{n } \\ & = & { { { \mathcal e}}(k_1)}+ { \bigl[{{{{\mathcal e}}(g_1)}^\nearrow}\bigr]_{n_1 - 1 } } + \cdots + { \bigl[{{{{\mathcal e}}(g_k)}^\nearrow}\bigr]_{n_k-1 } } \end{aligned}\ ] ] which by observation [ ob_e_equiv ] ( [ eqline2 ] ) gives us @xmath1033_{n_1 - 1 } + \cdots + \bigl[{{\mathcal e}}(g_k)\bigr]_{n_k-1 } \bigr).\ ] ] by applying proposition [ p_l1 ] to each term @xmath1034_{n_i-1}$ ] we obtain @xmath1035_{n_i-1 } \cup \bigl[{{{\mathcal e}}_v^{+}}(g_i)\bigr]_{n_i-1 } \bigr ) \bigr ) .\ ] ] for any @xmath1036 we will define @xmath1037_{n_i-1}.\ ] ] this gives us @xmath1038 we divide the @xmath1039 choices for @xmath1040 into two cases : either @xmath1041 is `` @xmath1042 '' for some @xmath1043 , or @xmath1044 is `` @xmath1045 '' for all @xmath182 . in the first case , by proposition [ p_del_equiv ] we have @xmath1046_{n_1 - 1 } + \cdots \\ & & \cdots + \bigl[{{\mathcal e}}(g_j - v ) + \{(1,1)\ } \big]_{n_j-1 } + \cdots \\ & & \cdots + \bigl[{{\mathcal e}}_v^{\epsilon_k}(g_k)\bigr]_{n_k-1}. \end{aligned}\ ] ] we wish to show that this is a subset of @xmath1047 . for every @xmath182 besides @xmath938 , we have @xmath1048_{n_i-1 } \subseteq { { \mathcal e}}(g_i - v)$ ] by proposition [ p_l2 ] . the remaining terms we regroup as @xmath1049_{n_j-1 } \ ! & = & \ ! { { { \mathcal e}}(k_1)}+ \bigl[{{\mathcal e}}(g_j - v)\bigr]_{n_j-3 } + \{(1,1)\ } \\ & \subseteq & \ ! { { { \mathcal e}}(k_1)}+ \bigl[{{\mathcal e}}(g_j - v)\bigr]_{n_j-2 } + \{(1,1)\}. \end{aligned}\ ] ] observation [ ob_e_equiv ] and proposition [ cor_settheoretic ] give us @xmath1050_{n_j-2 } = { { \mathcal e}}(g_j - v).\ ] ] we have thus shown that @xmath1051 and since @xmath1052_n,\ ] ] this gives us @xmath1053 in the case where @xmath1040 has at least one sign @xmath1054 `` @xmath1042 '' . this leaves the case where @xmath1040 has all signs @xmath1055 `` @xmath1045 '' . by observations [ ob_e_plus_equiv ] ( [ eqplusline1 ] ) and [ ob_settheoretic ] ( [ line5 ] ) , @xmath1056_{n_i-1}$ ] . starting with observation [ ob_e_plus_equiv ] ( [ eqplusline2 ] ) , applying lemma [ l_settheoretic ] both backwards and forwards , and finally using observation [ ob_e_plus_equiv ] ( [ eqplusline3 ] ) , we have @xmath1057_{n_1 - 1 } } + \cdots + { \bigl[{{{{{\mathcal e}}_v^{+}}(g_k)}^\nearrow}\bigr]_{n_k-1 } } \\ & = & \bigl[{{\{(0,0)\}}^\nearrow } + { { { { { \mathcal e}}_v^{+}}(g_1)}^\nearrow } + \cdots + { { { { { \mathcal e}}_v^{+}}(g_k)}^\nearrow } \bigr]_{n } \\ & = & \bigl [ \bigl[{{{{{\mathcal e}}_v^{+}}(g_1)}^\nearrow } + \cdots + { { { { { \mathcal e}}_v^{+}}(g_k)}^\nearrow } \bigr]_{n-1+k } \bigr]_n \\ & = & \bigl [ { \bigl[{{{{{\mathcal e}}_v^{+}}(g_1)}^\nearrow}\bigr]_{n_1 } } + \cdots + { \bigl[{{{{{\mathcal e}}_v^{+}}(g_k)}^\nearrow}\bigr]_{n_k } } \bigr]_n \\ & = & { q^{+}}. \end{aligned}\ ] ] the entire union thus collapses to @xmath1058 and the left and right hand expressions are equal . we can generalize the splitting of @xmath286 into @xmath1020 and @xmath1021 for non - elementary inertias : given a graph @xmath37 with vertex @xmath272 and @xmath111 , order the vertices of @xmath37 such that @xmath1059 and decompose @xmath18 as @xmath1060.\ ] ] if @xmath1061 is in the column space of @xmath58 , then say that @xmath1062 , and define @xmath1063 as @xmath1064_n$ ] . define @xmath1065 and @xmath1066 analogously . under these definitions we can uniformly replace @xmath1067 with @xmath795 or @xmath794 in observations [ ob_e_equiv ] , and [ ob_e_plus_equiv ] and in each of propositions [ p_21e ] , [ p_l1 ] , [ p_del_equiv ] , [ p_l2 ] , and [ p_l3 ] , and we claim that in every case the result still holds . we will not prove these statements , as we already have a proof of theorem [ th13 ] , but given those observations and propositions , the proof of theorem [ th14e ] demonstrates the same cut vertex formula for inertia sets . we now state and prove the main result of the section . [ th15 ] for any tree @xmath359 , @xmath1068 . let @xmath1069 . + if @xmath1070 , @xmath1071 and @xmath1072 } $ ] . since @xmath1073 is a bicolored span of @xmath1074 , the origin @xmath1075 is a color vector of @xmath43 and @xmath1076 } $ ] also . if @xmath1077 , then @xmath1078 , and @xmath1079}={{\mathcal e}}(k_2)$ ] . proceeding by induction , assume that @xmath1080 for all trees @xmath359 on fewer than @xmath1 vertices and let @xmath359 be a tree on @xmath1 vertices , @xmath1081 . let @xmath139 be a cut vertex of @xmath359 of degree @xmath1082 . write @xmath1083 , the vertex sum of @xmath1084 at @xmath139 . by theorem [ th13 ] , @xmath1085_n \\ & & \cup \ \bigl[{{\mathcal i}}(t_1-v)+\cdots+{{\mathcal i}}(t_k - v)+\{(1,1)\}\bigr]_n\end{aligned}\ ] ] and by theorem [ th14e ] , @xmath1086_n \\ & & \cup \ \bigl[{{\mathcal e}}(t_1-v)+\cdots+{{\mathcal e}}(t_k - v)+\{(1,1)\}\bigr]_n.\end{aligned}\ ] ] corresponding terms on the right hand side of these last two equations are equal by the induction hypothesis , so @xmath1087 . [ cor16 ] for any forest @xmath533 , @xmath1088 . by theorem [ th15 ] , @xmath1089 for every component @xmath43 of @xmath46 , and by additivity on components for both @xmath231 ( observation [ ob21 ] ) and @xmath286 ( proposition [ p_21e ] ) , @xmath1090 . claim 1 of theorem [ th_treemain ] , which says @xmath1091 for any forest @xmath46 , has now been verified . we restate theorem [ th_treemain ] compactly as let @xmath37 be a graph . then @xmath1092 if and only if @xmath37 is a forest . tabulating the full inertia set of a tree @xmath43 on @xmath38 vertices by means of theorem [ th_treemain ] appears , potentially , to require a lot of calculation : every integer @xmath64 in the range @xmath1093 with @xmath1094 gives a trapezoid ( possibly degenerate ) of elementary inertias , and the full elementary inertia set is the union of those trapezoids . ( one could also construct every possible bicolored span of the tree , which is even more cumbersome . ) in fact the calculation is quite straightforward once we have the first few values of @xmath381 . in this section we present the necessary simplifications and perform the calculation for a few examples . [ d19 ] for any graph @xmath0 on @xmath1 vertices and @xmath1095 , let @xmath1096 @xmath1097 since @xmath1098 for each @xmath64 , we will deal exclusively with @xmath1099 . the main simplification toward calculating the inertia set of a tree is the following : [ th17 ] let @xmath359 be a tree on @xmath1 vertices and let @xmath1100 . then @xmath1101 for @xmath64 in the given range , we can apply corollary [ cor_mdmonotone ] with @xmath1102 to obtain @xmath1103 and in particular @xmath1094 . we can thus apply the stars and stripes lemma to obtain @xmath1104 and hence @xmath1105 . it remains to prove , for @xmath1106 , @xmath1107 suppose by way of contradiction that @xmath650 with @xmath1108 and @xmath1109 . by theorem [ th_treemain ] , every element of @xmath458 is an elementary inertia , and thus there is some integer @xmath938 for which @xmath1110 , @xmath1111 , and @xmath1112 . this implies , for @xmath1113 , that @xmath1114 which contradicts corollary [ cor_mdmonotone ] . [ cor18 ] @xmath1115 is a strictly decreasing sequence . this follows from theorem [ th17 ] , corollary [ cor_mdmonotone ] ( with @xmath1116 ) , and theorem [ th9 ] . [ th19 ] let @xmath43 be a tree . then @xmath645 . in other words , every partial inertia of minimum rank is in the minimum - rank stripe already defined . we already have @xmath646 by theorem [ th9 ] , and @xmath1117 by definition . to show equality , it suffices by symmetry ( observation [ ob8 ] ) to show that for @xmath1118 , @xmath1119 . let @xmath1120 . if @xmath1121 , we are done . by observation [ ob_cmd ] , @xmath1122 but @xmath1123 for @xmath1124 . it follows by theorem [ th17 ] that @xmath1125 for @xmath1124 , which completes the proof . theorem [ th12 ] already gives a method for determining the inertia set @xmath458 for any tree @xmath43 , but with theorem [ th_treemain ] and the simplifications above there is a much easier method , which we summarize in the following steps : 1 . use the algorithm of observation [ ob_palgorithm ] to find @xmath452 . 2 . since @xmath43 is connected , @xmath1126 . if @xmath43 is a path then @xmath1127 ; otherwise @xmath1128 and @xmath1129 . continue to calculate higher values of @xmath381 until @xmath1130 , at which point @xmath647 . 3 . the defining southwest corners of @xmath458 are @xmath1131 and its reflection @xmath1132 , for @xmath1133 , together with the stripe @xmath643 of partial inertias from @xmath1134 to @xmath1135 . every other point of @xmath458 is a result of the northeast lemma applied to the defining southwest corners . we give three examples . [ ex13 ] let @xmath43 be the tree in example [ ex7 ] , whose inertia set we have already calculated . @xmath1136 here @xmath1137 and @xmath1138 . we have @xmath1139 so @xmath1140 , and from @xmath1141 we go immediately to @xmath643 , starting at height @xmath82 , which is the convex stripe of three partial inertias from @xmath1142 to @xmath1143 . ( 50,50 ) ( 42,4)(0,1)40 ( 40,6)(1,0)41 ( 40.75,39)(1,0)2.5 ( 42,39 ) ( 40.75,33.5)(1,0)2.5 ( 42,33.5 ) ( 47.5,33.5 ) ( 40.75,28)(1,0)2.5 ( 47.5,28 ) ( 53,28 ) ( 40.75,22.5)(1,0)2.5 ( 47.5,22.5 ) ( 53,22.5 ) ( 58.5,22.5 ) ( 40.75,17)(1,0)2.5 ( 53,17 ) ( 58.5,17 ) ( 64,17 ) ( 40.75,11.5)(1,0)2.5 ( 58.5,11.5 ) ( 64,11.5 ) ( 69.5,11.5 ) ( 47.5,4.75)(0,1)2.5 ( 53,4.75)(0,1)2.5 ( 58.5,4.75)(0,1)2.5 ( 64,4.75)(0,1)2.5 ( 69.5,4.75)(0,1)2.5 ( 69.5,6 ) ( 75,4.75)(0,1)2.5 ( 75,6 ) [ ex14 ] let @xmath43 be the tree @xmath1144 whose horizontal paths realize the path cover number @xmath1145 , so @xmath1146 . taking any vertex of degree @xmath851 we have @xmath1147 and taking the non - adjacent pair of degree-@xmath851 vertices we have @xmath1148 so @xmath1149 . starting as always from @xmath1150 , we need only one more value @xmath1151 before reaching the minimum - rank stripe @xmath643 from @xmath1152 to @xmath1153 . the complete set @xmath458 is : ( 50,70 ) ( 42,4)(0,1)58 ( 40,6)(1,0)58 ( 40.75,55.5)(1,0)2.5 ( 42,55.5 ) ( 40.75,50)(1,0)2.5 ( 42,50 ) ( 47.5,50 ) ( 40.75,44.5)(1,0)2.5 ( 47.5,44.5 ) ( 53,44.5 ) ( 40.75,39)(1,0)2.5 ( 47.5,39 ) ( 53,39 ) ( 58.5,39 ) ( 40.75,33.5)(1,0)2.5 ( 53,33.5 ) ( 58.5,33.5 ) ( 64,33.5 ) ( 40.75,28)(1,0)2.5 ( 53,28 ) ( 58.5,28 ) ( 64,28 ) ( 69.5,28 ) ( 40.75,22.5)(1,0)2.5 ( 58.5,22.5 ) ( 64,22.5 ) ( 69.5,22.5 ) ( 75,22.5 ) ( 40.75,17)(1,0)2.5 ( 64,17 ) ( 69.5,17 ) ( 75,17 ) ( 80.5,17 ) ( 40.75,11.5)(1,0)2.5 ( 75,11.5 ) ( 80.5,11.5 ) ( 47.5,4.75)(0,1)2.5 ( 53,4.75)(0,1)2.5 ( 58.5,4.75)(0,1)2.5 ( 64,4.75)(0,1)2.5 ( 69.5,4.75)(0,1)2.5 ( 75,4.75)(0,1)2.5 ( 80.5,4.75)(0,1)2.5 ( 86,4.75)(0,1)2.5 ( 86,11.5 ) ( 86,6 ) ( 91.5,4.75)(0,1)2.5 ( 91.5,6 ) the examples we have shown so far appear to exhibit some sort of convexity . for @xmath46 a forest we do at least have convexity of @xmath1154 on stripes of fixed rank , as stated in corollary [ cor_stripes ] . based on small examples one may be led to believe that , in addition , @xmath1155 is a convex function in the range @xmath1156 , or in other words that @xmath1157 however , this is not always the case , as seen in the following example : [ ex15 ] given @xmath1158 with @xmath272 a pendant vertex , let @xmath43 be the tree constructed as a vertex sum of four copies of the marked @xmath1158 : @xmath1159 here @xmath1160 and @xmath1161 . to find @xmath67 we always take a vertex of maximum degree ; here @xmath1162 for @xmath1163 we can either add the center of a branch or leave out the degree-@xmath115 vertex and take two centers of branches ; either choice gives us @xmath1164 at @xmath1165 something odd happens : to remove @xmath851 vertices and maximize the number of remaining components , we must not include the single vertex of maximum degree . taking the centers of three branches , we obtain @xmath1166 and finally taking all four vertices of degree @xmath851 gives us @xmath1167 so @xmath1168 . the sequence @xmath1155 thus starts @xmath1169 . as is the case with the stars @xmath476 and example [ ex8 ] , we here have a tree where the minimum - rank stripe @xmath643 is a singleton , in this case the point @xmath1170 . the full plot of @xmath458 is : ( 50,90 ) ( 22,4)(0,1)80 ( 20,6)(1,0)80 ( 20.75,77.5)(1,0)2.5 ( 22,77.5 ) ( 20.75,72)(1,0)2.5 ( 22,72 ) ( 27.5,72 ) ( 20.75,66.5)(1,0)2.5 ( 27.5,66.5 ) ( 33,66.5 ) ( 20.75,61)(1,0)2.5 ( 27.5,61 ) ( 33,61 ) ( 38.5,61 ) ( 20.75,55.5)(1,0)2.5 ( 27.5,55.5 ) ( 33,55.5 ) ( 38.5,55.5 ) ( 44,55.5 ) ( 20.75,50)(1,0)2.5 ( 33,50 ) ( 38.5,50 ) ( 44,50 ) ( 49.5,50 ) ( 20.75,44.5)(1,0)2.5 ( 38.5,44.5 ) ( 44,44.5 ) ( 49.5,44.5 ) ( 55,44.5 ) ( 20.75,39)(1,0)2.5 ( 38.5,39 ) ( 44,39 ) ( 49.5,39 ) ( 55,39 ) ( 60.5,39 ) ( 20.75,33.5)(1,0)2.5 ( 44,33.5 ) ( 49.5,33.5 ) ( 55,33.5 ) ( 60.5,33.5 ) ( 66,33.5 ) ( 20.75,28)(1,0)2.5 ( 44,28 ) ( 49.5,28 ) ( 55,28 ) ( 60.5,28 ) ( 66,28 ) ( 71.5,28 ) ( 20.75,22.5)(1,0)2.5 ( 55,22.5 ) ( 60.5,22.5 ) ( 66,22.5 ) ( 71.5,22.5 ) ( 77,22.5 ) ( 20.75,17)(1,0)2.5 ( 66,17 ) ( 71.5,17 ) ( 77,17 ) ( 82.5,17 ) ( 20.75,11.5)(1,0)2.5 ( 71.5,11.5 ) ( 77,11.5 ) ( 82.5,11.5 ) ( 88,11.5 ) ( 27.5,4.75)(0,1)2.5 ( 33,4.75)(0,1)2.5 ( 38.5,4.75)(0,1)2.5 ( 44,4.75)(0,1)2.5 ( 49.5,4.75)(0,1)2.5 ( 55,4.75)(0,1)2.5 ( 60.5,4.75)(0,1)2.5 ( 66,4.75)(0,1)2.5 ( 71.5,4.75)(0,1)2.5 ( 77,4.75)(0,1)2.5 ( 82.5,4.75)(0,1)2.5 ( 88,4.75)(0,1)2.5 ( 88,6 ) ( 93.5,4.75)(0,1)2.5 ( 93.5,6 ) while @xmath1155 is not a convex function over the range @xmath1171 in the last example , the calculated set @xmath458 does at least contain all of the lattice points in its own convex hull . to expect this convexity to hold for every tree would be overly optimistic , however : if we carry out the same calculation for the larger tree @xmath1172 ( on @xmath1173 vertices instead of @xmath1174 ) we find that the points @xmath1175 and @xmath1176 both belong to the inertia set , but their midpoint @xmath1177 does not . what is the computational complexity of determining the partial inertia set of a tree ? the examples above pose no difficulty , but they do show that the greedy algorithm for @xmath381 fails even for @xmath43 a tree . computing all @xmath38 values of @xmath63 for a general graph @xmath37 is np - hard because it can be used to calculate the independence number : @xmath1178 if and only if there is an independent set of size @xmath65 . in the next section we will consider more general graphs , rather than restricting to trees and forests , and we will see that even convexity of partial inertias within a single stripe can fail in the broader setting . in this section we investigate , over the set of all graphs , what partial inertia sets or more specifically , what complements of partial inertia sets can occur . once a graph @xmath37 is allowed to have cycles , we can no longer assume that @xmath1179 by diagonal congruence . it happens , however , that each of the results in this section is the same in the complex hermitian case as in the real symmetric case . for the two versions of each question we will therefore demonstrate whichever is the more difficult of the two , proving theorems over the complex numbers but providing counterexamples over the reals . a _ partition _ is a finite ( weakly ) decreasing sequence of positive integers . the first integer in the sequence is called the _ width _ of the partition , and the number of terms in the sequence is called the _ height _ of the partition . it is traditional to depict partitions with box diagrams . in order to agree with our diagrams of partial inertia sets , we choose the convention of putting the longest row of boxes on the bottom of the stack ; for example , the decreasing sequence @xmath1180 is shown as the partition . given a box diagram of height @xmath1181 and width @xmath530 , we index the rows by @xmath1182 from bottom to top and the columns by @xmath1183 from left to right . given a partition @xmath1184 , let @xmath1185 , and for @xmath1186 let @xmath1187 , i.e. the number of boxes in column @xmath182 of the box diagram of @xmath1188 . then @xmath1189 is called the conjugate partition of @xmath1188 . a partition @xmath1188 is _ symmetric _ if @xmath1190 . for example , we have @xmath1191 and @xmath1192 , so the partition with box diagram is symmetric . it is easy to recognize symmetric partitions visually , since a partition is symmetric if and only if its box diagram has a diagonal axis of symmetry . in this section we will describe @xmath231 , for a graph @xmath37 on @xmath38 vertices , in terms of its complement @xmath1193 . definition [ d19 ] gives us a natural way to describe the shape of this complement as a partition . we first extend to the hermitian case ( distinguishing from the real symmetric case as usual by prepending an ` @xmath1194 ' ) . [ dparti ] given a graph @xmath37 , let @xmath1198 and let @xmath1199 . then the _ inertial partition _ of @xmath37 , denoted @xmath1200 , is the partition @xmath1201 the _ hermitian inertial partition _ of @xmath37 , denoted @xmath1202 , is the partition @xmath1203 the northeast lemma ensures that the inertial partition and hermitian inertial partition of a graph are in fact partitions , and by observation [ ob8 ] the partitions @xmath1200 and @xmath1202 are always symmetric . this symmetry is the reason why @xmath1204 is the correct point of truncation : @xmath1205 , but @xmath1206 . [ rboxes ] if one starts with the entire first quadrant of the plane @xmath1207 and then removes everything `` northeast '' of any point belonging to @xmath231 , the remaining `` southwest complement '' has the same shape as the box diagram of @xmath1208 . the same applies of course to @xmath232 and @xmath1202 . the partial inertia sets @xmath231 and @xmath232 can be reconstructed from the partitions @xmath1200 and @xmath1202 , respectively , if the number of vertices of @xmath37 is also known . the addition of an isolated vertex to a graph @xmath37 does not change @xmath1200 . rather than examining all possible partial inertias for a particular graph , we are now examining what restrictions on partial inertias ( or rather excluded partial inertias ) may hold over the class of all graphs . the hermitian inertial partition classification problem is the same question with @xmath1202 in the place of @xmath1200 . while it is known that there are graphs @xmath37 for which @xmath231 is a strict subset of @xmath232 , it is not known whether there are partitions @xmath1188 that are inertial partitions but not hermitian inertial partitions , or vice versa . at the moment we are only able to give a complete answer to the inertial partition classification problem for symmetric partitions of height no greater than 3 . we first list examples for a few symmetric partitions that are easily obtained . of course , adding an isolated vertex to any example gives another example for the same partition . for simplicity we will identify the partition @xmath1208 with its box diagram . [ th_nosquares ] let @xmath37 be a graph and let @xmath1216 be a hermitian matrix with partial inertia @xmath1217 , @xmath1218 . then there exists a matrix @xmath1219 with partial inertia @xmath39 satisfying @xmath1220 and @xmath1221 . furthermore , if @xmath1222 is real then @xmath1223 can be taken to be real . [ cnosquares ] for any @xmath1218 , the square partition @xmath1224 of height @xmath64 and width @xmath64 is not the inertial partition of any graph @xmath37 , and is not the hermitian inertial partition of any graph @xmath37 . let @xmath37 be a graph on @xmath38 vertices and suppose that @xmath1216 is a hermitian matrix with partial inertia @xmath1217 . the matrix @xmath1225 $ ] is thus positive semidefinite of rank @xmath64 , and can be factored as @xmath1226 for some @xmath1227 complex matrix @xmath1228 $ ] . if @xmath1222 is real symmetric , then @xmath18 can be taken to be real . we wish to construct a matrix @xmath1219 with strictly fewer than @xmath64 positive eigenvalues and also strictly fewer than @xmath64 negative eigenvalues . by proposition [ p2 ] , we will have accomplished our purpose if we can find @xmath1229 matrices @xmath1230 $ ] and @xmath1231 $ ] such that @xmath1232 , with the requirement that @xmath58 and @xmath277 be real if @xmath1222 is real . we need to impose some mild general - position requirements on the first two rows of the matrix @xmath18 , which we accomplish by replacing @xmath18 by @xmath1233 , where @xmath1234 is a unitary matrix and where @xmath1234 is real ( and hence orthogonal ) in the case that @xmath18 is real . this is a permissible substitution because @xmath1235 . the first general - position requirement is that , for integers @xmath1236 , @xmath1237 and @xmath1238 unless column @xmath938 of @xmath18 is the zero column . the second requirement , which we will justify more carefully , is that the set of ratios @xmath1239 be disjoint from the set of conjugate reciprocals @xmath1240 , or equivalently @xmath1241 for any @xmath1242 where neither @xmath182 nor @xmath938 corresponds to a zero column . now we prove the existence of a unitary matrix @xmath1234 with the desired properties . to do so , we temporarily reserve the symbol @xmath1243 to represent a solution to @xmath1244 . for the duration of this argument , @xmath938 will represent any integer @xmath1245 such that column @xmath938 of @xmath18 is not the zero column . let @xmath1246 represent the vector @xmath1247 $ ] . if @xmath1248 represents the set of nonzero complex numbers , then our first general position assumption already guarantees @xmath1249 . we now define three functions @xmath1250 by @xmath1251 \right ) = p / q , \ \ \overline{z } \left ( \left [ \begin{array}{c } p \\ q \end{array } \right ] \right ) = \overline{p}/\overline{q } , \ \mbox { and } \ w \left ( \left [ \begin{array}{c } p \\ q \end{array } \right ] \right ) = \overline{q}/\overline{p}.\ ] ] our task is to find a unitary matrix @xmath1252 , orthogonal in the case that @xmath18 is real , such that the sets @xmath1253 and @xmath1254 are disjoint . in this case we can achieve the desired general position of @xmath18 by replacing it with @xmath1233 , where @xmath1255 . now consider the unitary matrices @xmath1256 \ \mbox { and } \ q = \frac{1}{\sqrt{2 } } \left [ \begin{array}{cc } 1 & i \\ i & 1 \end{array } \right ] .\ ] ] these matrices transform complex ratios as follows : @xmath1257 we have @xmath1258 as long as @xmath1259 and in particular as long as @xmath1260 is not pure imaginary , which is automatically true in the case @xmath18 is real . in the case where @xmath18 is not real , we can assume without loss of generality that no @xmath1260 is pure imaginary after uniformly multiplying on the left by an appropriate choice of @xmath1261 . given @xmath1262 and @xmath1263 in @xmath1264 such that neither @xmath1265 nor @xmath1266 is pure imaginary , @xmath1267 if and only if @xmath1268 . we have reduced the problem to that of finding a unitary matrix @xmath1252 , orthogonal in the case @xmath18 is real , such that the sets @xmath1269 and @xmath1270 are disjoint . in fact we will establish the stronger condition that the two finite subsets of the unit circle @xmath1271 are disjoint . let @xmath1272.\ ] ] then @xmath1252 is orthogonal , and @xmath1273 our general - position requirement for @xmath18 thus reduces to the following fact : given a finite subset @xmath869 of the unit circle , there is some @xmath1274 such that @xmath1275 is disjoint from its set of conjugates @xmath1276 and from the set @xmath1277 . to be concrete , if @xmath1278 is the minimum nonzero angle between any element of @xmath869 and any element of @xmath1279 or @xmath1277 , @xmath1280 will suffice . this concludes the argument justifying our assumption of general position for @xmath18 . now consider an arbitrary entry @xmath1287 of the matrix @xmath1288 ; this takes the form @xmath1289 which factors as @xmath1290 in case either column @xmath182 or column @xmath938 of @xmath18 is the zero column , we have @xmath1291 , and in all other cases we have , by the generic requirement @xmath1292 that @xmath1293 if and only if @xmath1294 . it follows that @xmath1223 is a matrix in @xmath12 , and by construction @xmath1223 has at most @xmath1295 positive eigenvalues and at most @xmath1295 negative eigenvalues . furthermore , if @xmath1222 is real symmetric then so is @xmath1223 . we have determined which inertial partitions occur for all partitions up to height @xmath851 except for one : the partition . perhaps surprisingly , there is indeed a graph , on @xmath1296 vertices , that achieves this non - inertia - balanced partition in both the real symmetric and hermitian cases . these 13 vectors give us the columns of a matrix @xmath1305 { $ m_{13 } = \left [ \raisebox{.15\totalheight}[-.3\totalheight ] { $ \begin{array}{rrrrrrrrrrrrc } x & y & z & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & \!-1 & 1 & 1 & \!-1 & \!-1 & 1 \rule{0pt}{3ex } \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 0 & \!-1 & \!-1 & 1 & \!-1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 & \!-1 & 1 & 0 & \!-1 & \!-1 & 1 & 1 \end{array } $ } \right ] $ } , \ ] ] which columns we index by the set of symbols @xmath1306 . the matrix @xmath1307 is real symmetric and positive semidefinite of rank @xmath851 , and thus has partial inertia @xmath1300 . we define @xmath1308 as the graph on 13 vertices ( labeled by the same 13 symbols ) for which @xmath1309 ; distinct vertices @xmath182 and @xmath938 of @xmath1308 are adjacent if and only if columns @xmath182 and @xmath938 of @xmath1310 are not orthogonal . we note in passing that the subgraph of @xmath1308 induced by vertices labeled 110 is the line graph of @xmath1311 , or the complement of the petersen graph . we now define the graph @xmath1297 ( as promised in theorem [ th_exists332 ] ) as the induced subgraph of @xmath1308 obtained by deleting the vertex labeled @xmath1312 . before proving theorem [ th_exists332 ] , we prove a lemma about a smaller graph @xmath1313 that is obtained from @xmath1297 by deleting the vertices labeled @xmath1314 and @xmath1315 ( while retaining the labels of the other vertices ) . the vertices of @xmath1313 are thus labeled @xmath1316 ( notice that this set skips index @xmath1314 ) . let @xmath1321 , @xmath1322 , and @xmath1323 be the first three diagonal entries of @xmath18 : @xmath1324 we show first that all three of these entries are nonzero . for this purpose it suffices to consider only the first six rows and columns of @xmath18 , corresponding to the graph @xmath1325 @xmath1326 sometimes called the supertriangle graph . the automorphism group of @xmath1327 realizes any permutation of the vertices @xmath1328 ( as do the automorphism groups of @xmath1308 and of @xmath1297 , but not that of @xmath1313 ) . the principal submatrix of @xmath18 on rows and columns @xmath1329 , like @xmath18 itself , has rank at most @xmath851 . suppose that we had @xmath1330 while @xmath1331 . then the @xmath1332 submatrix on rows @xmath1333 and columns @xmath1334 would be combinatorially nonsingular ( that is , permutation equivalent to an upper - triangular matrix with nonzero entries on the diagonal ) , contradicting that @xmath1335 . by the symmetries of @xmath1327 , we could have chosen any pair instead of @xmath1336 , and thus if any one of the three quantities @xmath1321 , @xmath1322 , or @xmath1323 is equal to zero , then all three must be . but if all three of the first diagonal entries were zero , then the @xmath1332 principal submatrix on rows and columns @xmath1337 would be combinatorially nonsingular . it follows that in the @xmath1338 rank-3 matrix @xmath18 , the first three diagonal entries @xmath1321 , @xmath1322 , and @xmath1323 are all nonzero . considering once more the full matrix @xmath18 , let @xmath1339 and @xmath1340 , so the first three rows of @xmath18 can be written : @xmath1341.\ ] ] since @xmath1321 , @xmath1322 , and @xmath1323 are nonzero and @xmath18 has rank at most @xmath851 , every other row of @xmath18 can be obtained from these first three rows by taking a linear combination , and the coefficients of the linear combination are determined by entries in the first three columns . every entry of @xmath18 is thus determined by the variables appearing in the @xmath1342 matrix above . for any @xmath182 and @xmath938 in the set @xmath1343 , we have @xmath1344 in those cases where @xmath1345 and @xmath1346 is not an edge of @xmath1313 , the entry @xmath1347 gives an equation on the entries of the first three rows . using several such equations , we deduce that @xmath1348 , as follows : 1 . the entries @xmath1349 and @xmath1350 give us the pair of equations @xmath1351 2 . combining the equations from @xmath1352 and @xmath1353 , we have @xmath1354 3 . combining the equations from @xmath1355 and @xmath1356 , we have @xmath1357 4 . combining the equations from @xmath1358 and @xmath1359 , we have @xmath1360 multiplying the first pair of equations and then substituting in each of the remaining equations in order , then canceling the nonzero term @xmath1361 , we arrive finally at @xmath1362 a positive quantity . this proves that , in any hermitian matrix @xmath1363 of rank no more than @xmath851 , the first two diagonal entries are nonzero and have the same sign . we review the definition of the graph @xmath1297 that will provide the claimed example : starting from the diagram of the cube with a labeled vector for every pair of faces , edges , or corners , we omit the vector @xmath1312 and connect pairs of vertices from the set @xmath1364 whenever their corresponding vectors are not orthogonal . letting @xmath1365 be the submatrix of @xmath1310 obtained by deleting the last column ( labeled @xmath1312 ) , we have @xmath1366 and thus @xmath1367 and @xmath1368 . it follows from theorem [ th_nosquares ] that the point @xmath1369 also belongs to @xmath1370 and @xmath1302 . to show that @xmath1298 , it suffices by the northeast lemma and symmetry to show that @xmath1371 . let @xmath18 be any matrix of rank @xmath851 in @xmath1372 , and let @xmath1321 , @xmath1322 , and @xmath1323 be the first three diagonal entries of @xmath18 . omitting rows and columns @xmath1314 and @xmath1315 gives us a matrix of rank no more than @xmath851 in @xmath1318 , and so lemma [ lg10 ] tells us that @xmath1321 and @xmath1322 are nonzero and have the same sign . however , the automorphism group of @xmath1297 inherits all the symmetries of a cube with one marked corner , and thus anything true of the pair of vertices @xmath1373 is also true of the pair @xmath1374 , so @xmath1322 and @xmath1323 are also nonzero and have the same sign . more explicitly , using the symmetry of a counterclockwise rotation of the cube around the corner marked @xmath1312 , we delete rows and columns @xmath115 and @xmath1375 ( instead of @xmath1314 and @xmath1315 ) and reorder the remaining rows and columns as @xmath1376 to yield a different matrix belonging to @xmath1318 , and invoke lemma [ lg10 ] again to obtain @xmath1377 , showing that the three diagonal entries @xmath1321 , @xmath1322 , and @xmath1323 all have the same sign . the principal submatrix of @xmath18 on rows and columns @xmath1328 has either three positive eigenvalues or three negative eigenvalues , and so by interlacing the partial inertia of @xmath18 must be either @xmath1300 or @xmath1378 . this completes the proof that the graph @xmath1297 achieves the inertial partition and hermitian inertial partition @xmath1299 , and is not inertia - balanced . of course the same argument also shows that @xmath1379 , but we are interested in the smallest possible graph that is not inertia - balanced . the following proposition justifies our claim that @xmath1297 is at least locally optimal . by theorem [ th_nosquares ] every graph @xmath37 with @xmath1380 is inertia - balanced and every graph @xmath37 with @xmath1381 is hermitian inertia - balanced . it thus suffices to show that for every proper induced subgraph @xmath46 of @xmath1308 other than @xmath1297 , @xmath1382 unless @xmath1383 . recall that @xmath1308 is defined by orthogonality relations between the columns of the matrix @xmath1384,\ ] ] corresponding to various axes of symmetry of a cube . in other words , @xmath1385 ( where the identity matrix @xmath1386 imposes the standard positive definite inner product on @xmath1303 ) and so @xmath1387 . the automorphism group of @xmath1308 has three orbits , corresponding to the faces ( @xmath1262 , @xmath1263 , and @xmath1388 ) , edges ( 1 , 2 , 3 , 4 , 5 , and 6 ) , and corners ( 7 , 8 , 9 , and 10 ) of the cube . the deletion of any corner yields @xmath1297 ( perhaps with a different labeling ) and there is , up to isomorphism , only one way to delete two corners . every proper induced subgraph of @xmath1308 other than @xmath1297 is thus isomorphic to an induced subgraph of @xmath1389 , of @xmath1390 , or of @xmath1391 . letting @xmath37 be each of these three graphs in turn , we exhibit for each a diagonal matrix @xmath48 with @xmath1392 and a real matrix @xmath1222 such that @xmath1393 . @xmath1394 : @xmath1395,\ \ m = \left [ \begin{array}{rrrrrrrrrrrrr } . & 0 & 2 & \!\!-1 & 1 & 1 & \!\!-1 & 0 & 1 & 0 & 1 & 0 & 1 \\ . & 1 & 0 & 1 & 0 & \!\!-1 & \!\!-1 & 0 & 1 & \!\!-2 & 3 & 2 & \!\!-3 \\ . & 0 & 3 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \end{array } \right]\ ] ] @xmath1396 : @xmath1397,\ \ m = \left [ \begin{array}{rrrrrrrrrrrrr } 1 & 0 & 0 & 0 & 2 & . & 0 & 1 & \!\!-1 & 1 & 4 & 1 & 2 \\ 0 & 1 & 0 & 2 & 0 & . & 1 & 0 & 1 & 4 & 1 & 1 & 2 \\ 0 & 0 & 1 & 1 & 1 & . & 2 & 2 & 0 & 2 & 2 & 2 & 1 \end{array } \right]\ ] ] @xmath1398 : @xmath1397,\ \ m = \left [ \begin{array}{rrrrrrrrrrrrr } 1 & 0 & 0 & 0 & 2 & 1 & 0 & 1 & \!\!-1 & . & . & 1 & 2 \\ 0 & 1 & 0 & 2 & 0 & 1 & 1 & 0 & 1 & . & . & 1 & 2 \\ 0 & 0 & 1 & 1 & 1 & 0 & 2 & 2 & 0 & . & . & 2 & 1 \end{array } \right]\ ] ] for each value of @xmath1222 and @xmath48 , the matrix @xmath1399 has partial inertia @xmath1301 and belongs to @xmath14 for the desired subgraph @xmath37 . if @xmath46 is a proper induced subgraph of @xmath1308 other than @xmath1297 , then @xmath46 is an induced subgraph of one of these three graphs . part ( a ) of proposition [ p22 ] allows us to delete vertices from any one of the three graphs and keep the partial inertia @xmath1301 as long as at least @xmath851 vertices remain , which gives us @xmath1382 unless @xmath1383 . theorems [ th_nosquares ] and [ th_exists332 ] only permit us to answer the inertial partition classification problem and hermitian inertial partition classification problem up to height @xmath851 . we have shown examples of constructing a graph whose minimum rank realization with a particular partial inertia is sufficiently `` rigid '' to prevent intermediate partial inertias of the same rank between the matrix and its negative . if the rank is allowed to increase , though , it is much less clear what restrictions can be made . the next difficult question appears to be whether @xmath1400 is an inertial partition . on the one hand , partial inertia @xmath1404 is of higher rank than partial inertia @xmath1405 , which means that any proof along the lines of theorem [ th_exists332]a proof that a particular arrangement of orthogonality relations of vectors in @xmath1406 could not be duplicated in @xmath1407 with an indefinite inner product of signature @xmath1404would have an extra degree of freedom to contend with . on the other hand , there seems to be little hope of constructing the matrix @xmath1223 directly from @xmath1222 using any sort of continuous map such as that employed in the proof of theorem [ th_nosquares ] . f.barioli , s.fallat and l.hogben , computation of minimal rank and path cover number for graphs , _ linear algebra and its applications _ , 392 : 289303 , 2004 . f.barioli , s.fallat and l.hogben , on the difference between the maximum multiplicity and path cover number for tree - like graphs , _ linear algebra and its applications _ , 409 : 1331 , 2005 . f.barioli , s.fallat and l.hogben , a variant on the graph parameters of colin de verdire : implications to the minimum rank of graphs , _ electronic journal of linear algebra _ , 13 : 387404 , 2005 . w.barrett , j.grout and r.loewy , the minimum rank problem over the finite field of order @xmath406 : minimum rank @xmath655 , + http://arxiv.org/abs/math.co/0612331 . r.brualdi , l.hogben and b.shader , aim workshop on spectra of families of matrices described by graphs , digraphs and sign patterns , final report : mathematical results ( revised * ) , 2007 . http://aimath.org/pastworkshops/matrixspectrumrep.pdf . w.barrett , h. van der holst and r. loewy , graphs whose minimal rank is two , _ electronic journal of linear algebra _ , 11 : 258280 , 2004 . w.barrett , h. van der holst and r. loewy , graphs whose minimal rank is two : the finite fields case , _ electronic journal of linear algebra _ , 14 : 3242 , 2005 . v.chvtal , tough graphs and hamiltonian circuits , _ discrete mathematics _ , 5 : 215228 , 1973 . m.fiedler , a characterization of tridiagonal matrices , _ linear algebra and its applications _ , 2 : 191197 , 1969 . o.h.hald , inverse eigenvalue problems for jacobi matrices , _ linear algebra and its applications _ , 14 : 6385 , 1976 . h. van der holst , graphs whose positive semi - definite matrices have nullity at most two , _ linear algebra and its applications _ , 375 : 111 , 2003 . hsieh , on minimum rank matrices having prescribed graph , _ ph . d. thesis _ , university of wisconsin - madison , 2001 . c.r.johnson and a.leal duarte , the maximum multiplicity of an eigenvalue in a matrix whose graph is a tree , _ linear and multilinear algebra _ , 46 : 139144 , 1999 . c.r.johnson and a.leal duarte , on the possible multiplicities of the eigenvalues of a hermitian matrix the graph of whose entries is a tree , _ linear algebra and its applications _ , 348 : 721 , 2002 . c.r.johnson , a.leal duarte and c.m.saiago , inverse eigenvalue problems and lists of multiplicities for matrices whose graph is a tree : the case of generalized stars and double generalized stars , _ linear algebra and its applications _ , 373 : 311330 , 2003 . c.r.johnson and c.msaiago , estimation of the maximum multiplicity of an eigenvalue in terms of the vertex degrees of the graph of the matrix , _ electronic journal of linear algebra _ , 9 : 2731 , 2002 . c.r.johnson and b.sutton , hermitian matrices , eigenvalue multiplicities , and eigenvector components , _ siam journal of matrix analysis and applications _ , 26 : 390399 2004 . p.m.nylen , minimum - rank matrices with prescribed graph , _ linear algebra and its applications _ , 248 : 303316 , 1996 . j.h.sinkovic , the relationship between the minimal rank of a tree and the rank - spreads of the vertices and edges , _ masters thesis _ , brigham young university , 2006 .

let @xmath0 be an undirected graph on @xmath1 vertices and let @xmath2 be the set of all real symmetric @xmath3 matrices whose nonzero off - diagonal entries occur in exactly the positions corresponding to the edges of @xmath0 . the inverse inertia problem for @xmath0 asks which inertias can be attained by a matrix in @xmath2 . we give a complete answer to this question for trees in terms of a new family of graph parameters , the maximal disconnection numbers of a graph . we also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs . finally , we give an example of a graph that is not inertia - balanced , and investigate restrictions on the inertia set of any graph . _ keywords : _ combinatorial matrix theory , graph , hermitian , inertia , minimum rank , symmetric , tree _ ams classification : _ 05c05 ; 05c50 ; 15a03 ; 15a57

high - energy emission from gamma - ray bursts ( grbs ) has been expected and various theoretical possibilities have been discussed by numerous authors ( see e.g. , * ? ? ? * and references there in ) . in fact , egret detected several grbs with gev emission ( e.g. , * ? ? ? * ) . recently , the _ fermi _ satellite was launched and the onboard large area telescope ( lat ) is widely expected to detect high - energy ( @xmath0 gev ) emission from a fraction of grbs . in addition , other space- and ground - based gamma - ray observatories such as agile , magic , veritas and hess also regard grbs as one of the main scientific targets . theoretically there are the two main classes as high - energy emission mechanisms , i.e. , leptonic and hadronic mechanisms . the leptonic mechanisms include synchrotron self - compton ( ssc ) emission and external inverse - compton emission , which are the most discussed scenarios for both the prompt and the afterglow emission components . high - energy ssc emission is produced by relativistic electrons that radiate seed synchrotron photons ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition , there are various possibilities for external inverse - compton emission . for example , prompt gamma - ray photons or the x - ray flare photons may act as seed photons for the relativistic electrons accelerated during the afterglow phase in the external shocks ( e.g. , * ? ? ? * ; * ? ? ? the hadronic mechanisms include synchrotron radiation of high - energy baryons , synchrotron radiation of the secondary leptons generated in photohadronic interactions , as well as the photons directly produced from @xmath4 decays . in order to see the baryon synchrotron radiation , sufficiently strong magnetic fields are typically required ( e.g. , * ? ? ? * ; * ? ? ? * ) . otherwise , photohadronic components would dominate over the baryon synchrotron component as long as the photon density is high enough . hadronic gamma rays can be observed only when the nonthermal baryon loading is large enough ( e.g. , * ? ? ? * ; * ? ? ? so far , both emission mechanisms have been widely considered in the standard scenario ( see reviews , e.g. , * ? ? ? * ; * ? ? ? * ) , i.e. , the internal shock model for the prompt emission and the external shock model for the afterglow emission , respectively . both mechanisms can in principle produce @xmath5 tev photons , although high - energy photons may not escape from the source due to two - photon pair production , especially during the prompt emission phase @xcite . even if such super - tev photons can escape from the source , they still suffer from pair creation due to the interaction with the cosmic infrared background ( cib ) or the cosmic microwave background ( cmb ) . in particular , the direct detection of tev photons would be difficult for grbs with redshift @xmath6 . on the other hand , the electron - positron pairs resulting from the pair creation are still energetic , so that they up - scatter numerous cmb photons via the inverse - compton process . such secondary photons are able to reach the observer in a longer duration than the duration of primary emission , and a significant fraction of them may be observed with a time delay due to several effects such as magnetic deflection and angular spreading . therefore , this emission is called `` pair echo '' emission , with a typical energy in the range of @xmath7 gev . this pair echo emission is not only indirect evidence of the intrinsic tev emission but also a clue to probe the weak intergalactic magnetic field ( igmf ) of @xmath8 g @xcite . the plaga s method is hitherto the only one to probe very weak magnetic fields of @xmath8 g. other methods utilizing faraday rotation or cosmic microwave background are sensitive to magnetic fields of order @xmath9 @xcite . the presence of very weak igmfs has been predicted by several mechanisms , such as inflation ( e.g. , * ? ? ? * ) , reionization ( e.g. , * ? ? ? * ) and density fluctuations ( e.g. , * ? ? ? * ; * ? ? ? observations of igmfs in voids would give important information on the origin of the galactic magnetic fields @xcite , although they may be contaminated by astrophysical sources such as galactic winds or quasar outflows @xcite . in this paper , we reinvestigate the observational effects of the possible pair echo emission of grb high - energy emission in the afterglow phase . three criteria should be satisfied to detect pair echo emission : ( 1 ) the object must emit @xmath1 tev gamma rays leading to pair echoes ; ( 2 ) the pair echo flux must be higher than the detector s flux sensitivity ; and ( 3 ) the pair echo emission component must not be masked by other emission components . concerning the point ( 1 ) , tev photons from grbs can be emitted during both the prompt and the afterglow phases . here we consider both as the primary emission components for the echoes , by acknowledging that during the prompt phase strong tev gamma rays are expected only for a small fraction of grbs due to the large @xmath10 optical depth , as has been studied by various authors @xcite . concerning the point ( 2 ) , we need to evaluate the pair echo flux quantitatively . this flux depends on the amount of the cib photons , the igmf strength , and the source distance . as for the cib , we use the acceptable cib models given by kneiske et al . ( 2002 , 2004 ) . in order to take into account of the effects of the igmf properly , we adopt the formulation developed by ichiki et al . ( 2008 ) , which enables us to calculate the time - dependent spectra better than the previous works @xcite . in addition , we have also taken into account up - scatterings of the cib photons as well as the cmb photons . this effect was neglected in the previous work for simplicity @xcite , but it can be also important @xcite . in this work , we focus on the detectability of the _ fermi _ lat , which is the most suitable one for our purpose , but also touch upon the capabilities of other ground based tev detectors such as magic and veritas . concerning the point ( 3 ) , we pay special attention to the high - energy afterglow emission , which is the main competitor of the pair echoes , and compare the its strengths with respect to the echo components . such a comparison was not done for previous researchers who studied the pair echo . at present , a detailed comparison between the pair echoes and high - energy afterglows is highly uncertain , as both have never been clearly detected . since various predictions of high - energy emission rely on many model assumptions , they should be tested by observations of _ fermi _ , magic , veritas and other detectors . despite of these uncertainties , we think it would be interesting and important to study effects of pair echoes that can affect high - energy emission , especially in the late phase @xcite . for a typical long - duration grb , prompt gamma - ray emission is observed in a duration of @xmath11 s. the typical isotropic energy is around @xmath12 . the observed specific flux spectrum is well approximated by a broken power - law , @xmath13 for @xmath14 and @xmath15 for @xmath16 , where @xmath17 is the break energy which is typically @xmath18 kev . @xmath19 and @xmath20 are the low- and high - energy photon indices , respectively . in this work , we extrapolate this spectrum to higher energies and adopt @xmath21 for @xmath22 , where @xmath23 is the intrisic cutoff energy which is typically determined by the opacity of pair production . whether tev gamma rays can escape from the source strongly depends on the lorentz factor and the emission radius . only when these quantities are large , do we expect tev gamma rays escaping from the source , i.e. , @xmath24 tev . notice that although the ssc or possible hadronic mechanism leads to more complicated spectra ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , this simplification is sufficient for calculating the pair echo ( e.g. , * ? ? ? the pair echo is a kind of regenerated processes , which is composed of up - scattered cmb and cib photons . the resulting pair echo spectrum sensitively depends on the intrinsic cutoff energy , while it is not so sensitive to source electron spectral indices of @xmath25 for a given @xmath26 @xcite . when the intrinsic cutoff energy is low enough , the resulting spectrum basically reflects the seed cmb and cib spectra , the inverse - compton spectrum is expected as @xmath27 below the peak . however , the pair spectrum is strongly affected by the cib field , and is proportional to @xmath28 , where @xmath29 is the optical depth of photons with @xmath30 emitted at the redshift @xmath31 . since the pair echo spectrum is rather sensitive to the igmf and the cib spectrum , it is not easy to know a source electron spectral index @xmath32 . ] , which roughly leads to the spectral peak of @xmath33 . here , @xmath34 k is the local cmb temperature . on the other hand , when the intrinsic cutoff energy is high enough , high - energy secondary photons are re - absorbed , and the resulting spectrum has the cutoff due to cmb / cib absorption . as the intrinsic cutoff energy is higher , the cascade effect becomes more and more significant , i.e. , repeating the pair creation and inverse - compton scattering is important . it affects the resulting spectrum , erasing the memory of the primary spectrum in the high energies . rather , the radiation energy output above tev is important for the pair echo flux , and we normalize the primary flux through the isotropic radiation energy above 0.1 tev , @xmath35 . the prompt emission is followed by the afterglow phase , during which the relativistic ejecta is decelerated by a circumburst medium . a pair of external shocks ( forward and reverse ) form , from which electrons ( and possibly baryons ) are accelerated and radiate afterglow photons . high - energy emission during this phase was predicted by many authors in both of the reverse and forward shock models . ( see * ? ? ? * and references there in ) . tev emission in the external shocks has a smaller optical depth for pair production , and hence , can escape the source more easily . for the forward shock , the characteristic energies for the ssc emission are given by ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) @xmath36 where @xmath37 and @xmath38 are the fractions of the shock energy transferred to the downstream magnetic fields and nonthermal electrons , respectively . @xmath39 is a numerical factor , which is expressed as @xmath40 for @xmath41 and the typical value for @xmath42 is @xmath43 . @xmath44 is the isotropic kinetic energy of the ejecta , @xmath45 is the circumburst medium density , and @xmath46 is the compton parameter . for @xmath47 , we roughly have is expected when only the first ssc component is important . in fact , the second ssc component is typically negligible due to the klein - nishina suppression in the optically thin synchrotron scenario . ] @xmath48 ( e.g. , * ? ? ? * ; * ? ? ? * ) , and the high - energy emission spectrum is written as @xmath49 for @xmath50 , @xmath51 for @xmath52 and @xmath53 for @xmath54 , where @xmath55 is the spectral index of the accelerated electrons . here @xmath56 is the cutoff energy determined either by the pair - creation opacity or the klein - nishina limit ( e.g. , * ? ? ? * ) . the energy flux at the ssc peak ( for @xmath42 ) is evaluated as @xmath57 by which we can normalize the ssc spectrum . the above temporal behavior is typically valid from the break time of @xmath58 s to the next break time of @xmath59 s during the so - called normal decay phase of x - ray afterglow . afterglow light curves of some grbs are steepened after @xmath60 , which is often interpreted as a jet break when the lorentz factor @xmath61 becomes the inverse of the jet opening angle @xmath62 ( rhoads 1999 ; sari et al . the temporal behavior after the jet break @xmath60 is expected as @xmath63 , @xmath64 , @xmath65 and @xmath66 . the afterglow behavior before @xmath67 can not be interpreted by the standard afterglow model . as observed by _ swift _ , a good fraction of x - ray afterglow has a shallow decay phase lasting from @xmath68 s ( at which the shallow decay emission becomes dominant in x rays ) to @xmath69 s ( see , e.g. , * ? ? ? * ; * ? ? ? * ) , which has a decay slope of @xmath70 . several models have been proposed for explaining this phase ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and one of the mostly discussed interpretations is continuous energy injection into the forward shock . here we consider the modified forward shock model with the energy injection of the form @xmath71 , where @xmath72 parameterizes the energy injection and @xmath73 corresponds to the case of no energy injection . such modified forward shock models are supported by the lack of spectral evolution across @xmath67 and the compliance of the closure relations " in the normal decay phase after @xmath67 @xcite . during this phase , the temporal behavior of various parameters are @xmath74 , @xmath75 , @xmath76 and @xmath77 @xcite . we have calculated the high energy light curves of the ssc emission during this phase . similar calculations were performed by e.g. , gou & mszros ( 2007 ) , wei & fan ( 2007 ) , and fan et al . ( 2008 ) . pair echoes are the up - scattered cmb and cib photons by the electron - positron pairs produced via the attenuation of the primary tev photons by the cib . for a given primary spectrum , the total fluence of the pair echo emission is determined by the @xmath10 optical depth of the cib , and does not depend on the igmf as long as the deflection angle is much smaller than the jet opening angle . primary photons with energy @xmath30 are converted to pairs with lorentz factor @xmath78 in the local cosmological rest frame , which then up - scatter cmb and cib photons . cmb photons are boosted to energies @xmath79 gev . to evaluate the pair echo flux , we must consider various time scales involved in the process , such as the angular spreading time , and the delay time due to magnetic deflections ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? these can be estimated as follows @xcite . the angular spreading time is @xmath80 , where @xmath81 is the local @xmath82 mean free path in terms of the local cib photon density @xmath83 , and @xmath84 is the local ic cooling length in term of the local cmb energy density @xmath85 . at the energies of our interest , @xmath86 so that @xmath87 . for sufficiently small deflections in weak igmfs with the present - day amplitude @xmath88 and coherence length @xmath89 , the magnetic deflection angle is @xmath90 $ ] , where @xmath91 is the larmor radius of the electrons or positrons in murase et al . the minimum " is correct rather than the maximum " . the calculations were performed properly . ] . the delay time due to magnetic deflection is @xmath92 . for coherent magnetic fields with @xmath93 , we have @xmath94 $ ] . note that the deflection angle due to successive ic scattering @xmath95 is usually very small , where @xmath96 is the number of scatterings and @xmath97 is the ic scattering mean free path . we have also assumed that both @xmath98 and @xmath99 do not exceed @xmath100 ; otherwise a significant fraction of photons or pairs will be deflected out of the line of sight and the echo flux is greatly diminished . in order to calculate the pair echo flux , we adopt the formalism developed by ichiki et al . ( 2008 ) , which enables us to calculate the time - dependent spectra in a more satisfactory manner , particularly at late times , accounting properly for the geometry of the pair echo process . in previous works , explicit descriptions of the time - dependent spectra were not possible without some ad hoc modifications @xcite . in this section , we present our results and compare the pair echo emission with the afterglow emission . the detectability by the _ fermi_/lat detector and the ground - based magic telescope are also discussed . one main uncertainty stems from the cib models , which can affect not only the pair echo fluence but also the time scales for angular spreading and magnetic deflection at all redshifts . recent high - energy observations of tev blazars point to a low - ir cib model , close to the lower limit from the galaxy count data ( e.g. , * ? ? ? * ) ( but see , e.g. , stecker & scully 2008 ) . hence , we here adopt the low - ir cib model presented by kneiske et al . ( 2002 , 2004 ) . more detailed discussion on the effects of the cib is found in murase et al . ( 2007 ) . as for the afterglow parameters in the forward shock model , we adopt @xmath101 ergs , @xmath102 , @xmath103 , @xmath104 and @xmath105 . we also assume the energy injection index @xmath106 before @xmath107 s , and take the jet break time as @xmath108 s. ergs and @xmath109 , plotted at @xmath110 s ( blue ) , @xmath111 s ( green ) and @xmath112 s ( red ) , for the case of @xmath113 g , @xmath114 mpc , and @xmath115 . fermi_/lat and magic ii sensitivities ( with the duty factor of 20 % ) are also overlayed ( carmona et al . note that the sensitivity curves in the sky survey mode are used for the long time observations , although the possible continuous observations by lat may improve the detectability by a factor of 3 - 5 ( e.g. , gou & mszros 2007 ) . ] ergs and @xmath109 , compared with the lat sensitivity at 1 gev ( thick ) and 10 gev ( thin ) , for the case of @xmath113 g with with @xmath114 mpc and @xmath116 g with @xmath117 kpc . the source redshift is @xmath115 . ] ergs and @xmath118 . ] ergs , for the cases of @xmath109 and @xmath118 , respectively . light curves at 1 gev ( thick ) and 10 gev ( thin ) are shown for the case of @xmath119 g with @xmath114 mpc . the source redshift is @xmath115 . ] in figs . 1 and 2 , we show the resulting spectra and light curves of the afterglow - induced pair echo and the primary afterglow emission . we can see that the echo component is out - shined by the afterglow component during the shallow and normal decay phases . this result is consistent with ando ( 2004 ) , who argued that observed emission is unaffected by the pair echo . the situation changes dramatically after the jet break . the pair echo emission lasts for a long time because of the igmf deflection of the pairs , and it can dominate the afterglow after the jet break by as much as an order of magnitude . it can be observed only for nearby grbs with @xmath120 for our afterglow parameters . if a grb is very nearby and energetic , we may detect many photons at @xmath1 gev energies and even observe tev photons during the afterglow phase . in such a case , in principle a non - detection of the high energy pair echo would allow us to obtain the lower limit on the igmf . this is because if @xmath121 one would expect an excess of the echo flux @xmath122 over the primary flux @xmath123 . the non - detection of the echo emission can then be attributed to the effect of a finite igmf , which deflects the secondary pairs to reduce the secondary echo flux to be @xmath124 , where @xmath125 is the detector sensitivity @xcite . the expected lower bound with our afterglow parameters ( @xmath126 ergs and @xmath109 ) for a grb with @xmath115 is estimated as @xmath127 } > { 10}^{-21}~{\rm g}~{\rm mpc}^{1/2}.\ ] ] in general the result depends on the source distance and the afterglow parameters which should be determined from observational properties . in any case , the expected lower bounds are comparable to those derived for blazar flares @xcite . similar to the case of blazar flares , one expects that whether the afterglow pair echo dominates over the primary emission depends on the high - energy afterglow spectrum . in figs . 3 and 4 , we show the case of @xmath118 , corresponding to @xmath128 . obviously , such steeper indices make it more difficult to see the afterglow - induced pair echo emission . this is just because steeper indices imply the smaller tev flux compared to the gev flux as for the afterglow emission . hence , the electron spectral index is one of the uncertainties that are closely relevant to whether the afterglow - induced pair echoes are detectable . also , it is clear that brighter afterglows are favorable for detections . since the pair echo can be dominant over the afterglow itself only after the jet break , we need to observe kind of energetic afterglows with @xmath129 ergs for @xmath115 ( see figs . 1 - 4 ) . s ( blue ) , @xmath111 s ( green ) and @xmath112 s ( red ) , for the case of @xmath113 g with @xmath114 mpc . fermi_/lat and magic ii sensitivities ( with the duty factor of 20 % ) also plotted for comparison . the prompt emission spectrum at @xmath130 s is shown , with @xmath131 ergs assumed . the canonical afterglow spectrum is also shown for the case of @xmath126 ergs and @xmath109 , the source redshift is @xmath115 . ] g and @xmath116 g with @xmath117 kpc . here @xmath132 ergs is assumed . the source redshift is @xmath115 . ] s ( blue ) , @xmath111 s ( green ) and @xmath112 s ( red ) , for the case of @xmath113 g with @xmath114 mpc . fermi_/lat and magic ii sensitivities ( with the duty factor of 20 % ) also plotted for comparison . the prompt emission spectrum at @xmath130 s is shown , with @xmath133 ergs assumed . the canonical afterglow spectrum is also shown for the case of @xmath134 ergs and @xmath109 , the source redshift is @xmath115 . in order to demonstrate the effect of up - scattered cib ( uscib ) photons ( solid ) , curves without up - scattering of cib photons are also shown ( dot - dashed ) . ] g and @xmath116 g with @xmath117 kpc . here the relevant parameters of the prompt emission and afterglow are the same as those used in fig . 7 . the source redshift is @xmath115 . ] in figs . 5 and 6 , we show the resulting spectra and light curves of the prompt - induced pair echo . the parameters for the primary prompt emission are taken as the following : @xmath131 ergs , @xmath135 and @xmath136 tev . the duration in the local rest frame is set to @xmath137 . for comparison we also show the afterglow spectra / light curves . we notice that the prompt - induced pair echo has been discussed by several authors before , but the comparison with the afterglow flux was never done previously . we find that the pair echo is observable only when grbs are strong tev emitters , i.e. @xmath138 ergs for our afterglow parameters ( where @xmath139 ergs ) . this is a strong requirement for the grbs with canonical afterglows . for weak but non - zero igmfs , the pair echo lasts for a longer time although its maximum flux is lower than the case of @xmath140 . then , the echo could still dominate over the afterglow at late times after the jet break , since its light curve is shallower than that of the afterglow . in figs . 7 and 8 , we show the more optimistic cases where brighter prompt emission and dimmer afterglow emission are assumed . in those cases , the observed behavior of high - energy afterglows is quite different from the predicted one from the afterglow theory , since the pair echo emission is dominant for a long time . a weak but non - zero igmf with @xmath141 g can even make the pair halo out - shine the shallow decay emission . in figs . 7 and 8 , we also show the effect of up - scattered cib photons . as is easily seen , their effect is important at high energies above 10 - 100 gev , which can be crucial for detections through the magic and veritas telescopes . note that this effect becomes important when the intrinsic cutoff energy is not so high , as pointed out in murase et al . otherwise , the up - scattered cib component is masked by the up - scattered cmb component . in fact , it is typically difficult to see the former for afterglow - induced pair echoes , where the pair echo spectrum at @xmath142 is composed of the up - scattered cmb photons produced by the primary photons emitted at different times from the source . similar to what has been discussed in the previous subsection , one may obtain the lower bound on the igmf for non - detection of the prompt - induced pair echo . however , the relative importance of the prompt - induced pair echo with respect to the afterglow emission is complicated , which strongly depends on the ratio of the prompt tev emission energy and the electron energy in the afterglow ( @xmath143 ) . in addition , the afterglow - induced pair echo would also contaminate the prompt - induced pair echo . here , for a conservative estimate , let us consider the epochs of @xmath144 . assuming that tev emission is detected , a non - detection of the pair echo would lead to @xmath127 } > { 10}^{-19.5}~{\rm g}~{\rm mpc}^{1/2},\ ] ] for our prompt and afterglow parameters used in fig . 7 . s ( blue ) , @xmath111 s ( green ) and @xmath112 s ( red ) , for the case of @xmath113 g with @xmath114 mpc . fermi_/lat and magic ii sensitivities ( with the duty factor of 20 % ) also plotted for comparison . the prompt emission spectrum at @xmath130 s is also shown , with @xmath145 ergs assumed . the source redshift is @xmath115 . ] g and @xmath116 g with @xmath117 kpc . here @xmath146 ergs is assumed . the source redshift is @xmath115 . ] as seen in the previous subsection ( see figs . 5 and 6 ) , afterglow emission may significantly mask a pair echo ( for both of long and short grbs ) . hence , of special interest are the grbs whose intrinsic high energy afterglow emission is weak and whose prompt tev emission is strong . since almost all the long grbs accompany afterglows , the possible candidates of such bursts are likely to be a fraction of short grbs that do not show conventional x - ray afterglows ( only show a steep decay phase as the tail of prompt emission spectrum ) . in fact , @xmath147 of short grbs ( e.g. , grb 050906 , 051210 , 070209 , 070810b and 080121 ) are such naked " bursts maybe due to the low density of the circumburst medium ( e.g. , la parola et al . 2006 ) . since these bursts are spectrally hard and less energetic ( than their long brethren ) , they may have prompt emission extending to the tev range ( e.g. , gupta & zhang 2007 ) . these bursts could therefore be the best targets to detect the pair echoes or to use non - detections to constrain the igmf . in figs . 9 and 10 , we show the resulting spectra and light curves of the prompt - induced pair echo from a nearby , rather energetic short grb . the parameters for the primary prompt emission are taken as the following : @xmath145 ergs , @xmath135 and @xmath136 tev . the duration is set to @xmath148 . for naked grbs , we expect that the primary emission decays according to the curvature effect , which typically drops as @xmath149 . for instance , when @xmath150 during the burst , we have @xmath151 at @xmath152 s. hence , we omit the afterglow spectra / light curves in figs . 9 and 10 . as is seen in fig . 10 , the igmf of @xmath153 leads to the detectable flux at @xmath154 , which should be observed as extended high - energy emission from short grbs . note that , when @xmath155 g , the pair echo duration is determined by the angular spreading time , @xmath156 . therefore , it may typically be difficult for pair echoes to explain gev emission whose time scale is shorter ( e.g. , grb 081024b and see also discussions in zou , fan , & piran 2008 ) , but they may also generate the high - energy extended emission . for non - detections , one may obtain a constraint as @xmath157 } > { 10}^{-21.5}~{\rm g}~{\rm mpc}^{1/2},\ ] ] for our optimistic prompt parameters . we need to observe primary tev emission for this purpose , but it is more difficult to make follow - up observations for short grbs with magic and veritas , compared to long grbs . note that significant and non - tentative tev signals have not been observed so far for both of the long and short grbs @xcite . this may be because a part of grbs can be tev emitters due to the small optical thickness for pair creation and tev photons from distant sources are significantly attenuated by the cib . in this paper , we have calculated the time - dependent spectra of the secondary pair echoes from the grb prompt and afterglow tev emission components that are attenuated by the cib , applying a recently developed formalism to properly describe the temporal evolution of the pair echoes . we have compared the flux of the pair echoes to that of the afterglow , taking into account up - scattering of the cib photons . in particular , we have demonstrated ( 1 ) that afterglow - induced pair echoes can be important after the jet break for long grbs with a canonical afterglow ; and ( 2 ) that prompt - induced pair echoes may also outshine the afterglow emission , if the prompt tev emission is intense , typically with @xmath158 . weak but non - zero igmfs can be crucial for detectability , since they make the duration of the pair echo emission much longer than the time scale of primary emission ( see figs . 2 , 4 , 6 , and 8) . although the detectability itself also depends on both of the spectral evolution of the primary emission and detector sensitivities , such non - zero igmfs can make it easier to detect secondary photons at late times when the pair echo emission remains shallow compared to the afterglow emission . concerning with the detection of pair echo signals , naked " ( short ) grbs without a significant afterglow emission could be more promising . the pair echo should be observed as extended emission with the time scale of @xmath159 s. the observational prospects of such pair echoes are quite interesting for the recently launched _ successful detections may be possible for nearby , bright events , and would open a new window to study the poorly unknown igmf . even in the case of non - detections , lower limits on the igmf of @xmath160 } \sim { 10}^{-20}-{10}^{-21}~{\rm g}~{\rm mpc}^{1/2}$ ] may be obtained . the main caveat in hunting afterglow - induced pair echoes and pair echoes from short grbs is that nearby bright grbs do not seem frequent . although there is large uncertainty on the nearby burst rate , the rate of bursts occurring within @xmath161 is estimated as @xmath1 a few events per year ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the actual detection rate also depends on several factors such as the detector sensitivity and field of view ( e.g. , @xmath162 sr for the _ fermi_/lat detector ) , so that only a fraction of them would be detected . if all the bursts are ideal tev emitters , we can expect pair echoes for these bursts in the near future . however , it is unlikely that all the bursts are bright tev emitters ( and it seems more plausible for prompt emission due to significant attenuation by the pair creation ) . although it is currently impossible to predict how many bursts can be bright tev emitters in both the prompt and afterglow phases , the expected detection rate for @xmath163 bursts would be at most " @xmath164 events per year . there may be further complications about nearby grbs . some of the nearby long bursts detected so far seem somewhat dimmer than classical grbs occurring at @xmath6 , but their local rate may be higher than the estimated local rate of classical grbs ( e.g. , * ? ? ? * ; * ? ? ? hence , we may have more nearby bursts that can be detected in the kev - mev band by detectors with better sensitivities ( e.g. , _ exist _ ) . but , since the typical luminosity of such low luminosity bursts seems small , it is not so easy to see pair echoes from them . in addition , energetic short grbs assumed in figs . 9 and 10 would also be rare , whose radiation energy is larger than the typical one ( @xmath165 ergs ) . nevertheless , possible detections of pair echoes would bring us a big impact in understanding grb physics and igmf , even though the bright tev grbs that can lead to such detections are rare the current on - orbit _ fermi _ satellite is suitable for such a purpose . magic and veritas can also provide valuable data via follow - up observations , since the pair echo emission can last for a long duration of time . in the near future , some constraints on the models may be achieved even for non - detections . we must also beware of the uncertainties in the intrinsic primary spectra since the pair echo flux depends on the amount of tev photons . as for afterglow emission , we only consider the conventional forward shock model with energy injection . although other parameter sets or other models such as the varying @xmath38 model can be considered , we expect that the qualitative features of the pair echoes themselves will not be changed significantly , as long as the light curve of high - energy emission is similar to that of x - rays and the amount of tev photons is not too different from that invoked in our case . as for the prompt emission , possible uncertainties may come from the intrinsic emission properties such as @xmath26 , as discussed in murase et al . ( 2007 ) . the contamination by other high - energy emission components might complicate the picture further . there are many possibilities of high - energy gamma ray emission during the afterglow phase ( see , e.g. , * ? ? ? * ; * ? ? ? * and references therein ) . for example , high - energy emissions associated with x - ray flares are expected at @xmath1 gev energies . gev photons can be produced by both of the leptonic mechanisms ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the hadronic mechanisms @xcite . in addition , the reverse shock electrons can also provide high - energy photons during the early afterglow phase . nonetheless , it is in principle possible to distinguish the pair echo emission from other possibilities , given an ideal broad - band ( optical , x - ray , mev and gev ) observational campaign . km and kt are supported by a grant - in - aid for the jsps fellowship . bz acknowledges nasa nng05gb67 g , nnx08an24 g , and nnx08ae57a for support . sn is supported in part by grants - in - aid for scientific research from the ministry of e.c.s.s.t . 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high - energy emission from gamma - ray bursts ( grbs ) is widely expected but had been sparsely observed until recently when the _ fermi _ satellite was launched . if @xmath0 tev gamma rays are produced in grbs and can escape from the emission region , they are attenuated by the cosmic infrared background photons , leading to regeneration of @xmath1 gev - tev secondary photons via inverse - compton scattering . this secondary emission can last for a longer time than the duration of grbs , and it is called a pair echo . we investigate how this pair echo emission affects spectra and light curves of high energy afterglows , considering not only prompt emission but also afterglow as the primary emission . detection of pair echoes is possible as long as the intergalactic magnetic field ( igmf ) in voids is weak . we find ( 1 ) that the pair echo from the primary afterglow emission can affect the observed high - energy emission in the afterglow phase after the jet break , and ( 2 ) that the pair echo from the primary prompt emission can also be relevant , but only when significant energy is emitted in the tev range , typically @xmath2 . even non - detections of the pair echoes could place interesting constraints on the strength of igmf . the more favorable targets to detect pair echoes may be the naked " grbs without conventional afterglow emission , although energetic naked grbs would be rare . if the igmf is weak enough , it is predicted that the gev emission extends to @xmath3 s. [ firstpage ] gamma rays : bursts magnetic fields radiation mechanisms : nonthermal

in recent years , a concept of perpendicular magnetic recording has been introduced in order to produce a new generation of high - density storage devices . l1@xmath11-ordered intermetallic magnetic alloys , such as equiatomic fepd , fept and copt , are the most promising candidates for this purpose , due to their uniaxial magnetic anisotropy in [ 001 ] direction @xcite . a possible industrial application of these materials strongly depends on the level of complication in producing ( 001)-textured and chemically well - ordered alloy . the perfect ordered single crystal bulk fepd alloy has aucu - i structure type ( space group p4/mmm ) @xcite . in such a structure fe atoms are placed in @xmath12 and @xmath13 wyckoff sites ( @xmath14 and @xmath15 positions in the unit cell ) , and pd occupy @xmath16 sites ( @xmath17 and @xmath18 positions ) . the lattice parameters of bulk alloy are @xmath19 and @xmath20 , which result in small lattice distortion in [ 001 ] crystallographic direction ( axial ratio @xmath21 ) . in order to obtain l1@xmath11-ordered alloy it is necessary to tune the composition to approx . @xmath22 at.% of both components @xcite . the most popular method of obtaining the l1@xmath11-ordered and ( 001)-oriented fepd thin alloy films is the epitaxial growth . however , from the technical point of view , due to the specific growth conditions , this process is not convenient for mass production of the data storage devices . the other method for fepd alloy formation is the alternating deposition or codeposition of the constituent materials . then , in order to produce alloys with l1@xmath11 structure and proper magnetic properties , the as deposited systems are annealed by various post - deposition thermal processes . crystallographic texture and chemical order could be induced by rapid thermal annealing @xcite , conventional long time annealing @xcite , or ion beam irradiation @xcite . however , a lot of attention was also paid to investigate different approaches , such as annealing in high magnetic field @xcite , combining ion irradiation and thermal treatment @xcite , and growing films on heated substrates @xcite . it was found that the addition of various impurities to the l1@xmath11 system can facilitate the formation of an ordered alloy , and many papers consider the dependence of chemical order , crystallite sizes , and magnetic properties on impurity additives . the formation of chemically ordered l1@xmath11 phase of fept alloys doped by cu @xcite , ag @xcite , and c @xcite was observed . also , the influence of zr , w , ti and tio@xmath6 or ta@xmath6o@xmath4 additions on fept alloy structure was reported in @xcite , but there have been only few reports @xcite on the doped l1@xmath11-fepd thin films or nanoparticles so far . the definition of the chemical order parameter for binary alloys with l1@xmath11 crystallographic structure can be found in @xcite . this approach was successfully applied in studies concerning epitaxial growth of binary alloys @xcite , as well as in investigations of ion irradiation induced ordering in polycrystalline films @xcite or l1@xmath11-ordered nanoparticles @xcite . however , the definition of chemical order parameter introduced in these papers is correct only for binary alloys . in case of ternary alloys we have to take into account the problem of site occupation by dopant atoms @xcite . in this paper , we present the results of the studies on crystallographic texture and chemical order , carried out on fepd thin alloy films with @xmath0 at.% of copper , investigated by x - ray diffraction and pole figure measurements . the initial cu / fe / pd multilayers were deposited on four different substrates , and then transformed into the alloy by application of two thermal annealing procedures . since the evaporation and annealing conditions were kept constant , it was possible to determine the influence both of substrate type as well as applied thermal treatment on the crystallographic properties of the fepd : cu alloys . in order to obtain chemical order degree and quantitative information about crystallographic texture , we have introduced the new approach , which takes into account the problem of lattice site occupation by cu addition atoms . it was found that these two quantities are dependent on the substrate type . furthermore we will show qualitative analysis of the crystallographic texture which is also substrate - dependent . the multilayer systems were prepared in an ultra - high vacuum chamber by thermal evaporation at working pressure in the range of @xmath23 pa . to provide @xmath24 stoichiometry between fe and pd atoms samples had [ cu(@xmath1 nm)/fe(@xmath2 nm)/pd(@xmath3 nm)]@xmath4 composition . the reference samples with composition [ fe(@xmath2 nm)/pd(@xmath3 nm)]@xmath4 were also prepared . the single layers were deposited sequentially and the thickness of each layer was monitored _ in - situ _ using quartz crystal microbalance . multilayer systems were evaporated on four substrates : polished mgo(100 ) , si(100 ) , si(111 ) and si(100 ) covered by @xmath5 nm thick layer of amorphous sio@xmath6 ( further denoted as si(100)/sio@xmath6 ) . before the deposition , the substrates were ultrasonically cleaned in acetone and ethanol and rinsed in deionised water . samples were deposited at room temperature , and the evaporation rates were @xmath25 nm / min for fe and pd layers and @xmath1 nm / min for cu layers . the chemical composition was checked _ ex - situ _ by rutherford backscattering ( rbs ) . nominal thicknesses of the single layers as well as total thickness of multilayer systems were confirmed _ ex - situ _ by x - ray reflectivity ( xrr ) measurements . xrr measurements also showed , that on the top of the si(100 ) and si(111 ) substrates the layer of native silicon oxide with thickness of about @xmath26 nm was formed . all samples initially had size of @xmath9 cm per @xmath9 cm . after the rbs and xrr measurements the samples were cut into four pieces , and two of them were annealed with different annealing procedures . in order to transform the initial multilayer system into the ordered alloy , the rapid thermal annealing method ( further denoted as rta ) the samples were annealed in an atmosphere of flowing nitrogen at @xmath7c for @xmath8 s , with a heating rate of @xmath27c / s . this procedure resulted in creation of the fepd : cu alloy . it should be noted that due to the small size of sample ( 5 mm per 5 mm ) the turbulences of flowing gas at the border of the samples could create the temperature gradient between the edge of the sample and its center . in addition to the rta annealing the conventional long annealing ( called la ) was applied at @xmath10c for @xmath9 hour in vacuum of @xmath28 pa . the heating rate was @xmath29c / min . the x - ray diffraction ( xrd ) experiments in @xmath30/@xmath31 geometry were performed on cristal beamline at soleil synchrotron ( france ) . the soleil synchrotron storage ring has operated at @xmath32 gev electron beam energy with a beam current of @xmath33 ma . at the cristal beamline the in - vacuum u20 undulator insertion device was used as the source of x - ray radiation . the beamline was equipped with set of double - si(111 ) single crystal flat monochromators with sagittal focusing . for the purpose of our measurements , x - ray radiation with energy of @xmath34 kev was used , which corresponds to the wavelength of @xmath35 nm . at this wavelength , the longitudinal coherence length given by the @xmath36 expression was equal to @xmath37 @xmath38 m . the horizontal and vertical size of the beamline spot was about @xmath39 @xmath38 m per @xmath5 @xmath38 m . for data collection , the proportional counter was used . all the measurements were carried out with angle @xmath31 changing from @xmath29 to @xmath40 , the instrumental step of angle @xmath31 was @xmath41 with the counting time @xmath9 s per step . the pole figures were measured using panalytical xpert pro laboratory diffractometer , equipped with three - circle euler cradle . the cu @xmath42 radiation operated at @xmath43 kv , and @xmath44 ma was used . the optics of incident beam consisted of @xmath45 divergence slit , parabolic graded w / si x - ray mirror with an equatorial divergence of less than @xmath41 , @xmath46 rad soller slit collimator , and @xmath47 mm mask for restricting the axial width of incident beam . the diffracted signal was collected by solid state stripe xcelerator detector with graphite monochromator working in receiving slit mode . the diffracted beam path was also equipped with @xmath48 mm high antiscatter slit and @xmath46 rad soller slit collimator . the pole figures were measured using regular @xmath49 x @xmath49 grid in polar @xmath50 and azimuthal @xmath51 angles , with @xmath50 changed from @xmath52 to @xmath53 and @xmath51 from @xmath52 to @xmath54 . the counting time was set for @xmath8 s per step . the xpert pro diffractometer was also used for xrd measurements of fepd alloy on si(100)/sio@xmath6 . all the hardware settings were the same as for pole figure measurements . the @xmath31 range was choosen in a way corresponding to the range measured by synchrotron radiation . the stripe detector was working in scanning mode with active length of @xmath55 . instrumental step was @xmath41 and the counting time was @xmath56 s per point . all the xrd and pole figures measurements were carried out at ambient conditions . the xrd patterns , collected for fepd and fepd : cu alloys after rta and combined rta+la treatment , are presented in fig . [ xrd_patterns ] . the positions of the ( 001 ) , ( 111 ) and ( 002 ) reflections for l1@xmath11-ordered bulk fepd alloy are marked in the figure with vertical dashed lines . the positions of the ( 001 ) , ( 111 ) and ( 002 ) reflections for the l1@xmath11-ordered bulk fepd alloy are indicated with vertical dashed lines . the mgo and the si substrate reflections are also indicated . all patterns were corrected for instrumental background and normalized to the intensity of the maximum intense sample reflection.,width=321 ] for fepd alloys only two reflections are observed in the patterns : the strong one at angle @xmath57 and very weak at @xmath58 . in all patterns collected for fepd : cu alloys , the set of three bragg reflections at angles @xmath59 , @xmath60 and @xmath61 are observed . the positions of the first and third peak are shifted towards higher angles in comparison the the bulk reflections , but the position of the second peak is the same as the position of bulk reflection . additionally , for fepd : cu alloy on si(100)/sio@xmath6 substrate , after rta , a strong reflection at @xmath62 appeared . it is readily observed that the intensity of the reflection at the same angular position in different samples is different and depends on sample type and annealing treatment . changes of the integral intensities ratios between different reflections will be discussed in detail later . by comparing the bulk reflection positions and the positions of the observed peaks we can identify the reflections observed at angles @xmath31 @xmath63 , @xmath60 , and @xmath61 as coming from ( 001 ) , ( 111 ) and ( 002 ) crystallographic planes of the l1@xmath11-ordered alloys . such result directly suggests the presence of only [ 111]-oriented crystallographic grains in fepd alloys and [ 001 ] and [ 111 ] orientations in case of fepd : cu alloys . this is true for all cases , except of fepd : cu alloy deposied on si(100)/sio@xmath6 substrate , which exhibited additional ( 200 ) peak at @xmath64 . the presence of mostly [ 001 ] and [ 111]-oriented crystallites in case of fepd : cu alloys is related to the impact of two main driving forces . the cu addition , as well as the strain - inducing rta annealing , introduce the grain orientation with c - axis perpendicular to the substrate plane . on the other hand , in metals and alloys with fcc or fct structure , the ( 111 ) plane is densly packed plane and has the smallest surface energy . therefore , due to surface energy minimalization , the crystallites will tend to orient with this plane by being aligned parallel to surface . the lack of crystallites with [ 110 ] or [ 101 ] orientations is related to the fact that they have larger surface energy than ( 111 ) , ( 001 ) or ( 100 ) planes and , consequently , are not energetically favoured . in case of fepd alloys the lack of cu addition results in creation of [ 111]-oriented crystallographic grains according to the surface energy minimalization rule . in order to obtain information about the values of lattice parameters @xmath65 and @xmath66 , the integral intensities @xmath67 and full width at half maximum ( @xmath68 ) of the ( 001 ) , ( 111 ) and ( 002 ) reflections , xrd data were fitted using the sum of pseudo - voigt line profiles . next , the values of lattice parameters @xmath65 and @xmath66 were calculated , based on the precise peak positions , from the standard definition of interplanar spacing @xmath69 . for fepd alloys values of lattice parameters @xmath65 and @xmath66 and axial ratio @xmath70 are very close to the bulk values . the obtained values of lattice parameter @xmath66 and axial ratio @xmath70 for all fepd : cu samples are shown in fig . [ lattice_parameters ] . ( full symbols , left scale ) and @xmath70 axial ratios ( open symbols , right scale ) calculated from the xrd patterns . the values of lattice parameter @xmath66 for alloys on si(100 ) and sio@xmath6 are the same after both annealing procedures . the bulk values of lattice parameter @xmath66 and axial ratio @xmath70 are marked with horizontal dashed lines ( upper and lower lines , respectively).,width=321 ] the shift of ( 001 ) and ( 002 ) peak positions towards higher angles is reflected by the smaller values of lattice paramater @xmath66 in comparison to bulk value . the lack of significant changes in ( 111 ) peak positions results in larger values for the lattice parameter @xmath65 than for the bulk material . the increase in values , together with simultaneous decrease of @xmath66 values , leads to larger tetragonal lattice distortion , visible as smaller values of axial ratios @xmath70 than for bulk material . the similar observation of lattice parameter changes was already reported in studies concerning cu - doped fept thin alloy films @xcite . it is worth noticing that the lattice parameter values change in such a way that the crystallographic @xmath71-spacing remains always the same and is the same as the bulk value . for fepd alloys the @xmath72-spacing does not change since the lattice parameters have the same values as for bulk . in case of fepd : cu alloys the lack of changes in the @xmath71-spacing can be also explained by considering the energetic conditions . as it was mentioned before the ( 111 ) plane is a densely packed plane and has the lowest surface energy . the cu addition to the fepd alloy causes reduction of the lattice parameter @xmath66 and disturbs the energetic equilibrium of the fepd l1@xmath11 structure . on the other hand , the structure tends to energy minimum , which is related to the constant value of @xmath71-spacing . therefore , the lattice compression in [ 001 ] direction must be balanced by stretching in the [ 100 ] and [ 010 ] directions . the l1@xmath11 ( 001 ) plane has the fourfold symmetry , so the stretching is equal in both directions and reflects in the expansion of the lattice parameter @xmath65 . taking into account the definition given by warren @xcite , the long - range order parameter @xmath73 for alloys with l1@xmath11 crystallographic structure can be expressed as corrected squared ratio between integral intensities of superstructure ( 001 ) and fundamental ( 002 ) bragg reflections : @xmath74 where @xmath75 are structure factors , @xmath76 are measured integral intensities , and @xmath77 , @xmath78 , @xmath79 and @xmath80 are geometry , lorentz , polarization and absorption factors , respectively ( exact expressions are presented in @xcite ) . the factors @xmath77 , @xmath78 and @xmath79 depend only on angle @xmath30 , but factor @xmath80 depends also on linear absorption coefficient @xmath38 . the value of @xmath38 was calculated as the weighted sum of the absorption coefficients of fe , pd and cu elements , with weights of their percentage mass contributions @xcite . the order parameter @xmath73 changes from @xmath81 for the lack of chemical order to @xmath9 for perfect chemical order in l1@xmath11 alloy . since the fepd alloys exhibit only ( 111 ) reflection the forthcoming analysis will be carried out only for fepd : cu alloys . the structure factors @xmath82 and @xmath83 in l1@xmath11 structure are following : @xmath84 and @xmath85 where @xmath86 , @xmath87 , @xmath88 and @xmath89 are atomic form factors associated with atoms placed in @xmath12 , @xmath13 and two @xmath16 wyckoff sites in the unit cell . for perfect l1@xmath11-ordered fepd alloy , the @xmath12 and @xmath13 positions are occupied by fe atoms , and in @xmath90 and @xmath91 sites pd atoms are placed . however , in case of ternary fepd : cu alloy it is necessary to consider where copper atoms are located in the alloy structure . in our previous studies , about the local structure of the fepd : cu alloy @xcite , it was shown that cu atoms substitute both fe and pd sites . with the assumption of a random distribution of copper atoms in l1@xmath11 structure the expressions for structure factors change to : @xmath92 \\ \nonumber - \left [ \left ( 0.9 f^{\mathrm{pd}}_{2e'}+0.1 f^{\mathrm{cu}}_{2e ' } \right)+\left ( 0.9 f^{\mathrm{pd}}_{2e''}+0.1 f^{\mathrm{cu}}_{2e '' } \right ) \right],\end{aligned}\ ] ] and @xmath93 \\ \nonumber + \left [ \left ( 0.9 f^{\mathrm{pd}}_{2e'}+0.1 f^{\mathrm{cu}}_{2e ' } \right)+\left ( 0.9 f^{\mathrm{pd}}_{2e''}+0.1 f^{\mathrm{cu}}_{2e '' } \right ) \right].\end{aligned}\ ] ] the @xmath94 , @xmath95 and @xmath96 are atomic form factors of fe , pd and cu atoms , respectively . based on this model , the values of @xmath73 parameter were calculated , and the results are shown in fig . [ s_parameter ] . for fepd : cu alloys on various substrates after different annealing procedures.,width=321 ] the largest @xmath73 value , close to @xmath97 @xmath2 , was found for alloys deposited on mgo(100 ) and si(100 ) substrates . alloys deposited on si(111 ) exhibited slightly smaller value of @xmath73 . in case of these three substrates , the differences in chemical order , caused by different annealing procedures , are relatively small . however , it is worth noting , that in the case of si(100 ) and si(111 ) substrates , the addition of the second annealing stage led to a small decrease in the chemical order , in contrary to the sample on mgo(100 ) substrate , where another annealing stage did not change the parameter @xmath73 value . on the other hand , concerning the fepd : cu alloy on the si(100)/sio@xmath6 substrate , it is well seen , that rta - annealed sample is not well - ordered ( @xmath98 ) . the significant increase of chemical order was observed after additional la treatment ( @xmath99 ) . these results could be explained in such a way that , in case of single - crystal substrates , the chemical ordering process starts during rta annealing with the intermixing of the ingredient layers in the multilayer system . additional long annealing does not influence the chemical ordering process . for the fepd : cu alloy on the si(100)/sio@xmath6 substrate , the amorphous @xmath5 nm thick sio@xmath6 layer forces a two - step process of intermixing and ordering . the first annealing stage results in transformation from the multilayer system into alloy , then the second stage improves the chemical order . the presence of strong ( 111 ) and very weak ( 200 ) reflections in the xrd patterns for fepd alloys is the evidence of ( 111 ) texture presence . the lack of ( 001 ) and ( 002 ) reflections suggests that the two - stage annealling of fepd alloy does not create the texture with @xmath66-axis perpendicular to the substrate plane . in case of the fepd : cu alloys the ( 111 ) and both ( 001 ) and ( 002 ) reflections are present in the patterns which is related to the existence of two texture components . for the quantitative analysis of the relation between these components let us consider two fundamental ( 002 ) and ( 111 ) bragg reflections of l1@xmath11 structure . the integral intensity of the ( hkl ) reflection can be expressed as @xcite : @xmath100 where @xmath101 is the multiplicity of the reflection , @xmath102 is the texture factor associated with the corresponding orientation , and @xmath103 is an instrumental factor related mostly to counting time and used setup ( in present case is always constant due to the same experimental conditions ) . factors @xmath75 , @xmath77 , @xmath78 , @xmath79 and @xmath80 were already described . in case of ( 002 ) and ( 111 ) reflections multiplicity @xmath104 equals to @xmath26 and @xmath105 , respectively . the texture parameter @xmath102 can be described as the fraction of the total sample volume , connected with ( hkl)-oriented crystallographic grains . considering the same model of cu atom location in the unit cell as for chemical order parameter calculations , the expression for @xmath106 is the same as expression [ f_002_cu ] . from expression [ integral_intensity ] the ratio between texture factors @xmath107 can be expressed as : @xmath108 the expression @xmath109 describes the ratio between total volumes of [ 001]- and [ 111]-oriented crystallites , and can have the following values : * for the lack of texture @xmath110 , * for [ 001 ] preferred orientation of grains @xmath111 , * for [ 111 ] preferred orientation of grains @xmath112 . the described approach is valid only for two texture components in the sample . the presence of only \{001 } and \{111 } reflections in the measured xrd patterns allows to use this model . the exception is the rta - annealed alloy deposited on si(100)/sio@xmath6 , where a strong ( 200 ) reflection appeared , however this case was treated as the others . the values of @xmath113 ratio for fepd : cu alloys on different substrates are shown in fig . [ preferred_orientation ] . . the value for the lack of texture is marked with horizontal dashed line.,width=321 ] it is clearly seen that , for rta - annealed fepd : cu alloys deposited on si(100 ) and si(111 ) substrates , the strong ( 001 ) crystallographic texture appeared . moreover , in case of these two substrates , the addition of the second annealing stage does not lead to a significant change in the @xmath109 ratio . a different effect can be observed in case of samples on mgo(100 ) substrate , where , after rta , a strong ( 111 ) texture was recorded . the supplementary la treatment led to a drastic reorientation of the crystallographic grains and strong ( 001 ) texture formation . after both annealing procedures , no well - defined crystallographic texture was found for si(100)/sio@xmath6/fepd : cu samples . let us discuss the problem of the ( 200 ) reflection presence in the pattern for rta - annealed alloy on si(100)/sio@xmath6 substrate . the thin film was deposited on the @xmath5 nm thick amorphous sio@xmath6 and we could expect , that the noncrystalline substrate might induce the formation of alloy without preferred grain orientation . this assumption is consistent with @xmath113 values , close to unity after both type of thermal treatment , which showed the same volume of [ 001 ] and [ 111]-oriented grains . the long annealing introduced the grain reorientation process which has evidence in the absence of [ 100]-oriented crystallites and lack of ( 200 ) reflection in the xrd pattern . in addition to preferred grain orientation determination , the \{001 } and \{111 } pole figure measurements were carried out for all rta+la - annealed samples . pole figures were obtained using fepd : cu l1@xmath11 ( 001 ) and ( 111 ) reflections , and the results are shown in fig . [ pole_figures ] ( for precise determination of the peak positions the samples were re - measured in @xmath30/@xmath31 geometry with cu @xmath114 radiation using laboratory diffractometer ) . the \{001 } pole figures for fepd : cu alloys on mgo(100 ) and si(100 ) substrates are nearly the same , with only one narrow peak in the centre of the patterns . however , the significant differences were found concerning the \{111 } pole figures , where two different types of signal were observed for @xmath50 about @xmath115 . for the alloy on mgo(100 ) , apart from the sharp central peak , four well - defined poles azimuthally separated by @xmath116 were recorded ( fig . [ pole_figures]b ) . in the case of the si(100)/fepd : cu system , the sharp central pole is surrounded by a well - defined diffraction ring ( fig . [ pole_figures]d ) . the observed signals are related to the [ 001]-oriented crystallographic grains in the alloy , and we can distinguish two types of crystallographic texture categories . for sample on mgo(100 ) the sharp ( 001 ) sheet texture is present , in contrast to the si(100 ) substrate , where the sharp ( 001 ) fiber texture is observed with the [ 001 ] crystallographic direction as the fiber axis . the \{001 } pole figure pattern observed for sample on si(111 ) substrate consists of the central pole , surrounded by the @xmath50-dependent poles density distribution ( fig . [ pole_figures]e ) . the distribution maximum is observed for @xmath50 about @xmath115 , and suggests the presence of the ( 111 ) texture component with the [ 001 ] fiber axis . in case of the 111 pole figure , a similar pattern was measured , with a central pole and symmetrical poles distribution , with the maximum also at @xmath50 about @xmath115 ( fig . [ pole_figures]f ) . the ring - shaped pole distribution is connected to the presence of the [ 001]-oriented grains with the [ 111 ] fiber axis . for si(100)/sio@xmath6/fepd : cu system , the recorded \{001 } pole figure is similar to \{001 } pattern for the alloy on si(111 ) substrate , indicating the ( 001 ) texture component with a [ 111 ] fiber axis . however , the central pole is not well - defined and the pole ring is more intense then for the si(111)/fepd : cu alloy . in the 111 pole figure , the only well - defined signal comes from the central pole ( fig . [ pole_figures]g ) . the pole figures show , that in case of annealed fepd : cu thin alloy films the crystallographic texture category depends on the substrate type ; the sheet texture was found for alloy on mgo(100 ) , and silicon - based substrates provide fiber crystallographic ordering . another aspect is that , for the mgo(100 ) and si(100 ) substrates , the texture consists of two sharp ( 001 ) and ( 111 ) components . the results for alloys on the remaining substrates indicate blurred polar - dependent pole density distribution . the appearance of [ 001]-oriented crystallites in fepd alloy is related mostly to two reasons : the cu addition in the structure , and the fast and strain - inducing rta annealing . however , the above mentioned reasons do not clarify the differences in the texture category for fepd : cu alloys deposited on the various substrates . in the case of the fepd : cu alloy on mgo(100 ) , the single crystal substrate is perfectly ordered and oriented with the [ 100 ] direction perpendicular to the surface . such an orientation , together with the symmetry of the mgo structure , causes the substrate surface to have a specific orientation defined by the [ 001 ] and [ 010 ] crystallographic directions . during the annealing process , the fepd : cu crystallites tend to choose the direction , which minimizes the total energy of the system . since the mgo substrate forces the two above mentioned directions , the ( 001 ) plane of l1@xmath11 alloy is not oriented randomly , which is an explanation of the sheet texture of the fepd : cu alloy on the mgo substrate . the described process is supported by the lattice misfit analysis . the mgo substrate has the same crystallographic symmetry as the l1@xmath11 ( 001 ) plane . taking into account the mgo and rta+la annealed fepd : cu l1@xmath11 ( 001 ) lattice parameters , the lattice misfit is about @xmath117% ( mgo lattice parameter ( @xmath118 ) . this relatively small value can be the next indication , that fepd : cu alloy crystallites tend to order along crystallographic directions of the mgo substrate . in the case of the si(100 ) and si(111 ) substrates , the fepd : cu alloy is formed on the disordered native si - oxide layer . for such layer , there are no well - defined directions as in the case of the mgo substrate . during the annealing process , the fepd : cu crystallites can orient randomly their ( 001 ) plane around the [ 001 ] direction , because , in the absence of substrate orientation , each of these configurations is equally favorable energetically . therefore , the lack of defined crystallographic orientation of the substrate surface is the reason for the fiber texture of fepd : cu alloys . for the same reason , the fiber texture is observed for fepd : cu alloy on the si(100)/sio@xmath6 substrate , but in this case the si crystal was intentionally covered by amorphous si - dioxide . it is clearly seen in fig . [ pole_figures ] that among the samples with fiber texture there are also significant differences in the sharpness of the orientation distribution . the reason of this effect is that the oxidation process of the si ( 100 ) and ( 111 ) surfaces proceeds in a different way @xcite and results in the creation of the si - oxide layers with slightly different properties . moreover , the native oxide layers are not completely amorphous , which causes the fepd : cu alloys deposited on these two substrates to reveal a larger @xmath109 ratio and better crystallographic order than those deposited on the si(100)/sio@xmath6 substrate . in order to obtain information about the dependence of crystallographic grain size on the annealing treatment and substrate type the scherrer equation was used : @xmath119 where @xmath120 is a dimensionless scherrer factor and its value is dependent on the crystallite shape @xcite , @xmath121 is the radiation wavelength , @xmath68 is a full width at half maximum for a ( hkl ) reflection at diffraction angle @xmath30 . for the purpose of these studies the value of @xmath122 was applied , as the most common used in similar studies . the calculations were carried out using ( 002 ) and ( 111 ) l1@xmath11 fundamental reflections , indicating crystallite size along [ 001 ] and [ 111 ] crystallographic directions perpendicular to the substrate plane . the results of the calculations are shown in fig . [ grain_sizes ] . the main influence on the grain size has the type of annealing treatment . for both the [ 001 ] and [ 111 ] directions and for all substrates , crystallites obtained after rta have sizes in the range of about @xmath117 nm to @xmath0 nm . after la treatment , crystallite size increases to @xmath0 @xmath123 nm and @xmath0 @xmath124 nm , for the [ 001 ] and [ 111 ] directions , respectively . the grain sizes for samples after rta+la are also dependent on the substrate the smallest values were recorded for the alloy on mgo(100 ) and si(100 ) substrates , and the largest for alloy on si(100)/sio@xmath6 . moreover , in case of the si(100)/sio@xmath6/fepd : cu system , the significant differences between grain size along the [ 001 ] and [ 111 ] directions were found , and crystallites along the [ 111 ] axis are larger . it is seen that rta annealing results in formation of nanocrystalline fepd alloy with grains of the size of a few nm at any type of substrate . additional long time annealing increases the size of grains , and this effect is particularly strong for amorphous si(100)/sio@xmath6 substrate . this might be due to the different character of the recrystallization process on an amorphous substrate , but for definite conclusion the microscopy measurements are necessary . we have investigated the crystallographic texture and the degree of the chemical order of the fepd : cu thin alloy films , obtained by two different thermal treatments . the fepd thin alloy films were used as reference samples . the alloys were deposited on four different substrates , allowing the determination of the substrate influence on the structural properties of studied alloys . based on the detailed xrd studies and pole figure measurements , we have found that the cu addition changes lattice parameters . the lattice parameter @xmath66 decreases with simultaneous increase of lattice parameter @xmath65 , leading to the larger lattice distortion , but the crystallographic @xmath71-spacing stays always the same . the calculations of the chemical order parameter @xmath73 were carried out with the assumption of the random distribution of copper atoms in the fepd l1@xmath11 lattice sites . the largest values of parameter @xmath73 were obtained for the fepd : cu alloys on mgo(100 ) and si(100 ) substrates . there is no relationship between chemical order and preferential orientations of grains . we have observed that the degree of ordering is very similar for samples deposited on mgo and si(100 ) , most likely due to the same symmetry of the substrate surfaces as the symmetry of l1@xmath11 phase . in addition to it , the strong contribution of [ 001 ] texture was observed for all alloys deposited on single crystalline substrates although the different sheet and fiber characters of texture were observed for mgo and si , respectively . this can be related to epitaxial - like growth of films on mgo , in contrary to nonepitaxial growth on si . for alloys on the si - based substrates , also the differences in the crystallographic texture were found . the alloy deposited on si(100 ) exhibits two well defined [ 001 ] and [ 111 ] components , whereas the signals for alloys deposited on si(111 ) and si(100)/sio@xmath6 were blurred , indicating the degradation of crystallographic orientation distribution . as could be expected the amorphous si / sio@xmath6 surface did not induced any preferential orientation of films . the scherrer analysis showed that the annealing treatment has a strong influence on crystallite sizes . in all samples , after rta annealing , grains are the size of a few nanometers . the additional long annealing leads to grain growth , except for the alloy deposited on mgo substrate . the largest change in grain size was found for the alloy on the si(100)/sio@xmath6 substrate , where the grain growth mostly in [ 111 ] direction was recorded . according to the presented studies , the mgo(100 ) and si(100 ) substrates were found to be the most suitable for preparation of the chemically ordered and ( 001)-textured polycrystalline fepd : cu thin alloy films . the investigations were partially supported by daad - polish ministry of science and higher education program of polish - german cooperation under the contract 329/n - daad/2008/0 and polish ministry of science and higher education project n@xmath125 n n507 500338 . the support of dr . erik elkaim from soleil synchrotron by working on proposal n@xmath125 20090217 and dr . denys makarov from institute for integrative nanosciences ifw dresden is also acknowledged . a. polit , m. krupinski , m. perzanowski , a. zarzycki , d. makarov , m. kac , y. zabila , j. zukrowski , m. albrecht , m. marszalek , the influence of copper on local structure of fepd : cu thin alloy films , in preparation .

fepd thin films have been recently considered as promising material for high - density magnetic storage devices . however , it is necessary to find a proper method of fabrication for the ( 001)-textured and chemically well - ordered alloy . in this paper , we present the detailed investigations of lattice parameters , chemical order degree , grain sizes and crystallographic texture , carried out on fepd alloys with @xmath0 at.% of cu addition . the initial [ cu(@xmath1 nm)/fe(@xmath2 nm)/pd(@xmath3 nm)]@xmath4 multilayers were thermally evaporated in an ultra - high vacuum on mgo(100 ) , si(100 ) , si(111 ) and si(100 ) covered by @xmath5 nm thick layer of amorphous sio@xmath6 . in order to obtain homogeneous fepd : cu alloy , the multilayers were annealed in two different ways . first , the samples were rapidly annealed in nitrogen atmosphere at @xmath7c for @xmath8 seconds . next , the long annealing in a high vacuum for @xmath9 hour at @xmath10c was done . this paper focuses on quantitative investigations of the chemical order degree and crystallographic texture of ternary fepd : cu alloys deposited on four different substrates . in order to obtain both quantities we have taken a novel approach to consider the problem of dopant atoms located in the fepd structure . the studies of the structure were done using x - ray diffraction ( xrd ) performed with synchrotron radiation and pole figures measurements . we have found that the addition of cu changes the fepd lattice parameters and lattice distortion . we have also shown , that using different substrates it is possible to obtain a fepd : cu alloy with different chemical order and texture . moreover , it was observed that texture category is substrate dependent . marcin perzanowski , yevhen zabila , michal krupinski , arkadiusz zarzycki , aleksander polit , marta marszalek _ department of materials science , institute of nuclear physics polish academy of sciences , radzikowskiego 152 , 31 - 342 krakow , poland , tel . ( + 48 ) 12 662 8145 _

the concept of path - width and tree - width were defined and played central role in the graph minors project by robertson and seymour . one important feature of path - width and the first result of this type is that excluding a tree in an infinite sequence of finite graphs results in a class of bounded path - width @xcite . similarly , a forbidden planar graph implies that the class has bounded tree - width @xcite . these results are complemented with the theorem that graphs of bounded tree - width are well - quasi - ordered @xcite giving a prototype of the deep and lengthy graph minor theorem . the proof for bounded tree - width is comprehensible and now well digested . this is one of the reasons that path - width is sometimes considered too simple or less valuable . since bounded tree - width implies bounded path - width , there is no direct proof that the graphs of bounded path - width are well - quasi - ordered . if path - width is `` so simple '' , then there should be a nash - williams type proof for this latter result @xcite . however , we believe that such a result is still unknown . another dogma we would like to attack is that graphs of large path - width are more important than that of small path - width . our point is that raising the connectivity and path - width simultaneously gives structural information . this idea has also been exploited in @xcite for algorithmic use . the number of excluded minors of this kind seems to increase mildly opposite to the number of excluded trees , which grows super - exponentially @xcite . we consider finite , simple graphs except in section [ 2elof ] , where allowing double edges makes the list of excluded minors more compact . a graph @xmath1 is a _ minor of a graph @xmath2 , denoted as @xmath3 , if @xmath1 can be obtained from a subgraph of @xmath2 by contracting edges . contraction of an edge might lead to double edges . when we consider 2-edge - connected graphs , it is natural to allow double edges . it will make our discussion more comfortable . otherwise , we keep the graph simple by removing multiple edges after contraction . _ in this paper , we focus on the following well - known graph parameter . a path - decomposition of a graph @xmath2 is a pair @xmath4 , where @xmath5 is a path , and @xmath6 is a family of subsets of @xmath7 , satisfying + @xmath8 @xmath9 , and every edge of @xmath2 has both ends in some @xmath10 , and + @xmath11 if @xmath12 and @xmath13 lies on the path from @xmath14 to @xmath15 , then @xmath16 . the width of a path - decomposition is @xmath17 , and the path - width of @xmath2 is the minimum width of all path - decompositions of @xmath2 . it follows from the definition that path - width is minor - monotone , that is , if @xmath3 , then @xmath18 . we will denote the class of graphs with path - width at most 2 by pw2 . it is clear that if @xmath19pw2 , then @xmath20 . therefore , any graph in pw2 is a planar graph . we will not make any use of it , but path - width is equivalent to a cops - and - robber game @xcite . being familiar with the game might help the reader s intuition throughout the discussion . a _ rooted graph _ @xmath21 is a graph @xmath2 with a specific node @xmath22 , that is called the _ root _ of @xmath2 . the rooted graph @xmath23 is a rooted minor of @xmath21 , denoted as @xmath24 , if @xmath25 is mapped to @xmath26 , when @xmath2 is mapped to @xmath1 under the minor operation . similarly , a two - rooted graph @xmath27 is a graph with two specified nodes . rooted minor of a two - rooted graph is defined analogously . with a slight abuse of notation , @xmath28 will be used for both rooted and two - rooted minors . let @xmath2 be a graph and let @xmath29 be a set of vertices . then @xmath30 is the set of edges incident to any element of @xmath29 . the graph @xmath31 is obtained by deleting the vertices in @xmath29 . the graph @xmath32 is the subgraph of @xmath2 induced by @xmath29 . let @xmath33 be a set of edges . then @xmath34 is the set of vertices incident to at least one member of @xmath33 . the graph @xmath35 is obtained from @xmath2 by deleting the edges in @xmath33 . the graph @xmath36 is the subgraph induced by the edge set @xmath33 , i.e. its vertex set is @xmath34 and its edge set is @xmath33 . it is well - known , that `` being identical or being on the same cycle '' is an equivalence relation on the edges of a graph . its equivalence classes span the so - called _ blocks _ of the graph . the one - element classes are the cut edges . let @xmath2 be a connected graph , and let @xmath37 be a cycle of @xmath2 . consider @xmath38 , and let @xmath39 be the vertices of a component of @xmath38 . then @xmath40 is a _ bridge _ of @xmath37 ( in @xmath2 ) , as well as any edge connecting two nodes of @xmath37 . the _ legs of a bridge @xmath41 _ are the common vertices of @xmath41 and @xmath37 . the set of legs are denoted by @xmath42 . let @xmath43 and @xmath44 be two - element subsets of @xmath45 . the two pairs are _ crossing _ if and only if they are disjoint , and along the cycle , the @xmath46-vertices alternate with the @xmath47-vertices . let @xmath29 and @xmath48 be subsets of @xmath45 . we say that @xmath29 and @xmath48 are _ crossing _ , if there are two nodes of @xmath29 and two nodes of @xmath48 such that the two pairs are crossing . _ two bridges are crossing _ if and only if their set of legs are crossing . a bridge is _ simple _ if and only if it is a path ( and hence it has two legs , the end - vertices of the path ) . a bridge is called trivial , if it consists of one edge or two edges . we describe a special class of graphs called tracks . they turn out to have path - width at most two and play fundamental role in our discussion . let @xmath49 and @xmath50 be two vertex disjoint paths . this is a slight abuse of notation , sometimes @xmath51 and @xmath5 is a path of length @xmath52 . we define two types of connections between the two paths . firstly , there might be edges connecting a vertex of @xmath5 to a vertex of @xmath53 . we call these edges _ short chords_. secondly , we allow paths of length two connecting @xmath5 and @xmath53 . we call these paths _ long chords_. a long chord has three nodes , a @xmath54 , a middle node @xmath55 , and a @xmath56 . the degree of @xmath55 is @xmath0 in the graph @xmath2 . that is , different long chords must have different middle nodes , in particular long chords are edge - disjoint . if we do not want to specify whether we talk about a short- or a long chord we refer to it as a chord . for brevity , we call a @xmath57-chord an @xmath58-chord . an @xmath58- and an @xmath59-chord are not crossing if and only if @xmath60 . a @xmath61-chord or a @xmath62-chord is called an _ end - chord_. if there are several candidates for an end - chord , then one of them has to be selected . a graph @xmath2 is called a _ track _ if and only if it can be represented by two vertex disjoint paths @xmath63 and noncrossing chords as above such that @xmath64 and @xmath65 are connected by a chord as well as @xmath66 and @xmath67 . in this way , a track is set to be @xmath0-connected . alternatively , we can look at a track as a graph that can be obtained from special outerplanar graphs by certain operations . let @xmath2 be an outerplanar graph with outer cycle @xmath37 . a graph @xmath2 is called _ multichordal _ outerplanar if we allow multiple edges inside @xmath37 . if we have a drawing of such a graph , then there might be some @xmath0-faces . if the dual of the interior of @xmath2 is a path , then @xmath2 is called _ series_. consider the simple graphs arising from a series multichordal outerplanar graph by subdividing some of its internal edges by a single vertex . these graphs are tracks for the following reason . we can select two paths @xmath68 and @xmath69 of the outer cycle @xmath37 such that * each of @xmath70 and @xmath71 is an edge or a path with two edges ; * the cycle @xmath72 ; * all chords have one end on the path @xmath68 and the other end on @xmath69 . the vertices @xmath73 ( that is @xmath74 in the original definition ) will be called _ corners_. we imagine @xmath46 and @xmath75 to be on the left - hand side , @xmath47 and @xmath76 on the right - hand side . therefore , @xmath46 and @xmath47 will be called _ opposite _ corners , as well as @xmath75 and @xmath76 . in the general case , a track has four corners . but some of them may coincide . if @xmath77 , then the node @xmath46 is called a degenerate side of the track . if we have a drawing of a track , then the inner faces can be listed as @xmath78 according to the dual path . then @xmath46 and @xmath75 are on face @xmath79 and @xmath47 and @xmath76 are on face @xmath80 . notice that the selection of the corners are not unique . also the drawing of a track is not unique . this can make our proofs more complicated , but only technically . a graph @xmath2 is called a partial track if and only if it is a subgraph of a track . it follows from the definition that tracks are 2-connected and have path - width at most two . we put this observation in a broader context . [ 2conntrack ] the following three statements are equivalent . 1 . the graph @xmath2 is @xmath0-connected and @xmath81 . the graph @xmath2 is a track . the graph @xmath2 is @xmath0-connected and @xmath82 . two implications are immediate : @xmath83 and @xmath84 . we show that @xmath85 implies @xmath86 . in a 2-connected graph any two vertices are contained in a cycle . let @xmath37 be a longest cycle of @xmath2 . consider the bridges of @xmath37 . since @xmath87 , there is no @xmath88-bridge . since @xmath89 , there is no internal edge in any bridge . therefore the bridges are trivial . since @xmath87 , the bridges are pairwise equivalent or avoiding . since @xmath90 , the graph @xmath2 is a track . note that the proof gives guidelines to find @xmath91 but usually these four vertices are not well - defined . even if the proof points out certain vertices , the chords , that play the role of end - chords , might be ambiguous . our goal is to prove an excluded minor characterization of 2-edge - connected pw2-graphs . we have to focus on the blocks , and how they are glued together . let @xmath19pw2 be 2-edge - connected . we know from basic graph theory that @xmath2 is built up from its blocks pasted together along vertices . we call these attachment vertices _ multiple points_. we collect some information about the blocks and the multiple points in four statements . when we say that a node @xmath25 can be a corner of a track @xmath92 , we literally mean that @xmath25 can be fixed to be corner @xmath46 , and the other three corners of @xmath92 can be selected in such a way that we get back the definition of a track with corners @xmath93 . similarly , we say that @xmath25 can be a degenerate side of a track @xmath92 meaning that after fixing @xmath94 , we can select the other two corners properly . the proofs are based on the following visualization of a track : it is a 2-connected graph , consider its longest cycle @xmath37 . since the chords are non - crossing bridges of @xmath37 , they can be linearly ordered . this order is not well - defined , because equivalent bridges can be interchanged . if the order is fixed , the position of @xmath25 is crucial , and we focus on that . [ s1 ] let @xmath95 be a track with a specified node . if @xmath96 , then @xmath25 can be a corner of @xmath92 . let @xmath37 be a longest cycle through @xmath25 in @xmath92 . consider the bridges of @xmath37 . since @xmath92 is a track , there is no bridge with three legs . since @xmath97 , the bridges avoiding @xmath25 are chords . assume there is a bridge @xmath1 adjacent to @xmath25 and @xmath26 . if there was a path of length three from @xmath25 to @xmath26 in @xmath1 , then @xmath98 , a contradiction . that means @xmath1 is a single vertex or empty . the vertices @xmath25 and @xmath26 cuts the cycle @xmath37 into two parts , let us say left and right from @xmath25 . if there is a chord avoiding @xmath25 on both sides , then @xmath99 . otherwise all chords are on one side , left from @xmath25 say . we can now select the two paths @xmath5 and @xmath53 in the definition of a track . let @xmath25 be a corner , @xmath46 say . the right neighbor of @xmath25 will be @xmath75 . starting a path from @xmath25 going to the left , we detect the legs of the chords in order . there is a unique moment , when two consecutive legs @xmath100 and @xmath101 belong to the same chord . if there were two such chords , then @xmath102 , a contradiction . therefore @xmath100 is a good choice for @xmath47 and the left neighbor of @xmath100 for @xmath76 . the case when there are no bridges adjacent to @xmath25 is very similar . going on @xmath37 to the left and to the right from @xmath25 , we find the legs @xmath103 and @xmath104 of the same chord . continuing to the left from @xmath103 we detect another place where two consecutive legs @xmath100 and @xmath101 belong to the same chord . this is unique , since @xmath105 . now @xmath25 can play the role of @xmath46 , the right neighbor of @xmath25 can be @xmath75 . we select @xmath100 to play the role of @xmath47 and the left neighbor of @xmath100 can be @xmath76 . [ s2 ] let @xmath106 be a track with two specified nodes such that any of @xmath25 and @xmath107 can be a corner of @xmath92 . if @xmath108 , then @xmath25 and @xmath107 can be two opposite corners . let @xmath37 be a longest cycle through @xmath25 and @xmath107 in @xmath92 . consider the bridges of @xmath37 . since @xmath92 is a track , there is no bridge with three legs . the vertices @xmath25 and @xmath107 cut the cycle @xmath37 into two parts , let us say left side and right side . if there is a bridge of @xmath37 with two legs on the same side , then @xmath109 , a contradiction . if there is a bridge with legs @xmath25 and @xmath107 , then the bridge is trivial , as otherwise @xmath98 . also all other bridges are trivial and adjacent to @xmath25 or @xmath107 . since @xmath110 , there are no two bridges adjacent to @xmath107 such that one of the other legs are on the left , and one is on the right from @xmath25 . therefore , there might be bridges @xmath111 adjacent to @xmath25 with legs @xmath112 on the right side , and bridges @xmath113 adjacent to @xmath107 with legs @xmath114 on the left side . in this case @xmath25 and @xmath107 can be corners as follows : we select @xmath115 and @xmath116 , and the path @xmath5 is the part of @xmath37 on the right side and path @xmath53 is the part of @xmath37 on the left side . therefore , @xmath75 is the left neighbor of @xmath46 and @xmath47 is the right neighbor of @xmath76 . if there is no bridge with legs @xmath25 and @xmath107 , then assume there is a bridge with legs crossing @xmath25 and @xmath107 . since @xmath117 , this bridge is trivial . there might be many non - crossing chords like that . finally , there might be some chords on one side adjacent to @xmath25 and some on the other side adjacent to @xmath107 . in this case the same scenario works as before . we select @xmath115 and @xmath5 to be the left side of @xmath37 , and @xmath116 and @xmath53 to be the right side of @xmath37 . ] [ s3 ] let @xmath95 be a track with a specified node . if @xmath118 , then @xmath25 can be a degenerate side of @xmath92 . let @xmath37 be a longest cycle through @xmath25 in @xmath92 . since @xmath118 , there is no bridge avoiding @xmath25 . if there is a nontrivial bridge with leg @xmath25 and other leg @xmath119 , then the length of @xmath37 between @xmath25 and @xmath26 is at least three on both sides , and therefore @xmath98 , a contradiction . so @xmath92 consists of @xmath37 and some trivial bridges , all of them adjacent to @xmath25 . these graphs satisfy the claim . we also need the following observation . [ s4 ] let @xmath95 be a @xmath0-edge - connected graph in which each block is a track , and @xmath25 is a specified node . if @xmath120 , then the blocks of @xmath92 are pasted along @xmath25 . we are done with all preparation for the main result , that is an excluded minor characterization of @xmath0-edge - connected pw2-graphs . [ fo ] the following three statements are equivalent . 1 . the graph @xmath2 is a @xmath0-edge - connected partial track . 2 . the @xmath0-edge - connected graph @xmath121 shown in figure @xmath122 . ( the blocks of @xmath2 are tracks glued together in a path - like fashion according to the following scheme ) the blocks of @xmath2 are tracks and can be listed as @xmath123 such that for any @xmath124 the block @xmath125 is a block with two ( possibly identical ) multiple nodes @xmath126 and @xmath127 . + if @xmath128 , then @xmath126 and @xmath127 can play the role of opposite corners in @xmath125 . + if @xmath129 , then this multiple node can play the role of a degenerate side in @xmath125 . + for @xmath130 or @xmath131 the block @xmath125 has a multiple node @xmath126 that can play the role of a corner . notice that the above numbering of blocks is not unique . two implications are easy to see : @xmath132 and @xmath133 . we show that @xmath86 implies @xmath85 . in the block structure of @xmath2 , the excluded minors @xmath134 and @xmath135 imply that no block can contain more than two multiple nodes . the excluded minors @xmath136 and @xmath137 together with lemma [ s1 ] imply that the block structure of @xmath2 satisfy that @xmath126 can play the role of a corner in @xmath125 , for any @xmath138 . if there is a block @xmath41 with two distinct multiple points @xmath55 and @xmath139 , then @xmath140 and @xmath141 together with lemma [ s2 ] imply that @xmath55 and @xmath139 can play the role of opposite corners in @xmath41 . assume that in the block structure of @xmath2 , there are blocks / tracks @xmath142 adjacent to the same multiple node @xmath55 . lemma [ s3 ] and lemma [ s4 ] together with the excluded minors @xmath143 imply that @xmath55 is the only multiple node in @xmath144 and @xmath55 can be a degenerate side in @xmath144 with at most two exceptions . these claims together prove the validity of @xmath85 . -edge - connected graphs ] we sketch how to prove along the above ideas an excluded minor characterization of pw2-graphs . the result is not new , since the list has been obtained by kinnersley and langston @xcite making a computer search . our proof outline explains certain similarities among the excluded minors . also , our proof will associate a task to each excluded minor ( as it happened in the 2-edge - connected case ) that explains the role of that specific excluded minor . we know from basic graph theory that @xmath2 is built up from its blocks and trees ( consisting of edges , that are cut - edges of @xmath2 ) pasted together along vertices in a tree - like fashion . a subgraph of @xmath2 which is a tree and consists only of cut - edges ( and hence it is an induced subgraph ) is called _ a tree - part of @xmath2_. one of the major reasons of difficulty in the discussion below is that these tree - like parts are not well - defined . a tree - like part can be considered as its tree - like edges are glued together . although the maximal tree - like parts are uniquely defined , we can not allow ourselves to consider the tree - like parts to be the maximal ones . let @xmath92 be a tree with two distinguished nodes @xmath145 and @xmath25 . we call @xmath146 a _ bond - tree _ if and only if there exists a track @xmath147 such that the unique @xmath148 path in @xmath92 can be identified with a side of @xmath149 . the nodes @xmath145 and @xmath25 are opposite corners of the bond - tree . a subtree @xmath92 of @xmath2 is called a _ tree - frippery _ rooted at @xmath25 if and only if 1 . the node @xmath25 is a cutnode of @xmath2 , where @xmath92 and @xmath150 are glued together , 2 . the subtree @xmath92 is a subgraph of a track @xmath149 such that @xmath92 is disjoint from one side of @xmath149 , and @xmath25 can play the role of a corner . a special kind of tree - frippery is important in the structure theorem . let @xmath95 be a tree - frippery . it is called an _ edge - frippery _ or _ hair _ if it has only one edge . therefore , one of its endpoints has degree @xmath151 , and the other endpoint @xmath25 belongs to the rest of @xmath2 . blocks ( tracks ) and bond - trees can be glued together as in the case of 2-edge - connected graphs . in addition , there might be fripperies attached to the tracks . the exact gluing pattern is described in the third part of the following theorem . let @xmath2 be a connected graph . the following three statements are equivalent . 1 . the graph @xmath2 has path - width at most @xmath0 . 2 . the graph @xmath2 does not contain any minor from the kinnersley langston list @xcite . the graph @xmath2 can be separated into its blocks and some tree parts , that are classified as bond - trees , tree - fripperies and hairs . the spine of @xmath2 can be glued together from @xmath0-connected tracks ( blocks ) and bond - trees in a path - like fashion according to the following scheme . the blocks and bond - trees are enumerated as @xmath123 and there is a corresponding sequence of nodes @xmath152 ( consecutive nodes possibly coincide ) such that for any @xmath124 the nodes @xmath126 and @xmath127 are opposite corners of @xmath125 . hence , @xmath153 are multiple nodes of the separation . the corners of blocks that are not multiple nodes are called _ free corners_. + there can be at most one tree - frippery attached to each free corner . + finally , there might be hairs attached to the sides of the blocks . so a hair can not be rooted at the middle node of a long chord , but several hairs may be rooted at a side vertex of a track . before the proof sketch , we must clarify an important point . suppose there is a @xmath125 with degenerate side @xmath154 such that @xmath145 is a free corner . that means @xmath25 is a free corner too , somewhat invisible . there might be a tree - frippery attached to @xmath145 , and another one attached to @xmath25 . therefore , if a maximal tree part @xmath92 is attached to the track at @xmath145 , that is a free corner , then we must separate @xmath92 into a left and a right tree - frippery in order to recognize the structure in ( iii ) . that is why the conditional form in ( iii ) is essential . as before , the implications @xmath155 are straightforward . the heart of the theorem is the implication @xmath156 . this can be proven by reversing the easy implications ( see the original proof in @xcite for the equivalence of ( i ) and ( ii ) ) or by analyzing the cases when the gluing pattern in ( iii ) is not identifyable . this proof method sheds light on the role of the excluded minors . it is tedious , since the number of excluded minors is large . the advantages of our method are already presented in the previous sections on 2-connected and 2-edge - connected graphs . we do not want to fatigue the reader by the long case analysis , we only want to exhibit the possibility of this alternative proof . this explains that the large number of excluded minors is caused by the subtlety in the structure described in ( iii ) , due to the tree - like parts of @xmath2 . let us demonstrate a few steps of the long road . the tree - like lemmas are without proofs . [ hair ] let @xmath92 be a tree with a distinguished node @xmath25 . the graph @xmath92 can not be a bucket of hairs rooted at the same node @xmath25 if and only if @xmath157 . [ fripp ] let @xmath92 be a tree with a distinguished node @xmath25 . the graph @xmath92 can not be a tree - frippery rooted at @xmath25 if and only if @xmath158 . [ fold ] let @xmath92 be a tree with a distinguished node @xmath25 . the graph @xmath92 can not be a disjoint union of two tree - fripperies rooted at the same node @xmath25 if and only if @xmath159 . we outline a couple of steps . * there must be a graph on the list of excluded minors guaranteeing that a track can not have more than four points , where non - hair trees are attached . the graph @xmath160 appears . * assume that four trees are attached to a track . there must be an excluded minor showing that at most two of them are not tree - fripperies . see @xmath161 . * assume there are three nodes where the attached trees are not tree - fripperies . then at least one of them must have an attachment that is a disjoint union of two tree - fripperies rooted at the same node . this phenomenon is described by the excluded minors @xmath162 . * we obtain a wider class of excluded minors if we take into account the possible connected tracks to our initial one ( this fills the @xmath163-class in @xcite ) . * further classes arise when some inner structure is known about the track where we want to see the neighborhood as described in ( iii ) . * we need more excluded minors to enforce the right positions of hairs . * finally the pure tree case should be discussed . we believe that @xmath88-connected pw3-graphs can be similarly characterized . we achieved some results in this direction and plan to complete those efforts . it looks substantially more difficult to continue along this line to exact description of @xmath164-connected pw@xmath164-graphs . we can expect asymptotic results rather than a precise one . we raise the following questions : * what is the number of excluded minors for @xmath164-connected pw@xmath164-graphs in terms of @xmath164 ? * is there a lower bound , that is greater than polynomial in @xmath164 ? * is there a good upper bound ? the first two authors initiated the study of pw2 in 1998 , when the first author started his phd studies . we were unaware of @xcite and looked for a characterization by paper and pencil . the results we obtained were written in two article submissions in 1999 . some version appeared in bart s phd thesis @xcite . one article was published @xcite , and the other was rejected @xcite . while this other paper was rewritten , the last two authors submitted their results @xcite for publication in 2003 . the first author was asked to referee the paper , and this is where our roads crossed . the four authors agreed to unify their forces . due to various difficulties in space and time , the process lasted longer than it should have . finally , we agreed to publish the present version , which is seemingly rather different from any of @xcite and @xcite . therefore , we decided to leave those versions available on our home page .

nancy g. kinnersley and michael a. langston has determined the excluded minors for the class of graphs with path - width at most two by computer . their list consisted of 110 graphs . such a long list is difficult to handle and gives no insight to structural properties . we take a different route , and concentrate on the building blocks and how they are glued together . in this way , we get a characterization of @xmath0-connected and @xmath0-edge - connected graphs with path - width at most two . along similar lines , we sketch the complete characterization of graphs with path - width at most two .

in this paper i study the exotic hadron @xmath0 ( narrow hadron resonance of 1862 mev decaying into a @xmath8 ) very recently discovered by the na49 collaboration at the cern sps @xcite . the simultaneous discovery of a resonance with similar mass and width decaying into a @xmath9 provides evidence for a @xmath4 iso - quadruplet . this completes a missing link of the anti - decuplet @xcite which includes the recently discovered @xmath10(1540 ) @xcite . moreover pentaquark structures have been observed in the lattice both with parity + and with parity - @xcite . the @xmath1 is an extremely exciting state , because it may be the first exotic hadron to be discovered , with quantum numbers that can not be interpreted as a quark and an anti - quark meson or as a three quark baryon . in what concerns the quark structure it was known for a long time that the simplest and lightest s - wave multiquarks would be repulsive and very unstable . for instance we recently computed @xcite the mass of the groundstate s - wave @xmath11 , @xmath12 @xmath13 pentaquark , and we checked that it would have a mass of 1535 mev , close to @xmath14 . however in this channel we find a purely repulsive exotic @xmath15 hard core s - wave interaction @xcite . this suggests why pentaquarks have been hard to find , and at the same time this indicates that pentaquarks should include an excitation . because excited multiquark systems are indeed difficult to study , different perspectives of the pentaquarks are welcome to fully address these new states . exotic multiquarks are expected since the early works of jaffe @xcite . soon after the @xmath1 was observed , jaffe and wilczek @xcite , and karliner and lipkin @xcite proposed that the pentaquarks are arranged in microscopic coloured diquarks or triquarks , connected by a string in a p - wave state . this is a very appealing structure , in particular the p - wave system tends to have a narrow width because the decay is only possible if the diquarks overlap . this model needs a novel cancellation of some of the mass of diquarks , to compensate the large p - wave excitations . the exotic anti - decuplet was first predicted , with the correct @xmath1 mass and similar decay width , by diakonov , petrov and polyakov @xcite . these authors interpret the exotic anti - decuplet as a rotational excitation of the chiral topological soliton @xcite . this approach suggests that chiral symmetry and p - wave excitations are crucial to understand the pentaquarks . however the skyrme chiral soliton has some difficulty to reproduce the short range repulsion in n - n interactions . soon after the @xmath16 was discovered , we proposed another pentaquark model @xcite which estimates a mass of 1.9 gev for the s= -2 , q = -2 state . importantly , this predicted mass is quite close to the observed one . to motivate our proposal let me first review the five possible excitations of the quark model , in decreasing energy shift order . the first excitation is the radial one , with an energy shift of @xmath17mev . the second excitation is the angular one , with an energy shift of @xmath18mev . the third excitation is the spin one , with an energy shift of @xmath19mev . the fourth excitation is the flavour one , with an energy shift of @xmath20mev or @xmath21mev . the fifth excitation is the quark - antiquark pair creation one , with an energy shift of @xmath22mev . the light mass @xmath23mev of the @xmath24 pentaquark suggests that it either has a flavour excitation or a @xmath25 pair creation , and not a p - wave excitation . because the production processes show no evidence for a weaker flavour changing reaction , we explored @xcite the @xmath25 creation hypothesis . moreover , when a flavor singlet quark - antiquark pair @xmath26 is created in the pentaquark @xmath27 , the resulting crypto - heptaquark @xmath28 is a state with an opposite parity to the original @xmath27 , where the reversed parity occurs due to the intrinsic parity of fermions and anti - fermions . in this sense the new heptaquark @xmath28 can be regarded as the chiral partner of @xmath27 . and , because @xmath28 is expected to have the lowest possible mass , it is naturally rearranged in a baryon belonging to the s - wave baryon octet and in two pseudoscalar mesons belonging to the s - wave meson octet . in this approach the mass of the heptaquark @xmath28 is simply expected to be slightly lower than the exact sum of these standard hadron masses due to the negative binding energy . for instance we recently suggested @xcite that the @xmath1 is probably a @xmath29 molecule with binding energy of 30 mev , a borromean three body s - wave boundstate of a @xmath30 , a @xmath31 and a @xmath32 @xcite , with positive parity @xcite and total isospin @xmath11 . we also addressed the s= -2 , q = -2 state @xmath0 , suggesting that it is a @xmath33 molecule . the na49 result for the iso - quadruplet of 1.862 gev is consistent with a binding energy of 60 mev for the hadronic molecule . in this paper i extend the quantitative techniques used in our first publication for the @xmath1 , @xcite to the remaining of the anti - decuplet . i start by reviewing , in section [ framework ] , the standard quark model ( qm ) , and the resonating group method ( rgm ) @xcite which is adequate to study states where several quarks overlap . using the rgm , i show that the corresponding exotic baryon - meson short range s - wave interaction is repulsive in exotic channels and attractive in the channels with quark - antiquark annihilation . in most iso - multiplets , except in the @xmath34 iso - multiplet the short range repulsion contradicts a possible pentaquark with a narrow width . section [ binding ] proceeds with the study of the heptaquarks in the exotic anti - decuplet , which are bound by the attractive non - exotic @xmath35 and @xmath36 interactions . i compute the masses of the exotic anti - decuplet . in section [ coupling ] the coupling and decays to p - wave channels are addressed . finally the conclusion is presented in section [ conclusion ] . our hamiltonian is the standard qm hamiltonian , @xmath37 where each quark or antiquark has a kinetic energy @xmath38 with a constituent quark mass . the colour dependent two - body interaction @xmath39 includes the standard qm confining term and a hyperfine term . the quark - antiquark annihilation - creation potential @xmath40 is necessary when the potential complies with chiral symmetry , including the light pion mass and the adler zero @xcite . the rgm provides an accurate framework @xcite to compute the effective multiquark energy using the matrix elements of the quark - quark interactions . any multiquark state can be decomposed in antisimmetrised combinations of simpler colour singlets , the baryons and mesons . for the purpose of this paper the details of the potentials in eq . ( [ hamiltonian ] ) are unimportant , only their matrix elements , extracted from the baryon spectroscopy , matter @xmath41 the detailed calculations are similar to the ones in reference @xcite , and lead to the attraction / repulsion criterion , + - _ whenever the two interacting hadrons have quarks ( or antiquarks ) with a common flavour , the repulsion is increased by the pauli principle , + - when the two interacting hadrons have a quark and an antiquark with the same flavour , the attraction is enhanced by the quark - antiquark annihilation_. + in the particular case of one nucleon interacting with anti - kaons and with kaons , this implies that the short range @xmath42 and @xmath43 interactions are repulsive , while the short range @xmath44 and @xmath45 interactions are attractive . quantitatively @xcite , the effective potentials computed for the channels relevant to this paper , are @xmath46 where @xmath47 are the isospin matrices , normalised with @xmath48 . in the chiral limit one would expect that the @xmath49 cancels with the @xmath50 , and this is confirmed by eq . ( [ overlap kernel ] ) . to arrive at eq.[overlap kernel ] we used a harmonic oscillator basis @xmath51 for the multiquark wave - function , where the inverse hadronic radius @xmath52 is the only free parameter in this framework . the simplest pentaquarks are not expected to bind due to the attraction / repulsion criterion . for instance the @xmath0 can not be a @xmath53 pentaquark . the possible elementary color singlets @xmath54 or @xmath55 are repelled because the elementary color singlets share the same flavour @xmath56 or @xmath57 . this also implies that the @xmath58 and @xmath59 systems are unbound . then the only way to have attraction consists in adding at least one quark - antiquark pair to the system . however including an extra pion in the fundamental configurations is not possible , except in the @xmath1 . because the pion is very light it is not expected to bind into a narrow resonance @xcite , except in the @xmath1 where it may be attracted both by a @xmath32 and a @xmath31 in a borromean structure . moreover the @xmath32 and @xmath31 are repelled in this framework , and the narrow @xmath1 is not supposed to exist unless it includes a pion to bind it . nevertheless , in the other iso - multiplets , where a @xmath60 exits , binding is possible because the @xmath60 is attracted both by the @xmath32 and the @xmath31 . therefore , although the @xmath61 system is repulsive , the @xmath62 , @xmath35 and @xmath63 systems are expected to bind . it is then convenient to build the anti - decuplet like a combinatoric newton pyramid , starting by the summit , @xmath64 , i=0 , @xmath1 . to reach any of the other three iso - multiplets in the anti - decuplet , one simply needs to add respectively one , two or three @xmath6 , @xmath60 , ( @xmath65 , @xmath66 ) to the @xmath1 . altough we advocate @xcite that @xmath1 is a @xmath29 linear molecule , let us consider for the flavour purpose , that it has the flavour of a i=0 , @xmath42 system . the next @xmath6 , @xmath67 iso - multiplet ( @xmath68 , @xmath69 ) can be obtained combining a @xmath6 , @xmath60 , ( @xmath65 , @xmath66 ) with the @xmath1 . a possible binding structure is a @xmath62 linear molecule . the next @xmath7 , @xmath70 iso - multiplet ( @xmath71 , @xmath72 , @xmath73 ) can be obtained combining a @xmath6 , @xmath60 , ( @xmath65 , @xmath66 ) with the @xmath67 . the simplest exotic binding structure is pentaquark with the flavour of a @xmath35 system . finally the last @xmath4 , @xmath74 iso - multiplet ( @xmath75 , @xmath76 , @xmath77 , @xmath53 ) can be obtained combining a @xmath6 , @xmath60 , ( @xmath65 , @xmath66 ) with the @xmath70 . in this case the simplest exotic binding structure is a @xmath63 linear molecule . the different multiquarks are summarised in table [ heptaquark candidates ] . i now compute the energy of the simplest state , the i=1 iso - triplet @xmath70 , as a pentaquark with the quantum numbers of a @xmath35 system . the other @xmath45 system , the iso - singlet @xmath11 @xmath78 has been studied in detail in the literature . nevertheless the @xmath7 system is also attractive , moreover the @xmath35 binding is relevant to the @xmath4 , @xmath74 . in this case the reduced mass is @xmath79 mev , and the potential @xmath80 is attractive and separable , where @xmath81 was computed in eq . ( [ overlap kernel ] ) and where @xmath52 is a free parameter . in the @xmath82 case , binding exists if @xmath83 304 mev . however for a binding energy of the order of 60 mev , close to the one proposed for the the exotic @xmath0 and consitent with a crypto - exotic @xmath70(1385 ) a too small @xmath84 would be required . therefore binding is expected in this system , although an accurate prediction of the binding energy of this @xmath70 system would need technical improvements of our method , see section [ conclusion ] for details . . proposed list of multiquark states , candidates to the exotic pentaquark and heptaquark anti - decuplets . [ cols="^,^,^",options="header " , ] i now use an adiabatic hartree method to study the stability of the linear @xmath85 molecule . while the @xmath7 pentaquark is not a molecule since the five quarks and antiquarks overlap in a s - wave state , here the large @xmath43 repulsion , with @xmath86 mev , prevents the overlap of all the quarks . essentially the wave - function of the @xmath31 is centred between the two @xmath60 , and the two @xmath60 only overlap with the nucleon , but not with each other . this results in a linear molecule . i solve a schrdinger equation for a @xmath60 in the potential produced by a nucleon placed at the origin and by the other @xmath60 placed at a distance @xmath87 of the nucleon . the potential of the @xmath31 is produced by a @xmath60 anti - kaon at the point @xmath87 and another anti - kaon at the point @xmath88 . the potentials are respectively , @xmath89 where the sub - index denotes the position of the potential . this produces three binding energies @xmath90 , and three wave - functions . in the hartree method the total energy is the sum of these energies minus the matrix elements of the potential energies , @xmath91 this is easily computed once the two schrdinger equations are respectively solved with the potentials ( [ potential k ] ) and ( [ potential n ] ) . the total energy is a function of the distance @xmath92 , and i minimise it as a function of @xmath93 . the energy minimum is obtained for @xmath94 . the centres of the two @xmath60 are separated by a distance of @xmath95 , and therefore they essentially do not overlap . i first verify that binding is easy to produce in this system , although an @xmath52 consistent with the expected nucleon radius @xmath96 fm would produce a small binding energy . then the case of the excessively small @xmath84 , that provides a large binding for the @xmath35 system , is investigated . this would already correspond to an extremely large nucleon mean radius @xmath97 , nevertheless let us check that a larger binding can be obtained for the @xmath98 . in this case i find that all the three one hadron binding energies are similar to -40 mev . when the potential energies are subtracted in eq . ( [ hartree energy ] ) , the final binding energy results in @xmath99 -40mev which is consistent with a @xmath0 mass of @xmath100 gev . this remains larger than a mass of @xmath101 gev for the @xmath0 , however a complete computation of the effect of coupled channels on the binding energy remain to be estimated . the length @xmath102 of the @xmath98 linear molecule is larger than @xmath103 fm , even when we have a small nucleon , with @xmath104 mev . a similar binding energy , and a similar mass of the order of @xmath105 gev can be computed for the @xmath106 system in the non - exotic iso - doublet . in the case of the @xmath107 , the mass is already 400 mev higher than the mass of the @xmath108 system , and 200 mev higher than the mass of the @xmath109 system . in this case the p - wave excitation may also be relevant , as suggested by the chiral soliton model and by the diquark model . moreover the decay channels are also p - wave channels . this motivates the study of the coupling of the s - wave crypto - heptaquark system to the p - wave pentaquark systems . the simplest model to couple a s - wave crypto - heptaquark to a p - wave pentaquark consists in assuming the standard @xmath110 quark - antiquark creation / annihilation potential . the coupling form factor is computed with the overlap of the s - wave crypto - heptaquark @xmath111 with the p - wave pentaquarks @xmath112 ( or @xmath113 ) plus a flavour singlet @xmath110 quark - antiquark pair . between these two wave - functions we must sandwich the quark antisimmetriser . after separating the center of mass coordinates , the coupling is equivalent to an overlap of harmonic oscillator wave - functions of the six jacobi coordinates depicted in fig . [ jacobi coordinates ] . the ribeiro graphical @xcite rules , and the harmonic oscillator energy conservation , require that at least one of the heptaquark coordinates should include a radial excitation . the @xmath61 relative coordinate is expected to be partly excited because the @xmath61 are repelled . if we compare our coupling with the decay of the rho into two pions we find a suppression in the coupling by a factor of 1/3 . the decay width is proportional to the square of the coupling and this factor already explains the small decay width of heptaquarks which are found in all experiments to be smaller than 20 mev . moreover the ribeiro rules show that overlaps with excited wave - functions are further suppressed . however , for a consistent computation , the probability for a @xmath110 annihilation to occur in a pentaquark remains to be consistenly studied . nevertheless the p - wave pentaquark components should essentially not affect the mass of the crypto - heptaquark . the relative comparison of the @xmath1 and @xmath0 decay widths can be estimated with a better precision than the full computation of the decay . the decay widths depend on the phase space , and they are proportional to @xmath114 in the non - relativistic case and to @xmath115 in the ultra - relativistic case , when the momentum is smaller that the inverse radius @xmath52 . therefore i expect that the partial decay width of the @xmath0 to a p - wave @xmath116 is larger than the @xmath1 decay to a p - wave @xmath117 by a factor of @xmath118 , and i expect that the partial decay width of the @xmath0 to a p - wave @xmath119 is larger than the @xmath1 decay to a p - wave @xmath117 by a factor smaller than @xmath120 . thus it is expected that the total decay width of the @xmath0 is larger by a factor @xmath121 than the decay width of the @xmath1 . nevertheless , because the decay width of the @xmath1 is quite small , the coupling of the crypto - heptaquark to the p - wave pentaquark is not expected to be large . to conclude , in this paper i address the exotic anti - decuplets with a standard quark model hamiltonian , where the quark - antiquark annihilation is constrained by the spontaneous breaking of chiral symmetry . i first derive a criterion showing that the @xmath1 and @xmath0 hadrons very recently discovered can not be an s - wave or p - wave pentaquark . it is plausible that the @xmath0 is a linear @xmath122 molecule , a heptaquark state . the @xmath11 , @xmath1 and the @xmath6 , @xmath67 are also heptaquarks , or linear meson - meson - baryon molecules . in these linear molecules the two external hadrons do not overlap . all these heptaquarks have a positive parity . the only pentaquark is the iso - triplet @xmath70 , with the quantum numbers of a tightly bound s - wave @xmath82 system , a negative parity system . the spectrum of the pentaquark and heptaquark antidecuplets is depicted in figure [ triangle ] , assuming that the experimental @xmath123 gev and @xmath124 gev are correct . importantly , the @xmath0(1862 ) is expected to decay both to the @xmath116 and to the @xmath125 p - wave channels . for a future improvement of this work it is important to discuss the size parameter @xmath97 . in our previous paper @xcite we used a quite large @xmath126 fm@xmath127 , for the potential involving the pion . the pion is expected to couple with a shorter range interaction than other hadrons , because the adler zero suppresses the low momentum part of the pionic couplings . nevertheless the very short range pion potential and the longer range kaon potential suggest that the hadron - 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i study the very recently discovered s=-2 resonance @xmath0 ( 1886 ) by the na49 collaboration at the cern sps . this resonance was already predicted , with a mass close to 1.9 gev , in a recent publication mostly dedicated to the s=1 resonance @xmath1(1540 ) . to confirm this recent prediction , i apply the same standard quark model with a quark - antiquark annihilation constrained by chiral symmetry . this method also explained with success the repulsive hard core of nucleon - nucleon , kaon - nucleon exotic scattering , and the short range attraction present in pion - nucleon and pion - pion non - exotic scattering . i find that repulsion excludes the @xmath0 as a @xmath2 s - wave pentaquark . i explore the @xmath0 as a heptaquark , equivalent to a @xmath3 linear molecule , with positive parity and total isospin @xmath4 . i find that the kaon - kaon repulsion is cancelled by the attraction existing in the kaon - nucleon channels . in our framework this state is easier to bind than the @xmath1 described by a @xmath5 borromean bound - state . the remaining @xmath6 doublet and @xmath7 triplet of the exotic anti - decuplet are also studied , and the coupling to p - wave decay channels is briefly addressed .

the precision of measurements may in principle be improved using quantum mechanical effects , viz . , the quantum aided metrology @xcite . a convenient way of quantifying the precision in parameter estimation , for instance , is via the quantum fisher information ( qfi ) @xcite . also , non - classical states of light have become a promising resource for the improvement of parameter estimation @xcite . amongst the quantum states of light ( probe states ) which might bring advantages to phase estimation and that have been already discussed in the literature , we may cite the continuous - variable states , either one - mode ( squeezed ) states @xcite or two - mode ( entangled coherent ) states @xcite . interestingly , schemes employing continuous - variable states in the low photon number regime have a superior performance when compared to schemes using other types of non - classical states , for instance , the noon " states @xcite . it has also been demonstrated the optimality of squeezed states in ideal phase estimation , as well as the fact that the qfi scales quadratically with the mean photon number if squeezed states are employed in place of coherent states ( linear scaling ) @xcite . it is therefore of importance to seek other probe states that could lead to more efficient protocols . we remark that in the above mentioned works , features such as squeezing and entanglement have been considered separately in different protocols . thus , we may ask ourselves : could we have any advantage if we use continuous - variable entangled states also exhibiting squeezing ? in this work we investigate the adequacy , for phase estimation purposes , of interpolated quasi - bell squeezed states , which are continuous - variable entangled states having squeezed coherent states @xmath0 as component states . it should be noted that , under more realistic conditions , if one considers external unwanted influences such as noise @xcite or even a unitary disturbance " @xcite , the accuracy of the estimation may be considerably degraded . accordingly , we will analyze the phase estimation using quasi - bell squeezed states also taking into account a linear unitary disturbance in the derivation of the qfi . our manuscript is organized as follows : in section 2 we present the calculations of the qfi and study the quantum phase estimation using quasi - bell squeezed states . we first consider the ideal case and also investigate the effect of a linear disturbance . the role of entanglement in the phase estimation is also discussed in that section . in section 3 present our conclusions . a class of interesting continuous variable states are the quasi - bell squeezed states , defined as @xcite @xmath1 is the squeezing parameter , with @xmath2 and @xmath3 . the overlap between the different component states is given by @xmath4\right\}$ ] . we have also introduced the auxiliary interpolating parameter @xmath5 @xmath6 , and @xmath7 is the normalization factor @xmath8 . for our purposes we are going to consider only the input states @xmath9 . we would like to remark that the interpolation parameter @xmath5 is intimately related to the entanglement of the state @xmath9 and this will allow us to identify the role of entanglement in the phase estimation process without compromising the other parameters involved . we have that , for @xmath10 the state @xmath9 is a maximally entangled state , while for @xmath11 the state is partially entangled in general , becoming maximally entangled only in the limit of @xmath12 . for @xmath13 we have a product state , and mode @xmath14 of @xmath9 is reduced to the squeezed coherent state @xmath0 . we consider the following phase rotation transformation @xmath15 applied to mode @xmath14 of the entangled state @xmath9 : given that the involved state is pure and the transformation is unitary , the qfi is given by : @xmath16\ , . \label{qfimain}\ ] ] expanding the expression in eq . ( [ qfimain ] ) we obtain : @xmath17\,,\end{aligned}\ ] ] where the mean photon number in mode @xmath14 of state @xmath9 is @xmath18\ , , \label{nina}\ ] ] with @xmath19 being the `` average photon number in the component state @xmath20 '' and @xmath21\right\}$ ] . the most challenging terms to compute are of the kind @xmath22 ( the @xmath14 subscripts have been omitted ) and can be obtained through @xmath23 where @xmath24 is the operator @xmath25 after the transformation @xmath26 . in this way , the qfi can be computed straightforwardly term by term . after some manipulations we obtain : @xmath27- \left(n_{\mathsf{in},a}\right)^{2}\right\ } \,,\ ] ] where we have defined @xmath28+\frac{1}{2}\left(4\alpha^{2}-1\right)\cosh(2r)+\frac{3}{8}\cosh(4r)+\frac{1}{8}\,,\end{aligned}\ ] ] and @xmath29\\ & = & \frac{1}{4}\kappa\left\ { 2\alpha^{2}\left[\alpha^{2}\left(1 + 2\sinh^{2}(4r)\cos(2\theta)+3\cosh(8r)\right)-2(\sinh(2r)+3\sinh(6r))\cos\theta\right.\right.+\\ & & \left.-6\cosh(6r)\right]+4\cosh(2r)\left[4\alpha^{2}\left(2\alpha^{2}\cosh(4r)+1\right)\sinh(2r)\cos\theta-\alpha^{2}-1\right]+\\ & & \left.+\left(8\alpha^{2}+3\right)\cosh(4r)+1\right\ } \,.\end{aligned}\ ] ] naturally , if we let @xmath13 we re - obtain the following equation derived by monras @xcite @xmath30 + \cosh(4r)-1\,.\label{qfilzero}\ ] ] we now re - parametrize the qfi as a function of @xmath31 and the squeezing fraction of the component state @xmath20 " , @xmath32 . in this approach , we should not interpret the parameters @xmath31 and @xmath33 as having any specific physical meaning ; they are just two auxiliary parameters that will be useful to compare the results of this work with the non - entangled case @xcite . the energy of the input state depends on the parameters @xmath31 and @xmath33 of the component state @xmath20 as well as on the interpolating parameter @xmath5 . for this reason , we represent the qfi as a function of the input average photon number @xmath34 [ eq . ( [ nina ] ) ] in mode @xmath14 of the entangled state @xmath9 and as a function of the parameter @xmath33 . it is not an easy task to algebraically invert eq . ( [ nina ] ) to obtain @xmath31 explicitly as function of @xmath34 , and we do this numerically , adjusting the value of @xmath31 in order to get the desired input photon number @xmath34 . in fig . [ fig : qfi - zero - eta ] ( _ top _ ) we plot the qfi as a function of the squeezing fraction @xmath33 . the optimal probe state is the one capable of reconciling the gains due to entanglement without loosing the gains due to squeezing . we notice that when @xmath35 we have a `` squeezing fraction '' @xmath36 for which the qfi is greater than the one obtained with non - entangled states . however , when @xmath37 , although we have a state that may be maximally entangled when @xmath38 , we notice that we do not have any increase for the qfi . thus we conclude that the best strategy is to spend all the energy in squeezing the state . to understand this phenomenon , we analyse the parameter @xmath31 that the component state @xmath20 must have in order to let the input photon number be the available value @xmath34 ( fig . [ fig : qfi - zero - eta](_bottom _ ) ) . because the energy @xmath34 increases for @xmath38 , we must reduce @xmath31 to keep the energy @xmath34 fixed while we change @xmath5 . this implies a reduction of the qfi when @xmath38 , unless we make @xmath39 and the optimal state is not entangled . [ cols="^,^,^ " , ] we have verified that if the parameter @xmath5 is negative , the optimal input state is not an entangled state , and the mode @xmath14 of this state corresponds to what has been previously found @xcite , i.e. , the qfi is given by @xmath40 we remind that in the case of phase estimation with single - mode states we have @xmath41 . if @xmath5 increases from @xmath42 to @xmath43 , though , the qfi increases if there is enough energy . in this range of values for the energy , increasing @xmath33 past @xmath44 leads to a reduction of entanglement . this is because the components of the quasi - bell state become two squeezed vacuum states , and the overlap @xmath45 increases @xcite . for this reason , @xmath46 is not equal to @xmath43 , which reconciles the gains due to entanglement with the gains due to squeezing for the phase estimation . in fig . [ fig : emaranhamento - sonda - otima ] we plot the entanglement of the optimal probe state for various interpolation parameters @xmath5 . we draw attention to the fact that even when we take @xmath47 , entanglement is not imposed , because the state is not entangled if @xmath41 , as it happens for @xmath37 . moreover , in the case of a single mode squeezed state ( when there was no additional parameter @xmath5 ) @xmath41 was indeed the optimal value for the squeezing fraction " . this means that the phase estimation with linear unitary disturbance could be upgraded by the use of entangled states . the qfi is gradually increased when we allow the state to be more and more entangled ( increasing @xmath5 ) , so it seems natural to look for a more direct relation between the entanglement of the quasi - bell states and the resulting qfi . when we found the optimal parameters @xmath33 and @xmath48 for the input state by maximizing the qfi , we removed the dependence of the qfi on those parameters . we are now able to observe the dependence of the qfi on the interpolation parameter @xmath5 , which fixes the maximal entanglement of the input state . because both the qfi and the entanglement are monotonic functions of @xmath5 , for @xmath35 , we can represent the qfi directly as a function of the entanglement for this ( positive ) @xmath5 semi - axis . in fig . [ fig : qfi_functionof_e ] we notice that the qfi increases monotonically as the entanglement of the probe state increases , showing that entanglement is a resource for quantum phase estimation even if there is a unitary disturbance in the system . the qfi is an increasing function of the parameter @xmath49 of the disturbance because there is an energy increase parameterized by @xmath49 during the transformation . the average photon number in the mode a of the output state may be calculated in the following way : @xmath50 using again eq . ( [ auxiliareq ] ) we obtain , in the limit of @xmath51 : @xmath52\,,\ ] ] where we have defined @xmath53\\ & = & 2\kappa\left\ { \eta^{2}+\sinh^{2}r-2\alpha^{2}\left[\sinh\left(4r\right)\cos\theta+\cosh\left(4r\right)\right]\right\ } \,.\end{aligned}\ ] ] we notice that , as expected , for @xmath54 and @xmath51 , @xmath55 reduces to @xmath56 in order to perform a more precise analysis of how the qfi depends on the disturbance , we plot in fig . [ fig : qfi - functionof - nout - entanglement ] the qfi as a function of the average photon number in mode @xmath14 for @xmath57 ( similarly to what is done in @xcite ) . of course for @xmath13 we re - obtain the previous results for single - mode gaussian states @xcite . we note that the presence of the disturbance @xmath49 affects the phase estimation also when the probe state is entangled , if the total available energy ( input state + transformation ) is taken into account . however , we observe that the qfi may attain larger values than in the non - entangled case @xcite , showing once more the advantages of using entangled states for quantum phase estimation even in the presence of a unitary disturbance . finally , we may analyze the feasibility of the adjustment of the parameters that were optimized along this work . firstly we remark that the plot in fig . [ fig : qfi - functionof - nout - entanglement ] corresponds to the behavior of the left ends ( @xmath57 ) of the plots in fig . [ fig : qfi - functionof - l ] ( _ bottom _ ) , when we consider higher energies . for @xmath57 , the value of @xmath46 is well defined , but it may be hard to reach it for a total input photon number larger than @xmath58 . this is because , in this case , we have @xmath59 and the sensitivity of the qfi upon a very small variation of @xmath33 around @xmath46 is very high . the plot in fig . [ fig : qfi - functionof - nout - entanglement ] represents then only a theoretical indication of how entanglement may be useful for phase estimation . in this work we have analyzed the use of the interpolated quasi - bell states as input probes for quantum phase estimation . we have compared the qfi in the ideal case with the qfi when a unitary disturbance is included in the hamiltonian which determines the evolution . we have found that the use of continuous variable entangled states based on coherent squeezed states makes possible to increase the precision for phase estimation , specially when the total average photon number is not negligible ( @xmath60 ) . we have verified that the qfi is an increasing function of the interpolation parameter @xmath5 ( for @xmath35 ) , which is related to the entanglement contained in the optimal probe state . we have also observed that the larger the unitary disturbance parameter @xmath49 , the larger will be the energy of the output state , which increases the qfi . however when we consider all the energy spent in the process ( including the energy used in the transformation ) we found that the disturbance actually impairs the phase estimation . we also highlight that for input states with higher energy , the `` optimal squeezing fraction '' parameter @xmath46 must be finely adjusted in order to maximize the qfi when @xmath11 . d.d.s . acknowledges fundao de amparo pesquisa do estado de so paulo ( fapesp ) grant no 2011/00220 - 5 , brazil . a.v.b . acknowledges partial support from conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) , inct of quantum information , grant no 2008/57856 - 6 , brazil . yonezawa , h. , nakane , d. wheatley , t.a . , iwasawa , k. , takeda , s. , arao , h. , ohki , k. , tsumura , k , berry , d.w . , ralph , t.c . , wiseman , h.m , huntington , e.h . , and furusawa , a. : quantum - enhanced optical - phase tracking . science 337 , 1514 ( 2012 ) .

in this paper we analyze the quantum phase estimation problem using continuous - variable , entangled squeezed coherent states ( quasi - bell states ) . we calculate the quantum fisher information having the quasi - bell states as probe states and find that both squeezing and entanglement might bring advantages , increasing the precision of the phase estimation compared to protocols which employ other continuous variable states , e.g. , coherent or non - entangled states . we also study the influence of a linear ( unitary ) perturbation during the phase estimation process , and conclude that entanglement is a useful resource in this case as well . * quantum phase estimation with squeezed quasi - bell states * douglas delgado de souza and a. vidiella - barranco + +

[ [ volume - conservation - and - kinetic - constraints . ] ] volume conservation and kinetic constraints . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we begin by considering an ideal wound - healing geometry where the dimensions of the cell layer are by the instantenous height @xmath2 ( in the z - direction ) , x - length @xmath3 , and a y - length @xmath104 . we further consider a thin film , letting @xmath150 , such that the changes in @xmath104 are negligible compared to @xmath94 and @xmath95 and assume that there is translational invariance along the y - direction . if the initial height of the cell layer is @xmath47 and the length is @xmath46 , we define vertical and horizontal strains as , @xmath151 , and @xmath152 . volume conservation implies , @xmath153 constant . this leads to the following ( related ) kinetic constraints , the homogeneous steady - state of eqs . ( 1 - 2 ) in the main text is given by @xmath70 and @xmath71 . this is a quiescent steady - state when the cell layer does not spread . to determine the inhomogeneous steady - state describing an expanded cell monolayer , we seek solutions of eqs . ( 1 - 2 ) by setting @xmath158 and @xmath159 . to make analytical progress , we first consider a cell monolayer of length much larger than the length scale of variations in the polarization , i.e @xmath160 . in this limit the spreading force can be described by a signum function , @xmath161 for @xmath162 . next we linearize the active stress by considering small deviation of @xmath28 from its rest - state @xmath31 , @xmath163 . the steady - state solution for the local stress is given by , @xmath164 where @xmath165 and @xmath166 are respectively the length and the height of the cell monolayer at @xmath167 , given by @xmath168 and @xmath169 . the spatial average of the steady - state strain @xmath170 is given by , @xmath171 combining eq . with the equation for @xmath172 we obtain a second order ordinary differential equation in @xmath173 whose analytical solution is , where @xmath175 is the effective elastic modulus renormalized by active contractility and @xmath176 is a characteristic length scale associated with the spatial variation of the active agents . the steady - state solution for the average strain is , @xmath177 the condition for the cell monolayer to expand is given by @xmath178 which leads to a critical value for the force density , @xmath179 above which the cell monolayer spreads . in this section we examine the linear stability of the homogeneous steady - state . we render our system of equations dimensionless by letting @xmath180 and @xmath181 . then new dimensionless parameters are @xmath182 , @xmath183 , @xmath184 , @xmath185 , @xmath186 and @xmath187 . in these units , stress is measured in the units of @xmath17 and the concentration of contractile agents are expressed in units of @xmath31 . in the following we drop the tilde notation over the dimensionless parameters for simplicity . letting @xmath188 and @xmath189 , the linearized equations for the strain and the concentration fields are given by , taking two spatial derivatives in eq . and substituting the expression for @xmath191/\beta$ ] ( obtained from eq . [ eq : delta_strain ] ) , we obtain the effective dynamics of strain fluctuations , @xmath192 where @xmath81 and @xmath82 . the dynamics of strain fluctuations are isomorphic to the dynamics of an driven damped kelvin - voigt material , with a characteristic frequency of oscillation @xmath193 given as a function of the wavelength @xmath194 , @xmath195 } \label{eq : freq}\;.\ ] ] we compare the analytical prediction for the time period , @xmath85 , with the time period determined numerically by performing a fast fourier transform on the solution for the strain rate at the center of the cell monolayer . the two values are in good agreement as shown in fig . 3a of the main text for @xmath196 . next we look for solutions in the form @xmath197 . the two eigenvalues controlling the dynamics of fluctuations are given by the following dispersion relations , @xmath198 ^ 2-\dfrac{4h_0}{\tau } \left[(1 + d\tau q^2)b+\alpha \beta\right]q^2}\ ] ] where @xmath199q^2 $ ] . if the coupling to the concentration of contractile elements is neglected ( @xmath200 ) , @xmath201 and @xmath202 ( we neglect the diffusion constant @xmath40 for simplicity assuming that its contribution is small ) . in this case the elastic deformations are stable and diffuse through the cell layer and no oscillations are observed . when the coupling to the concentration field is considered , we find a region in the parameter space , @xmath203>0 $ ] , where the linear fluctuations are unstable . the system is purely diffusive when @xmath204=0 $ ] and @xmath203<0 $ ] . the region in the parameter space defined by the complex values of the fourier modes , i.e @xmath204 \neq 0 $ ] , describe oscillatory solutions . furthermore , these propagating waves are stable when @xmath203<0 $ ] . the mean - field model admits an expanded steady - state solution for the gel with strain @xmath205 , and an average concentration of contractile agents @xmath206 . the strain fluctuations decay with similar dynamics as in eq . . with @xmath207 , we get for the dynamics of @xmath208 , @xmath209 where , @xmath210 and @xmath211 . the time - period for oscillations is given by , @xmath212 . since the mean - field model neglects diffusion , the effective viscosity @xmath213 characterizing the dissipation of strain - rate is less than @xmath214 . we thus add an additive correction @xmath215 to @xmath213 in order to accurately estimate the numerical value for the effective viscosity in our numerical analyses . thus the condition for oscillatory solutions is given by , @xmath216 ^ 2 $ ] . [ [ turnovers - in - contractility - are - essential - for - stress - wave - propagation . ] ] turnovers in contractility are essential for stress wave propagation . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + , ( c , d ) monolayer stress @xmath22 , ( e , f ) concentration of contractile elements , ( g , h ) traction stress @xmath217 and ( i , j ) strain rate @xmath218 in the non conserved case ( left column ) and conserved case ( right column ) of active units . parameters : @xmath57 pa , @xmath60 min , @xmath219 @xmath220 , @xmath62 @xmath29 m , @xmath63 @xmath29 m , @xmath221 pa , @xmath222 nn min/@xmath29m@xmath67 , @xmath68 @xmath29m@xmath69/min . for the magnitude of the contractile stress , we used @xmath58 pa for the non conserved case and @xmath117 pa for the conserved case . ] is increased ( left to right ) . ( a , d , g ) x - waves , @xmath142 nn/@xmath29 m ; ( b , e , h ) traveling stress pulse , @xmath143 nn/@xmath29 m ; ( c , f , i ) complex oscillatory patterns , @xmath144 nn/@xmath29 m . parameters : @xmath145 @xmath29m / min , @xmath146 @xmath29m / min , @xmath147 @xmath29m@xmath69/min , @xmath223 min , @xmath224 min . other parameter values are the same as in main text . ] in the main text , we describe an effective one - dimensional model for a spreading cell monolayer by assuming a translational invariance along y - direction . here we formulate the dynamics of the spreading cell layer in two dimensions . the cell layer is modeled as an elastic medium whose local deformations are characterized by a displacement field @xmath225 about its initially undeformed state . contractile activity in the constituent cells are described in terms of a scalar concentration field @xmath226 of actomyosin stress fibers . in addition , we describe local cell polarization by a vector field @xmath227 , accounting for orientation of actin stress fibers . in the thin - film limit , where monolayer height is negligible compared to its planar dimensions , the thickness - averaged dynamics of the deformation field @xmath225 is given by , @xmath228 where @xmath229 is the mechanical stress tensor given by a sum of elastic stress ( @xmath230 ) and the active stresses ( @xmath231 ) due to contractility and stress fiber polarization , @xmath232 . assuming linear elasticity , the elastic stress tensor is simply given by , @xmath233 where @xmath25 and @xmath234 are the compressional and shear elastic moduli , respectively , and @xmath235 is the symmetrized strain tensor , @xmath236 . the active stress comes from cell contractility , cell proliferation and polarization fluctuations . it is given by , @xmath237 where @xmath238 is an internal pressure due to cell proliferation , @xmath33 describes the mean contractile stress exerted by actomyosin units . the parameters @xmath239 and @xmath240 describe tensions induced by local gradients in the polarization field . the dynamics of the concentration field @xmath226 is given by , @xmath241 where @xmath40 is the diffusion constant ; @xmath36 is the timescale of turnovers in contractility ; @xmath242 and @xmath243 are the rates of production of @xmath28 due to compressive mechanical and polarization strains . the dynamics of the polarization field @xmath227 is given by , @xmath244 where the first two terms ( with @xmath126 ) allow for the onset of a homogeneously polar state , @xmath245 , when @xmath127 . local cost of fluctuations in polarization is characterized by an isotropic stiffness @xmath128 ; @xmath246 , @xmath247 and @xmath248 describe strength of alignment of cell polarization with gradients of elastic strain and the concentration field . together , eqs . , and describe the planar dynamics of the cell monolayer . while the model parameters are cell - type dependent , their values are chosen to quantitatively capture available experimental data in mdck colonies @xcite . the model parameters are tuned to capture the experimental data on velocity , strain rates , traction stress and intercellular stresses , while the remaining values are chosen within the order of magnitudes reported in prior literature . specifically , we choose an initial length of the monolayer @xmath249 m , cross - sectional area @xmath250 , spreading force @xmath251 and friction @xmath252 . the timescale to reach a spread steady - state is thus @xmath253 . the timescale controlling activity turnover , @xmath36 , and the timescale controlling the strain - concentration feedback , @xmath254 , are chosen smaller than @xmath255 but of the same order . we assume that diffusion is negligible and tend to represent the tendency of neighboring cells to equalize the concentration of active agents . in our simulation we set the diffusion length to be @xmath256 m ( one grid size ) per minute .

coordinated motion of cell monolayers during epithelial wound healing and tissue morphogenesis involves mechanical stress generation . here we propose a model for the dynamics of epithelial expansion that couples mechanical deformations in the tissue to contractile activity and polarization in the cells . a new ingredient of our model is a feedback between local strain , polarization and contractility that naturally yields a mechanism for viscoelasticity and effective inertia in the cell monolayer . using a combination of analytical and numerical techniques , we demonstrate that our model quantitatively reproduces many experimental findings [ nat . phys . * 8 * , 628 ( 2012 ) ] , including the build - up of intercellular stresses , and the existence of traveling mechanical waves guiding the oscillatory monolayer expansion . + pacs numbers : 87.10.ca , 87.18.fx , 87.18.gh many developmental processes , such as embryogenesis @xcite , tissue morphogenesis @xcite , wound healing @xcite and cancer metastasis @xcite , involve collective cell migration @xcite and long - scale force generation , which in turn rely on the interplay of cell - cell cohesion , cell adhesion to the extracellular matrix , as well as myosin based contractility @xcite . recent experiments reveal that unconstrained tissue expansion is accompanied by propagating mechanical waves and build - up of intercellular stresses @xcite . these waves are controlled by expressions of myosin activity , cell - cell adhesion and cytoskeletal remodeling . these findings pose a fundamental physical question : how do waves arise in over - damped active elastic media ? what are the underlying spatio - temporal patterns governing stress propagation in dense expanding cell layers ? active materials encompass a wide range of living and non - living systems with inborn mechanical stresses regulated by chemical reactions . generic descriptions of the dynamics of such materials predict a broad class of non - equilibrium states including spontaneous flow , wave propagation and pattern formation @xcite . while the dynamics of active fluids have been extensively studied , quantitative descriptions of active contractile materials are much less developed . recent work has suggested that a polarized elastic medium driven by chemical agents can exhibit finger - like protrusions and internal stress accumulation during expansion @xcite . it remains unclear , however , how cell contractility , polarization or tissue cohesion influence stress generation and wave propagation . earlier work by two of us and others showed that the coupling of mechanical and chemical degrees of freedom can lead to an effective inertia and sustained propagation of waves @xcite . related models also emphasize that turnovers in actomyosin activity are essential to capture spontaneous oscillations in cell cytoskeleton @xcite . in this letter , we propose a new mechanism of stress propagation in multicellular materials based on a local feedback between elastic deformations and cell contractility . we consider a minimal model for an expanding cell monolayer , described as an elastic continuum coupled to an internal degree of freedom , the concentration of active contractile units . the assumption of elasticity is supported by experimental evidence that in cohesive cell layers stress and strain tend to be in phase , as in elastic materials @xcite . the contractile units represent actomyosin assemblies that locally generate contractile stresses in the cells . we propose that tissue expansion promotes the rate of assembly of these contractile units , leading to larger contractile forces that can compete with propulsion forces . this mechano - chemical feedback successfully captures the experimentally observed stress waves @xcite . the steady state of such a system is described by polarization being largest at the edges and lowest at the center . a scaling model for the expanding cell layer captures the mechanical oscillations and predicts self - sustained periods of stiffening and fluidization in the tissue . [ [ continuum - model - for - spreading - cell - layer . ] ] continuum model for spreading cell layer . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we consider a thin film of cell monolayer spreading in the @xmath0-@xmath1 plane , with height @xmath2 and length @xmath3 at time @xmath4 ( fig . [ fig : stress]a , inset ) . in the absence of external forces , in - plane force - balance gives @xmath5 , where @xmath6 is the stress tensor and the latin indices denote in - plane coordinates @xmath7 . for @xmath8 @xmath9 , the @xmath0 and @xmath1 linear extensions of the cell layer , we average the force - balance equation across the z - direction to obtain @xmath10 , where @xmath11 , assuming that the top layer ( @xmath12 ) is stress free . the shear stress at the cell - substrate interface is the traction stress exerted by the cell on the substrate . it is given by , @xmath13 , with @xmath14 the friction density , @xmath15 the elastic displacement field , @xmath16 the cell polarization and @xmath17 the propulsion force per unit cross - sectional area . the term @xmath18 is supported by the experimental observation that the local velocity of expanding monolayers is generally not aligned with traction , requiring the existence of an internally generated driving force associated with cell motility @xcite . both @xmath14 and @xmath17 are controlled by integrin - mediated cell - environment interactions . we further simplify the model by assuming translational invariance along the y - direction . the equation of motion governing the displacement field , @xmath19 , of the cell layer is ( @xmath20 ) , @xmath21 where @xmath22 is the internal stress in the monolayer , @xmath23 . it is given by the sum of an internal pressure ( @xmath24 ) , an elastic stress , with @xmath25 the compressional elastic modulus and @xmath26 the strain field , and an active stress @xmath27 that depends on the concentration @xmath28 of active contractile units , such as phosphorylated myosins interacting with actin filaments . the constant pressure @xmath24 accounts for internal growth due to cell proliferation which is assumed negligible without loss of generality . the active stress is proportional to the chemical potential of the active species , @xmath29 , which we take proportional to the logarithm of the concentration of the species . we thus have @xmath30 , where @xmath31 is the concentration of contractile elements in equilibrium ( @xmath32 ) and @xmath33 the magnitude of the contractile stress . the dynamics of the concentration field @xmath34 is given by , @xmath35 where @xmath36 is the timescale of turnover of the contractile elements , @xmath37 is the rate of production of @xmath28 due to local extension ( or degradation due to contraction ) and @xmath38 is the current responsible for transport of these active units . this is in contrast to our earlier works @xcite , where the strain field enters the dynamics of @xmath28 through the decay rate . the total current is a sum of diffusive and convective fluxes , @xmath39 , where @xmath40 is an effective diffusion constant , describing the tendency of neighboring cells to equalize activity levels . together eqs . and define the dynamics of the spreading monolayer , given the form of @xmath41 , the boundary and initial conditions . we first consider the case of constant but non - uniform propulsion force given by @xmath42 where @xmath43 is a length scale controlling the width of the transition zone from left moving to right moving cells at the center of the monolayer ( see fig . [ fig : stress]a ) . the length of the spreading layer at time @xmath4 is given by , @xmath44 , and the height is determined by the condition of volume conservation , @xmath45 , with @xmath46 and @xmath47 the initial length and height of the monolayer , respectively . the boundary of the monolayer is stress free , i.e. , @xmath48 at all times . we assume that the monolayer is initially undeformed , @xmath49 , with an equilibrium concentration of contractile elements , @xmath50 , and choose a no - flux boundary condition for @xmath28 , @xmath51 . ) are indicated by arrows and the colormap denotes local magnitude of monolayer stress . bottom : profile of cell polarization . ( b ) time - evolution of the internal stress @xmath22 in the monolayer . ( c ) time - evolution of the concentration of contractile units , @xmath28 , normalized by its equilibrium value . ( d ) midline stress @xmath52 ( blue solid ) , midline strain @xmath53 ( blue dashed ) and midline strain rate @xmath54 ( red solid , units @xmath55 s@xmath56 ) as functions of time . parameters : @xmath57 pa , @xmath58 pa , @xmath59 , @xmath60 min , @xmath61 min@xmath56 , @xmath62 @xmath29 m , @xmath63 @xmath29 m , @xmath64 pa , @xmath65 m , @xmath66 nn min/@xmath29m@xmath67 , @xmath68 @xmath29m@xmath69/min . ] [ [ propagating - waves . ] ] propagating waves . + + + + + + + + + + + + + + + + + + in the absence of propulsion force ( @xmath32 ) , the cell layer is in a quiescent homogeneous state , with @xmath70 and @xmath71 . when @xmath72 , the cell layer spreads and reaches a steady - state at long times . we have integrated numerically eqs . ( [ eq : u],[eq : c ] ) with the given initial and boundary conditions , using the runge - kutta - fehlberg method . the model parameters are chosen to quantitatively describe the available experimental data for mdck colonies @xcite . the phase diagram shown in fig . [ fig : waves]a displays three dynamical regimes in terms of contractile activity @xmath73 and compressional modulus @xmath25 ( controlled by cell - cell adhesion ) : a region where fluctuations are stable and diffusive at low contractility , an intermediate region where the system supports propagating waves , and a region where the propagating waves become unstable at high contractility . there is good agreement between the boundaries obtained via numerical solution of the full nonlinear equations ( red diamonds ) and those determined by the linear instability of fluctuations about the equilibrium , undeformed state @xcite and about the long - time solution of the mean - field model in eqs .. in the region of propagating waves , the stress initially shows a few local maxima ( fig . [ fig : stress]b ) , which evolve towards a single maximum at the center of the monolayer , as observed in experiments @xcite . the concentration of contractile elements also oscillates and builds up at the center of the monolayer ( fig . [ fig : stress]c ) . the stress waves propagate nearly in phase with the strain field , whereas the strain rate fluctuates nearly out of phase with the stress ( fig . [ fig : stress]d ) . thus the response of the material is dominated by elastic relaxation with dissipation induced by turnovers in contractility on a timescale @xmath36 . the waves span the entire length of the monolayer and consist of a strain rate wavefront that propagates inwards from the edge , and then travels back to the edge , resembling an x - pattern , as observed experimentally @xcite . with the given parameter values our numerical simulations capture the mechanical waves as evident in the kymographs of stress , strain rate and concentration of contractile units ( fig . [ fig : waves]b - d ) . and the horizontal axis is the compressional modulus @xmath25 . three behaviors are observed : stable diffusive , stable propagating waves , and oscillatory instability . the red squares are obtained from the numerical solutions of the full nonlinear model , the black solid lines are the results of the linear stability analysis ( lsa ) of the equilibrium state ( at @xmath74 ) @xcite , and the dashed green lines refer to the lsa of the mean - field model given in eqs . . kymographs of ( b ) the monolayer stress field , ( c ) strain rate @xmath75 , and ( d ) @xmath76 . the parameter values are taken to be the same as in fig . [ fig : stress ] . ] to understand the origin of wave propagation and estimate the wave frequency , it is useful to examine the linear fluctuations in the strain field , @xmath77 and the concentration field @xmath78 , about the quiescent homogeneous state , @xmath70 , @xmath71 and no spreading force . using eqs . and , one can then eliminate @xmath78 from such linearized equations to obtain the linearized dynamics of strain fluctuations , @xmath79 the above equation shows that the coupling of strain to concentration field yields an effective mass density ( inertia ) , @xmath80 , and viscoelasticity characterized by an effective elastic modulus , @xmath81 , and an effective viscosity @xmath82 . the dynamics of strain fluctuations resembles a damped kelvin - voigt oscillator with a characteristic frequency of oscillations , @xmath83 , with @xmath84 the wavevector . the estimate for the time period @xmath85 agrees well with the time period determined from numerics for @xmath86 ( see fig . [ fig : meanfield]a ) and with the value measured in recent experiments @xcite . finally , we note that if the concentration @xmath28 is conserved ( @xmath87 ; @xmath88 ) , stable propagating waves are spontaneously generated for @xmath89 . if diffusion is slow compared to elastic relaxation , @xmath90 , stable propagating waves are not observed @xcite . in the opposite limit of infinitely fast turnovers in contractility ( @xmath91 ) , strain fluctuations decay diffusively at a rate @xmath92 . [ [ mean - field - model . ] ] mean field model . + + + + + + + + + + + + + + + + + the mean field limit of the continuum model is obtained by neglecting spatial variations in @xmath28 and @xmath93 and it is formulated in terms of the length ( @xmath94 ) , height ( @xmath95 ) , and the average concentration of contractile elements , @xmath96 , with [ mf ] @xmath97 with @xmath98 the propulsion force , @xmath99 the friction , @xmath100 the cross - sectional area , @xmath101 the strain and @xmath102 the internal stress given by @xmath103 . the height is determined using the incompressibility condition , with the size in the @xmath1 direction , @xmath104 , fixed . the steady state solution is @xmath105 , @xmath106 and @xmath107 , with @xmath108 the net compressive strain in the @xmath109-direction . for a given value of elastic modulus @xmath25 , the mean - field model predicts oscillatory solutions for @xmath110 , where @xmath111 defines the phase boundary in @xmath112 plane separating the regions of propagating waves and diffusive spreading ( dashed line in fig . [ fig : waves]a ) . for @xmath113 the monolayer diffusively approaches the steady state @xmath114 . , [ eq : c ] ) ( red squares ) , obtained from eq . ( black solid circles ) , and as predicted by the mean - field model ( green open circles ) for various @xmath73 and @xmath25 . ( b ) mean - filed elastic modulus @xmath115 of the cell monolayer as a function of time , showing oscillatory stiffening / fluidization for @xmath116 pa ( solid ) and steady stiffening for @xmath117 pa ( dashed ) . parameters : @xmath118 pa , @xmath60 min , @xmath119 min , @xmath120 nn , @xmath121 nn min/@xmath29 m , @xmath122 . ] this simple mean - field approach allows us to study the material response of the monolayer characterized by an effective elastic modulus , @xmath123 . the oscillatory regime ( @xmath110 ) exhibits sustained oscillations in the material rigidity , @xmath115 , with a slow period of stiffening followed by a sharp turnover ( see fig . [ fig : meanfield]b ) . for @xmath113 , the material gradually stiffens with @xmath115 asymptotically approaching the value @xmath124 . these oscillations reflect self - sustained turnovers in the cytoskeleton with periodic reinforcement and fluidization on different timescales , which was invoked to be the underlying mechanism of wave propagation in ref . @xcite . [ [ time - dependent - propulsion - forces . ] ] time - dependent propulsion forces . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + finally , we consider time variations of the propulsion force , as arising from the dynamics of cell polarization @xmath41 given by @xmath125 where the first two terms with @xmath126 allow for the onset of a homogeneous polarized state when @xmath127 . the stiffness constant @xmath128 characterizes the cost of local deformations in the polarization . the last two terms in eq . define active couplings of @xmath129 to the strain and the concentration field , with @xmath130 , such that @xmath129 aligns with the gradient of monolayer density and the concentration field . in other words , cell polarization is enhanced in the direction opposite to that of elastic restoring forces . additionally , polarization gradients can induce mechanical stresses , and the stress tensor is modified to read , @xmath131 , where @xmath132 is a contractile tension induced by polarization gradients . we assume a no - flux boundary condition , @xmath133 . for @xmath134 and if @xmath135 , such that @xmath136 , the solution is essentially time - independent , and can be approximated as , @xmath137 , with @xmath138 . when the coupling of polarization to strain and contractility is turned on , various spatiotemporal patterns emerge as the active tension @xmath139 is varied . for small @xmath139 , the stress patterns are qualitatively similar to fig . [ fig : waves]b ( with time - independent propulsion ) , and @xmath129 asymptotically approaches @xmath140 with initial oscillations near the midline ( fig . [ fig : pol ] a , d ) . for intermediate @xmath139 , a traveling stress pulse emerges in the layer and the location of stress maxima oscillate around the midline ( fig . [ fig : pol]b ) . this is accompanied by large amplitude oscillations of net polarity that attenuate in time to generate a symmetric steady state polarization profile ( fig . [ fig : pol]e ) . these traveling pulses persist even in the case @xmath141 . for even higher values of @xmath139 complex oscillatory patterns emerge in the monolayer stress and polarization ( fig . [ fig : pol]c , f ) . [ [ discussions . ] ] discussions . + + + + + + + + + + + + we have developed a simple yet rich dynamic model for an active spreading gel , based on a linear feedback between local strain and contractility . a local increase in length due to spreading promotes the assembly of active elements that in turn induce contraction . we propose that a finite turnover rate in the active contractile elements can yield an effective inertia and viscoelasticity in the gel that vanishes for infinitely fast turnover rates . this simple mechano - chemical model allows us to capture the experimentally observed propagating stress waves during tissue expansion without invoking nonlinear elasticity @xcite . these stress waves are characterized by strain rate wavefronts that initiate from the leading edge and periodically travel into and away from the midline of the monolayer . our findings also elucidate that the effective material rigidity of the tissue undergoes sustained periods of stiffening and softening as the waves propagate . we emphasize that spreading is not crucial for wave propagation and that oscillations can also occur under confinement . however in contrast to our model , ref . @xcite recently proposed that oscillatory modes in confined layers can also be generated by stochastic motion of cells . experimental tests that inhibit myosin based contractility or cell directionality can help discriminate between these different models . is increased ( left to right ) . ( a , d ) x - waves , @xmath142 nn/@xmath29 m ; ( b , e ) traveling stress pulse , @xmath143 nn/@xmath29 m ; ( c , f ) complex oscillatory patterns , @xmath144 nn/@xmath29 m . parameters : @xmath145 @xmath29m / min , @xmath146 @xmath29m / min , @xmath147 @xmath29m@xmath69/min , @xmath148 min@xmath56 , @xmath149 min@xmath56 . other parameter values are the same as in fig . [ fig : stress ] . see supplemental material @xcite for kymographs of strain rate , velocity and the traction stress . ] we thank jeffrey fredberg , james butler , jacob notbohm and michael kpf for useful discussions . the work at syracuse university was supported by the national science foundation ( nsf ) awards dmr-1305184 and dge-1068780 . mcm also acknowledges support from the simons foundation and from nsf award phy11 - 25915 at the kitp of the university of california , santa barbara , and thanks kitp for its hospitality during completion of some of this work . sb gratefully acknowledges support from kadanoff - rice fellowship through nsf materials research science and engineering center at the university of chicago . 23ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop @noop * * , ( ) @noop * * ( )

neutrinos do not interact with photons in the standard model ( sm ) . however , minimal extension of the sm with massive neutrinos yields neutrino - photon and neutrino - two photon interactions through radiative corrections @xcite . despite the fact that minimal extension of the sm induces very small couplings , there are several models beyond the sm predicting relatively large neutrino - photon and neutrino - two photon couplings . electromagnetic properties of the neutrinos have important implications on particle physics , astrophysics and cosmology . therefore probing electromagnetic structure of the neutrinos at colliders is important for understanding the physics beyond the sm and contributes to the studies in astrophysics and cosmology . the large hadron collider ( lhc ) generates high energetic proton - proton collisions with a high luminosity . it is commonly believed that it will help to answer many fundamental questions in particle physics . recently a new phenomenon called exclusive production was observed in the measurements of cdf collaboration @xcite and its physics potential has being studied at the lhc @xcite . complementary to proton - proton interactions , studies of exclusive production might be possible and opens new field of studying very high energy photon - photon and photon - proton interactions . therefore it is interesting to investigate the potential of lhc as a photon collider to probe the electromagnetic properties of the neutrinos . atlas and cms collaborations have a program of forward physics with extra detectors located at distances of 220 m and 420 m from the interaction point @xcite . range of these 220 m and 420 m detectors overlap and they can detect protons in a continuous range of @xmath2 where @xmath2 is the proton momentum fraction loss defined by the formula , @xmath3 . here @xmath4 is the momentum of incoming proton and @xmath5 is the momentum of intact scattered proton . atlas forward physics ( afp ) collaboration proposed an acceptance of @xmath6 . there are also other scenarios with different acceptances of the forward detectors . cms - totem forward detector scenario spans @xmath7 @xcite . one of the well - known application of the forward detectors is the high energy photon - photon fusion . this reaction is produced by two quasireal photons emitted from protons . since the emitted quasireal photons have a low virtuality they do not spoil the proton structure . therefore scattered protons are intact and forward detector equipment allows us to detect intact scattered protons after the collision . the photon - photon fusion can be described by equivalent photon approximation ( epa ) @xcite . in the framework of epa , emitted photons have a low virtuality and scattered with small angles from the beam pipe . therefore they are almost real and the cross section for the complete process @xmath8 can be obtained by integrating the cross section for the subprocess @xmath9 over the effective photon luminosity @xmath10 @xmath11 where w is the invariant mass of the two photon system and the effective photon luminosity is given by the formula @xmath12 with @xmath13 here @xmath14 is the energy of one of the emitted photons from the proton , @xmath15 and @xmath16 are the functions of the equivalent photon spectra . equivalent photon spectrum of virtuality @xmath17 and energy @xmath18 is given by @xmath19\end{aligned}\ ] ] where @xmath20 here e is the energy of the incoming proton beam and @xmath21 is the mass of the proton . the magnetic moment of the proton is taken to be @xmath22 . @xmath23 and @xmath24 are functions of the electric and magnetic form factors . in the above epa formula , electromagnetic form factors of the proton have been taken into consideration . in this work we investigate the potential of exclusive @xmath0 and @xmath1 reactions at the lhc to probe @xmath25 and @xmath26 couplings . we obtain model independent bounds on these couplings considering dirac neutrinos . during numerical analysis we assume that center of mass energy of the proton - proton system is @xmath27 tev . non - standard @xmath25 interaction can be described by the following dimension 6 effective lagrangian @xcite @xmath28 where @xmath29 is the magnetic moment of @xmath30 and @xmath31 @xmath32 is the transition magnetic moment . in the above effective lagrangian new physics energy scale @xmath33 is absorbed in the definition of @xmath29 . the most general dimension 7 effective lagrangian describing @xmath26 coupling is given by @xcite @xmath34 where @xmath35 , @xmath36 , @xmath37 and @xmath38 are dimensionless coupling constants . current experimental bounds on neutrino magnetic moment are stringent . the most sensitive bounds from neutrino - electron scattering experiments with reactor neutrinos are at the order of @xmath39 @xcite . bounds derived from solar neutrinos are at the same order of magnitude @xcite . bounds on magnetic moment can also be derived from energy loss of astrophysical objects . these give about an order of magnitude more restrictive bounds than reactor and solar neutrino probes @xcite . neutrino - two photon coupling has been less studied in the literature . current experimental bounds on this coupling are derived from rare decay @xmath40 @xcite and the analysis of @xmath41 conversion @xcite . lep data on @xmath40 decay sets an upper bound of @xcite @xmath42 ^ 6 \sum_{i , j , k}\left(|\alpha^{ij}_{rk}|^2+|\alpha^{ij}_{lk}|^2\right)\leq2.85\times10^{-9}\end{aligned}\ ] ] the analysis of the primakoff effect on @xmath41 conversion in the external coulomb field of the nucleus @xmath43 yields about two orders of magnitude more restrictive bound than @xmath44 decay @xcite . in the presence of the effective interactions ( [ nunuphoton ] ) and ( [ nunuphotonphoton ] ) , @xmath45 scattering is described by three tree - level diagrams . the polarization summed amplitude square is given by the following simple formula @xmath46 where s , t and u are the mandelstam variables and we omit the mass of neutrinos . in the above amplitude we also neglect interference terms between interactions ( [ nunuphoton ] ) and ( [ nunuphotonphoton ] ) . neutrinos are not detected directly in the central detectors . instead , their presence is inferred from missing energy signal . therefore statistical analysis has to be performed with some care . any sm process with final states which are not detected by the central detectors can not be discerned from @xmath47 . atlas and cms have central detectors with a pseudorapidity coverage @xmath48 . the sm processes with final state particles scattered with very large angles from the beam pipe may exceed the angular range of the central detectors . hence any sm process with final states in the interval @xmath49 should be accepted as a background for @xmath45 . there are also other sources of backgrounds . the one which may affect our results is the instrumental background due to calorimeter noise . the calorimeter noise can be effectively suppressed by imposing a cut on the transverse energy of the jets . according to ref.@xcite the calorimeter noise is negligible for jets with a transverse energy greater than 40 gev ( @xmath50 gev ) . of course the calorimeter noise is not the only factor which affects the jet efficiency . based on @xcite we take into account a global efficiency of 0.6 . this is actually a prudent value for the global efficiency . we have considered the following background processes : @xmath51 our backgrounds can be arranged into three classes : ( 1)- @xmath52 , @xmath53 and @xmath54 with @xmath49 for all final charged particles . ( 2)-@xmath53 and @xmath55 with @xmath49 for final charged leptons and @xmath48 , @xmath56 gev for final quarks , i.e. , final quarks with @xmath56 gev are assumed to be missing even for @xmath48 . ( 3)-we assume that 40% of the number of events from reactions @xmath53 and @xmath57 with @xmath49 for final charged leptons and @xmath48 , @xmath50 gev for final quarks is missing . among all the background processes the biggest contribution comes from electron - positron production with @xmath49 . the reason originates from the fact that the cross section is highly peaked in the forward and backward directions due to small mass of the electron and @xmath58 pole structure . the sum of all background cross sections is @xmath59 pb for @xmath7 and @xmath60 pb for @xmath6 . we have estimated 95% confidence level ( c.l . ) bounds using one - parameter @xmath61 test without a systematic error . sm prediction about missing number of events is @xmath62 . here @xmath63 is the integrated luminosity , 0.9 is the qed two - photon survival probability and @xmath64 is the sum of all background cross sections . in order to test sm prediction we use the following @xmath61 function : @xmath65 where @xmath66 is the cross section for the process @xmath67 and @xmath68 is the statistical error . in tables [ tab1 ] and [ tab2 ] , we show 95% c.l . upper bounds of the couplings @xmath69 and @xmath70 . as we have mentioned , diagonal elements @xmath29 are strictly constrained by the experiments . if we assume that @xmath31 is diagonal then our limits are many orders of magnitude worse than the current experimental limits . on the other hand we see from the tables that our bounds on @xmath71 are approximately at the order of @xmath72 . it is 7 orders of magnitude more restrictive than the lep bound . as we have mentioned , during statistical analysis we consider the background processes given in ( [ background ] ) . the main contribution is provided by the processes @xmath73 with @xmath49 . of course there are other backgrounds that we have not taken into account . but these others are expected to give relatively small contributions . furthermore , even a large background does not spoil our limits significantly . for instance , if we assume that background cross section is 4 times larger than the sum of all backgrounds that we have considered , our limits are spoiled only a factor of 2 . they are still at the order of @xmath72 . the subprocess @xmath74 is described by 8 tree - level diagrams containing effective @xmath25 and @xmath26 couplings ( fig.[fig1 ] ) . the analytical expression for the amplitude square is quite lengthy so we do not present it here . but it depends on the couplings of the form ; @xmath75 , @xmath76 and @xmath77 where we assume that @xmath78 ; @xmath79 . @xmath80 is absent in the sm at the tree - level . sm contribution is originated from loop diagrams involving 5 vertices . since the sm contribution is very suppressed it is appropriate to set bounds on the couplings using a poisson distribution . the expected number of events has been calculated considering the leptonic decay channel of the z boson as the signal @xmath81 , where @xmath82 or @xmath83 . we also place a cut of @xmath48 for final state @xmath84 and @xmath83 . upper bounds of the couplings @xmath85 , @xmath86 and @xmath69 are presented in tables [ tab3 ] and [ tab4 ] . we observe from the tables that the subprocess @xmath74 provides approximately an order of magnitude more restrictive bounds on @xmath87 coupling with respect to @xmath88 . on the other hand both processes have almost same potential to probe the coupling @xmath89 . forward detector equipments allow us to study lhc as a high energy photon collider . by use of forward detectors we can detect intact scattered protons after the collision . therefore deep inelastic scattering which spoils the proton structure , can be easily discerned from the exclusive photo - production processes . this provides us an opportunity to probe electromagnetic properties of the neutrinos in a very clean environment . we show that exclusive @xmath90 and @xmath91 reactions at the lhc probe neutrino - two photon couplings with a far better sensitivity than the current limits . former reaction improves the sensitivity limits by up to a factor of @xmath92 and latter improves a factor of @xmath93 with respect to lep limits . v. castellani and s. deglinnocenti , astrophys . j. * 402 * , 574 ( 1993 ) . m. catelan , j. a. d. pacheco and j. e. horvath , astrophys . j. * 461 * , 231 ( 1996 ) [ arxiv : astro - ph/9509062 ] . a. ayala , j. c. dolivo and m. torres , phys . d * 59 * , 111901 ( 1999 ) [ arxiv : hep - ph/9804230 ] . r. barbieri and r. n. mohapatra , phys . lett . * 61 * , 27 ( 1988 ) . j. m. lattimer and j. cooperstein , phys . lett . * 61 * , 23 ( 1988 ) . a. heger , a. friedland , m. giannotti and v. cirigliano , astrophys . j. * 696 * , 608 ( 2009 ) [ arxiv:0809.4703 [ astro - ph ] ] . cms collaboration , _ cms physics technical design report volume i _ , editor : d. acosta , cern / lhcc 2006 - 001 . upper bounds of the couplings @xmath89 and @xmath71 for the process @xmath94 . we consider various values of the integrated lhc luminosities . forward detector acceptance is @xmath7 . @xmath33 is taken to be 1 gev and limits of @xmath89 is given in units of bohr magneton . [ tab1 ] [ cols="^,^,^,^,^",options="header " , ]

exclusive production of neutrinos via photon - photon fusion provides an excellent opportunity to probe electromagnetic properties of the neutrinos at the lhc . we explore the potential of processes @xmath0 and @xmath1 to probe neutrino - photon and neutrino - two photon couplings . we show that these reactions provide more than seven orders of magnitude improvement in neutrino - two photon couplings compared to lep limits .

the development of a new generation of multi - object spectrographs , exemplified by the ` two degree field ' , or 2df , multi - fibre spectrograph on the anglo - australian telescope ( aat ) , has opened up whole new areas of astronomical survey science . one particular area , which we discuss in this paper , is the opportunity to make a truly complete spectroscopic survey of a given area on the sky , down to well determined , faint limits , irrespective of image morphology or any other preselection of target type . the _ fornax spectroscopic survey _ , or , seeks to exploit the huge multiplexing advantage of 2df by surveying a region of 12 square degrees centred on the fornax cluster of galaxies . it will encompass both cluster galaxies , of a wide range of types and magnitudes , and background and foreground galaxies ( over a similarly wide range of morphologies ) , as well as galactic stars , qsos and any unusual or rare objects . although many surveys of nearby clusters have been made over the past 20 years or more , these are all limited in several crucial aspects . spectroscopic surveys exist , but typically only of the few dozen brightest cluster galaxies ( and any background interlopers in the top few magnitudes of the cluster luminosity function ) . photometric surveys , of course , go much deeper , but such studies must be of a statistical nature ( e.g. subtracting off the expected background numbers ; smith et al . 1997 ) , or rely on subjective judgements of likely cluster membership based on morphology , surface brightness or colour ( e.g. ferguson 1989 ) . of particular concern is the surface brightness ; low surface brightness galaxies ( lsbgs ) seen towards a cluster are conventionally assumed to be members , while apparently faint , but high surface brightness galaxies ( hsbgs ) are presumed to be luminous objects in the background ( e.g. sandage , binggeli , & tammann 1985 ) . the failure of either assumption , i.e. the existence of large background lsbgs ( such as the serendipitously discovered malin 1 ; bothun et al . 1987 ) or of a population of high surface brightness ( compact ) dwarfs in the cluster ( drinkwater & gregg 1998 ) , can have a dramatic effect on our perception of the galaxy population as a whole . furthermore , it is possible that a population of extremely compact galaxies ( either in the cluster or beyond ) could masquerade as stars and hence be missed altogether from galaxy samples . examples have previously been found in , for example , qso surveys , but again these are serendipitous discoveries and hard to quantify ( see drinkwater et al . 1999a = paper ii , and references therein ) . few previous attempts at all - object surveys have been made . the one most similar to ours is probably that of morton and tritton in the early 1980s . they obtained around 600 objective prism spectra and 100 slit spectra for objects in a 0.31 square degree region of background sky ( i.e. no prominent cluster ) over the course of a 5 year period ( morton , krug & tritton 1985 ) . more recently colless et al . ( 1993 ) obtained spectra of about 100 objects in a small area of sky and small magnitude range in order to investigate the completeness of faint galaxy redshift surveys . our overall survey will therefore represent a huge increase in the volume of data and in addition will give a uniquely complete picture of a cluster of galaxies . it is worth noting that the huge galaxy surveys planned , with 2df ( folkes et al . 1999 ; colless 1999 ) or the sloan digital survey ( gunn 1995 ; loveday & pier 1998 ) will not address such problems , since their galaxy samples will be pre - selected from photometric surveys and will only include objects classified as galaxies and not of too low surface brightness , thus removing both ends of any potentially wide range of galaxy parameters . in the present paper we discuss the design and aims of our all - object _ fornax spectroscopic survey _ and present initial results on the velocity distributions . section [ sec_survey ] gives a technical definition of the survey , describing the relevant features of the 2df spectrograph , the selection of our target catalogue and the calibration of this input catalogue . in section [ sec_science ] we discuss the scientific aims of the survey and summarise the types and numbers of objects we expect to observe . in section [ sec_obs ] we discuss the spectroscopic observations and observational strategy . we describe the technique we have developed to identify and classify objects automatically from the 2df spectra and give some examples from our initial observations . section [ sec_initsci ] gives the initial redshift results and section [ sec_summary ] summarises the survey work to date . in this section we describe the basic parameters of the . we start with the relevant technical details of the 2df spectrograph , and then discuss our selection of targets from the digitised photographic sky survey plates and the calibration of our input catalogues . the 2df facility ( taylor , cannon & parker 1998 ) is probably the most complex ground - based astronomical instrument built to date . via a straightforward ` top - end ' change the capability of the 3.9-manglo - australian telescope is transformed into an unique wide - field multi - fibre spectroscopic survey instrument . up to 400 fibres are available at any one time for rapid configuration over the full two - degree diameter focal surface via a highly accurate robotic positioner mounted in situ . each 2 diameter fibre can be placed to an accuracy of 0.3 in less than 10 seconds . the input target positions must be accurate to 0.3 r.m.s . or better over the whole two - degree field to avoid vignetting of the fibre entrance apertures . this requirement is only for relative positions ; the absolute accuracy of a complete set of targets and guide stars need only be 12 as the guide stars will then centre all the targets accurately . the wide field is provided by a highly sophisticated multi - component corrector with in - built atmospheric dispersion compensator . in a novel arrangement 2df can simultaneously observe 400 target objects on the sky at the same time that a further 400 fibres are being configured using the robotic positioner on one of the two available ` field plates ' ( focal surfaces ) . once observations and configurations have been completed ( usually over the same timescale ) a tumbling mechanism allows the newly configured field plate to point at the sky whilst the previously observed field can be re - configured for the next target field . in this way rapid field inter - change is provided for an extremely efficient observing environment . each set of 400 fibres feeds two spectrographs which accept 200 fibres each . these are mounted on the 2df top end ring and can produce low to medium resolution spectra on the dedicated @xmath4 tek ccds . the 2df is now operating at close to the original specifications anticipated for this most complex of instruments ( lewis , glazebrook & taylor , 1998 ) . field configuration times of about 1 hour for 400 fibres permit rapid cycling of target fields and have enabled excellent progress to be made with our complete survey . we chose the fornax cluster for this study because it is a well - studied , compact , nearby southern galaxy cluster suited to this type of survey . we and several other groups have made photometric or small - scale spectroscopic surveys of the region ( e.g. ferguson 1989 ; davies et al . 1988 ; drinkwater & gregg 1998 , hilker et al . 1999 ) . the published spectroscopic samples have either been very small or have concentrated on the brighter cluster galaxies ( jones & jones 1980 ; drinkwater et al . 2000a ) . a search of ned in our central 2df field ( number 1 in table [ tab_fields ] ) found only 42 objects brighter than @xmath5 with measured redshifts : 30 cluster galaxies , 6 background galaxies and 6 qsos . with 2df we can now measure the redshifts of some 700900 galaxies and quasars in this same field . the fornax cluster is concentrated within one united kingdom schmidt telescope ( ukst ) sky survey plate and our survey will comprise four separate 2df fields which are listed in table [ tab_fields ] . we show the distribution of our fields on the sky in fig . [ fig_sky ] compared to the positions of galaxies classified as likely cluster members by ferguson ( 1989 ) . our first field is centred on the large galaxy ngc 1399 at the centre of the cluster . in order both to cover a large number of targets and to go significantly deeper than previous spectroscopic surveys we chose to limit our survey at a magnitude of 19.7 . this is then essentially the same depth as the large scale 2df galaxy redshift survey ( grs ) of ellis , colless and collaborators ( e.g. colless 1999 ; folkes et al . 1999 ) . this combination of survey area and magnitude limit will optimise our measurement of the cluster galaxies ( see section [ sec_cluster ] ) . .fornax spectroscopic survey fields [ cols="<,^,^,<",options="header " , ] in the second stage of the identification process we check each identification interactively using the rvsao package to display the best cross - correlation and a plot of the object spectrum with common spectral features plotted at the corresponding redshift . when the redshift is obviously wrong ( e.g. with the calcium h & k lines clearly present but misidentified ) , it is flagged as being wrong or in some cases is recalculated . the recaculation most commonly involves repeating the template cross - correlation on a restricted wavelength range chosen to avoid the red end of the spectrum affected by poorly removed sky features . in extreme cases the object may be a qso : these are distinguished by strong broad emission lines and are measured using a composite qso spectrum ( francis et al.1991 ) . objects still not identified at this stage are flagged to be reobserved . a third , supplementary stage is used for any spectra measured with good signal ( a signal - to - noise ratio @xmath610 in each 4.3 wide pixel ) but no obvious features in their 2df spectra : these are flagged as ` strange ' and scheduled for detailed follow - up observations with conventional slit spectrographs . once the spectroscopic redshift measurements are complete , they are corrected to heliocentric values . we checked the accuracy of the redshift measurements by comparing the results for 66 objects with repeated measurements . the r.m.s . scatter of the velocity differences is 90 . this uncertainty is consistent with the combined error estimates for the same measurements produced by rvsao : the mean predicted error was 92 . note that this implies a measurement error of a single observation of @xmath7 . we also compared our results to redshifts of 44 galaxies found in a search of the literature using ned ( most were from hilker et al . the comparison gave a mean velocity difference of ( @xmath8 ) and an r.m.s . scatter of 111 , entirely consistent with our internal calibration . the 2df spectra , although of low resolution and unfluxed , are useful for more detailed analysis than simple redshift measurements and object classifications ( c.f . tresse et al . we defer any detailed analysis of the spectra to later papers dealing with specific object classes , but note here that , even for the lowest luminosity galaxies , they can be used to measure emission line equivalent widths , and hence star formation rates , line widths ( limited by the resolution of 900 ) , emission line ratios ( e.g. / and / ) , absorption line indices ( e.g. h+/k and /fei4045 ) and even ages and metallicities from these balmer and the metal absorption lines ( paper ii ; folkes et al . 1999 ) . although this present paper is mainly concerned with the principles behind the , we already have a considerable amount of 2df data for the project from commissioning observations in 1996/1997 , and scheduled time in december 1997/january 1998 and november 1998 . we have nearly completed our observations of the first field having observed 92% of all targets in the range @xmath9 to 19.7 and successfully obtained redshifts for 94% of those observed . for resolved objects ( galaxies ) the success rate of our redshift measurements is a function of surface brightness . in fig . [ fig05_lim ] we plot the numbers of galaxies observed and identified as a function of central surface brightness . we have attempted to optimise the exposure times to the surface brightnesses of the objects , using exposures up to 3.75 hours for the lower surface brightness images . the identification rate runs at 78% or better to a limit of 23 . fainter than this limit ( corresponding to a mean surface brightness inside the detection threshold @xmath10 ) , the identifications drop off rapidly . the unresolved objects at higher surface brightness ( mostly stars ) have an identification rate of 95% in our target magnitude range of @xmath9 to 19.7 . in fig . [ fig06_spc ] we show example spectra from our initial observations of the various types of object discussed above . the first two spectra are galactic stars , an m - dwarf and a white dwarf . the next two spectra are of normal low - redshift galaxies , one with an absorption line spectrum and one with an emission line spectrum . the remaining four spectra are all of objects that were unresolved ( i.e. classified as stars in the target catalogues ) , but have been identified as various types of galaxy . the first is a compact emission line galaxy ( celg ; see section [ sec_celg ] ) , the second is a normal , optically selected qso and the third is an x - ray source . the final spectrum is of a fainter radio - loud quasar . the number of galaxies observed in the fornax cluster itself is not yet large enough to allow a detailed study of the cluster population : this will require results from the remaining three 2df spectrograph fields to achieve the statistical samples needed . however , we have ample data to delineate clearly the velocity structure in the direction of fornax . in particular , we have determined accurately the velocity distribution of galactic stars , as well as the galaxy distribution in redshift space behind the cluster . the radial velocity distribution of galactic stars is revealed in the existing _ fss _ results , despite the modest resolution of the spectra compared to those conventionally used in kinematic surveys of the galaxy . a total of 2467 objects in field 1 of table [ tab_fields ] have reliable radial velocities @xmath11 . the estimated standard errors in the velocities are typically @xmath12 , sufficiently small to reveal the contributions of different galactic components . figure [ fig_is_starvels ] shows the distribution of all fss field 1 objects with reliable ( @xmath13 ) velocities less than 750 kms@xmath14 . the field has galactic co - ordinates @xmath15 , so we are sampling a sight - line looking diagonally ` down ' through the galactic plane between the anti - centre and anti - rotation directions . the component of the motion of the local standard of rest in this direction is @xmath16 kms@xmath14 . for our chosen magnitude range , the survey will sample predominantly disc , thick disc and halo main sequence stars , with some contribution from halo giants and disc white dwarfs ( gilmore & reid 1983 ) . the results can be compared with dedicated spectroscopic studies of faint stars in high - latitude fields ( e.g. kuijken & gilmore 1989 ; croswell et al . 1991 ; majewski 1992 ) . the contribution of the various galactic components can be demonstrated by considering subsamples of the stars defined by colour . basic colour information can be derived from the blue and red magnitudes given in the apm catalogue . figure [ fig_is_starcols ] shows the distribution of these @xmath17 colours for the field 1 objects with velocities @xmath18 kms@xmath14 . the form of the distribution is similar to that obtained in dedicated studies of the properties of faint stars ( e.g. reid & majewski 1993 ) . we divide the stars into three samples : relatively blue stars having @xmath19 ; moderately red stars having @xmath20 ; and very red stars having @xmath21 . the sharp decline in numbers bluewards of @xmath22 is the result of the main sequence cut - off for moderately old stellar populations ; the blue sample extends to this limit . these limits at @xmath23 and 1.7 correspond to @xmath24 and 1.1 the moderately red stars are expected to include g and k dwarfs in the thick disc and halo , g and k giants in the halo , and disc k dwarfs . the halo component , being dynamically pressure supported , has a broad radial velocity distribution which is displaced relative to the solar motion by the component of the solar rotation velocity towards fornax ( freeman 1987 ; gilmore , wyse & kuijken 1987 ; majewski 1993 ) . disc and thick disc stars , being rotationally supported , have a zero or small asymmetric drift and a modest intrinsic velocity dispersion : their velocity distributions will be centred closer to zero heliocentric velocity . the moderately - red star sample is therefore expected to have a broad velocity distribution with the halo component contributing a high velocity tail , consistent with the velocity distributions shown in figure [ fig_is_starvelscols ] . in contrast , the very red star sample will be rich in disc late k and m dwarfs and will include halo late k and m giants . it is therefore expected to have only a modest net drift with respect to the local standard of rest but with a tail to high velocity , as observed in figure [ fig_is_starvelscols ] . the blue stars include local ( disc ) white dwarfs and halo horizontal branch stars . they are therefore expected to have a broad range of velocities , consistent with the results here . of particular interest is the high velocity tail at @xmath25 km s@xmath14 . if the extreme examples are confirmed by higher resolution spectroscopy they will provide useful constraints on the mass of the galaxy ( e.g. carney , latham & laird 1988 ; croswell et al . 1991 ; majewski , munn & hawley 1996 ; freeman 1999 , private communication ) . a gap is present in the velocity distribution between the cut - off in galactic stars at @xmath26 kms@xmath14 and the fornax cluster at @xmath27 . no objects are found in this intermediate velocity range among the results from the first 2df field . the low velocity limit of the cluster velocity distribution is therefore defined without ambiguity . it is of interest to determine whether there are any galaxies in the foreground to the fornax cluster having heliocentric radial velocities @xmath28 which might be overlooked given the very large number of galactic stars in this velocity range . the apm catalogue ( used as the input database for the survey ) provides a classification for each image from the blue and red sky survey plates . of the 2467 objects having @xmath28 and cross - correlation @xmath29 parameter @xmath30 , 14 are classified as being galaxies in both blue and red . all 14 were inspected visually on the digitised sky survey and again on a supercosmos measuring machine ( miller et al . 1992 ) scan of film or17818 taken on tech pan emulsion with the ukst . the tech pan data provided higher resolution and greater depth than the digitised sky survey ( e.g. phillipps & parker 1993 ) . all foreground galaxy candidates appeared to be compact images merged with another , fainter image . most were unambiguously galactic stars merged with either another star or with a background galaxy . none of the 14 candidates had the extended appearance expected of a nearby dwarf galaxy . to extend the search , the visual inspection was repeated on the five images with reliable velocities having the largest apm @xmath31 parameter on the blue sky survey plates . the @xmath31 parameter measures the degree to which an image differs from a point - spread function and is a convenient indicator of a non - stellar light profile . of the five images having a blue @xmath32 , none had the appearance expected of a nearby dwarf galaxy : all were found to be compact ( star - like ) images merged with either another star or a faint galaxy . a third and final search for foreground galaxies was performed using large exponential scale lengths as a indicator of extended images . the surface photometry described in section [ ssec_resolvedobjs ] derived a scale length from the low surface brightness regions of each image . only five images with reliable velocities @xmath33 had scale lengths @xmath34 . none had the appearance expected of a nearby dwarf galaxy on the digitised sky survey or the scan of the tech pan film : all objects were again found to be merged images . we conclude that no foreground galaxies were found with star - like velocities . we therefore have no galaxies with heliocentric in field 1 of table [ tab_fields ] within our magnitude range . among the brighter galaxies ( @xmath35 ) in the whole cluster region , jones & jones ( 1980 ) previously found a small number with such low velocities ( ngc 1375 , ngc 1386 , ngc 1396 ( @xmath36 g75 ) , and ngc 1437a ) , though the exact number depends on the accuracy of their redshift determinations . a search of the _ nasa extragalactic database _ ( _ ned _ ) identifies the same four galaxies . of these , ngc 1375 , ngc 1386 and ngc 1396 lie in our field 1 . figure [ fig_is_clusterfss ] shows the velocity distribution of fornax cluster galaxies from the _ fss_. the mean heliocentric radial velocity from the _ fss _ data is @xmath37 ( 26 galaxies ) . this compares with @xmath38 from jones & jones . recall that the jones & jones galaxies are much brighter than ours ( roughly @xmath39 as against @xmath40 ) and are spread over a much larger area , 6 degrees or about 1.6 mpc across compared to our 2 degrees or 0.5 mpc . a velocity dispersion can be estimated fairly unambiguously as there are no galaxies with velocities less than 900 or between 2300 and @xmath41 . our 26 galaxies give an observed radial velocity dispersion of @xmath42 , compared to the @xmath43 of jones & jones . the _ fss _ velocity distribution can also be compared with the equivalent distribution compiled from all published redshift data . figure [ fig_is_clusterned ] presents the velocity data from _ ned_. these give a mean heliocentric radial velocity of @xmath44 ( 32 galaxies ) , and a velocity dispersion of @xmath45 , entirely consistent with the _ fss _ results . the _ ned _ results generally , though not entirely , apply to the brighter cluster galaxies . fornax is an apparently well relaxed , regular cluster as judged by its central density concentration and low spiral content . it would require a very much larger sample of redshifts over the other fields in order to explore properly any dynamical differences between different galaxy populations . these initial results do not reveal any difference in the dynamics between the bright and faint ( giant and dwarf ) members of the cluster , although a wide - field study of brighter galaxies ( drinkwater et al . 2000a ) does suggest such a difference . figure [ fig_is_backgndfss ] shows our redshift distribution behind the cluster . immediately beyond the cluster , as noted by jones & jones ( 1980 ) and phillipps & davies ( 1992 ) , there is a large void , extending some 40 mpc ( from the cluster mean redshift to about @xmath46 assuming @xmath47 ) . beyond this `` fornax void '' , we see the ubiquitous ` spiky ' distribution ( broadhurst et al . 1990 ) showing more distant walls and filaments . figure [ fig_is_backgndned ] shows the distribution of background galaxies taken from the _ ned_. the difference in depth between the two data sets is immediately apparent : the results probe to much greater distances on account of the fainter magnitudes of the galaxies . nevertheless , the first two main features in our distribution clearly match the two peaks seen in the _ ned _ data ( i.e. in the brighter galaxies ) . a standard cone diagram is shown in figure [ fig_is_conefssra2 ] , illustrating the skeleton of the large scale 3-d structure beyond fornax . the median redshift of the entire galaxy sample is 0.15 . this compares with a mean of 0.11 in the preliminary data from the 2df galaxy redshift survey ( colless 1999 ) . the data continue to map structure out to @xmath48 , where there are still significant numbers of galaxies . the cluster j1556.15bl identified by couch et al . ( 1991 ) lies in field 1 at @xmath49 , but the density of _ fss _ galaxies at this redshift is too small to show the cluster . in addition to the general galaxy population , figure [ fig_is_conefssra2 ] also shows ( as large solid points ) the compact galaxies discussed in paper ii . these objects have star - like images on schmidt survey plates but the 2df spectroscopy showed them to be compact star - forming galaxies at redshifts @xmath50 . the figure also shows low surface brightness galaxies having intrinsic ( cosmologically corrected ) central surface brightnesses fainter than @xmath51 , plotted as open circles . despite their low surface brightnesses , these objects are sufficiently distant that they are too luminous to be dwarfs given the apparent magnitude limits of the survey ( they have @xmath52 to @xmath53 for @xmath47 ) . many authors ( e.g. phillipps & shanks 1987 ; eder et al . 1989 ; thuan et al . 1991 ; loveday et al . 1995 ; mo , mcgaugh & bothun 1994 ) have discussed whether or not low luminosity and/or low surface brightness galaxies follow the same structures as the brighter component . although , as stated earlier , this sample is not yet complete so we can not use strictly objective measures such as the galaxy correlation function ( phillipps & shanks 1987 ) we do have enough information in our distribution to see that the low surface brightness galaxies ( shown as open circles in figures [ fig_is_zsb ] and [ fig_is_conefssra2 ] ) do trace the same large scale structure and are not seen `` filling the voids''(dekel & silk 1986 ) . the present data extend this comparison of the distribution of lsbg with that of normal galaxies to significantly lower surface brightnesses than most other studies ( or , indeed , will be possible with the standard sloan or grs samples ) . similarly , the compact ( high surface brightness ) galaxies , which are also likely to be missing from other surveys , again follow the same overall large scale structure in figure [ fig_is_conefssra2 ] as the general galaxy population . this is unlike the suggestions from some previous emission line galaxy surveys ( e.g. salzer 1989 ) that such objects can appear in very low density regions . in this paper we have presented an overview of the , the first complete , all - object spectroscopic survey to cover a large area of sky . this project has only been made possible by the advent of the 400-fibre two - degree field spectrograph on the anglo - australian telescope . in total we hope to observe some 14,000 objects to a magnitude limit of = 19.7 both ` stars ' and ` galaxies ' in a 12 area of sky centred on the fornax cluster . the main technical challenges of the project concern the preparation of the target catalogue and the analysis of the resulting spectra . our input catalogues are based on uk schmidt sky survey plates digitised by the apm facility . we have demonstrated that the apm image catalogues provide sufficiently accurate target positions and photometry for the unresolved sources . for the resolved sources our photometry is derived by fitting exponential profiles to the image parameters measured by the apm . we have tested our calibration with new ccd observations . we use a semi - automated procedure to classify our spectra and measure radial velocities based on cross - correlation comparison with a set of stellar spectra , two emission - line galaxy spectra and one qso spectrum . this procedure successfully identifies stars , galaxies and qsos completely independently of their image morphology . when the is complete we will have a unique , complete , sample of galactic stars , fornax cluster galaxies , field galaxies and distant agn . we have discussed some of the scientific questions that can be addressed with such a sample . the principal objective is to obtain an unbiased sample of cluster members , which includes compact galaxies and low surface brightness dwarfs , independent of a membership classification based on morphological appearance . redshift / velocity distributions are presented here based on spectroscopic results from the first of four 2df fields . the velocity distribution of galactic stars can be understood in terms of a conventional three - component model of the galaxy . the fornax cluster dwarf galaxies in the first 2df field have a mean heliocentric radial velocity of @xmath37 and a radial velocity dispersion of @xmath42 . the fornax cluster is well - defined dynamically , with a low density of galaxies in the foreground and immediate background . beyond @xmath54 , the large - scale structure behind the fornax cluster is clearly delineated out to a redshift @xmath48 . the compact galaxies found behind the cluster by drinkwater et al . ( 1999a ) are found to follow the structures delineated by the general galaxy population , as are background low surface brightness galaxies . some more detailed initial results have already been presented elsewhere ( drinkwater et al . 1999a , 1999b ) . this project would not be possible without the superb 2df facility provided by the aao and the generous allocations of observing time we have received from patt and atac . mjd is grateful for travel support from the university of bristol and the international astronomical union . sp acknowledges the support of the royal society via a university research fellowship . jbj and jhd are supported by the uk pparc . sp , jbj and rms acknowledge the hospitality of the school of physics , university of new south wales . part of this work was done at the institute of geophysics and planetary physics , under the auspices of the u.s . department of energy by lawrence livermore national laboratory under contract no . w-7405-eng-48 . drinkwater m.j . , phillipps s. , jones j.b . , 1999b , in : the low surface brightness universe , proc . iau colloq . 171 , eds . davies , c.d . impey , s. phillipps , astron . pacific , san francisco , p. 120 lasker b.m . , sturch c.r . , lopez c. , mallama a.d . , mclaughlin s.f . , russell j.l . , wisniewski w.z . , gillespie b.a . , jenkner h. , siciliano e.d . , kenny d. , baumert j.h . , goldberg a.m. , henry g.w . , kemper e. , siegel m.j . , 1988 , apjs , 68 , 1 taylor k. , cannon r.d . , parker q.a . , 1998 , in : new horizons from multi - wavelength sky surveys , proc . iau symposium 179 , eds . mclean , d.a . golombek , j.j.e . hayes , h.e . payne , kluwer academic publishers , p. 135

the _ fornax spectroscopic survey _ will use the two degree field spectrograph ( 2df ) of the anglo - australian telescope to obtain spectra for a complete sample of all 14000 objects with @xmath0 in a 12 square degree area centred on the fornax cluster . the aims of this project include the study of dwarf galaxies in the cluster ( both known low surface brightness objects and putative normal surface brightness dwarfs ) and a comparison sample of background field galaxies . we will also measure quasars and other active galaxies , any previously unrecognised compact galaxies and a large sample of galactic stars . by selecting all objects both stars and galaxies independent of morphology , we cover a much larger range of surface brightness and scale size than previous surveys . in this paper we first describe the design of the survey . our targets are selected from uk schmidt telescope sky survey plates digitised by the automated plate measuring ( apm ) facility . we then describe the photometric and astrometric calibration of these data and show that the apm astrometry is accurate enough for use with the 2df . we also describe a general approach to object identification using cross - correlations which allows us to identify and classify both stellar and galaxy spectra . we present results from the first 2df field . redshift distributions and velocity structures are shown for all observed objects in the direction of fornax , including galactic stars , galaxies in and around the fornax cluster , and for the background galaxy population . the velocity data for the stars show the contributions from the different galactic components , plus a small tail to high velocities . we find no galaxies in the foreground to the cluster in our 2df field . the fornax cluster is clearly defined kinematically . the mean velocity from the 26 cluster members having reliable redshifts is @xmath1 . they show a velocity dispersion of @xmath2 . large - scale structure can be traced behind the cluster to a redshift beyond @xmath3 . background compact galaxies and low surface brightness galaxies are found to follow the general galaxy distribution . astrometry galaxies : active galaxies : statistics stars : statistics surveys techniques : spectroscopic

despite more than a decade of study , the star formation processes within low surface brightness ( lsb ) galaxies , those objects with central surface brightness at least 1 magnitude fainter than the night sky ( uncorrected for inclination ) , remain enigmatic . the general properties of lsb galaxies blue colors , high gas mass - to - luminosity ratios , and low metallicities lead to the conclusion that lsb systems are under - evolved compared to their high surface brightness ( hsb ) counterparts . when combined with the low gas density ( typically @xmath3 ) and low baryonic - to - dark matter content typical of lsb systems , the question can be raised not of why lsb galaxies are under - evolved , but instead of how lsb systems form stars at all ( oneil , schinnerer , & hofner 2003 , and references therein ) . one of the primary methods for studying the star formation rate and efficiency in galaxies is through study of the galaxies interstellar medium ( ism ) , and one mechanism for studying a galaxy s ism is through observing its co content . until recently , all attempts at detecting co in lsb systems had been unsuccessful @xcite , leading to speculation as to why lsb galaxies appear to lack molecular gas . however , in the past three years the first detections of co in lsb galaxies have been made , giving the first look into the ism of lsb systems @xcite . while the three detections to date of co gas in lsb galaxies are of considerable importance , it is clear further study is paramount to understand lsb systems knowing the co content of only three lsb galaxies does not provide enough information to understand the molecular gas content of lsb systems as a whole . to aid in our understanding , we undertook co observations of three additional lsb galaxies , with properties similar in most respects to the properties of the lsb galaxies with detected co. the results of those observations are described herein . lccccccccc ugc 04144 & sc & 24.4@xmath4 & -20.0 & 42 & 83 & 9.9@xmath5 & 9795@xmath5 & 494@xmath5 & 1 + ugc 05440 & sd & 25.7@xmath4 & -20.5 & 96 & 65 & 10.8@xmath5 & 18932@xmath5 & 531@xmath5 & 0 + ugc 06124 & s & 26.0@xmath4 & -19.9 & 81 & 82 & 10.3@xmath5 & 13970@xmath5 & 583@xmath5 & 0 + ugc 01922 & s ? & @xmath6 & -19.8 & 59 & 38 & 10.33@xmath5 & 10894@xmath5 & 1120@xmath7 & 2 + ugc 12289 & sd & 23.3 & -19.7 & 57 & 22 & 10.13@xmath5 & 10160@xmath5 & 488@xmath5 & 2 + p06 - 1 & sd & 23.2 & -18.6 & 29 & 70 & 9.87 & 10882 & 458 & 1 + ugc 06968 & sc & @xmath6 & -21.1 & 48 & 71 & 10.30 & 8232 & 574 & 3 + lsbc f582 - 2 & sbc & @xmath6 & & 41 & 66 & 9.99 & 7043 & 310 & 1 + malin 1 & s & 26.4 & -21.4 & 240 & 20 & 10.6 & 24733 & 710 & 0 + lccccc ugc 04144 & 07:59:27.3 & 07:26:30.0 & 3.4 & 2.0 + ugc 05440 & 10:05:35.9 & 04:16:45.0 & 1.7 & 3.4 + ugc 06124 & 11:03:39.5 & 31:51:30.0 & 2.0 & 2.6 + as mentioned above , the three sources observed for this project have properties similar to those of the three lsb galaxies which previously have been detected in co. that is , all three galaxies observed fall into the category of massive lsb ( mlsb ) galaxies , or ` malin 1 ' cousins ( named after the largest and most famous of the mlsb galaxies ) , with m@xmath8 , w@xmath9 450 , m@xmath1018.5 , and d@xmath1150 kpc . a complete description of the previously known properties of all mlsb galaxies observed at co , including those observed for this paper are given in table [ tab : props ] . the co j(10 ) and j(21 ) rotational transitions of the galaxies were observed using the telescope in the period from 7 - 8 march , 2003 . table [ tab : galobs ] lists the adopted positions ( determined using the digitized palomar sky survey plates and accurate to 2 - 3 ) and heliocentric velocities @xcite for our target sources . the beams ( 22 at 110 ghz ) were centered on the nucleus of each galaxy . this resulted in a coverage of the inner 14 , 27 , and 24 kpc for ugc 04144 , ugc 05440 , and ugc 06124 , respectively . pointing and focus were checked every hour and pointing was found to be within the telescope limits ( better than @xmath12 ) . for each source both transitions were observed simultaneously with two receivers . both back ends were set using with 510 mhz bandwidth and 1.25 mhz resolution , resulting in an unsmoothed resolutions of 3.36 and 1.68 for the 3 mm and 1 mm observations , respectively . for data reduction , the lines were smoothed to 26.9 resolution . each target was observed on - source for a total of 66 minutes . all observations used the wobbling secondary with the maximal beam throw of @xmath13 . the image side band rejection ratios were measured to be @xmath14db for the @xmath15 mm sis receivers and @xmath16db for the @xmath17 mm sis receivers . the data were calibrated using the standard chopper wheel technique @xcite and are reported in main beam brightness temperature t@xmath18 . typical system temperatures during the observations were 170190k and 350450k in the @xmath15 mm and @xmath17 mm band , respectively . all data reduction was done using class the continuum and line analysis single - dish software developed by the observatoire de grenoble and iram @xcite . two of the three galaxies observed , ugc 04144 and ugc 06124 , were detected in co while the third galaxy , ugc 05440 , was not detected , with an upper limit of @xmath19 0.82 k and m@xmath20(figure [ fig : u4144 ] ) . due to its distance ( v@xmath21=18932 ) , the upper m@xmath1 limit placed on ugc 05440 was too high for any significance . however , the limit placed on m@xmath1/m@xmath2 is low for a galaxy of its luminosity . table [ tab : co ] lists the co properties of all three objects , as determined by our observations . for comparison with previous studies ( oneil , schinnerer , & hofner 2003 , and references therein ) we used a standard co @xmath22 h@xmath23 conversion factor ( x ) of @xmath24 adopted from @xcite . as discussed in @xcite , this assumption does not include dependence based on the structure of the ism , metallicity , etc . a discussion of any errors which may arise due to this assumption can be found in @xcite . of the three galaxies observed , ugc 04144 is the most nearby , making it unsurprising that ugc 04144 s j(1@xmath00 ) flux is considerably higher than that found for the other galaxies . however , it is notable that both ugc 04144 and ugc 06124 have m@xmath1/m@xmath2 values considerably higher than is found for the other mlsb galaxies with co detections . before these observations , the average value of m@xmath1/m@xmath2 for the mlsb galaxies was 0.08 . in contrast , ugc 06124 has m@xmath1/m@xmath2 = 0.14 and ugc 04144 has m@xmath1/m@xmath2 = 0.38 . comparing these galaxies with the other mlsb galaxies observed , both those with and without co detections , shows very little differences in their properties . all galaxies observed have similar morphologies ( sc / sd ) , colors ( from u through k ned ) , masses , and total ( dynamic ) masses . two of the mlsb galaxies have both 1.4 ghz continuum and iras detections ugc 04144 ( f@xmath25=4.7 mjy , f@xmath26=0.40 jy ) and ugc 01922 ( f@xmath25=38.5 mjy ) , f@xmath26=0.33 jy @xcite , but ugc 01922 has m@xmath1/m@xmath2 of only 0.07 . similarly , the number of neighboring galaxies does not seem to alter the quantity of molecular in within these galaxies . ugc 04144 has one nearby neighbor ( ngc 02499 , at a distance of 350 kpc and @xmath27=185 ) , while ugc 06124 has no galaxies within a 750 kpc/2,000 radius ( table [ tab : props ] ) . the single quantity which does appear to distinguish ugc 04144 and ugc 06124 from the other mlsb galaxies observed is their high inclination ( _ i_=83@xmath28 and 82@xmath28 for ugc 04144 and ugc 06124 , respectively , versus _ i_=22@xmath28 , 38@xmath28 , and 70@xmath28 for the other three galaxies with co detections ) . however , the error for the inclination measurements is 510@xmath28 , making the inclination of ugc 04144/ugc 06124 comparable to that of [ obc97 ] p06 - 1 ( _ i_=70@xmath28 ) , ugc 06968 ( _ i_=71@xmath28 ) . as a result , while it is possible the high inclination angle has contributed to the higher m@xmath1/m@xmath2 values seen for ugc 04144 and ugc 06124 , it is unlikely this is the only explanation . follow - up and co imaging should help resolve this question . with the results in this paper we have added two more measurements of molecular gas in lsb galaxies , bringing the total number of detections up to five , out of a total of nine mlsb ( and 37 lsb galaxies of any type ) . figure [ fig : mbh2 ] compares the findings in this paper with all other lsb galaxy co studies and with a sample of measurements from a variety of other galaxy studies . these include ` standard ' hsb disk galaxy studies @xcite , dwarf galaxy studies @xcite , and a study of extreme late - type spiral galaxies @xcite . in all cases a conversion factor of @xmath29 was used to allow ready comparison between the results . as can be seen in figure [ fig : mbh2 ] , both the detected co and upper limits placed on the non - detections ( and by inference h@xmath23 detections and upper limits ) for lsb galaxies fall within the ranges typically found for high surface brightness objects . ( the one exception to this , ugc 06968 , is described in detail in oneil , schinnerer , & hofner 2003 . ) using the data from @xcite , @xcite , and @xcite gives @xmath30=0.51@xmath310.78 for all hsb galaxies and @xmath30=0.53@xmath310.80 for hsb galaxies with m@xmath32 . these numbers , though , are skewed due to the presence of a few galaxies with @xmath33 . looking instead at the median value for the hsb galaxies with m@xmath32 is @xmath34=0.27 . the mlsb galaxies with co detections have @xmath35=0.07 0.5 , with @xmath34=0.09 , within the range of the values for the hsb galaxies , albeit a bit lower . as no correction has been made to the mlsb galaxy data to account for surveying only the central 10 - 25 kpc of each galaxy for co , the fact that the mlsb galaxies m@xmath1/m@xmath2 appears to be somewhat lower than that found for the hsb galaxies can not be considered significant . it is also clear from figure [ fig : mbh2 ] that the only lsb galaxies which have been detected at co are the massive lsb galaxies . unlike their less massive counterparts which often have little or no central concentration of matter , the higher gravitational potential at the center of mlsb galaxies typically results in a dense central bulge . @xcite speculate that it is within this high density region that the star formation history of massive lsb galaxies most readily mimics that of hsb galaxies , resulting in an overall higher star formation rate , and producing the molecular gas detected . this speculation is given considerable more weight with our most recent observations . previously , detection of molecular gas in lsb galaxies seemed like an impossible task , with the first co detection occurring only after 10 years of searching . yet in this paper we described observing only three lsb galaxies and detecting co in two of the three a 67% detection rate . as all three sources were chosen using the criteria described in @xcite lsb galaxies with high dynamical masses , m@xmath36 , and large central bulges it would appear oneil et al.s speculation has merit . the high detection rate shows we are now able to reliably find co gas within the central region of mlsb galaxies . lcccccc ugc 04144 & 1@xmath00 & 3.56 & 9797 & 430 & 7.3 & 0.38 + ugc 04144 & 2@xmath01 & 7.17 & 9763 & 429 & 9.0 & 0.20 + ugc 05440 & 1@xmath00 & @xmath370.50 & & & @xmath375.7 & + ugc 05440 & 2@xmath01 & @xmath370.59 & & & @xmath375.6 & + ugc 06124 & 1@xmath00 & 1.72 & 13940 & 517 & 8.5 & 0.14 + ugc 06124 & 2@xmath01 & @xmath370.78 & & & @xmath377.9 & + ugc 01922@xmath38 & 1@xmath00 & 1.38 & 10795 & 404 & 9.2 & 0.07 + ugc 01922@xmath38 & 2@xmath01 & 2.96 & 10802 & 403 & 8.9 & 0.04 + ugc 12289@xmath38 & 1@xmath00 & 1.16 & 10162 & 200 & 9.0 & 0.07 + ugc 12289@xmath38 & 2@xmath01 & 0.69 & 10185 & 201 & 8.2 & 0.01 + p06 - 1@xmath39 & 1@xmath00&0.95 & 10904 & 302 & 8.8 & 0.09 + p06 - 1@xmath39 & 2@xmath01&1.14 & 10903 & 216 & 8.3 & 0.03 + ugc 06968@xmath38 & 1@xmath00&@xmath370.21 & & & @xmath377.9 & @xmath370.004 + ugc 06968@xmath38 & 2@xmath01&@xmath370.58 & & & @xmath377.8 & @xmath370.003 + lsbc f582 - 2@xmath39 & 1@xmath00&@xmath370.54 & & & @xmath379.2 & @xmath370.2 + malin 1@xmath39 & 1@xmath00&@xmath370.15 & & & @xmath379.4 & @xmath370.06 + malin 1@xmath39 & 2@xmath01&@xmath370.35 & & & @xmath378.7 & @xmath370.01 +

to date , the only low surface brightness ( lsb ) galaxies which have been detected in co are the massive lsb ( mlsb ) galaxies . in 2003 , oneil , schinnerer , & hofner hypothesized that is the prominent bulge component in mlsb galaxies , not present in less massive low surface brightness galaxies , which gives rise to the detectable quantities of co gas . to test this hypothesis , we have used the iram 30 m telescope to obtain three new , deep co j(1@xmath00 ) and j(2@xmath01 ) observations of mlsb galaxies . two of the three galaxies observed were detected in co one in the j(1@xmath00 ) line and the other in both the j(1@xmath00 ) and j(2@xmath01 ) lines , bringing the total number of mlsb galaxies with co detections to 5 , out of a total of 9 mlsb galaxies observed at co to date . the third object had no detection to 2 mk at co j(1@xmath00 ) . comparing all mlsb galaxy co results with surveys of high surface brightness galaxies , we find the mlsb galaxies m@xmath1 and m@xmath1/m@xmath2 values fall within the ranges typically found for high surface brightness objects , albeit at the low end of the distribution , with the two mlsb galaxies detected at co in this survey having the highest m@xmath1/m@xmath2 values yet measured for any lsb system , by factors of 23 .

the brst theory provides the most powerful approach to the quantization of gauge systems @xcite . it includes the batalin - vilkovisky ( bv ) formalism for lagrangian gauge systems and its hamiltonian counterpart known as the batalin - fradkin - vilkovisky ( bfv ) formalism . usually , the two formalisms are developed in parallel starting , respectively , from the classical action or the first - class constraints on the phase space of the system . in either case one applies the homological perturbation theory ( hpt ) to obtain the master action or the classical brst charge at the output . a relationship between both the pictures of gauge dynamics is established through the dirac - bergmann ( db ) algorithm , which allows one to generate the complete set of first - class constraints by the classical action . all these can be displayed diagrammatically as follows : @xmath0\txt{\small \;lagrangian gauge theory\;\;\\\small with action $ s_0 $ } \ar[rr]^-{\txt{\small \textit{hpt}}}\ar[dd]|{\txt{\small \textit{db algorithm } } } & & * + [ f]\txt{\small master action \\\small \;\ ; $ s = s_0+\cdots\;\;$}\ar@{.>}[dd]^{?}\\&&\\ * + [ f]\txt{\small hamiltonian theory with\\\small the $ 1$-st class constraints $ t_a$ } \ar[rr]^-{\txt{\small \textit{hpt}}}&&*+[f]\txt{\small brst charge \\\small $ \omega = c^at_a+\cdots$ } } \ ] ] looking at this picture it is natural to ask about the dotted arrow making the diagram commute . the arrow symbolizes a hypothetical map or construction connecting the bv and bfv formalisms at the level of generating functionals . as we show below such a map really exists . by making use of the variational tricomplex @xcite , we propose a direct construction of the classical brst charge from the bv master action . the construction is explicitly covariant ( even though we pass to the hamiltonian picture ) and generates the full spectrum of bfv ghosts immediately from that of the bv theory . we also derive a covariant poisson bracket on the extended phase space of the theory , with respect to which the classical brst charge obeys the master equation . the construction of the covariant poisson bracket is similar to that presented in @xcite , except that our poisson bracket is defined off shell . finally , it should be noted that the first variational tricomplex for gauge systems was introduced in @xcite as the koszul - tate resolution of the usual variational bicomplex for partial differential equations . using this tricomplex , the authors of @xcite were able to relate various lie algebras associated with the global symmetries and conservation laws of a classical gauge system . our tricomplex is similar in nature but involves the full brst differential , and not its koszul - tate part . in modern language the classical fields are just the sections of a locally trivial , fiber bundle @xmath1 over an @xmath2-dimensional space - time manifold @xmath3 . the typical fiber @xmath4 of @xmath5 is called the _ target space of fields_. in case the bundle is trivial , i.e. , @xmath6 , the fields are merely the smooth mappings from @xmath3 to @xmath4 . for the sake of simplicity , we restrict ourselves to fields associated with vector bundles . in this case the space of fields @xmath7 has the structure of a real vector space . bearing in mind gauge theories as well as field theories with fermions , we assume @xmath8 to be a @xmath9-graded supervector bundle over the ordinary ( non - graded ) smooth manifold @xmath3 . the grassmann parity and the @xmath9-grading of a homogeneous object @xmath10 will be denoted by @xmath11 and @xmath12 , respectively . it should be emphasized that in the presence of fermionic fields there is no natural correlation between the grassmann parity and the @xmath13-grading . since throughout the paper we work exclusively in the category of @xmath14-graded supermanifolds , we omit the boring prefixes `` super '' and `` graded '' whenever possible . for a quick introduction to the graded differential geometry and some of its applications we refer the reader to @xcite , @xcite , @xcite . in the local field theory , the dynamics of fields are governed by partial differential equations . the best way to account for the local structure of fields is to introduce the variational bicomplex @xmath15 on the infinite jet bundle @xmath16 associated with the vector bundle @xmath8 . here @xmath17 and @xmath18 denote the horizontal and vertical differentials in the bigraded space @xmath19 of differential forms on @xmath16 , where @xmath20 and @xmath21 refer to the vertical and horizontal degrees , respectively . a brief account of the concept of a vatiational bicomplex can be found in @xcite , @xcite . the free variational bicomplex represents thus a natural kinematical basis for defining local field theories . in order to specify dynamics two more geometrical ingredients are needed . these are the classical brst differential and the brst - invariant ( pre)symplectic structure on @xmath16 . let us give the corresponding definitions . by a _ @xmath22-form on @xmath16 we understand an element @xmath23 satisfying an element of the quotient space @xmath24 and its representative in @xmath25 . the sign @xmath26 means equality modulo @xmath27 . ] @xmath28 the form @xmath29 is assumed to be homogeneous , so that we can speak of an odd or even presymplectic structure of definite @xmath9-degree . the triviality of the relative `` @xmath18 modulo @xmath17 '' cohomology in positive vertical degree ( see ( * ? ? ? * sec . 19.3.9 ) ) implies that any presymplectic @xmath22-form is exact , namely , there exists a homogeneous @xmath30-form @xmath31 such that @xmath32 . the form @xmath31 is called the _ presymplectic potential _ for @xmath29 . clearly , the presymplectic potential is not unique . if @xmath33 is one of the presymplectic potentials for @xmath29 , then setting @xmath34 we get @xmath35 in other words , any presymplectic form has a @xmath18-closed representative . denote by @xmath36 the space of all evolutionary vector fields @xmath37 on @xmath16 that fulfill the relation is called _ evolutionary _ if @xmath38 , where @xmath39 is the operation of contraction of @xmath37 with differential forms . ] @xmath40 a presymplectic form @xmath29 is called non - degenerate if @xmath41 , in which case we refer to it as a _ symplectic form_. an evolutionary vector field @xmath37 is called _ hamiltonian _ with respect to @xmath29 if it preserves the presymplectic form , that is , @xmath42 obviously , the hamiltonian vector fields form a subalgebra in the lie algebra of all evolutionary vector fields . eq . ( [ xom ] ) is equivalent to @xmath43 again , because of the triviality of the relative @xmath18-cohomology , we can write @xmath44 for some @xmath45 . we refer to @xmath46 as a _ hamiltonian form _ ( or _ hamiltonian _ ) associated with @xmath37 . sometimes , to indicate the relationship between the hamiltonian vector fields and forms , we will write @xmath47 for @xmath37 . in general , the relationship is far from being one - to - one . the space @xmath48 of all hamiltonian @xmath49-forms can be endowed with the structure of a lie algebra . the corresponding lie bracket is defined as follows : if @xmath50 and @xmath51 are two hamiltonian vector fields associated with the hamiltonian forms @xmath10 and @xmath52 , then @xmath53 the next proposition shows that the bracket is well defined and possesses all the required properties . [ 2.1 ] the bracket ( [ pb ] ) is bilinear over reals , maps the hamiltonian forms to hamiltonian ones , enjoys the symmetry property @xmath54 and obeys the jacobi identity @xmath55 an odd evolutionary vector field @xmath56 on @xmath16 is called _ homological _ if @xmath57=2q^2=0\ , , \qquad \deg\,q=1\,.\ ] ] the lie derivative along the homological vector field @xmath56 will be denoted by @xmath58 . it follows from the definition that @xmath59 . hence , @xmath58 is a differential of the algebra @xmath60 increasing the @xmath9-degree by 1 . moreover , the operator @xmath58 anticommutes with the coboundary operators @xmath17 and @xmath18 : @xmath61 this allows us to speak of the tricomplex @xmath62 , where @xmath63 in the physical literature the homological vector field @xmath56 is known as the _ classical brst differential _ and the @xmath9-grading is called the _ ghost number_. these are the two main ingredients of all modern approaches to the covariant quantization of gauge theories . in the bv formalism , for example , the brst differential carries all the information about equations of motions , their gauge symmetries and identities , and the space of physical observables is naturally identified with the group @xmath64 of `` @xmath58 modulo @xmath17 '' cohomology in ghost number zero . for general non - lagrangian gauge theories the classical brst differential was systematically defined in @xcite , @xcite . the equations of motion of a gauge theory can be recovered by considering the zero locus of the homological vector field @xmath56 . in terms of adapted coordinates @xmath65 on @xmath16 the vector field @xmath56 , being evolutionary , assumes the form represents the set of symmetric covariant indices and @xmath66 . the _ order _ of the multi - index is given by @xmath67 . ] @xmath68 then there exists an integer @xmath69 such that the equations @xmath70 define a submanifold @xmath71 . the standard regularity condition implies that @xmath72 fibers over @xmath73 for each @xmath74 . this gives the infinite sequence of projections @xmath75 & \sigma^{l+3}\ar[r]&\sigma^{l+2}\ar[r]&\sigma^{l+1}\ar[r]&\sigma^l}\rightarrow m\,,\ ] ] which enables us to define the zero locus of @xmath56 as the inverse limit @xmath76 in physics , the submanifold @xmath77 is usually referred to as the _ shell_. the terminology is justified by the fact that the classical field equations as well as their differential consequences can be written as @xmath78 in other words , the field @xmath79 satisfies the classical equations of motion iff @xmath80 . by a _ gauge system _ on @xmath16 we will mean a pair @xmath81 consisting of a homological vector field @xmath56 and a @xmath56-invariant presymplectic @xmath22-form @xmath29 . in other words , the vector field @xmath56 is supposed to be hamiltonian with respect to @xmath29 , so that @xmath82 . the last relation implies the existence of forms @xmath83 , @xmath46 , and @xmath84 such that @xmath85 as was mentioned in sec.[prst ] , we can always assume that @xmath86 for some presymplectic potential @xmath31 , so that @xmath87 . then applying @xmath18 to the second equality in ( [ des ] ) and using the first one , we find @xmath88 . on account of the exactness of the variational bicomplex , the last relation is equivalent to @xmath89 thus , @xmath83 is a presymplectic @xmath90-form on @xmath16 coming from the presymplectic potential @xmath84 . furthermore , the form @xmath83 is @xmath56-invariant as one can easily see by applying @xmath58 to the first equality in ( [ des ] ) and using once again the fact of exactness of the variational bicomplex . let @xmath91 denote the hamiltonian for @xmath56 with respect to @xmath83 , i.e. , @xmath92 given the pair @xmath93 , we call @xmath83 the _ descendent presymplectic structure _ on @xmath16 and refer to @xmath94 as the _ descendent gauge system_. the next proposition provides an alternative definition for the descendent hamiltonian of the homological vector field . [ p2 ] let @xmath29 be a @xmath18-closed representative of a presymplectic @xmath22-form on @xmath16 and @xmath95 , then @xmath96 [ c1 ] @xmath46 is a maurer - cartan element of the lie algebra @xmath97 , that is , @xmath98 [ cl ] the hamiltonian form @xmath91 is @xmath17-closed on - shell . in particular , for @xmath99 it defines a conservation law . [ p5 ] suppose that the @xmath56-invariant presymplectic form @xmath29 of top horizontal degree has the structure @xmath100 and @xmath46 is the hamiltonian of @xmath56 with respect to @xmath29 . then the presymplectic potential for the descendent presymplectic ( 2,n-1)-form @xmath101 is defined by the equation @xmath102 the above construction of the descendent gauge system @xmath94 can be iterated producing a sequence of gauge systems @xmath103 , where the @xmath74-th presymlectic form @xmath104 is the descendant of @xmath105 . the minimal @xmath74 for which @xmath106 gives a numerical invariant of the original gauge system @xmath93 . in this section , we apply the construction of the variational tricomplex for establishing a direct correspondence between the bv formalism of lagrangian gauge systems and its hamiltonian counterpart known as the bfv formalism . we start from a very brief account of both the formalisms in a form suitable for our purposes . for a systematic exposition of the subject we refer the reader to @xcite . the starting point of the bv formalism is an infinite - dimensional manifold @xmath107 of gauge fields that live on an @xmath2-dimensional space - time @xmath3 . depending on a particular structure of gauge symmetry the manifold @xmath107 is extended to an @xmath108-graded manifold @xmath109 containing @xmath107 as its body . the new fields of positive @xmath108-degree are called the _ ghosts _ and the @xmath108-grading is referred to as the _ ghost number_. let us collectively denote all the original fields and ghosts by @xmath110 and refer to them as fields . at the next step the space of fields @xmath109 is further extended by introducing the odd cotangent bundle @xmath111\mathcal{m}$ ] . the fiber coordinates , called _ antifields _ , are denoted by @xmath112 . these are assigned with the following ghost numbers and grassmann parities : @xmath113 thus , the total space of the odd cotangent bundle @xmath111\mathcal{m}$ ] becomes a @xmath9-graded supermanifold . the canonical poisson structure on @xmath111\mathcal{m}$ ] is determined by the following odd poisson bracket in the space of functionals of @xmath114 and @xmath115 : @xmath116 here @xmath117 is a volume form on @xmath3 and the subscripts @xmath69 and @xmath118 refer to the standard left and right functional derivatives . in the physical literature the above bracket is usually called the _ antibracket _ or the _ bv bracket_. the functionals of the form @xmath119 where @xmath120 and @xmath121 , are called _ local_. under suitable boundary conditions for @xmath122 s the map @xmath123 defines an isomorphism of vector spaces , which gives rise to a pulled - back poisson bracket on @xmath124 . this last bracket is determined by the symplectic structure @xmath125 according to ( [ pb ] ) . by definition , @xmath126 and @xmath127 . the central goal of the bv formalism is the construction of a _ master action _ @xmath128 on the space of fields and antifields . this is defined as a proper solution to the _ classical master equation _ @xmath129 the local functional @xmath128 is required to be of ghost number zero and start with the action @xmath130 of the original fields to which one couples vertices involving antifields . all these vertices can be found systematically from the master equation ( [ bv_meq ] ) by means of the so - called _ homological perturbation theory _ @xcite . the classical brst differential on the space of fields and antifields is canonically generated by the master action through the antibracket : @xmath131 because of the master equation for @xmath128 and the jacobi identity for the antibracket ( [ abr ] ) , the operator @xmath56 squares to zero in the space of smooth functionals . the physical quantities are then identified with the cohomology classes of @xmath56 in ghost number zero . when restricted to the subspace of local functionals the classical brst differential ( [ clbrstd ] ) induces a homological vector field on the total space of the jet bundle @xmath16 . the hamiltonian formulation of the same gauge dynamics implies a prior splitting @xmath132 of the original space - time into space and time ; the factor @xmath133 can be viewed as the physical space at a given instant of time . the initial values of the original fields are then considered to form an infinite - dimensional manifold @xmath134 . to allow for possible constraints on the initial data of fields the manifold @xmath134 is extended to an @xmath108-graded supermanifold @xmath135 by adding new fields , called ghosts , of positive @xmath108-degree . then the space of original fields and ghosts is doubled by introducing the cotangent bundle @xmath136 endowed with the canonical symplectic structure . if we denote the local coordinates on @xmath135 by @xmath137 and the linear coordinates in the cotangent spaces by @xmath138 , then the canonical poisson bracket in the space of functionals of @xmath137 and @xmath138 reads @xmath139 here @xmath140 stands for a volume form on @xmath133 . by the definition of the cotangent bundle of a graded manifold @xmath141 again , the space of local functionals , i.e. , functionals of the form @xmath142 appears to be closed w.r.t . the even poisson bracket ( [ epb ] ) and the map @xmath143 induces an even poisson bracket on @xmath144 . the latter is determined by the even symplectic form @xmath145 of ghost number zero . the gauge structure of the original dynamics is encoded by the _ classical @xmath146 charge _ this is given by an odd , local functional of ghost number @xmath148 satisfying the classical master equation @xmath149 the classical brst differential in the extended space of fields and momenta is given now by the hamiltonian action of the brst charge : @xmath150 it is clear that @xmath151 . the group of @xmath56-cohomology in ghost number zero is then naturally identified with the space of physical observables . upon restriction to the space of local functionals the variational vector field ( [ bdif ] ) induces a homological vector field on the total space of the infinite jet bundle . it must be clear from the discussion above that any gauge system in the bfv formalism may be viewed as the descendant of the same system in the bv formalism . more precisely , we can define the even presymplectic structure @xmath83 on the phase space of a gauge theory as the descendant of the odd symplectic structure ( [ ops ] ) : @xmath152 the corresponding classical brst charge is given by @xmath153 where @xmath154 is a space - like , cauchy hypersurface and @xmath155 is the hamiltonian of the classical brst differential @xmath156 w.r.t . the descendent presymplectic form @xmath83 , i.e. , @xmath157 it is clear that @xmath158 . in virtue of corollary [ c1 ] , the functional @xmath147 obeys the classical master equation @xmath159 with respect to the even poisson bracket associated with @xmath83 . according to corollary [ cl ] the form @xmath160 represents a conserved current , the brst current . formally , this means that the `` value '' of the odd charge @xmath161 does not depend on the choice of @xmath133 provided that @xmath162 . since the canonical symplectic structure ( [ ops ] ) on the space of fields and antifields is @xmath18-exact , we can give an equivalent definition for @xmath160 in terms of the antibracket ( [ abr ] ) . for this end , consider the dynamics of fields in a domain @xmath163 bounded by two cauchy hypersurfaces @xmath164 and @xmath165 . the fields and antifields are assumed to vanish on space infinity together with their derivatives . by proposition [ p2 ] , @xmath166=\omega_{n_2}-\omega_{n_1}\,.\ ] ] in the bv formalism , the free electromagnetic field in @xmath167-dimensional minkowski space is described by the master action @xmath168 here @xmath169 is the strength tensor of the electromagnetic field , @xmath170 is the antifield to the electromagnetic potential @xmath171 , and @xmath172 is the ghost field associated with the standard gauge transformation @xmath173 . since the gauge symmetry is abelian , the master action ( [ med ] ) does not involve the ghost antifield @xmath174 . the odd symplectic structure ( [ ops ] ) on the space of fields and antifields assumes the form @xmath175 and the action of the classical brst differential is given by @xmath176 the variation of the lagrangian density reads @xmath177 one can easily check that @xmath178 . by proposition [ p5 ] the form @xmath179 defines the potential for the descendent presymplectic form @xmath180 ( of course , one could arrive at this expression by considering the brst variation @xmath181 of the original symplectic structure . ) applying the brst differential to the form @xmath83 yields one more descendent presymplectic form @xmath182 this last form , being `` absolutely '' invariant under the brst transformations ( [ brst - t ] ) , leaves no further descendants . the @xmath183-form of the conserved brst current @xmath160 associated to the brst symmetry transformations ( [ brst - t ] ) is determined by eq . ( [ j ] ) . we find @xmath184 once we identify @xmath185 with time in the hamiltonian formalism , the antifield @xmath186 plays the role of ghost momentum canonically conjugate to @xmath172 with respect to the presymplectic structure ( [ wc ] ) . the on - shell conservation of the corresponding brst charge @xmath187 expresses nothing but the gauss law @xmath188 . voronov , _ graded manifolds and drinfeld doubles for lie bialgebroids _ , in : quantization , poisson brackets and beyond , theodore voronov , ed . , * 315 * , amer . soc . , providence , ri , 2002 , 131 - 168 .

by making use of the variational tricomplex , a covariant procedure is proposed for deriving the classical brst charge of the bfv formalism from a given bv master action .

you are asked to complete a task @xmath5 drawn according to a pmf @xmath6 from a finite set of tasks @xmath7 . you do not get to see @xmath5 but only its description @xmath8 , where @xmath9 in other words , @xmath5 is described to you using @xmath10 bits . you know the mapping @xmath11 and you promise to complete @xmath5 based on @xmath8 , which leaves you no choice but to complete every task in the set @xmath12 in the interesting case where @xmath13 , you will sometimes have to perform multiple tasks , of which all but one are superfluous . ( we use @xmath14 to denote the cardinality of sets . ) given @xmath15 , the goal is to design @xmath11 so as to minimize the @xmath0-th moment of the number of tasks you perform @xmath16 } = \sum_{x\in{\mathcal{x } } } p(x ) { { \lvertf^{-1}(f(x))\rvert}}^\rho,\ ] ] where @xmath0 is some given positive number . this minimum is at least one because @xmath5 is in @xmath17 ; it decreases as @xmath15 increases ; and it is equal to one when @xmath18 . our first result is a pair of upper and lower bounds on this minimum as a function of @xmath15 . the bounds are expressed in terms of the _ rnyi entropy of @xmath5 of order @xmath1 _ : @xmath19 throughout @xmath20 stands for @xmath21 , the logarithm to base @xmath22 . for typographic reasons we henceforth use the notation @xmath23 [ thm : oneshot ] let @xmath24 . 1 . for all positive integers @xmath15 and every @xmath25 , @xmath26 } \geq 2^{\rho(h_{{\tilde{\rho}}}(x)-\log m)}.\ ] ] 2 . for every integer @xmath27 there exists @xmath28 such that @xmath26}\\ < 1 + 2^{\rho(h_{{\tilde{\rho}}}(x)-\log \widetilde{m})},\ ] ] where @xmath29 . a proof is provided in section [ sec : proof ] . the lower bound is essentially ( * ? ? ? * lemma iii.1 ) . theorem [ thm : oneshot ] is particularly useful when applied to the case where a sequence of tasks is produced by a source @xmath30 with alphabet @xmath7 and the first @xmath2 tasks @xmath31 are jointly described using @xmath3 bits : @xmath32 we assume that the order in which the tasks are performed matters and that every @xmath2-tuple of tasks in the set @xmath33 must be performed . the total number of performed tasks is therefore @xmath34 , and the ratio of the number of performed tasks to the number of assigned tasks is @xmath35 . [ thm : main ] let @xmath30 be any source with finite alphabet @xmath7 . 1 . if @xmath36 , then there exist encoders @xmath37 such that stands for @xmath38 . ] @xmath39 } = 1.\ ] ] 2 . if @xmath40 , then for any choice of encoders @xmath41 , @xmath39 } = \infty.\ ] ] on account of theorem [ thm : oneshot ] , for all @xmath2 large enough so that @xmath42 , @xmath43}\\ < 1 + 2^{n\rho\bigl(\frac{h_{{\tilde{\rho}}}(x^n)}{n } - r+\delta_n\bigr)},\end{gathered}\ ] ] where @xmath44 as @xmath45 . when it exists , the limit @xmath46 is called the _ rnyi entropy rate of order @xmath47_. it exists for a large class of sources , including time - invariant markov sources @xcite . theorem [ thm : main ] generalizes ( * ? ? ? * theorem iv.1 ) from iid sources to sources with memory and furnishes an operational characterization of the rnyi entropy rate for all orders in @xmath48 . note that for iid sources the rnyi entropy rate reduces to the rnyi entropy because in this case @xmath49 . the proof of the lower bound in theorem [ thm : oneshot ] hinges on the following simple observation . [ prop : count_lists ] if @xmath50 is a partition of a finite set @xmath7 into @xmath15 nonempty subsets ( i.e. , @xmath51 and @xmath52 if , and only if , @xmath53 ) , and @xmath54 is the cardinality of the subset containing @xmath55 , then @xmath56 @xmath57 note that the reverse of proposition [ prop : count_lists ] is not true in the sense that if @xmath58 satisfies @xmath59 then there need not exist a partition of @xmath7 into @xmath60 subsets such that the cardinality of the subset containing @xmath55 is at most @xmath61 . a counterexample is @xmath62 with @xmath63 , @xmath64 , and @xmath65 . in this example , @xmath66 , but we need 3 subsets to satisfy the cardinality constraints . however , as our next result shows , allowing a slightly larger number of subsets suffices : [ prop : sufficiency ] if @xmath7 is a finite set , @xmath67 and @xmath68 ( with the convention @xmath69 ) , then there exists a partition of @xmath7 into at most @xmath70 subsets such that @xmath71 where @xmath54 is the cardinality of the subset containing @xmath55 . proposition [ prop : sufficiency ] is the key to the upper bound in theorem [ thm : oneshot ] . combined with proposition [ prop : count_lists ] it can be considered an analog of the kraft inequality ( * ? ? ? * theorem 5.5.1 ) for partitions of finite sets . a proof is given in section [ sec : proposition ] . the construction of the encoder in the derivation of the upper bound in theorem [ thm : oneshot ] requires knowledge of the distribution @xmath6 of @xmath5 ( see section [ sec : upper_bound ] ) . in section [ sec : divergence ] we consider a mismatched version of this direct part where the construction is carried out based on the law @xmath72 instead of @xmath6 . we show that the penalty incurred by the mismatch between @xmath6 and @xmath72 can be expressed in terms of the divergence measures @xmath73 where @xmath47 can be any positive number not equal to one . ( we use the convention @xmath74 and @xmath75 if @xmath76 . ) this family of divergence measures was proposed by sundaresan @xcite , who showed that it plays a similar role in the massey - arikan guessing problem @xcite . the proof of the lower bound is inspired by the proof of ( * ? ? ? * theorem 1 ) . fix an encoder @xmath77 , and note that it gives rise to a partition of @xmath7 into the @xmath15 subsets @xmath78 let @xmath79 denote the number of nonempty subsets in this partition . also note that for this partition the cardinality of the subset containing @xmath55 is @xmath80 recall hlder s inequality : if @xmath81 , @xmath82 and @xmath83 , then @xmath84 rearranging gives @xmath85 substituting @xmath86 , @xmath87 , @xmath88 and @xmath89 in , we obtain @xmath90 where follows from , , and proposition [ prop : count_lists ] ; and where follows because @xmath91 . since hlder s inequality holds with equality if , and only if , ( iff ) @xmath92 is proportional to @xmath93 , it follows that the lower bound in theorem [ thm : oneshot ] holds with equality iff @xmath94 is proportional to @xmath95 . we derive the upper bound in theorem [ thm : oneshot ] by constructing a partition that approximately satisfies this relationship . to this end , we use proposition [ prop : sufficiency ] with @xmath96 in and @xmath97 where we choose @xmath98 just large enough to guarantee the existence of a partition of @xmath7 into at most @xmath15 subsets satisfying . this is accomplished by the choice @xmath99 ( this is where we need @xmath100 . ) indeed , @xmath101 and hence @xmath102 let then the partition @xmath103 with @xmath91 be as promised by proposition [ prop : sufficiency ] , and construct @xmath77 by setting @xmath104 if @xmath105 . for this encoder , @xmath106 where the strict inequality follows from and the inequality @xmath107 which is easily checked by considering separately the cases @xmath108 and @xmath109 . we describe a procedure for constructing a partition of @xmath7 with the desired properties . since the labels do not matter , we may assume for convenience of notation that @xmath110 and @xmath111 the first subset in the partition we construct is @xmath112 if @xmath113 , then the construction is complete and and are clearly satisfied . otherwise we follow the steps below to construct additional subsets @xmath50 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ step @xmath114 _ : if @xmath115 then we complete the construction by setting @xmath116 and @xmath117 . otherwise we set @xmath118 and go to step @xmath22 . + _ step @xmath119 _ : if @xmath120 then we complete the construction by setting @xmath121 and @xmath122 . otherwise we let @xmath123 contain the @xmath124 smallest elements of @xmath125 , i.e. , we set @xmath126 and go to step @xmath127 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we next verify that is satisfied and that the total number of subsets @xmath128 does not exceed . clearly , @xmath129 for every @xmath130 , so to prove we check that @xmath131 for every @xmath132 . it is clear that @xmath133 for all @xmath134 . let @xmath135 denote the smallest element in the subset containing @xmath55 . then @xmath136 for all @xmath137 by construction , and since @xmath138 , we have @xmath139 by the assumption , and hence @xmath131 for all @xmath140 . it remains to check that @xmath128 does not exceed . this is clearly true when @xmath117 , so we assume that @xmath141 . since @xmath142 for all @xmath143 , we have on account of proposition [ prop : count_lists ] @xmath144 fix an arbitrary @xmath145 and let @xmath146 be the set of indices @xmath147 such that there is an @xmath148 with @xmath149 . we next argue that @xmath150 . to this end , enumerate the indices in @xmath146 as @xmath151 . for each @xmath152 select @xmath153 such that @xmath154 . then @xmath155 note that if @xmath156 and @xmath157 and @xmath158 , then @xmath159 . thus , @xmath160 because @xmath161 and @xmath162 , and @xmath163 . consequently , @xmath164 iterating this argument shows that @xmath165 and since @xmath166 for @xmath167 by , it follows that @xmath150 . continuing from with @xmath168 , @xmath169 where the first inequality follows because @xmath170 for @xmath171 , and where the second inequality follows from the hypothesis of the proposition . since @xmath128 is an integer and @xmath145 is arbitrary , it follows from that @xmath128 is upper - bounded by . the key to the upper bound in theorem [ thm : oneshot ] was to use proposition [ prop : sufficiency ] with @xmath172 as in and to obtain a partition of @xmath7 for which the cardinality of the subset containing @xmath55 is approximately proportional to @xmath95 . evidently , this construction requires knowledge of the distribution @xmath6 of @xmath5 . in this section , we derive the penalty when @xmath6 is replaced with @xmath72 in and . since it is then still true that @xmath173 proposition [ prop : sufficiency ] guarantees the existence of a partition of @xmath7 into at most @xmath15 subsets satisfying . constructing @xmath11 from this partition as in section [ sec : upper_bound ] and proceeding similarly as in to , we obtain @xmath174 where @xmath175 is as in and @xmath176 is as in theorem [ thm : oneshot ] . ( note that @xmath177 only if the support of @xmath6 is contained in the support of @xmath72 . ) the penalty in the exponent when compared to the upper bound in theorem [ thm : oneshot ] is thus given by @xmath175 . to reinforce this , further note that @xmath178 where @xmath179 and @xmath180 are the @xmath2-fold products of @xmath6 and @xmath72 . consequently , if the source @xmath30 is iid @xmath6 and we construct @xmath181 similarly as above based on @xmath180 instead of @xmath179 , we obtain the bound @xmath182}<1 + 2^{n\rho(h_{{\tilde{\rho}}}(x_1 ) + \delta_{{\tilde{\rho}}}(p||q ) - r + \delta_n)},\ ] ] where @xmath183 as @xmath45 . the rhs of tends to one provided that @xmath184 . thus , in the iid case @xmath175 is the rate penalty incurred by the mismatch between @xmath6 and @xmath72 . we conclude this section with some properties of @xmath185 . properties 13 ( see below ) were given in @xcite ; we repeat them here for completeness . note that rnyi s divergence ( see , e.g. , @xcite ) @xmath186 satisfies properties 1 and 3 but none of the others in general . property 2 follows by inspection of . properties 35 follow by simple calculus . as to property 1 , consider first the case where @xmath193 . in view of property 2 , we may assume that @xmath198 . inequality with @xmath201 and @xmath202 gives @xmath203 the conditions for equality in hlder s inequality imply that equality holds iff @xmath191 . consider next the case where @xmath145 . by hlder s inequality with @xmath204 and @xmath205 , @xmath206 with equality iff @xmath191 .

a task is randomly drawn from a finite set of tasks and is described using a fixed number of bits . all the tasks that share its description must be performed . upper and lower bounds on the minimum @xmath0-th moment of the number of performed tasks are derived . the key is an analog of the kraft inequality for partitions of finite sets . when a sequence of tasks is produced by a source of a given rnyi entropy rate of order @xmath1 and @xmath2 tasks are jointly described using @xmath3 bits , it is shown that for @xmath4 larger than the rnyi entropy rate , the @xmath0-th moment of the ratio of performed tasks to @xmath2 can be driven to one as @xmath2 tends to infinity , and that for @xmath4 less than the rnyi entropy rate it tends to infinity . this generalizes a recent result for iid sources by the same authors . a mismatched version of the direct part is also considered , where the code is designed according to the wrong law . the penalty incurred by the mismatch can be expressed in terms of a divergence measure that was shown by sundaresan to play a similar role in the massey - arikan guessing problem .

recently , there has been a lot of interest in the study of the proton drip - line nucleus @xmath1b which is perhaps the most likely candidate for having a proton halo structure @xcite , since its last proton has a binding energy of only 137 kev . several measurements reported lately do seem to provide evidence in favor of this possibility . for example , the electric quadrupole moment of @xmath1b is found to be twice as large as the value predicted by the shell model , which can be explained with a single particle wave function corresponding to a matter density of root mean square ( rms ) radius of 2.72 fm @xcite . the observed narrow longitudinal momentum ( @xmath6 ) distribution of the @xmath2be fragment emitted in the breakup reaction of 1.47 gev / nucleon @xmath1b on @xmath7 target has been interpreted in terms of a greatly extended proton distribution in @xmath1b @xcite . the significantly enhanced reaction cross sections of @xmath1b measured at beam energies between 20 - 60 mev / nucleon are shown to be consistent with the large matter radius of @xmath1b required to explain its quadrupole moment @xcite . nevertheless , the existence of a proton halo in @xmath1b is still an open issue . nakada and otsuka @xcite have shown that @xmath8 shell model calculations can reproduce the measured large quadrupole moment of @xmath1b without invoking a proton halo structure . the interaction cross sections measured by tanihata et al . @xcite at 790 mev / nucleon are consistent with a normal size of @xmath1b . the @xmath6 distribution of the @xmath2be fragment emitted in the breakup reaction of @xmath1b at 41 mev / nucleon has been found to be dependent on the target mass in ref . @xcite where it is argued that in contrast to the situation in the neutron halo nuclei @xcite , the assumption of an unusually extended spatial distribution is not necessary to explain the narrow @xmath6 distribution in case of @xmath1b ; the reaction mechanism plays an important role here . breakup reactions in which the halo particle(s ) is(are ) removed from the projectile in the coulomb and nuclear fields of the target nucleus , have played a significant role in probing the neutron halo structure in some light neutron drip - line nuclei ( see e.g. @xcite for a recent review ) . the enhanced total coulomb breakup cross sections @xcite , narrow longitudinal momentum distributions of the heavy fragments @xcite , and sharply forward peaked angular distributions of the valence neutron(s ) @xcite are some of the pivotal observations through which the neutron halo structure has been well manifested . apart from the @xmath6 distributions of the @xmath2be fragment , some data on the total cross section of the breakup reaction @xmath1b + a @xmath9 @xmath2be + x on low mass targets have also become available recently @xcite . the theoretical studies reported so far have used either the serber type @xcite of models @xcite or the diffraction dissociation picture @xcite developed by sitenko and co - workers @xcite . both these approaches are essentially semi - classical in nature , hence a more microscopic calculation within the quantum mechanical scattering theory is needed to interpret the data properly . a proper understanding of the nuclear breakup of @xmath1b is also important in the context of the extraction of the astrophysical @xmath10-factor for the radiative capture reaction @xmath2be(p,@xmath11)@xmath1b from the coulomb dissociation of @xmath1b @xcite . in this paper , we present calculations of the cross sections for the breakup reaction @xmath1b @xmath12 @xmath0si @xmath9 @xmath2be @xmath12 x within a direct fragmentation model ( dfm ) , which is formulated in the framework of the post form distorted wave born approximation ( pfdwba ) @xcite . as the target nucleus involved in this reaction is very light , we shall consider only the nuclear breakup process . however , the coulomb breakup can also be calculated in this theory on the same footing ( see e.g. @xcite ) . in the next section we present the details of our formalism . in section 3 , our results are presented and discussed . the conclusions of our work are described in section 4 . the nuclear breakup cross section consists of two components : the elastic breakup ( elb ) ( also known as `` diffraction dissociation '' ) where x corresponds to the target nucleus a in its ground state and proton , and inelastic breakup ( inelb ) ( also known as `` diffraction stripping '' ) where x can be any other channel of the a @xmath12 p system . the triple differential cross section for the elastic breakup reaction @xmath13 ( e.g for our case , @xmath14 = @xmath1b , @xmath15 = @xmath2be , @xmath16 = p ) , is defined as @xmath17 where the transition amplitude @xmath18 is given by @xmath19 in eqs . ( 1 ) and ( 2 ) , @xmath20 is the orbital angular momentum for relative @xmath15 + @xmath16 system and @xmath21 are the spherical harmonics . @xmath22 represents the interaction between constituents @xmath15 and @xmath16 while @xmath23 is the wave function for their relative motion in the projectile ground state . @xmath24 ( @xmath25 ) , @xmath26 ( @xmath27 ) and @xmath28 ( @xmath29 ) are the momenta ( reduced masses ) of the particles @xmath14 , @xmath15 and @xmath16 respectively . @xmath30 s denote the scattering wave functions which are generated by the appropriate optical potentials in respective channels . the system of coordinates used are the same as that given in ref . @xcite . the transition amplitude is a six dimensional integral . by making a zero range approximation ( zra ) this integral is reduced to three dimensions @xcite , although its calculation is still a major problem as it involves a product of three scattering waves which converge very slowly . in the zra the details of the internal structure of the projectile appear in the amplitude only through an overall normalization constant and the values of @xmath20 other than zero are necessarily excluded . because of the relative p - state between @xmath2be and the proton in the ground state of @xmath1b the zra , therefore , is not suitable for this case . however , we introduce a constant range approximation ( cra ) which reduces the integral in eq . ( 2 ) to three dimensions and at the same time allows the non - zero values of @xmath20 to enter in the calculations . we assume that the breakup reaction is strongly peripheral and that only those configurations where ( 1 ) the proton is in the collinear position between the target nucleus and @xmath2be and ( 2 ) the relative separation between proton and @xmath2be is constant ( say @xmath31 @xmath32 ) , contribute to the transition amplitude . we can then write @xmath33 where @xmath31 is taken to be the distance between origin and the maximum in the @xmath1b ground state wave function . approximations similar to the cra have been used earlier by phlhoffer et al . @xcite and kubo and hirata @xcite to study the @xmath34 transfer reactions induced by @xmath35li and @xmath2li projectiles within the dwba . these authors have found a reasonable agreement between the calculated and measured angular distributions particularly for momentum matched transitions . furthermore , as long as the reaction is not sensitive to the smaller distances , the results obtained with eq . ( 3 ) are in agreement with those of the full finite range calculations . substituting eq . ( 3 ) in eq . ( 2 ) , making partial wave expansion for @xmath36 and using eq . ( 4.13 ) of @xcite , we get @xmath37 where the reduced amplitude @xmath38 is given by @xmath39 in eq . ( 5 ) we have @xmath40 and @xmath41 is the radial part of the wave function @xmath36 . ( 5 ) is similar to the amplitude obtained with the zra and can be evaluated by using the method described in e.g. @xcite . the expressions for the zero range amplitudes are retrieved from eq . ( 4 ) - eq . ( 6 ) by assuming @xmath20 and @xmath31 equal to zero . the transition amplitude for the inelastic breakup reaction @xmath42 , where @xmath43 is some final state of the system @xmath44 , is given by @xcite @xmath45 where @xmath46 is the form factor , which is obtained by taking the overlap of the wave function for the channel @xmath43 ( which incorporates all the possible reactions initiated by the interaction between nuclei @xmath47 and @xmath16 ) , with the wave function describing the internal states of the target and projectile nuclei . ( 3 ) and other steps as described above , an expression similar to eq . ( 4 ) can be derived for @xmath48 with the reduced amplitude given by @xmath49 where @xmath50 is the radial part of the form factor . its calculation simplifies greatly if we introduce the so called `` surface approximation '' and write @xmath50 in terms of its asymptotic form @xmath51 where @xmath52 , with @xmath53 and @xmath54 being the regular and irregular coulomb wave functions . this equation can be rewritten in terms of the elastic scattering wave function @xmath41 ( see eq . ( 5 ) ) as @xmath55 where @xmath56 are the @xmath10 matrix elements for the elastic channel corresponding to the angular momentum @xmath57 . the validity of the surface approximation has been tested by kasano and ichimura @xcite who found it to be well fulfilled even for the deuteron . we use eq . ( 11 ) for the form factor @xmath58 also in the interior region in eq . ( 9 ) , which is not expected to be a serious approximation as this region contributes very little to the whole dwba integral . in order to calculate the inelastic breakup cross section one has to sum over all the channels @xmath59 , which can be easily done by using the unitarity of the @xmath10 matrix as all the dependence on channel @xmath43 in the transition amplitude rests in the @xmath10 matrix @xmath60 @xmath61 thus the inelastic breakup cross section can be written as @xmath62 where @xmath63 is the same as the amplitude defined in eq . ( 4 ) with the wave function @xmath41 replaced by the regular coulomb function . the partial reaction and elastic cross sections @xmath64 and @xmath65 are related to the scattering matrix elements @xmath56 by their usual definitions . it may be noted that the quantities required to calculate the inelastic breakup are the same as those already calculated in the case of elastic breakup . in order to study the impact parameter dependence of the breakup cross section , we define the `` probability of breakup '' @xmath66 as @xmath67 in eq . ( 14 ) , @xmath68 is the sum of the elastic and inelastic breakup cross sections given by eqs . ( 1 ) and ( 13 ) . the optical potentials in the entrance and outgoing channels and the constants @xmath69 and @xmath31 are required as input in our numerical calculations . although some elastic scattering data for the @xmath1b , @xmath2be + @xmath70c systems at the beam energy of 40 mev / nucleon are available @xcite , the usual optical model fits to them is unfortunately not reported . unless otherwise stated , our calculations have been performed with the following set of optical potentials , @xmath71 = 123.0 mev , @xmath72 = 0.75 fm , @xmath73 = 0.80 fm , @xmath74 = 65.0 mev , @xmath75 = 0.78 fm , @xmath76 = 0.80 fm , with real and imaginary volume woods - saxon terms . this potential , which is similar to that used in the recent analysis of @xmath77li and @xmath78li elastic scattering @xcite , has been used for both @xmath1b and @xmath2be . the @xmath79 convention was followed to get the radii from the radius parameters . the global becchetti - greenlees parameterization @xcite was used for the proton - target potential . the constants @xmath69 ( see e.g. @xcite ) and @xmath31 have been determined with a @xmath1b ground state wave function obtained by assuming it to be a pure @xmath80 proton single particle state , with separation energy 0.137 mev , calculated in a central woods - saxon potential of geometry , @xmath81 = 1.54 fm , and @xmath82 = 0.52 fm @xcite . this gives @xmath69 = -39.0 mev @xmath83 and @xmath31 = 1.8 fm , which has been used in all the calculations described in this paper . the radius and diffuseness parameters used by barker @xcite ( @xmath81 = 1.25 fm , and @xmath82 = 0.65 fm ) , and esbensen and bertsch ( @xmath81 = 1.25 fm , and @xmath82 = 0.52 fm ) @xcite lead to @xmath69 = -40.4 and -41.7 mev @xmath83 and @xmath31 = 1.8 and 1.7 fm respectively . on the other hand , with more elaborate rpa models for the @xmath2be + p overlap wave function @xcite , the values of @xmath69 and @xmath31 are found to be -38.0 mev @xmath83 and 1.8 fm respectively . hence , the constants @xmath69 and @xmath31 do not show any marked dependence on the nuclear structure model of @xmath1b . for the sake of comparison with other approaches the simplified potential model as used by us seems to be adequate at this stage of the present theory . 1 shows the calculated elastic ( dotted line ) , inelastic ( dashed line ) and total ( solid line ) breakup cross sections for the @xmath1b @xmath12 @xmath0si @xmath9 @xmath2be @xmath12 x reaction as a function of beam energy together with the data taken from @xcite . we see that the measured total breakup cross sections are well reproduced by our calculations although the contributions of the elastic and inelastic breakup modes are slightly over- and under - predicted respectively . the breakup cross sections decrease with beam energy up to 20 mev / a and after that they are almost constant ( although the inelastic breakup mode still shows a tendency of decreasing somewhat ) . the nuclear breakup cross sections of @xmath1b as reported in @xcite show a similar type of energy dependence in this beam energy regime although their increase below 20 mev / nucleon is less pronounced than that seen in fig . 1 . clearly , more measurements are needed to clarify this point . a striking feature of the results shown in fig . 1 is that the contribution of the inelastic breakup mode to the total breakup cross section is limited only to about 30@xmath3 , which is in agreement with the experimental data @xcite . this is in marked contrast to the situation in stable nuclei where this mode of breakup makes up about 75 - 80 @xmath3 of the total @xmath84 breakup cross section ( see e.g. @xcite ) . to understand this difference , we show in fig . 2 the breakup probability ( @xmath85 ) ( defined by eq . ( 14 ) ) for the reaction @xmath0si(@xmath1b , @xmath2be ) as a function of the impact parameter @xmath86 ( @xmath87 ) . it can be seen that @xmath85 peaks at about 8 fm , which is in remarkable agreement with that obtained in ref . @xcite from the semi - classical arguments . this is quite large in comparison to the sum of the radii ( @xmath88 ) of @xmath1b and @xmath0si ( @xmath89 6 fm ) . furthermore , most of the contribution to @xmath85 comes from the impact parameters beyond 8 fm , while those from distances @xmath90 @xmath88 are strongly suppressed . this clearly shows that the breakup of @xmath1b takes place far away from the nuclear surface which reduces the probability of the inelastic breakup process ; large impact parameters favor the elastic breakup mode . in contrast , for stable isotopes , the breakup probability peaks around @xmath88 ( where the inelastic breakup mode is maximum ) and the drop from the peak value for @xmath91 is much faster than that seen in fig . 2 @xcite . this , explains to some extent the difference in the nature of the inelastic breakup cross section of @xmath1b as compared to that of the stable isotopes . the fact that the breakup of @xmath1b is dominated by contributions coming from a large range of impact parameters @xmath92 , is in agreement with the observation made earlier in the case of neutron halo nuclei @xmath78li and @xmath78be @xcite . this could provide an indirect evidence for a larger spatial extension of the proton in the ground state of @xmath1b . it should , however , be stressed that a @xmath80 configuration for the p - @xmath2be relative motion already leads to a @xmath1b matter density with a larger rms radius @xcite . nevertheless , in the present calculations the @xmath1b structure input largely affects only the absolute magnitudes of the cross sections ( through the constant @xmath69 ) ; the peak position in fig . 2 is mostly decided by the reaction dynamics . it is possible , in principle , to include other components ( eg . @xmath93 , @xmath94 ) in the @xmath1b wave function within this formalism . however , as the @xmath80 component carries by far the largest spectroscopic weight ( @xmath95 90 @xmath3 ) , this is unlikely to alter our results much . as a side remark , we point out that the relative contributions of the elastic and inelastic breakup modes are independent of the uncertainties in the values of @xmath69 and @xmath31 as the same constants enter in all the cross sections . in fig . 3 , we show the contributions of elastic and inelastic breakup modes to the energy distribution of the @xmath2be fragment at the beam energy of 30 mev / nucleon . one notes that while the elastic breakup mode dominates in the peak region , its contribution is very weak towards the high energy end ( where the proton energy @xmath96 ) ; total cross section is made up mostly of the inelastic breakup mode in this region . the threshold behavior of the breakup cross section can be easily understood from that of the phase - shift @xmath97 of the scattering of the proton from the target . it can be shown that @xcite in the limit @xmath96 , the elastic and inelastic breakup cross sections are proportional to @xmath98 and @xmath99 ( where @xmath100 is the radius of the nuclear potential ) respectively . therefore , in this limit the elastic breakup cross section tends to zero even for the @xmath101 wave a + p interaction while the inelastic one to a finite value , which explains the observation made in fig . 3 . it would be very interesting therefore , to perform measurements for the energy spectra of the fragment @xmath2be to confirm the @xmath96 behavior . it may help in fixing the absolute magnitude of the inelastic breakup cross section in the breakup experiments . to conclude , we have presented for the first time a fully quantum mechanical calculation of the elastic and inelastic modes of the nuclear breakup of @xmath1b on the si target . we employed a direct fragmentation model which is formulated within the framework of the post form distorted wave born - approximation . this is a definite improvement over the semi - classical models of the breakup reactions used so far for this purpose . we obtain a good overall description of the experimental data measured recently . the inelastic breakup mode is found to contribute only up to 30@xmath3 to the total breakup cross section for the @xmath0si(@xmath1b,@xmath2be ) reaction , which is in contrast to the breakup of the stable isotopes . most of the contributions to the breakup cross section come from the distances far beyond the nuclear surface which favors the elastic breakup mode . the energy spectra of the @xmath2be fragment is dominated by the inelastic breakup mode towards the high energy end where the proton energy goes to zero . this observation which is beyond the scope of the semi - classical models , is a natural outcome of the pfdwba theory of the breakup reactions and it should be verified experimentally . we must stress that the lack of the precise knowledge about the parameters of the optical potential for the @xmath1b,@xmath2be @xmath12 @xmath0si system is a potential source of uncertainty in our calculations . therefore , the analysis of the existing data ( taken at around 40 mev / nucleon ) for this system @xcite in terms of the conventional optical model would be extremely useful . moreover , similar studies at other beam energies are also clearly needed . the present calculations are not very sensitive to the nuclear structure models of @xmath1b . the direct fragmentation model should be improved further so that more detail of the projectile wave function can enter in the calculations explicitly . once such an extended reaction description is available , the use of more elaborate nuclear structure models of the projectile will be meaningful . it may then become possible to use the breakup data to distinguish the wave functions of @xmath1b obtained from a 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brown , a. csoto and r. sherr , nucl . phys.a * 597 * , 66 ( 1996 ) baur80 g. baur , r. shyam , f. rsel and d. trautmann , helv . acta , * 53 * , 506 ( 1980 ) ; g. baur , proc . varna int . summer school on nuclear pysics , sept . 22 - oct . 1 , 1985 , varna , romania ( unpublished ) . * total cross section for the breakup reaction of @xmath1b @xmath12 @xmath0si @xmath9 @xmath2be @xmath12 x as a function of the beam energy . the contributions of the elastic and inelastic breakup modes are shown by dotted and dashed lines respectively while their sum is depicted by the solid line . the experimental data for the total ( solid circles ) , elastic ( solid squares ) and inelastic ( open circles ) breakup cross sections are taken from @xcite . * breakup probability ( @xmath102 ) ( as defined by eq . ( 14 ) ) for the reaction studied in fig . 1 at the beam energy of 40 mev / nucleon as a function of the impact parameter . * energy distribution of the @xmath2be fragment emitted in the breakup of @xmath1b on @xmath0si target at the beam energy of 30 mev / nucleon . the solid , dashed and dotted lines have the same meaning as in fig . 1 .

we calculate the cross sections of the elastic and inelastic breakup modes for the inclusive breakup reaction @xmath0si(@xmath1b,@xmath2be ) at beam energies between 10 - 40 mev / nucleon within a direct fragmentation model formulated in the framework of the post form distorted - wave born - approximation . in contrast to the case of the stable isotopes , the inelastic breakup mode is found to contribute only up to 30@xmath3 to the total breakup cross section , which is in agreement with the recently measured experimental data . however , the high energy tail of the energy spectra of @xmath2be fragment is dominated by the inelastic breakup mode . the breakup amplitude is found to be dominated by contributions from distances well beyond the nuclear surface . keywords : nuclear breakup of @xmath1b on @xmath0si , elastic and inelastic breakup cross sections , proton halo in @xmath1b . pacs no . 25.60.gc , 24.10.-i , 25.70.mn psfig = 24 true pt nuclear breakup of @xmath1b in a direct fragmentation model + * r. shyam@xmath4 and h. lenske@xmath5 * + @xmath4_saha institute of nuclear physics , calcutta - 700 064 , india _ + @xmath5_institut fr theoretische physik , universitt giessen , d-35392 giessen , germany _ +

the harmonic oscillator plays the central role in classical and modern science , starting from elementary pendulum in xvii century , when ideas of johannes kepler signify important intermediate step between previous magic - symbolical and modern quantitative - mathematical description of nature @xcite . at that time the birth of classical physics was caused by application of an abstract idea of periodicity to concrete variety of problems @xcite . wave theory and fourier series are some tools developed from this concept in classical physics . but only in modern science we observe essentially new qualitative transition from dynamics to the oscillation theory . it was emphasized by l. mandelstam in lectures on oscillation theory @xcite , that the difference between standard dynamics and the oscillation theory is that in dynamics we are interested in description of what is going on in a given place at a given time . while in the oscillation theory - in the motion of the system as a whole . in classical mechanical picture of the world , positions and velocities are primary objects , while oscillations are the secondary ones . however , starting from quantum mechanics , in which our knowledge of simultaneous positions and velocities is restricted by the heisenberg uncertainty relations , this point of view has been drastically changed . the wave mechanics affirms that the wholeness of the quantum process is something of the same primacy as position of a particle . and every particle is associated with some stationary oscillation process . essential to note that knowledge of phase of the particle and knowledge of the stationary state are excluding each other . the phase of the particle ( trajectory ) in a single stationary state does not exist , since any attempt to derive this phase switches the system to another stationary state @xcite . this way the basic characteristic of oscillation theory , as a consideration of the process as a whole , becomes underlie of fundamental questions of mechanics in quantum world . this point of view also influenced the modern theory of nonlinear dynamical systems , studying the qualitative behavior of a system in the phase space , its integrability properties and chaos @xcite . in recent development of quantum integrable systems the concept of quantum group as a deformation of the lie group with deformation parameter @xmath0 was discovered . the notion of quantum @xmath0-oscillator as a @xmath0-deformed harmonic oscillator was introduced in studies on quantum heisenberg - weyl group @xcite , @xcite , @xcite . this oscillator is related with so called symmetric @xmath0-calculus @xcite , while another version of @xmath0-oscillator @xcite , @xcite with non - symmetric @xmath0-calculus . the difference is in definition of @xmath0-number and several generalizations of these numbers and oscillators with different basis were found . in the large stream of articles devoted to @xmath0-oscillator here we like to emphasize the set of papers published by v. i. manko and coauthors , which are not widely known @xcite , @xcite , @xcite . in these papers the physical approach to q - oscillator as a nonlinear oscillator was developed . it was shown that it is an oscillator with frequency depending on its energy in the form of hyperbolic cosine function of the energy . the classical motion of this nonlinear oscillator becomes descriptive of the motion of a q - oscillator and the frequency of oscillations is increasing exponentially with energy . by standard quantization of this nonlinear system , the authors got a quantum q - oscillator as a nonlinear quantum oscillator with anharmonicity described by power series in energy . then they generalized the approach to an arbitrary energy dependence of frequency and called it as the @xmath1-oscillator . in fact , in addition to frequency , any constant parameters in some exactly solvable system can be replaced by integrals of motion . this idea in some sense continues dirac s approach to fundamental constants as a simple functions slowly varying in time . the slow variation of parameters can be implemented by adiabatic invariants and integrals of motion , with corresponding quantization . then starting from any integrable system one can replace parameters of the system by integrals of motion and get the hierarchy of integrable systems from the given one . in the present paper we develop several ideas as a variations on such approach to @xmath0- and @xmath1-oscillators as the main theme . in section 2 we show that an arbitrary one dimensional integrable model in action - angle variables has natural description as a classical and quantum @xmath1-oscillator . as an example we consider semi - relativistic oscillator , related with landau levels for relativistic electron in magnetic field . in section 3 we review some results on symmetric @xmath0-oscillator and in the next section 4 we study linear q - schrodinger equation with dispersion relation of this oscillator . the symmetry operators and polynomial solutions with moving zeros are described , as well as nonlinearization of the model by complex cole - hopf - madelung transformation . the symmetry of the linear equation then is rewritten as bcklund transformation for the nonlinear q - burgers equation . in section 6 , following the general procedure developed in @xcite , we construct the nonlinear schrdinger equation with symmetric @xmath0-dispersion . this q - nls equation is an integrable model from nls hierarchy with lax representation , infinite number of integrals of motion , soliton solutions etc . in the limit @xmath3 it reduces to nls , which is one of the universal soliton equations and for @xmath4 provide higher order corrections in dispersion and nonlinearity . peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around @xmath2 . as a next variation on q - calculus application in section 4 we consider hydrodynamic flow in bounded circular wedge domain . we formulate general two circular wedge theorem and show that the flow is determined by q - periodic functions . for real @xmath5 it is the flow in annular concentric circles domain , while for wedge with angle @xmath6 , @xmath0 is the primitive root of unity . for a vortex problem in such domains we describe full set of vortex images as a kaleidoscope in terms of @xmath0-elementary functions . as an application we describe the point vortex motion in annular domain as a nonlinear oscillator with frequency depending on radius of motion . then we find the @xmath1-oscillator form of this model and discuss corresponding quantization . in the last section 10 by introducing matrix form of binet formula for fibonacci numbers , we solve so called golden oscillator and find corresponding coherent states and fock - bargman type representation . details of some proofs are given in appendix . i apologize that in this paper would be not able to give complete list of references in this very wide field of research . instead of this i try to represent some basic ideas in a pedagogical way . representation of an integrable model by the action - angle variables allows one to interpret the model as a set of nonlinear oscillators . for simplicity here we illustrate the idea only for the one degree of freedom hamiltonian system . due to conservation of energy this system is integrable . let @xmath7 is hamiltonian function with canonical variables @xmath8 . the action and angle variables @xmath9 are introduced by generating function , @xcite @xmath10 as the abbreviated action , in the following way @xmath11 we suppose that the motion is finite and integral is taken along the full period of oscillations . hamilton s equations of motion in these variables are @xmath12 the first equation implies that @xmath13 is function of @xmath14 only , independent of @xmath15 ( cyclic coordinate ) , and the action variable , as well as the energy @xmath16 is an integral of motion . the second equation determines nonlinear frequency @xmath17 and solution @xmath18 the change of @xmath19 and @xmath15 in the period is @xmath20 and trajectory of the system represents a curve on the cylindrical surface in phase space @xmath21 endowed with time axis @xmath22 @xcite . then , original variable @xmath8 as functions of @xmath9 are periodic and can be decomposed to fourier series as the spectral decomposition @xmath23 as an example , the linear oscillator @xmath24 in action variables is @xmath25 with the following solution @xmath26 the spectral decomposition has only harmonics with @xmath27 and @xmath28 appearance of other modes in the decomposition ( [ spectraldecomposition ] ) means anharmonicity in oscillations . quantization of action - angle variables implies that the action variable @xmath14 is replaced by hermitian operator @xmath29 , while instead of @xmath15 one should use the unitary operator @xmath30 @xcite , @xcite . this suggests that proper variables for quantization of the system in action - angle variables @xmath31 are complex functions @xmath32 with canonical bracket @xmath33 since @xmath34 , the hamiltonian of the linear oscillator ( [ linearoscillator ] ) in these variables is just @xmath35 ( for simplicity we put @xmath36 ) and generic nonlinear hamiltonian is function of @xmath37 : @xmath38 it determines evolution equation as a nonlinear oscillator @xmath39 with frequency @xmath40 depending on value of amplitude , where the last one is an integral of motion . for generic @xmath41 as in ( [ generich ] ) , by introducing complex variables @xmath42 where @xmath43 , we express the model in the form of the f - oscillator @xcite , @xmath44 with the poisson bracket @xmath45 and evolution determined by the same frequency as in ( [ evolution1 ] ) @xmath46 quantization of this system replaces complex variables @xmath47 , @xmath48 with bosonic operators @xmath49 = 1 $ ] , @xmath50 . then the hamiltonian of the corresponding quantum f- oscillator is @xcite , @xmath51 where we defined new operators as @xmath52 so that @xmath53 and @xmath54 = h(n+i ) - h(n).\ ] ] it gives the hamiltonian @xmath55\ ] ] with discrete spectrum @xmath56.\ ] ] as an example here we consider the semi - relativistic harmonic oscillator with energy @xmath57 in non - relativistic limit @xmath58 becoming the usual harmonic oscillator @xmath59 such type of dispersion appears in several physical applications . first is the problem of a relativistic electron in a magnetic field : the relativistic landau levels problem @xcite . another class of applications is the relativistic harmonic oscillator model of hadrons to explain magnetic moments of baryons and hadron spectroscopy @xcite . in terms of action variables the hamiltonian function is @xmath60 and in terms of complex variables ( [ complexcoordinate ] ) we have nonlinear oscillator @xmath61 with nonlinear frequency @xmath62 in the non - relativistic limit it reduces to the linear oscillator frequency @xmath63 . introducing @xmath64 with poisson brackets @xmath65 we get the classical f - oscillator form of the model @xmath66 the evolution is determined by equation @xmath67 where the frequency is the nonlinear function depending on amplitude @xmath68 when @xmath58 it gives @xmath69 so that the linear oscillator is recovered at @xmath70 . the next order correction in @xmath71 is just a quadratic nonlinearity in amplitude . such type of nonlinearity is very specific and is described by cubic nonlinear schrodinger equation ( nls ) as generic integrable nonlinear envelope soliton equation @xcite . it means that the first relativistic correction to the nonlinear dispersion ( [ dispersion1 ] ) will produce nls equation . introducing operators @xmath72 we find relations @xmath73 and commutator @xmath74 = \sqrt{m^2c^4 + 2 m c^2 \omega_0 ( n+1 ) } - \sqrt{m^2c^4 + 2 m c^2 \omega_0 n}.\ ] ] then the hamiltonian is @xmath75,\ ] ] with discrete spectrum @xmath76.\ ] ] here we briefly reproduce main formulas for the classical and quantum @xmath0-oscillator @xcite , @xcite . in terms of complex variables @xmath77 the hamilton function is @xmath78 with poisson bracket @xmath79 and evolution equations @xmath80 the action - angle variables are defined as @xmath81 with bracket @xmath82 , so that the hamiltonian function depends only on @xmath14 , @xmath83 then the nonlinear frequency depends exponentially on amplitude and the energy @xmath84 the standard creation and annihilation operators @xmath85 and @xmath86 with commutator @xmath49 = 1 $ ] in the fock basis @xmath87 , @xmath88 determines the number operator @xmath50 and symmetric q - number operator @xmath89_{\tilde q } = \frac{q^n - q^{-n}}{q - q^{-1 } } = \frac{\sinh \lambda n}{\sinh \lambda},\ ] ] where @xmath90 . by transformation @xcite , @xcite , @xcite @xmath91_{\tilde q}}{n}}=\sqrt{\frac{[n+i]_{\tilde q}}{n+i } } a,\,\,\,\,\ , a_f^+ = a^+\sqrt{\frac{[n+i]_{\tilde q}}{n+i } } = \sqrt{\frac{[n]_{\tilde q}}{n } } a^+ \label{ftransformation}\ ] ] so that @xmath92_{\tilde q } , \,\,\,\,a_f a^+_f = [ n+i]_{\tilde q}\ ] ] and @xmath54 = [ n+i]_{\tilde q } - [ n]_{\tilde q}\ ] ] the hamiltonian is represented in terms of symmetric @xmath0-number operators @xmath93_{\tilde q } + [ n+i]_{\tilde q})\ ] ] and the spectrum is @xcite , @xcite , @xcite @xmath94_{\tilde q } + [ n+1]_{\tilde q } ) = \frac{1}{2 } \frac{\sinh ( ( n+\frac{1}{2})\lambda)}{\sinh \frac{\lambda}{2}}.\ ] ] the first quantization of the @xmath0-oscillator is described by the linear schrdinger equation with hamiltonian @xmath95 for free q - particle , when the oscillator coupling constant @xmath96 , we have the hamiltonian @xmath97 which determines the linear schrodinger equation with nonlinear dispersion @xmath98 we can call it as a q - schrodinger equation . operators of the time and space translations @xmath99 commute with the schrdinger operator @xmath100 @xmath101 = 0 $ ] , @xmath102 . the boost operator @xmath103 is also commuting with @xmath19 , @xmath104 = 0 $ ] . commuting it with space and time translations we obtain the following algebra of symmetry operators @xmath105 = 0,\,\,[p_1,k ] = - \rmi\hbar,\,\,[p_0 , k ] = -\frac { \lambda}{m \sinh \lambda } \cosh \frac{\lambda}{2}\left ( \frac{p_1}{m } \right ) p_1.\label{symmetryalgebra}\ ] ] an operator commutative with schrodinger operator determines the dynamical symmetry @xcite . if @xmath106 is a solution of ( [ schrodinger ] ) and @xmath107 is an operator , so that @xmath108 = 0 $ ] then @xmath109 is also a solution of ( [ schrodinger ] ) . then as follows @xmath110 operator can generate new solutions of ( [ schrodinger ] ) . in the limit @xmath111 the boost operator ( [ boost ] ) reduces to the galilean boost @xmath112 and @xmath0-deformed symmetry algebra ( [ symmetryalgebra ] ) to the usual non - relativistic algebra of the galilean group @xcite . for given classical dispersion @xmath113 we define the plane wave solution as a generating function of @xmath0-kampe de feriet polynomials @xmath114 , @xmath115 , @xmath116 the polynomials @xmath117 are solutions of ( [ schrodinger ] ) with initial value @xmath118 . from commutativity @xmath119 = 0 $ ] , operator @xmath110 evolves according to the heisenberg equation @xmath120\ ] ] and has the form @xmath121 from this follows that operator @xmath110 generates an infinite hierarchy of polynomial solutions according to the recursion @xmath122 and first few polynomials are @xmath123 in the limit @xmath3 or @xmath111 we have @xmath124 , and the above polynomials reduce to the schrodinger polynomials @xcite @xmath125 let @xmath126 x^n$ ] are the standard kampe de feriet polynomials , then @xmath127 or in terms of the hermit polynomials @xmath128 we notice that for @xmath129 only coefficients in @xmath130 becomes deformed , while starting from @xmath131 a new term , vanishing in the limit @xmath111 appears . motion of n zeros for polynomial solutions ( [ qkf ] ) is determined by the system of ordinary differential equations @xmath132 \cdot 1,\ ] ] k=1, ... ,n , describing system of n interacting particles in a line . for @xmath111 the polynomials of complex argument , correspond to holomorphic extension of schrodinger equation in ( 2 + 1)-dimensional chern - simons theory @xcite and zeros of these polynomials describe motion of point vortices in plane . then for @xmath133 , consideration of holomorphic q - schrodinger equation could describe a q - deformed vortex dynamics in the plane . using schrdinger s log @xmath106 transform : @xmath134 and identity @xmath135 the q - schrdinger equation ( [ schrodinger ] ) can be rewritten in the form @xmath136 for a given complex function @xmath137 we introduce a new complex function with dimension of velocity @xmath138 where @xmath139 is the complex potential , with real and imaginary parts as the classical and quantum velocities @xmath140 then ( [ schrodinger3 ] ) becomes the quantum q - hamilton - jacobi equation : @xmath141 in the classical ( dispersionless ) limit @xmath142 the quantum velocity @xmath143 vanishes and the complex potential reduces to the real velocity potential @xmath144 , so that ( [ chj ] ) becomes the classical q - hamilton - jacobi equation for action @xmath19 : @xmath145 differentiating ( [ schrodinger3 ] ) we have nonlinear complex q - burgers type equation for the complex velocity ( complex q - madelung equation ) @xmath146 in the classical limit it gives the newton equation in the hydrodynamic form @xmath147 which is just differentiation of the classical hamilton - jacobi equation ( [ clhj ] ) . equation ( [ hf ] ) has implicit general solution @xmath148 where @xmath1 is an arbitrary function , and develops shock at a critical time when derivative @xmath149 is blowing up . as we have seen , using boost transformation ( [ boost ] ) from a given solution @xmath150 of the q - schrdinger equation ( [ schrodinger ] ) we can generate another solution as @xmath151 \psi_1.\ ] ] by using identity @xmath152 for complex velocities @xmath153 , @xmath154 , for complex q - burger equation ( [ schrodinger4 ] ) we obtain the bcklund transformation @xmath155.\ ] ] it is noted that in the classical limit @xmath142 , @xmath156 the above bcklund transformation reduces to the trivial identity @xmath157 . in previous sections we have studied q - schrdinger equation determined by hyperbolic sine dispersion . by adding the oscillator coupling term it becomes exactly solvable q - oscillator equation ( [ qoscillatorschrodinger ] ) . here we show that from this dispersion it is possible to construct also nls type integrable evolution equation . this q - nls equation in the limit @xmath3 or @xmath111 reduces to the nls equation . it is interesting to note that by expansion in powers of @xmath158 then we have an infinite set of nls type equations integrable at arbitrary order of deformation @xmath0 . dispersion formula for linear q - schrdinger equation ( [ linearqschrodinger ] ) expanded to series in @xmath159 is @xmath160 this may be used to construct a linear q - schrdinger equation as a formal power series @xmath161 \psi \end{aligned}\ ] ] combining two complex conjugate versions of this equation together we have the system @xmath162 following the general procedure described in appendix a one may proceed further : by replacing the derivative operator @xmath163 as a momenta to the full recursion operator @xmath164 ( [ recursion ] ) , one obtains an _ integrable _ nonlinear q - schrodinger equation ( q - nls ) @xmath165 applying general result of appendix a to the above q - deformed non - relativistic dispersion , we have the next linear problem ( the lax representation ) for equation ( [ qnls ] ) @xmath166 @xmath167 where @xmath168 @xmath169 and the spectral parameter @xmath170 has meaning of the classical momentum . the model is an integrable nonlinear q - schrodinger equation with q - deformed dispersion as well as the nonlinear terms . in the limit @xmath111 it reduces to standard nls model ( [ nls ] ) . the last one as a universal integrable equation describing envelope modulation has appeared in many applications , especially in nonlinear optics . it admits n - soliton solutions , higher symmetries etc.@xcite remarkable property of our model ( [ qnls ] ) is that it generalizes the nls model in a very special way . if we expand it in @xmath158 , then at every order of @xmath158 we get an integrable system . it means that we have integrable corrections to the nls equation at any order of @xmath158 . due to the symmetry @xmath171 of ( [ qnls ] ) only even powers of @xmath158 appear in the expansion . then the lowest corrections are given by integrable nls model with six order dispersion @xmath172 and up to 13-th order nonlinearity @xmath173 \left(\begin{array}{clcr } \psi\\ \bar\psi \end{array } \right ) + o(\lambda^4 ) . \label{qcorrectionnls}\ ] ] as a next variation on the subject here we consider hydrodynamical problem related with motion in double connected domain . this problem can naturally be formulated in terms of q - calculus and as we will see , for the point vortex motion in annulus it provides an example of nonlinear oscillator , formulated as an @xmath1-oscillator . for incompressible and irrotational flow in a domain bounded by curve @xmath174 , the problem is to find analytic function @xmath175 with boundary condition @xmath176 where @xmath177 is the stream function . then the normal velocity vanishes at boundary @xmath178 . for a given flow in plane with complex potential @xmath179 , introduction of boundary circle at the origin @xmath174 : @xmath180 produces the flow with complex potential @xcite @xmath181 for annular domain , @xmath182 , between two concentric circles @xmath183 , @xmath184 the complex potential is @xcite @xmath185 where @xmath186 , @xmath187 is flow in even annular image domain , @xmath188 is the flow in odd annular image domain . from ( [ twocircle ] ) follows that @xmath189 , which implies that the complex potential is q - periodic function @xmath190 . depending on number and position of vortices or other objects , we fix singularity of this function in terms of q - elementary functions @xcite,@xcite . here we are going to formulate some new theorems in wedge domain . for a given flow in plane with complex potential @xmath179 , introduction of boundary wedge with angle @xmath191 , where @xmath192 is a positive even number , produces the flow with complex potential @xmath193 or shortly @xmath194 where @xmath195 is the primitive root of unity : @xmath196 . for proof of this theorem see appendix b. here we notice that the complex potential ( [ wedge2 ] ) is @xmath197 periodic analytic function @xmath198 , while the complex velocity @xmath199 is the scale invariant analytic function @xmath200 . as an example , we consider single vortex in the wedge at point @xmath201 : @xmath202 then applying the above wedge theorem we get complex potential @xmath203 describing a kaleidoscope of @xmath204 vortices with positive strength at points @xmath205 @xmath206 and with negative strength at points @xmath207 @xmath208 . positive images are just rotations of original vortex position @xmath201 on angles @xmath209 , while negative images are rotations of the reflected vortex position @xmath210 on the same angles . this expression can be drastically simplified due to the next identity @xmath211 this identity is valid for @xmath212 as the primitive @xmath213-th root of unity and has been considered long time ago by e. kummer . simplest proof follows from factorization of polynomial @xmath214 by roots of unity . as a result , finally we get the following compact expression for the vortex flow in the wedge ( we can call it as the kummer kaleidoscope of vortices ) @xmath215 for @xmath216 we have vortex at @xmath201 in upper half plane , with one image at @xmath210 @xmath217 for @xmath218 the vortex at @xmath201 in first quadrant produces images at @xmath219 @xmath220 here we consider circular wedge with angle @xmath191 , bounded by lines @xmath221 : @xmath222 and @xmath223 : @xmath224 and the circular boundary @xmath225 @xmath226 , @xmath227 . then , the flow bounded by such domain is @xmath228 + \sum^{n-1}_{k=0 } [ \bar f(\frac{r^2}{q^{2 k}z } ) + f(\frac{r^2}{q^{2 k}z})].\ ] ] the proof is similar to the above one and shows that imaginary part of @xmath175 vanishes at boundaries @xmath221 , @xmath223 and @xmath229 . the theorem could be considered as combination of milne - thomson s one circle theorem with the wedge theorem . for single vortex in circular wedge after some calculations and simplifications we obtain @xmath230 comparing with vortex kaleidoscope ( [ kummer ] ) we observe doubling of images by reflection in circle @xmath231 . now we consider more general region , the double circular wedge , bounded by two lines @xmath221 : @xmath222 and @xmath223 : @xmath224 and two circular boundaries @xmath225 @xmath232 , @xmath227 , and @xmath233 @xmath234 , @xmath227 . by combination of two circle theorem with the wedge theorem we have the flow @xmath235 where @xmath236 in explicit form we obtain @xmath237\ ] ] it is noticed that in this case we have q - calculus with two different basis . the first one @xmath238 is a real number relating an infinite number of reflections in both circular boundaries . the second one @xmath212 is a complex unitary number with finite number of reflections @xmath213 . the complex potential here is a double @xmath239 - periodic analytic function : @xmath240 for single vortex in the double circular wedge we get result @xmath241 this function describes self - similar kaleidoscope of infinite set of vortices on @xmath242 geometric lattice . it generalizes expression for single vortex images in concentric annular domain considered in @xcite . for @xmath243 , @xmath244 , and we have single vortex in upper half - plane of annular domain @xmath245 for @xmath218 , @xmath246 we have single vortex in first quadrant of annulus and @xmath247 or @xmath248 this formula demonstrates how vortices are reflected for every value of @xmath249 . as an application of above formulas here we consider the point vortex problem in annular domain as a nonlinear oscillator , or as the specific form of the f - oscillator . for a single vortex in annular domain the complex velocity at the vortex position is determined by @xcite , @xmath250 where in the complex velocity @xmath251\nonumber\\ = \sum_{n= \pm 1}^{\pm\infty}\frac{\rmi \kappa}{z - z_0 q^n}- \sum_{n= \pm 1}^{\pm\infty}\frac{\rmi \kappa}{z- \frac{r_1 ^ 2}{\bar z_0 } q^n},\nonumber\end{aligned}\ ] ] where @xmath252 . contribution of the vortex on itself is excluded . if we take into account that q - harmonic series @xmath253 } = - ln_q 0\ ] ] converges for @xmath254 , then at @xmath255 the first two terms cancel each other and we get the following equation of motion @xmath256.\label{vortexnonlinearoscillator}\ ] ] here to avoid manipulations with infinite sums we introduced q - logarithm function @xmath257},\,\,|x| < q , \,\,q>1 , \label{def2}\ ] ] where the q - number @xmath258 \equiv 1 + q + q^2 + ... + q^{n-1 } = \frac{q^{n}-1}{q-1}\ ] ] for any positive integer @xmath213 . equation ( [ vortexnonlinearoscillator ] ) is a nonlinear oscillator @xmath259 with frequency depending on amplitude ( and energy ) @xmath260.\ ] ] in addition to the energy , another conserved quantity in this problem is an angular momentum @xmath261 . in hamiltonian form we have @xmath262 with canonical bracket @xmath263 and hamiltonian function @xmath264 where the jackson q - exponential function is defines as @xmath265!}.\label{qexp1}\ ] ] it is entire in @xmath266 if @xmath267 , and admits infinite product representation @xmath268 showing that zeros of this function are ordered in geometric progression with ratio @xmath0 . by introducing the action - angle variables @xmath9 @xmath269 with canonical bracket @xmath270 we get the hamiltonian function in terms of action variables @xmath14 only @xmath271 and the frequency of rotation @xmath272.\label{frequency}\ ] ] the angular momentum then is just @xmath273 . to represent our model as an @xmath1- oscillator we introduce complex functions @xmath274 then we have simply f - oscillator @xmath275 with poisson bracket @xmath276 and evolving with the frequency ( [ frequency ] ) @xmath277 in semiclassical approach , quantization is implemented by bohr - zommerfeld quantization rule of replacing @xmath278 . then the energy spectrum is @xmath279 the f - oscillator quantization of this system , replaces complex variables @xmath201 with bosonic operators @xmath49 = 1 $ ] , @xmath50 . corresponding f- oscillator is given by ( [ fhamilton ] ) , ( [ ftrans ] ) for hamiltonian ( [ hamilton ] ) h(n ) and spectrum is @xmath56.\ ] ] in a similar way one can study n vortex polygon rotation in annular domain @xcite as a nonlinear or f - oscillator . as a final variation here we consider two golden ratio bases quantum q - oscillator . we define creation and annihilation operators @xmath280 and @xmath281 in the fock basis @xmath87 , @xmath88 represented by infinite matrices @xmath282 where @xmath283 are fibonacci numbers . by introducing the fibonacci operator as a matrix binet formula , @xmath284 where @xmath50 is the standard number operator , and @xmath285 , @xmath286 are solutions of @xmath287 , @xmath288 - is the golden ratio , we find that in the fock basis the eigenvalues are just fibonacci numbers @xmath289 and in the matrix form @xmath290 it satisfies fibonacci recursion rule @xmath291 . then we have @xmath292 and the commutator is @xmath293 = f_{n+i } - f_{n } = f_{n - i}.\ ] ] from definition of @xmath294 we get the matrix identity @xmath295 where @xmath296 and the following deformed commutation relations @xmath297 the hamiltonian @xcite @xmath298 is diagonal @xmath299 and gives the energy spectrum as the fibonacci sequence @xmath300 these energy levels satisfy the fibonacci three term relations @xmath301 and the difference between levels @xmath302 is growing as fibonacci sequence . then the relative distance @xmath303 for asymptotic states @xmath304 is given just by the golden ratio @xmath305 this behavior drastically differs from the harmonic oscillator and shows that there is no simple classical limit for this golden oscillator . in fact , the hamiltonian function for corresponding classical system becomes complex valued . by transformation @xmath306 one can show that eigenstates @xmath307 coincide with the fock states @xmath308 . then we have @xmath309 we define the golden coherent states as eigenstates @xmath310 expanding these states in the fock space @xmath311 we find recurrence relation @xmath312 giving @xmath313 we fix @xmath314 by normalization condition @xmath315 so that @xmath316 where we have introduced the fibonacci exponential function @xmath317 which as easy to see is entire function of @xmath266 . as a result we get normalized coherent state @xmath318 with the scalar product @xmath319 for an arbitrary state from the fock space @xmath320 by projection @xmath321 we find the analytic wave function @xmath322 in the golden fock - bargman representation . by simple calculation it is easy to see that operators @xmath280 and @xmath281 in this representation are given by @xmath323 where the binet - fibonacci complex derivative we define as @xmath324 action of this derivative on monomial gives just fibonacci numbers @xmath325 and for the fibonacci exponential function we have @xmath326 . then the fibonacci number operator is represented as @xmath327 if an analytic function @xmath328 is scale invariant @xmath329 , then it satisfies equation @xmath330 or @xmath331 this eigenvalue problem is just the golden fock - bargman representation of the fibonacci operator eigenvalue problem ( [ fibnumbereigenvalue ] ) , where eigenfunctions @xmath332 are scale invariant . however if we look for general solution of ( [ fdifferenceequation ] ) , then it is of the form @xmath333 where @xmath334 is an arbitrary golden - periodic analytic function @xmath335 . such a structure characterizes the quantum fractals @xcite,@xcite and requires additional studies . we consider the nls hierarchy @xmath336 where @xmath337 , @xmath338 is an infinite time hierarchy . here @xmath339 is the matrix integro - differential operator - the recursion operator of the nls hierarchy - @xmath340 and @xmath341 - the pauli matrix . for the first few members of the hierarchy n = 1,2,3,4 this gives @xmath342 @xmath343 @xmath344 @xmath345 in the linear approximation , when @xmath346 , the recursion operator is just the momentum operator @xmath347 and the nls hierarchy ( [ nlshierarchy ] ) becomes the linear schrodinger hierarchy @xmath348 written in the madelung representation it produces the complex burgers hierarchy so that this representation plays the role of the complex cole - hopf transformation @xcite . every equation of the hierarchy ( [ nlshierarchy ] ) is integrable . the linear problem for the @xmath349-th equation is given by the zakharov - shabat problem @xmath350 for the space evolution , and @xmath351 for the time part . coefficient functions @xmath352 and @xmath353 are @xcite , @xmath354 to write this expression in a compact form , by analogy with q - calculus it is convenient to introduce notation of nonsymmetric q - number operator @xmath355_q,\ ] ] where @xmath0 is a linear operator . hence with operator @xmath356 we have the finite laurent part in the spectral parameter @xmath170 @xmath357_{{\cal{r}}/p}.\ ] ] then we have shortly @xmath358_{{\cal{r}}/p}\left ( \begin{array}{c } \psi\\ \bar \psi \end{array}\right).\label{cn}\ ] ] in a similar way @xmath359 and due to ( [ cn ] ) @xmath360_{{\cal{r}}/p}\left ( \begin{array}{c } \psi\\ \bar \psi \end{array}\right).\label{an}\ ] ] equations ( [ zstn]),([cn ] ) and ( [ an ] ) give the time part of the linear problem ( the lax representation ) for the n - th flow of nls hierarchy ( [ nlshierarchy ] ) . for the time @xmath22 determined by the formal series @xmath361 where @xmath362 are arbitrary constants , the general nls hierarchy equation is @xcite @xmath363 integrability of this equation is associated with the zakharov - shabat problem ( [ zs11 ] ) and the time evolution @xmath364 where @xmath365_{{\cal{r}}/p}\left ( \begin{array}{c } \psi\\ \bar \psi \end{array}\right).\label{ch}\ ] ] in the last equation we have used that for @xmath366 , @xmath367 . then we have @xmath368 the above equation ( [ gnlshierarchy ] ) gives integrable nonlinear extension of a linear schrdinger equation with general analytic dispersion . let us consider the classical particle system with the energy - momentum relation @xmath369 then the corresponding time - dependent schrdinger wave equation is @xmath370 where the hamiltonian operator results from the standard substitution for momentum @xmath371 in the dispersion ( [ dispersion ] ) . equation ( [ schr ] ) together with its complex conjugate can be rewritten as a system @xmath372 the momentum operator here is just the recursion operator ( [ reco ] ) in the linear approximation @xmath373 . hence ( [ mschr ] ) can be rewritten as the linear schrdinger equation with an arbitrary analytic dispersion @xmath374 then the nonlinear integrable extension of this equation appears as ( [ gnlshierarchy ] ) , which corresponds to the replacement @xmath375 , ( @xmath376 ) , so that @xmath377 from this point of view the standard substitution for classical momentum @xmath378 or equivalently @xmath379 for the equation in spinor form , gives quantization in the form of the linear schrdinger equation . while substitution @xmath380 gives `` nonlinear quantization '' and the nonlinear schrdinger hierarchy equation . the related lax representation for equation ( [ gnlshierarchy1 ] ) is given by ( [ ch ] ) , ( [ ah ] ) . using definition of q - derivative @xmath381 for operator @xmath382 we have relation @xmath383_{{\cal{r}}/p}\ , p^{n-1}.\ ] ] then equation ( [ ch ] ) can be rewritten as @xmath384_{{\cal{r}}/p}\left ( \begin{array}{c } \psi\\ \bar \psi \end{array}\right)= \sum_{n=1}^\infty e_n d_{{\cal{r}}/p}^{(p ) } \ , p^n\left ( \begin{array}{c } \psi\\ \bar \psi \end{array}\right ) \label{cdh1}\ ] ] or using linearity of ( [ qderivative ] ) and dispersion ( [ dispersion ] ) @xmath385 due to definition ( [ qderivative ] ) it gives simple formula @xmath386 where @xmath387 then for @xmath388 we obtain @xmath389 equations ( [ cdh3]),([ah3 ] ) give the lax representation of the general integrable nls hierarchy model ( [ gnlshierarchy1 ] ) . it is worth to note here that special form of the dispersion @xmath390 is fixed by physical problem . in @xcite we have constructed semi - relativistic nls equation . in section 4 as another application we discuss symmetric q - nls equation . proof : first we show that @xmath391 , where @xmath392 is the real axis . substituting @xmath393 to ( [ wedge2 ] ) and using identities for even powers @xmath394 we find @xmath395 where in the second sum we have changed summation order by substitution @xmath396 . this shows that on the real line the complex potential is pure real and imaginary part vanishes . now we show that @xmath397 , where @xmath398 is the second boundary line . substituting we have @xmath399 by identities for odd powers @xmath400 and as follows @xmath401 , we rewrite the sum @xmath399 after changing order of summation in the second sum and shifting summation index we finally get @xmath402 which shows that the stream function vanishes at the boundary . this completes proof of the wedge theorem . 9 pauli w 1955 the influence of archetypal ideas on the scientific theories of kepler in _ the interpretation of nature and the psyche _ bollingen series * li * ( new york : pantheon books)(english translation ) whitehead a n 1956 _ science and modern world _ lowell lectures 1925 ( new american library ) mandelstam l i 1972 _ lectures on vibration theory _ ( moscow : nauka ) pauli w 1952 _ the general principles of wave mechanics _ ( berkeley : univ of california ) andronov a a , witt a a and haikin s e 1981 _ vibration theory _ ( moscow : nauka ) biedenharn l c 1989 _ j. phys . a. _ * 22 * l873 macfarlane a j 1989 _ j. phys . a. _ * 22 * 4581 sun c p and fu h c 1989 _ j. phys . a. _ * 22 * l983 arik m and coon d d 1976 _ j. math . * 17 * 524 kuryshkin v v 1980 _ ann fond louis de broglie_*5 * 111 kac v and cheung p 2002 _ quantum calculus _ ( new york : springer ) manko v i , marmo g , solimeno s and zaccaria f 1993 _ int j mod phys _ 3577 manko v i , marmo g , solimeno s and zaccaria f 1993 _ phys lett_a176 173 manko v i , marmo g , sudarshan e c g and zaccaria f 1997 _ physica scripta_55 528 sagdeev r z and zaslavsky g m 1988 _ introduction to nonlinear physics : from the pendulum turbulenz and chaos_(moscow : nauka ) dirac p a m 1926 _ proc . * a111 * 279 rabi i i 1928 _ zeit f physik _ * 49 * 507 bannur v m 2009 _ mod phys lett _ * a24 * 3183 novikov s , manakov s v , pitaevskii l p and zakharov v e 1984 _ theory of solitons : the inverse scattering method_(springer ) song x c 1990 _ j. phys . a. _ * 23 * l821 kulish p p and damaskinsky e v 1990 _ j. phys . a. _ * 23 * l415 pashaev o k and yilmaz o 2008 _ j. phys . a. math theor _ * 41 * 135207 pashaev o k and yilmaz o 2011 _ j. phys . a. math theor _ * 44 * 185501 milne - 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we present several ideas in direction of physical interpretation of @xmath0- and @xmath1-oscillators as a nonlinear oscillators . first we show that an arbitrary one dimensional integrable system in action - angle variables can be naturally represented as a classical and quantum @xmath1-oscillator . as an example , the semi - relativistic oscillator as a descriptive of the landau levels for relativistic electron in magnetic field is solved as an @xmath1-oscillator . by using dispersion relation for @xmath0-oscillator we solve the linear q - schrdinger equation and corresponding nonlinear complex q - burgers equation . the same dispersion allows us to construct integrable q - nls model as a deformation of cubic nls in terms of recursion operator of nls hierarchy . peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around @xmath2 . as another variation on the theme , we consider hydrodynamic flow in bounded domain . for the flow bounded by two concentric circles we formulate the two circle theorem and construct solution as the q - periodic flow by non - symmetric @xmath0-calculus . then we generalize this theorem to the flow in the wedge domain bounded by two arcs . this two circular - wedge theorem determines images of the flow by extension of @xmath0-calculus to two bases : the real one , corresponding to circular arcs and the complex one , with @xmath0 as a primitive root of unity . as an application , the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f - oscillator is studied . extending idea of q - oscillator to two bases with the golden ratio , we describe fibonacci numbers as a special type of @xmath0-numbers with matrix binet formula . we derive the corresponding golden quantum oscillator , nonlinear coherent states and fock - bargman representation . the spectrum of it satisfies the triple relations , while the energy levels relative difference approaches asymptotically to the golden ratio and has no classical limit . october 2014 _ keywords _ : q - oscillator , nls hierarchy , nonlinear oscillator , circle theorem , fibonacci numbers

transient lunar phenomena are defined for the purposes of this investigation as localized ( smaller than a few hundred km across ) , transient ( up to a few hours duration , and probably longer than typical impact events - less than 1s to a few seconds ) , and presumably confined to processes near the lunar surface . how such events are manifest is summarized by cameron ( 1972 ) . in paper i we study the systematic behavior ( especially the spatial distribution ) of tlp observations - particularly their significant correlations with tracers of lunar surface outgassing , and in paper ii some simple , theoretical predictions of other , not - so - obvious aspects that might be associated with tlps and outgassing events . in this paper we suggest several ways that more information might be gleaned to determine the true nature of these events . at several points we emphasize the importance of timely implementation of these approaches . tlps are infrequent and short - lived , and this is the overwhelming fact of their study that must be surmounted . it is our goal to design a nested system of observations which overcomes the problems that this fact has produced , a largely anecdotal and bias - ridden data set , and replace it with another data set with _ a priori _ explicit , calculable selection effects . this might seem a daunting task , since the data set we used in paper i was essentially the recorded visual observations of the entire human race since the invention of the telescope , and even somewhat before . with modern imaging and computer technology , however , we can overcome this . another problem that becomes clear in paper ii is the many , complex means by which outgassing can interact with the regolith . in the case of slow seepage , gases may take a long time to work their way through the regolith . if the gases are volcanic , there may be interactions along the way , and if water vapor is involved , it and perhaps others of these gases may remain trapped in the regolith . these factors must be remembered in designing our future investigations . we can make significant headway , however . the various factors which complicate our task due to the paucity of information about tlps also leave open avenues that modern technology can exploit . the many methods detailed in this paper are summarized in table 1 . there has been no areal - encompassing , digital image monitoring of the near side with appreciable time coverage using modern software techniques to isolate transients . there are no published panspectral maps at high spectral / spatial resolution of the near side surface , beyond what is usually called multispectral imaging . ( to some degree this will be achieved by the moon mineralogy mapper onboard _ chandrayaan-1 _ , but not before other relevant missions such as @xmath0 have passed ) . there are numerous particle detection methods that are of use . the relevant experiments on apollo were of limited duration , either of a week or less , or 5 - 8 years in the case of alsep . furthermore the _ clementine _ and _ lunar prospector _ missions were also of relatively short duration . all of these limitations serve as background to the following discussions . by necessity the monitoring of optical transients from the vicinity of earth must be limited to the near side . as detailed in paper i , however , all physical correlations tied to tlps likewise strongly favor the near side e.g. , @xmath1rn outgassing ( 4 of 4 episodes being nearside , as well as nearly all @xmath1rn residual ( seen as @xmath2po ) and mare edges ( @xmath385% nearside , somewhat depending on one s definition , even more so if low - contrast albedo features such as aitken basin are not included ) . remote sensing in the optical / ir is limited in spatial resolution either by the diffraction limit of the telescope or by atmospheric seeing . one arcsecond , a typical value for optical imaging seeing fwhm , corresponds to 1.8 - 2.0 km on the lunar surface , and is the diffraction limit of a 12 cm diameter telescope at @xmath4 nm . the best , consistent imaging resolution will come from the @xmath5 @xmath6 @xmath7 with @xmath8 arcsec fwhm , and indeed images of the moon have been obtained with the @xmath9/advanced camera for surveys combination ( garvin et al . @xmath9 observations of the moon turn out to be relatively expensive in terms of spacecraft time due to setup time complicated by the relative motions of the target and spacecraft , and inefficiency due to exposure setup times of @xmath380s for each exposure of typically 1s . altogether @xmath10 - 1 h of spacecraft time is needed to successfully image a small region in one filter band ( due in part to several overlapping exposures needed for complete coverage avoiding masks and other obstructions on the hrc detector , as well as to reject cosmic ray signals ) . at least until the @xmath5 servicing mission 4 , the guiding of @xmath9 and the state of acs will allow no further such observations . a competing method for producing high - resolution imaging is the `` lucky exposures '' ( le , also `` lucky imaging '' ) technique which exploits occasionally superlative imaging quality among a series of rapid exposures , then sums the best of these with a simple shift - and - add algorithm ( fried 1978 , tubbs 2003 ) . the technique requires a high - speed , linear - response imager , and can be accomplished only with great difficulty using a more conventional astronomical ccd system . nonetheless , many amateur setups have achieved excellent results with this technique , and the cambridge group ( law , mackay & baldwin 2006 ) have achieved diffraction - limited imaging on a 2.5-meter telescope , very close to @xmath9 angular resolution . in practice , only about 1 - 10% of exposures , hence less than 1% of observing time , survive image quality selection , but for the moon this amounts to a small investment of telescope time ( a few minutes ) . we have attempted this ourselves and encountered some minor problems : image quality must be selected in terms of a fourier decomposition of the image rather than inspection of the point - spread function of a reference star , and shift - and - add parameters must be similarly defined , by image cross - correlation rather than by centroiding a bright star . we will present results from these efforts when they succeed more usefully . unlike adaptive optics approaches , le does not depend on a bright reference star to define the incoming wavefront , but le improvements are still limited to an angular area of the isoplanatic patch determined by atmospheric turbulence , @xmath31000 arcsec@xmath11 . covering the entire nearside moon would be challenging ( @xmath33000 fields needed - at least 20 nights on a moderate - sized telescope ) . likewise , the acs hrc on @xmath9 , covering 750 arcsec@xmath11 at a time , can not be used practically to map the entire near side . the greater flexibility of an le program , in terms of choice of epoch and wavelength coverage , provides many advantages ; acs hrc , on the other hand , would provide consistent - quality results , albeit at great expense . high resolution imaging can be used to monitor small , specific areas over time , or in a one - shot application comparing a few exposures to imaging from another source . currently , the best full - surface comparison map in the optical is the @xmath12 uvvis ccd map ( eliason et al . 1999 ) , 5 bands at 415 - 1000 nm , with typically 200 m resolution , a good match to le and @xmath9 resolutions . unfortunately , neither @xmath12 uvvis , or infrared cameras nir ( 1100 - 2800 nm ) or lwir ( 8000 - 9500 nm ) cover some of the more interesting bands for our purposes ( for example , the regolith hydration bands at 2.9 and 3.4 @xmath13 m ) . in the future , we will be able to make comparisons to the extensive map of the moon mineralogy mapper ( pieters et al . 2005 ) on chandrayaan-1 , with 140 m and 20 nm fwhm spatial and wavelength resolution , respectively , over 0.4 - 3.0 @xmath13 m . the 3 @xmath13m - reflectance hydration features in asteroidal regolith have been studied ( lebofsky et al . 1981 , rivkin et al . 1995 , 2002 , volquardsen et al . there is little written about the spectroscopic reaction of lunar regolith to hydration ; however , it is apparent that the reflectance features near 3 @xmath13 m do not appear immediately in lunar samples subjected to the terrestrial atmosphere ( akhmanova et al . 1972 ) , but do after several years ( markov et al . 1980 , pieters et al . 2006 ) . at least in the latter , samples lose this hydration reflectance effect within a few days of exposure to a dry environment . this issue could easily be studied with further lunar sample experiments . the prime technique for detecting changes between different epochs in similar images will involve image subtraction . this technique is well - established in studying supernovae , microlensing and variable stars , and produces photon poisson noise - limited performance ( tomaney & crotts 1996 ) . this technique is well matched to ccd or cmos detectors , and at 1 - 2 arcsec fwhm resolution , these can cover the whole moon with 10 - 20 mpixels , as is available for conventional detectors . for proper image subtraction , one needs at least 2 pixels per fwhm diameter , or else non - poisson residuals tend to dominate , driving up the variable source detection threshold . to illustrate how image subtraction would work , we present data of the kind that might be produced by a monitor to detect tlps . while the image shown in figure 1 is taken on a 0.9-meter telescope with 24 @xmath13 m pixels , the data are similar to that would be produced by a smaller , 1-arcsec diffraction - limited telescope with typical commercially - available digital - camera pixels e.g. , 6 @xmath13 m on a 20-cm telescope . image subtraction delivers nearly photon - noise level accuracy in the residual images taken in a ground - based time series , and this is demonstrated in figures 2 - 4 . we introduce an artificial `` tlp '' signal that is a 8% enhancement over the background in the peak pixel of an unresolved source - a signal at or below the threshold of a visual search . the tlp is detected convincingly even in a single image , once subtracted from a reference image e.g. , the average of a time series . the subtraction gives a very flat residual subtracted image ( except for the simulated tlp and a few `` cosmic rays '' of much smaller area and amplitude ) . the only exception is in the complex image region of the highlands near the global terminator . more meaningful , perhaps , is the signal - to - noise ratio of residual sources , shown in figure 3 . this shows the tlp clearly and unambiguously , but there are some false detections in the highland local terminator region at the level of 10 - 20% of the tlp ; we would like to improve on this . one alternative to reduce this noise is to consider applying an edge filter to supply a weighting function to suppress regions where the image structure is too complex . figure 3 shows the result from processing the raw image with a roberts edge enhancement filter ( @xmath14 , where @xmath15 is the raw count in the pixel @xmath16 and @xmath17 is the function shown in figure 3 ) . when the signal difference from figure 2 is divided by figure 3 , the result ( figure 4 ) uniquely and clearly shows the tlp . we would like to avoid this edge filter strategy if possible , relying completely on simple image subtraction , since it may be that some tlps are associated with local terminators on the lunar surface . our group has automated a tlp monitor on the summit of cerro tololo that should be producing regular lunar imaging data as of mid-2007 ( crotts , hickson & pfrommer 2007 ) . this will cover the entire moon at 1 arcsec resolution , and we expect to be able to process the images at a rate of one per 10s . this is sufficient to time - sample nearly all reported tlps ( see paper i ) . in addition we plan to add a second imaging channel on a video loop ; this will retain a continuous record of imaging of sufficient duration so that an alert to a tlp event from the image subtraction processing pipeline will allow one to query the image cache of the video channel record and reconstruct the event at finer time resolution . the image subtraction channel will include a neutral - density filter to allow the exposure time to nearly equal the image cycle time , hence even short tlps ( or meteorite impacts ) will be detected , albeit at a sensitivity reduced by a factor roughly proportional to the square - root of the event duration . the presence of a lunar imaging monitor opens many possibilities for tlp studies . for the first time , this will produce an extensive , objective , digital record of changes in the appearance of the moon , at a sensitivity level much finer than the capability of the human eye . while we will see the true frequency of tlps soon enough , paper i indicates that perhaps one tlp per month might be visible to a human observer observing at full duty cycle . an automated system should be able to distinguish changes in contrast at the level of 1% or slightly better , whereas this is perhaps 10% for a point source observed by the human eye ( based on our tests ) . even augmented human - eye surveys ( such as project moon blink or the corralitos observatory tlp survey - see paper i ) would be at least several times less sensitive than a purely digital survey . the resulting frequency of tlp detections at higher sensitivity depends on the event luminosity distribution function , poorly defined even at brighter limits and completely unknown at the level that will now be accessible . it might be reasonable to assume that a single monitor might detect several tlps per month of observing time . over several years , monitors at a range of terrestrial longitudes might detect of order 100 or more tlps , providing a well - characterized sample that will avoid many of the selection problems of the anecdotal visual data base and approach similar sample sizes . our plan eventually is to run two or more such monitors independently . not only does this increase the likely tlp detection rate , but allows us to perform simultaneous imaging in different bands , or in different polarization states . dollfus ( 2000 ) details tlps evident as polarimetric anomalies . the timescales involved are not tightly constrained , between 6 min and 1 d. other transient polarimetric events ( dzhapiashvili & ksanfomaliti 1962 , lipsky & pospergelis 1966 ) are even less constrained temporally ; however , the fact that we can observe the same event with two monitors simultaneously ( while observing the rest of the moon ) , means that there is little systematic doubt concerning the degree of polarization due to variability of the source while the apparatus is switching polarizations . presumably , since these are likely due to simple scattering effects on linear polarization , we should align the e - vector of one monitor s polarizer parallel to the sun - moon direction on the sky , and the second perpendicular to it . in the case of three or four monitors operating simultaneously , we can reconstruct stokes parameters for linear polarization conventionally by orienting polarizer e - vectors every @xmath18 or @xmath19 , respectively . the total flux from two or more monitors can be obtained by summing in quadrature signals from the different polarizations . a tlp imaging monitor will also open new potential as an alert system for other observing modes . a monitor detection can trigger le imaging in a specific active area . a qualitatively unique possibility is using the monitor to initiate spectroscopic observations , which much better than imaging will provide information about non - thermal processes and perhaps betray the gas associated with the tlp . tlp spectroscopy has its challenges . in order to detect a change , we must make comparisons over a time series of spectroscopic observations . this is essentially a four - dimensional independent - variable problem , therefore : two spatial dimensions of the lunar surface , plus wavelength implying a data cube , plus time . whereas `` hyperspectral '' imaging usually refers to a resolving power @xmath20 , where @xmath21 is the fwhm wavelength resolution , the emission lines from tlps might conceivably be many times more narrow than this , thereby diluted if higher resolution is not employed . it is not currently conceivable to monitor the whole near side in this way ( at @xmath22 gpixel s@xmath23 for @xmath24 and an exposure every 10s ) , but this is unnecessary . a practical approach may be to set up the reduction pipeline of the tlp monitor to alert to an event during its duration e.g. , in under 1000s , and then to bring a larger telescope with an optical or ir spectrograph to bear on the target , which our experience shows might be accomplished in @xmath3300s . we are working to implement this in 2007 . there are reasons to prepare an @xmath25 data cube in advance of a tlp campaign for reasons beyond simply having a `` before '' image of the moon prior to an event . for instance , in the ir there are regolith hydration bands near 2.9 and 3.4 @xmath13 m , the latter with substructure on the scale of @xmath320 nm , which will be degraded unless the instrumental resolution is @xmath26 . while there are fewer narrow features in the optical / near - ir , the surface fe@xmath27 feature at 950 nm of pyroxene ( which requires only @xmath28 to be resolved ) , shows compositional shifts in wavelength centroid and width on the scale of @xmath310 nm ( hazen , bell & mao 1978 ) , which requires @xmath29 to be studied in full detail . likewise , differentiating pyroxenes from iron - bearing glass ( farr et al . 1980 ) requires @xmath30 . this fe@xmath27 band ( and the corresponding band near 1.9@xmath13 m ) are useful for lunar surface age - determination since they involve surface states that are degraded by micrometeorites and solar wind in agglutinate formation ( adams 1974 , charette et al . it appears that overturn of fresh material can also be monitored with enhanced blue optical broadband reflectivity ( buratti et al . 2000 ) . such datasets are straightforward to collect , as are their reduction ( although requiring of some explanation ) . observations involve scanning across the face of the moon with a long slit spectrograph , which greatly improves the contrast of an emission - line source relative to the background ( figure 5 , showing recent data from the mdm observatory 2.4-meter / ccd spectrograph ) . since the spectral reflectance function of the lunar surface is largely homogenized by impact mixing of the regolith , more than 99% of the light in such a spectrum can be simply `` subtracted away '' by imposing this average spectrum and looking for deviations from it ( figure 6 ) . if a tlp radiates primarily in line emission , this factor along with our ability to reject photons outside the line profile yields a contrast as high as 10,000 times better than the human eye observing the moon through a telescope . this could also be done farther into the infrared , for instance we are preparing to observe the l - band ( 2.9 - 4.3 @xmath13 m ) using spex on the nasa infrared telescope facility in single - order mode , which can deliver @xmath31 . in general observations of this kind might be useful in the infrared for wider band emission , which is repeatable based primarily on temperature ( versus ionizing excitation as in paper ii , appendix 1 ) . using the hitran database to compute vibrational / rotational states for different molecules , one can see these starting in the infrared ( or smaller wavenumbers for h@xmath32o , nh@xmath33 , co and ch@xmath34 ) , and extending into the optical for h@xmath32o but at least to k - band for nh@xmath33 ( and intermediate bands for co@xmath32 , co and ch@xmath34 ) . at least for these molecules , the band patterns are strong and highly distinct . to be clear , this latter idea requires having an ir spectrograph available at several minutes notice to follow up on an alert of a tlp ( probably found in imaging ) . on a longer timescale , ir spectroscopy might also be useful for the l - band hydration test outlined above , especially on some of the narrower spectral features near 3.4 @xmath13 m that imaging might overlook , even through narrow - band filters . the data cube described above can be sliced in any wavelength to construct a map of lunar features in narrow or broad bands . figure 7 shows that specific surface features can be reconstructed in good detail and fidelity . given the constraints on imaging from the vicinity of earth , it is interesting to consider the limits and potentials of imaging monitors closer to the moon . in general , we will not be proposing special - purpose missions in space - based remote sensing , and indeed will only mention dedicated missions related to in - situ exploration of areas affected by volatiles , where special - purpose investment seems unavoidable . with in - situ cases , we would perform a more extensive study , so will largely postpone these discussions to later work concentrating on close - range science . here we propose experiments and detectors which might ride on other platforms , either preceding or in concert with human exploration , and which will accommodate the same orbits and other mission parameters which might be chosen for other purposes . some of these purposes are not designated priorities for planned missions , but might prove useful and probably should be considered in the future . in some cases , we will give rough estimates of project costs based on our prior experience with similar spacecraft . these are for discussion only and would need to be re - estimated in detail to be taken with greater credibility . an instance of such joint use : does exploration of the moon imply establishment of a communications network with line - of - sight visibility from essentially all points on the lunar surface ( excepting those within deep craters , etc . ) ? if so , these platforms might also serve as suitable locations for comprehensive imaging monitoring . a minimal example of such a network might have a tetrahedral configuration ( with each point typically 60000 km above the surface ) with a single platform at earth - moon lagrange point l1 , covering most of the nearside moon , and three points in wide halo orbits around l2 , each covering their respective portion of the far side plus a portion of the limb as seen from earth . no single satellite will be capable of covering the entire far side , especially if operation of farside radio telescopes there require a policy of solely high - frequency communications e.g. , via optical lasers . a single l2 satellite will cover at most 97% of the far side ( subtending @xmath35 , selenocentrically ) ; full coverage ( not to mention some communications system redundancy ) will require three satellites , plus some means of covering the near side . with this configuration , the farthest points from each satellite will be typically @xmath36 ( in selenocentric angle ) , hence forshortened due to proximity to the limb by @xmath37 times . extensive discussion is underway of using a facility at l1 to aid in transfer orbits throughout the solar system ( lo 2004 , ross 2006 ) ; in that case we should also consider placing an imaging monitor at l1 . an imaging monitor to improve significantly on earth - vicinity capabilities might need to be an ambitious undertaking . for instance , to acheive 100 m fwhm resolution at the sub - satellite point on the face of the moon requires an imager of about 4 gpixels , an aperture @xmath38 m , and a field - of - view of 3@xmath39 . each such monitor , separate from power , downlink , attitude control and other infrastructure requirements will cost perhaps $ 100 m . a stand - alone facility might cost several times more , at each of the several stations . perhaps the system could be cut to a single farside monitor , in a narrow halo orbit extending beyond the moon s earth - shadow , plus some nearside monitoring , which together could still cover perhaps 95% of the lunar surface , albeit with some extreme limb foreshortening . we also need to ask ourselves at some point if the essential research and resource exploitation might be confined to the near side . this is an expensive undertaking , and one that must probably be combined with other reasons to establish platforms near l1 and l2 . in the meantime , we should accomplish what is possible from the ground . if the goal is to discover the source of volatiles for the sake of further scientific exploration or resource exploitation , however , an investment in remote sensing , in terms of spatial resolution ( or spectral resolution to discover the substances involved , or temporal resolution to define the behavoir of the source ) makes in - situ reconnaissance and exploration much less problematic . a human mission , or a sophisticated robotic mission , could conceivably cost $ 1b , and remote sensing could inform this effort as to where to look in detail , when dangerous eruptions might occur , and what is the material goal . without such information , these investigation is likely to be more time - consuming , problematic , and perhaps more hazardous . we concentrate further on remote sensing , even if the proposed expense might be significant . as explained in paper ii , an expectation of water vapor seepage from the lunar interior should be an ice layer within the regolith about 15 m below the lunar surface . a remote means of studying this feature would be ground - penetrating radar , either from the ground or spacecraft platforms . one should realize that there is significant heritage and as well as plans involving lunar radar . the lunar sounder experiment ( lse ) on _ apollo 17 _ ( brown 1972 , porcello 1974 ) operated in both a high - frequency and penetrating radar mode ( 5 , 16 and 260 mhz ) . also planned are the lunar radar sounder ( lre ) aboard selene ( ono & oya 2000 : at 5 mhz ( with an option at 1 mhz and 15 mhz ) , and mini - rf on the lunar reconnaissance orbiter , operating at 3 ghz and @xmath310 ghz . finally , of note for comparison s sake in the martian case is marsis ( mars advanced radar for subsurface and ionosphere sounding at 1.8 , 3.0 , 4.0 , and 5.0 mhz : porcello et al . 2005 ) . at 5 mhz ( @xmath40 m ) the depth of penetration is many kilometers below the lunar surface , but the spatial resolution is necessarily coarse . to study the regolith and shallow bedrock , we should choose a frequency closer to 100 - 300 mhz . the apollo lse operated for only a few orbits and only close to the equator . the selene lre runs at lower frequency . a higher frequency mode is desirable . the ground - based alternative is useful ; lunar radar maps have been made at 40 mhz , 430 mhz , and 8 ghz ( thompson & campbell 2005 ) , also 2.3 ghz ( stacy 1993 , campbell et al . 2006a , b ) . at 8 ghz we are only studying structure of several centimeters within a meter of the surface . for 430 mhz we see perhaps @xmath41 m inside , and at 40 mhz , 100 m towards the interior ( with attenuation lengths of roughly 10 - 30 wavelengths ) . in practice , better angular resolution at higher frequencies is possible e.g. , 20 m ( campbell et al . 2006a , b ) . of course from earth only the nearside is accessible , and larger angles of incidence e.g. , @xmath42 , imply echoes dominated by diffuse scattering in a way which can not be modulated . use of circular polarization return measurements can be used to test for water ice ( nozette 1996 , 2001 ) but have been questioned ( simpson 1998 , campbell et al . we will not review this debate here , but application of the idea to subsurface ice is problematic . it is unclear that this could be accomplished at frequencies of hundreds of mhz required to penetrate to depths of @xmath315 m , and the more standard technique ( at 13 cm ) only performs to depths @xmath431 m , where ice sublimation and diffusion rates are almost certainly prohibitive of accumulation . finding subsurface ice has its challenges . for instance , the dielectric constant @xmath44 for both regolith and water ice ( which is slightly higher ) , as it is for many relevant mineral powders of comparable specific gravity e.g. , anorthosite and various basalts . ice and these substances have similar attenuation lengths , as well . on the strength of net radar return signal alone , it will be difficult to distinguish ice from any usual regolith by their mineral properties . however , in terrestrial situations massive ice bodies reflect little internally e.g. , moorman , robinson & burgess ( 2003 ) . one might expect ice - bearing regions to be relatively dark in radar images , if lunar ice - infused volumes homogenize or `` anneal '' in this way , either by forming a uniform slab or by binding together regolith into a single , uniform @xmath45 bulk . on the other hand , hydrated regolith samples have @xmath45 values much higher than unhydrated ones ( by up to an order of magnitude ) , as well as attenuation lengths even more than an order of magnitude shorter ( chung 1972 ) . this hydration effect is largest at lower frequencies , even below 100 mhz . one might suspect that significant water ice might perturb the chemistry of the regolith significantly , which might even increase charge mobility as in a solution , which appears to invariably drive up @xmath45 , and conductivity even more , increasing the loss tangent : conductivity divided by @xmath45 ( and the frequency ) . one should expect a reflection passing into this high-@xmath45 zone , but this depends strongly on the details of the suddenness of the transition interface . of particular interest is the radar map at 430 mhz ( ghent et al . 2004 ) of the aristarchus region , site of roughly 50% of tlp and radon reports . the 43-km diameter crater is surrounded by a low radar - reflectivity zone some 150 km across , particularly in directions downhill from the aristarchus plateau onto oceanus procellarum . in general the whole plateau is relatively dark in radar , occasionally interrupted by bright crater pock - marks and vallis schrteri . in contrast the dark radar halo centered on aristarchus itself is uniquely smooth , indicating that it was probably formed or modified by the impact itself , a few hundred million years ago . this darkness might be interpreted as higher loss tangent , consistent with the discussion in the previous paragraphs , or simply fewer scatterers ( ghent et al . 2004 ) i.e. , rocks of approximately meter size ; it is undemonstrated why the latter would be true in the ejecta blanket of a massive impact especially given the bright radar halo within 70 km of the aristarchus center . ghent et al . ( 2005 ) show that other craters , some comparable in size to aristarchus , have dark radar haloes , but none so extended . the region around aristarchus has characteristics that might be expected from subsurface ice redistributed by impact melt : dark , smooth radar - return , spreading downhill but otherwise centered on the impact ; this should be expected to be confused , at least , with the dark halo effect seen around some other impacts . it seems well - motivated to search for similar dark radar areas around other likely outgassing sites , particularly ones not associated with recent impacts ; unfortunately , the foremost candidate for such a signature is competing with such an impact , aristarchus , which can be expected to produce its own confusing effect . we would propose that radar at frequencies near hundreds of mhz be considered for future missions , in a search for subsurface ice . this is a complex possibility that we will not detail here , that must be weighed against the potential of future ground - based programs . in particular , the near side has been mapped at about 1 km resolution for 70 cm wavelength ( campbell et al . 2007 ) , this could be improved with an even more intensive ground - based program , or from lunar orbit . orbital missions can be configured to combine with higher frequencies and different reception schemes to provide better spatial resolution , deal with ground clutter , and varying viewing angles . a lunar orbiter radar map would be less susceptible to interference speckle noise , which will likely require long series of pointings to be reduced from the ground . in combination with an optical monitor , a ghz - frequency radar might produce detailed maps in which changes due to tlps might be sought , and might be then correlated with few - hundred mhz maps to aid in interpretation in terms of volatiles . at shorter wavelengths one should consider mapping possible changes in surface features due to explosive outgassing , which paper ii hints might occur frequently on scales excavated over tens of meters , and expelled over hundreds or thousands of meters . again , earth - based observations suffer from speckle , but planned observations by the _ lunar reconnaissance orbiter ( lro ) _ mini - rf ( mini radio - frequency technology demonstration - chin et al . 2007 ) at 4 and 13 cm might easily make valuable observations of this kind . both modes scan in a swath @xmath35 km wide , which would make comprehensive mapping difficult , but would mesh well with the event resolution from a ground - based optical monitor . a `` before '' and `` after '' radar sequence meshed with an optical monitoring program would likely be instructive as to how outgassing and optical transients actually interact with the regolith . several upcoming missions will carry high - resolution optical imagers , each of which will be capable of mapping nearly the entire lunar surface e.g. , _ change-1 _ ccd imager ( yue et al . 2007 ) , _ selene _ spectrometer / multiband imager ( lism / mi ) ( ohtake et al . 2007 ) , _ lro _ camera ( lroc ) ( robinson et al . 2005 ) , and _ chandrayaan-1 _ moon mineralogy mapper ( mmm ) ( pieters et al . 2006 ) , typically at tens to hundreds of meters resolution . in particular the mi / sp will usefully observe at 20 m resolution the pyroxene near - ir band that can indicate the exposure of fresh surface , as can the mmm ( albeit at 280 m resolution ) . all of these are sensitive at blue wavelengths which can also indicate surface age . the lroc and mmm will repeatedly map each point on the moon , not in any way sufficient to be considered realtime monitoring of transients , but sufficient to allow frequent sampling on timescales of a lunation . this allows an interesting synergy with ground - based monitors since they can highlight sites of activity for special analysis . furthermore , lroc has a high resolution pointed mode which might provide sub - meter information in areas where tlps have been recently detected , hence excellent sampling on the scales that we suspect will be permanently effected , perhaps in a `` before '' and `` after '' sequence . at any given time , any these four spacecraft have a roughly 10% chance of at least one of them being in view of a particular site above its horizon ; it would be fascinating ( but perhaps too logistically difficult ) if a program could be implemented wherein spacecraft could be alerted to image at high resolution a tlp site in real time during an event . in order to study outgassing directly , we need instruments at or near the lunar surface . in the case of @xmath1rn , the thermal velocity is typically @xmath46 m s@xmath23 , so typical ballistic free flight occurs over @xmath47 km . over its half - life of 3.8 d , a @xmath1rn atom travels typically 50000 km in a random walk that wanders from the source only a few hundred km before decaying ( or sticking to a cold surface ) . thus the alpha particles must be detected in much less than a day after outgassing , or the @xmath1rn signal disperses by an amount that makes superfluous placing the detector less than a few hundred km above the lunar surface , except for @xmath48 sensitivity considerations . three alpha - particle spectrometers have observed the surface of the moon , but for relatively brief periods of time . the latitude coverage was severely limited on _ apollo 15 _ ( @xmath49 for 145 hours ) and _ apollo 16 _ ( @xmath50 , 128 h ) . _ lunar prospector s _ alpha particle spectrometer covered the entire moon , over 229 days spanning 16 months , but was partially damaged ( one of five detectors ) upon launch and suffered a sensitivity drop due solar activity ( binder 1998 ) . _ apollo 15 _ observed two outgassing events ( from aristarchus and grimaldi ) , _ apollo 16 _ none , and _ lunar prospector _ two sources ( aristarchus and kepler ) , although the signals from these last sources were integrated over the mission duration . in addition , apollo and _ lunar prospector _ instruments detected an enhancement at mare / highlands boundaries from daughter product @xmath2po , indicating @xmath1rn leakage over approximately the previous century . the expected detection rate for a single alpha - particle spectrometer in a polar orbit and without instantaneous sensitivity problems , might be grossly estimated from these data . the _ apollo 16 _ instrument covered a sufficiently small fraction ( @xmath51% ) of the lunar surface so that we will not consider it , whereas _ apollo 15 _ covered about 37% . these missions were in orbit @xmath36 d apiece , and considering the @xmath1rn lifetime thereby were sensitive to events ( at @xmath5210% full sensitivity ) for @xmath53 d. _ lunar prospector _ covered the entire lunar surface every 14 d , hence caught events typically at 28% instantaneous full strength ( minimum 8% ) , however , by averaging over the mission diluted this by an factor @xmath320 - 30 . these data are consistent with a picture in which aristarchus produces an outgassing event 1 - 2 times per month at the level detectable by _ apollo 15 _ , and by _ lunar prospector _ when integrated over the mission . apparently other sites such as grimaldi and kepler collectively are about equally active as aristarchus , together all sites might produce 2 - 4 events per month at the sensitivity level of _ apollo 15_. this level of activity is consistent with the statistics of tlps constrained in paper i. a new orbiting alpha - particle spectrometer with a lifetime of a year or more and an instantaneous sensitivity equal to that of _ apollo 15 _ s detector would likely produce a relatively detailed map of where outgassing occurs on the lunar surface , separate from any optical manifestation . this is likely an important test for many of the procedures mentioned above , which are critically dependent on the outgassing / optical correlation . this must be examined in further detail , because there are many ways in which one might imagine that gas issues from the interior , thereby producing radon , without a visible manifestation , either due on one extreme to such rapid outgassing that previous events have cleared the area of regolith that might interact with gas on its way to the vacuum , or due to seepage sufficiently slow to trap water ( and perhaps other gasses by reaction ) in the regolith , and too slow to perturb dust at the surface . radon , an inert gas that will not freeze or react on its way to the surface , is more likely to escape the regolith to be detected , regardless . the alpha - ray detector ( ard ) onboard _ selene _ ( nishimura et al . 2006 ) promises to be @xmath3 25 times more sensitive than the apollo alpha particle spectrometers , with a mission lifetime of one year or more , in a polar orbit . this , in conjunction with an aggressive optical monitoring program ( as in section 2.1 ) , holds the prospect of extending the tlp/@xmath1rn - outgassing correlation test from paper i to a dataset of order 10 times larger . this would likely serve as a significant advance in understanding their connection , but it is probably best to consider what a following generation alpha - particle spectrometer study might entail . to insure better sensitivity coverage two such detectors in complementary orbits would cover the lunar surface every 1.8 half - lives of @xmath1rn . this may nearly double the detected sample . unless the alpha - particle detectors are constructed with a veto for solar wind particles , it is best to avoid active solar intervals . we will exit the solar minimum probably by year 2008 , with the next starting by about 2016 . on the other hand , some of the lack of sensitivity to lunar alpha particles and elevated solar particle background count on _ lunar prospector _ was due in part to it being spin - stabilized . if detectors on a future mission were kept oriented towards the lunar surface and shielded from solar wind to the extent possible , the apollo results indicate that prompt @xmath1rn outburst detection at good sensitivity is possible . beyond this , extending the mission(s ) , of course , will help , and the best approach might be to develop a small alpha - spectrometer package that might easily fly on any extended low - orbital mission . the radioactive decay delay in alpha - particle detection insures that a reasonable number of orbiting detectors can have near unit efficiency . this is not the case for prompt detection of outgassing e.g. , by mass spectrometers . an instantaneous outburst seen 100 km away will undergo a dispersion of only a few tens of seconds in arrival time . the detectors must either be very sensitive or densely spaced , and prepared to measure and analyze what they can in these short time intervals . this is a problem for apollo - era instruments e.g. , the _ apollo 15 _ orbital mass spectrometer experiment ( omse - hoffman & hodges 1972 ) required 62s to scan through a factor of 2.3 in mass ( 12 to 28 , or 28 to 66 amu ) . total amount of outgassing is in the range of many tons per year , and with perhaps tens of outbursts per year , the mass fluence of particles from a single outburst seen at a distance of 1000 km is approaching @xmath54 @xmath55 amu . while a burst on the opposite side of the moon will not be detected and/or properly interpreted , one that can be seen by a few detectors would be very well constrained . the specific operational strategies of these detectors is paramount . for example consider an event at 1000 km distance , which will spread over @xmath56s in event duration . a simple gas pressure gauge will not be overwhelmingly sensitive , in that even with an ambient atmosphere that is not unusual e.g. , number density @xmath57 @xmath58 ( varying day / night e.g. , hodges , hoffman & johnson 2000 ) , the background rate of collisions over 500 s amounts to an order of magnitude or more than the particle fluence than for a typical outgassing outburst , assuming @xmath59 amu particles in the outburst . since interplanetary solar proton densities can change by amount of order unity in an hour or less ( e.g. , mcguire 2006 ) , pressure alone is not likely to be a useful event tracer . a true mass spectrometer is useful in part by subdividing the incoming flux , in mass , obviously , but also in direction , thus decreasing the effective background rate . the disadvantage of this approach in the past has been that it can not cover the entire parameter range of this subdivision at once , so must scan in atomic mass or direction , or must always accept a significantly limited range . for a short burst , this means that mass components may not be examined during the event , or that events might be missed due to detectors pointing in the wrong direction . for apollo - era detectors , these problems , particularly the former , were significant . we would prefer to operate a mass spectrometer operating continuously over a significant mass range , with ballistic trajectory reconstruction over a large incoming acceptance solid angle . we will return to this concept below . first , let us discuss low - orbit platforms . we will not propose special purpose probes of the atmosphere alone , but there are other reasons for dense constellations of lunar satellites , most prominently a lunar global positioning system ( gps ) . terrestrial systems in operation ( gps ) and planned ( galileo , beidou and glonass : global navigation satellite system ) are typically 25 - 30 satellites at orbital radii @xmath325000 km . around the moon this could be much lower , @xmath60 km , and with fewer satellites , @xmath312 , which would put satellites within @xmath37000 km of a surface outburst . this is compared to @xmath61 km for apollo . scaling the sensitivity of the _ apollo 15 _ omse ( hodges et al . 1973 ) , a detector on a gps would be sensitive ( at the 5@xmath62 level ) to an instantaneous outburst of about 50000 kg ( and more depending on the details of non-@xmath48 propagation effects ) . this is insufficient sensitivity to detect outgassing events . one needs a lower orbit ( or much more sensitive detectors , by three orders of magnitude ) . it is unclear if a lower - orbit gps system , while more favorable for an add - on mass spectrometer array , would serve its navagational purpose . a gps / mass spectrometer constellation only 1000 km above the lunar surface could likely be made sufficiently sensitive for gas outburst monitoring , nearly continuously . such a low orbit makes gps more difficult , require several more satellites , and increasing the effects of mascons on their orbit . this requires further modelling . nonetheless , we should consider other science instrumentation on a lunar gps . high - resolution imaging from @xmath60 km radius could be 10@xmath63 finer ( @xmath64 m ) than platforms at l1 or near l2 . covering the moon at this resolution would require @xmath65 pixels , which might allow mapping occasionally , but only crude monitoring temporally . still , if one - third of lunar gps platforms were equipped with a prompt , high - resolution imager , any portion of the lunar surface could be imaged during the course of a surface event . if an event is observed from the ground or from l1/l2 , it could be detailed at 10 m or even higher resolution . this imager network should establish an atlas of global maps ( at various illumination conditions ) to serve as a `` before '' image in this comparison ( as well as allowing a wealth of other studies ) . by allowing transient events to be studied at @xmath66 m resolution , this sets the stage for activity to be isolated at a sufficiently fine scale for in - situ investigations that would thereby be targetted and efficient in localization . returning to mass spectrometry , it is clear that there are two separate modes for gas propagation above the lunar surface , neutral and ionized , and that a significant amounts are seen in both ( vondrak , freeman & lindeman 1974 , hodges et al . 1972 ) , at a rate of one to hundreds of tonne y@xmath23 for each process . there is some possibility that a large portion of the ionized fraction might be molecular in nature ( vondrak et al . 1974 ) . for neutral atoms more massive than h or he , their thermal escape lifetime is sufficiently long that they have ample time to migrate across the lunar surface until they stick in a shadowed cold - trap . furthermore , the ionized component will predominently follow the electric field embedded in the solar wind , which tends to be oriented perpendicular to the sun - moon vector and hence frequently pointing from the sunrise terminator into space . for these two reasons the best location to monitor outgassing is a point above the sunrise terminator , presumably on a low - orbit platform . note that there is some degeneracy between the timing information recorded by a particle detector on such a satellite between the episodic behavior of particle outgassing versus the motion of the spacecraft at @xmath67 km s@xmath23 . the ideal situation would be to triangulate such signals with more than one platform . such an experiment is not trivial , but there are alternatives , explored below . for a low lunar orbit to be `` low maintenance '' i.e. , require few corrections due to mascon perturbations , it should be at one of several special `` frozen orbit '' inclination angles @xmath68 , @xmath69 , @xmath70 or @xmath71 ( e.g. , ramanan & adimurthy 2005 ) . however , we want to maintain a position over the terminator , using a sun - synchronous orbit , which requires a precession rate @xmath72 rad s@xmath23 . natural precession due to lunar oblateness is determined by the gravitational coefficient @xmath73 ( konopliv et al . 1998 ) according to @xmath74 , where @xmath75 is the lunar radius , @xmath76 the orbital angular speed , @xmath77 the lunar mass and @xmath78 the orbital radius . ( the precession caused by earth is 1000 times smaller , and 60000 times smaller for the sun . ) one can not effectively institute both conditions , however , since the maximum inclination orbit with @xmath79 s@xmath23 occurs at @xmath80 ( or else the orbit is below the surface ) . while an orbit at @xmath68 is stable ( at @xmath81 km , 138 km above the surface ) and has the correct precession rate , it spends most of its time away from the terminator . in contrast , at @xmath82 , @xmath83 s@xmath23 , and the spacecraft needs to accelerate continuously only @xmath84 mm s@xmath85 to place it into sun - synchronous precession . this is nearly the same as the thrust provided by the hall - effect ion engine on _ ( and corresponds to an area per mass of 330 cm@xmath11 g@xmath23 under the influence of solar radiation pressure . ) while it is not apparent that an ion engine would be the best choice for a platform with mass and ion spectrometers , this illustrates the small amount of impulse need to maintain this favorable orbit , comparable to station - keeping in many non - frozen orbits . in truth , the most efficient location to apply this acceleration is only near the poles , so a slightly more powerful thruster might be needed . since , time - averaged , this perturbed orbit still lands in a frozen - orbit zone , it should still be relatively stable in terms of radius . we would propose that a instrumented platform in this driven , sun - synchronous polar orbit would be ideal for studying outgassing signals near the terminators . there is an interesting synergy between this outgassing monitor platform and another useful investigation from a similar satellite(s ) , although not necessarily simultaneously . an outstanding problem is gravitational potential structure of the moon , particularly the far side ( where satellite orbits can not be monitored from earth ) . with the inclusion of the the 562-day _ lunar prospector _ data set ( konopliv et al . 2001 ) the error is typically 80 milligals on the far side ( corresponding to surface height errors of about 25 m ) versus 10 milligals in the near - side potential . also the limiting harmonic is of order 110 approximately on the near side , and only order 60 on the far side ( @xmath87 200 km resolution ) . in contrast , the grace ( gravity recovery and climate experiment ) can define the geodesy of earth at much better field and spatial resolution , a few milligals at about order 200 ( tapley et al . 2005 - one year of data ) , using a double satellite at @xmath3500 km above the earth in polar orbit , with the separation ( @xmath3200 km ) between the two components carefully monitored ( by laser interferometer for the proposed grace follow - on mission - watkins et al . 2006 , or in the microwave k - band for grace itself ) . such a satellite pair in lunar orbit would improve our knowledge of the farside field by orders of magnitude , determined independent of earth - based tracking measurements , and in general make the accuracy and detail of lunar potential mapping much closer in quality to mineralogical mapping already in hand . one interesting question this might address is whether mascons extend to much smaller scales than currently known . while this mapping is underway , one could use outgassing monitors on board to look for outbursts , and when the geodetic mission is complete , drive the satellites into a polar , sun - synchronous orbit above the terminator . depending on the type of monitors imployed , forcing sun - synchronous precession by chemical , ion or even solar - sail propulsion may or may not interfere ; neutral - gas spectrometers may be compatible with ion drives while charged species trajectories might be perturbed , for instance . maintaining @xmath84 mm s@xmath23 for a 100 kg spacecraft requires 20 kg month@xmath23 of chemical propellant ( exhaust velocity of 4000 m s@xmath23 ) versus 2.5 kg month@xmath23 of ion propellant ( 30000 m s@xmath23 ) . for a 100 kg spacecraft a solar sail about 30 m in radius would be required . none of these solutions are so easy that they do not inspire a search for alternatives , and their non - gravitational acceleration would mean that they could take place only after ( or before ) any geodesic mission phase . furthermore , ion propulsion and probably chemical propulsion would tend to interfere with mass spectrometry . these should be traded against other possibilites e.g. , several small probes on various orbital planes at @xmath82 , rather than one or two sun - synchronous platforms . the fact that there would be an outgassing detectors on each platform would make temporal / spatial location of specific outbursts more unambiguous , aided by differences in timing and signal strength at the two moving platforms , at least for neutral species . the timing difference will give an indication of the distance difference to the sources , with the source confined to the hyperboloid @xmath88 where @xmath89 is the distance along the line connecting the two satellites , with the origin at the half - way point between them , and @xmath90 is the distance perpendicular to this line . the distance between the two satellites is given by @xmath91 and the difference in distance between the source and the first satellite versus the source and the second is @xmath92 . there is still a left / right ambiguity in event location to be resolved by detector directionality , and better directional sensitivity would add a helpful overconstraint on the measurement . our research group is developing ways to efficiently transfer the insight gained from a program of remote sensing to a program of in - situ research involving the lunar surface . i would like to emphasize a few key points already becoming apparent . the neutral fraction from lunar outgassing need not respect the correlation with lunar sunrise ; a detector giving enough prompt information about outgassing might be invaluable . neutral gas emitted on the day side is free to bounce ballistically until either sticking to a cold surface or escaping ( either due to ionization or by reaching the high - velocity maxwellian tail ) . a highly desirable monitor of this activity would be a mass spectrometer capable of simultaneously accepting particles in a wide range of masses e.g. , @xmath93 a.m.u . , and reconstructing incoming particle trajectories and velocities to allow the locus of outgassing to be reconstructed ( at least within hundreds or thousands of km ) . in addition to tracking the sunrise terminator outgassing signal , such a mass spectrometer would be able to monitor wide areas of the moon for prompt neutral outburst signals from point sources , and therefore the instrument should be placed in the vicinity of known outgassing sites to establish which species succeed in propagating to the regolith surface . the suggested ground - based approaches provides this rough localization , buttressed by the low - orbital outgassing detectors . at some point the identification of a good tracer gas to act as a proxy for endogenous emission would be highly valuable in simplification of outgassing alert monitors not required to scan entire mass ranges . now it is unclear what that gas should be . it is true that @xmath1rn seems to be highly correlated with optical transients , but the relationship between radiogenic gas emission and that of volcanic emission is uncertain . besides , while usefully radioactive , radon is a very minor constituent . radiogenic @xmath94ar is more abundant , and episodic , but its relation to volcanic gas is uncertain ( as is its correlation to optical transients ) . the most reliable observed molecular atmospheric component is ch@xmath34 , but it is likely to derive in large part from cometary / meteoritic impacts and is somewhat unnatural to expect from the oxygen - rich interior . water suffers from the situation described in paper ii in which a large fraction of any large , endogenous source might never propagate gas to the surface , making it an unreliable tracer . even while endogenous water of nearly certain volcanic origin has been found in glasses likely derived for the deep interior ( saal et al . 2007 ) , co@xmath32 is absent . the limits on co are more unclear , as are those for oxides of nitrogen . the first mass spectrometer probes should be designed to clarify this situation . to place these monitors on the surface , one may exploit human exploration sorties , which will be relatively infrequent and potentially concentrated in sites of just a very few bases . i reiterate that another concern is the contamination that each of the missions will produce , concentrated primarily near the landing site itself . it is evident that by the deployment of lace on the final apollo landing that the outgassing environment was contaminated by a large contribution of anthropogenic gas , and that these vehicles in a new epoch of human exploration will deliver many tens of tons per mission of gases to the lunar surface of composition relevent to species suspected from a potential endogenous volcanic component , a level of contamination comparable to the potential annual output of such gases from endogenous sources . the constellation spacecraft consist of orion , carrying about 10 tonne of n@xmath32o@xmath34 ( nitrogen teroxide ) and ch@xmath33n@xmath32h@xmath33 ( monomethyl hydrazine ) propellant , and lsam , propelled by liquid oxygen and nitrogen . the orion fuel mix produces n@xmath32 , co@xmath32 and h@xmath32o and the lsam exhausts water . depending on the orientations and trajectories of the spacecraft when thrusting they will deposit about 20 tonnes of mostly water to the surface , where most will remain for days ( up to about one lunation ) . during the course of the return to the moon , measurements of at least these three product molecules will be suspect , since in fact their signal will disappear completely over successive lunations . in many respects the surface layer of regolith should be considered as a planet - sized sorption pump coupling the atmosphere , across which gases are free to propagate ( and exit the system if they are ionized or low - mass ) , and the lower regolith , which is cold ( @xmath95k ) and relatively impermeable . gas in the atmosphere can be delivered to the surface where , if it penetrates a few cm , enters a region in which particle mobility slows considerably and where it essentially becomes entrained in the time - averaged signal of endogenous gas ( radiogenic or volcanic ) that is leaking from greater depth . ( indeed , since the temperature increases inwards , gas reaching this colder zone preferential migrates to greater depths . ) furthermore , once gas from the interior reaches the outer few cm of regolith subject to large temperature swings , it is likely to escape into the vacuum . there is a scientific premium , therefore , to delivering surface monitors to their site without delivery of many tons of anthropogenic gas , annd for this purpose one might consider small , parasitic landing rockets that deliver an experiment package from the orion or lsam human exploration vehicles to the vicinity of the surface , but transition to a low - contamination soft lander system such as an airbag . this is an established , low - cost technology with extensive heritage ( from the ranger block 2 lunar probes to the highly successful mars exploration rovers ) and might easily be the landing technique of choice for small lunar surface packages . on small ( @xmath43 tens of km ) scales , robotic rovers are less prone to sowing contamination when delivering detector packages across the surface . when human exploration turns towards study of lunar outgassing sites the primary challenge may be converting the lower spatial resolution information obtained at earth or lunar orbit into meter or 10-meter scale intelligence regarding where to initiate in - situ exploration . the transitional technologies to bridge this gap consist of local networks of sensors that map area on the scale of a 1 km or 100 m to resolutions of 1 - 10 m using various techniques : local ground - penetrating radar , local seismic arrays , directional and ground - sniffing mass - spectrometers that work to localize , and another technique we propose to investigate : intensive laser grids that densely populate the space above the patch of surface in question with lines of sight sampling strong transitions of some predominant species e.g. , an infrared vibrational / rotational transition if molecules are discovered in quantity . the details of the ideas promulgated in this section are beyond the scope of the current paper and will be presented in a larger document currently in preparation . if the reader will allow a personal statement , i am not easily swayed into writing research papers based on data of the uncertain quality of those seen in papers i and ii , but this is the nature of the field . it has been the purpose of this investigation not only to clarify the implications of existing data , which i think it has done , but also to understand the range of interesting possibilities of phenomena consistent with these data and ask how we should proceed to investigate them , cognizant that many of our actions have implications in terms of disturbing the environment that we care to assay . we need to access which interesting questions need to be addressed , given the state of our ignorance , and consider how to proceed . i hope and intend that these works have advanced the discussion significantly . the phenomena that we have been studying are subtle , and many important aspects may be highly covert . the above - surface signals of outgassing of radiogenic endogenous sources is fairly clear , but gas of more magmatic origin , while possibly present , needs further study to be absolutely confirmed . activity associated with apollo landings easily dominated with anthropogenic gas production the activity in molecular species that might trace residual lunar magnatism . apollo - era and later data were insufficiently sensitive to establish the level of outgassing beyond @xmath1rn , @xmath96ne and isotopes of ar , plus he , presumably , but did detect molecular gas , particularly ch@xmath34 , but of uncertain origin . it is important to assess how we can advance the apollo - era understanding . consistent with these molecular gas outflows , and perhaps traced by optical transients , there is a range of possible phenomena that have interesting possible scientific consequences and might easily be useful in terms of resource exploitation for human exploration . while this amount of volatile production is inconsequential on the scale of the geology of the moon as a whole , and is poorly constrained by any measurement of current or previous volatiles , even in returned surface samples , it is still capable of massively altering the environment locally in ways which should be investigated in a timely way . we could learn a great deal from the current production of volatiles and their accumulation over geologic timescales in an extraterrestrial environment so easily explored . the salient facts from the above treatment is that for many years yet monitoring for optical transients will still be best done from the earth s surface , even considering the important contributions that will be made by lunar spacecraft probes in the next several years . these spacecraft will be very useful in evaluating the nature of transient events in synergy with ground - based monitoring , however . given the likely behavior of outgassing events , it is unclear that in - situ efforts alone will necessarily isolate their sources within significant winnowing of the field by remote sensing . early placement of capable mass spactrometers of the lunar surface , however , might prove very useful in refining our knowledge of outgassing composition , in particular a dominant component that could be used as a tracer to monitor outgassing activity with more simple detectors . this must take place before significant pollution by large spacecraft , which will produce many candidate tracer gasses in their exhaust . we do not know enough now to discuss the potential implications of this line of research in terms of resources for human exploration , or even in terms of prebiologic chemistry on the moon and for tenuous endogenous outgassing and atmospheric interactions with the regolith on other bodies , but all of these are interesting , new avenues of such research . it is crucial that exploration of these issues progress while we have a pristine lunar surface as our laboratory . i would much like to thank alan binder and james applegate , as well as daniel savin , daniel austin and the other members of aeolus ( `` atmosphere as seen from earth , orbit and lunar orbit '' ) for helpful discussion . * references : * adams , j.b . 1974 , jgr , 79 , 4829 . akhmanova , m.v . , dementev , b.v . , markov , m.n . & sushchinskii , m.m . 1972 , cosmic research , 10 , 381 binder , a.b . 1998 , science , 281 , 1475 ; also see video interview , http://lunar.arc.nasa.gov/results/alres.htm brown , w.e . , jr . 1972 , earth moon plan . , 4 , 133 . buratti , b.j . , mcconnochie , t.h . , calkins , s.b . , hillier , j.k . & herkenhoff , k.e . 2000 , icarus , 146 , 98 . campbell , b.a . , campbell , d.b . , margot , j .- , ghent , r.r . , nolan , m. , carter , l.m . , stacy , n.j.s . 2007 , eos , 88 , 13 . campbell , d.b . , campbell , b.a . , carter , l.m . , margot , j .- l . & stacy , n.j.s . 2006 , nature , 443 , 835 . campbell , b.a . , carter , l.m . , campbell , d.b . , hawke , b.r . , ghent , r.r . & margot , j .- 2006 , lun . plan . conf . , 37 , 1717 . charette , m.p . , adams , j.b . , soderblom , l.a . , gaffey , m.j . & mccord , t.b . 1976 , lun . 7 , 2579 . chin , g. , et al . 2007 , lun . plan . 38 , 1764 . chung , d.h . 1972 , earth moon & plan . , 4 , 356 . crotts , a.p.s . 2007 , icarus , submitted ( paper i ) . crotts , a.p.s . & hummels , c. 2007 , apj , submitted ( paper ii ) . crotts , a.p.s . 2007 , et al . 2007 , lun . plan . conf . , 28 , 2294 . dollfus , a. 2000 , icarus , 146 , 430 . dzhapiashvili , v.p . & ksanfomaliti , l.v . 1962 , the moon , iau symp . 14 , ( academic press : london ) eliason , e.m . , et al . 1999 , lun . plan . , 30 , 1933 farr , t.g . , bates , b. , ralph , r.l . & adams , j.b . 1980 , lun . plan . , 11 , 276 . fried , d.l . 1978 , opt . j. , 68 , 1651 . garvin , j. , robinson , m. , skillman , d. , pieters , c. , hapke , b. & ulmer , m. 2005 , @xmath9 proposal go 10719 . ghent , r.r . , leverington , d.k . , campbell , b.a . , hawke , b.r . & campbell , d.b . 2004 , lun . plan . conf . , 35 , 1679 . ghent , r.r . , leverington , d.k . , campbell , b.a . , hawke , b.r . & campbell , d.b . 2005 , jgr . 110 , doi : 10.1029/2004je002366 . hazen , r.m . , bell , p.m. & mao , h.k . 1978 , lun . plan . , 9 , 483 . hodges , r.r . , jr . , hoffman , j.h . & johnson , f.s . 1973 , lun . conf . , 4 , 2855 . hodges , r.r . , jr . , hoffman , j.h . & johnson , f.s . 1974 , icarus , 21 , 415 . hodges , r.r . , jr . , hoffman , j.h . , yeh , t.t.j . & chang , g.k . 1972 , jgr , 77 , 4079 . hoffman , j.h . & hodges , r.r . , jr . 1972 , lun . conf . , 3 , 2205 . konopliv , a.s . , asmar , s.w . , carranza , e. , sjogren , w.l . & yuan , d.n . 2001 , icarus , 150 , 1 konopliv , a.s . , binder , a.b . , hood , l.l . , kucinskas , a.b . , sjogren , w.l . & williams , j.g . 1998 , science , 281 , 1476 law , n.m . , mackay , c.d . & baldwin , j.e . 2006 , a&a , 446 , 739 . lebofsky , l.a . , feierberg , m.a . , tokunaga , a.t . , larson , h.p . & johnson , j.r . 1981 , icarus , 48 , 453 lipsky , yu.n . & pospergelis , m.m . 1966 , astronomicheskii tsirkular , 389 , 1 . lo , m.w . 2004 , in `` proc . internatl lunar conf . 2003 , ilewg 5 '' ( adv . in astronaut . , sci . & tech . 108 ) , eds . durst et al . ( univelt : sandiego ) , p. 214 . markov , m.n . , petrov , v.s . , akhmanova , m.v . & dementev , b.v . 1979 , in _ space research , proc . open mtgs . working groups _ ( pergamon : oxford ) , p. 189 . mcguire , r.e . 2006 `` space physics data facility '' - nasa goddard space flight center : http://lewes.gsfc.nasa.gov/cgi-bin/cohoweb/selector1.pl?spacecraft=omni moorman , b.j . , robinson , s.d . & burgess , m.m . 2003 , permafrost & periglac . , 14 , 319 . nishimura , j. , et al . 2006 , adv . space res . , 37 , 34 . nozette , s. et al . 1996 , science , 274 , 1495 . nozette , s. et al . 2001 , jgr , 106 , 23253 . ono , t. & oya , h. 2000 , earth plan . space , 52 , 629 . picardi , g. , et al . 2005 , science , 310 , 1925 . pieters , c.m . , et al . 2005 , in _ space resources roundtable viii _ , lun . plan . contrib . , 1287 , 73 . pieters , c.m . , et al . 2005 , http://moonmineralogymapper.jpl.nasa.gov/science/volatiles/ porcello , l.j , et al . 1974 , proc . ieee , 62 , 769 . ramanan , r.v . & adimurthy , v. 2005 , j. earth syst . rivkin , a.s . , howell , e.s . , britt , d.t . , lebofsky , l.a . , nolan , m.c . branston , d.d . 1995 , icarus , 117 , 90 rivkin et al . 2002 , asteroids iii , 237 ross , s.d . 2006 , am . , 94 , 230 . saal , a.e . , hauri , e.h . , rutherford , m.j . & cooper , r.f . 2007 , lun . plan . , 38 , 2148 . simpson , r.a . 1998 , in `` workshop on new views of the moon , '' eds . b.l . jolliff & g. ryder ( lpi : houston ) , p. 61 . stacy , n.j.s . 1993 , ph.d . thesis ( cornell u. ) . tapley , b.j . , et al . 2005 , j. geodesy , 79 , 467 . thompson , t.w . & campbell , b.a . 2005 , lun . plan . conf . , 36 , 1535 . tomanry , a.b . and crotts , a.p.s . 1996 , aj , 112 , 2872 . tubbs , r.n . 2003 , ph.d . thesis ( university of cambridge ) . vondrak , r.r . , freeman , j.w . & lindeman , r.a . 1974 , lun . plan . conf . , 5 , 2945 . watkins , m. , folkner , w.m . , nerem , r.s . & tapley , b.d . 2006 , in _ proc . grace science meeting , 2006 dec . 8 - 9 _ , in press ( http://www.csr.utexas.edu/grace/gstm/2006/a1.html ) . table 1 : summary of basic experimental / observational techniques detailed here .... all methods are earth - based remote sensing unless specified otherwise . -------------------------------------------------------------------------------- goal detection method channel advantages difficulties ------------- ----------------------- ------- -------------- --------------- map of tlp imaging monitor , entire optical schedulability nearside only , activity nearside , ~2 km resol . comprehensive ; limited resol . more sensitive than human eye polarimetric compare reflectivity in optical easy to requires use study of dust two monitors with schedule ; two monitors perpendicular polarizers further limits dust behavior changes in adaptive optic imaging , 0.95 " on demand " undemonstrated , small , active ~100 m resolution micron , given good depends on areas etc . conditions seeing ; covers ~50 km dia . max " lucky imaging , " 0.95 on demand low duty cycle , ~200 m resolution micron , given good depends on etc . conditions seeing hubble space telescope , 0.95 on demand currently ~100 m resolution micron , given advance unavailable ; etc . notice low efficiency clementine / lro/ 0.95 existing or limited epochs ; chandrayaan-1 imaging , micron , planned survey low flexibility ~100 m resolution etc . selene / chang'e-1 0.95 existing or limited epochs ; imaging , higher resol . micron , planned survey low flexibility etc . tlp spectrum scanning spectrometer nir , may be best requires alert map , then spectra taken optical method to find from tlp image during tlp event composition & monitor ; limit tlp mechanism to long events regolith nir hydration bands 2.9,3.4 directly probe requires alert hydration seen before / after tlp micron regolith / water from monitor , measurement in nir imaging chemistry ; flexible detect water scheduling scanning spectrometer 2.9,3.4 directly probe requires alert map , then spectra taken micron regolith / water from monitor , soon after tlp chemistry ; flexible detect water scheduling relationship simultaneous monitoring rn-222 refute / confirm optical monitor between tlps for optical tlps and by alpha & tlp / outgassing only covers & outgassing selene for rn-222 alpha optical correlation ; nearside ; more particles find gas loci monitors better subsurface penetrating radar ~430mhz directly find ice signal is water ice subsurface ice easily confused with existing with others technique penetrating radar from ~300mhz better resol . ; ice signal is lunar orbit can study easily confused sites of lower with others ; activity more expensive surface radar from > 1ghz better resol . ; redundant with lunar orbit study tlp site high resol . surface change imaging ? high resol . imagers at / near l1 , l2 optical map tlps with expensive , but tlp activity covering entire moon , greater resol . could piggyback map at 100 m resolution & sensitivity , communications entire moon network comprehensive two rn-222 alpha rn-222 map outgassing expensive ; even rn-222 alpha detectors in polar alpha events at full better response particle map orbits 90 degrees apart sensitivity w/ 4 detectors comprehensive two mass spectrometers ions & map outgassing expensive ; even map of outgas adjacent polar orbits neutral events & find better w/ more components composition spectrometers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in situ , surface experiments : we refer the reader to work in preparation by aeolus collaboration . abbreviations used : dia . = diameter , max = maximum , nir = near infrared , resol . = resolution _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .... 0.00 in figure 5 - a ) left : spectrum of an 8-arcmin slit intersecting aristarchus ( bright streak just above center ) and extending over oceanus procellarum , and covering wavelengths 5500 - 10500 , taken by the mdm 2.4-meter telescope ; * b ) * right : the residual spectrum once a model consisting of the outer product the one - dimensional average spectrum from figure 3a times the one - dimensional albedo profile from figure 3a . the different spectral reflectance of material around aristarchus is apparent ( at a level of about 7% of the initial signal ) , with r.m.s . deviations of about 0.5% , dominated by interference fringing in the reddest portion , which can be reduced . figure 6 - a ) left : a b - band image of the region around aristarchus ; b ) right : an image of aristarchus in a 3 - wide centered near 6000 , constructed by taking a vertical slice through figure 3a and other exposures from the same sequence of spectra scanning the surface . any such band between 5500 and 10500 can be constructed in the same manner , with resolution of about 1 km and 3 .

in paper i of this series , we show that transient lunar phenomena ( tlps ) correlate with lunar outgassing , geographically , based on surface radon release episodes versus the visual record of telescopic observers ( the later prone to major systematic biases of unspecified nature , which we were able to constrain in paper i ) . in paper ii we calculate some of the basic predictions that this insight implies , in terms of outgassing / regolith interactions . in this paper we propose a path forward , in which current and forthcoming technology provide a more controlled and sensitive probe of lunar outgassing . many of these techniques are currently being realized for the first time . given the optical transient / outgassing connection , progress can be made by earth - based remote sensing , and we suggest several programs of imaging , spectroscopy and combinations thereof . however , as found in paper ii , many aspects of lunar outgassing seem likely to be covert in nature . tlps betray some outgassing , but not all outgassing produces tlps . some outgassing may never appear at the surface , but remain trapped in the regolith . as well as passive remote sensing , we also suggest more intrusive techniques , from radar mapping to in - situ probes . understanding these volatiles seems promising in terms of their exploitation as a resource for human presence on the moon and beyond , and offers an interesting scientific goal in its own right . this paper reads , therefore , as a series of proposed techniques , some in practice , some which might be soon , and some requiring significant future investment ( some of which may prove unwise pending results from predecessor investigations ) . these point towards enhancement of our knowledge of lunar outgassing , its relation to other lunar processes , and an increase in our understanding of how volatiles are involved in the evolution of the moon . we are compelled to emphasize certain ground - based observations in time for the flight of _ selene , lro _ and other robotic missions , and others before extensive human exploration . we discuss how study of the lunar atmosphere in its pristine state is pertinent to understanding the role of anthropogenic volatiles , at times a significant confusing signal . 6.5 in 8.5 in 0.0 in 0.0 in

the ricci flow is an evolution equation for a curve of riemannian metrics on a manifold . in recent years , the ricci flow has proven to be a very important tool . many strong results , not only in riemannian geometry , have been proven by using this equation . the objective of this paper is to study the ricci flow for solvmanifolds whose lie algebra has an abelian ideal of codimension one and get similar results to those obtained by j. lauret in @xcite in the case of nilmanifolds . let @xmath1 be a solvmanifold , i.e. a simply connected solvable lie group @xmath2 endowed with a left - invariant metric @xmath3 assume that the lie algebra of @xmath2 has an abelian ideal of codimension one . consider the ricci flow starting at @xmath4 , that is , @xmath5 the solution @xmath6 is a left - invariant metric for all @xmath7 , thus each @xmath6 is determined by an inner product on the lie algebra . we will follow the approach in @xcite to study the evolution of these metrics by varying lie brackets instead of inner products . more precisely , let @xmath8 be a lie bracket on @xmath9 with an abelian ideal of codimension one . we may assume that @xmath8 is determined by @xmath10 where @xmath11 and @xmath12 is the abelian ideal , and so it will be denoted by @xmath13 each @xmath14 determines a riemannian manifold @xmath15 where @xmath16 is the simply connected lie group with lie algebra @xmath17 and @xmath18 is the left - invariant metric determined by @xmath19 the canonical inner product on @xmath20 every solvmanifold whose lie algebra has an abelian ideal of codimension one is isometric to some @xmath14 ( see section [ pre ] ) . by ( * theorem 3.3 ) , the ricci flow solution is given by @xmath21 , where @xmath22 is a family of lie brackets solving a ode called the bracket flow , and @xmath23 is the lie group isomorphism with derivative @xmath24 and @xmath25 satisfies @xmath26 in our case , we see that @xmath27 where @xmath28 is the solution to the following ode , @xmath29-\tfrac{1}{2}\tr(a)[a , a^{t } ] , \quad a(0)=a,\ ] ] and then we study the evolution of the matrix @xmath30 . the main results in this paper can be summarized as follows : * the ricci flow solution @xmath6 is defined for all @xmath31 where @xmath32 and if @xmath33 then @xmath6 is a type - iii solution ( see proposition [ a ] and proposition [ type3 ] ) . * the scaling - invariant functional @xmath34\|}{\|a(t)\|^2}$ ] is strictly decreasing unless @xmath35 is an algebraic soliton , in which case it is constant ( see lemma [ func ] ) . this happens precisely when @xmath30 is either normal or nilpotent of a special kind ( see proposition [ solmua ] ) . * for any sequence @xmath36 there exists a subsequence of @xmath37 which converges in the pointed topology to a flat manifold , up to local isometry ( see corollary [ subconv ] ) . * if @xmath38 ( i.e. @xmath39 unimodular ) , then @xmath40 converges to a matrix @xmath41 as @xmath42 ( see lemma [ cociente ] and remark [ nochaos ] ) . * for any sequence @xmath36 there exists a subsequence of @xmath43 which converges in the pointed topology to @xmath44 ( up to local isometry ) , which is an algebraic soliton . in addition , @xmath44 is non - flat , unless every eigenvalue of @xmath30 is purely imaginary ( see theorem [ convnorsol ] ) . * if @xmath39 admits a negatively curved left - invariant metric , then there exists @xmath45 such that @xmath6 is negatively curved for all @xmath46 ( see theorem [ to2 ] ) . this is not true in general for solvmanifolds ( see example [ ejsol ] ) . _ _ i wish to express my deep gratitude to my advisor , jorge lauret , for his invaluable guidance . i am also grateful to ramiro lafuente and roberto miatello for helpful observations . let @xmath47 be a riemannian manifold . the ricci flow starting at @xmath47 is the following partial differential equation : @xmath48 where @xmath6 is a curve of riemannian metrics on @xmath49 and @xmath50 the ricci tensor of the metric @xmath6 . a complete riemannian metric @xmath4 on a differentiable manifold @xmath49 is a ricci soliton if its ricci tensor satisfies @xmath51 where @xmath52 denotes the space of differentiable vector fields on @xmath49 and @xmath53 the usual lie derivative in the direction of the field @xmath54 equivalently , ricci solitons are precisely the metrics that evolve along the ricci flow only by the action of diffeomorphisms and scaling ( i.e. @xmath55 ) , giving geometries that are equivalent to the starting point , for all time @xmath7 ( see @xcite for more information about ricci solitons ) . [ type3 ] a ricci flow solution @xmath6 is said to be of type - iii if it is defined for @xmath56 and there exists @xmath57 such that @xmath58 where @xmath59 is the riemann curvature tensor of the metric @xmath60 we fix @xmath61 with @xmath62 an inner product on @xmath12 and we define @xmath63 @xmath64 and @xmath65 the adjoint representation of @xmath66 ( i.e. @xmath67 ) . then , @xmath68 acts on @xmath69 by @xmath70 each @xmath66 defines a lie group endowed with a left - invariant riemannian metric , @xmath71 where @xmath72 is the simply connected lie group with lie algebra @xmath73 endowed with the left - invariant riemannian metric determined by the inner product @xmath74 often , we will denote this metric by @xmath75 note that @xmath76 may be viewed as a metric on @xmath77 in fact , @xmath72 is diffeomorphic to @xmath78 geometrically , each @xmath79 determines a riemannian isometry @xmath80 by exponentiating the lie algebra isomorphism @xmath81 . thus the orbit @xmath82 parameterizes the set of all left - invariant metrics on @xmath83 [ solal ] let @xmath1 be a lie group with a left - invariant riemannian metric ; @xmath4 is called an algebraic soliton if @xmath84 where @xmath85 is the ricci operator of @xmath4 and @xmath86 is the lie algebra of @xmath87 any homogeneous simply connected algebraic soliton is a ricci soliton ( see ( * ? ? ? * proposition 3.3 ) ) . let @xmath1 be a simply connected lie group endowed with a left - invariant riemannian metric . then , if we fix @xmath62 an inner product on the lie algebra of @xmath88 @xmath1 is isometric to @xmath89 for some @xmath90 in this case , the equation of the ricci flow ( [ fr ] ) is equivalent to the following ordinary differential equation ( see ( * ? ? ? * section 3 ) ) : @xmath91 where @xmath92 and @xmath93 is the identity of @xmath83 in subsection [ vlb ] , we have observed that @xmath82 parameterizes the set of all left invariant riemannian metrics on @xmath94 then it is very natural to ask : how is the behavior of the ricci flow in @xmath69 ? [ bflg ] given @xmath95 the bracket flow starting at @xmath8 is the following ordinary differential equation : @xmath96 where @xmath97 , @xmath98 , @xmath99 . let us consider @xmath6 the ricci flow solution flow starting at @xmath76 , and @xmath22 the solution of the bracket flow starting at @xmath100 by @xcite , we know that @xmath6 and @xmath22 are related in the following way . [ rel ] ( * ? ? ? * theorem 3.3 ) there exists time - dependent diffeomorphisms @xmath101 moreover , if we identify @xmath102 , then @xmath103 can be chosen as the equivariant diffeomorphism determined by the lie group isomorphism between @xmath72 and @xmath104 with derivative @xmath105 , where @xmath106d@xmath107 is the solution to any of the following systems of ordinary differential equations : 1 . @xmath108 , @xmath109 2 . @xmath110 , @xmath109 the following conditions hold : * @xmath111 * @xmath112 . in this paper , theorem [ rel ] has only been stated in the case of lie groups , however , in @xcite it is stated and proved in the general homogeneous case . so , the ricci flow @xmath6 can be obtained from the bracket flow @xmath22 by solving ( 2 ) and applying part ( 3 ) . in the same way , we can obtain @xmath22 solving ( 1 ) and replacing in ( 4 ) . in particular , both flows are defined in the same interval of time . for more information , see @xcite . we now recall some results proved by j. lauret in @xcite about the ricci flow for simply connected nilmanifolds . @xcite[nilva ] let @xmath22 be the solution bracket flow starting at @xmath113 and @xmath6 the ricci flow starting at @xmath75 then * @xmath22 is defined for all @xmath114 * @xmath6 is a type - iii solution for a constant @xmath115 that only depends on the dimension @xmath116 * @xmath117 as @xmath118 moreover , @xmath119 converges in @xmath120 to the flat metric @xmath121 * @xmath122 converges in @xmath120 to an algebraic soliton @xmath123 uniformly on compact sets in @xmath124 as @xmath118 in this section , we study the bracket flow for a metric solvable lie algebra with an abelian ideal of codimension one . we consider @xmath125 with @xmath62 the canonical inner product on @xmath20 if the dimension of the lie algebra is @xmath126 then up to isomorphism , we can assume that the lie bracket has the following form with respect to the canonical basis @xmath127 @xmath128 where we think of an @xmath129 as an operator acting on @xmath124 the subspace generated by @xmath130 ( i.e. the codimension - one abelian ideal ) . from now on , we denote these algebras by @xmath131 or simply , @xmath132 if @xmath133 then the bracket flow starting at @xmath134 is given by @xmath135 @xmath136 where @xmath137 satisfies @xmath138-\tfrac{1}{2}\tr(a)[a , a^{t } ] , \quad a(0)=a_0.\ ] ] by using the formula for the ricci operator of a solvmanifold ( see for instance ( * ? ? ? * section 4 ) ) , we obtain that the ricci operator of @xmath139 with respect to the basis @xmath140 is represented by the matrix @xmath141-\tr(a)s(a ) \\ \end{array } \right),\ ] ] where @xmath142 is the symmetric part of the matrix @xmath30 and @xmath143 is the trace . then , @xmath144 e_i\\ & = & \left(-\tr(s(a)^{2})a+\tfrac{1}{2}[a,[a , a^{t}]]-\tfrac{1}{2}\tr(a)[a , a^{t}]\right)e_i , \end{array}\ ] ] and , on the other hand , we have that @xmath145 for all @xmath146 as @xmath147 so , @xmath148-\tfrac{1}{2}\tr(a)[a , a^{t}].\ ] ] then , this family of lie algebras is invariant under the bracket flow , which is equivalent to ( [ a ] ) . in addition , the maximal interval of time where @xmath149 exists is of the form @xmath150 for some @xmath151 since is an ode . so , given a matrix @xmath152 we have that the bracket flow starting at @xmath134 is equivalent to an evolution equation for a curve of matrices with initial condition @xmath153 in what follows , we will often think of the bracket flow as this evolution . [ flat ] note that the only fixed points of the system ( [ a ] ) are the skew - symmetric matrices , which are precisely the flat solvmanifolds of the form @xmath14 , since by ( [ ricmu_a ] ) they are precisely the ricci - flat ones ( see @xcite and @xcite ) . [ solmua ] for any @xmath133 the following conditions are equivalent : * @xmath134 is an algebraic soliton . * @xmath154 is either a normal matrix or @xmath154 is a nilpotent matrix such that @xmath155= c a_0,$ ] for some @xmath156 moreover , the evolution of the bracket flow is respectively given by @xmath157 assuming part ( i ) , we have two cases : * if the nilradical of @xmath134 has dimension @xmath158 then @xmath154 is a normal matrix ( see ( * ? ? ? * theorem 4.8 ) ) . * if the nilradical of @xmath134 has dimension @xmath126 then @xmath134 is nilpotent and so @xmath154 is a nilpotent matrix . in addition , from ( [ ricmu_a ] ) , we have that @xmath159 \\ \end{array } \right ) = \ricci_{\mu_{a_0}}= ci+d,\ ] ] and it follows that @xmath160 also , we know that @xmath161=-\ad_{\mu_{a_0}}(d(e_0)),$ ] so @xmath162 \\ \end{array } \right ) = [ \ad_{\mu_{a_0}}(e_0),\ricci_{\mu_{a_0 } } ] = [ \ad_{\mu_{a_0}}(e_0),d ] = -\lambda \ad_{\mu_{a_0}}(e_0 ) = -\lambda \left ( \begin{array}{cc } 0 & 0 \\ 0 & a_0 \\ \end{array } \right).\ ] ] conversely , if @xmath154 is a normal matrix , then @xmath134 is an algebraic soliton ( see ( * ? ? ? * theorem 4.8 ) ) and if @xmath154 is a nilpotent matrix which satisfies @xmath155= c a_0,$ ] then @xmath163 \\ \end{array } \right ) = \tfrac{c-\|a_0\|^2}{2}i+\left ( \begin{array}{cc } -\tfrac{c}{2 } & 0 \\ 0 & -\tfrac{c}{2}i+\tfrac{1}{2}\|a_0\|^2i+\tfrac{1}{2}[a_0,{a_0}^t ] \\ \end{array } \right),\ ] ] and it is easy to see that @xmath164 \\ \end{array } \right)$ ] is a derivation of @xmath165 and so ( i ) is proved . finally , if @xmath134 is an algebraic soliton , then the family @xmath166 is invariant under the flow . therefore , we have that * if @xmath154 is a normal matrix , then the bracket flow is equivalent to the following differential equation for @xmath167 @xmath168 and so the solution is @xmath169 * if @xmath154 is a nilpotent matrix , then the bracket flow is equivalent to @xmath170 and so the solution is @xmath171 the first natural question that arises is related with the maximal time interval of the solution @xmath172 an important point to observe here is that @xmath173 since @xmath174 always has non - positive scalar curvature ( see ( [ ricmu_a ] ) ) . [ a ] @xmath137 is always defined for all @xmath175 ( i.e. @xmath176 ) . by using ( [ a ] ) , we get @xmath177 \rangle- 2\langle a,\tfrac{1}{2}\tr(a)[a , a^{t}])\rangle.\ ] ] but since @xmath178\rangle=-\|[a , a^{t}]\|^{2}$ ] and @xmath179\rangle=0,$ ] it follows that @xmath180\|^{2 } \leq 0.\ ] ] therefore , @xmath181 decreases and so @xmath137 is defined for all @xmath182 as the solution remains in a compact subset . by theorem [ rel ] and the previous proposition , we obtain that the ricci flow starting at any of these solvmanifolds @xmath183 is defined for @xmath182 often called an immortal solution . in what follows , we introduce a positive , non - increasing function along the normalized bracket flow @xmath184 which is strictly decreasing unless @xmath185 is an algebraic soliton . the advantage of having this function lies in the fact that it will allow us to prove that for any sequence @xmath186 there exists a subsequence in which the normalized bracket flow always converges to an algebraic soliton . [ func ] let @xmath149 be the bracket flow starting at @xmath134 and set @xmath187 then @xmath188\|^2 $ ] is a positive , non - increasing function along the flow . moreover , @xmath189 is an algebraic soliton . we consider @xmath190 @xmath191\|^2.\ ] ] then @xmath192\|^2}{\|a\|^4}= \tfrac{1}{\|a\|^8 } ( \|a\|^4 \tfrac{d}{dt } \|[a , a^t]\|^2 - \|[a , a^t]\|^2 \tfrac{d}{dt}\|a\|^4).\ ] ] by using the bilinearity of the inner product and the lie bracket we obtain that @xmath193\|^2=-4\tr(s(a)^2 ) \|[a , a^t]\|^2 - 2 \|[a,[a , a^t]]\|^2,\ ] ] and from ( [ norma ] ) @xmath194\|^{2}.\ ] ] then , if we consider @xmath195 it follows that @xmath196\|^4- \|b\|^2 \|[b,[b , b^{t}]]\|^{2 } ) \leq 0,\ ] ] by using the cauchy - schwarz inequality . moreover , if there exists @xmath197 such that @xmath198 then the cauchy - schwarz equality holds and there exists @xmath199 such that @xmath200 = c b(t_0).\ ] ] we have two cases : * if @xmath201 then @xmath202=0,$ ] and so @xmath203b(t_0)^t)=0 $ ] , this implies that @xmath204\|^2=0 $ ] , i.e. @xmath205 is normal and @xmath206 is an algebraic soliton ( see proposition [ solmua ] ) . on the other hand , by using ( [ a ] ) and ( [ norma ] ) , it is easy to see that @xmath207 - \tr(a ) [ a , a^t ] + \tfrac{\|[a , a^t]\|^2}{\|a\|^2}a \right)\\ & = & \tfrac{\|a\|^2}{2 } \left ( [ b,[b , b^t ] ] - \tr(b ) [ b , b^t ] + \|[b , b^t]\|^2b \right ) , \end{array}\ ] ] so , @xmath208 for all @xmath209 since @xmath205 is a fixed point of ( [ anorm ] ) . it follows that @xmath210 for all @xmath211 * if @xmath212 then by using ( [ cse ] ) , we obtain that @xmath213 and @xmath214 since @xmath215b(t_0)^k)\\ & = & \tr([b(t_0)^k , b(t_0)][b(t_0),b(t_0)^t])\\ & = & 0 . \end{array}\ ] ] therefore , @xmath205 is a nilpotent matrix that satisfies ( [ cse ] ) , so by proposition [ solmua ] we have that @xmath206 is an algebraic soliton . in addition , @xmath205 is a fixed point of ( [ anorm ] ) , so , @xmath210 for all @xmath211 conversely , if @xmath216 is an algebraic soliton , then by using ( [ f ] ) , we have that @xmath217 [ notafunc ] let @xmath149 be the bracket flow starting at @xmath134 and set @xmath187 then for any sequence @xmath186 there exists a subsequence of @xmath43 converging in the pointed topology to an algebraic soliton @xmath218 every sequence @xmath219 has a convergent subsequence , i.e. after passing to a subsequence , @xmath219 converges to a matrix @xmath220 then @xmath221 is an algebraic soliton by lemma [ func ] , as @xmath222 is a fixed point of the flow . from now on , our purpose is to study the ode ( [ a ] ) . we emphasize that our aim is not to solve the ode , we are interested in understanding the qualitative behavior of the solution along the time , which is not trivial to predict even when @xmath223 is very small . in the next lemma we study how it evolves . [ ffc ] the bracket flow @xmath149 starting at @xmath134 has the form @xmath224 where @xmath225 is a positive , non - increasing , real valued function , and @xmath226 for each @xmath211 if @xmath227 , with @xmath228 a real function and @xmath226 , then @xmath229-\tr(a(t))s(a(t)))\varphi_t \\ \end{array } \right).\ ] ] the map @xmath230 given in part ( 2 ) of theorem [ rel ] has therefore the form @xmath231 and it follows from ( 4 ) of the same theorem that @xmath232 in addition , @xmath233 so , we have that @xmath234 is a positive , non - decreasing function . it follows that if @xmath235 then @xmath225 is a positive , non - increasing function . in what follows , @xmath135 @xmath236 will be the bracket flow solution starting at @xmath134 and we will denote it simply by @xmath172 [ spe ] assume that @xmath237 for some sequence @xmath238 then @xmath239 @xmath240 @xmath241 for some @xmath242 here @xmath243 denotes the unordered set of @xmath223 complex eigenvalues of the matrix @xmath244 we know that @xmath245 by lemma [ ffc ] , therefore @xmath246 then , as @xmath237 we have that @xmath247 where @xmath248 ( recall that from lemma [ ffc ] , @xmath225 is a positive , non - increasing function ) . [ sa ] @xmath249 is strictly decreasing if @xmath154 is not skew - symmetric . moreover , @xmath250 as @xmath251 . recall that @xmath252 and so @xmath253 then , as in proposition [ a ] we have already studied @xmath254 we will only analyze @xmath255 by using ( [ a ] ) , we obtain @xmath256 therefore , it follows from ( [ norma ] ) and ( [ tra2 ] ) that @xmath257\|^{2 } \leq 0,\ ] ] and if there exists @xmath258 such that @xmath259 then @xmath260 is a skew - symmetric and so @xmath261 for all @xmath211 conversely if @xmath154 is skew - symmetric , we have that @xmath262 so , @xmath263 is strictly decreasing if @xmath154 is not skew - symmetric . in addition , @xmath264 and then @xmath263 is dominated by @xmath265 which is a solution of @xmath266 therefore @xmath267 as @xmath268 recall that if @xmath16 is the simply connected solvable lie group with lie algebra @xmath269 then @xmath18 denotes the left - invariant riemannian metric on @xmath16 such that @xmath270 , where @xmath93 is the identity of the group @xmath16 and @xmath271 is the canonical inner product on @xmath9 . [ subconv ] if @xmath272 as @xmath273 then @xmath274 is a skew - symmetric matrix and for any sequence @xmath186 there exists a subsequence of @xmath37 which converges in the pointed topology to a manifold locally isometric to @xmath275 , which is flat . by proposition [ sa ] , we know that @xmath276 as @xmath277 , therefore @xmath274 is skew - symmetric and then @xmath275 is flat ( see remark [ flat ] ) . finally , since @xmath278 by ( * ? ? ? * corollary 6.20 ) , for any sequence @xmath186 there exists a subsequence of @xmath279 which converges in the pointed topology to a manifold locally isometric to @xmath275 , which is flat , as shown above . in the following proposition , we prove that under an additional hypothesis , the convergence is actually smooth . [ conva ] if @xmath280 and @xmath281 as @xmath273 then @xmath282 smoothly on @xmath20 for each @xmath283 we define @xmath284 by @xmath285 where @xmath286 is the lie exponential of @xmath287 let @xmath288)$ ] be the linear transformation such that @xmath289 and @xmath290 , @xmath291 where @xmath292 and @xmath293 , @xmath294 . then @xmath295 is an isomorphism of lie algebras . therefore , under the isomorphism @xmath295 , we have that @xmath296 where @xmath297 is the exponential function of matrices . then @xmath298 but @xmath299 and @xmath300 , therefore @xmath301 it is easy to see that if @xmath302 or @xmath303 then @xmath304 is a diffeomorphism . so , as @xmath305 we have that @xmath306 and @xmath307 or @xmath308 by proposition [ spe ] , and therefore we have that @xmath309 smoothly on @xmath12 ( see ( * ? ? ? * remark 6.11 ) ) . in particular , if @xmath134 is completely solvable ( @xmath310 for all @xmath311 ) , then the convergence is smooth . this also follows by using proposition [ spe ] and ( * ? ? ? * corollary 6.20 ) , since @xmath149 is completely solvable for all @xmath211 recall that if the norm of the riemann tensor decays at least as fast as @xmath312 where @xmath313 is a constant , then the solution of the ricci flow is a type - iii solution ( see definition [ type3 ] ) . [ type3 ] for every @xmath134 with @xmath314 , the ricci flow @xmath6 with @xmath315 is a type - iii solution , for some constant @xmath316 that only depends on the dimension @xmath317 in proposition [ a ] , we proved that @xmath149 is defined for @xmath114 we observe that , by using ( [ tra2 ] ) , if @xmath314 then @xmath318 for all @xmath211 further , in proposition [ sa ] , we prove that @xmath319 therefore , by using ( [ trsa ] ) , we have that @xmath320 where @xmath313 is the maximum of the continuous function @xmath321 restricted to the unit sphere of @xmath322 the question that naturally arises is whether the flow converges . the following section is devoted to study such question . in this section , we analyze the @xmath0-limit of the bracket flow @xmath149 ( i.e. the set of limit points of sequences under the bracket flow ) . to do this , we consider two cases : when @xmath323 ( i.e. , @xmath134 is unimodular ) and when @xmath324 let us first suppose that @xmath325 we consider the functional @xmath326\|^{2},$ ] which is , in fact , the square norm of the moment map of the conjugation action of the real reductive group @xmath327 on the vector space @xmath328 and we compute its gradient : @xmath329\|^{2}\\ & = & \tfrac{d}{dt}|_{t=0 } \la [ a+tb , a^t+tb^{t } ] , [ a+tb , a^t+tb^{t}]\ra\\ & = & 2 \la [ a , a^t ] , \tfrac{d}{dt}|_{t=0 } [ a+tb , a^t+tb^{t}]\ra\\ & = & 2 \la [ a , a^t ] , [ b , a^t]+[a , b^t ] \ra\\ & = & 4 \la [ a , a^t ] , [ b , a^t ] \ra = -4 \la [ a,[a , a^t ] ] , b \ra . \end{array}\ ] ] thus , @xmath330 $ ] and the negative gradient flow of @xmath331 is given by @xmath332.\ ] ] observe that @xmath333 is a decreasing function . indeed , @xmath334,\bar a \ra = -8 \|[\bar a,\bar a^{t}]\|^{2},\ ] ] as @xmath335 , \bar a \rangle = -\|[\bar a,\bar a^{t}]\|^{2}.$ ] so , @xmath336 has a limit point @xmath337 and then we have that there exists the limit of @xmath338 as @xmath42 and it is unique ( see ( * ? ? ? * introduction ) ) . in addition , if @xmath339 we have two cases : * if @xmath340 then @xmath341 exists and @xmath342 * if @xmath343 then by ( * ? ? ? * theorem 7.1 ) , @xmath341 exists . if @xmath154 is nilpotent , then @xmath134 turns out to be nilpotent and so the bracket flow starting at @xmath134 has been studied in @xcite ( see theorem [ nilva ] ) . therefore , we assume that @xmath154 is not nilpotent . [ cociente ] assume that @xmath323 and @xmath154 is not nilpotent . let @xmath149 be the bracket flow starting at @xmath134 and let @xmath344 be the negative gradient flow ( [ abarra ] ) starting at @xmath153 then the limit of @xmath345 exists and @xmath346 we prove that , up to scaling and reparameterization of the time , the bracket flow @xmath137 starting at @xmath154 is @xmath347 the solution of ( [ abarra ] ) starting at @xmath152 i.e. we want to show that there exist @xmath348 and @xmath349 such that @xmath350 let @xmath348 and @xmath349 be solutions of the following system of differential equations with initial conditions : @xmath351 it is easy to see that @xmath348 and @xmath349 are defined for all @xmath209 and with a simple calculation it is easy to verify that @xmath352 is a solution of the equation ( [ a ] ) , therefore by uniqueness @xmath353 if @xmath354 then @xmath355 we suppose that @xmath356 , @xmath357 as @xmath358 , then @xmath359 this implies that @xmath360 is an algebraic soliton , since it is the limit of a normalized bracket flow ( see ( * ? ? ? * proposition 4.1 ) ) . as , @xmath154 is not nilpotent and @xmath361 is conjugated to @xmath152 for each @xmath209 we have then @xmath360 is normal ( see proposition [ solmua ] ) , i.e. @xmath362 is normal . so , @xmath363 for all @xmath364 by ( [ abarra ] ) . therefore , @xmath365 as was to be shown . [ nochaos ] it follows from lemma [ cociente ] and ( * ? ? ? * section 7 ) that if @xmath134 is unimodular , i.e. @xmath366 then the @xmath0-limit of @xmath367 is a single point . [ trneq0 ] if @xmath368 then @xmath369 as @xmath370 we know that @xmath371 by lemma [ ffc ] , therefore @xmath372 if @xmath373 , then @xmath374 so , as @xmath368 we have that @xmath375 on the other hand , @xmath376 and so @xmath377 then @xmath378 since @xmath274 is a skew - symmetric matrix . by using the two previous lemmas , we can prove the following theorem , which provides information about the @xmath0-limit of @xmath165 for any @xmath379 the @xmath0-limit of @xmath134 is a single point , for any @xmath379 by lemma [ trneq0 ] , we have that if @xmath368 then @xmath380 as @xmath118 if @xmath323 , we know by remark [ nochaos ] that @xmath381 then , we have that @xmath382 as @xmath370 indeed , the norm of @xmath137 decreases and therefore @xmath383 if @xmath384 then @xmath385 and if @xmath386 we have that @xmath387 which completes the proof . all results obtained so far can be summarized in the following theorem . [ summ ] given @xmath388 consider the bracket flow @xmath149 starting at @xmath134 and @xmath6 the ricci flow starting at @xmath389 . * @xmath6 is defined for @xmath31 where @xmath390 * the @xmath0-limit of @xmath134 is a single point . * for any sequence @xmath36 there exists a subsequence of @xmath37 which converges in the pointed topology to a manifold locally isometric to @xmath391 , which is flat . * if @xmath305 then @xmath309 smoothly on @xmath78 * if @xmath392 the ricci flow @xmath6 with @xmath315 is a type - iii solution , for some constant @xmath316 that only depends on the dimension @xmath317 [ asecir ] let @xmath393 it is easy to see that the family of matrices of this kind is invariant under the flow ( [ a ] ) , which is equivalent to the following ode system for the variables @xmath394 @xmath395 @xmath396 [ fcxy ] the phase plane for this system is displayed in figure [ figure 1 ] , as computed in maple . it is easy to see that it is enough to assume @xmath397 since if @xmath398 is the solution starting at @xmath399 then @xmath400 is the solution starting at @xmath401 regarding the interval of definition , the solutions remain in a compact subset and so they are defined in @xmath402 the solutions converge to the points @xmath403 which are precisely the fixed points of the system and correspond to skew - symmetric matrices ( which in turn correspond to flat metrics ) . also , we observe that points of the form @xmath404 @xmath405 and @xmath406 correspond to algebraic solitons ( they are symmetric or special nilpotent matrices ) . despite the fact that the solutions in the upper half - plane converge to @xmath407 we can see from the figure that they are approaching the soliton line @xmath408 , so considering a suitable normalization we may be able to obtain convergence of those solutions to a non - flat algebraic soliton . this will be the topic of the next section . according to theorem [ summ ] ( iii ) , for any sequence @xmath186 there exists a subsequence in which the ricci flow converge in the pointed topology to a flat manifold . in order to avoid this type of convergence and get a more interesting limit , we consider different normalizations of the flow . in this section , we study the normalized bracket flow by the bracket norm , i.e. if @xmath149 is the bracket flow starting at @xmath165 we will study @xmath409 we use the positive , non - increasing function obtained in section [ nue ] to determine which limits correspond to flat manifolds . before stating the theorem of convergence , we demonstrate the following technical lemma . from now on , let @xmath410 [ derivadas ] the following evolution equations along the normalized flow by the bracket norm hold : * @xmath411\|^2 \tr(b),$ ] * @xmath412\|^2\tr(b^2).$ ] to prove ( i ) , we use ( [ a ] ) and ( [ norma ] ) . part ( ii ) follows from ( [ tra2 ] ) and ( [ norma ] ) . [ convnorsol ] for any sequence @xmath36 there exists a subsequence of @xmath43 converging in the pointed topology to an algebraic soliton @xmath413 moreover , the following conditions are equivalent : * @xmath414 * @xmath44 is flat . as @xmath415 every sequence has a convergent subsequence , i.e. , @xmath219 converge to @xmath41 which is an algebraic soliton ( see corollary [ notafunc ] ) . by using ( [ fc ] ) , we have that @xmath416 if @xmath417 then @xmath418 and so by proposition [ derivadas ] ( ii ) , we have that @xmath419 for all @xmath209 and @xmath420 is a decreasing function . it follows that @xmath421 and then @xmath222 is normal , as @xmath222 is an algebraic soliton ( see proposition [ solmua ] ) . so , by ( [ normspe ] ) , we have that @xmath422 and so @xmath222 is a skew - symmetric matrix . conversely , if @xmath222 is skew - symmetric , then @xmath422 , so , by using ( [ normspe ] ) , we have that @xmath414 here again , we wonder ourselves what happens with the @xmath0-limit of @xmath409 recall that in section [ punlim ] we saw that if @xmath366 then the @xmath0-limit of @xmath345 is a single point . in the following proposition we analyze the case @xmath324 [ omelim ] if @xmath423 and @xmath424 for some sequence @xmath36 then the @xmath0-limit of @xmath345 is contained in @xmath425 let @xmath154 be such that @xmath426 and we suppose that @xmath427 and @xmath428 we want to see that @xmath429 and @xmath430 are conjugate by an orthogonal matrix . * if @xmath431 then @xmath432 and by proposition [ derivadas ] ( i ) , @xmath433 and therefore @xmath434 is a decreasing function and it follows that @xmath435 for all @xmath211 * if @xmath436 then @xmath437 and by proposition [ derivadas ] ( i ) , @xmath438 and therefore @xmath434 is an increasing function and it follows that @xmath439 for all @xmath211 then , @xmath440 and @xmath441 furthermore , the function @xmath434 is either increasing or decreasing . so , @xmath442 from this and ( [ fc ] ) it follows that @xmath443 and @xmath444 finally , we observe that @xmath429 and @xmath430 are normal matrices , since @xmath445 and @xmath446 are algebraic solitons ( see corollary [ notafunc ] ) , and so , @xmath429 and @xmath430 are normal or nilpotent ( see proposition [ solmua ] ) . as @xmath440 and @xmath447 they are not nilpotent matrices . then , we have two normal matrices with the same spectrum , from which it follows that they are conjugate by an orthogonal matrix ( see @xcite ) . in this section , we are interested in how the curvature evolves along the ricci flow . we define the sectional curvature @xmath448 of @xmath449 a lie algebra endowed with an inner product , as the sectional curvature of @xmath450 where @xmath2 is the simply connected lie group with lie algebra @xmath86 and @xmath4 is the left - invariant metric in @xmath2 such that @xmath451 in the case of @xmath452 we simply denote it by @xmath453 we say that a riemannian manifold has negative curvature , and denote it by @xmath454 if all sectional curvatures are strictly negative . next , we enunciate two results proved by heintze in @xcite . theorems [ solk ] and [ adneg ] give necessary and sufficient conditions for certain solvable lie algebras with an inner product to have negative curvature and for a solvable lie algebra to admit an inner product with negative curvature , respectively . * theorem 1)[solk ] let @xmath455 be a solvable lie algebra with an inner product such that the derived algebra is abelian ( i.e. , @xmath456 $ ] abelian ) . then @xmath457 if and only if the following conditions hold : * @xmath458 * there exists a unit vector @xmath459 orthogonal to @xmath460 such that @xmath461 is positive definite , where @xmath462 is the symmetric part of @xmath463 * if @xmath464 is the skew - symmetric part of @xmath465 , then @xmath466|_{\ggo'}$ ] is also positive definite . [ notathm2 ] we observe that in the case of @xmath283 the assumption that the derived algebra is abelian is always true . furthermore , @xmath467 if and only if conditions ( a ) - ( c ) hold . if in addition @xmath30 is normal and invertible , then @xmath467 if and only if ( b ) holds , since condition ( a ) is satisfied as @xmath30 is invertible and condition ( c ) follows from ( b ) . [ to2 ] let @xmath473 be a solvable lie group that admits a left - invariant metric with negative curvature . if @xmath149 is the bracket flow starting at @xmath165 then there exists @xmath474 such that @xmath475 for all @xmath476 it is sufficient to prove that the theorem holds for @xmath477 i.e. there exists @xmath197 such that @xmath478 for all @xmath479 indeed , for each @xmath209 @xmath149 and @xmath480 differ only by scaling . assume that , after passing to a subsequence , @xmath219 converges to @xmath41 as @xmath483 then , arguing as in proposition [ omelim ] , we have that @xmath222 is normal and @xmath484 so , either @xmath485 or @xmath486 then @xmath487 is either positive or negative definite . it follows by remark [ notathm2 ] that @xmath488 thus , there exists @xmath489 such that @xmath490 for all @xmath491 finally , there must exist @xmath258 such that @xmath478 for all @xmath492 otherwise we would be able to extract a convergent subsequence @xmath493 whose sectional curvatures are not strictly negative , and this contradicts the previous paragraph . [ ejsol ] we consider @xmath494 defined as follows : @xmath495 and @xmath62 the inner product for which @xmath496 is an orthonormal basis . by * theorem 4.8 ) , we know that @xmath494 is an algebraic soliton if and only if @xmath497 we consider the @xmath498-dimensional plane @xmath499 and we compute its sectional curvature : @xmath500 so , @xmath501 we observe that if @xmath502 then @xmath503 and so @xmath504 is a matrix such that @xmath505 then , theorem [ adneg ] said that if @xmath502 then @xmath506 with @xmath507 admits an inner product with negative curvature . on the other hand , since @xmath494 is an algebraic soliton , if @xmath22 is the bracket flow starting at @xmath508 then @xmath509 has planes with curvature bigger than or equal to zero . let @xmath455 be a solvable lie algebra with an inner product such that ( a ) - ( c ) of the theorem [ solk ] hold . then , we have a orthogonal decomposition @xmath510}.$ ] for @xmath511 let @xmath512 be the lie algebra with the same inner product that @xmath513 but with the following modification in the lie bracket @xmath514_{\alpha}:= \alpha [ a_0,x ] , \mbox { para todo } x \in \ggo'=\ggo_{\alpha}'.\ ] ] ( * ? ? ? * theorem 2)[solub ] let @xmath455 be a solvable lie algebra with an inner product and assume that ( a)-(c ) hold . then there exists @xmath515 such that @xmath512 has negative curvature for all @xmath516 we return to example [ ejsol ] . let @xmath517 be fixed and we consider the bracket flow @xmath22 starting at @xmath518 then @xmath22 is given by @xmath519 with @xmath520 and @xmath521 that satisfy the following differential equations : @xmath522 where @xmath523 furthermore , solving the equations we obtain that @xmath524 and @xmath525 clearly , in this case , the bracket flow converge to a flat metric , but for fixed @xmath209 we have that @xmath526 then , @xmath527 further , @xmath528 so , if @xmath502 there exists @xmath258 such that @xmath529 let @xmath517 be such that @xmath502 and we consider @xmath530 then @xmath531 is a solvable lie algebra with an inner product that satisfies ( a ) - ( c ) . by theorem [ solub ] , we know that there exists @xmath515 such that @xmath532 has negative curvature . then , @xmath533 has a negative curvature . on the other hand , we know that if @xmath22 is the bracket flow starting at @xmath534 there exists @xmath258 such that @xmath535 @xmath509 has planes with curvature bigger than or equal to zero .

in this paper , we study the ricci flow of solvmanifolds whose lie algebra has an abelian ideal of codimension one , by using the bracket flow . we prove that solutions to the ricci flow are immortal , the @xmath0-limit of bracket flow solutions is a single point , and that for any sequence of times there exists a subsequence in which the ricci flow converges , in the pointed topology , to a manifold which is locally isometric to a flat manifold . we give a functional which is non - increasing along a normalized bracket flow that will allow us to prove that given a sequence of times , one can extract a subsequence converging to an algebraic soliton , and to determine which of these limits are flat . finally , we use these results to prove that if a lie group in this class admits a riemannian metric of negative sectional curvature , then the curvature of any ricci flow solution will become negative in finite time .

the optical spectra of fullerene superconductors in the normal state were found to exhibit some unusual features @xcite . the optical conductivity , @xmath0 , deviates considerably from the simple drude behavior expected for conventional metals : the spectral weight of the drude peak is reduced by about an order of magnitude and transfered to a mid - infrared ( mir ) region around @xmath3 ev . this suggests that the strong interaction effects due to the coulomb and electron - phonon interactions should be important in the optical spectra of the fullerenes . understanding this unusual behavior in the optical conductivty , therefore , could reveal important information about the fullerenes and contribute to understanding other physical properties of the material . the optical conductivity @xmath0 represents the rate at which electrons absorb the incident photons at energy @xmath6 , and is a useful probe in determining electronic characteristics of the material under study . for an ideal free electron gas , where the interactions between the electrons , and between the electrons and phonons are neglected , and the impurity scattering rate @xmath7 , @xmath0 collapses to a delta - function , @xmath8 , where the coefficient @xmath9 represents the total spectral weight . in this case , the optical conductivity sum rule , @xmath10 , where @xmath11 is the density and @xmath12 is the mass of the electrons , is exhausted entirely by the delta function drude contribution alone . when the material becomes dirtier , the drude peak of the optical conductivity acquires the lorentzian shape with the width of @xmath13 . the conductivity sum rule is still exhausted by the drude part alone when only the impurity scatterings are present in the system . the total weight @xmath9 , however , can change as the inpurity scatterings or other interactions are introduced when we consider a finite bandwidth , because the projection to a restricted basis set disregards all excitations to higher energy than the bandwidth . when other interactions are present , the free - carrier drude weight is reduced by the quasiparticle renormalization factor @xmath14 such that @xmath15 , and the missing spectral weight from the drude part is transfered to a higher energy region of @xmath0 reflecting the excitation of incoherent scatterings . the experimentally measured @xmath0 in the normal state @xmath16 shows a remarkable reduction of drude weight and , concomitantly , a pronounced mir absorption below the inter - band absorption peak : degiorgi @xmath17 found a pronounced mir peak around 0.06 ev and analysized that the drude weight is reduced to about @xmath18 of the total intra - band spectral weight @xcite , while iwasa @xmath17 observed the mir absorption peak around 0.4 ev and determined that the drude weight is reduced to about 0.6 of the total intra - band spectral weight @xcite . although their results show somewhat different drude weight and mir absorption energy each other , the pronounced suppression of the drude weight and the accompanying mir absorption imply strong electron - phonon and/or electron - electron interactions in this material . in order to understand this unusual feature in @xmath0 of doped fullerenes , gunnarsson @xmath19 studied the effects of the electron - phonon interaction on @xmath0 assuming that the migdal theorem is valid @xcite . they showed that the electron - phonon interaction leads to a narrowing of the drude peak by the factor @xmath20 , where @xmath21 is the dimensionless electron - phonon coupling constant , and a transfer of the depleted drude weight to a mir region at somewhat larger energies than the phonon energy . their results , however , are far from sufficient to describe experimental observations . therefore , they hinted that the coulomb interaction between conduction electrons , which is neglected in their study , could lead to futher reduction of the drude weight and more pronounced mir absorption . on the other hand , one of the present authors recently found , by studying the nmr coherence peak supression in the fullerene superconductors , that the coulomb interaction between conduction electrons , characterized by @xmath22 , where @xmath23 is the effective coulomb interaction and @xmath24 is the density of states ( dos ) at the fermi level , should be included in addition to the electron - phonon interaction to understand the various experimental observations in fullerenes in a coherent way @xcite . we , therefore , included the electron - electron as well as electron - phonon interactions at the presence of the impurity scatterings in the present paper , to better understand the experimentally obsered unusual features in the optical spectra of the fullerene superconductors in the normal state . for fullerene superconductors , the fermi energy @xmath25 ev and the average phonon frequency @xmath26 ev , where @xmath27 is the bandwidth . therefore , @xmath28 for fullerenes unlike conventional metals , where @xmath29 . when @xmath28 , the phonon vertex correction becomes important because the migdal theorem does not hold @xcite , and the frequency dependence of the effective coulomb interaction , @xmath30 , should be considered because the frequency scale at which @xmath31 varies is comparable with that of electron - phonon interaction @xcite . in this present work , concerned with the effects of the coulomb and electron - phonon interactions on the optical spectra in the narrow band fullerene superconductors , the vertex correction is incorporated in calculating the electron self - energy @xcite . the coulomb interaction , modelled in terms of the onsite hubbard repulsion , is included on an equal footing with the electron - phonon interaction , and considered fully self - consistently in calculating the effective electron - electron interaction @xcite . the effective electron - electron interaction becomes frequency dependent through the screening . the impurity effects are included with the @xmath32-matrix approximation . through the relation ( ip ) = g^-1(ip ) - g_0 ^ -1(ip ) , [ s1 ] one obtain the electron self - energy @xmath33 in the mastubara frequency , which gives @xmath34 in the real frequency after the analytic continuation . @xmath35 and @xmath36 are , respectively , the bare and renormalized electron green s functions . @xmath34 or @xmath37 , where the renormalization function @xmath37 is given by @xmath38 , defines the single - particle green s function of an interacting system as g^-1=-_k-()=z()-_k , [ s2 ] where @xmath39 is the electron energy measured from the chemical potential , @xmath40 . then , the optical conductivity can be obtained by calculating the current - current correlation function , @xmath41 , using the renormalized green s function obtained from solving eq.@xmath42 ( [ s1 ] ) self - consistently . the calculated optical conductivity shows a strong reduction of drude weight and a broad mir absorption , although the mir feature around 0.06 ev is less pronounced and broader compared with experimental observations . this paper is organized as follows : in the following section , we present the eliashberg - type formalism in the matsubara frequency to calculate the renormalized green s function with the impurity , electron - phonon , and coulomb interactions included self - consistently . we then describe the analytic continuation procedure to obtain the renormalization function @xmath37 in the real frequency . the optical conductivity calculated with the renormalized green s function is presented in sec.@xmath42 iii . we will discuss how the drude part and the mir absorption of @xmath0 are affected as the impurity scattering rate , the electron - phonon and electron - electron interactions are varied . these result will then be compare with the experimental observations . finally , sec.@xmath42 iv is for the summary and some concluding remarks . the optical conductivity is calculated from the current - current correlation function , @xmath43 , as @xmath44 @xcite . we use the approximation where the electron self energy is momentum independent . in this case , it can be shown that the vertex correction in the current - current correlation function vanishes for @xmath45 @xcite . this leads to @xmath46 where @xmath47 and @xmath48 are , respectively , fermion and boson mastubara frequencies , where @xmath49 is the temperature , @xmath12 and @xmath11 are the integers . @xmath50 , and @xmath51 is the volume . the evaluation of eq.@xmath42 ( [ pi ] ) using eq.@xmath42 ( [ s2 ] ) produces ( i)=_ip in the mastubara frequency . after performing the analytic continuation of @xmath52 , to the real frequency , the optical conductivity is given by ( ) & = & _ -^ d + & & re , [ rsigma ] where @xmath53 is the fermi distribution function , and @xmath54 $ ] . the finite conduction bandwidth @xmath27 with a constant dos is explicitly considered through the factor of @xmath55 , which is @xmath56 for the usual case of infinite bandwidth metal . in order to calculate the optical conductivity from eq.@xmath42 ( [ rsigma ] ) we need @xmath37 which defines single - particle interacting green s function @xmath57 . this can be obtained by solving eq.@xmath42 ( [ s1 ] ) self - consistently . the electron self - energy is obtained by calculating the exchange diagram of the renormalized electron green s function and the effective electron - phonon and coulomb interactions with the vertex correction included via the method of nambu . the coulomb interaction , modelled in terms of the onsite hubbard repulsion for simplicity , is included on an equal footing with the electron - phonon interaction . the impurity effects are included with the @xmath32-matrix approximation . the eliashberg - type equation can be written in the mastubara frequency as @xmath58 2\theta_m \gamma + \frac{1}{\pi \tau } \theta_n , \label{eli}\end{aligned}\ ] ] where @xmath59 , and @xmath60}$ ] is the electron - phonon interaction kernel . @xmath61 and @xmath62 are , respectively , the interactions in the charge and spin channels due to the hubbard repulsion . they are determined self - consistently as _ ch(k ) & = & un_f \ { - _ n + _ n^2 } , [ lch ] + _ sp(k ) & = & un_f \ { + _ n + _ n^2 } , where @xmath63 is the dimensionless susceptibility given by _ n(k ) = _ l_l _ l+k . the @xmath64 on the right hand side of eq.@xmath42 ( [ eli ] ) represents the vertex correction satisfying the ward - identity@xcite . when we neglect the vertex correction , @xmath65 . if we assume a weak frequency dependence of @xmath64 , the vertex function @xmath64 reduces to @xmath66 . in this work , we treat the vertex correction exactly , and @xmath64 is given by = . solving eq.@xmath42 ( [ eli ] ) self - consistently yields @xmath67 in the mastubara frequency . in order to calculate @xmath0 , analytic continuation of @xmath68 should be performed to get @xmath37 in real frequency . the numerically exact analytic continuation of standard eliashberg equation is usually performed by the iterative method developed by marsiglio , shossmann , and carbotte ( msc ) using a mixed - representation @xcite . but when we include the vertex function exactly , the msc method can not be applied because it needs a specific form of equation . here , in order to consider vertex correction exactly , we do the alanytic continuation by employing the iterative method extended by takada @xcite . in this case , eq.@xmath42 ( [ eli ] ) is transformed to a mixed representation as fallow : z ( ) & = & ( ) + _ 0^dp ( ) [ n_b ( ) + n_f(+ ) ] + & & g(+ ) + & + & [ n_b ( ) + n_f(- ) ] g(- ) + & & , [ elir ] where ( ) & = & 1 + _ m_0^dp ( ) ( - ) + & & g(ip_m ) + , + p ( ) & = & -im ( ) + ( ) & = & _ ch ( ) - _ ph ( ) - _ sp ( ) + g(ip_m ) & = & 2(ip_m ) , g^r ( ) = 2i ( ) . @xmath69 of eq.@xmath42 ( [ elir ] ) represents the renormalization function obtained by substituting @xmath70 to @xmath71 @xmath72 the frequency summation . the second term is the correction to @xmath69 to yield the correct retarded renormalization function @xmath37 one would have obtained if the analytic continuation were performed @xmath73 the frequency summation . putting the solutions of eq.@xmath42 ( [ eli ] ) , @xmath67 , into the @xmath74 eq.@xmath42 ( [ elir ] ) yields a self - consistent eliashberg - type equation in the real frequency . then , @xmath37 can be obtained by computing iteratively eq.@xmath42 ( [ elir ] ) . in order to model fullerene superconductors , three truncated - lorentization functions were used to represent @xmath75 as follow @xcite : ^2 f ( ) & = & _ = 1 ^ 3 _ ^2f _ ( ) , + f _ ( ) & = & \ { l , for |- _ |_c , + 0 , otherwise , . [ alpha ] where @xmath76 is the truncated lorentizian centered at @xmath77 with the width of @xmath78 , @xmath79 is the cutoff frequency of @xmath80 , and @xmath81 is normalization constant such that @xmath82 . various theoretical and experimental estimates do not agree well each other in terms of distribution of coupling strength @xmath83 among different modes . these estimates show , however , that the phonon frequency derived from intramolecular @xmath84 and @xmath85 modes are distributed over @xmath86 ev with the total @xmath21 in the range of @xmath87 ev . in view of this , we represent the phonon modes with three groups centered around @xmath88 ev , and @xmath89 , respectively , for @xmath90 . note that @xmath91 . the @xmath92 sets the strength of @xmath75 and @xmath93 is independent of @xmath92 . for infinite bandwidth superconductors , @xmath21 is equal to @xmath92 in the limit @xmath94 . for a finite bandwidth system , however , @xmath21 is reduced from @xmath92 because the available states to and from which quasiparticles can be sccattered are restricted as the bandwidth is reduced . the self - consistent equation of eq.@xmath42 ( [ s1 ] ) is solved numerically as described in the previous section to obtain @xmath37 . then , the optical conductivity is calculated from the eq.@xmath42 ( [ rsigma ] ) . fig.@xmath42 1 shows the optical conductivity @xmath0 as @xmath21 is varied when the coulomb interaction @xmath23 is set to 0 for a reference . here , the fermi energy @xmath95 , temperature @xmath49 and impurity scattering rate @xmath13 are set to @xmath96 ev , respectively . this result shows quite a similar behavior to van den brink @xmath17 calculation . as @xmath21 is increased , the width of drude peak becomes narrower and it s weight is transferred to a mid - infrared spectrum . however , the reduction of drude weight is less than the factor of ( @xmath97 ) , because of the finite bandwidth . the inset shows a mir absorption spectra obtained by extracting drude part from the total optical conductivity . in determining the drude weight , fitting procedure was carefully employed and confirmed by examining zero frequency extrapolation in the mastubara frequency which is proposed by scalapino @xmath98 @xcite . the three lorentizian peaks of @xmath99 in the electron - phonon paring kernel are attributted to the development of these mir peaks . but , the mir peaks are broadened and move to slightly high frequencies . fig.@xmath42 2 shows the mir absorption due to the coulomb interaction . the mir part is also extracted by fitting as shown in the inset . in order to focus on how @xmath23 affects the total optical conductivity , @xmath21 is set to 0 . @xmath95 , @xmath49 and @xmath13 are same as in fig.@xmath42 1 . the coulomb interaction induces the strong @xmath6 dependence of renormalization function @xmath37 , and the low frequency strong @xmath6 dependence of @xmath37 distorts the drude part of optical conductivity and induces the mir absorption in the fairly low frequency region . as the impurity effect is enhanced , the mir absorption due to coulomb interactions tends to shift to higher frequency and finally merge together with the mir peaks developed by electron - phonon interaction , as shown in fig.@xmath42 3 for @xmath100 and @xmath101 . note that the position of this merged mir peak in fig.@xmath42 3 is around and above @xmath3 ev which is experimentally observed value of degiorgi . fig.@xmath42 4 is @xmath0 of doped fullerenes with @xmath102 ev , @xmath103 ev , @xmath104 ev , @xmath100 , and @xmath101 , which is to be compared with the experimental observations . the drude weight is reduced to 0.467 of the total intra - band optical weight . the reduction factor of the drude weight by electron phonon interaction is @xmath97 , and the finite bandwidth futher restricts the reduction factor . it therefore seems unlike that the drude weight less than about 0.6 of the total intra - band spectral weight can be explained without the coulomb interactions , when we take @xmath105 . the coulomb interaction suppresses the drude part substantially by inducing @xmath6 dependence of the renormalization function @xmath37 in the low frequency region . we think that the large reduction of drude weight like the degiorgi experiment is a result of the strong coulomb interaction between conduction electrons in addtion to the electron - phonon interaction . however , our results are still not sufficient to explain experimentally founded results : ( a ) the drude weight is about 0.46 of the total intra - band optical weight with a resonable set of parameter values while degiorgi found @xmath106 . ( b ) the mir absorption is very broad which begins around 0.02 ev , has a peak around 0.07 ev and extends well over the fermi energy . in this paper , we tried to give an explanation for the unusual behavior of optical conductivity in the normal state @xmath16 . it is generally accepted that the fullerene superconductor could be characterized by the phonon - mediated @xmath107-wave superconductor @xcite . however , a few experiments like optical conductivity still remain not understood by the electron - phonon scattering together with the disorder effects . our motivation lies in that the fullerene superconductors have such a narrow bandwidth that the phonon frequency , the coulomb interaction , and the fermi energy are all comparable , @xmath108 . in order to consider properly the coulomb interaction and the narrow bandwidth of fullerene superconductors , the self - consistent eliashberg - type coupled equations are solved to obtain the renormalized green s function . the theory includes the frequency dependent screened coulomb interaction together with the electron - phonon interaction and includes the vertex correction via nambu s method . in order to treat the vertex function exactly , analytic continuation is performed via the iterative method of mixed repersentation which is developed by takada . once we get renormalization function @xmath37 in real frequency , we can calculate optical conductivity in normal states . as we expected , the electron - phonon interaction is not suffficient to resolve the substantial reduction of drude weight and pronounced mir peak . the strong coulomb interaction induces @xmath6 dependence in renormalization function @xmath37 . as a result , the drude form in optical spectra is distorted accompanying the reduction of drude weight . when the impurity effect is enhanced , the mir absopption induced by strong coulomb interaction merge together with the mir peaks due to electron - phonon scattering showing large reduction of drude weight and mir peak around @xmath3 ev . although it is not sufficent to explain experimentally founded results , our result is close to degiorgi s experiment . we improve gunnarsonn s calculation by considering the electron - electron interaction and finite bandwidth effects explicitly . in conclusion , the unusual behavior of optical conductivity of the normal state @xmath109 reveals the fact that both the coulomb interaction and electron - phonon interaction are important in examining dynamical properties of fullerene superconductors . l. degiorgi , g. briceno , m. s. fushrer , a. zettel , and p. wachter , nature * 359 * , 541 ( 1994 ) ; l. degiorgi , e.j nicol , o. klein , g. gr@xmath110ner , p. wachter , s .- huang , j. wiley , and r.b . kaner , phys . b * 49 * , 7012 ( 1994 ) ; l. degiorgi , mod . b * 9 * , 445 ( 1995 ) . y. iwasa and t. kaneyasu , phys . b * 51 * , 3678 ( 1995 ) . j. van den brink , o. gunnarsson , and v. eyert , phys . b * 57 * , 2163 ( 1998 ) . h. y. choi , phys . lett . * 81 * , 441 ( 1998 ) . a. b. migdal , sov . phys . jept * 7 * , 996 ( 1958 ) . d. j. scalapino , in _ superconductivity _ , ed . by d. r. park ( dekker , n. y. ) , vol . 1 , 449 ( 1969 ) . p. b. allen and b. mitrovi@xmath111 , in _ solid state physics _ 37 , 1 ( 1982 ) . o. gunnarsson , rev . 69 * , 575 ( 1997 ) . y. nambu , phys . rev . * 117 * , 648 ( 1960 ) . h. y. choi ( unpublished ) . g. d. mahan , _ many - particle physics _ ( plenum , n. y. ) ( 1981 ) . a. khurana , phys . . lett . * 64 * , 1990 ( 1990 ) . s. engelsberg and j. r. schrieffer , phys . rev . * 131 * , 993 ( 1969 ) . f. marsiglio , m. schossmann , and j. p. carbotte , phys . b * 37 * , 4965 ( 1988 ) . y. takada , phys . b * 52 * , 12708 ( 1995 ) . d. j. scalapino , s. r. white and s. c. zhang phys . rev * 47 * , 7995 ( 1993 ) . figure 1 . the optical conductivity as a function of @xmath6 for various electron - phonon coupling constants @xmath21 when @xmath112 ev , @xmath113 ev and @xmath114 ev . @xmath23 is set to 0 for a referrence . as @xmath21 is increased , the width and the weight of the drude peak are reduced . the inset shows a decomposition of the total conductivity into the drude and mir parts for @xmath101 . the mir spectra induced by the coulomb interaction when @xmath112 ev , @xmath113 ev , @xmath114 ev and @xmath115 . the inset shows a decomposition of the total conductivity , as in fig.@xmath42 1 , into the drude and mir parts for @xmath116 . the mir spectra as the impurity scattering rates @xmath13 are varied when @xmath117 ev , @xmath113 ev , @xmath118 and @xmath119 . when @xmath114 ev the lower peak is mainly from the coulomb interaction while the other peaks are from the electron - phonon interaction . as @xmath13 is increased , these peaks are merged altogether and finally evolve into a single broad peak around @xmath120 ev . figure 4 . the total optical conductivity with it s drude and mir parts for @xmath117 ev , @xmath113 ev , @xmath121 ev , @xmath118 and @xmath119 . the drude part in the low frequency region is substantially suppressed due to the strong coulomb interaction . consequently , the missing spectral weight is transfered to the broad mir peak , which peaks around 0.07 ev and extends well into the higher energy region . the ratio of the mir spectral weight to the total intra - band spectral weight is 0.533 .

we calculate the optical conductivity , @xmath0 , in the normal state fullerene superconductors by self - consistently including the impurity scatterings , the electron - phonon and electron - electron coulomb interactions . the finite bandwidth of the fullerenes is explicitely considered , and the vertex corection is included @xmath1 @xmath2 nambu in calculating the renormalized green s function . @xmath0 is obtained by calculating the current - current correlation function with the renormalized green s function in the matsubara frequency and then performing analytic continuation to the real frequency at finite temperature . the drude weight in @xmath0 is strongly suppressed due to the interactions and transfered to the mid - infrared region around and above @xmath3 ev which is somewhat less pronounced and much broader compared with the expermental observation by degiorgi @xmath4 @xmath5 . 2

we use the projectors corresponding to the results @xmath66 and @xmath67 as the computational basis , so that using the projector @xmath68 and rotations from @xmath66 to @xmath69 and from @xmath67 to @xmath70 conveniently parameterized by @xmath71 we obtain under the assumption of independent errors at equal rate , the ch operator including efficiency is @xmath73 local realism bounds the expectation value of this measurement below zero , so any positive eigenvalue to the operator will give a violation . the eigenvalues are the solutions of the characteristic polynomial @xmath74 \\ + 2(\eta-1)\eta^3\left[st(\eta^2-\eta)-1\right]\lambda \\ -\eta^2(4\eta-5)\lambda^2 - 2\eta(\eta-2)\lambda^3+\lambda^4=0 . \label{eq:10}\end{gathered}\ ] ] seeking a maximum violation , we need to find the parameter values @xmath75 and @xmath76 that gives this maximum . only the sum @xmath77 and product @xmath78 occur above , so we can also view this as finding values of @xmath77 and @xmath78 that gives the maximum . the combination @xmath77 occurs only in the constant term in the polynomial ( with a positive coefficient if @xmath79 ) so that , for a given value of @xmath78 , the maximum highest solution is obtained when @xmath77 is minimal , i.e. , when @xmath80 . in this case , @xmath81 \\ + 2(\eta-1)\eta^3\left[t^2(\eta^2-\eta)-1\right]\lambda \\ -\eta^2(4\eta-5)\lambda^2 - 2\eta(\eta-2)\lambda^3+\lambda^4=0 . \label{eq:11}\end{gathered}\ ] ] changing to the bell basis ( in the order @xmath82 , @xmath83 so that the singlet is last ) , the ch operator becomes @xmath84 and it is immediately clear that the singlet state is an eigenvector to this operator . we can read off the corresponding eigenvalue @xmath85 , which is always negative . the remaining three eigenvalues can be obtained by solving the equation @xmath86\frac{\lambda^2}{\eta^2 } \\ + \left[\eta^2(t^2 - 2t)+2\eta(t-1)+2\right]\frac\lambda\eta \\ + \eta^3t^3 - 3\eta^2 t^2 + 2\eta t^2=0 . \label{eq:2}\end{gathered}\ ] ] solving with a general method ( using complex numbers ) will give the right answer . when the roots are real , the trigonometric method gives the following expression : @xmath87+\frac{2}{3 } \eta\sqrt{3 - 6\eta+(4 + 2t-2t^2)\eta^2}\\ & \times \cos\big[\frac{1}{3 } \arccos\big[\eta\big(9 - 18\eta+8\eta^2 -10t^3\eta^2 + 3t\big(3 - 6\eta+2\eta^2\big)\\ & \phantom{\times\cos\big[}-3t^2\big(9 - 18\eta+4\eta^2\big)\big ) \big/{\sqrt{(3 - 6\eta+2(t-2)(t+1)\eta^2)^3}}\big]\big ] , \end{split } \label{eq:5}\ ] ] the next step would be to differentiate @xmath90 with respect to @xmath76 and solve for 0 , but that is not so easy . a simpler way is to use the equation as an implicit definition of @xmath5 as a function of @xmath76 and do implicit differentiation , as in @xmath91 since we seek the maximum , @xmath92 and we arrive at @xmath93 back substitution and elimination of some simple factors gives a fourth - degree polynomial equation for @xmath76 , @xmath94 this is solvable , but an analytic solution is very long . reproducing that here is not very helpful ; it is better to insert the efficiency @xmath65 and solve for @xmath76 to obtain the rotation between the two measurement setups . this can be used to obtain @xmath5 , and the eigenvectors are then simple to find . + + [ [ detection - efficiency - vs .- entanglement - in - the - i_3322-inequality ] ] detection efficiency vs. entanglement in the @xmath51 inequality ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the results in the main text shown that , for chained bell inequalities of any number of settings , and in the absence of noise , product states are the most nonlocal ones . this result also holds for the @xmath51 inequality @xcite . [ gfig ] shows the threshold detection efficiency and the violation of the @xmath51 inequality as a function of the degree of entanglement ( measured by @xmath48 ) . we observe that , in the absence of noise , almost product states are again those that require lower detection efficiencies . we also observe that the detection threshold for the @xmath51 is lower than the one for the @xmath58 inequality , which has the same number of local settings . for two - qubit systems , the maximum violation is @xmath95 , and can be achieved with a maximally entangled state , as was previously shown in @xcite . however , the minimum required efficiency with maximally entangled states is @xmath96 , not @xmath97 as reported in n. brunner and n. gisin , phys . a * 372 * , 3162 ( 2008 ) . this value of @xmath96 can indeed be obtained analytically . for a maximally entangled state , it is easy to show that @xmath98 .

we show that , for any chained bell inequality with any number of settings , nonlocality and entanglement are not only essentially different properties but opposite ones . we first show that , in the absence of noise , the threshold detection efficiency for a loophole - free bell test increases with the degree of entanglement , so that the closer the quantum states are to product states , the harder it is to reproduce the quantum predictions with local models . in the presence of white noise , we show that nonlocality and entanglement are simultaneously maximized only in the presence of extreme noise ; in any other case , the lowest threshold detection efficiency is obtained by reducing the entanglement . . introduction._nonlocality and entanglement are two core concepts in quantum information . if @xmath0 is the joint probability that alice obtains @xmath1 and bob @xmath2 on a system prepared in state @xmath3 , nonlocality is the impossibility of expressing @xmath0 as @xmath4 , where @xmath5 are preestablished classical correlations @xcite . entanglement is the impossibility of writing down a quantum state as a convex combination of separable states . nonlocality and entanglement are related concepts in the sense that , to have nonlocality , entanglement is needed @xcite . the difference between both concepts has been pointed out before : first , noticing that there are entangled states which do not violate specific bell inequalities @xcite , and then pointing out that states with lower entanglement lead to larger violations of some specific bell inequalities @xcite . the difficulty in reaching a general conclusion about their relationship is that of finding a general scenario where incontrovertible measures of nonlocality and entanglement can be compared . a bipartite scenario has the advantage that any of the many measures of entanglement assigns zero entanglement to product states and maximum entanglement to maximally entangled states @xcite . nonlocality is a more delicate issue in that different restrictions on the number of settings usually lead to different measures of nonlocality . this suggests that any conclusion should be based on a bipartite scenario in which the parties can perform an arbitrary number of measurements . the aim of this letter is to show that , in such scenario , and assuming some natural measures , nonlocality and entanglement actually have have _ opposite _ behaviors . in @xcite , braunstein and caves ( bc ) introduced a generalization of the clauser - horne - shimony - holt ( chsh ) @xcite and clauser - horne ( ch ) @xcite bell inequalities , known as chained bell inequalities , in which alice and bob choose among @xmath6 settings . chained bell inequalities have some interesting applications : case @xmath7 fixes a loophole that occurs in some experiments based on the chsh inequality @xcite . besides , it reduces the number of trials needed to rule out local hidden variable theories @xcite , and improves the security of some quantum key distribution protocols @xcite . in the case in which @xmath8 tends to infinity , the inequality allows one to discard nonlocal hidden variable theories with a nonzero local fraction @xcite . chained bell inequalities have been experimentally tested using pairs of photons , with @xmath7 @xcite , @xmath9 @xcite , and @xmath10 @xcite . a natural measure of nonlocality is the minimum detection efficiency required for a loophole - free violation @xcite . the idea is simple : when one observes a violation of a bell inequality with perfect detection efficiency , this means that no local model can reproduce the joint probabilities . if the critical detection efficiency is @xmath11 , this means that no local model exists , even if one locally rejects a fraction @xmath12 of the events . therefore , the smaller @xmath11 , the harder it is to reproduce the joint probabilities with local models ; the smaller @xmath11 , the larger nonlocality . the minimum detection efficiency required for a loophole - free violation of chained bell inequalities for any @xmath6 using maximally entangled states has been obtained in @xcite . the fact that the maximum quantum violation of chained bell inequalities is always achieved with maximally entangled states @xcite might suggest that the minimum detection efficiency occurs for maximally entangled states , but no proof exists of whether the detection efficiency for the chained bell inequalities can indeed be reduced when one considers more general classes of entangled states . indeed , for case @xmath13 , corresponding to the ch inequality , the minimum detection efficiency occurs for almost product states @xcite . in the following , we will show that , in the absence of noise ( e.g. , considering pure states ) , the minimum detection efficiency for any chained bell inequality is obtained for _ almost product states _ for any @xmath8 . additionally , we will show that , if the state is affected by a ( small ) amount of white noise , then the largest nonlocality ( i.e. , the lowest detection efficiency ) requires only a ( small ) amount of entanglement . _ detection efficiency for chained bell inequalities._the version of the chained bell inequalities introduced in @xcite , which is symmetric under the permutation of alice and bob , reads @xmath14 where @xmath15 \nonumber \\ & - p(a_1b_1)-\sum^{m}_{k=2}\left[p(a_k)+p(b_k)\right ] , \label{sbch } \end{aligned}\ ] ] and @xmath16 is the expectation value of @xmath17 in the state @xmath3 . here , @xmath18 ( @xmath19 ) with @xmath20 represents dichotomic observables , and @xmath21 is the joint probability of obtaining @xmath22 . assuming the same detection efficiency for every party and setting , i.e. , @xmath23 , the value of @xmath17 becomes @xmath24,\ ] ] where @xmath25 is the expectation value of @xmath26 in the state @xmath3 . therefore , inequality is violated when @xmath27 , with @xmath28 } { ( s_m)_\rho+\sum^{m}_{k=2}\left[p_\rho(a_k)+p_\rho(b_k)\right]}.\ ] ] any generic two - qubit pure states @xmath29 , can be written ( in a suitable basis ) as @xmath30 with @xmath31 and @xmath32 . let us consider the following eigenstates : @xmath33 and choose @xmath34 such that @xmath35 . then , @xmath36 and the critical efficiency becomes @xmath37 which , when @xmath34 tends to zero ( i.e. , _ when the state tends to a product state _ ) , tends to @xmath38 the important point here is that this value is _ smaller _ than the minimum value of @xmath39 for maximally entangled states @xcite , namely , @xmath40 moreover , for bell inequalities of the form , the value in the right hand side of is the minimum detection efficiency needed for any quantum state and choice of settings . _ proof : _ inserting the assumption of independent errors , the critical efficiency of inequality becomes @xmath41 } { p(a_{m}b_{m})+\sum_{k=2}^{m}\left[p(a_kb_{k-1})+p(a_{k-1}b_k)\right]-p(a_1b_1)}. \label{eq:9b}\ ] ] clearly , @xmath42,\ ] ] and the lowest possible bound is obtained when @xmath43 and @xmath44 for @xmath45 and @xmath46 not both equal to one . we obtain @xmath47 which can not be achieved exactly , but arbitrarily close with an appropriate quantum state , as shown by equation . this proves that , in the absence of noise , almost product states give the best detection efficiency for chained bell inequalities for any value of @xmath8 . we have also obtained the critical efficiency as a function the degree of entanglement measured by @xmath48 and compared it with the corresponding maximal violation of the bell inequality . by using the method of conjugate gradient , we have numerically found @xmath39 and the maximum values of @xmath17 as a function of @xmath48 . the results for @xmath49 are shown in fig . [ fig : etacritk ] . the analytic expression for the state giving the largest violation for a given detection efficiency for the case @xmath13 can be found in the supplementary material . ( a ) minimum @xmath39 as a function of the degree of entanglement measured by @xmath48 . ( b ) maximum violation of @xmath50 for a given @xmath48.,scaledwidth=45.0% ] we observe that larger violations of @xmath17 do not correspond to lower critical detection efficiencies . the value of @xmath17 is not a good measure of nonlocality . for any number of settings , the lowest threshold efficiency occurs when the state is almost a product state rather than when it is a maximally entangled state . indeed , as it is shown in the supplementary material , this conclusion also holds for less symmetrical bell inequalities like the @xmath51 inequality @xcite . _ noise._how does noise affect this conclusion ? in the presence of white noise , the state becomes @xmath52 and the threshold detection for the chained bell inequalities efficiency is changed to @xmath53+\frac{q}{1-q}(m-1 ) } { ( s_m)_\rho+\sum^{m}_{k=2}\left[p_\rho(a_k)+p_\rho(b_k)\right]+\frac{q}{2(1-q)}(m-1)}.\ ] ] in fig . [ fig : etawithnoise ] we show , for three different values of noise ( @xmath54 , @xmath55 , and @xmath56 ) , the dependence of @xmath39 and the maximum values of @xmath57 and @xmath58 with the degree of entanglement of the initial pure state . we observe that , when the noise is different from @xmath59 , the best quantum state giving the lowest threshold is not an almost separable state , but a nonmaximally entangled state depending on @xmath60 . however , the lower the noise @xmath60 , the smaller the entanglement required to obtain the optimal threshold . ( a ) values of @xmath39 and ( b ) maximum violation of the chained bell inequality for different number of settings and different degree of noise ( @xmath60).,scaledwidth=46.0% ] furthermore , in fig . [ fig : etawithnoise](b ) we observe that , the lower @xmath8 is , the more resistant to noise is the violation of the bell inequality . in fact , it is possible to calculate the maximum tolerated noise to violate the chained bell inequalities . given @xmath48 and the maximal violation of @xmath17 defined as @xmath61 , the maximum tolerated noise is @xmath62 . using the method of conjugate gradient to minimize eq . ( [ etanoise ] ) , it is also possible to obtain the threshold and the required entanglement for any value of the noise @xmath60 . the results are shown in fig . [ fig : etavsq ] . we observe that , for chained bell inequalities , nonlocality and entanglement are simultaneously maximized _ only in case of extreme noise_. a better threshold detection efficiency is obtained by lowering the noise and suitably decreasing the entanglement . from this we conclude that nonlocality and entanglement are synonymous only for extremely noisy scenarios . _ discussion._in a general two - party @xmath8-setting bell scenario ( for any @xmath63 finite ) , nonlocality and entanglement are , in the absence of noise , opposite properties in the following sense . we have argued that the critical detection efficiency @xmath64 is a good measure of nonlocality , since it marks the border where local hidden variable descriptions becomes possible : the smaller @xmath64 , the harder it is to express the joint probabilities with local models . therefore , the result can be summarized by saying that , in a noiseless scenario , larger nonlocality requires smaller entanglement ; in the absence of noise , almost product states are the most nonlocal ones . when the noise is turned on , the most nonlocal states acquire some amount of entanglement ; however , the smaller the noise is , the lower their entanglement becomes . this emphasizes that nonlocality and entanglement are not only different but , in many cases , opposite concepts . this work was supported by the micinn project no.fis2008-05596 , fondecyt project nos . 11085055 and nc10 - 030-f , conicyt , and the wenner - gren foundation . 1 j. s. bell , physics * 1 * , 195 ( 1964 ) . n. gisin , phys . lett . a * 154 * , 201 ( 1991 ) . r. f. werner and m. m. wolf , phys . rev . a * 61 * , 062102 ( 2000 ) . a. acn , t. durt , n. gisin , and j. i. latorre , phys . rev . a * 65 * , 052325 ( 2002 ) . a. acn , r. gill , and n. gisin , phys . rev . lett . * 95 * , 210402 ( 2005 ) . m. junge and c. palazuelos , . t. vidick and s. wehner , phys . rev . a * 83 * , 052310 ( 2011 ) . w. k. wootters , phys . rev . lett . * 80 * , 2245 ( 1998 ) . s. l. braunstein and c. m. caves , in _ bell s theorem , quantum theory , and conceptions of the universe _ , edited by m. kafatos ( kluwer , dordrecht , 1989 ) , p. 27 . s. l. braunstein and c. m. caves , ann . phys . 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high - velocity outflows are an important part of the quasar phenomenon . they are often studied via broad absorption lines ( bals ) in the rest - frame uv that reveal outflow speeds from a few thousand to tens of thousands of km s@xmath11 @xcite . some studies suggest that bal outflows play an important role in feedback " to the quasar s host galaxy evolution contributing to galaxy - scale blowouts of gas and dust , disrupting star formation in the galaxy hosts , and regulating the growth of the central supermassive black hole ( smbh , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . nonetheless , many aspects of quasar outflows remain poorly understood , including their basic physical conditions and acceleration mechanism(s ) . bal outflows are believed to arise from quasar accretion disks , driven out by radiation pressure @xcite or magneto - hydrodynamic or magneto - centrifugal forces @xcite . radiation pressure is expected to be important in luminous quasars , e.g. , at high accretion rates relative to eddington @xcite . however , the intense radiation available to push the outflows can also over - ionize the gas and make it too transparent for radiative driving . @xcite and @xcite proposed to solve the over - ionization problem by noting that a highly - ionized and radiatively thick absorbing region should develop naturally at the base of bal outflows , near the quasar s intense source of ionizing radiation . this additional absorbing medium is itself too ionized and too transparent for radiative driving , but it serves critically as a shield to block ionizing radiation and thereby allow the bal gas behind it to reach sufficient opacities for radiative acceleration . the general picture of bal outflows behind a thick radiative shield has become a mainstay of theoretical models @xcite . even the magneto - hydrodynamic models invoke large column densities of shielding gas to launch bal outflows with moderate ionizations from the strong gravity environment near the central smbh @xcite . the shielding hypothesis is also supported by bal quasar observations that reveal strong x - ray absorption with typical neutral - equivalent column densities @xmath12 @xmath1 @xcite . moreover , sources with stronger x - ray absorption tend to have larger outflow speeds and stronger @xmath13@xmath131548,1550 absorption lines @xcite . these results suggest that x - ray shielding is important for the development of bal outflows and , specifically , that a thicker radiative shield leads to more efficient radiative acceleration . however , this picture is complicated by observations of other quasars with narrow absorption line ( nal ) outflows and so - called `` mini - bals , '' which have smooth bal - like profiles but velocity widths below the standard bal threshold ( fwhm @xmath7 2000 km s@xmath11 , * ? ? ? * ; * ? ? ? * ; * ? ? ? these narrow line outflows are more common than bals @xcite with high speeds and degrees of ionization ( typified by and @xmath131032,1038 absorption ) similar to bals , but they have dramatically less x - ray absorption ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and 4.2 below ) . the similar properties of bal , nal , and mini - bal outflows suggest that they arise from the same general outflow phenomenon , while orientation or temporal / evolution effects might explain their important differences . in one popular orientation - based scheme , bals form in the main part of the outflow near the accretion disk plane while nals and mini - bals form along sightlines at higher latitudes that ( perhaps ) skim the ragged edges of the bal flow farther above the disk @xcite . this geometry is broadly consistent with the theoretical models mentioned above ( also * ? ? ? * ) and it can explain the weak or absent x - ray absorption in nal and mini - bal quasars if the x - ray absorber resides primarily near the plane of the accretion disk @xcite . the difficulty arises when we consider that nal and mini - bal outflows achieve the same high speeds and moderate degrees of ionization as bals _ without _ the protection of a radiative shield . this suggests that the shield is not a critical feature of the winds . it appears to undermine the main premise of all current radiative acceleration models that a strong shield is essential to moderate the outflow ionizations and launch gas behind the shield to high speeds . it might also require us to abandon current models with smooth continuous flows in favor of earlier schemes that involve small dense clouds with an overall small volume filling factor ( see 6 below , also * ? ? ? * ) . in this paper , we present new x - ray and rest - frame uv observations of 7 quasars with extreme - velocity mini - bal outflows , supplemented by archival data for a similar quasar , pg 2302 + 029 @xcite . the outflow speeds in these quasar are in the range 0.1c to 0.2c , which is 2 - 3 times larger than typical bals or mini - bals in previous x - ray studies . the extreme speeds require favorable conditions for the outflow acceleration . we aim to test whether these conditions involve a strong radiative shield . we might expect stronger shielding in extreme - velocity outflows because they should originate from small accretion disk radii , near the ionizing uv emission source . in particular , the maximum flow velocities , @xmath14 ( at infinity ) , are expected to scale roughly with the gravitational escape speed , @xmath15 , at the launch radius , @xmath16 , such that @xmath17 ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * also 4.1 below ) . therefore , the extreme outflow speeds in our mini - bal sample , 2 - 3 times larger than previous studies , might correspond to launch radii that are 4 - 9 times smaller where the ionizing flux is 16 - 81 times greater . we will show that the x - ray absorption is weak or moderate and radiative shielding is not important in these outflows . 2 provides information about the quasar sample . 3 describes our new uv and x - ray observations . 4 presents some basic analysis of the mini - bal variabilities with constraints on the outflow locations , as well as measures of the x - ray absorption and absorber properties . 5 describes cloudy photoionization simulations that place quantitative constraints on the amount and importance of shielding across the full uv to x - ray spectrum . 6 discusses the implications for quasar outflow models , and 7 provides a brief summary . throughout this paper , we adopt a cosmology with @xmath18 km s@xmath11 mpc@xmath11 , @xmath19 and @xmath20 . our study focusses on 8 quasars known to have broad outflow lines at extreme velocities @xmath210.1@xmath22 . six of them were discovered by @xcite in spectra from the sloan digital sky survey ( sdss , * ? ? ? the other two have been studied more extensively ( e.g. , j093857 + 412821 by @xcite ; pg 2302 + 029 by @xcite ) . all 8 quasars are radio quiet @xcite range from @xmath230.3 for the visibly brightest source ( j093857 + 412821 ) to @xmath230.8 for the faintest ( j090508 + 074151 ) . ] based on non - detections in the _ first _ radio survey @xcite . figure 1 shows the outflow lines from the studies cited above plus own new measurements at the mdm observatory ( see 3.2 below ) . these outflow features have velocities in the range @xmath24 km s@xmath11 and widths @xmath25 km s@xmath11 . some of the lines are broad enough to qualify as bals , with fwhm @xmath21 2000 km s@xmath11 , except that their speeds are too high to register a balnicity index bi @xmath21 0 @xcite . thus they are not bals by this formal definition and , in any case , they are narrower and weaker than the majority of bona fide bals studied previously at lower speeds . we will refer to all of the outflow lines in our sample as mini - bals " ( see * ? ? ? * for more discussion ) . table 1 lists some basic data for the quasars and mini - bals . the emission line redshifts , @xmath26 , are from @xcite and the @xmath27 magnitudes are from the sdss for all sources except pg2302 + 029 , for which the redshift and @xmath28 magnitude are from @xcite . the mini - bal centroid velocities , @xmath29 , and fwhms are from @xcite , except for pg 2302 + 029 from an hst spectrum obtained by @xcite in 1994.58 and for j093857 + 412821 ( also called pg 0935 + 417 ) from a 1996.2 spectrum obtained at the lick observatory 3.0 m shane telescope ( * ? ? ? * ; * ? ? ? * also figure 1 ) . a second weaker mini - bal at @xmath30 km s@xmath11 in the lick spectrum of j093857 + 412821 is marked in figure 1 but not listed in table 1 . for j090508 + 074151 , we re - examined the sdss spectrum measured by @xcite and chose to use a lower , more conservative continuum placement . this yields a weaker and narrower absorption line that now appears questionable as a mini - bal . the value of fwhm @xmath31 740 km s@xmath11 in table 1 is largely due to the doublet separation ( @xmath31500 km s@xmath11 ) . columns 68 in table 1 provide the observation dates for the sdss , chandra , and mdm data in decimal years . [ cols="<,^,^,^,^,^,^,^ " , ] @xmath32 and @xmath33 have units @xmath1 and km s@xmath11 , respectively . the line center optical depths apply to the short wavelength components of the doublets @xmath131548 , @xmath131239 , @xmath131032 , and @xmath13770 with contributions from the other doublet component if there is blending at large @xmath33 . hardness ratios , hr , are listed for redshifts @xmath34 and @xmath35 . hr and @xmath36 are given in pairs for incident spectra with @xmath37 and @xmath38 . for comparison , unabsorbed spectra with @xmath37 and @xmath38 have hr = 0.85 and 1.81 , respectively . the top two panels in figure 4 show the transmitted spectra for models noc4b100 and noc4b1000 ( blue curves ) compared to the incident spectrum ( black ) . strong absorption is evident across far - uv to soft x - ray wavelengths . this is caused by bound - free opacities plus many blended metal lines . the absorption is stronger in model noc4b1000 because the broader lines i ) directly absorb more of the continuum flux , and ii ) permit a lower degree of ionization and thus larger continuous opacities under the same @xmath39 constraint . table 3 lists the predicted hardness ratios , @xmath36 ( measured relative to the incident spectrum ) , and line - center optical depths for several important lines in the model spectra . these values of hr and especially @xmath36 are substantially larger ( more negative for @xmath36 ) than the observations ( table 2 and 4.2 ) , indicating that the models produce much more x - ray absorption than actual mini - bal outflows . nonetheless , these model absorbers are not effective shields for and because they do not significantly suppress the flux near the c@xmath40 and o@xmath41 ionization edges ( figure 4 ) . the models also predict very strong absorption lines of and , with optical depths @xmath936 and @xmath956 respectively . measurements or upper limits on these high ionization lines in actual quasars would place even stronger constraints on the overall amounts of shielding . model ` noo6b1000 ' in table 3 and figure 4 illustrates this point for an absorber with the same total column density , @xmath42 , but now with the @xmath131032 line center optical depth constrained to the value @xmath43 for @xmath44 km s@xmath11 . this limit on @xmath45 requires a higher degree of ionization compared to the models constrained by @xmath39 , resulting in weaker absorption at all wavelengths and completely negligible shielding . nonetheless , the x - ray absorption at @xmath312 kev is still stronger ( more negative @xmath36 ) than most of the observations ( table 2 and 4.2 ) . model ` noc4b1000xr ' roughly maximizes the far - uv shielding consistent with our data by requiring @xmath39 for @xmath44 km s@xmath11 ( as in model noc4b1000 above ) with an added constraint on the total column to yield @xmath46 , which is at the high end of measured values in mini - bal quasars ( 4.2 ) . the resulting spectrum shown in the lower right panel of figure 4 shows that the strength of x - ray absorption is similar to a neutral absorber with @xmath47 ( see also @xmath36 in table 3 ) . this model again predicts a strong saturated and lines near the rest velocity , which violates previous observations of j093857 + 412821 @xcite and some other well - measured cases of nal and mini - bal outflows @xcite . nonetheless , there are still only modest amounts of far - uv and soft x - ray absorption that are again not sufficient to shield the mini - bal gas . we can quantify the failure of the model absorbers in 5.1 to be effective shields by examining their influence on the ionization of gas behind them . first , we consider hypothetical mini - bal regions exposed directly to the quasar continuum described above , with no shielding at all . from the definition of the ionization parameter , @xmath48 where @xmath49 is the total emitted luminosity of hydrogen - ionizing photons ( # /s ) , we have this general relationship between the gas density , @xmath50 , and its radial distance , @xmath51 , from the continuum source , @xmath52 where the luminosity @xmath53 ergs s@xmath11 is roughly typical of our sample and @xmath54 pc is a reasonable guess for the location of the mini - bal gas ( 4.1 ) . if the mini - bal gas is optically thin throughout the lyman continuum , this equation and the continuum shape fully describe its ionization . the value of @xmath55 is conservatively high for and absorption , slightly favoring to be consistent with mini - bal observations ( see refs . above ) . the specific ion fractions are c@xmath40/c @xmath56 , o@xmath41/o @xmath57 , and c@xmath40/o@xmath58 for solar abundances . we conclude from equation 2 that high densities of order @xmath59 @xmath60 are needed to keep the ionization low enough for absorption if a shield is not present . figure 5 shows similar results for more realistic situations with non - negligible column densities in the mini - bal region . in particular , the solid curves show the values of @xmath61 and @xmath50 that produce ( black ) or ( blue ) mini - bals with @xmath62 and @xmath44 km s@xmath11 for different total column densities , @xmath32 . these curves thus define the minimum densities ( and maximum @xmath61 ) needed for significant / measurable and mini - bals in outflow regions without a shield . the densities shown in this figure are for the particular distance @xmath63 pc , but they can be scaled to other distances by multiplying by the factor ( 2 pc/@xmath51)@xmath64 , as in equation 2 . , and ionization parameters , @xmath61 , needed to produce or mini - bals in outflow regions with different total column densities , @xmath32 . the specific densities shown are for @xmath63 pc , but they can be scaled to any distance by the factor @xmath65 . the solid bold / black and thin / blue curves show the conditions needed for @xmath62 mini - bals of / ( for @xmath44 km s@xmath11 ) in outflow gas illuminated by the unattenuated quasar spectrum ( figure 4 ) . densities above these curves lead to stronger / measurable mini - bals . the dashed bold / black and thin / blue curves mark @xmath62 absorption by / in mini - bal regions behind a maximum defined by model noc4b1000xr ( table 3 ) . the @xmath61 values appropriate for the dashed curves are @xmath310.12 larger than shown on the right - hand axis ( because @xmath49 for the transmitted shield spectrum is slightly reduced , eqn . next we insert the strongest shield consistent with our data ( model noc4b1000xr ) between the quasar continuum source and the mini - bal gas . we do this using the transmitted spectrum from model noc4b1000xr ( lower right panel in fig . 4 ) to illuminate the hypothetical mini - bal clouds described above . in the optically thin case ( eqn . 2 ) , the ionization in the mini - bal gas decreases to yield c@xmath40/c @xmath66 , o@xmath41/o @xmath67 , and c@xmath40/o@xmath68 . ionization conditions similar to the unshielded situation can now occur at a factor of @xmath313 lower density compared to equation 2 . for the cases with significant column densities in the mini - bal region , the dashed curves in figure 5 show the revised values of @xmath50 and @xmath61 needed for just - measurable and mini - bals with @xmath69 behind the maximum shield . these results are approximate because the putative shield is not offset in velocity from the mini - bal gas . nonetheless , the displacement between the dashed and solid curves in figure 5 indicates that the addition of the _ maximum _ shield allowed by our data reduces the density thresholds by factors of @xmath313 to @xmath3110 . these changes are significant but not nearly large enough to affect the structure of the flow behind the shield or facilitate the acceleration of outflow gas that would otherwise be too ionized for radiative driving ( see also 6 below ) . all of the models just described , including noc4b100 and noc4b1000 which grossly over - predict the x - ray absorption , fail to be effective shields because there is not sufficient continuous opacity near the ionization energies of c@xmath40 and o@xmath41 ( figure 4 ) . notably absent are significant absorption edges of h@xmath70 ( 13.6 ev ) , he@xmath70 ( 24.6 ev ) , or he@xmath71 ( 54.4 ev ) because the gas is too ionized . other candidates for suppressing the flux near the c@xmath40 and o@xmath41 edges are the c@xmath40 and o@xmath41 ions themselves . these edges are also missing because the shields are not allowed to form or absorption lines . this is evident from the relationship between @xmath72 and the optical depth at the c@xmath40 edge , @xmath73 , namely , @xmath74 and similarly for @xmath45 related to @xmath75 at the o@xmath41 edge @xmath76 using atomic data from @xcite and @xcite . weak and especially weak lines in the near - uv lead to very weak absorption at critical far - uv energies below @xmath31200 ev . conversely , an effective shield that _ does _ have strong absorption at these energies will produce very strong and/or absorption lines , which can be tested by uv observations . the absence of these lines near @xmath77 points to the absence of a viable shield at @xmath77 . we can not exclude the possibility that the mini - bal regions themselves have large column densities leading to some amount of `` self - shielding . '' however , existing data do not support large mini - bal or nal outflow column densities are available for nal and especially mini - bal outflows in quasars . our own analysis of j093857 + 412821 in the current sample ( * ? ? ? * as measured in 1996 ) crudely indicates @xmath78 @xmath1 in the mini - bal gas , which is similar to other estimates for lower speed outflow lines in quasars ( e.g. , * ? ? ? * ; * ? ? ? @xcite find @xmath79 @xmath1 for some weaker and much lower speed nal outflows . @xcite favored larger values , @xmath80 @xmath1 , for one particular mini - bal system that might have unusual properties . ] . a self - shielding mini - bal region would also imply large columns of highly ionized gas already moving at speeds @xmath210.1c . this gas would be too transparent for radiative driving , which begs the question of how it was accelerated to high speeds without a separate shield ( at @xmath77 , 1 ) . moreover , within reasonable @xmath32 limits , a self - shielding mini - bal region would not change the main conclusion that high densities are needed to explain the observed ionizations low enough for absorption lines ( e.g. , figure 5 , which considers mini - bal column densities up to @xmath81 @xmath1 ) . we discuss the implications these density results in 6 below . here we address an important caveat : shielding gas might cover only portions of the spatially - stratified emitting regions . some authors have argued that the strong x - ray absorption in bal quasars is caused by a compact , high column density medium that covers the x - ray emission source ( near the central smbh ) without covering the near - uv and visible continuum regions ( farther out in the accretion disk , * ? ? ? * ; * ? ? ? this picture is appealing for extreme cases like felobals where the x - ray absorber appears to be compton thick even though the quasars are bright in the near - uv and visible ( * ? ? ? * ; * ? ? ? * also lou et al . 2013 , apj , in press ; hamann et al . , in prep . ) . it might also explain sources where the outflow velocities measured in the uv and x - rays are dramatically different ( e.g. , * ? ? ? * ; * ? ? ? the minimum requirement for a shield in any geometry is that it covers ( at least ) the far - uv emission source as seen from the outflow to prevent over - ionization . in bal quasars , the observed high column density x - ray absorbers might reasonably cover the far - uv emission source without also covering the more extended near - uv and visible emission regions . this type of geometry might allow there to be significant shielding at rest in the quasar frame with no near - uv absorption lines near @xmath77 ( e.g. , for bals with detached troughs ) . however , in nal and mini - bal outflows the required geometry seems implausible because the shield would need to cover _ only _ the unobserved far - uv continuum source ( essential for shielding ) while not covering the near - uv and x - ray emitting regions ( as seen by the observer ) . one can imagine geometries that might do this , but they all appear to have serious problems . a thorough discussion of these possibilities is beyond the scope of the present paper . the main result from 5 is that nal and mini - bal outflows are not significantly shielded in the far - uv and , therefore , the outflow ionizations are not moderated by the presence of a shield . this disagrees with bal outflow models that rely on shielding at the base of the outflow to maintain sufficient opacities for radiative driving ( 1 ) . evidently , nal and mini - bal outflows are accelerated to high speeds like bals without the benefits of a radiative shield . one might suppose that this conclusion could be avoided by variable shielding , e.g. , due to transverse motions of a patchy shielding medium near the continuum source @xcite . in that situation , the acceleration could occur during a past period of strong shielding , under bal - like outflow conditions , while today we see the flow already at high speeds without a significant shield . however , this scenario does not explain the moderate degrees of ionization ( ) that we observe in the outflows now . as soon as the shield goes away , the resulting over - ionization of the outflow should appear ( to an observer ) roughly instantaneously because the absorption line gas lies along our lines of sight to the ionizing continuum source . another possibility is that the nal and mini - bal gas was accelerated in a location with more shielding ( e.g. , nearer the accretion disk plane ) before moving to its observed location ( farther above the disk ) where shielding is negligible @xcite . this would be consistent with suggestions ( based on variability ) that mini - bals form in small blobs or filaments along the ragged upper boundary of the main bal outflow @xcite . however , it is not clear what vertical force would be available to push these mini - bal blobs away from the disk plane _ after _ they have been accelerated to speeds @xmath210.1c . radiative forces can only drive the flow away from the uv continuum source , directly toward the observer . the flow trajectories might be curved by strong magnetic fields , but that describes a different model where magnetic forces also dominate the acceleration @xcite . the essential point is that nal and mini - bal outflows have moderate degrees of ionization , similar to bals , even though they are not behind a radiative shield . if a shield is not needed to maintain these moderate outflow ionizations , then it is also not needed to facilitate the acceleration . the tendency for over - ionization can be seen by considering a smooth and continuous flow ( volume filling factor unity ) with radial thickness of order @xmath82 at @xmath83 pc . if this flow has a conservatively large total column density with @xmath84 @xmath1 , the volume density would be only @xmath85 @xmath60 . more realistic outflow columns with @xmath3 @xmath1 would imply @xmath86 @xmath60 . if there is no shielding , the ionization parameter in this gas would be conservatively @xmath87 ( ignoring minor effects of geometric dilution , eqn . 2 ) . figure 5 shows that these flow densities would lead to complete over - ionization , with no detectable or absorption , even if the flow resides behind the maximum shield allowed by our data . this gas would have no possibility for radiative acceleration to high speeds because it is too transparent ( * ? ? ? * hamann et al . , in prep . ) . we conclude that the outflow ionizations are kept moderate by high densities in discrete clouds or substructures . the required densities are shown in figure 5 . if the mini - bal regions have conservatively @xmath88 @xmath1 at our fiducial radius @xmath83 pc , then the minimum densities needed to produce a mini - bal are @xmath89 @xmath60 if the gas is not shielded or @xmath90 @xmath60 behind a maximum shield . the total radial extents of these mini - bal clouds are only @xmath91 cm or @xmath92 cm in the unshielded and shielded cases , respectively . more realistic mini - bal column densities @xmath93 @xmath1 require higher densities resulting in total cloud thicknesses @xmath94 cm or @xmath7@xmath95 cm and radial filling factors of only @xmath96 or @xmath7@xmath8 for the unshielded and maximum shield situations , respectively . these inferred cloud sizes and densities clearly depend on the outflow radius . at smaller radii , nearer the expected launch point of the outflows ( 4.1 ) , the total cloud thicknesses and filling factors are much smaller because the minimum densities scale like @xmath97 ( eqn . 2 ) . at large radii , much higher densities are still needed to moderate the ionizations compared to what is expected from the simple scaling @xmath98 for continuous flows . in particular , the derived radial filling factor @xmath99 scales like @xmath100 . even the recent models that place some felobal outflows at @xmath31kpc distances require small clouds with radial filling factors @xmath101 @xcite . in the transverse direction , the outflow sizes are constrained by the size of the quasar continuum source . mini - bal troughs that reach @xmath915% below the continuum imply absorbing regions that cover @xmath915% of the continuum source in projected area . for the quasar luminosities in our sample , the uv continuum region at @xmath102 should have radius @xmath103 cm ( intermediate between the predictions of standard accretion disk models and the larger numbers from microlensing observations , e.g. , * ? ? ? this means that the transverse width of the mini - bal flow is at least @xmath104 cm , where @xmath105 is our viewing angle of the disk measured from the polar axis and the numerical result assumes @xmath106 . altogether these estimates imply that the outflow regions are thin and wide like `` pancakes '' viewed face on , or they occupy larger volumes like a fine spray of tiny clouds with a very small volume filling factor ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . these arguments for small dense clouds exactly parallel early photoionization studies of the broad emission line regions ( where densities @xmath107 @xmath60 are required , e.g. , * ? ? ? * ) and studies of bal outflows before the shielding model was introduced @xcite . small dense clouds present a major theoretical challenge to understand how they are created and maintained @xcite . they also pose an observational challenge to explain the smooth appearance of broad absorption and broad emission line profiles . if the clouds individually have only thermal velocity dispersions ( roughly 15 km s@xmath11 for hydrogen in a photoionized gas with @xmath108 k ) , then the cloud numbers required for smooth line profiles are estimated to be @xmath9@xmath109 in bals @xcite and @xmath21@xmath110 or @xmath21@xmath111 for the broad emission lines @xcite . the shielding model introduced by @xcite and @xcite was designed to avoid these problems for quasar outflows by permitting moderate ionization levels at low gas densities in smooth continuous outflow streams . however , the absence of significant shielding in nal and mini - bal outflows means that tiny dense clouds are once again required . if we accept the prevailing view that bals , nals , and mini - bals are all part of the same general outflow phenomenon , based on their many observational similarities ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , then we need to consider that dense substructures and small volume filling factors are a common characteristic of all quasar outflows . these substructures might resemble the small - scale clumps inferred from fluctuations in the line - of - sight covering fraction in one well - measured bal outflow @xcite . the most promising theoretical scheme to explain dense substructured outflows is probably magnetic confinement in a magnetic disk wind ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? some type of confinement seems necessary because the cloud dissipation times are less than a characteristic flow time , e.g. , @xmath112 yr for @xmath113c and @xmath83 pc . for example , in the extreme case of outflows comprised of a single bullet " cloud the thickness would be @xmath5 cm or @xmath7@xmath114 cm for maximum shielding and mini - bal regions with @xmath3 @xmath1 or @xmath7@xmath115 @xmath1 , respectively , based the minimum densities in figure 5 . if the internal velocity dispersion is similar to the measured line widths , @xmath116 km s@xmath11 , then the dissipation times @xmath117 would be only @xmath70.2% or @xmath72% , respectively , of the flow time given above . in the more likely event that the flows are comprised of many clouds with individual sizes @xmath31@xmath118 , where @xmath119 is the number of clouds , the dissipation times can not be longer than a sound crossing time or @xmath710%/@xmath119 or @xmath7160%/@xmath119 of the flow time ( for the same two cases of maximally shielded mini - bal clouds with @xmath108 k , see eqn . 2 in * ? ? ? * ) . in the @xcite model , which draws upon earlier work on quasar broad emission line regions @xcite , many small clouds with a low volume filling factor are driven out by radiative forces while being confined by magnetic pressure . one advantage of magnetic confinement is that individual clouds can have super - thermal velocity dispersions and , therefore , fewer clouds are needed to explain the observed smooth line profiles ( see * ? ? ? another advantage of magnetic confinement is that the clouds can maintain a roughly constant density and ionization across the acceleration region @xcite . this can be essential to preserve sufficient opacities for radiative driving . in contrast , simple mass continuity in steady - state continuous flows ( with volume filling factor unity and expanding into a fixed solid angle ) requires @xmath120 where @xmath121 is the mass loss rate and @xmath122 is the mass density . while too simple to describe real quasar outflows , this relationship compared to equation 1 shows that the ionization parameter in smooth flows scales approximately with the outflow velocity , i.e. , @xmath123 . thus the ionization conditions needed for radiative acceleration are very fragile : at the locations where flow material is just distant enough or just shielded enough to have low - enough ionizations and sufficient opacities for radiative driving , the subsequent acceleration leads to a dramatic drop in the density ( eqn . 5 ) , a rise in the ionization parameter ( eqn . 1 ) , and a sharp decrease in the acceleration . as a result , smooth flows can easily `` fail '' in the sense of not reaching the gravitational escape speed or not reaching speeds high enough to match observations ( see also * ? ? ? * hamann et al . in prep . ) . it is not clear if current theoretical models with radiative driving have fully solved this problem , even with radiative shielding . it is clear , however , that the problem is avoided altogether if the outflows are composed of small dense clouds confined by external forces . finally , we note that none of this discussion answers the question of why the outflows in our quasar sample reach extreme speeds @xmath210.1@xmath22 . if higher flow speeds require smaller launch radii ( e.g. , by factors of a few compared to normal bal and mini - bal outflows , 1 and 4.1 ) , then what different physical conditions allow that to happen ? one possibility is that extreme speeds are aided by unusually soft far - uv continua ( e.g. , * ? ? ? this might help maintain lower degrees of ionization ( favorable for radiative driving ) closer to the central smbh . a quick inspection of the sdss spectra indicates that the quasars in our sample have weak or absent heii @xmath131640 emission , consistent with softer far - uv continua and higher outflow speeds observed in bals ( * ? ? ? * and a. baskin , private comm . ) . however , reasonable changes to the emitted far - uv spectra can not solve the over - ionization problem discussed above . small dense flow structures are still needed if there is not substantial shielding . we describe rest - frame uv and x - ray spectra of 8 quasars with mini - bal outflows at extreme speeds in the range 0.1c to 0.2c . we constrain basic outflow properties and test the hypothesis that extreme speeds require a strong radiative shield for extreme radiative acceleration . our main results are the following : \1 ) at least 5 of the 7 bona fide mini - bals ( 2 ) varied significantly between observations separated by @xmath312 - 3 years in the quasar rest frame ( 4.1 ) . in one well - studied case , the variability time is @xmath71 yr @xcite . these results are consistent with bal variability studies and , in particular , with an observed trend for greater variability at higher speeds ( e.g. , * ? ? ? \2 ) we interpret the mini - bal variations in terms of outflow clouds crossing our lines of sight ( 4.1 ) . this implies radial distances @xmath124 pc from the central smbh for variability times @xmath73 yr , or @xmath125 pc for times @xmath71 yr @xcite . \3 ) the x - ray absorption is typically weak or moderate , with total neutral - equivalent column densities , @xmath126 @xmath1 , consistent with previous studies of mini - bal and nal outflows but substantially less than bal quasars ( 4.2 ) . thus we find no evidence for strong radiative shielding related to the extreme outflow speeds in our mini - bal sample . \4 ) cloudy photoionization models show that the maximum amounts of shielding consistent with our data are not sufficient to control the outflow ionizations ( 5 ) and , therefore , not important for the acceleration ( 6 ) . these results apply to shields that fully cover the emission source at all wavelengths . however , shielding in more complex geometries seems unlikely for nal and mini - bal outflows because the alleged shield would need tuning to cover the far - uv emission source ( essential for shielding ) while _ not _ blocking the x - ray emission near the central smbh and _ not _ covering the near - uv source farther out in the accretion disk ( 5.3 ) . \5 ) we propose that the outflow ionizations are maintained at moderate levels ( low enough for and absorption lines ) by high gas densities in small outflow substructures ( 6 ) . if the mini - bal gas is at @xmath83 pc with column density @xmath3 @xmath1 behind a maximum shield , then the cloud densities must be @xmath127 @xmath60 corresponding to total radial extents @xmath128 cm and filling factors @xmath129 . if the shielding is negligible , then the cloud properties are @xmath130 @xmath60 , @xmath94 cm , and @xmath96 . compared to transverse absorber sizes @xmath9@xmath10 cm ( based on measured line depths ) , the outflows have geometries like thin pancakes " viewed face on or they occupy much larger volumes like a mist of many dense clouds with a very small volume filling factor . \6 ) in the context of popular models that have bals , mini - bals , and ( some ) outflow nals forming in the same general outflow phenomenon , our results suggest that all types of quasar accretion disk outflows are composed of small dense substructures and that radiative shielding is not important for the acceleration . the best theoretical scheme to explain substructured flows is probably with magnetic confinement in magnetic disk winds ( e.g. , * ? ? ? * ; * ? ? ? we are grateful to alexei baskin , gary ferland , and ari laor for helpful discussions , and to gary ferland for making the cloudy photoionization code available as a public resource . we thank the staffs of the chandra and mdm observatories for their willing help with the observations . this work was supported by nasa through the smithsonian astrophysical observatory award go1 - 12146a / b / c . fh also acknowledges support from the usa national science foundation grant ast-1009628 . me and jc acknowledge support from nsf grant ast-0807993 . g. , eracleous m. , misawa t. , giustini g. , charlton j. , 2012 , in chartas g. , hamann f. , leighly k. m. , eds , agn winds in charleston vol . 460 of astronomical society of the pacific conference series . p. 37 g. p. , bautz m. w. , ford p. g. , nousek j. a. , ricker jr . g. r. , 2003 , in truemper j. e. , tananbaum h. d. , eds , society of photo - optical instrumentation engineers ( spie ) conference series vol . 4851 , advanced ccd imaging spectrometer ( acis ) instrument on the chandra x - ray observatory . p. 28 f. , barlow t. , cohen r. d. , junkkarinen v. , burbidge e. m. , 1997a , in arav n. , shlosman i. , weymann r. j. , eds , mass ejection from active galactic nuclei vol . 128 of astronomical society of the pacific conference series , p. 19 v. , cohen r. d. , barlow t. a. , hamann f. , 2002 , in crenshaw d. m. , kraemer s. b. , george i. m. , eds , mass outflow in active galactic nuclei : new perspectives vol . 255 of astronomical society of the pacific conference series , p. 167 s. , green p. j. , arav n. , brotherton m. , crenshaw m. , dekool m. , elvis m. , goodrich r. w. , hamann f. , hines d. c. , kashyap v. , korista k. , peterson b. m. , shields j. c. , shlosman i. , van breugel w. , voit m. , 2000 , apj , 533 , l79 d. , kurosawa r. , 2010 , in maraschi l. , ghisellini g. , della ceca r. , tavecchio f. , eds , accretion and ejection in agn : a global view vol . 427 of astronomical society of the pacific conference series , p. 41 g. t. , lacy m. , storrie - lombardi l. j. , hall p. b. , gallagher s. c. , hines d. c. , fan x. , papovich c. , vanden berk d. e. , trammell g. b. , schneider d. p. , vestergaard m. , york d. g. , jester s. , anderson s. f. , budavri t. , szalay a. s. , 2006 , apjs , 166 , 470 p. , hamann f. , eracleous m. , capellupo d. , charlton j. , shields j. , 2012 , in chartas g. , hamann f. , leighly k. m. , eds , agn winds in charleston vol . 460 of astronomical society of the pacific conference series , p. 93

quasar accretion disk winds observed via broad absorption lines ( bals ) in the uv produce strong continuous absorption in x - rays . the x - ray absorber is believed to serve critically as a radiative shield to keep the outflow ionizations low enough for radiative driving . however , previous studies have shown that mini - bal " and narrow absorption line ( nal ) outflows have dramatically less x - ray absorption than bals . here we examine x - ray and rest - frame uv spectra of 8 mini - bal quasars with outflow speeds in the range 0.1c to 0.2c to test the hypothesis that these extreme speeds require a strong shield . we find that the x - ray absorption is weak or moderate , with neutral - equivalent column densities @xmath0 @xmath1 , consistent with mini - bals at lower speeds . we use photoionization models to show that the amount of shielding consistent with our data is too weak to control the outflow ionizations and , therefore , it is not important for the acceleration . shielding in complex geometries also seems unlikely because the alleged shield would need to extinguish the ionizing far - uv flux while avoiding detection in x - rays and the near - uv . we argue that the outflow ionizations are kept moderate , instead , by high gas densities in small clouds . if the mini - bals form at radial distances of order @xmath2 pc from the central quasar ( broadly consistent with theoretical models and with the mini - bal variabilities observed here and in previous work ) , and the total column densities in the mini - bal gas are @xmath3 @xmath1 , then the total radial extent of outflow clouds is only @xmath4 cm in cases of no / weak shielding or @xmath5 cm behind the maximum shield allowed by our data . this implies radial filling factors @xmath6 or @xmath7@xmath8 for the unshielded or maximally shielded cases , respectively . compared to the transverse sizes @xmath9@xmath10 cm ( based on measured line depths ) , the outflows have shapes like thin pancakes " viewed face - on , or they occupy larger volumes like a spray of many dense clouds with a small volume filling factor . these results favor models with magnetic confinement in magnetic disk winds . to the extent that bals , mini - bals , and nals probe the same general outflow phenomenon , our result for dense substructures should apply to all three outflow types . [ firstpage ] galaxies : active quasars : general quasars : absorption lines

a key step in the star formation process is the production of cold dense cores of molecular gas and dust @xcite . cores which do not contain a stellar object are referred to as _ starless _ ; an important subset of these consists of _ prestellar _ cores , i.e. , those cores which are gravitationally - bound and therefore present the initial conditions for protostellar collapse . observations of cold cores are best made in the submillimetre regime in which they produce their peak emission , and observations made with ground - based telescopes have previously helped to establish important links between the stellar initial mass function ( imf ) and the core mass function ( cmf ) @xcite . with the advent of _ herschel _ @xcite , however , these cores can now be probed with high - sensitivity multiband imaging in the far infrared and submillimetre , and hence the cmf can be probed to lower masses than before . one of the major goals of the _ herschel _ gould belt survey @xcite is to characterise the cmf over the densest portions of the gould belt . this survey covers 15 nearby molecular clouds which span a wide range of star formation environments ; preliminary results for aquila have been reported by @xcite . key programme , hobys ( _ herschel _ imaging survey of ob young stellar objects " ) @xcite , is aimed at more massive dense cores and the initial conditions for high - mass star formation , and preliminary results have been presented by @xcite . the taurus molecular cloud is a nearby region of predominantly non - clustered low - mass star formation , at an estimated distance of 140 pc @xcite , in which the stellar density is relatively low and objects can be studied in relative isolation . its detailed morphology at _ herschel _ wavelengths is discussed by @xcite . the region is dominated by two long ( @xmath7 ) , roughly parallel filamentary structures , the larger of which is the northern structure . early results from _ herschel _ regarding the filamentary properties have been reported by @xcite . in this paper we focus on the starless core population of the field with particular interest in core structure and star - forming potential . our analysis is based on observations of the western portion of the northern filamentary structure , designated as n3 in @xcite , which includes the lynds cloud l1495 and contains barnard clouds b211 and b213 as prominent subsections of the filament . our analysis involves a sample of 20 cores which we believe to be representative of relatively isolated cores in this region . the principal aims of the study are as follows : 1 . accurate mass estimation based on models which take account of radial temperature variations and which use spatial and spectral data ; 2 . a comparison of these results with those from simpler techniques commonly used for estimating the core mass function in order to provide a calibration benchmark for such techniques ; 3 . investigation of processes such as heating of the dust by the interstellar radiation field ( isrf ) and the effect of temperature gradients on core stability ; 4 . examination of the results in the context of other observations of the same cores where possible , particularly with regard to gaining insight into the relationship between the dust and gas . the estimation of the core density and temperature structures is achieved using our newly developed technique , corefit , complementary in some ways to the recently used abel transform method @xcite . before discussing corefit and its results in detail , we first describe our observations and core selection criteria . the observational data for this study consists of a set of images of the l1495 cloud in the taurus star - forming region , made on 12 february , 2010 and 78 august , 2010 , during the course of the _ herschel _ gould belt survey ( hgbs ) . the data were taken using pacs @xcite at 70 @xmath0 m and 160 @xmath0 m and spire @xcite at 250 @xmath0 m , 350 @xmath0 m , and 500 @xmath0 m in fast - scanning ( 60 arcsec / s ) parallel mode . herschel _ observation ids were 1342202254 , 1342190616 , and 1342202090 . an additional pacs observation ( i d 1342242047 ) was taken on 20 march 2012 to fill a data gap . calibrated scan - map images were produced in the hipe version 8.1 pipeline @xcite using the scanamorphos @xcite and naive " map - making procedures for pacs and spire , respectively . a detailed description of the observational and data reduction procedures is given in @xcite . the first step in our core selection procedure consists of source extraction via the _ getsources _ algorithm @xcite which uses the images at all available wavelengths simultaneously . these consist of the images at all five _ herschel _ bands plus a column density map which is used as if it were a sixth band , the purpose being to give extra weight to regions of high column density in the detection process . the column density map itself is obtained from the same set of spire / pacs images , using the procedure described by @xcite which provides a spatial resolution corresponding to that of the 250 @xmath0 m observations . the detection list is first filtered to remove unreliable sources . this is based on the value of the global goodness " parameter @xcite which is a combination of various quality metrics . it incorporates the quadrature sums of both the detection significance " and signal to noise ratio ( @xmath8 ) over the set of wavebands , as well as some contrast - based information . the detection significance " is defined with respect to a spatially bandpass - filtered image , the characteristic spatial scale of which matches that of the source itself . at a given band , the detection significance is then equal to the ratio of peak source intensity to the standard deviation of background noise in this image . the @xmath8 is defined in a similar way , except that it is based on the observed , rather than filtered , image . for present purposes we require a global goodness " value greater than or equal to 1 . a source satisfying this criterion may be regarded as having an overall confidence level @xmath9 and can therefore be treated as a robust detection . classification as a core for the purpose of this study then involves the following additional criteria : 1 . detection significance ( as defined above ) greater than or equal to 5.0 in the column density map ; 2 . detection significance greater than or equal to 5.0 in at least _ two _ wavebands between 160 @xmath0 m and 500 @xmath0 m ; 3 . detection significance _ less than _ 5.0 for the 70 @xmath0 m band and no visible signature on the 70 @xmath0 m image , in order to exclude protostellar cores , i.e. , those cores which contain a protostellar object ; 4 . ellipticity less than 2.0 , as measured by _ getsources _ ; 5 . source not spatially coincident with a known galaxy , based on comparison with the nasa extragalactic database . this procedure resulted in a total of 496 cores over the observed @xmath10 region . the total mass , 88 @xmath11 , of the detected cores represents approximately 4% of the mass of the l1495 cloud , estimated to be 15002700 @xmath11 @xcite . from this set , 20 cores were selected for detailed study . the main goal of the final selection process was to obtain a list of relatively unconfused cores , uniformly sampled in mass according to preliminary estimates obtained via sed fitting as outlined in the next section . cores which were multiply peaked or confused , based on visual examination of the 250500 @xmath0 m images , were excluded . the mass range 0.022.0 @xmath11 was then divided into seven bins , each of which spanned a factor of two in mass , and a small number of objects ( nominally three ) selected from each bin . the selection was made on a random basis except for a preference for objects for which previously - published data were available , thus facilitating comparison of deduced parameters . [ fig1 ] shows the locations of the 20 selected cores on a spire 250 @xmath0 m image of the field . m image of the l1495 region . the green circles represent the locations of the 20 cores selected for modeling . the other symbols represent previously published core locations at other wavelengths ; _ red squares _ : h@xmath3co@xmath4 @xcite ; _ blue triangles _ : n@xmath6h@xmath4 @xcite ; _ yellow cross _ : 850 @xmath0 m @xcite . the image is shown on a truncated intensity scale in order to emphasize faint structure ; the display saturates at 200 mjy sr@xmath12 which corresponds to 100% on the greyscale.,width=317 ] preliminary values of core masses and dust temperatures are estimated by fitting a greybody spectrum to the observed spectral energy distribution ( sed ) constructed from the set of five - wavelength _ getsources _ fluxes . for this computation , sources are assumed to be isothermal and have a wavelength variation of opacity of the form @xcite : @xmath13})^2\qquad{\rm cm}^2\,{\rm g}^{-1 } \label{eq1}\ ] ] although obtained observationally , the numerical value of the coefficient in this relation is consistent with a gas to dust ratio of 100 . to obtain better estimates of core mass and other properties , a more detailed model fit is required . for this purpose we have developed a new procedure , corefit , which involves maximum likelihood estimation using both spatial and spectral information . the fitting process involves calculating a series of forward models , i.e. , sets of model images based on assumed parameter values , which are then compared with the data . the models are based on spherical geometry , in which the radial variations of volume density and temperature are represented by parametrized functional forms . for a given set of parameters , a model image is generated at each of the five wavelengths by calculating the emergent intensity distribution on the plane of the sky and convolving it with the instrumental point spread function ( psf ) at the particular wavelength . the parameters are then adjusted to obtain an inverse - variance weighted least squares fit to the observed images . in this procedure the wavelength variation of opacity is assumed to be given by eq . ( [ eq1 ] ) and the the radial variations of volume density and dust temperature are assumed to be described by plummer - like @xcite and quadratic forms , respectively . specifically we use : @xmath14 \label{eq2 } \\ t(r ) & = & t_0 + ( t_1\!-\!t_0\!-\!t_2)r / r_{\rm out } + t_2(r / r_{\rm out})^2 \label{eq3}\end{aligned}\ ] ] where @xmath15 represents the number density of h@xmath6 molecules at radial distance @xmath16 , @xmath17 represents the radius of an inner plateau , and @xmath18 is the outer radius of the core , outside of which the core density is assumed to be zero . also , @xmath1 is the central core temperature , @xmath19 is the temperature at the outer radius , and @xmath20 is a coefficient which determines the curvature of the radial temperature profile . in relating @xmath15 to the corresponding profile of mass density we assume a mean molecular weight of 2.8 @xcite . the set of unknowns then consists of : @xmath21 , @xmath17 , @xmath18 , @xmath22 , @xmath1 , @xmath19 , @xmath20 , @xmath23 , @xmath24 , where the latter two variables represent the angular coordinates of the core centre . representing this set by an 9-component parameter vector , * p * , we can write the measurement model as : @xmath25 where @xmath26 is a vector representing the set of pixels of the observed image at wavelength @xmath27 , @xmath28 represents the model core image for parameter set * p * , and @xmath29 is the measurement noise vector , assumed to be an uncorrelated zero - mean gaussian random process . also , @xmath30 represents the local background level , estimated using the histogram of pixel values in an annulus surrounding the source . this measurement model assumes implicitly that the core is optically thin at all wavelengths of observation . in principle , the solution procedure is then to minimise the chi squared function , @xmath31 , given by : @xmath32 ^ 2 / \sigma_\lambda^2 \label{eq5}\ ] ] where subscript @xmath33 refers to the @xmath33th pixel of the image at the given wavelength and @xmath34 represents the standard deviation of the measurement errors , evaluated from the sky background fluctuations in the background annulus . in practice , two difficulties arise : 1 . an unconstrained minimisation of @xmath31 is numerically unstable due to the fact that for a given total number of molecules , @xmath21 in eq . ( [ eq2 ] ) becomes infinite as @xmath35 . it results in near - degeneracy such that the data do not serve to distinguish between a large range of possible values of the central density . to overcome this , we have modified the procedure to incorporate the constraint @xmath36 , where @xmath37 is equal to one quarter of the nominal angular resolution , which we take to be the beamwidth at 250 @xmath0 m . the estimate of central density then becomes a beam - averaged " value over a resolution element of area @xmath38 . for a distance of 140 pc , @xmath37 corresponds to about 600 au . most cores show some degree of asymmetry . this can degrade the quality of the global fit to a spherically - symmetric model , causing the centre of symmetry to miss the physical centre of the core . some negative consequences include an underestimate of the central density and an overestimate of the central temperature . to alleviate this , we estimate the @xmath39 location of the core centre ahead of time using the peak of a column density map , constructed at the spatial resolution of the 250 @xmath0 m image . the maximum likelihood estimation is then carried out using a 7-component parameter vector which no longer involves the positional variables . having performed the position estimation and constrained chi squared minimisation , the core mass is then obtained by integrating the density profile given by eq . ( [ eq2 ] ) , evaluated using the estimated values of @xmath21 , @xmath17 and @xmath22 . evaluation of the uncertainties in parameter estimates is complicated by the nonlinear nature of the problem which leads to a multiple - valley nature of @xmath31 . the usual procedure , in which the uncertainty is evaluated by inverting a matrix of 2nd derivatives of @xmath31 @xcite , then only provides values which correspond to the width of the global maximum and ignores the existence of neighbouring peaks which may represent significant probabilities . we therefore evaluate the uncertainties using a monte carlo technique in which we repeat the estimation procedure after adding a series of samples of random noise to the observational data and examine the effect on the estimated parameters . we have also implemented an alternate version of corefit , referred to as corefit - ph , " in which the dust temperature profile is based on a radiative transfer model , phaethon @xcite , rather than estimating it from the observations . in this model , the radial density profile has the same form as for the standard corefit ( eq . ( [ eq2 ] ) ) but with the index , @xmath22 , fixed at 2 . the temperature profile is assumed to be determined entirely by the heating of dust by the external isrf ; the latter is modeled locally as a scaled version of the standard isrf @xcite using a scaling factor , @xmath40 , which represents an additional variable in the maximum likelihood solution . we now compare the results obtained using the two approaches , both for synthetic and real data . we have tested both corefit and corefit - ph against synthetic data generated using an alternate forward model for dust radiative transfer , namely modust ( bouwman et al . , in preparation ) . using the latter code , images at the five wavelengths were generated for a set of model cores and convolved with gaussian simulated psfs with full width at half maxima ( fwhm ) corresponding to the _ herschel _ beamsizes . the models involved central number densities of @xmath41 @xmath42 , @xmath43 @xmath42 , and @xmath44 @xmath42 with corresponding @xmath17 values of 2500 au , 4000 au , and 1000 au , respectively , and @xmath18 values of @xmath45 au , @xmath46 au , and @xmath47 au , respectively . the corresponding core masses were 0.59 @xmath11 , 18.37 @xmath11 , and 3.11 @xmath11 , respectively . the synthesized images and corresponding gaussian psfs were then used as input data to the inversion algorithms . the results are presented in table 1 . it is apparent that corefit gave masses and central temperatures in good agreement with the true model . while corefit - ph reproduced the central temperatures equally well , it underestimated the masses of these simulated cores by factors of 0.7 , 0.5 , and 0.5 , respectively . the reason for these differences is that even though the two radiative transfer codes ( phaethon and modust ) yield central temperatures in good agreement with each other for a given set of model parameters , they produce divergent results for the dust temperatures in the outer parts of the cores , due largely to differences in dust model opacities . since the outer parts comprise a greater fraction of the mass than does the central plateau region , this can lead to substantially different mass estimates given the same data . this problem does not occur for corefit since the latter obtains the temperature largely from the spectral variation of the data rather than from a physical model involving additional assumptions . these calculations thus serve to illustrate the advantages of simultaneous estimation of the radial profiles of dust temperature and density . table 2 shows the complete set of corefit parameter estimates for each of the taurus cores . also included are the assumed values of the inner radius of the annulus used for background estimation , equal to the _ getsources _ footprint size . table 3 shows a comparison of the mass and temperature estimates amongst the different techniques , which include corefit and corefit - ph as well as the sed fitting discussed in section 3 . to facilitate comparison between the corefit temperatures and the mean core temperatures estimated from the spatially integrated fluxes used in the sed fits , we include the spatially averaged corefit temperature , @xmath48 , defined as the density - weighted mean value of @xmath49 for @xmath50 . the corefit - ph results include the values of the isrf scaling factor , @xmath40 , the median value of which is 0.33 . the fact that this is noticeably less than unity can probably be attributed to the fact that these cores are all embedded in filaments and hence the local isrf is attenuated by overlying filamentary material . as an example of the fitting results , figs . [ fig2 ] and [ fig3 ] show the estimated density and temperature profiles for core no . 2 in table 2 , based on corefit and corefit - ph , respectively . [ fig4 ] shows that the two techniques yield consistent estimates of masses , but the radiative transfer calculations produce central temperatures which are , on average , @xmath51 k lower than the corefit estimates . although the difference is not significant in individual cases ( the standard deviation being 1.4 ) , it is clear from fig . [ fig4 ] that a systematic offset is present ; the mean temperature difference , @xmath52 , is @xmath53 k. based on the results of testing with synthetic data , this difference seems too large to be explained by systematic errors associated with dust grain models , although we can not rule out that possibility . one could also question whether our @xmath40 values are spuriously low . we do , in fact , find that by forcing the latter parameter to a somewhat larger value ( 0.5 ) , the median @xmath52 can be reduced to zero with only a modest increase in the reduced chi squared , @xmath54 ( 0.85 as opposed to 0.83 for the best fit ) . the observations are completely inconsistent with @xmath55 , however . as an additional test , we can take the corefit estimate of the radial density distribution for each core and use the standalone phaethon code to predict the central temperature for any assumed value of @xmath40 . we thereby obtain consistency with the corefit estimates with @xmath56 . however , this consistency comes at significant cost in terms of goodness of fit ( the median @xmath54 increases to 2.27 ) , and therefore does _ not _ serve to reconcile the corefit results with the expectations of radiative transfer . in summary , the corefit results are not entirely consistent with our assumed model for dust heating by the isrf , but further work will be necessary to determine whether the differences are model - related or have astrophysical implications . so at this stage we have no evidence to contradict the findings of @xcite who considered various heating sources ( the primary and secondary effects of cosmic rays and heating of dust grains by collisions with warmer gas particles ) and concluded that heating by the isrf dominates over all other effects . how do the corefit estimates of temperature and mass compare with the preliminary values estimated from the _ getsources _ seds ? in the case of temperature , the relevant comparison is between the sed - derived value and the spatially averaged corefit value ; the data in table 3 then give a mean corefit minus sed " difference of -0.2 k , with a standard deviation of 1.1 for individual cores . the temperature estimates are thus consistent . with regard to mass , fig . [ fig5 ] shows that sed fitting under the isothermal assumption yields masses that are systematically smaller than the corefit values ; the mean ratio of corefit mass to sed - based mass is 1.5 , with a standard deviation of 1.0 in individual cases . since the internal temperature gradient increases with the core mass , one might expect that the correction factor for sed - derived masses would increase with mass , although fig . [ fig5 ] has too much scatter to establish this . it may be evident when the results are averaged for a much larger statistical sample of cores , although the correction may well depend on environmental factors such as the intensity of the local isrf . [ fig6 ] shows a plot of estimated central temperature as a function of core mass . linear regression indicates that these quantities are negatively correlated with a coefficient of -0.64 . this correlation can be explained quite naturally as a consequence of increased shielding of the core , from the isrf , with increasing core mass . this being the case , one would expect an even stronger correlation with peak column density and this is confirmed by fig . [ fig7 ] , for which the associated correlation coefficient is -0.86 . fig . [ fig8 ] shows a plot of @xmath22 versus mass , where @xmath22 is the index of radial density variation as defined by eq . ( [ eq2 ] ) , and the masses are the corefit values . given the relatively large uncertainties , the @xmath22 values are , for the most part , consistent with values expected for bonnor - ebert spheres , whereby @xmath57 provides an accurate empirical representation at radial distances up to the instability radius @xcite , and that @xmath22 decreases to its asymptotic value of 2 beyond that . core in l1495 ( no . 2 in table 2 ) . the solid lines indicate maximum likelihood estimates of the profiles of relative volume density and dust temperature . the dashed lines provide a measure of the uncertainty in the estimated density and temperature . they represent the results of a monte carlo simulation in which the estimation procedure is repeated 10 times after adding synthetic measurement noise to the observed images ; the standard deviation of the added noise corresponds to the estimated measurement noise of the observed images.,width=332 ] . in this variant of the estimation procedure , the dust temperature profile is modeled using a radiative transfer code ( phaethon ) instead of estimating it from the observations . , width=332 ] versus mass , where @xmath52 represents the difference in estimated temperature ( corefit minus corefit - ph).,width=332 ] , as a function of core mass.,width=332 ] , as a function of peak column density of h@xmath6 molecules.,width=332 ] , as a function of core mass.,width=332 ] the general consistency with the bonnor - ebert model is supported by the fact that when the maximum likelihood fitting procedure is repeated using the constraint @xmath58 , the chi squared values are , in most cases , not significantly different from the values obtained when @xmath22 is allowed to vary . two exceptions , however , are cores 2 and 13 , both of which are fit significantly better by density profiles steeper than bonnor - ebert ( @xmath59 and 2.8 , respectively ) , as illustrated by fig . [ fig9 ] for the former case . specifically , the chi squared , was taken as the total number of resolution elements contained within the fitted region for all five input images ; @xmath60 is then @xmath61 and @xmath62 for the two cases , respectively . ] differences ( 17.2 and 7.5 , respectively ) translate into relative probabilities , for the @xmath63 " hypothesis , of @xmath64 and 0.02 , respectively . if confirmed , such behaviour may have some important implications for core collapse models ; a steepening of the density distribution in the early collapse phase is , in fact , predicted by the model of @xcite in which the collapsing core begins to detach from its outer boundary . . _ solid line : _ observed profile ; _ dashed line : _ best fitting model ( @xmath59 ) convolved with the corresponding psf at each wavelength ) ; _ dotted line : _ same , except for the constraint @xmath65 . note that the latter model results in a poor fit in the central region.,width=359 ] assessments of core stability are frequently made using sed - based estimates of core mass and temperature and observed source size , assuming that cores are isothermal and can be described as bonnor - ebert spheres @xcite . using the sed - based data in table 3 in conjunction with the _ getsources _ estimates of core size , we thereby find that the estimated core mass exceeds the bonnor - ebert critical mass for 10 of our 20 cores , suggesting that half of our cores are unstable to gravitational collapse . our corefit parameter estimates enable us to make a more detailed assessment of core stability based on a comparison with the results of hydrostatic model calculations that take account of the non - isothermal nature of the cores . this is facilitated by the modified bonnor - ebert ( mbe ) sphere models of @xcite . adopting their model curves , based on the @xcite grains which best reproduce our estimated core temperatures , the locus of critical non - isothermal models on a density versus mass plot is shown by the solid line on fig . [ fig10 ] . also plotted on this figure , for comparison , are the corefit estimates of those quantities . the seven points to the right of this curve represent cores that we would consider to be gravitationally unstable based on the modified bonnor - ebert models . although this is somewhat less than the 10 that were classified as unstable based on the sed fits , the difference is probably not significant given that several points on the plot lie close to the stability " line . molecules as a function of core mass . the circles represent the corefit estimates for l1495 ; filled symbols designate the subset of cores whose preliminary assessment of dynamical status suggests that they are gravitationally bound , based on _ getsources _ fluxes and sizes in conjunction with the standard model of isothermal bonnor - ebert spheres . for comparison , the solid line represents the locus of critically - stable non - isothermal bonnor - ebert spheres ( sipil , harju & juvela ( 2011 ) ; points to the right of this line represent cores which are unstable to gravitational collapse according to that model.,width=332 ] the consistency between the above two procedures for stability assessment is illustrated by the fact that the mbe stability line in fig . [ fig10 ] provides a fairly clean demarcation between the cores classified as stable ( open circles ) and unstable ( filled circles ) from the simpler ( sed - based ) procedure . these results therefore suggest that prestellar cores can be identified reliably as such using relatively simple criteria . the bonnor - ebert model also provides a stability criterion with respect to the centre - to - edge density contrast , whereby values greater than 14 indicate instability to gravitational collapse , both for the isothermal and non - isothermal cases @xcite . however the outer boundaries are not well defined for the present sample of cores , and consequently the contrast values are uncertain in most cases . two exceptions are cores 2 and 13 , both of which have contrast estimates whose significance exceeds @xmath66 . in both cases the mass exceeds the bonnor - ebert critical mass ( by ratios of 1.2 and 6.0 , respectively ) , and the centre - to - edge contrast values ( @xmath67 and @xmath68 , respectively ) are in excess of 14 . so for those two cores , at least , the core stability deduced from the density constrast is thus consistent with that assessed from total mass . [ cols=">,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] considering first the h@xmath3co@xmath4 data , comparison of observed peak locations with dust continuum source positions from corefit shows a distinct lack of detailed correspondence . this behaviour is apparent in fig . [ fig1 ] and from table 4 which includes the angular distance ( labeled as offset " in the table ) between each of the h@xmath3co@xmath4 source locations and the corresponding dust continuum core location . the median distance is @xmath69 , considerably larger than the spatial resolution of either the h@xmath3co@xmath4 observations ( @xmath70 ) or the _ herschel _ data ( @xmath71 at 250 @xmath0 m ) . these offsets are somewhat surprising , since previous comparisons between h@xmath3co@xmath4 and dust continuum maps have shown good correspondence @xcite . one could question whether they are due to gridding errors in the h@xmath3co@xmath4 data , in view of the fact that the observations were made on a relatively coarse grid ( the eight cores of @xcite in table 4 are split evenly between @xmath72 and @xmath73 grid spacings ) . however , the measured offsets show no correlation with the grid spacing the mean offset is approximately @xmath74 in either case ; this argues against gridding error as an explanation . the most likely reason for the systematic offsets is that the h@xmath3co@xmath4 is frozen out at the low ( @xmath75 k ) temperatures of the core centres @xcite . detailed comparison of the dust continuum core locations with the h@xmath3co@xmath4 maps ( four examples of which are given in fig . [ fig10 ] ) shows that the majority of sources are elongated and/or double and that in some cases ( onishi core no . 3 in particular ) the dust continuum source falls between the pair of h@xmath3co@xmath4 components . in other cases ( e.g. onishi core no . 16a ) , the dust continuum peak falls on a nearby secondary maximum of the h@xmath3co@xmath4 emission . in the latter case , surprisingly , the main peak of h@xmath3co@xmath4 falls in a local _ minimum _ of dust emission . comparisons between h@xmath3co@xmath4 images and their 250 @xmath0 m counterparts show that , in general , the elongation and source alignment in h@xmath3co@xmath4 is along the filament , so we have a rod - like , rather than spherical , geometry . the picture which thus emerges is that when a core forms in a filament @xcite , we see the core centre in dust continuum emission and the warmer ( but still dense , @xmath76 @xmath42 ) h@xmath3co@xmath4 gas on either side of it in a dumbbell - like configuration . the median separation of the dust continuum and h@xmath3co@xmath4 sources then corresponds to the typical radius of the depletion zone . for an ensemble of randomly oriented filaments , the mean projected separation is @xmath77 times the actual separation , which means that our estimated median separation of @xmath69 corresponds to an actual separation of @xmath78 , or about @xmath79 au at the distance of l1495 . this is similar to the radius of the dark - cloud chemistry zone in which carbon - bearing molecules become gaseous @xcite . co@xmath4 emission . the estimated locations of cores 1 , 13 , 14 and 15 ( corresponding to onishi core nos . 3 , 8 , 9 , and 13a , respectively ) are superposed on h@xmath3co@xmath4 maps taken from @xcite ( b1950 coordinates ) . in each case , the location of peak dust column density is indicated by a red cross . the green cross in onishi field ( 9 ) represents a secondary peak of dust emission . the black cross in onishi field ( 13 ) represents a protostar location.,width=332 ] comparing the estimated masses , table 4 shows that the values estimated from dust continuum observations are , in most cases , much smaller than those based on h@xmath3co@xmath4 . the discrepancy ranges from a factor of @xmath51 to more than an order of magnitude . based on the mass and positional discrepancies it is clear that h@xmath3co@xmath4 and submillimetre continuum are not mapping the same structures . nevertheless , it remains to explain why so much of the expected dust emission from the h@xmath3co@xmath4 emitting gas is apparently not being seen in the submillimetre continuum . it is unlikely to be a result of the background subtraction in corefit since the corefit mass estimates match the sed - based values from _ getsources _ fluxes within @xmath80% and the only background that was subtracted during the latter processing corresponded to the natural spatial scale of the broader underlying emission structure . the most likely explanation for the discrepancy is an overestimation of the virial mass of the gas component due , in part , to the assumption by @xcite of uniform velocity dispersion . specifically , the velocity dispersion of the relatively cool gas being probed by dust emission is likely to be at least a factor of two lower than that of h@xmath3co@xmath4 , as suggested by the n@xmath6h@xmath4 observations of @xcite , and since the estimated virial mass depends on the square of that dispersion , it could have been overestimated by a factor of up to 4 . two additional effects , both of which are likely to have led to overestimation of the virial mass are : 1 . the @xcite virial mass was based on assumed spherical shape as opposed to the filamentary geometry observed , and hence the source volumes may have been overestimated ; 2 . @xcite assumed a constant density value for each core . however , virial models involving this assumption are likely to lead to overestimates of mass in cases where the actual density decreases outwards @xcite . while the h@xmath3co@xmath4 peaks do not correlate well with the dust continuum , the situation is different for n@xmath6h@xmath4 . this behaviour can be seen from table 4 which includes the positional offsets between n@xmath6h@xmath4 and dust continuum peaks ; the median offset is only @xmath81 , i.e. , only a third of the corresponding value for h@xmath3co@xmath4 even though the resolution of the n@xmath6h@xmath4 observations ( @xmath73 ) was much coarser . thus the positional data provide no evidence for n@xmath6h@xmath4 freeze - out , and this is supported by the fact that the n@xmath6h@xmath4 detections seem preferentially associated with the coldest cores ( the seven n@xmath6h@xmath4 detections include four of our five lowest - temperature cores , all of which are cooler than 7 k ) . however , at higher resolution the situation may be different , since interferometric observations of @xmath82 oph have shown that the correspondence between dust emission and n@xmath6h@xmath4 clumps does indeed break down on spatial scales @xmath83 @xcite . theoretical models have , in fact , shown that within @xmath84 au of the core centre , complete freeze - out of heavy elements is likely @xcite . core profiling based on dust emission thus promises to make an important contribution to investigations of core chemistry by providing an independent method for estimating temperatures in the centres of cores . finally , our core no . 16 has been observed previously in the 850 @xmath0 m continuum by @xcite and given the designation jcmtsf_041950.8 + 271130 . while the quoted 850 @xmath0 m source radius of 0.019 pc is close to the corefit @xmath17 value of @xmath85 pc , the estimated masses are significantly different . the estimate of @xcite is based on the observed 850 @xmath0 m flux density and an assumed dust temperature of 13 k ; this yielded 0.22 @xmath11 which is a factor @xmath86 smaller than our corefit value and most likely an underestimate resulting from too high an assumed temperature . this illustrates the large errors in mass which can occur in the absence of temperature information , as has been noted by others @xcite . the principal conclusions from this study can be summarised as follows : 1 . for this sample of cores , the dust temperatures estimated from sed fits , using spatially integrated fluxes and an isothermal model , are consistent with the spatially averaged temperatures derived from the corefit profiles . however , the masses obtained from the sed fits are systematically lower ( by a factor of @xmath87 ) than those obtained from detailed core profiling . the present statistical sample , however , is insufficient to obtain a definitive correction factor , the latter of which is likely to be dependent on mass and possibly environment ( isrf ) also . estimates of central dust temperature are in the range 612 k. these temperatures are negatively correlated with peak column density , consistent with behaviour expected due to shielding of core centre from the isrf , assuming that the latter provides the sole heating mechanism . the model core temperatures obtained from radiative transfer calculations are , however , systematically @xmath51 k lower than the corefit estimates ; it is not yet clear whether that difference has an astrophysical origin or is due to errors in model assumptions . the radial falloff in density is , in the majority of cases , consistent with the @xmath88 variation expected for bonnor - ebert spheres although exceptions are found in two cases , both of which appear to have steeper density profiles . since both involve cores which are gravitationally unstable based on bonnor - ebert criteria , such behaviour may have implications for models of the early collapse phase . 4 . the reliability of core stability estimates derived from isothermal models is not seriously impacted by the temperature gradients known to be present in cores . thus the preliminary classification of cores as gravitationally bound or unbound can be based on relatively simple criteria , facilitating statistical studies of large samples . core locations do not correspond well with previously published locations of h@xmath3co@xmath4 peaks , presumably because carbon - bearing molecules are frozen out in the central regions of the cores , most of which have dust temperatures below 10 k. the results suggest that the h@xmath3co@xmath4 emission arises from dense gas in the filamentary region on either side of the core itself , in a dumbbell - like geometry , and that the radius of the sublimation zone is typically @xmath5 au . the coldest cores are mostly detected in n@xmath6h@xmath4 , and the n@xmath6h@xmath4 core locations are consistent with those inferred from dust emission , albeit at the relatively coarse ( @xmath89 ) resolution of the n@xmath6h@xmath4 data . our data therefore do not show evidence of n@xmath6h@xmath4 freeze - out . we thank t. velusamy , d. li , and p. goldsmith for many helpful discussions during the early development of the corefit algorithm . we also thank the reviewer for many helpful comments . spire has been developed by a consortium of institutes led by cardiff univ . ( uk ) and including : univ . lethbridge ( canada ) ; naoc ( china ) ; cea , lam ( france ) ; ifsi , univ . padua ( italy ) ; iac ( spain ) ; stockholm observatory ( sweden ) ; imperial college london , ral , ucl - mssl , ukatc , univ . sussex ( uk ) ; and caltech , jpl , nhsc , univ . colorado ( usa ) . this development has been supported by national funding agencies : csa ( canada ) ; naoc ( china ) ; cea , cnes , cnrs ( france ) ; asi ( italy ) ; mcinn ( spain ) ; snsb ( sweden ) ; stfc , uksa ( uk ) ; and nasa ( usa ) . pacs has been developed by a consortium of institutes led by mpe ( germany ) and including uvie ( austria ) ; ku leuven , csl , imec ( belgium ) ; cea , lam ( france ) ; mpia ( germany ) ; inafifsi/ oaa / oap / oat , lens , sissa ( italy ) ; iac ( spain ) . this development has been supported by the funding agencies bmvit ( austria ) , esa - prodex ( belgium ) , cea / cnes ( france ) , dlr ( germany ) , asi / inaf ( italy ) , and cicyt / mcyt ( spain ) . 99 andr , p. , ward - thompson , d. , motte , f. 1996 , a&a , 314 , 625 andr , p. , menshchikov , a. , bontemps , s. , et al . 2010 , a&a , 518 , l102 caselli , p. 2011 , in jos cernicharo & rafael bachiller , eds . iau symposium no . 280 , 19 elias , j. h. 1978 , apj , 224 , 857 evans , n. j. ii , rawlings , j. m. c. , shirley , y. l. & mundy , l. g. 2001 , apj , 557 , 193 friesen , r. k. , di francesco , j. , shimajiri , y. & takakuwa , s. 2010 , apj , 708 , 1002 giannini , t. , elia , d. , lorenzetti , d. et al . 2012 , a&a , 539 , a156 griffin , m. j. , abergel , a. , abreu , a. , et al . 2010 , a&a , 518 , l3 griffin , m. j. , north , c. e. , amaral - rogers , a. et al . 2013 , mnras , 434 , 992 hacar , a. , tafalla , m. , kauffmann , j. & kovcs , a. 2013 , a&a , 554 , 55 hildebrand , r. h. 1983 , qjras , 24 , 267 hill , t. , pinte c. , minier , v. , burton , m. g. , cunningham , m. r. 2009 mnras , 392 , 768 hill , t. , longmore , s. n. , pinte , c. , cunningham , m. r. , burton , m. g. , minier , v. 2010 , mnras , 402 , 2682 kirk , j. m. , ward - thompson , d. , palmeirim , p. et al . 2013 , mnras , 432 , 1424 knyves , v. , andr , p. , menshchikov , a. , et al . 2010 , a&a , 518 , l106 kramer , c. & winnewisser , g. 1991 , a&as , 89 , 42 lada , c. j. , muench , a. a. , rathborne , j. , alves , j. f. , & lombardi , m. 2008 , apj , 672 , 410 li , a. , draine , b. t. 2001 , apj , 554 , 778 lutz , d. 2012 , _ herschel _ document picc - me - tn-033 maclaren , i. , richardson , k. m. , wolfendale , a. w. 1988 , apj , 333 , 821 menshchikov , a. , andr , ph . , didelon , p. et al . 2012 , a&a , 542 , 81 motte , f. , andr , neri , r. 1998 , a&a 336 , 150 motte , f. , zavagno , a. , bontemps , s. , et al . 2010 , a&a 518 , l77 onishi , t. , mizuno , a , kawamura , a et al . 2002 , apj 575 , 950 ott , s. , 2010 , in y. mizumoto , k .- morita , & m. ohishi ed . , astronomical data analysis software and systems xix vol . 434 of asp conference series , the _ herschel _ data processing system hipe and pipelines up and running since the start of the mission , p. 139 mizuno , a , kawamura , a et al . 2002 , apj 575 , 950 palmeirim , p. , andr , ph . , kirk , j. et al . 2013 , a&a , 550 , a38 pilbratt , g. l. , riedinger , j. r. , passvogel , t. , et al . 2010 , a&a 518 , l1 plummer , h. c. 1911 , mnras , 71 , 460 poglitsch , a. , waelkens , c. , geis , n. , et al . 2010 , a&a , 518 , l2 rousel , h. 2013 , arxiv:1205.2576 ; pasp ( in press ) roy , a. , andr , ph . , palmeirim , p. , et al . 2013 , a&a , submitted sadavoy , s. i. , di francesco , j. , bontemps , s. , et al . 2010 , apj , 710 , 1247 sipil , o. , harju , j. , juvela , m. 2011 , a&a , 535 , 49 stamatellos , d. & whitworth , a. p. 2003 , a&a , 407 , 941 stamatellos , d. , whitworth , a. p. , ward - thompson , d. 2007 , mnras , 379 , 1390 tafalla , m. , myers , p. c. , caselli , p. , & walmsley , c. m. 2004 , a&a , 416 , 191 umemoto , t. , kamazaki , t. , sunada , k. , kitamura , y. & hasegawa , t. 2002 , in proc . iau 8th asian - pacific regional meeting , asj , vol . ii , 229 vorobyov , e. i. & basu , s. , 2005 , mnras , 360 , 675 walmsley , c. m. , flower , d. r. & pineau des forts , g. 2004 , in s. pfalzner et al . the dense interstellar medium in galaxies , proc . 4th cologne - bonn - zermatt symposium , springer proceedings in physics , 91 , 467 ward - thompson , d. , scott , p. f. , hills , r. e. , andr , p. , 1994 , mnras , 268 , 276 whalen , a. d. 1971 , detection of signals in noise " ( new york : academic press )

the density and temperature structures of dense cores in the l1495 cloud of the taurus star - forming region are investigated using _ herschel _ spire and pacs images in the 70 @xmath0 m , 160 @xmath0 m , 250 @xmath0 m , 350 @xmath0 m and 500 @xmath0 m continuum bands . a sample consisting of 20 cores , selected using spectral and spatial criteria , is analysed using a new maximum likelihood technique , corefit , which takes full account of the instrumental point spread functions . we obtain central dust temperatures , @xmath1 , in the range 612 k and find that , in the majority of cases , the radial density falloff at large radial distances is consistent with the asymptotic @xmath2 variation expected for bonnor - ebert spheres . two of our cores exhibit a significantly steeper falloff , however , and since both appear to be gravitationally unstable , such behaviour may have implications for collapse models . we find a strong negative correlation between @xmath1 and peak column density , as expected if the dust is heated predominantly by the interstellar radiation field . at the temperatures we estimate for the core centres , carbon - bearing molecules freeze out as ice mantles on dust grains , and this behaviour is supported here by the lack of correspondence between our estimated core locations and the previously - published positions of h@xmath3co@xmath4 peaks . on this basis , our observations suggest a sublimation - zone radius typically @xmath5 au . comparison with previously - published n@xmath6h@xmath4 data at 8400 au resolution , however , shows no evidence for n@xmath6h@xmath4 depletion at that resolution . [ firstpage ] stars : formation stars : protostars ism : clouds submillimetre : ism methods : data analysis techniques : high angular resolution .

the recent discovery of the superconductivity in hydrated lamellar cobaltate na@xmath0coo@xmath6h@xmath1o @xcite raised tremendous interest in the nature and symmetry of the superconductive pairing in these materials . the phase diagram of this compound , with varying electron doping concentration @xmath7 and water intercalation @xmath8 , is rich and complicated ; in addition to superconductivity , it exhibits magnetic and charge orders , and some other structural transitions @xcite . the parent compound , na@xmath0coo@xmath1 , is a quasi - two - dimensional system with co in coo@xmath1 layers forming a triangular lattice where the co - co in - plane distance is two times smaller than the inter - plane one . na ions reside between the coo@xmath1 layers and donate additional @xmath7 electrons to the layer , lowering the co valence from co@xmath9 ( 3@xmath10 configuration ) to co@xmath11 ( 3@xmath12 configuration ) upon changing @xmath7 from 0 ( coo@xmath1 ) to 1 ( nacoo@xmath1 ) . the hole in the @xmath13-orbital occupies one of the @xmath14 levels , which are lower than @xmath15 levels by about 2 ev @xcite . the degeneracy of the @xmath2 levels is partially lifted by the trigonal crystal field distortion which splits the former into the higher lying @xmath16 singlet and the lower two @xmath17 states . first principles lda ( local density approximation ) and lda+u band structure calculations predict na@xmath0coo@xmath1 to have a large fermi surface ( fs ) centered around the @xmath18 point with mainly @xmath19 character and six hole pockets near the @xmath20 points of the hexagonal brillouin zone of mostly @xmath17 character for a wide range of @xmath7 @xcite . at the same time , recent surface sensitive angle - resolved photo - emission spectroscopy ( arpes ) experiments @xcite reveals a doping dependent evolution of the fermi surface , which shows no sign of hole pockets for a wide range of na concentrations , i.e. ( @xmath21 . instead , the fermi surface is observed to be centered around the @xmath22 point and to have mostly @xmath16 character . furthermore , a dispersion of the valence band is measured which is only half of that calculated within the lda . this indicates the importance of the electronic correlations in na@xmath0coo@xmath1 . shubnikov - de haas effect measurements revealed two well - defined frequencies in na@xmath23coo@xmath1 , suggesting either the existence of na superstructures or the presence of the @xmath17 pockets @xcite . last possibility was found to be incompatible with existing specific heat data . also , within the lda scheme the na disorder was shown to destroy the small @xmath17 pockets in na@xmath24coo@xmath1 because of their tendency towards the localization @xcite . the hole pockets are absent in the lsda+u ( local spin density approximation + hubbard u ) calculations @xcite . however , in this approach the insulating gap is formed by a splitting of the local single - electron states due to spin polarization , resulting in a spin polarized fermi surface with an area twice as larger as arpes observes . the dynamical character of the strong electron correlations has been taken into account within dynamical mean field theory ( dmft ) calculations @xcite and , surprisingly has led to an enhancement of the area of the small fermi surface pockets , in contrast to the experimental observations . at the same time , the use of the strong - coupling gutzwiller approximation within the multiorbital hubbard model with fitting parameters @xcite yields an absence of the hole pockets at the fermi surface . according to these findings , the bands crossing the fermi surface have @xmath16 character . concerning the magnetic properties , lsda predicts na@xmath0coo@xmath1 to have a weak intra - plane itinerant ferromagnetic ( fm ) state for nearly all na concentrations , @xmath25 @xcite . on contrary , neutron scattering finds a - type antiferromagnetic order at @xmath26k with an inter(intra)-plane exchange constant @xmath27mev and with ferromagnetic ordering within co - layer _ only _ for @xmath28 @xcite . in this paper we derive an effective low - energy model describing the bands crossing the fermi level on the basis of the lda band structure calculations . due to the fs topology , inferred from lda band structure , the magnetic susceptibility @xmath29 reveals two different regimes . for @xmath30 the susceptibility shows pronounced peaks at the antiferromagnetic ( afm ) wave vector @xmath31 resulting in a tendency towards in - plane @xmath32 afm order . for for @xmath33 the susceptibility is peaked at small momenta near @xmath34 . this clearly demonstrates the tendency of the system towards an itinerant in - plane fm state . we find that the formation of the electron pocket around the @xmath3 point is crucial for the in - plane fm ordering at high doping concentrations . we further analyze the role of the many - body effects calculated within the fluctuation - exchange ( flex ) , gutzwiller , and hubbard - i approximations . the paper is organized as follows . in section [ section : tbmodel ] the lda band structure and tight - binding model parameters derivation are described . the doping dependent evolution of the magnetic susceptibility within the tight - binding model is presented in section [ section : chi0 ] . the role of strong electron correlations is analyzed in section [ section : sec ] . the last section summarizes our study . the band structure of na@xmath35coo@xmath1 was obtained within the lda @xcite in the framework of tb - lmto - asa ( tight binding approach to the linear muffin - tin - orbitals using atomic sphere approximation ) @xcite computation scheme . this compound crystallizes at 12k in the hexagonal structure ( @xmath36 symmetry group ) with a=2.83176and c=10.84312@xcite . a displacement of na atoms from its ideal sites @xmath37 @xmath38 on about 0.2are observed in defected cobaltates for both room and low temperatures . this is probably due to the repulsion of the randomly distributed na atoms , locally violating hexagonal symmetry @xcite . in this study na atoms were shifted back to the high symmetry @xmath37 sites . oxygen was situated in the high - symmetry @xmath37-position @xmath39 . the obtained co - o distance is 1.9066which agrees well with experimentally observed one 1.9072(4)@xcite . this unit cell was used for all doping concentrations . the effect of the doping was taken into account within the virtual crystal approximation where each co site has six nearest neighbor virtual atoms with fractional number of valence electrons @xmath7 and a core charge @xmath40 instead of randomly located na . note , that all core states of the virtual atom are left unchanged and corresponds to na ones . we have chosen co @xmath41 states , @xmath42 states of o and @xmath43 states of na as the valence states for the tb - lmto - asa computation scheme . the radii of atomic spheres are 1.99 a.u . for co , 1.61 a.u . for oxygen , and 2.68 a.u . two classes of empty spheres ( pseudo - atoms without core states ) were added in order to fill the unit cell volume . the contribution of co-@xmath16 states is denoted by the vertical broadening ( in red ) of the bands with thickness proportional to the weight of the contribution . the crosses indicate the dispersion of the bands obtained by projection on the @xmath2 orbitals . the horizontal line at zero energy denotes the fermi level . ] coo@xmath1 with hopping notations within the first three coordination spheres ( c.s . ) . ( b ) lda - calculated fermi surface with cylindrical part ( violet ) having mostly @xmath16 character and six hole pockets ( red ) having mostly @xmath44 character . @xmath45 and @xmath46 coordinates of the symmetry points are given in units of @xmath47 with @xmath48 being the in - plane lattice constant . ] in order to find an appropriate basis the occupation matrix was diagonalized and its eigenfunctions were used as the new local orbitals . this procedure takes into account the real distortion of the crystal structure . the new orbitals are not pure trigonal @xmath16 and @xmath17 orbitals but we still use the former notations for the sake of simplicity . 288 @xmath49-points in the whole brillouin zone were used for the band structure calculations ( 12x12x2 mesh for @xmath45 , @xmath46 , and @xmath50 , correspondingly ) . the bands crossing the fermi level are shown in ( fig . [ fig : bands ] ) . one sees that they have mostly @xmath16 character , consistent with previous lda findings @xcite . note , the small fs pockets near the k point with @xmath17-symmetry present at @xmath51 [ see fig . [ fig : hopp](b ) ] disappear for higher doping concentrations because of the corresponding bands sink below the fermi level . the difference in the dispersion along @xmath52 and @xmath53 directions is due to non - negligible interaction between coo@xmath1 planes . a small gap between co-3@xmath13 and o-2@xmath54 states at about -1.25 ev present for x=0.61 disappears for x=0.33 due to the shift of the @xmath13-band to lower energy upon decreasing number of electrons . in the following we restrict ourselves to the model with the in - plane hoppings inside coo@xmath1 layer to describe the doping dependence of the itinerant in - plane magnetic order . hence we neglect bonding - antibonding ( bilayer ) splitting present in the lda - bands . this assumption seems to be justified since the largest interlayer hopping matrix element is an order of magnitude smaller than the intra - plane one ( 0.012 ev vs. 0.12 ev ) . c|c|| c|c|c|c|c|c|c|c|c|c & in - plane vector : & ( 0 , 1 ) & ( @xmath55 , @xmath56 ) & ( @xmath55 , -@xmath56 ) & ( @xmath57 , 0 ) & ( @xmath55 , @xmath58 ) & ( @xmath55 , -@xmath58 ) & ( 0 , 2 ) & ( @xmath57 , 1 ) & ( @xmath57 , -1 ) + @xmath59 & @xmath60 & @xmath61 & @xmath62 & @xmath63 & @xmath64 & @xmath65 & @xmath66 & @xmath67 & @xmath68 & @xmath69 & @xmath70 + + @xmath16 & 0.000 & @xmath71 & 0.123 & 0.123 & 0.123&-0.022 & -0.022&-0.021&-0.025 & -0.025 & -0.025 + & & @xmath72 & -0.044 & 0.089 & -0.044 & 0.010 & 0.010&-0.021 & -0.021 & 0.042 & -0.021 + @xmath73 & -0.053 & @xmath74 & -0.077 & 0.000 & 0.077 & 0.018 & -0.018 & 0.000 & -0.036 & 0.000 & 0.036 + & & @xmath75 & -0.069&-0.005 & -0.069 & 0.018 & 0.018&-0.026&-0.017 & -0.085 & -0.017 + @xmath76 & -0.053 & @xmath77 & 0.037 & 0.000 & -0.037&-0.026 & 0.026 & 0.000&-0.039 & 0.000 & 0.039 + & & @xmath78 & -0.026&-0.090 & -0.027&-0.011 & -0.011 & 0.033&-0.062 & 0.006 & -0.062 + + @xmath16 & 0.000 & @xmath71 & 0.110 & 0.110 & 0.110&-0.019 & -0.019&-0.019&-0.023&-0.023&-0.023 + & & @xmath72 & -0.050 & 0.100 & -0.050 & 0.008 & 0.008&-0.016&-0.017 & 0.035&-0.017 + @xmath73 & -0.028 & @xmath74 & 0.087 & 0.000 & -0.087&-0.014 & 0.014 & 0.000 & 0.030&-0.000&-0.030 + & & @xmath75 & -0.069&-0.031 & -0.069 & 0.015 & 0.015&-0.022&-0.016&-0.076&-0.016 + @xmath76 & -0.028 & @xmath77 & -0.022 & 0.000 & 0.022 & 0.021 & -0.021 & 0.000 & 0.035 & 0.000&-0.035 + & & @xmath78 & -0.044&-0.081 & -0.044&-0.009 & -0.009 & 0.027&-0.056 & 0.005&-0.056 + + @xmath16 & 0.000 & @xmath71 & 0.105 & 0.105 & 0.105&-0.018 & -0.018&-0.018&-0.022&-0.022&-0.022 + & & @xmath72 & -0.052 & 0.104 & -0.052 & 0.007 & 0.007&-0.015&-0.016 & 0.033&-0.016 + @xmath73 & -0.019 & @xmath74 & -0.090 & 0.000 & -0.090 & 0.013 & -0.013 & 0.000&-0.028 & 0.000 & 0.028 + & & @xmath75 & -0.068&-0.039 & -0.068 & 0.014 & 0.014&-0.020&-0.015&-0.073&-0.015 + @xmath76 & -0.019 & @xmath77 & 0.016 & 0.000 & -0.016&-0.020 & 0.020 & 0.000&-0.034 & 0.000 & 0.034 + & & @xmath78 & -0.048&-0.077 & -0.049&-0.009 & -0.009 & 0.026&-0.054 & 0.005&-0.054 + to construct the effective hamiltonian and to derive the effective co - co hopping integrals @xmath79 for the @xmath2-manifold we apply the projection procedure @xcite . here , @xmath80 denotes a pair of orbitals , @xmath16 , @xmath73 or @xmath76 . the indices @xmath81 and @xmath82 correspond to the co - sites on the triangular lattice . the obtained hoppings and the single - electron energies are given in table [ table : params ] for three difference doping concentrations . a comparison between the bands obtained using projection procedure and the lda bands is shown in fig . [ fig : bands ] confirming the co-@xmath2 nature of the near - fermi level bands @xcite . for simplicity we have enumerated site pairs with @xmath83 , @xmath84 ( see fig . [ fig : hopp](a ) and the correspondence between in - plane vectors and index @xmath85 in table [ table : params ] ) . due to the @xmath86 symmetry of the lattice , the following equalities apply : @xmath87 , @xmath88 , @xmath89 . in addition @xmath90 for @xmath91 hoppings , which , however , does not hold for @xmath92 orbitals . since the hybridization between the @xmath16 and the @xmath44 bands is not small , a simplified description of the bands crossing the fermi level in terms of the @xmath16 band only ( neglecting @xmath44 band and the corresponding hybridizations , see for example ref . @xcite ) may lead to an incorrect result due to a higher symmetry of the @xmath16-band . in summary , the free electron hamiltonian for coo@xmath1-plane in a hole representation is given by : @xmath93 where @xmath94 ( @xmath95 ) is the annihilation ( creation ) operator for the hole with momentum @xmath96 , spin @xmath97 and orbital index @xmath59 , @xmath98 , and @xmath99 is the fourier transform of the hopping matrix element , @xmath60 is the single - electron energies , and @xmath100 is the chemical potential . introducing matrix notations , @xmath101 and @xmath102 , the hoppings matrix elements in the momentum representation are given by : @xmath103 where @xmath104 , @xmath105 , @xmath106 . note , the parameters do not change significantly upon changing the doping concentration . in fig . [ fig : diffparams ] we show two results of the rigid - band approximation with the hamiltonian ( [ eq : h0 ] ) and the hopping values obtained in lda calculation for two different doping concentrations , @xmath51 and @xmath107 ( see table [ table : params ] ) . the doping concentration used to calculate the chemical potential @xmath100 was fixed to be @xmath107 for both hamiltonians . although one finds the pronounced differences in the dispersion around the @xmath108-point , they are small around the fs . since most of the physical quantities are determined by the states lying close to the fermi level , we can safely ignore the small differences of the band structure and describe the doping evolution of the na@xmath0coo@xmath1 by simply varying the chemical potential . in the following we will use _ ab initio _ parameters calculated for @xmath51 and change the chemical potential to achieve different doping concentrations . within the rigid band approximation the @xmath17 hole pockets are well below the fermi level for @xmath109 . most important however , we find the local minimum of the band dispersion around the @xmath3 point ( see fig . [ fig : diffparams ] ) to yield an inner fs contour centered around this point . the area of this electron fs pocket increases upon increasing the doping concentration @xmath7 . as we will show later , the main reason for the local minimum around the @xmath3 point is the presence of the next - nearest - neighbor hopping integrals which enter our tight - binding dispersion . although this minimum is not yet directly observed by arpes experiments , note that the inner fs contour would reduce the total fs volume and therefore may explain why the volume of the fs observed in arpes so far is larger than it follows from luttinger s theorem @xcite . furthermore , an emergence of this pocket would influence the hall conductivity at high doping concentrations which is interesting to check experimentally . note , the appearance of the inner contour of the fs around the @xmath3 point for large doping concentrations is not unique to our calculations , previously it has been obtained within the lda calculations for a single co - layer per unit cell@xcite . coo@xmath1 within the rigid - band approximation with _ ab initio _ parameters for @xmath107 ( the solid blue curve ) and for @xmath51 ( red dashed curve ) . the horizontal ( green ) line denotes the chemical potential @xmath100 for @xmath107 . ] to analyze the possibility of the itinerant magnetism we calculate the magnetic susceptibility @xmath29 based on the hamiltonian @xmath110 . the doping - dependent evolution of the peaks in @xmath111 is shown in fig . [ fig : chi0 ] . at @xmath112 the @xmath17 bands are below the fermi level , and the fs has the form of the rounded hexagon . it results in a number of nesting wave vectors around the antiferromagnetic wave vector @xmath113 . the corresponding broad peaks in the @xmath114 appear around @xmath115 , indicating the tendency of the electronic system towards an @xmath32 afm sdw ordered state @xcite . upon increasing doping , the fermi level crosses the local minimum at the @xmath3 point , resulting in an almost circle inner fs contour . as soon as this change of the fs topology occurs , the scattering at the momentum @xmath115 is strongly suppressed at @xmath116 . simultaneously , a new scattering vector , @xmath117 , at small momenta appears . correspondingly , the magnetic susceptibility peaks at small momenta , indicating the tendency of the magnetic system towards itinerant sdw order with small momenta . the relevance of the local minimum around the @xmath3 point for the formation of the scattering at small momenta was originally found in ref . @xcite . as a function of the momentum in units of @xmath118 ( left ) , and the fermi surface for corresponding doping concentration @xmath7 ( right ) . the arrows indicate the scattering wave vectors @xmath119 as described in the text . ] for large @xmath7 the area of the inner fs contour increases leading to a further decrease of the @xmath117 . observe that for @xmath120 , the fs topology again changes yielding six distant fs contours that moves @xmath117 further to zero momenta . the scattering at small momenta seen in the bare magnetic susceptibility for @xmath121 is qualitatively consistent with the ferromagnetic ordering at @xmath34 , observed in the neutron scattering experiments @xcite . it is important to understand the impact of electronic correlations on the magnetic instabilities obtained within the rigid band approximation . since obtained magnetic susceptibility depends mostly on the topology of the fs one expects that the behavior shown in fig . [ fig : chi0 ] will be valid even if one consider an rpa susceptibility with an interaction term @xmath122 taken into account , at least in the case if the only interaction is the on - site hubbard repulsion @xmath123 . the only difference would be a shift of the critical concentrations @xmath4 , at which the fs topology changes and tendency to the afm order changes towards the tendency to the fm ordered state . similar to refs . @xcite we add the on - site coulomb interaction terms to eq.([eq : h0 ] ) . at present , it is not completely clear to which extent the electronic correlations governs the low - energy properties in na@xmath0coo@xmath1 due to multi - orbital effects in this compound which complicates the situation . therefore , in the following we discuss three different approximations valid for different @xmath5 ratio . to analyze the regime of strong electron correlations we project the doubly occupied states out and formulate an effective model equivalent to the hubbard model with an infinite value of @xmath123 . this approximation could be justified by the large ratio of the on - site coulomb interaction on the coo@xmath1 cluster @xmath123 with respect to the bandwidth @xmath124 . in the atomic limit the local low - energy states on the co sites are the vacuum state @xmath125 and the single - occupied hole states @xmath126 , @xmath127 , @xmath128 . coo@xmath1 . here @xmath129 stands for number of holes , @xmath130 enumerates single - particle excitations . the filling factor of the corresponding state upon changing the doping concentration @xmath7 is given in square brackets . ] the single - particle hole excitations and local atomic states are shown in fig . [ fig : localstates ] . the simplest way to describe the quasiparticle excitations between these states is to use the projective hubbard @xmath131-operators that take the no - double occupancy constraint into account automatically @xcite : @xmath132 , where index @xmath133 enumerates quasiparticles . there is a simple correspondence between the fermionic - like @xmath131-operators and single - electron creation - annihilation operators : @xmath134 , where @xmath135 determines the partial weight of a quasiparticle @xmath136 with spin @xmath97 and orbital index @xmath59 . in these notations the hamiltonian of the hubbard model in the limit @xmath137 has the form : @xmath138 coo@xmath1 for @xmath139 . the dashed ( red ) , solid ( blue ) and dash - dotted ( cyan ) curves represent the results of the rigid - band , the gutzwiller , and the hubbard - i approximations , respectively . the horizontal ( green ) line denotes the position of the chemical potential @xmath100 . ] to study a quasiparticle energy spectrum of the system and its thermodynamics we use the fourier transform of the two - time retarded green function in the frequency representation , @xmath140 . this can be rewritten as : @xmath141 , where @xmath142 is the matrix green function in the @xmath131-operators representation . using the diagram technique for hubbard x - operators @xcite one obtains the generalized dyson equation @xcite : @xmath143^{-1 } \hat{p}({\bf k},e ) . \label{eq : d}\ ] ] here , @xmath144 stands for the ( exact ) local green function , @xmath145 $ ] . in the hubbard - i approximation the self - energy @xmath146 is equal to zero and the strength operator @xmath147 is replaced by the sum of the occupation factors , @xmath148 $ ] , @xmath149 . here @xmath150 stands for the usual thermodynamic average . thus , one obtains : @xmath151^{-1 } \hat{p}. \label{eq : d0}\ ] ] in the paramagnetic phase the occupation factors are : @xmath152 , @xmath153 , @xmath154 which yields the diagonal form of the strength operator , @xmath155 . therefore , the quasiparticle bands formed by the @xmath156 hoppings will be renormalized by the @xmath157 factor , while the quasiparticle bands formed by the @xmath17 hopping elements will be renormalized by @xmath7 . in fig . [ fig : approximations_bands ] the quasiparticle spectrum , the dos , and the fs are displayed in different approximations . within hubbard - i approximations one finds the narrowing of the bands with lowering the doping concentration @xmath7 due to doping dependence of the quasiparticle s spectral weight introduced by the strength operator @xmath158 . however , the doping evolution of the fs is qualitatively similar to that in the rigid - band picture . namely , the bandwidth reduction and the spectral weight renormalization do not change the topology of the fs . as a result , the presence of the strong electronic correlations within hubbard - i approximation do not change qualitatively our results for the bare susceptibility . quantitatively , the critical concentration @xmath4 shifts towards higher values of the doping and becomes @xmath159 . the reason for this shift is the band narrowing and the renormalization of the quasiparticle s spectral weight , which enters the equation that determines the position of the chemical potential @xmath100 . luttinger s theorem , which holds for a perturbative expansion of the green s function in terms of the interaction strength is violated within the hubbard - i approximation . this violation is due to the renormalization of the spectral weight of the green function by the occupation factors in the strength operator in eq . ( [ eq : d ] ) . this is the reason why in spite of the @xmath17 band narrowing the @xmath17 hole pockets at the fermi surface are still present at @xmath51 . the gutzwiller approximation @xcite for the hubbard model provides a good description for the correlated metallic system . its multiband generalization was formulated in ref . @xcite . in this approach , the hamiltonian describing the interacting system far from the metal - insulator transition for @xmath160 @xmath161 with @xmath110 being the free electron hamiltonian ( [ eq : h0 ] ) , is replaced by the effective non - interacting hamiltonian : @xmath162 here , @xmath163 is the renormalized hopping , @xmath164 , @xmath165 is the orbital s filling factors , @xmath166 is the equation for the chemical potential . @xmath167 are the lagrange multipliers yielding the correlation induced shifts of the single - electron energies . the constant @xmath168 is determined from the condition that the ground state energy is the same for both hamiltonians @xmath169 where @xmath170 is the wave function of the free electron system ( [ eq : heff ] ) , and @xmath171 is the gutzwiller wave function for the hamiltonian ( [ eq : hg ] ) . the lagrange multipliers are determined by minimizing the energy , @xmath172 with respect to the orbital filling factors @xmath173 . here @xmath174 , as determined from eq . ( [ eq : psi0psig ] ) . this results in the following expression for the single - electron energy renormalizations : @xmath175 it is this energy shift that forces the @xmath17 fs hole pockets to sink below the fermi energy @xcite , which is clearly seen in the doping - dependent evolution of the quasiparticle dispersion and the fs as obtained within gutzwiller approximation ( fig . [ fig : approximations_bands ] ) . although the narrowing of the bands due to strong correlations is similar to the one found in the hubbard - i approximation , the fs obeys luttinger s theorem . note , in contrast to the hubbard - i approximation the relative positions of the @xmath2-bands are also renormalized by @xmath176 . at the same time , for @xmath177 the topology of the fs in the gutzwiller approximation is qualitatively the same as in the rigid - band picture . the also yields similar results for the bare susceptibility s doping dependence discussed in section [ section : chi0 ] . the only effect of the strong correlations for @xmath178 is the observed shift of the critical concentration towards higher values , @xmath179 . this is due to combined effect of the bands narrowing and the doping dependence of the @xmath16 and @xmath17 band s relative positions , determined by the eq . ( [ eq : vareps ] ) . note , for @xmath180 , due to different fs topology that occurs in the gutzwiller approximation , the bare susceptibility differs from that obtained in ref . @xcite where the strong renormalization of the electronic bands removing @xmath17 pockets away from the fs was neglected . a certain disadvantage of the gutzwiller and hubbard - i like approximations is that the dynamic character of electronic correlations is not taken into account within these approaches . at the same time , the momentum and frequency dependencies of the self - energy @xmath181 play a crucial role , in particular , for determining the low - energies excitations close to the fermi level . in this subsection we focus on the @xmath16-band with nearest and next - nearest hopping integrals only and employ the single - band fluctuation exchange approximation ( flex ) @xcite which sums all particle - hole(particle ) ladder graphs for the generating functional self - consistently valid for the intermediate strength of the correlations . the flex equations for the single - particle green function @xmath182 , the self - energy @xmath183 , the effective interaction @xmath184 , the bare ( @xmath185 ) and renormalized spin ( @xmath186 ) and charge ( @xmath187 ) susceptibilities read @xmath188^{-1 } , \\ \sigma_{\bf k}(\omega_n ) & = & \frac{t}{n } \sum\limits_{{\bf p } , m } v_{\bf k - p}(\omega_n - \omega_m ) g_{\bf p}(\omega_m ) , \\ v_{\bf q}(\nu_m ) & = & u^2 \left [ \frac{3}{2 } \chi^s_{\bf q}(\nu_m ) + \frac{1}{2 } \chi^c_{\bf q}(\nu_m ) - \chi^0_{\bf q}(\nu_m ) \right ] , \\ \chi^0_{\bf q}(\nu_m ) & = & - \frac{t}{n } \sum\limits_{{\bf k } , n } g_{\bf k+q}(\omega_n + \nu_m ) g_{\bf k}(\omega_n ) , \\ \chi^{s , c}_{\bf q}(\nu_m ) & = & \frac{\chi^0_{\bf q}(\nu_m)}{1 \mp u \chi^0_{\bf q}(\nu_m)},\end{aligned}\ ] ] where @xmath189 and @xmath190 . in the last equation the @xmath191 sign in the denominator corresponds to the @xmath192 , while the @xmath193 sign corresponds to the @xmath194 . we compute the matsubara summations using almost real contour technique of ref . i.e. , the contour integrals are performed with a finite shift @xmath195 ( @xmath196 ) into the upper half - plane . all final results are analytically continued from @xmath197 onto the real axis @xmath198 by pad approximation . the following results are based on flex solutions using a lattice of 64x64 sites with 4096 equidistant @xmath199-points in the energy range of @xmath200 $ ] . the temperature has been kept at @xmath201 , where @xmath202 is the hopping amplitude to the nearest neighbors for the @xmath16 band corresponding to @xmath203 . the hubbard repulsion was set to @xmath204 . ( in units of @xmath202 , relative to @xmath100 ) within flex approximation for ( a ) @xmath205 and ( b ) @xmath206 and for two doping concentrations . ] for ( a ) @xmath205 and ( b ) @xmath206 . notice that for large @xmath207 the commensurate peak at @xmath208 point is absent at a very low @xmath7 . ] near the fs in direction @xmath209 for ( a ) @xmath205 and ( b ) @xmath206 . ] previously , the flex approximation has been applied successfully to the study of superconductivity as well as spin and charge excitations in na@xmath0coo@xmath1 @xcite . complementary , we will focus on the quasiparticle dispersion and study the impact of the momentum and frequency dependencies of the @xmath210 , and the role played by the next - nearest hopping integral , @xmath211 , corresponding to @xmath212 . the quasiparticle dispersion @xmath213 , which is determined from equation @xmath214 , is shown in fig . [ fig : flexek ] for @xmath205 and @xmath206 , in units of @xmath202 . first , observe that the local minimum around the @xmath3 point appears only if the next - nearest - neighbor hopping @xmath211 is included which agrees with our previous findings . in addition , we obtain a pronounced mass enhancement of the order of unity at the fs crossings - the so - called kink structure . this enhancement is due to low - energy spin fluctuations which are present in @xmath215 @xcite . to shed more light onto the two - dimensional spin correlations , in fig . [ fig : flexchi ] we display the static spin structure factor @xmath216 from the flex for two different doping concentrations . as doping increases from @xmath217 towards @xmath218 the maximum in the spin susceptibility @xmath219 moves towards the @xmath208-point of the first bz and develops into a sharp and commensurate peak at @xmath115 and the incommensurate spin fluctuations are suppressed . one may also note that the commensurate peak is @xmath220 larger for @xmath206 than for @xmath205 . these results are consistent with those obtained in a previous sections . we further notice smooth evolution of the quasiparticle dynamics with doping in na@xmath0coo@xmath1 showing no sign of unusual behavior at @xmath217 . the frequency dependence of the imaginary part of the quasiparticle self - energy , i.e. @xmath221 , near the fs is shown in fig . [ fig : flexsigma ] . we find the self - energy to be nearly isotropic along the fs with only a weak maximum occurring into the direction of the commensurate spin fluctuations . near the fermi energy the self - energy is clearly proportional to @xmath222 at low energies for all dopings shown , which is indicative of the normal fermi - liquid behavior . this is in sharp contrast to the flex analysis of the hubbard model on the square lattice close to half - filling . there one typically finds marginal fermi - liquid behavior with @xmath223 over a wide range of frequencies @xcite . therefore , along this line one is tempted to conclude that the normal state of the superconducting cobaltates is more of conventional metallic nature than in the high - t@xmath224 cuprates . this is even more so , if one realizes from fig . [ fig : flexsigma ] that the quasiparticle scattering rate displays its smallest curvature for @xmath218 , which implies the quasiparticles to be rather well defined there . for lower @xmath7 proximity of the fs to the van hove singularity ( see flat region of dispersion in fig . [ fig : flexek ] ) enhances both the absolute value of @xmath225 as well as the curvature . this effect is most pronounced for @xmath205 . to conclude , we have calculated the doping dependent magnetic susceptibility in the tight - binding model with _ ab - initio _ calculated parameters . we find , that at a critical doping concentration , @xmath4 , electron pocket develops on the fs in the center of the brillouin zone . for @xmath226 , the system shows a tendency towards an @xmath32 afm ordered state , while for @xmath121 a peak in the magnetic susceptibility forms at small wave vectors indicating a strong tendency towards an itinerant fs state . within a tight - binding model we have estimated @xmath4 to be approximately @xmath227 . analyzing the influence of strong coulomb repulsion and the corresponding reduction of the bandwidth and the quasiparticle spectral weight in the strong - coupling hubbard - i and gutzwiller approximations , we have shown that the critical concentration changes to @xmath159 and @xmath179 , respectively . at the same time , the underlying physics of the formation of the itinerant fm state remains the same . we neglected the bonding - antibonding splitting due to the 3-dimensionality in the non - intercalated compounds . this splitting was taken into account in ref . @xcite , where within the flex approximation the single @xmath16-band hubbard model was considered . the results obtained also suggest a tendency to fm fluctuations for high doping concentrations . the presence of a local band minimum around the @xmath3 point played a crucial role , similar to our present study . to analyze the low - energy quasiparticle properties at low doping concentrations we have employed the single - band hubbard model within the flex approximation . we have found a significant fs mass enhancement of order unity due to quasiparticle scattering from spin fluctuations . in contrast to the hubbard model on the square lattice we have found the quasiparticle scattering rate to display a conventional fermi - liquid type of energy dependence . we have also shown that the static spin structure factor exhibits a large commensurate peak at wave vector @xmath115 for doping concentrations of @xmath228 . this response was found to be significantly enhanced by the next - nearest - neighbor hopping , emphasizing its significance . we would like to thank g. bouzerar , p. fulde , s.g . ovchinnikov , n.b . perkins , d. singh , ziqiang wang , and v. yushankhai for useful discussions , i. mazin for critical reading of the manuscript , and s. borisenko for sharing with us the experimental results prior to publication . m.m.k . acknowledge support form intas ( ys grant 05 - 109 - 4891 ) and rfbr ( grants 06 - 02 - 16100 , 06 - 02 - 90537-bnts ) . a.s . and v.i.a . acknowledge the financial support from rfbr ( grants 04 - 02 - 16096 , 06 - 02 - 81017 ) , and nwo ( grant 047.016.005 ) . i. terasaki , y. sasago , and k. uchinokura , phys . b * 56 * , r12685 ( 1997 ) . y. wang , n. rogado , r.j . cava , and n.p . ong , nature * 423 * , 425 ( 2003 ) . foo , y. wang , s. watauchi , h.w . zandbergen , t. he , r.j . cava , and n.p . ong , phys . lett . * 92 * , 247001 ( 2004 ) . sales , r. jin , k.a . affholter , p. khalifah , g.m . veith , and d. mandrus , phys . b * 70 * , 174419 ( 2004 ) . hasan , y .- d . chuang , d. qian , y.w . li , y. kong , a. kuprin , a.v . fedorov , r. kimmerling , e. rotenberg , k. rossnagel , z. hussain , h. koh , n.s . rogado , m.l . foo , and r.j . cava , phys . 92 * , 246402 ( 2004 ) . yang , s .- c . wang , a.k.p . sekharan , h. matsui , s. souma , t. sato , t. takahashi , t. takeuchi , j.c . campuzano , r. jin , b.c . sales , d. mandrus , z. wang , and h. ding , phys . 92 * , 246403 ( 2004 ) . yang , z .- h . pan , a.k.p . sekharan , t. sato , s. souma , t. takahashi , r. jin , b.c . sales , d. mandrus , a.v . fedorov , z. wang , and h. ding , phys . * 95 * , 146401 ( 2005 ) . d. qian , d. hsieh , l. wray , y .- d . chuang , a. fedorov , d. wu , j.l . luo , n.l . wang , l. viciu , r.j . cava , and m.z . hasan , phys . lett . * 96 * , 216405 ( 2006 ) . boothroyd , r. coldea , d.a . tennant , d. prabhakaran , l.m . helme , and c.d . frost , phys . * 92 * , 197201 ( 2004 ) . bayrakci , i. mirebeau , p. bourges , y. sidis , m. enderle , j. mesot , d.p . chen , c.t . lin , and b. keimer , phys . lett . * 94 * , 157205 ( 2005 ) . helme , a.t . boothroyd , r. coldea , d. prabhakaran , d.a . tennant , a. hiess , and j. kulda , phys . lett . * 94 * , 157206 ( 2005 ) . n. marzari and d. vanderbilt , phys . b. * 56 * , 12847 ( 1997 ) . anisimov , d.e . kondakov , a.v . kozhevnikov , i.a . nekrasov , z.v . pchelkina , j.w . allen , s .- k . kim , p. metcalf , s. suga , a. sekiyama , g. keller , i. leonov , x. ren , and d. vollhardt , phys . b. * 71 * , 125119 ( 2005 ) . johannes , d.a . papaconstantopoulos , d.j . singh , and m.j . mehl , europhys . * 68 * , 433 ( 2004 ) . zaitsev , sov . jetp * 41 * , 100 ( 1975 ) . izumov and b.m . letfullov , j. phys . : condens . matter * 3 * , 5373 ( 1991 ) . s.g . ovchinnikov and v.v . valkov , _ hubbard operators in the theory of strongly correlated electrons _ ( imperial college press , london , 2004 ) bickers , d.j . scalapino , and s.r . white , phys . lett . * 62 * , 961 ( 1989 ) ; j. altmann , w. brenig , and a.p . kampf , eur . j. b * 18*,429 ( 2000 ) . j. schmalian , m. langer , s. grabowski , and k.h . bennemann , computer phys . comm . * 93 * , 141 ( 1996 ) .

based on the _ ab - initio _ band structure for na@xmath0coo@xmath1 we derive the single - electron energies and the effective tight - binding description for the @xmath2 bands using projection procedure . due to the presence of the next - nearest - neighbor hoppings a local minimum in the electronic dispersion close to the @xmath3 point of the first brillouin zone forms . correspondingly , in addition to a large fermi surface an electron pocket close to the @xmath3 point emerges at high doping concentrations . the latter yields the new scattering channel resulting in a peak structure of the itinerant magnetic susceptibility at small momenta . this indicates dominant itinerant in - plane ferromagnetic fluctuations above certain critical concentration @xmath4 , in agreement with neutron scattering data . below @xmath4 the magnetic susceptibility shows a tendency towards the antiferromagnetic fluctuations . we further analyze the many - body effects on the electronic and magnetic excitations using various approximations applicable for different @xmath5 ratio .

one of the fundamental objects associated to a hyperplane arrangement @xmath3 is the module @xmath4 of logarithmic one - forms with pole along the arrangement or ( dually ) the module @xmath5 of derivations tangent to the arrangement . both are graded @xmath6 modules ; @xmath7 is defined via : @xmath8 for all @xmath9 such that @xmath10 . over a field of characteristic zero , @xmath11 , where @xmath12 is the euler derivation and @xmath13 corresponds to the module of syzygies on the jacobian ideal of the defining polynomial of @xmath14 . when @xmath15 or @xmath16 , an elegant theorem of terao relates the freeness of the module @xmath5 to the poincar polynomial of @xmath17 . in this note , we restrict to @xmath18 , but broaden the class of curves which make up the arrangement . in particular , suppose @xmath19 where each @xmath20 is a smooth rational plane curve ; call such a collection a conic - line ( cl ) arrangement . [ firstex ] for the cl arrangement below , @xmath21 . we begin with some facts about hyperplane arrangements ; for more information see orlik and terao @xcite . a hyperplane arrangement @xmath14 is a finite collection of codimension one linear subspaces of a fixed vector space v. @xmath14 is _ central _ if each hyperplane contains the origin * 0 * of v. the intersection lattice @xmath22 of @xmath14 consists of the intersections of the elements of @xmath14 ; the rank of @xmath23 is simply the codimension of @xmath24 . v is the lattice element @xmath25 ; the rank one elements are the hyperplanes themselves . @xmath14 is called _ essential _ if rank @xmath26 dim @xmath27 . the mbius function @xmath28 : @xmath29 is defined by @xmath30 we now restrict to the case that @xmath27 is complex . a foundational result is that the poincar polynomial of @xmath31 is purely combinatorial ; in particular @xmath32 an arrangement @xmath33 is _ free _ if @xmath34 ; the @xmath35 are called the _ exponents _ of @xmath14 . terao s famous theorem @xcite is that if @xmath34 , then @xmath36 . if @xmath37 is central , then @xmath38 also defines a set of lines in @xmath39 , and obviously @xmath40 , where @xmath41 is the complement of the corresponding arrangement of lines in @xmath18 . hence @xmath42 it follows from terao s theorem that if @xmath43 , then @xmath44 . this can be generalized @xcite to line arrangements which are not free , using the chern polynomial . the motivating question of this paper is : _ what happens if the arrangement of lines is replaced with a cl arrangement ? _ in @xcite , cogolludo - agustn studies the complement of an arrangement of rational curves in @xmath39 , where the individual curves can have singularities , and can meet non - transversally . the main result is that the cohomology ring of the complement to a rational curve arrangement is generated by logarithmic @xmath45 and @xmath46-forms and its structure depends on a finite number of invariants of the curve . one fact is that if @xmath41 is the complement of an arrangement of @xmath47 irreducible curves in @xmath39 , then @xmath48 @xmath49 where @xmath50 is the number of branches passing thru @xmath51 , and @xmath52 is the normalization of @xmath20 . since we are assuming that all the @xmath20 are smooth and rational , we have that @xmath53 where the intersection poset @xmath54 is defined precisely as for a linear arrangement ( typically , @xmath54 is only a poset , not a lattice ) . a crucial distinction between line and curve arrangements , even in our simple setting , is the difference between the milnor and tjurina numbers at a singularity . let @xmath55 be a reduced ( but not necessarily irreducible ) curve in @xmath56 , let @xmath57 , and let @xmath58 denote the ring of convergent power series . the milnor number of @xmath59 at @xmath60 is @xmath61 to define @xmath62 for an arbitrary point @xmath51 , we translate so that @xmath51 is the origin . the tjurina number of @xmath59 at @xmath60 is @xmath63 a singularity is quasihomogeneous iff there exists a holomorphic change of variables so the defining equation becomes weighted homogeneous ; @xmath64 is weighted homogeneous if there exist rational numbers @xmath65 such that @xmath66 is homogeneous . in @xcite , reiffen proved that if @xmath67 is a convergent power series with isolated singularity at the origin , then @xmath67 is in the ideal generated by the partial derivatives if and only if @xmath68 is quasihomogeneous ( see @xcite for a generalization ) . as noted earlier , for a line arrangement with defining polynomial @xmath69 , @xmath70 consists of the syzygies on the jacobian ideal @xmath71 of @xmath69 . if @xmath72 is a reduced curve , then after a change of coordinates , we may assume that @xmath73 has no singularities on the line @xmath74 . dehomogenizing so that @xmath75 yields : @xmath76/\langle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y } , f \rangle = \!\!\!\!\sum\limits_{p \in \text{sing}(c)}\!\!\!\!\tau_p(f).\ ] ] it follows that if all the singular points are quasihomogeneous then @xmath77 for a line arrangement , the singularities are always quasihomogeneous , but this is not the case for cl arrangements : let @xmath78 be as below : @xmath79 has five singular points , all ordinary . when @xmath51 is an ordinary singularity and @xmath59 has @xmath47 distinct branches at @xmath51 , then @xmath80 , so the sum of the milnor numbers is 20 . however , @xmath81 ; at @xmath82 we have @xmath83 but @xmath84 . the first criterion for the freeness of @xmath85 is @xmath33 is free exactly when there exist @xmath86 elements @xmath87 such that the determinant of the matrix @xmath88 $ ] is a nonzero constant multiple of the defining polynomial of @xmath33 . saito s criterion holds for an arrangement of reduced hypersurfaces @xmath89 ; let @xmath90 where @xmath91 , and @xmath92 . by induction , @xmath93 so we have @xmath94 any arrangement of ( reduced ) hypersurfaces will have a singular locus of codimension two . as for a linear arrangement , @xmath95 , with @xmath96 , so freeness is equivalent to @xmath97 ( so also equivalent to @xmath71 cohen - macaulay ) . by the hilbert - burch theorem ( @xcite ) , any codimension two cohen - macaulay ideal @xmath98 with @xmath99 generators is generated by the maximal minors of an @xmath100 matrix @xmath101 , whose columns generate the module of first syzygies on @xmath98 . so when @xmath102 , appending a column vector @xmath103 $ ] to @xmath101 and taking the determinant yields a multiple of @xmath69 , by euler s formula . saito s criterion is most useful when an explicit set of candidates for the generating set of @xmath104 is known . there are two other fundamental tools that can be used to prove that a line arrangement is free . the first method is based on an inductive operation known as deletion - restriction : given an arrangement @xmath14 and a choice of hyperplane @xmath105 , set @xmath106 the collection @xmath107 is called a _ triple _ , and a triple yields ( see proposition 4.45 of @xcite ) a left exact sequence @xmath108 for a triple with @xmath109 , more is true ( see @xcite ) : after pruning the euler derivations and sheafifying , there is an exact sequence @xmath110 where @xmath111 ; @xmath112 in @xcite , terao showed that freeness of a triple is related via : [ thm : teraoad]@xmath113addition - deletion@xmath114 let @xmath107 be a triple . then any two of the following imply the third * @xmath115 * @xmath116 * @xmath117 theorem [ thm : teraoad ] applies in general , not just to arrangements in @xmath39 . a smooth conic is intrinsically a @xmath118 , so it is natural to ask if cl arrangements which admit a short exact sequence similar to ( [ eq : ses ] ) have an addition - deletion theorem ; we tackle this in the next two sections . a second criterion for freeness is special to the case of line arrangements ; to state it we need to define freeness for _ multiarrangements_. a multiarrangement @xmath119 is an arrangement together with a multiplicity @xmath120 for each hyperplane . the module of derivations consists of @xmath121 such that @xmath122 . as shown by ziegler in @xcite , freeness of multiarrangements is not combinatorial ; for recent progress see @xcite . [ thm : yoshi]@xmath113yoshinaga s multiarrangement criterion @xcite@xmath114 @xmath123 is free iff @xmath124 and @xmath125 the multiarrangement @xmath126 has minimal generators in degree @xmath127 and @xmath128 . the main results of this paper ( theorems [ thm : adline ] and [ thm : adconic ] ) show that an addition - deletion construction holds for cl arrangements with quasihomogeneous singularities ; the freeness of example [ firstex ] is explained by our results . as one application , we show that a free cl arrangement , when restricted to different lines , can yield multiarrangements with different exponents ; hence any version of theorem [ thm : yoshi ] for cl arrangements will be quite subtle . an addition - deletion theorem for multiarrangements has recently been proven by abe - terao - wakefield in @xcite ; our results are the first ( to our knowledge ) to give an inductive criterion for freeness for nonlinear arrangements . let @xmath129 be a triple of cl arrangements in @xmath130 , where @xmath131 , @xmath132 and @xmath133 _ is a line_. we begin by examining some examples : let @xmath134 be the union of : @xmath135 @xmath136 is free with exponents @xmath137 , and the degree of the jacobian ideal is @xmath138 , which is equal to the sum of the milnor numbers at the intersection points . therefore at each singular point @xmath139 . if we restrict to any line , the corresponding multiarrangement has two points of multiplicity 3 , and it follows from @xcite that the exponents are @xmath140 . hence the obvious generalization of yoshinaga s criterion does not hold . let @xmath141 and let @xmath142 . the degree of the jacobian ideal is @xmath143 , which is equal to the sum of milnor numbers at the points . it will follow from our results that @xmath144 is free with exponents @xmath145 . let @xmath146 and let @xmath147 . then @xmath148 is free with exponents @xmath149 . the degree of the jacobian ideal is @xmath150 , whereas the sum of the milnor numbers is 38 ; the singularity at @xmath82 has @xmath84 and @xmath83 . for cl arrangements similar to @xmath151 , there is an addition - deletion theorem : a triple @xmath129 of cl arrangements is called quasihomogeneous if @xmath139 at each singular point of @xmath134 and @xmath79 . [ thm : adline ] let @xmath152 be a quasihomogeneous triple with @xmath153 . the following are equivalent : 1 . @xmath134 is free with exponents @xmath154 . @xmath79 is free with exponents @xmath155 . examples 2.1 and 2.2 illustrate the theorem ; before giving the proof of theorem [ thm : adline ] , we need some preliminaries . [ lem : derexact ] let @xmath156 . then the maps @xmath157 , @xmath158 and @xmath159 , @xmath160 are well defined and yield an exact sequence : @xmath161 let @xmath162 be the defining polynomial of @xmath79 , where @xmath163 is the defining polynomial of @xmath134 . then the defining polynomial of @xmath164 is @xmath165 . if @xmath166 , then @xmath167 for some @xmath168 ; @xmath169 so @xmath51 is well defined and injective . let @xmath170 . then @xmath171 , so @xmath172 . if @xmath173 , then @xmath174 and @xmath175 , hence @xmath176 , where @xmath177 . because @xmath178 , @xmath179 . since @xmath24 and @xmath163 are relatively prime , we get that @xmath180 , which implies that @xmath181 . it remains to show is that @xmath182 is well defined . for suitable @xmath183 and @xmath184 we have that @xmath185 let @xmath186 be a line in @xmath134 defined by the vanishing of @xmath187 for some @xmath188 and @xmath189 , and let @xmath170 . then @xmath190 , so evaluating at @xmath191 and using the earlier observation that @xmath172 , we find @xmath192 . now suppose @xmath59 is a conic in @xmath134 ; after a change of coordinates we may assume @xmath59 intersects @xmath156 in the points @xmath82 and @xmath193 . then @xmath194 and @xmath195 , where @xmath196 is some linear form . we have @xmath197 evaluating at @xmath191 and again using that @xmath172 we find @xmath198 since @xmath199 and @xmath200 are relatively prime we obtain @xmath201 this shows that for each factor @xmath202 of @xmath203 , @xmath204 so the map @xmath182 is well defined . it follows that @xmath205(-(k-1))$ ] , where @xmath206 . a similar argument works if @xmath59 is tangent to @xmath207 . [ lem : admilnors ] let @xmath208 and @xmath209 be two reduced plane curves with no common component , meeting at a point @xmath51 . then @xmath210 where @xmath211 is the intersection number of @xmath208 and @xmath209 at @xmath51 . see @xcite , theorem 6.5.1 ; the point is that the milnor fiber is a connected curve , and the result follows from using the additivity of the euler characteristic and the interpretation of @xmath62 as the first betti number of the milnor fiber . [ prop : sesline ] let @xmath129 be a quasihomogeneous triple . then @xmath212 is also right exact . it follows from lemma [ lem : derexact ] that quotienting by the euler derivation and sheafifying yields the left exact sequence above ; so it will suffice to show that @xmath213 , where @xmath214 denotes the hilbert polynomial . for an cl arrangement @xmath79 with @xmath215 lines and @xmath47 conics , let @xmath216 . we have an exact sequence : @xmath217 where @xmath218 $ ] and @xmath219 is the jacobian ideal of the defining polynomial of @xmath79 . since @xmath220 @xmath221 we find that @xmath222 by the assumption that @xmath129 is a quasihomogeneous triple , @xmath223 let @xmath224 be the sum of milnor numbers of points off @xmath207 , so @xmath225 since @xmath226 , by lemma [ lem : admilnors ] , the above is @xmath227 as @xmath228 and @xmath229 , we obtain : @xmath230 by bezout s theorem , @xmath231 so @xmath232 , hence @xmath233 since @xmath234 , this yields the result . a coherent sheaf @xmath235 on @xmath236 is @xmath237regular iff @xmath238 for every @xmath239 . the smallest number @xmath215 such that @xmath235 is @xmath237regular is @xmath240 . [ lem : regbounds ] for a quasihomogeneous triple with @xmath241 , @xmath242 immediate from proposition [ prop : sesline ] ( see @xcite ) . [ lem : lineexact ] if @xmath243 , then there is an exact sequence of @xmath244-modules : @xmath245 for all @xmath246 , @xmath247 , so the long exact sequence in cohomology arising from proposition [ prop : sesline ] and the vanishing of @xmath248 yield an exact sequence : @xmath249 theorem a.4.1 of @xcite relates a graded module to its sheaf and local cohomology ( at the maximal ideal @xmath250 ) modules : @xmath251 this is true also for @xmath252 and @xmath253 . by @xcite , a.4.3 , @xmath254 if @xmath255 . lemma 2.1 of @xcite gives the desired bound on depth for the modules of derivations , which concludes the proof . the next two lemmas prove the two implications @xmath256 and @xmath257 of theorem [ thm : adline ] . in what follows , @xmath129 is a quasihomogeneous triple , with @xmath207 a line and @xmath258 . [ lem : lines1 ] if @xmath134 is free with @xmath259 , then @xmath79 is free with @xmath260 . first , if @xmath261 $ ] , and @xmath262 is a free graded @xmath244-module with generator in degree @xmath188 , then the hilbert series satisfies @xmath263 if @xmath134 is free with exponents @xmath154 , then @xmath264 . it follows from the proof of proposition [ prop : sesline ] that @xmath265 . so by lemma [ lem : lineexact ] and the additivity of hilbert series on an exact sequence , @xmath266 since @xmath267 , @xmath268 . by lemma [ lem : regbounds ] , if @xmath269 , then @xmath270 ; and if @xmath271 , then @xmath272 . if @xmath271 , then a free resolution for @xmath273 is of the form : @xmath274 from regularity constraints , @xmath128 must be at most @xmath275 . as this is a minimal free resolution , and it is impossible to have a syzygy on a single generator , the only situation which can actually arise occurs when @xmath276 : @xmath277 let @xmath278 be two independent derivations in @xmath273 of degrees @xmath279 and @xmath280 ; our computation of the hilbert series , combined with the fact that @xmath281 means such derivations must exist . let @xmath282 be a basis for @xmath283 with @xmath284 and @xmath285 , and @xmath12 the euler derivation . now note that @xmath286 , for otherwise in @xmath273 there would be an element of degree @xmath127 . so @xmath287 . since @xmath288 , then @xmath289 and @xmath290 , where @xmath291 is a constant , @xmath292 , @xmath293 and @xmath294 . for a resolution as above , @xmath295 , where @xmath207 is a linear form and @xmath296 . hence @xmath297 and since @xmath282 is a basis we find that @xmath291 vanishes and @xmath298 . but @xmath299 and @xmath287 . since @xmath300 , @xmath24 must divide @xmath68 , and so @xmath301 for some @xmath302 . since @xmath295 , we obtain @xmath303 , a contradiction . if @xmath304 , simply switch the roles of @xmath127 and @xmath275 above . [ lem : lines2 ] if @xmath79 is free with @xmath260 , then @xmath134 is free with @xmath259 . in order to obtain an appropriate vanishing , we need to dualize . apply @xmath305 to the exact sequence @xmath306 the vanishing of @xmath307 and @xmath308 yield an exact sequence : @xmath309 the free @xmath310 resolution for @xmath311 is : @xmath312 so @xmath313 . since @xmath314 , combining this with the long exact sequence in cohomology yields a regularity bound @xmath315 and the exact sequence of @xmath316modules : @xmath317 with @xmath318 . so : @xmath319 an argument as in the proof of lemma [ lem : lines1 ] shows that @xmath320 , hence @xmath136 is free with exponents @xmath154 . a free cl arrangement , when restricted to a line , can yield different multiarrangements . in example 2.2 , add the line @xmath321 , where @xmath322 . then @xmath207 passes through @xmath323 , and the choices for @xmath224 ensure that @xmath207 is not tangent to any conic , and misses all singularities save @xmath323 . the new arrangement is quasihomogeneous , and @xmath207 meets @xmath324 in six points . by theorem [ thm : adline ] , the new arrangement is free with exponents @xmath325 . restrict this new arrangement to the line @xmath326 . after a change of coordinates , we obtain a multiarrangement with defining polynomial @xmath327 this is exactly ziegler s example from @xcite : @xmath328 gives exponents @xmath329 , and for @xmath330 , the exponents are @xmath331 . let @xmath129 be a triple of cl arrangements in @xmath130 , where @xmath59 is a conic in @xmath79 , and @xmath332 , @xmath333 . we begin with some examples . suppose @xmath79 is as in example 2.2 , so @xmath79 has quasihomogeneous singularities , and is free with exponents @xmath145 . if we delete one of the conics , the resulting arrangement @xmath334 is free and quasihomogeneous , with exponents @xmath335 . when @xmath275 is odd , the situation is more complicated : let @xmath134 be the braid arrangement @xmath336 , and @xmath337 , where the conic @xmath338 . @xmath134 is a free arrangement with exponents @xmath335 , and @xmath339 . @xmath2 is also quasihomogeneous , but not free . let @xmath79 be the quasihomogeneous cl arrangement with defining polynomial @xmath340 . @xmath341 is free with exponents @xmath342 . deleting the conic yields a free line arrangement with exponents @xmath343 . [ thm : adconic ] let @xmath129 be a quasihomogeneous triple , with @xmath344 . if @xmath345 then the following are equivalent : 1 . @xmath134 is free with @xmath346 . @xmath79 is free with @xmath347 . if @xmath348 then : 1 . @xmath349 . if @xmath350 with @xmath351 then @xmath79 is not free . 3 . if @xmath352 with @xmath351 then @xmath134 is not free . we begin with some preliminaries . after an appropriate change of coordinates , we may suppose that @xmath353 . let @xmath188 be the composition of the maps @xmath354 where @xmath355 , and let @xmath356 be the composite map : @xmath357 \stackrel{\phi}{\longrightarrow}\mathbb k[s^2,st , t^2 ] \hookrightarrow \mathbb k[s , t],\ ] ] where @xmath358 .1 in let @xmath359 be a derivation . then @xmath360 , which means @xmath361 for some @xmath168 . via the map @xmath356 this translates into @xmath362 so there exist @xmath363 $ ] such that @xmath364 .1 in if @xmath365 is a ring map and @xmath101 is an @xmath366module , let @xmath367 denote the @xmath316module obtained by restriction of scalars . there is an exact sequence of @xmath316modules @xmath368 where @xmath369 for every @xmath370 and @xmath371 are defined as above ; and @xmath372 is the arrangement of the reduced points @xmath373 in @xmath374 . it is easy to check that @xmath375 is a homomorphism . for exactness , note : @xmath376 it remains to show that the image of @xmath375 is in @xmath377 . suppose @xmath378 is a line of @xmath79 . let @xmath379 . then @xmath380 for some @xmath381 . therefore @xmath382 which implies @xmath383 this means that @xmath384 $ ] . since @xmath385 is the defining polynomial of the two points @xmath386 in @xmath374 , we get that @xmath387 is a derivation on the arrangement of these two points . .1 in suppose @xmath388 is a conic in the cl arrangement @xmath79 . let @xmath359 . computations as above show that @xmath389 @xmath390.\ ] ] since @xmath391 is the defining polynomial of the four points @xmath392 in @xmath374 , we get that @xmath387 is a derivation on the arrangement of these four points . similar arguments work in the case of tangencies . let @xmath393 such that @xmath394 . then @xmath395 with @xmath396 . thus @xmath397 and @xmath121 is the euler derivation in @xmath341 . so quotienting by the euler derivations yields an exact sequence : @xmath398 since @xmath399 , after sheafifying , @xmath400 , and hence the sheafification of @xmath401 is @xmath402 . [ lem : hpconic ] @xmath403 . case 1 : @xmath345 . let @xmath12 be the divisor of the reduced @xmath275 points @xmath404 . then the ideal sheaf @xmath405 , where @xmath406 $ ] of degree @xmath345 . there exists @xmath407 , unique modulo @xmath408 , such that @xmath409 . clearly @xmath410 can not divide @xmath411 , otherwise @xmath412 , so the ideal of the reduced @xmath275 points on @xmath59 is @xmath413 . hence @xmath414 as an ideal of @xmath415 . as an @xmath316module , it has free resolution @xmath416 which yields : @xmath417 .1 in case 2 : @xmath348 . let @xmath12 be the divisor of the reduced @xmath275 points @xmath404 . then the ideal sheaf @xmath405 , where @xmath418 $ ] of degree @xmath348 . let @xmath419_1 $ ] be two independent linear forms which do not divide @xmath68 , and let @xmath420 . since @xmath421 , then @xmath422 defines the same ideal sheaf on @xmath374 as @xmath423 . so @xmath424 . both @xmath425 and @xmath426 are of even degree @xmath427 . so there exist @xmath428 $ ] of degree @xmath99 such that @xmath429 . next we show that @xmath430 is the ideal of the reduced points @xmath431 on @xmath59 . to see this , note that if @xmath432 , then @xmath433 . so @xmath434 , and hence @xmath435 . clearly @xmath410 does not divide @xmath436 , otherwise @xmath437 is identically zero . also , suppose @xmath438 , where @xmath439 is a constant . then @xmath440 , i.e. @xmath441 ; a contradiction . so @xmath219 is the ideal of @xmath442 points on the conic @xmath443 . by the hilbert - burch theorem , such an ideal is minimally generated by the @xmath444 minors of @xmath445 $ ] where both @xmath224 and @xmath446 have degree @xmath215 . so indeed @xmath447 , and @xmath448 . as an @xmath316module it has free resolution @xmath449 } } { \longrightarrow } s^2(-1-m)\longrightarrow \langle \bar{g_1},\bar{g_2 } \rangle \longrightarrow 0,\ ] ] so for the odd case we find that @xmath450 @xmath451 [ prop : sheafconicexact ] for a quasihomogeneous triple @xmath452 , the sequence @xmath453 is exact , where @xmath454}{\longrightarrow } { \mathbb p}^2 $ ] . we have @xmath455 , where @xmath456 is the degree of the defining polynomial @xmath68 of @xmath79 and @xmath163 is the defining polynomial of @xmath134 . since @xmath129 is a quasihomogeneous triple , bezout s theorem and lemma [ lem : admilnors ] imply that @xmath457 and hence @xmath458 by lemma [ lem : hpconic ] , this is exactly the hilbert polynomial of the sheaf @xmath459 associated to @xmath460 . [ lem : moduleconicexact ] for a quasihomogeneous triple such that @xmath134 is free with exponents @xmath461 , @xmath462 is exact . as we ve seen , @xmath463 . with the assumption on @xmath134 , @xmath464 vanishes for all @xmath246 , and exactness follows as in the proof of lemma [ lem : lineexact ] . theorem [ thm : adconic ] will follow from the next two lemmas . [ lem : conic1 ] let @xmath129 be a quasihomogeneous triple , with @xmath344 . if @xmath134 is free with exponents @xmath461 , then 1 . if @xmath345 then @xmath79 is free with @xmath347 . 2 . if @xmath348 and @xmath465 , then @xmath79 is free with @xmath466 . 3 . if @xmath348 and @xmath467 , then @xmath79 is not free . it follows from the computations in the proof of lemma [ lem : hpconic ] that * if @xmath348 , then @xmath468 . * if @xmath345 , then @xmath469 . combining these results yields the hilbert series of @xmath273 . case 1 : @xmath345 . by lemma [ lem : moduleconicexact ] , @xmath470 since @xmath281 , this means that there exist minimal generators @xmath471 with @xmath472 and @xmath473 . suppose @xmath474 basis for @xmath283 with @xmath12 the euler derivation and @xmath475 . we now use that @xmath288 . * @xmath476 . since @xmath477 , @xmath478 for some @xmath479 . then @xmath480 is a basis for @xmath283 , so by saito s criterion @xmath481 is a basis for @xmath482 . * . then @xmath484 , where @xmath485 constants , not both zero . if @xmath486 , then @xmath487 is a basis for @xmath283 , so by saito s criterion @xmath488 is a basis for @xmath482 . * . then @xmath490 , where @xmath491 is a constant and @xmath492 is a linear form , not both zero . if @xmath493 then @xmath494 . since @xmath59 is irreducible , then @xmath495 , and so @xmath496 is of degree @xmath497 . this is inconsistent with the hilbert series of @xmath273 . so @xmath498 , and so @xmath480 is a basis for @xmath283 , and again by saito s criterion @xmath481 is a basis for @xmath482 . . then @xmath500 , where @xmath501 is a constant and @xmath502 is a quadratic form , not both zero and @xmath503 , where @xmath504 is a constant and @xmath505 is a quadratic form , not both zero . if @xmath506 , then either @xmath507 or @xmath496 , a contradiction ( because @xmath508 ) . therefore @xmath509 , contradicting the fact that @xmath510 are minimal generators of @xmath273 . so if @xmath511 , then @xmath512 is a basis for @xmath283 , and so by saito s criterion @xmath513 is a basis for @xmath482 . * @xmath514 . then @xmath500 , where @xmath501 is a constant and @xmath502 is a polynomial , not both zero and @xmath503 , where @xmath504 is a constant and @xmath505 is a quadratic form , not both zero . if @xmath506 , then @xmath515 and @xmath516 , @xmath517 nonzero constant , and the argument used above yields a contradiction . so @xmath498 or @xmath486 . applying saito s criterion yields the desired result . .1 in case 2 : @xmath518 . by lemma [ lem : moduleconicexact ] , @xmath519 this implies there exist degree @xmath99 minimal generators @xmath520 . suppose @xmath474 is a basis for @xmath283 where @xmath12 is the euler derivation and @xmath521 . so @xmath522 and @xmath523 , where @xmath524 are linear forms , and for any @xmath525 , @xmath526 can not be simultaneously zero . hence @xmath527 . but @xmath528 is in @xmath136 and @xmath529 , else @xmath530 is nonzero , which is inconsistent with the hilbert series . hence @xmath531 , where @xmath491 is a constant . if @xmath532 , then @xmath533 or @xmath534 , where @xmath535 are constants , and that @xmath536 and @xmath537 . if @xmath533 , and @xmath538 we get @xmath539 . since @xmath540 ( else @xmath541 ) then @xmath542 implies @xmath543 , yielding a degree @xmath215 derivation in @xmath544 , a contradiction . if @xmath545 , then @xmath121 and @xmath28 are not minimal generators , also a contradiction . if @xmath546 , then we find @xmath547=cc\det[e,\theta_1,\theta_2]$ ] , and saito s criterion shows that @xmath548 is a basis for @xmath341 . .1 in case 3 : @xmath549 . by lemma [ lem : moduleconicexact ] , @xmath550 since @xmath551 , there is no cancellation in the numerator , hence @xmath273 can not be free . [ lem : conic2 ] let @xmath129 be a quasihomogeneous triple , with @xmath344 . if @xmath79 is free , then 1 . if @xmath552 and @xmath553 , then @xmath134 is free with @xmath554 . 2 . if @xmath555 and @xmath556 , then @xmath134 is free with @xmath557 . 3 . if @xmath558 and @xmath559 with @xmath560 , then @xmath134 is not free . as in lemma [ lem : lines2 ] , apply @xmath305 to the exact sequence @xmath561 since @xmath562 is supported on the conic @xmath59 , @xmath563 . the assumption that @xmath273 is free implies that @xmath564 . this yields an exact sequence : @xmath565 as @xmath273 free with known exponents , so also is @xmath566 , and the hilbert series is known . the proof of lemma [ lem : hpconic ] provides a free resolution of @xmath567 , which allows us to compute @xmath568 . combining everything yields the hilbert series of @xmath569 , and the result follows as in the previous analysis . we close with a pair of examples which show that in the cl case , terao s conjecture that freeness is a combinatorial invariant of an arrangement is false . [ exm : last ] let @xmath324 be given by @xmath570 @xmath571 is tangent to both @xmath572 and @xmath573 at the point @xmath574 ; @xmath572 and @xmath573 have two other points in common . adding the line @xmath575 passing through @xmath576 to @xmath324 yields a quasihomogeneous , free cl arrangement @xmath79 , with @xmath577 the line @xmath578 passes through @xmath576 , and misses the other singularities of @xmath324 . the cl arrangement @xmath579 is combinatorially identical to @xmath79 , but @xmath134 is not quasihomogeneous , and not free : @xmath580 [ exm : noterao ] let @xmath581 be the union of the five smooth conics : @xmath582 @xmath581 has 13 singular points , all ordinary . at 10 of these points only two branches of @xmath581 meet , while at the points @xmath583 , all five conics meet . the milnor and tjurina numbers agree at all singularities except @xmath82 , where @xmath84 and @xmath83 . adding lines @xmath584 connecting @xmath583 yields a free cl arrangement @xmath79 , with @xmath585 . .05 in next , let @xmath586 be the union of the following five smooth conics : @xmath587 @xmath586 is combinatorially identical to @xmath581 , but at the points @xmath588 where all the branches meet , @xmath84 and @xmath83 . adding the lines connecting these three points yields an cl arrangement @xmath589 which is combinatorially identical to @xmath79 but _ not free _ ; the free resolution of @xmath590 is : @xmath591 as was pointed out by the referee , the complements of arrangements @xmath33 and @xmath592 are homeomorphic ( via a cremona transformation centered on the three multiple intersection points ) to the complements of a pair of line arrangements consisting of eight lines in general position . the moduli space of such objects is connected , so the complements are rigidly isotopic , hence homeomorphic . so freeness is also not a topological invariant . 1 . as noted in 1.2 , for the complement @xmath208 of an cl arrangement in @xmath593 the betti numbers @xmath594 and @xmath595 depend only on the combinatorics , and so if @xmath208 is quasihomogeneous and free , there is a version of terao s theorem , which we leave for the interested reader . 2 . in the examples above , the jacobian ideals are of different degrees , so are not even members of the same hilbert scheme . do there exist cl arrangements with isomorphic intersection poset _ and _ singularities which are locally isomorphic , one free and one nonfree ? do there exist counterexamples where all singularities are quasihomogeneous ? as shown by example 2.3 , quasihomogenity is not a necessary condition for freeness of cl arrangements . however , without this assumption , the sequences in propositions 2.8 and 3.8 may not be exact , which means that any form of addition - deletion will require hypotheses on higher cohomology . * acknowledgments * : macaulay2 computations were essential to our work . we also thank an anonymous referee for many useful suggestions , in particular for pointing out that we should remove one of our original conditions ( that @xmath596 has only ordinary singularities ) .

let @xmath0 be a collection of smooth rational plane curves . we prove that the addition - deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves , giving an inductive tool for understanding the freeness of the module @xmath1 of logarithmic differential forms with pole along @xmath2 . we also show that the analog of terao s conjecture ( freeness of @xmath1 is combinatorially determined if @xmath2 is a union of lines ) is false in this setting . [ section ] [ defn0]proposition [ defn0]conjecture [ defn0]theorem [ defn0]lemma [ defn0]corollary [ defn0]example

the question of first - cause has been troubling to philosophers and scientists alike for over two thousand years . aristotle found this sufficiently troubling that he proposed avoiding it by having the universe exist eternally in both the past and future . that way , it was always present and one would not have to ask what caused it to come into being . this type of model has been attractive to modern scientists as well . when einstein developed general relativity and applied it to cosmology , his first cosmological model was the einstein static universe , which had a static @xmath1 spatial geometry which lasted forever , having no beginning and no end @xcite . as we shall discuss , since the big bang model s success , models with a finite beginning have taken precedence , even when inflation and quantum tunneling are included . so the problem of first - cause reasserts itself . the big question appears to be how to create the universe out of nothing . in this paper we shall explore the idea that this is the wrong question . a remarkable property of general relativity is that it allows solutions that have closed timelike curves ( ctcs ) @xcite ( for review see @xcite ) . often , the beginning of the universe , as in vilenkin s tunneling model @xcite and hartle and hawking s no - boundary model @xcite , is pictured as being like the south pole of the earth and it is usually said that asking what happened before that is like asking what is south of the south pole @xcite . but , suppose the early universe contains a region of ctcs . then , asking what was the earliest point might be like asking what is the easternmost point on the earth . you can keep going east around and around the earth there is no eastern - most point . in such a model every event in the early universe would have events that preceded it . this period of ctcs could well have ended by now , being bounded by a cauchy horizon . some initial calculations of vacuum polarization in spacetimes with ctcs indicated that the renormalized energy - momentum tensor diverged at the cauchy horizon separating the region with ctcs from the region without closed causal curves , or at the polarized hypersurfaces nested inside the cauchy horizon @xcite . some of these results motivated hawking @xcite to propose the chronology protection conjecture which states that the laws of physics do not allow the appearance of ctcs . but , a number of people have challenged the chronology protection conjecture by giving counter - examples @xcite . in particular , li and gott @xcite have recently found that there is a self - consistent vacuum in misner space for which the renormalized energy - momentum tensor of vacuum polarization is zero everywhere . ( cassidy @xcite has independently given an existence proof that there should be a quantum state for a conformally coupled scalar field in misner space , for which the renormalized energy - momentum tensor is zero everywhere , but he has not shown what state it should be . li and gott @xcite have found that it is the `` adapted '' rindler vacuum . ) in this paper we give some examples to show how it is possible in principle to find self - consistent vacuum states where the renormalized energy - momentum tensor does not blow up as one approaches the cauchy horizon . to produce such a region of ctcs , the universe must , at some later time , be able to reproduce conditions as they were earlier , so that a multiply connected solution is possible . interestingly , inflation is well suited to this . a little piece of inflationary state expands to produce a large volume of inflationary state , little pieces of which resemble the starting piece . also there is the possibility of forming baby universes at late times where new pieces of inflating states are formed . farhi , guth , and guven @xcite , harrison @xcite , smolin @xcite , and garriga and vilenkin @xcite have considered such models . if one of those later inflating pieces simply turns out to be the inflating piece that one started out with , then the universe can be its own mother . since an infinite number of baby universes are created , as long as the probability of a particular multiple connection forming is not exactly zero , then such a connection might be expected , eventually . then the universe neither tunneled from nothing , nor arose from a singularity ; it created itself ( fig . [ f1 ] ) . before discussing this approach to the first - cause problem , let us review just how troublesome this problem has been . as we have noted , einstein @xcite initially tried to avoid it by siding with aristotle in proposing a model which had an infinite past and future . the einstein static universe appears to be the geometry einstein found _ a priori _ most aesthetically appealing , thus presumably he started with this preferred geometry and substituted it into the field equations to determine the energy - momentum tensor required to produce it . he found a source term that looks like dust ( stars ) plus a term that was proportional to the metric which he called the cosmological constant . the cosmological constant , because of its homogeneous large negative pressure , exerts a repulsive gravitational effect offsetting the attraction of the stars for each other ; allowing a static model which could exist ( ignoring instabilities , which he failed to consider ) to the infinite past and future . if one did not require a static model , there would be no need for the cosmological constant . friedmann @xcite calculated models without it , of positive , negative or zero curvature , all of which were dynamical . when hubble @xcite discovered the expansion of the universe , einstein pronounced the cosmological constant the biggest blunder of his life . but now there was a problem : all three friedmann models ( @xmath2 , @xmath3 , and @xmath4 ) that were expanding at the present epoch had a beginning in the finite past ( see e.g. @xcite ) . in the friedmann models the universe began in a singularly dense state at a finite time in the past . the equations could not be pushed beyond that finite beginning singularity . furthermore , if today s hubble constant is @xmath5 , then all of the friedmann models had ages less than @xmath6 . the universe thus began in a big bang explosion only a short time ago , a time which could be measured in billions of years . the universe was not infinitely old . gamow @xcite and his colleagues alpher and herman @xcite calculated the evolution of such a big bang cosmology , concluding correctly that in its early phases it should have been very dense and very hot , and that the thermal radiation present in the early universe should still be visible today as microwave radiation with a temperature of approximately @xmath7 . penzias and wilson s discovery of the radiation with a temperature of @xmath8 @xcite cinched the case for the big bang model . the cobe results which have shown a beautifully thermal spectrum @xcite and small fluctuations in the temperature @xmath9 @xcite , fluctuations that are of approximately the right magnitude to grow into the galaxies and clusters of galaxies that we see at the present epoch , have served to make the big bang model even more certain . with the big bang model in ascendancy , attention focused on the initial singularity . hawking and penrose proved a number of singularity theorems @xcite showing that , with some reasonable constraints on the energy - momentum tensor , if einstein s equations are correct and the expansion of the universe is as observed today , there is no way to avoid an initial singularity in the model ; that is , initial singularities would form even in models that were not exactly uniform . so the initial singularity was taken to be the first - cause of the universe . this of course prompted questions of what caused the singularity and what happened before the singularity . the standard answer to what happened before the big bang singularity has been that time was created at the singularity , along with space , and that there was no time before the big bang . asking what happened before the big bang was considered to be like asking what is south of the south pole . but particularly troublesome was the question of what caused the initial singularity to have its almost perfect uniformity for otherwise the microwave background radiation would be of vastly different temperatures in different directions on the sky . yet the initial singularity could not be exactly uniform , for then we would have a perfect friedmann model with no fluctuations which would form no galaxies . it needed to be almost , but not quite perfectly uniform a remarkable situation how did it get that way ? these seemed to be special initial conditions with no explanation for how they got that way . another problem was that singularities in physics are usually smeared by quantum effects . as we extrapolated back toward the initial singularity ( of infinite density ) , we would first reach a surface where the density was equal to the planck density and at this epoch classical general relativity would break down . we could not extrapolate confidently back to infinite density , we could only say that we would eventually reach a place where quantum effects should become important and where classical general relativity no longer applied . since we do not have a theory of quantum gravity or a theory - of - everything we could honestly say that the singularity theorems only told us that we would find regions in the early universe where the density exceeded the gut or planck densities beyond which we did not know what happened rather much like the terra incognita of old maps . we could not then say how our universe formed . so , questions about how the initial big bang singularity was formed and what preceded it remained . the closed friedmann model , popular because it is compact and therefore needs no boundary conditions , re - collapses in a finite time in the future to form a big crunch singularity at the end . singularity theorems tell us that in a collapsing universe the final big crunch singularity can not be avoided . classical general relativity tells us that a closed universe begins with a singularity and ends with a singularity , with nothing before and nothing after . nevertheless , many people speculated that there could be more than one connected cycle after all , the singularities only indicated a breakdown of classical general relativity and the quantum terra incognita at the planck density might allow a cosmology collapsing toward a big crunch to bounce and make another big bang @xcite . in support of this is the fact that de sitter space ( representing the geometry of a false vacuum an inflationary state as proposed by guth @xcite with a large cosmological constant ) looks like a spatially closed @xmath1 universe whose radius as a function of proper time is @xmath10 , where @xmath11 is the radius of the de sitter space and @xmath12 is the cosmological constant ( throughout the paper we use units @xmath13 ) , which is a collapsing cosmology which bounces and turns into an expanding one . thus if quantum gravitational effects make the geometry look like de sitter space once the density reaches the planck density as some have suggested @xcite , then a big crunch singularity might be avoided as the closed universe bounced and began a big bang all over again . this bouncing model avoids the first - cause problem . the answer to what caused our universe in this model is `` the collapse of the previous universe '' , and so on . an infinite number of expansion and contraction cycles make up the universe ( note the capital u in this paper this denotes the ensemble of causally connected universes ) which consists of an infinite number of closed big bang models laid out in time like pearls on a string . the universe ( the infinite string of pearls ) has always been in existence and will always be in existence , even though our cycle , our standard closed big bang cosmology ( our pearl ) has a finite duration . so we are back to aristotle , with an eternal universe , and close to einstein with just an oscillating ( rather than static ) closed universe that has infinite duration to the past and future . thus in this picture there is no first - cause because the universe has existed infinitely far back in the past . the oscillating universe was thought to have some problems with entropy @xcite . entropy is steadily increasing with time , and so each cycle would seem to be more disordered than the one that preceded it . since our universe has a finite entropy per baryon it was argued , there could not be an infinite number of cycles preceding us . likewise it was argued that each cycle of the universe should be larger than the preceding one , so if there were an infinite number preceding us , our universe would have to look indistinguishable from flat ( i.e. , closed but having an infinite radius of curvature ) . the real challenge in this model is to produce initial conditions for our universe ( our pearl ) that were as uniform and low entropy as observed . cobe tells us that our universe at early times was uniform to one part in a hundred thousand @xcite . at late times we expect the universe at the big crunch to be very non - uniform as black hole singularities combine to form the big crunch . in the early universe the weyl tensor is zero , whereas at the big crunch it would be large @xcite . how does the chaotic high - entropy state at the big crunch get recycled into the low - entropy , nearly uniform , state of the next big bang ? if it does not , then after an infinite number of cycles , why are we not in a universe with chaotic initial conditions ? entropy and the direction of time may be intimately tied up with this difference between the big bang and the big crunch . maxwell s equations ( and the field equations of general relativity ) are time - symmetric , so why do we see only retarded potentials ? wheeler and feynman addressed this with their absorber theory @xcite . they supposed that an electron shaken today produces half - advanced - half - retarded fields . the half - advanced fields propagate back in time toward the early universe where they are absorbed ( towards the past the universe is a perfect absorber ) by shaking charged particles in the early universe . these charged particles in turn emit half - advanced - half - retarded fields ; their half - retarded fields propagate toward the future where they : ( a ) perfectly cancel the half - advanced fields of the original electron , ( b ) add to its retarded fields to produce the electron s full retarded field , and ( c ) produce a force on the electron which is equal to the classical radiative reaction force . thus , the electron only experiences forces due to fields from other charged particles . this is a particularly ingenious solution . it requires only that the early universe is opaque which it is and that the initial conditions are low - entropy ; that is , there is a cancelation of half - advanced fields from the future by half - retarded fields from the past , leaving no `` signals '' in the early universe from later events a state of low - entropy . ( note that this argument works equally well in an open universe where the universe may not be optically thick toward the future all that is required is that the universe be a perfect absorber in the past , i.e. , toward the state of low - entropy . ) wheeler and feynman noted that entropy is time - symmetric like maxwell s equations . if you find an ice cube on the stove , and then come back and re - observe it a minute later , you will likely find it half - melted . usually an ice cube gets on a stove by someone just putting it there ( initial conditions ) , but suppose we had a truly isolated system so that the ice cube we found was just a statistical fluctuation . then if we asked what we would see if we had observed one minute _ before _ our first observation , we will also be likely to see a half - melted ice cube , for finding a still larger ice cube one minute before would be unlikely because it would represent an even more unlikely statistical fluctuation than the original ice cube . in an _ isolated _ system , an ( improbable ) state of low - entropy is likely to be both followed and preceded by states of higher - entropy in a time - symmetric fashion . given that the early universe represents a state of high order , it is thus not surprising to find entropy increasing after that . thus , according to wheeler and feynman @xcite , the fact that the retarded potentials arrow of time and the entropy arrow of time point in the same direction is simply a reflection of the low - entropy nature of the big bang . the big crunch is high - entropy , so time follows from past to future between the big bang and the big crunch . thus , in an oscillating universe scenario , we might expect entropy to go in the opposite direction with respect to time , in the previous cycle of oscillation . in that previous universe there would be only advanced potentials and observers there would sense a direction of time opposite to ours ( and would have a reversed definition of matter and anti - matter because of cpt invariance ) . thus the cycle previous to us would , according to _ our _ definition of time , have advanced potentials and would end with a uniform low - entropy big crunch and begin with a chaotic high - entropy big bang ( see gott @xcite for further discussion ) . thus , an infinite string of oscillating universes could have alternating high and low - entropy singularities , with the direction of the entropy ( and causality via electromagnetic potentials ) time - reversing on each succeeding cycle . every observer using the entropy direction of time would see in his `` past '' a low - entropy singularity ( which he would call a big bang ) and in his `` future '' a high - entropy singularity ( which he could call a big crunch ) . then the mystery is why the low - entropy big bangs exist they now look improbable . an oscillating universe with chaotic bangs and crunches and half - advanced - half - retarded potentials throughout would seem more likely . at this point anthropic arguments @xcite could be brought in to say that only low - entropy big bangs might produce intelligent observers and that , with an infinite number of universes in the string , eventually there would be by chance a sufficiently low - entropy big bang to produce intelligent observers . still , the uniformity of the early universe that we observe seems to be more than that required to produce intelligent observers , so we might wonder whether a random intelligent observer in such a universe would be expected to see initial conditions in his / her big bang as uniform as ours . ( among intelligent observers , the copernican principle tells us that you should not expect to be special . out of all the places for intelligent observers to be there are by definition only a few special places and many non - special places , so you should expect to be in one of the many non - special places @xcite . ) guth s proposal of inflation @xcite offered an explanation of why the initial conditions in the big bang should be approximately , but not exactly uniform . ( for review of inflation see @xcite . ) in the standard big bang cosmology this was always a puzzle because antipodal points on the sky on the last scattering surface at @xmath14 had not had time to be in communication with each other . when we see two regions which are at the same temperature , the usual explanation is that they have at some time in the past been in causal communication and have reached thermal equilibrium with each other . but there is not enough time to do this in the standard big bang model where the expansion of the scale factor at early times is @xmath15 . grand unified theories ( gut ) of particle physics suggest that at early times there might have been a non - zero cosmological constant @xmath12 , which then decayed to the zero cosmological constant we see today . this means that the early universe approximates de sitter space with a radius @xmath11 whose expansion rate at late times approaches @xmath16 . regions that start off very close together , and have time to thermally equilibrate , end up very far apart . when they become separated by a distance @xmath17 , they effectively pass out of causal contact if inflation were to continue forever , they would be beyond each other s event horizons . but eventually the epoch of inflation ends , the energy density of the cosmological constant is dumped into thermal radiation , and the expansion then continues as @xmath15 as in a radiation - dominated big bang cosmology . as the regions slow their expansion from each other , enough time elapses so that they are able to interchange photons once again and they come back into effective causal contact . as bill press once said , they say `` hello '' , `` goodbye '' , and `` hello again '' . when they say `` hello again '' they appear just like regions in a standard big bang cosmology that are saying `` hello '' for the first time ( i.e. , are just coming within the particle horizon ) except that with inflation these regions are already in thermal equilibrium with each other , because they have seen each other in the past . inflation also gives a natural explanation for why the observed radius of curvature of the universe is so large ( @xmath18 ; here @xmath19 km s@xmath20 mpc@xmath20 is the hubble constant ) . during the big bang phase , as the universe expands , the radius of the universe @xmath21 expands by the same factor as the characteristic wavelength @xmath22 of the microwave background photons , so @xmath23 . how should we explain this large observed dimensionless number ? inflation makes this easy . the energy density during the inflationary epoch is @xmath24 . let @xmath22 be the characteristic wavelength of thermal radiation which would have that density . even if @xmath21 started out of the same order as @xmath22 , by the end of the inflationary epoch @xmath25 , providing that the inflationary epoch lasts at least as long as @xmath26 , or @xmath27 @xmath28-folding times . at the end of the inflationary epoch when the inflationary vacuum of density @xmath24 decays and is converted into an equivalent amount of thermal radiation , the wavelength of that radiation will be @xmath22 and the ratio of @xmath29 is fixed at a constant value which is a dimensionless constant @xmath30 , retained as the universe continues to expand in the radiation and matter - dominated epochs . thus , even a short run of inflation , of @xmath27 @xmath28-folding times or more , is sufficient to explain why the universe is as large as it is observed to be . another success of inflation is that the observed zeldovich - peebles - yu - harrison fluctuation spectrum with index @xmath31 @xcite has been naturally predicted as the result of random quantum fluctuations @xcite . the inflationary power spectrum with cdm has been amazingly successful in explaining the qualitative features of observed galaxy clustering ( cf . the amount of large scale power seen in the observations suggests an inflationary cdm power spectrum with @xmath32 @xcite . gott @xcite has shown how an open inflationary model might be produced . the initial inflationary state approximates de sitter space , which can be pictured by embedding it as the surface @xmath33 in a five - dimensional minkowski space with metric @xmath34 @xcite . slice de sitter space along surfaces of @xmath35 , then the slices are three - spheres of positive curvature @xmath36 where @xmath37 . if @xmath38 measures the proper time , then @xmath39 and @xmath40 . this is a closed universe that contracts then re - expands at late times expanding exponentially as a function of proper time . if slices of @xmath41 are chosen , the slices have a flat geometry and the expansion is exponential with @xmath42 . if the slices are vertical ( @xmath43 ) , then the intersection with the surface is @xmath44 , a hyperboloid @xmath45 living in a minkowski space , where @xmath46 . this is a negatively curved surface with a radius of curvature @xmath21 . let @xmath38 be the proper time from the event e ( @xmath47 ) in the de sitter space . then the entire future of e can be described as an open @xmath4 cosmology where @xmath48 . at early times , @xmath49 , near e , @xmath50 , and the model resembles a milne cosmology @xcite , but at late times the model expands exponentially with time as expected for inflation . this is a negatively curved ( open ) friedmann model with a cosmological constant and nothing else . note that the entire negatively curved hyperboloid ( @xmath44 ) , which extends to infinity , is nevertheless causally connected because all points on it have the event e in their past light cone . thus , the universe should have a microwave background that is isotropic , except for small quantum fluctuations . at a proper time @xmath51 after the event e , the cosmological constant would decay leaving us with a hot big bang open ( @xmath4 ) cosmology with a radius of curvature of @xmath52 at the end of the inflationary epoch . if @xmath53 , then @xmath54 is a few tenths today ; if @xmath55 , then @xmath56 today @xcite . gott @xcite noted that this solution looks just like the interior of a coleman bubble @xcite . coleman and de luccia @xcite showed that if a metastable symmetric vacuum ( with the higgs field @xmath57 ) , with positive cosmological constant @xmath12 were to decay by tunneling directly through a barrier to reach the current vacuum with a zero cosmological constant ( where the higgs field @xmath58 ) , then it would do this by forming a bubble of low - density vacuum of radius @xmath59 around an event e. the pressure inside the bubble is zero while the pressure outside is negative ( equal to @xmath60 ) , so the bubble wall accelerates outward , forming in spacetime a hyperboloid of one sheet ( a slice of de sitter space with @xmath61 ) . this bubble wall surrounds and is asymptotic to the future light cone of e. if the tunneling is direct , the space inside the bubble is minkowski space ( like a slice @xmath62 in the embedding space , which is flat ) . the inside of the future light cone of e thus looks like a milne cosmology with @xmath63 and @xmath64 . gott @xcite noted that what was needed to produce a realistic open model with @xmath54 of a few tenths today was to have the inflation continue inside the bubble for about @xmath27 @xmath28-folding times . thus , our universe was one of the bubbles and this solved the problem of guth s inflation that in general one expected the bubbles not to percolate @xcite . but , from inside one of the bubbles , our view could be isotropic @xcite . it was not long before a concrete mechanism to produce such continued inflation inside the bubble was proposed . a couple of weeks after gott s paper appeared linde s @xcite proposal of new inflation appeared , followed shortly by albrecht and steinhardt @xcite . they proposed that the higgs vacuum potential @xmath65 had a local minimum at @xmath57 where @xmath66 . then there was a barrier at @xmath67 , followed by a long flat plateau from @xmath68 to @xmath69 where it drops precipitately to zero at @xmath69 . the relation of this to the open bubble universe s geometry is outlined by gott @xcite ( see fig . 1 and fig . 2 in @xcite ) . the de sitter space outside the bubble wall has @xmath57 . between the bubble wall , at a spacelike separation @xmath59 from the event e , and the end of the inflation at the hyperboloid @xmath44 which is the set of points at a future timelike separation of @xmath51 from e , the higgs field is between @xmath68 and @xmath69 , and @xmath51 is the time it takes the field ( after tunneling ) to roll along the long plateau [ where @xmath65 is approximately equal to @xmath24 and the geometry is approximately de sitter ] . after that epoch , @xmath58 where the energy density has been dumped into thermal radiation and the vacuum density is zero ( i.e. , a standard open big bang model ) . in order that inflation proceeds and the bubbles do not percolate , it is required that the probability of forming a bubble in de sitter space per four volume @xmath70 is @xmath71 where @xmath72 @xcite . in order that there be a greater than @xmath73 chance that no bubble should have collided with our bubble by now , so as to be visible in our past light cone , @xmath74 for @xmath75 , @xmath76 , @xmath77 today @xcite , but this is no problem since we expect tunneling probabilities through a barrier to be exponentially small . this model has an event horizon , which is the future light cone of an event e@xmath78 ( @xmath79 ) which is antipodal to e. light from events within the future light cone of e@xmath78 never reaches events inside the future light cone of e. so we are surrounded by an event horizon . this produces hawking radiation ; and , if @xmath17 is of order the planck length , then the gibbons - hawking thermal state @xcite ( which looks like a cosmological constant due to the trace anomaly @xcite ) should be dynamically important @xcite . if we observe @xmath80 and @xmath81 , then @xmath4 and we need inflation more than ever we still need it to explain the isotropy of the microwave background radiation and we would now have a large but _ finite _ radius of curvature to explain , which @xmath27 @xmath28-folds of inflation could naturally produce . when gott told this to linde in 1982 , linde said , yes , if we found that @xmath80 , he would still have to believe in inflation but he would have a headache in the morning ! why ? because one has to produce a particular amount of inflation , approximately @xmath27 @xmath28-folds . if there were @xmath82 @xmath28-folds or @xmath82 million @xmath28-folds , then @xmath54 currently would be only slightly less than 1 . so there would be what is called a `` fine tuning of parameters '' needed to produce the observed results . the single - bubble open inflationary model @xcite discussed above has recently come back into fashion because of a number of important developments . on the theoretical side , ratra and peebles @xcite have shown how to calculate quantum fluctuations in the @xmath44 hyperbolic geometry with @xmath48 during the inflationary epoch inside the bubble in the single bubble model . this allows predictions of fluctuations in the microwave background . bucher , goldhaber , and turok @xcite have extended these calculations , as well as yamamoto , sasaki and tanaka @xcite . importantly , they have explained @xcite that the fine tuning in these models is only `` logarithmic '' and , therefore , not so serious . linde and mezhlmian @xcite have shown how there are reasonable potentials which could produce such bubble universes with different values of @xmath54 . in a standard chaotic inflationary potential @xmath65 @xcite , one could simply build in a bump , so that one would randomly walk to the top of the curve via quantum fluctuations and then roll down till one lodged behind the bump in a metastable local minimum . one would then tunnel through the bump , forming bubbles that would roll down to the bottom in a time @xmath51 . one could have a two - dimensional potential @xmath83 , where @xmath84 is a constant and there is a metastable trough at @xmath85 with altitude @xmath86 with a barrier on both sides , but one could tunnel through the barrier to reach @xmath87 where @xmath88 has a true minimum , and at fixed @xmath59 , is proportional to @xmath89 @xcite . then individual bubbles could tunnel across the barrier at different values of @xmath90 , and hence have different roll - down times @xmath51 and thus different values of @xmath54 . with a myriad of open universes being created , anthropic arguments @xcite come into play and if shorter roll - down times were more probable than large ones , we might not be surprised to find ourselves in a model which had @xmath54 of a few tenths , since if @xmath54 is too small , no galaxies will form @xcite . a second reason for the renaissance of these open inflationary models is the observational data . a number of recent estimates of @xmath91 ( the present hubble constant in units of 100 km s@xmath20 mpc@xmath20 ) have been made ( i.e. , @xmath92 @xcite , @xmath93 @xcite , @xmath94 @xcite , and @xmath95 @xcite ) . ages of globular cluster stars have a @xmath96 lower limit of about 11.6 billion years @xcite , we require @xmath97 if @xmath98 , but a more acceptable @xmath99 if @xmath75 , @xmath81 . models with low @xmath54 but @xmath100 are also acceptable . also , studies of large scale structure have shown that with the inflationary cdm power spectrum , the standard @xmath98 , @xmath101 model simply does not have enough power at large scales . a variety of observational samples and methods have suggested this : counts in cells , angular covariance function on the sky , power spectrum analysis of 3d samples , and finally topological analysis , all showing that @xmath32 @xcite . if @xmath102 this implies @xmath103 , which also agrees with what one would deduce from the age argument as well as the measured masses in groups and clusters of galaxies @xcite . with the cobe normalization there is also the problem that with @xmath98 , @xmath104 and this would require galaxies to be anti - biased [ since for galaxies @xmath105 and would also lead to an excess of large - separation gravitational lenses over those observed @xcite . these things have forced even enthusiasts of @xmath2 models to move to models with @xmath80 and a cosmological constant so that @xmath106 and @xmath2 @xcite . they then have to explain the small ratio of the cosmological constant to the planck density ( @xmath107 ) . currently we do not have such a natural explanation for a small yet finite @xmath12 as inflation naturally provides for explaining why the radius of curvature should be a big number in the @xmath4 case . turner @xcite and fukugita , futamase , and kasai @xcite showed that a flat @xmath108 model produces about 10 times as many gravitational lenses as a flat model with @xmath98 , and kochanek @xcite was able to set a @xmath109 confidence lower limit of @xmath110 in flat models where @xmath106 , and a @xmath111 confidence lower limit @xmath112 in open models with @xmath113 . thus , extreme-@xmath12 dominated models are ruled out by producing too many gravitational lenses . data on cosmic microwave background fluctuations for spherical harmonic modes from @xmath114 to @xmath115 will provide a strong test of these models . with @xmath116 , the @xmath98 , @xmath113 model power spectrum reaches its peak value at @xmath117 ; an @xmath118 , @xmath119 model reaches its peak value also at @xmath117 @xcite ; while an @xmath75 , @xmath113 model reaches its peak value at @xmath120 @xcite . this should be decided by the map and planck satellites which will measure this range with high accuracy @xcite . for the rest of this paper we shall usually assume single - bubble open inflationary models for our big bang universe ( while recognizing that chaotic inflationary models and models with multiple epochs of inflation are also possible ; it is interesting to note that penrose also prefers an open universe from the point of view of the complex - holomorphic ideology of his twister theory @xcite ) . if the inflation within the bubble is of order 67 @xmath28-folds , then we can have @xmath54 of a few tenths ; but if it is longer than that , we will usually see @xmath54 near 1 today . in any case , we will be assuming an initial metastable vacuum which decays by forming bubbles through barrier penetration . the bubble formation rate per unit four volume @xmath70 is thus expected to be exponentially small so the bubbles do not percolate . inflation is thus eternal to the future @xcite . borde and vilenkin have proved that if the universe were infinitely old ( i.e. , if the de sitter space were complete ) then the bubbles would percolate immediately and inflation would never get started ( see @xcite and references cited therein ) . recall that a complete de sitter space may be covered with an @xmath1 coordinate system ( a @xmath3 cosmology ) whose radius varies as @xmath40 so that for early times ( @xmath121 ) the universe would be contracting and bubbles would quickly collide preventing the inflation from ever reaching @xmath122 . thus borde and vilenkin have proved that in the inflationary scenario the universe must have a beginning . if it starts with a three - sphere of radius @xmath17 at time @xmath122 , and after that expands like @xmath40 , the bubbles do not percolate ( given that the bubble formation rate per four volume @xmath70 is @xmath123 ) and the inflation continues eternally to @xmath124 producing an infinite number of open bubble universes . since the number of bubbles forming increases exponentially with time without limit , our universe is expected to form at a finite but arbitrarily large time after the beginning of the inflationary state . in this picture our universe ( our bubble ) is only 12 billion years old , but the universe as a whole ( the entire bubble forming inflationary state ) is of a finite but arbitrarily old age . but how to produce that initial spherical @xmath1 universe ? vilenkin @xcite suggested that it could be formed from quantum tunneling . consider the embedding diagram for de sitter space . de sitter space can be embedded as the surface @xmath33 in a five - dimensional minkowski space with metric @xmath34 . this can be seen as an @xmath1 cosmology with radius @xmath40 where @xmath39 and @xmath125 gives the geometry of @xmath1 . this solution represents a classical trajectory with a turning point at @xmath126 . but just as it reaches this turning point it could tunnel to @xmath127 where the trajectory may be shown as a hemisphere of the euclidean four - sphere @xmath128 embedded in a flat euclidean space with the metric @xmath129 and @xmath130 where @xmath125 and @xmath131 . the time - reversed version of this process would show tunneling from a point at @xmath132 to a three sphere at @xmath133 of radius @xmath17 which then expands with proper time like @xmath40 giving a normal de sitter space thus vilenkin s universe created from nothing is obtained @xcite . hawking has noted that in this case , in hartle and hawking s formulation , the point @xmath132 is not special , the curvature does not blow up there : it is like other points in the euclidean hemispherical section @xcite . however , this point is still the earliest point in euclidean time since it is at the center of the hemisphere specified by the euclidean boundary at @xmath133 . so the beginning point in the vilenkin model is indeed like the south pole of the earth @xcite . vilenkin s tunneling universe was based on an analogy between quantum creation of universes and tunneling in ordinary quantum mechanics @xcite . in ordinary quantum mechanics , a particle bounded in a well surrounded by a barrier has a finite probability to tunnel through the barrier to the outside if the height of the barrier is finite ( as in the @xmath134-decay of radioactive nuclei @xcite ) . the wave function outside the barrier is an outgoing wave , the wave function in the well is the superposition of an outgoing wave and an ingoing wave which is the reflection of the outgoing wave by the barrier . due to the conservation of current , there is a net outgoing current in the well . the probability for the particle staying in the well is much greater than the probability for the particle running out of the barrier . the energy of the particle in the well _ can not _ be zero , otherwise the uncertainty principle is violated . thus there is always a finite zero - point - energy . the vilenkin universe was supposed to be created from `` nothing '' , where according to vilenkin `` nothing '' means `` a state with no classical spacetime '' @xcite . thus this is essentially different from tunneling in ordinary quantum mechanics since in ordinary quantum mechanics tunneling always takes place from one classically allowed region to another classically allowed region where the current and the probability are conserved . but creation from `` nothing '' is supposed to take place from a classically forbidden ( euclidean ) region to a classically allowed ( lorentzian ) region , so the conservation of current is obviously violated . vilenkin obtained his tunneling universe by choosing a so - called `` tunneling boundary condition '' for the wheeler - dewitt equation @xcite . his `` tunneling from nothing '' boundary condition demands that when the universe is big ( @xmath135 where @xmath12 is the cosmological constant and @xmath21 is the scale factor of the universe ) there is only an outgoing wave in the superspace @xcite . if the probability and current are conserved ( in fact there does exist a conserved current for the wheeler - dewitt equation @xcite , and a classically allowed solution with @xmath127 and zero `` energy '' ) , there must be a finite probability for the universe being in the state before tunneling ( i.e. , @xmath127 ) and this probability is much bigger than the probability for tunneling . this implies that there must be `` something '' instead of `` nothing '' before tunneling . this becomes more clear if matter fields are included in considering the creation of universes . in the case of a cosmological constant @xmath12 and a conformally coupled scalar field @xmath90 ( conformal fields are interesting not only for their simplicity but also because electromagnetic fields are conformally invariant ) as the source terms in einstein s equations , in the mini - superspace model ( where the configurations are the scale factor @xmath21 of the @xmath1 robertson - walker metric and a homogeneous conformally coupled scalar field @xmath90 ) the wheeler - dewitt equation separates @xcite @xmath136 @xmath137\psi(a)=e\psi(a ) , \label{ea1}\end{aligned}\ ] ] where @xmath138 is the wave function of the universe [ @xmath139 , @xmath140 is the `` energy level '' of the conformally coupled scalar field , ( we use quotes because for radiation the conserved quantity is @xmath141 instead of the energy @xmath142 where @xmath143 is the energy density ) , and @xmath144 is a constant determining the operator ordering . ( [ ea1a ] ) is just the schrdinger equation of a harmonic oscillator with unit mass and unit frequency and energy @xmath140 , the eigenvalues of @xmath140 are @xmath145 where @xmath146 eq . ( [ ea1 ] ) is equivalent to the schrdinger equation for a unit mass particle with total energy @xmath147 in the one - dimensional potential @xmath148 it is clear that in the case of @xmath149 , there exist one classically forbidden region @xmath150 and two classically allowed regions @xmath151 and @xmath152 where @xmath153 $ ] ( fig . [ f2 ] ) . because @xmath154 is regular at @xmath127 , we expect that the wave function @xmath155 is also regular at @xmath127 . if @xmath156 and the conformally coupled scalar field is in the ground state with @xmath157 , we have @xmath158 , @xmath159 and the potential in region @xmath151 is @xmath160 like a harmonic oscillator . the quantum behavior of the universe in region @xmath151 is like a quantum harmonic oscillator . this may describe a quantum oscillating ( lorentzian ) universe without big bang or big crunch singularities , which has a finite ( but small ) probability [ @xmath161 to tunnel through the barrier to form a de sitter - type inflating universe . the existence of this tiny oscillating universe is due to the existence of a finite `` zero - point - energy '' ( @xmath162 ) of a conformally coupled scalar field and this `` zero - point - energy '' is required by the uncertainty principle . since a conformally coupled scalar field has an equation of state like that of radiation , the friedmann equation for @xmath163 is @xmath164 where @xmath165 and @xmath143 is the energy density of the conformally coupled scalar field . eq . ( [ ea3 ] ) is equivalent to the energy - conservation equation for a classical unit mass particle with zero total energy moving in the potential @xmath166 the difference between @xmath154 and @xmath167 is caused by the fact that in the integral of action the volume element contains a factor @xmath168 which is also varied when one makes the variation to obtain the dynamical equations . the potential @xmath167 is singular at @xmath127 and near @xmath127 we have @xmath169 . for @xmath156 and @xmath157 ( we take @xmath170 ) , the classical universe in region @xmath171 is radiation dominated . this universe expands from a big bang singularity , reaches a maximum radius , then re - collapses to a big crunch singularity : @xmath127 is a singularity in the classical picture . but from the above discussion , the wheeler - dewitt equation gives a regular wave function at @xmath127 . in such a case near @xmath127 the quantum behavior of the universe is different from classical behavior . this implies that , near @xmath127 , classical general relativity breaks down and quantum gravity may remove singularities . this case is like that of a hydrogen atom where the classical instability ( according to classical electrodynamics , an electron around a hydrogen nucleus will fall into the nucleus due to electromagnetic radiation ) is cured by quantum mechanics . anyway , it is _ not _ nothing at @xmath127 . there is a small classically allowed , oscillating , radiation dominated , closed , quantum ( by `` quantum '' we mean that its quantum behavior deviates significantly from its classical behavior ) friedmann universe near @xmath127 , which has a small probability to tunnel through the barrier to form an inflationary universe . ( if @xmath172 there is no classically forbidden region and thus no tunneling . ) so in this model the universe did not come from a point ( nothing ) but from a tiny classically allowed , oscillating , quantum friedmann universe whose radius is of order the planck magnitude . but where did this oscillating universe come from ? because it has a finite probability to tunnel ( each time it reaches maximum radius ) to a de sitter space , it has a finite `` half - life '' for decay into the de sitter phase and can not last forever . it could , of course , originate by tunneling from a collapsing de sitter phase ( the time - reversed version of the creation of a de sitter state from the oscillating state ) , but then we are back where we started . in fact , starting with a collapsing de sitter phase one is more likely to obtain an expanding de sitter phase by simply re - expanding at the classical turning point rather than tunneling into and then out of the tiny oscillating universe state . an alternative might be to have the original tiny oscillating universe created via a quantum fluctuation ( since it has just the `` zero - point - energy '' ) but here we are basically returning to the idea of tryon @xcite that you could get an entire friedmann universe of any size directly via quantum fluctuation . but quantum fluctuation of what ? you have to have laws of physics and a potential etc . hartle and hawking @xcite made their no - boundary proposal and obtained a model of the universe similar to vilenkin s tunneling universe . the no - boundary proposal is expressed in terms of a euclidean path integral of the wave function of the universe @xmath173 , \label{ea5}\end{aligned}\ ] ] where the summation is over compact manifolds @xmath174 with the prescribed boundary @xmath175 ( being a compact three - manifold representing the shape of the universe at a given epoch ) as the _ only _ boundary ; @xmath176 is the euclidean metric on the manifold @xmath174 with induced three - metric @xmath177 on @xmath175 , @xmath90 is the matter field with induced value @xmath68 on @xmath178 ; @xmath179 is the euclidean action obtained from the lorentzian action @xmath180 via wick rotation : @xmath181 . in the mini - superspace model the configuration space is taken to include the @xmath163 robertson - walker metric and a homogeneous matter field . in the wkb approximation the wave function is ( up to a normalization factor ) @xmath182 , \label{ea5a}\end{aligned}\ ] ] where @xmath183 is the euclidean action for the solutions of the euclidean field equations ( einstein s equations and matter field equations ) . the factor @xmath184 is the determinant of small fluctuations around solutions of the field equations @xcite . if the matter field is a conformally coupled scalar field @xmath185 ( which is the case that hartle and hawking @xcite discussed ) , @xmath186 is conserved where @xmath143 is the energy density of @xmath90 satisfying the field equations . then the friedmann equation is given by eq . ( [ ea3 ] ) . the corresponding euclidean equation is obtained from eq . ( [ ea3 ] ) via @xmath187 @xmath188 the solution to eq . ( [ ea6 ] ) is ( for the case @xmath189 ) @xmath190^{1/2 } , \label{ea7}\end{aligned}\ ] ] where @xmath191 . this is a euclidean bouncing space with a maximum radius @xmath192^{1/2}$ ] and a minimum radius @xmath193^{1/2}$ ] ( fig . [ f3 ] ) . if @xmath194 , we have @xmath195 , @xmath196 , and @xmath197 , one copy of this bouncing space is a four - sphere with the euclidean de sitter metric @xmath198 $ ] which is just a four - sphere embedded in a five - dimensional euclidean space @xmath199 with metric @xmath200 this is the solution that hartle and hawking used @xcite . but , as we have argued above , according to hartle and hawking @xcite and hawking @xcite , the wheeler - dewitt equation for @xmath201 [ eq . ( [ ea1a ] ) ] gives rise to a `` zero - point - energy '' for the conformally coupled scalar field : @xmath202 ( the state with @xmath194 violates the uncertainty principle ) . one copy of this bouncing euclidean space is _ not _ a compact four - dimensional manifold with no boundaries , but has two boundaries with @xmath203 ( see fig . [ f3 ] ) . if @xmath204 ( i.e. @xmath156 ) , we have @xmath205 , @xmath206 . penrose @xcite has criticized hawking s no - boundary proposal and the model obtained by gluing a de sitter space onto a four - sphere hemisphere by pointing out that there are only very few spaces for which one can glue a euclidean and a lorentzian solution together since it is required that they have both a euclidean and a lorentzian solution , but the generic case is certainly very far from that . here `` with a zero - point - energy '' we have have both a euclidean solution and a lorentzian solution , and they can be glued together . but the euclidean solution is not closed in any way ; that is , it does not enforce the no - boundary proposal . hartle and hawking argued that there should be a constant @xmath207 in @xmath140 which arises from the renormalization of the matter field , i.e. , @xmath140 should be @xmath208 @xcite . but there is _ no _ reason that @xmath207 should be @xmath209 to exactly cancel the `` zero - point - energy '' @xmath210 . ( as in the case of a quantum harmonic oscillator , we have no reason to neglect the zero - point - energy . ) in fact , since @xmath207 comes from the renormalization of the matter field ( without quantization of gravity ) , it should be much less than the planck magnitude , i.e. , @xmath211 , and thus @xmath207 is negligible compared with @xmath210 . in fact in @xcite hawking has dropped @xmath207 . in @xcite hartle and hawking have realized that for excited states ( @xmath212 ) , there are two kinds of classical solutions : one represents universes which expand from zero volume , to reach a maximum radius , and then re - collapse ( like our tiny oscillating universe ) ; the other represents the de sitter - type state of continual expansion . there are probabilities for a universe to tunnel from one state to the other . here we argue that for the ground state ( @xmath157 ) , there are also two such kinds of lorentzian universes . one is a tiny quantum oscillating universe ( having a maximum radius with planck magnitude ) . here `` quantum '' just means that the classical description fails ( so singularities might be removed ) . the other is a big de sitter - type universe . these two universes can be joined to one another through a euclidean section , which describes quantum tunneling from a tiny oscillating universe to an inflating universe ( or from a contracting de sitter - type universe to a tiny oscillating universe ) . during the tunneling , the radius of the universe makes a jump ( from the planck length to @xmath213 or _ vice versa _ ) . as hartle and hawking @xcite calculated the wave function of the universe for the ground state , they argued that , for the conformally coupled scalar field case , the path integral over @xmath21 and @xmath214 separates since `` not only the action separates into a sum of a gravitational part and a matter part , but the boundary condition on the @xmath215 and @xmath216 summed over do not depend on one another '' where @xmath217 is the conformal time . the critical point for the variable s separation in the path integral is that `` the ground state boundary conditions imply that geometries in the sum are conformal to half of a euclidean - einstein static universe ; i.e. , the range of @xmath217 is @xmath218 . the boundary conditions at infinite @xmath217 are that @xmath216 and @xmath215 vanish . the boundary conditions at @xmath219 are that @xmath220 and @xmath221 match the arguments of the wave function @xmath222 and @xmath223 '' @xcite . but this holds only for some specific cases , such as de sitter space . our solution ( [ ea7 ] ) does not obey hartle and hawking s assumption that @xmath217 ranges from @xmath224 to @xmath225 . for a general @xmath163 ( euclidean ) robertson - walker metric , @xmath226 is a functional of @xmath21 , and the action of matter ( an integral over @xmath217 ) is a functional of @xmath21 . therefore , the action _ can not _ be separated into a sum of a gravitational part and a matter part as hartle and hawking did . the failure of hartle and hawking s path integral calculation is also manifested in the fact that de sitter space is _ not _ a solution of the friedmann equation if the `` zero - point - energy '' of the conformally coupled scalar field is considered , whereas the semiclassical approximation implies that the principal contribution to the path integral of the wave function comes from the configurations which solve einstein s equations . one may hope to overcome this difficulty by introducing a scalar field with a flat potential @xmath65 ( as in the inflation case ) . but this does not apply to the quantum cosmology case since as @xmath227 the universe always becomes radiation - dominated unless the energy density of radiation is exactly zero ( but the uncertainty principle does not allow this case to occur ) . from the arguments in the last section , we find that the universe does _ not _ seem to be created from nothing . on the other hand , if the universe is created from _ something _ , that something could have been _ itself_. thus it is possible that the universe is its own mother . in such a case , if we trace the history of the universe backward , inevitably we will enter a region of ctcs . therefore ctcs may play an important role in the creation of the universe . it is interesting to note that hawking and penrose s singularity theorems do not apply if the universe has had ctcs . and , it has been shown that , if a compact lorentzian spacetime undergoes topology changes , there must be ctcs in this spacetime @xcite . [ basically there are two type of spacetimes with ctcs : for the first type , there are ctcs everywhere ( gdel space belongs to this type ) ; for the second type , the ctcs are confined within some regions and there exists at least one region where there are no closed causal ( timelike or null ) curves , and the regions with ctcs are separated from the regions without closed causal curves by cauchy horizons ( misner space belongs to this type ) . in this paper , with the word `` spacetimes with ctcs '' we always refer to the second type unless otherwise specified . ] while in classical general relativity there exist many solutions with ctcs , some calculations of vacuum polarization of quantum fields in spacetimes with ctcs indicated that the energy - momentum tensor ( in this paper when we deal with quantum fields , with the word `` the energy - momentum tensor '' we always refer to `` the renormalized energy - momentum tensor '' because `` the unrenormalized energy - momentum tensor '' has no physical meaning ) diverges as one approaches the cauchy horizon separating the region with ctcs from the region without closed causal curves . this means that spacetimes with ctcs may be unstable against vacuum polarization since when the energy - momentum tensor is fed back to the semiclassical einstein s equations ( i.e. einstein s equations with quantum corrections to the energy - momentum tensor of matter fields ) the back - reaction may distort the spacetime geometry so strongly that a singularity may form and ctcs may be destroyed . based on some of these calculations , hawking @xcite has proposed the chronology protection conjecture which states that the laws of physics do not allow the appearance of ctcs . ( it should be mentioned that the chronology protection conjecture does _ not _ provide any restriction on spacetimes with ctcs but no cauchy horizons since there is _ no _ any indication that this type of spacetime is unstable against vacuum polarization . in the next section we will show a simple example of a spacetime with ctcs but no cauchy horizons , where the energy - momentum tensor is finite everywhere . ) but , on the other hand , li , xu , and liu @xcite have pointed out that even if the energy - momentum tensor of vacuum polarization diverges at the cauchy horizon , it does _ not _ mean that ctcs must be prevented by physical laws because : ( 1 ) einstein s equations are local equations and the energy - momentum tensor may diverge only at the cauchy horizon ( or at the polarized hypersurfaces ) and be well - behaved elsewhere within the region with ctcs ; ( 2 ) the divergence of the energy - momentum tensor at the cauchy horizon does _ not _ mean that the cauchy horizon must be destroyed by the back - reaction of vacuum polarization , _ but _ instead means that near the cauchy horizon the usual quantum field theory on a prescribed classical spacetime background can not be used and the quantum effect of gravity must be considered . ( this is like the case that hawking and penrose s singularity theorems do _ not _ mean that the big bang cosmology is wrong but mean that near the big bang singularity quantum gravity effects become important @xcite . ) when hawking proposed his chronology protection conjecture , hawking @xcite and kim and thorne @xcite had a controversy over whether quantum gravity can save ctcs . kim and thorne claimed that quantum gravitational effects would cut the divergence off when an observer s proper time from crossing the cauchy horizon was the planck time , and this would only give such a small perturbation on the metric that the cauchy horizon could not be destroyed . but , hawking @xcite noted that one would expect the quantum gravitational cut - off to occur when the invariant distance from the cauchy horizon was of order the planck length , and this would give a very strong perturbation on the metric so that the cauchy horizon would be destroyed . since there does not exist a self - consistent quantum theory of gravity at present , we can not judge who ( hawking or kim and thorne ) is right . but in any case , these arguments imply that in the case of a spacetime with ctcs where the energy - momentum tensor of vacuum polarization diverges at the cauchy horizon , quantum gravity effects should become important near the cauchy horizon . li , xu , and liu @xcite have argued that if the effects of quantum gravity are considered , in a spacetime with ctcs the region with ctcs and the region without closed causal curves may be separated by a _ quantum barrier _ ( e.g. a region where components of the metric have complex values ) instead of a cauchy horizon generated by closed null geodesics . by quantum processes , a time traveler may tunnel from the region without closed causal curves to the region with ctcs ( or _ vice versa _ ) , and the spacetime itself can also tunnel from one side to the other side of the quantum barrier @xcite . in classical general relativity , a region with ctcs and a region without closed causal curves must be separated by a cauchy horizon ( compactly generated or non - compactly generated ) which usually contains closed null geodesics if it is compactly generated @xcite . but if quantum gravity effects are considered ( e.g. in quantum cosmology ) , they can be separated by a complex geometric region ( as a quantum barrier ) instead of a cauchy horizon @xcite . ( in the path integral approach to quantum cosmology , complex geometries are _ required _ in order to make the path integral convergent and to overcome the difficulty that in general situations a euclidean space _ can not _ be directly joined to a lorentzian space @xcite ) . and , using a simple example of a space with a region with ctcs separated from a region without closed causal curves by a complex geometric region , li , xu , and liu @xcite have shown that in such a space the energy - momentum tensor of vacuum polarization is finite everywhere and the chronology protection conjecture has been challenged . without appeal to quantum gravity , counter - examples to the chronology protection conjecture also exist . by introducing a spherical reflecting boundary between two mouths of a wormhole , li @xcite has shown that with some boundary conditions for geodesics ( e.g. the reflection boundary condition ) closed null _ geodesics _ [ usually the `` archcriminal '' for the divergence of the energy - momentum tensor as the cauchy horizon is approached ( see e.g. @xcite ) ] may be removed from the cauchy horizon separating the region with ctcs and the region without closed causal curves . in such a case the spacetime contains neither closed null _ geodesics _ nor closed timelike _ geodesics _ , though it contains both closed timelike _ non - geodesic _ curves and closed null _ non - geodesic _ curves . li @xcite has shown that in this spacetime the energy - momentum tensor is finite everywhere . following li @xcite , low @xcite has given another example of spacetime with ctcs but without closed causal _ geodesics_. recently , with a very general argument , li @xcite has shown that the appearance of an absorber in a spacetime with ctcs may make the spacetime stable against vacuum polarization . li @xcite has given some examples to show that there exist many collision processes in high energy physics for which the total cross - sections increase ( or tend to a constant ) as the frequency of the incident waves increases . based on these examples , li @xcite has argued that material will become opaque for waves ( particles ) with extremely high frequency or energy , since in such cases the absorption caused by various types of scattering processes becomes very important . based on calculation of the renormalized energy - momentum tensor and the fluctuation in the metric , li @xcite has argued that if an absorbing material with appropriate density is introduced , vacuum polarization may be smoothed out near the cauchy horizon so that the metric perturbation caused by vacuum fluctuations will be very small and a spacetime with ctcs can be stable against vacuum polarization . boulware @xcite and tanaka and hiscock @xcite have found that for sufficiently massive fields in gott space @xcite and grant space @xcite respectively , the energy - momentum tensor remains regular on the cauchy horizon . krasnikov @xcite has found some two - dimensional spacetimes with ctcs for which the energy - momentum tensor of vacuum polarization is bounded on the cauchy horizon . sushkov @xcite has found that for an automorphic complex scalar field in misner space there is a vacuum state for which the energy - momentum tensor is zero everywhere . more recently , cassidy @xcite and li and gott @xcite have independently found that for the real conformally coupled scalar field in misner space there exists a quantum state for which the energy - momentum tensor is zero everywhere . li and gott @xcite have found that this quantum state is the `` adapted '' rindler vacuum ( i.e. the usual rindler vacuum with multiple images ) and it is a self - consistent vacuum state because it solves the semiclassical einstein s equations exactly . li and gott @xcite have also found that for this `` adapted '' rindler vacuum in misner space , an inertial particle detector perceives nothing . in this paper , we find that for a multiply connected de sitter space there also exists a self - consistent vacuum state for a conformally coupled scalar field ( see section [ ix ] ) . thorne @xcite has noted that , even if hawking s argument that a quantum gravitational cut - off would occur when the geometric invariant distance from the cauchy horizon is of order the planck length is correct , by using two wormholes the metric fluctuations near the cauchy horizon can be made arbitrarily small so a spacetime with ctcs created from two wormholes can be stable against vacuum polarization . recently visser @xcite has generalized this result to the roman - ring case . the above arguments indicate that the back - reaction of vacuum polarization may _ not _ destroy the cauchy horizon in spacetimes with ctcs , and thus such spacetimes can be stable against vacuum polarization . in a recent paper , cassidy and hawking @xcite have admitted that `` back - reaction does not enforce chronology protection '' . on the other hand , cassidy and hawking @xcite have argued that the `` number of states '' may enforce the chronology protection conjecture since `` this quantity will always tend to zero as one tries to introduce ctcs '' . their arguments are based on the fact that for the particular spacetime with ctcs they constructed [ which is the product of a multiply connected ( via a boost ) three - dimensional de sitter space and @xmath228 the entropy of a massless scalar field diverges to minus infinity when the spacetime develops ctcs @xcite . however , whether this conclusion holds for general spacetimes with ctcs remains an open question and further research is required . and , from ordinary statistical thermodynamics we know that entropy is always positive , so the physical meaning of a _ negative _ entropy is unclear . the number of states in phase space is given by @xmath229 where @xmath230 , @xmath231 , @xmath232 ( @xmath233 ) is a canonical coordinate , @xmath234 is a canonical momentum , and @xmath235 is the number of degrees of freedom . the uncertainty principle demands that @xmath236 and thus we should always have @xmath237 . thus the `` fact '' that the number of states tends to zero as one tries to develop ctcs ( i.e. as one approaches the cauchy horizon ) may simply imply that near the cauchy horizon quantum effects of gravity can not be neglected , which is consistent with li , xu , and liu s argument @xcite . the entropy is defined by @xmath238 where @xmath239 is the number of states and @xmath240 is the boltzmann constant . when @xmath239 is small , quantization of the entropy becomes important ( remember that the number of states @xmath239 is always an integer ) . the entropy can not _ continuously _ tend to negative infinity ; it should _ jump _ from @xmath241 to @xmath242 , _ jump _ from @xmath242 to zero ( but in cassidy and hawking s arguments @xcite we have not seen such a jump ) , then the uncertainty principle demands that the entropy should stand on the zero value as one approaches the cauchy horizon . on the other hand , ordinary continuous thermodynamics holds only for the case with @xmath243 . thus , as one approaches the cauchy horizon the thermodynamic limit has already been violated and ordinary thermodynamics should be revised near the cauchy horizon . in other words , cassidy and hawking s results @xcite can not be extended to the cauchy horizon . based on the fact that the effective action density diverges at the polarized hypersurfaces of spacetimes with ctcs @xcite , cassidy and hawking @xcite have argued that the effective action `` would provide new insight into issues of chronology protection '' . but we should note that the effective action is only a _ tool _ for computing some physical quantities ( such as the energy - momentum tensor ) and the effective action itself has not much physical meaning . the divergence of the effective action may imply that the effective action is not a good _ tool _ as the polarized hypersurfaces are approached . our argument is supported by the fact that there exist many examples for which the energy - momentum tensor is finite everywhere , as mentioned above . recently , kay , radzikowski , and wald @xcite have proved two theorems which demonstrate that some fundamental quantities such as hadamard functions and energy - momentum tensors must be ill - defined on a compactly generated cauchy horizon in a spacetime with ctcs , as one extends the _ usual _ quantum field theory in a global hyperbolic spacetime to an acausal spacetime with a compactly generated cauchy horizon . basically speaking , their theorems imply that the _ usual _ quantum field theory can not be _ directly _ extended to a spacetime with ctcs @xcite . their theorems tell us that serious difficulties arise when attempting to _ define _ quantum field theory on a spacetime with a compactly generated cauchy horizon @xcite . the ordinary quantum field theory must be significantly changed or some new approach must be introduced when one tries to do quantum field theory on a spacetime with ctcs . a candidate procedure for overcoming this difficulty is the euclidean quantization proposed by hawking @xcite . quantum field theory is well - defined in a euclidean space because there are no ctcs in a euclidean space @xcite . in fact , even in simply connected minkowski spacetime , quantum field theory is _ not _ well - defined since the path integral does not converge . to overcome this difficulty , the technique of wick - rotation ( which is essentially equivalent to euclidean quantization ) is used . kay , radzikowski , and wald @xcite have also argued that their results may be interpreted as indicating that in order to create ctcs it would be necessary to enter a regime where quantum effects of gravity will be dominant ( see also the discussions of visser @xcite ) ; this is also consistent with li , xu , and liu s arguments @xcite . cramer and kay @xcite have shown that kay , radzikowski , and wald s theorems @xcite also apply to misner space ( for sushkov s automorphic field case @xcite and krasnikov s two - dimensional case @xcite , respectively ) where the cauchy horizon is not compactly generated , in the sense that the energy - momentum tensor must be ill - defined on the cauchy horizon itself . but we note that this only happens in a set of measure zero which does not make much sense in physics for if the renormalized energy - momentum tensor is zero everywhere except on a set of measure zero where it is formally ill - defined , then continuity would seem to require setting it to zero there also @xcite . perhaps a conclusion on the chronology protection conjecture can only be reached after we have a quantum theory of gravity . however , we can conclude that the back - reaction of vacuum polarization does _ not _ enforce the chronology protection conjecture , a point hawking himself also admits @xcite . ( originally the back - reaction of vacuum polarization was supposed to be the strongest candidate for chronology protection @xcite . ) a simple spacetime with ctcs is obtained from minkowski spacetime by identifying points that are related by time translation . minkowski spacetime is @xmath244 . in cartesian coordinates @xmath245 the lorentzian metric @xmath246 is given by @xmath247 now we identify points @xmath245 with points @xmath248 where @xmath249 is a positive constant and @xmath250 is any integer . then we obtain a spacetime with topology @xmath251 and the lorentzian metric . such a spacetime is closed in the time direction and has no cauchy horizon . all events in this spacetime are threaded by ctcs . ( this is the only acausal spacetime without a cauchy horizon considered in this paper . ) minkowski spacetime @xmath252 is the covering space of this spacetime . usually there is no well - defined quantum field theory in a spacetime with ctcs . ( kay - radzikowski - wald s theorems @xcite enforce this claim , though they do not apply directly to an acausal spacetime without a cauchy horizon . ) however , in the case where a covering space exists , we can do it in the covering space with the method of images . in fact in most cases where the energy - momentum tensor in spacetimes with ctcs has been calculated , this method has been used ( for the theoretical basis for the method of images see ref . @xcite and references cited therein ) . the method of images is sufficient for our purposes in this paper ( computing the energy - momentum tensor and the response function of particle detectors ) . thus in this paper we use this method to deal with quantum field theory in spacetimes with ctcs . for any point @xmath245 in @xmath253 , there are an infinite number of images of points @xmath248 in the covering space @xmath244 . for the minkowski vacuum @xmath254 of a conformally coupled scalar field ( by `` conformally coupled '' we mean that the mass of the scalar field is zero and the coupling between the scalar field @xmath90 and the gravitational field is given by @xmath255 where @xmath256 is the ricci scalar curvature ) in the minkowski spacetime , the hadamard function is @xmath257 here @xmath258 and @xmath259 . with the method of images , the hadamard function of the `` adapted '' minkowski vacuum ( which is the minkowski vacuum with multiple images ) in the spacetime @xmath253 is given by the summation of the hadamard function in ( [ e2 ] ) for all images @xmath260 the regularized hadamard function is usually taken to be @xmath261 the renormalized energy - momentum tensor is given by @xcite @xmath262 inserting eq . ( [ e4 ] ) into eq . ( [ e5 ] ) we get @xmath263 we find that this energy - momentum tensor is constant and finite everywhere and has the form of radiation . thus ctcs do not mean that the energy - momentum tensor must diverge . now let us consider a particle detector @xcite moving in this spacetime . the particle detector is coupled to the field @xmath90 by the interaction lagrangian @xmath264 $ ] , where @xmath265 is a small coupling constant , @xmath266 is the detector s monopole moment , @xmath267 is the proper time of the detector s worldline , and @xmath268 is the trajectory of the particle detector @xcite . suppose initially the detector is in its ground state with energy @xmath269 and the field @xmath90 is in some quantum state @xmath270 . then the transition probability for the detector to all possible excited states with energy @xmath271 and the field @xmath90 to all possible quantum states is given by @xcite @xmath272 where @xmath273 and @xmath274 is the response function @xmath275 which is independent of the details of the particle detector and is determined by the positive frequency wightman function @xmath276 ( while the hadamard function is defined by @xmath277 ) . the response function represents the bath of particles that the detector effectively experiences @xcite . the remaining factor in eq . ( [ e14a ] ) represents the selectivity of the detector to the field and depends on the internal structure of the detector @xcite . the wightman function for the minkowski vacuum is @xmath278 where @xmath279 is an infinitesimal positive real number which is introduced to indicate that @xmath280 is the boundary value of a function which is analytic in the lower - half of the complex @xmath281 plane . for the adapted minkowski vacuum in our spacetime @xmath253 , the wightman function is @xmath282 assume that the detector moves along the geodesic @xmath283 @xmath284 , @xmath285 , then the proper time is @xmath286 with @xmath287 . on the geodesic , the wightman function is reduced to @xmath288 inserting eq . ( [ e10 ] ) into eq . ( [ e7 ] ) , we obtain @xmath289 where @xmath290 and @xmath291 . the integration over @xmath292 is taken along a contour closed in the lower - half plane of complex @xmath292 . inspecting the poles of the integrand , we find that all poles are in the upper - half plane of complex @xmath292 ( remember that @xmath293 ) . therefore according to the residue theorem we have @xmath294 such a particle detector perceives no particles , though the renormalized energy - momentum tensor of the field has the form of radiation . another simple space with ctcs constructed from minkowski space is misner space @xcite . in cartesian coordinates @xmath295 in minkowski spacetime , a boost transformation in the @xmath296 plane ( we can always adjust the coordinates so that the boost is in this plane ) takes point @xmath295 to point @xmath297 where @xmath298 is the boost parameter . in rindler coordinates @xmath299 , defined by @xmath300 the minkowski metric can then be written in the rindler form @xmath301 the rindler coordinates @xmath302 only cover the right quadrant of minkowski space ( i.e. the region r defined by @xmath303 ) . by a reflection @xmath304 [ or @xmath305 @xmath306 , the rindler coordinates and the rindler metric can be extended to the left quadrant ( l , defined by @xmath307 ) . by the transformation @xmath308 the rindler coordinates can be extended to the future quadrant ( f , defined by @xmath309 ) and the past quadrant ( p , defined by @xmath310 ) . in region l the rindler metric has the same form as the metric in region r , which is given by eq . ( [ e56 ] ) . but in f and p the rindler metric is extended to be @xmath311 misner space is obtained by identifying @xmath295 with @xmath312 . under such an identification , point @xmath302 in r ( or l ) is identified with points @xmath313 in r ( or l ) , point @xmath314 in f ( or p ) is identified with points @xmath315 in f ( or p ) . clearly there are ctcs in r and l but there are no closed causal curves in f and p , and these regions are separated by the cauchy horizons @xmath316 , generated by closed null geodesics . misner space is not a manifold at the intersection of @xmath317 and @xmath318 . however , as hawking and ellis @xcite have pointed out , if we consider the bundle of linear frames over minkowski space , the corresponding induced bundle of linear frames over misner space is a hausdorff manifold and therefore well - behaved everywhere . the energy - momentum tensor of a conformally coupled scalar field in misner space has been studied in @xcite . hiscock and konkowski @xcite have calculated the energy - momentum tensor of the adapted minkowski vacuum . in rindler coordinates their results can be written as @xmath319 where the constant @xmath320 is @xmath321 eq . ( [ e67 ] ) holds only in region r [ because rindler coordinates defined by eq . ( [ e55 ] ) only cover r ] , but it can be analytically extended to other regions by writing @xmath322 in cartesian coordinates or by the transformations mentioned above . obviously for any finite @xmath298 , @xmath323 diverges as one approaches the cauchy horizon ( @xmath324 ) . this divergence is coordinate independent since @xmath325 also diverges as @xmath324 . this indicates that though the minkowski vacuum is a good and self - consistent vacuum for simply connected minkowski space , the adapted minkowski vacuum is _ not _ self - consistent for misner space ( i.e. it does not solve einstein s equations given the misner space geometry ) . this result has led hawking @xcite to conjecture that the laws of physics do not allow the appearance of ctcs ( i.e. , his chronology protection conjecture ) . li and gott @xcite have studied the adapted rindler vacuum in misner space . the hadamard function for the rindler vacuum is @xcite @xmath326 } , \label{e76}\end{aligned}\ ] ] where @xmath327 , @xmath328 , and @xmath329 is defined by @xmath330 the hadamard function for the adapted rindler vacuum in misner space is @xmath331}. \label{e78}\end{aligned}\ ] ] though @xmath332 and @xmath333 given by eq . ( [ e76 ] ) and eq . ( [ e78 ] ) are defined only in region r , they can be analytically extended to regions l , f , and p in minkowski and misner space . the regularized hadamard function for the adapted rindler vacuum is @xmath334 , where @xmath335 is the hadamard function for the minkowski vacuum given by eq . ( [ e2 ] ) . inserting this together with eq . ( [ e78 ] ) and eq . ( [ e2 ] ) into eq . ( [ e5 ] ) , we obtain the energy - momentum tensor for a conformally coupled scalar field in the adapted rindler vacuum @xcite @xmath336 \left(\begin{array}{cccc } -3&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array } \right ) , \label{e69}\end{aligned}\ ] ] which is expressed in rindler coordinates and thus holds only in region r but can be analytically extended to other regions with the method mentioned above for the case of the adapted minkowski vacuum . we @xcite have found that unless @xmath337 , @xmath338 blows up as one approaches the cauchy horizon ( @xmath324 ) ( as also does@xmath339 ) . but , if @xmath337 , we have @xmath340 which is regular as one approaches the cauchy horizon and can be regularly extended to the whole misner space , where it is also zero . in such a case , the vacuum einstein s equations without cosmological constant are automatically satisfied . thus this is an example of a spacetime with ctcs at the semiclassical quantum gravity level . we @xcite have called this vacuum the _ self - consistent vacuum _ for misner space , and @xmath337 is the _ self - consistent condition_. ( cassidy @xcite has also independently proven that for a conformally coupled scalar field in misner space there should exist a quantum state for which the energy - momentum tensor is zero everywhere . but he has not shown what quantum state it should be . we @xcite have shown that it is the adapted rindler vacuum . ) another way to deal with quantum fields in spacetimes with ctcs is to do the quantum field theory in the euclidean section and then analytically extend the results to the lorentzian section @xcite . for misner space the euclidean section is obtained by taking @xmath217 and @xmath298 to be @xmath341 and @xmath342 . the resultant space is the euclidean space with metric @xmath343 and @xmath344 and @xmath345 are identified where @xmath344 are cylindrical polar coordinates with @xmath346 the angular polar coordinate and @xmath347 the radial polar coordinate . the geometry at the hypersurface @xmath348 is conical singular unless @xmath349 . when extending that case to the lorentzian section , we get @xmath337 which is just the self - consistent condition . this may be the geometrical explanation of the self - consistent condition . by doing quantum field theory in the euclidean space , then analytically extending the results to the lorentzian section , we obtain the renormalized energy - momentum tensor in r ( or l ) region of the misner space . then we can extend the renormalized energy - momentum tensor in r ( or l ) to regions f ( or p ) . the results are the same as that obtained with the method of images . let us consider a particle detector moving in misner space with the adapted rindler vacuum . suppose the detector moves along a geodesic with @xmath350 , @xmath351 , and @xmath352 ( @xmath21 and @xmath353 are constants and @xmath21 is positive ) , which goes through the p , r , and f regions . the proper time of the detector is @xmath286 with @xmath287 . on this geodesic , the hadamard function in ( [ e78 ] ) is reduced to @xmath354 where @xmath329 is given by @xmath355 and @xmath356 is given by @xmath357 though this hadamard function is originally defined only in r , it can be analytically extended to f , p , and l. the wightman function is equal to @xmath162 of the hadamard function with @xmath38 replaced by @xmath358 and @xmath359 replaced by @xmath360 , where @xmath279 is an infinitesimal positive real number . then the response function is @xcite @xmath361 [ a^2-\zeta^2(t-{\delta\tau\over2}+{i\epsilon\over2\zeta})^2]}~\left\{- [ ( \eta-\eta^\prime)^++nb]^2+{\gamma^+}^2\right\ } } , \label{i4}\end{aligned}\ ] ] where @xmath362 , @xmath363 ; @xmath364 and @xmath365 are given by ( [ i2 ] ) and ( [ i3 ] ) with @xmath38 replaced by @xmath358 and @xmath359 replaced by @xmath360 . the integral over @xmath292 can be worked out by the residue theorem where we choose the integration contour to close in the lower - half complex-@xmath292 plane . the result is zero since there are no poles in the lower - half plane . therefore such a detector can not be excited and so it detects nothing @xcite . we @xcite have also calculated the response functions for detectors on worldlines with constant @xmath347 , @xmath366 , and @xmath367 and worldlines with constant @xmath368 , @xmath366 , and @xmath367 both are zero . in order to compare our model for the creation of the universe with vilenkin s tunneling universe , in this section we calculate the vacuum fluctuation of a conformally coupled scalar field in vilenkin s tunneling universe . the geometry of vilenkin s tunneling universe has been described in section iv . such a universe is described by a lorentzian - de sitter space joined to a euclidean de sitter space @xcite . the lorentzian section has the topology @xmath369 and the metric @xmath370 . \label{e13}\end{aligned}\ ] ] the euclidean section has the topology @xmath371 and the metric @xmath372 . \label{e14}\end{aligned}\ ] ] the lorentzian section and the euclidean section are joined at the boundary @xmath373 defined by @xmath374 . @xmath373 is a three - sphere with the minimum radius in de sitter space and the maximum radius in the euclidean four - sphere . the boundary condition for a conformally coupled scalar field @xmath90 is @xcite @xmath375 which is a kind of neumann boundary condition and indicates that the boundary @xmath373 is like a kind of reflecting boundary . the green functions ( including both the hadamard function and the wightman function ) should also satisfy this boundary condition @xmath376 the vacuum state of a conformally coupled scalar field in de sitter space is usually taken to be that obtained from the minkowski vacuum by the conformal transformation according to which de sitter space is conformally flat . ( the quantum state so obtained is usually called the conformal vacuum @xcite . ) such a vacuum is de sitter invariant and we call it the conformal minkowski vacuum . the hadamard function for this de sitter vacuum ( i.e. the conformal minkowski vacuum ) is @xcite @xmath377 where @xmath378 , @xmath379 , and @xmath380 is defined by @xmath381\}. \label{e18}\end{aligned}\ ] ] in vilenkin s tunneling universe , the hadamard function satisfying the boundary condition ( [ e16 ] ) is given by @xmath382 , \label{e19}\end{aligned}\ ] ] where @xmath383 is the image of @xmath378 with respect to the reflecting boundary @xmath373 . there are various schemes for obtaining the renormalized energy - momentum tensor for de sitter space ( e.g. @xcite ) . they all are equivalent to subtracting from the hadamard function a reference term @xmath384 to obtain a regularized hadamard function and then calculating the renormalized energy - momentum tensor by @xcite @xmath385 for the conformally coupled scalar field , the differential operator @xmath386 is @xmath387 where @xmath388 is the geodesic parallel displacement bivector @xcite . [ it is easy to show that if @xmath389 eq . ( [ e21 ] ) and eq . ( [ e22 ] ) are reduced to eq . ( [ e5 ] ) . ] the regularized hadamard function for the adapted conformal minkowski vacuum in vilenkin s tunneling universe is @xmath390 + g_{\rm cm}^{(1)}(x^-,x^\prime ) . \label{e20}\end{aligned}\ ] ] ( in this paper the exact form of @xmath384 is not important for us . ) substituting eqs . ( [ e17]-[e19 ] ) and eq . ( [ e20 ] ) into eq . ( [ e21 ] ) , we find that @xmath391 which shows that the boundary condition ( [ e15 ] ) does not produce any renormalized energy - momentum tensor ; but the action of @xmath386 on @xmath392 should give the energy - momentum tensor for the conformal minkowski vacuum in an eternal de sitter space @xcite @xmath393=- { 1\over960\pi^2 r_0 ^ 4 } g_{ab}. \label{e24}\end{aligned}\ ] ] therefore , the energy - momentum tensor of a conformally coupled scalar field in the adapted minkowski vacuum in vilenkin s tunneling universe is @xmath394 which is the same as that for an eternal de sitter space . now consider a particle detector moving along a geodesic with @xmath395 . the response function is given by eq . ( [ e7 ] ) but with the integration over @xmath267 and @xmath396 ranging from @xmath225 to @xmath397 . the wightman function is obtained from the corresponding hadamard function by the relation @xmath398 where @xmath279 is an infinitesimal positive real number . along the worldline of the detector , we have @xmath399 and @xmath400 then the response function is @xmath401 , \label{e30}\end{aligned}\ ] ] where @xmath402 and @xmath403 . it is easy to calculate the contour integral over @xmath292 . we find that the integration of the second term is zero and therefore , the result is the same as that for an inertial particle detector in an eternal de sitter space @xcite . thus we have @xmath404 which is just the response function for a detector in a thermal radiation with the gibbons - hawking temperature @xcite @xmath405 [ the factor @xmath17 over @xmath406 in eq . ( [ e31 ] ) is due to the fact that by definition @xmath407 is dimensionless . ] therefore such a detector perceives a thermal bath of radiation with the temperature @xmath408 . though the boundary between the lorentzian section and the euclidean section behaves as a reflecting boundary , a particle detector can not distinguish vilenkin s tunneling universe from an eternal de sitter space , and they have the same energy - momentum tensor for the conformally coupled scalar field . a time - nonorientable de sitter space can be constructed from de sitter space by identifying antipodal points @xcite . under such an identification , point @xmath378 is identified with @xmath409 . friedman and higuchi @xcite have described this space as a `` lorentzian universe from nothing '' ( without any euclidean section ) , although one could also describe it as always existing . friedman and higuchi have studied quantum field theory in this space but have not calculated the renormalized energy - momentum tensor @xcite . de sitter space is the covering space of this time - nonorientable model . using the method of images , the hadamard function of a conformally coupled scalar field in the time - nonorientable de sitter space with the `` adapted '' conformal minkowski vacuum can be constructed as @xmath410 \nonumber\\ & = & { 1\over4\pi^2 r_0 ^ 2}\left[{1\over1-z(x , x^\prime)}+ { 1\over1+z(x , x^\prime)}\right ] . \label{e33}\end{aligned}\ ] ] the regularized hadamard function is @xmath411+g_{\rm cm}^{(1)}(-x , x^\prime ) . \label{e34}\end{aligned}\ ] ] inserting eq . ( [ e33 ] ) and eq . ( [ e34 ] ) into eq . ( [ e21 ] ) , we find that the contribution of @xmath412 to the energy - momentum tensor is zero . therefore the renormalized energy - momentum tensor is the same as that in an eternal de sitter space , which is given by eq . ( [ e25 ] ) . suppose a particle detector moves along a worldline with @xmath413 . the response function is given by eq . . the wightman function is obtained from the hadamard function through eq . ( [ e26 ] ) . on the worldline of the particle detector , we have @xmath414 inserting this into eq . ( [ e7 ] ) we get @xmath415 which represents a thermal spectrum with a temperature equal to twice the gibbons - hawking temperature . therefore a particle detector moving along such a geodesic in this time - nonorientable spacetime perceives thermal radiation with temperature @xmath416 . for this time - nonorientable de sitter space , the area of the event horizon is one half that of an eternal de sitter space . this together with @xmath416 tells us that the first thermodynamic law of event horizons @xmath417 is preserved , where @xmath418 is the mass within the horizon , and @xmath320 is the area of the horizon @xcite . de sitter space is a solution of the vacuum einstein s equations with a positive cosmological constant @xmath12 , which is one of the maximally symmetric spacetimes ( the others being minkowski space and anti - de sitter space ) @xcite . de sitter space can be represented by a timelike hyperbolic hypersurface @xmath419 embedded in a five - dimensional minkowski space @xmath420 with the metric @xmath421 where @xmath11 @xcite . de sitter space has ten killing vectors four of them are boosts , and the other six are rotations . the global coordinates @xmath422 have been described in previous sections . static coordinates @xmath423 on de sitter space are defined by @xmath424 where @xmath425 , @xmath426 , @xmath427 , and @xmath428 . in these coordinates the de sitter metric is written as @xmath429 we divide de sitter space @xmath430 into four regions @xmath431 which are separated by horizons where @xmath432 and @xmath433 . ( see fig . it is obvious that the static coordinates defined by eq . ( [ e41 ] ) only cover region @xmath434 . however , similar to the rindler coordinates , these static coordinates can be extended to region @xmath435 by the complex transformation @xmath436 where @xmath437 and @xmath438 . in region @xmath435 , with the coordinates @xmath439 , the de sitter metric can be written as @xmath440 transforming the coordinate @xmath441 to the proper time @xmath267 by @xmath442 the de sitter metric in @xmath435 is written as @xmath443 ( see fig . [ f4 ] . ) the coordinates @xmath444 are related to @xmath199 by @xmath445 the universe with metric ( [ e50 ] ) is a type of kantowski - sachs universe @xcite . any hypersurface of @xmath446 has topology @xmath447 and has four killing vectors . similarly , the static coordinates can also be extended to @xmath448 and @xmath449 . another coordinate system which will be used in this paper is the steady - state coordinate system @xmath450 , defined by @xmath451 these coordinates cover regions @xmath452 and the horizon at @xmath453 . with these steady - state coordinates , the de sitter metric can be written in the steady - state form @xmath454 introducing the conformal time @xmath455 and spherical coordinates @xmath456 defined by @xmath457 , @xmath458 , and @xmath459 , the de sitter metric can be written as @xmath460 . \label{e54}\end{aligned}\ ] ] the de sitter metric is invariant under the action of the de sitter group . because the boost group in de sitter space is a sub - group of the de sitter group , the de sitter metric is also invariant under the action of the boost group . a boost transformation in the @xmath461 plane in the embedding five - dimensional minkowski space induces a boost transformation in the de sitter space . under such a transformation , point @xmath199 is taken to @xmath462 . in static coordinates in @xmath434 , point @xmath423 is taken to @xmath463 where @xmath464 . in coordinates @xmath465 in @xmath435 , point @xmath466 is taken to @xmath467 . similar to misner space , our multiply connected de sitter space is constructed by identifying points @xmath199 with @xmath468 on de sitter space @xmath430 . in regions @xmath434 , points @xmath423 are identified with @xmath469 ; in region @xmath435 , points @xmath466 are identified with @xmath470 . we denote the multiply connected de sitter space so obtained by @xmath471 , where @xmath472 denotes the boost group . under the identification generated by the boost transformation , clearly @xmath471 has ctcs in regions @xmath434 and @xmath449 , but has no closed causal curves in regions @xmath435 and @xmath448 . the boundaries at @xmath432 and @xmath433 are the cauchy horizons which separate the causal regions @xmath435 and @xmath448 from the acausal regions @xmath434 and @xmath449 and are generated by closed null geodesics ( fig . [ f4 ] ) . similar to the case of misner space , @xmath471 is not a manifold at the two - sphere defined by @xmath473 and @xmath474 . however , as in hawking and ellis s arguments for misner space @xcite , the quotient of the bundle of linear frames over de sitter space by the boost group is a hausdorff manifold and thus is well - behaved everywhere . it may not be a serious problem in physics that @xmath471 is not a manifold at the two - sphere mentioned above since this is a set of measure zero . it is well known that de sitter space is conformally flat . the de sitter metric is related to the minkowski metric by the conformal transformation @xmath475 it is easy to show this relation by writing the steady - state de sitter metric using conformal time [ see eq . ( [ e54 ] ) ] . however , in this paper it is more convenient to show this conformal relation by writing the de sitter metric in the static form and the minkowski metric in the rindler form , and using the transformation @xcite @xmath476 then the conformal factor @xmath477 is @xmath478 the conformal relations given by eq . ( [ e59 ] ) and eq . ( [ e60 ] ) define a _ conformal map _ between the static de sitter space and the rindler space . the horizon at @xmath479 in the static de sitter space coordinates corresponds to the horizon @xmath348 in rindler space , and the worldline @xmath480 in de sitter space corresponds to the worldline with @xmath481 and @xmath285 in rindler space . this conformal relation can also be extended to region @xmath435 in de sitter space and region f in minkowski space , where we have @xmath482 and @xmath483 eq . ( [ e59a ] ) and eq . ( [ e60a ] ) give a _ locally _ conformal map in the sense that in @xmath435 in de sitter space , the map given by eq . ( [ e59a ] ) and eq . ( [ e60a ] ) with a `` @xmath484 '' sign only covers @xmath485 , where @xmath486 ; the map given by eq . ( [ e59a ] ) and eq . ( [ e60a ] ) with a `` @xmath487 '' sign only covers @xmath488 . ( remember that in f in rindler space we have @xmath489 . ) this conformal map is singular at @xmath490 . however , since the hypersurfaces @xmath491 and @xmath492 are homogeneous , in a neighborhood of any point in region f , we can always adjust coordinates @xmath493 so that eq . ( [ e59a ] ) and eq . ( [ e60a ] ) hold , except for the points lying in region o defined by @xmath494 ( i.e. @xmath495 ) in f ; because as @xmath496 we have @xmath497 . this means that there always exists a _ locally _ conformal map between @xmath498 and f - o ( defined by @xmath499 in f ) , and future infinity ( @xmath496 ) in @xmath435 corresponds to the hyperbola @xmath500 ( i.e. @xmath501 ) in f. with the above conformal transformation , misner space is naturally transformed to the multiply connected de sitter space @xmath471 with @xmath502 for a conformally coupled scalar field in a conformally flat spacetime , the green function @xmath503 of the conformal vacuum is related to the corresponding green function @xmath504 in the flat spacetime by @xcite @xmath505 the renormalized energy - momentum tensors are related by @xcite @xmath506 , \label{e63}\end{aligned}\ ] ] where @xmath507 and for scalar field we have @xmath508 and @xmath509 @xcite . [ the sign before @xmath510 is positive here because we are using signature @xmath511 . for de sitter space we have @xmath512 , @xmath513 , and thus @xmath514 , @xmath515 . inserting them into eq . ( [ e63 ] ) , we have @xmath516 since the renormalized energy - momentum tensor for minkowski space in the minkowski vacuum is zero , we have @xmath517 , and thus for a conformally coupled scalar field in the conformal minkowski vacuum in a simply connected de sitter space @xmath430 @xmath518 which is just the expected result [ see eq . ( [ e25 ] ) ] . if we insert the energy - momentum tensor in eq . ( [ e66 ] ) into the semiclassical einstein s equations @xmath519 and recall that for de sitter space we have @xmath520 , we find that the semiclassical einstein s equations are satisfied if and only if @xmath521 if @xmath76 , the solutions to eq . ( [ e75b ] ) are @xmath522 and @xmath523 @xcite . gott @xcite has called the vacuum state in de sitter space with @xmath522 the self - consistent vacuum state ( it has a gibbons - hawking thermal temperature @xmath524 ) @xcite . in this self - consistent case , @xmath525 itself is the source term producing the de sitter geometry @xcite . this may give rise to inflation at the planck scale @xcite . ( in a recent paper of panagiotakopoulos and tetradis @xcite , inflation at the planck scale has been suggested to lead to homogeneous initial conditions for a second stage inflation at the gut scale . ) the second solution @xmath523 corresponds to minkowski space . these perhaps supply a possible reason that the effective cosmological constant is either of order unity in planck units or exactly zero . that is interesting because we observe @xmath526 today and a high @xmath527 is needed for inflation . if @xmath528 , we find that the solutions to eq . ( [ e75b ] ) are @xmath529 a de sitter space with @xmath17 given by eq . ( [ e75c ] ) automatically satisfies the semiclassical einstein s equations ( [ e75a ] ) . such a de sitter space and its corresponding vacuum are thus self - consistent . from eq . ( [ e65 ] ) we find that if we know the energy - momentum tensor of a conformally coupled scalar field in some vacuum state in misner space , we can get the energy - momentum tensor in the corresponding conformal vacuum in the multiply connected de sitter space . two fundamental vacuums in minkowski space are the minkowski vacuum and the rindler vacuum @xcite . the energy - momentum tensor of the conformally coupled scalar field in the adapted minkowski vacuum in misner space has been worked out by hiscock and konkowski @xcite ; their results are given by eq . ( [ e67 ] ) . inserting eq . ( [ e67 ] ) into eq . ( [ e65 ] ) , and using eqs . ( [ e59]-[e61 ] ) , we obtain the energy - momentum tensor of a conformally coupled scalar field in the adapted conformal minkowski vacuum in our multiply connected de sitter space @xmath471 . in static coordinates @xmath423 , it is written as @xmath530 where @xmath531 this result is defined in region @xmath434 , but it can be extended to region @xmath435 through the transformation in eq . ( [ e47 ] ) , and can also be extended to region @xmath449 and @xmath448 through similar transformations . similar to misner space , this energy - momentum tensor diverges at the cauchy horizon as @xmath532 for any finite @xmath353 ; and the divergence is coordinate independent since @xmath533 also diverges there . though the conformal minkowski vacuum is a good vacuum for simply connected de sitter space @xcite , it ( in the adapted version ) is not self - consistent for the multiply connected de sitter space @xmath471 . ( that is , it does not solve the semiclassical einstein s equations . ) in the case of an eternal schwarzschild black hole , there are the boulware vacuum @xcite and the hartle - hawking vacuum @xcite . the globally defined hartle - hawking vacuum bears essentially the same relationship to the boulware vacuum as the minkowski vacuum does to the rindler vacuum @xcite . for the boulware vacuum , the energy - momentum tensor diverges at the event horizon of the schwarzschild black hole , which means that this state is _ not _ a good vacuum for the schwarzschild black hole because , when one inserts this energy - momentum tensor back into einstein s equations , the back - reaction will seriously alter the schwarzschild geometry near the event horizon . for the hartle - hawking vacuum , however , the energy - momentum tensor is finite everywhere and a static observer outside the horizon sees hawking radiation @xcite . people usually regard the hartle - hawking vacuum as the reasonable vacuum state for an eternal schwarzschild black hole because , when its energy - momentum tensor is fed back into einstein s equations , the schwarzschild geometry is only altered slightly @xcite . therefore , in the case of misner space , li and gott @xcite have tried to find a vacuum which is also self - consistent and found that the adapted rindler vacuum is such a vacuum if @xmath337 . here we also try to find a self - consistent vacuum for our multiply connected de sitter space . let us consider the adapted conformal rindler vacuum in @xmath471 . the energy - momentum tensor of a conformally coupled scalar field in the adapted rindler vacuum in misner space is given by eq . ( [ e69 ] ) . inserting eq . ( [ e69 ] ) into eq . ( [ e65 ] ) and using eqs . ( [ e59]-[e61 ] ) , we obtain the energy - momentum tensor for the adapted conformal rindler vacuum of a conformally coupled scalar field in our multiply connected de sitter space @xmath534 \left(\begin{array}{cccc } -3&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array } \right)-{1\over960\pi^2 r_0 ^ 4}\delta_{\mu}^{~\nu } , \label{e73}\end{aligned}\ ] ] where the coordinate system is the static coordinate system @xmath423 . similarly , this result can also be analytically extended to the whole @xmath471 , though the static coordinates only cover region @xmath434 . we find that , if @xmath535 this energy - momentum tensor is regular on the whole space . ( [ e74 ] ) corresponds to @xmath337 via eq . ( [ e61 ] ) ] . otherwise both @xmath536 and @xmath537 diverge as the cauchy horizon is approached . for the case @xmath538 , the energy - momentum tensor is @xmath539 which is the same as the energy - momentum tensor for the conformal minkowski vacuum in the simply connected de sitter space . the euclidean section of our multiply connected de sitter space is a four - sphere @xmath371 embedded in a five dimensional flat euclidean space with those points related by an azimuthal rotation with angle @xmath540 being identified . there are conical singularities unless @xmath541 . this may be regarded as a geometrical explanation of the self - consistent condition in ( [ e74 ] ) . similarly , our multiply connected de sitter space solves the semiclassical einstein s equations with a cosmological constant @xmath12 and the energy - momentum tensor in eq . ( [ e75 ] ) ( and thus it is self - consistent ) if @xmath542 ( if @xmath76 , we have the two solutions @xmath543 and @xmath523 @xcite ) . it is well known that in the simply connected de sitter space , an inertial particle detector perceives thermal radiation with the gibbons - hawking temperature [ eq . ( [ e32 ] ) ] if the conformally coupled scalar field is in the conformal minkowski vacuum @xcite . now we want to find what a particle detector perceives in the adapted conformal rindler vacuum in our multiply connected de sitter space . the response function of the particle detector is still given by eq . ( [ e7 ] ) . the wightman function is obtained from the corresponding hadamard function by eq . ( [ e26 ] ) . the hadamard function for the conformally coupled scalar field in multiply connected de sitter space is related to that in misner space via eq . ( [ e62 ] ) [ with @xmath503 replaced by @xmath544 . the hadamard function for the adapted rindler vacuum in misner space is given by eq . ( [ e78 ] ) . inserting eq . ( [ e78 ] ) [ as @xmath545 into eq . ( [ e62 ] ) and using eqs . ( [ e59]-[e61 ] ) , we obtain the hadamard function for the adapted conformal rindler vacuum of the conformally coupled scalar field in our multiply connected de sitter space @xmath546 } , \label{e79}\end{aligned}\ ] ] where @xmath547 , @xmath548 , and @xmath329 is written in @xmath549 as @xmath550\right\}. \label{e80}\end{aligned}\ ] ] the wightman function is obtained from eq . ( [ e79 ] ) via eq . ( [ e26 ] ) . the hadamard function given by eq . ( [ e79 ] ) and the wightman function obtained from that are defined in region @xmath434 in the multiply connected de sitter space , but they can be analytically extended to region @xmath435 via the transformation in eq . ( [ e47 ] ) . however , it should be noted that as we make the continuation from @xmath434 to @xmath435 , @xmath551 should be continued to be @xmath552 instead of @xmath553 . this is because if we take @xmath554 , we should also take @xmath555 ( instead of @xmath556 ) ( @xmath367 and @xmath557 should be continued along the same path ) , thus @xmath558 . using similar transformations , the results can also be continued to regions @xmath448 and @xmath449 ( we do not write them out because we do not use them here ) . we consider particle detectors moving along three kinds of worldlines in our multiply connected de sitter space : _ 1 . a particle detector moving along a worldline with @xmath559 in @xmath434 . _ in such a case , on the worldline of the particle detector , @xmath329 is zero and the hadamard function is reduced to @xmath560 where @xmath561 is the proper time of the particle detector . the corresponding wightman function obtained from eq . ( [ e26 ] ) is @xmath562 where @xmath279 is an infinitesimal positive real number . inserting it into eq . ( [ e7 ] ) , obviously the integration over @xmath290 is zero since all poles of the integrand are in the upper - half plane of complex @xmath292 while the integration contour is closed in the lower - half plane . therefore the response function @xmath274 is zero and no particles are detected . all of these worldlines are accelerated , except for the one at @xmath480 . _ 2 . a particle detector moving along a geodesic with @xmath563 in region @xmath435 . _ in this region the hadamard function is @xmath564 } , \label{e83}\end{aligned}\ ] ] where @xmath565 is given by @xmath566\right\}. \label{e84}\end{aligned}\ ] ] [ eq . ( [ e83 ] ) and eq . ( [ e84 ] ) are obtained from eq . ( [ e79 ] ) and eq . ( [ e80 ] ) via the transformation in eq . ( [ e47 ] ) respectively . ] on the worldline of the particle detector , the hadamard function is reduced to @xmath567 and @xmath568 is reduced to @xmath569 using the proper time @xmath267 defined by eq . ( [ e49 ] ) , on the worldline of the particle detector @xmath568 and @xmath570 can be written as @xmath571 and @xmath572 where @xmath573 , @xmath574 , @xmath575 , @xmath407 , and @xmath576 . the wightman function is equal to one half of the hadamard function with @xmath292 replaced by @xmath577 [ eq . ( [ e26 ] ) ] . thus the response function is @xmath578 where @xmath579_{\delta\tau\rightarrow\delta\tau - i\epsilon}. \label{e91}\end{aligned}\ ] ] now we consider the poles in the complex @xmath292 plane of the integrand in the integral of @xmath580 . the poles are given by the equation @xmath581 ( it is easy to check that @xmath582 does not give any poles . ) from eq . ( [ e92 ] ) and eq . ( [ e88 ] ) , we have ( we neglect the term @xmath583 , and at the end of the calculation we return it back to the expressions ) @xmath584 solutions to eq . ( [ e93 ] ) are @xmath585 where @xmath586 where @xmath587 is the principal value of @xmath588 , and here it is real ( similarly for @xmath589 ) . we need to check if all @xmath590 are roots of eq . ( [ e92 ] ) , because the number of roots might increase as we go from eq . ( [ e92 ] ) to eq . ( [ e93 ] ) . [ e.g. , for any integer @xmath266 , @xmath591 solves the equation @xmath592 ; but , only @xmath593 solves the equation @xmath594 . ] @xmath595 is obviously a root of eq . ( [ e92 ] ) . the question is : as @xmath292 goes from @xmath595 to @xmath596 , does eq . ( [ e88 ] ) give the same @xmath565 which is a real value [ @xmath597 ; see eq . ( [ e92 ] ) ] ? ( remember that @xmath588 is a multi - valued complex function . ) to answer this question , let @xmath598 ( where @xmath599 is real ) . then from eq . ( [ e88 ] ) we have @xmath600 where we have used @xmath601 . the real components of @xmath602 and @xmath603 are respectively @xmath604 @xmath605 by eq . ( [ e95 ] ) , we find that @xmath606 and @xmath607 are always positive for any real @xmath599 . this means that as @xmath292 goes from @xmath595 to @xmath596 , the arguments ( the argument of a complex number @xmath608 is @xmath134 ) of @xmath602 and @xmath603 do not change , neither does the argument of @xmath609 . the value of @xmath565 remains in the same branch of @xmath610 as @xmath599 varies . thus , for all @xmath611 , we have @xmath612 and eq . ( [ e92 ] ) is satisfied . therefore all @xmath590 in eq . ( [ e94 ] ) are poles . the residues of the integrand in ( [ e91 ] ) at poles @xmath590 are ( here @xmath583 is returned to the expressions ) @xmath613 then by the residue theorem ( the contour for the integral is closed in the lower - half plane of complex @xmath292 ) we have @xmath614 and @xmath615 the @xmath616 factor in eq . ( [ e102 ] ) indicates that the @xmath617 terms contribution can be both positive ( absorption by the detector ) and negative ( emission from the detector ) . we see that the contribution of the @xmath157 term is just the hawking radiation with the gibbons - hawking temperature @xmath524 in the simply connected de sitter space . the contribution of the @xmath617 terms is a kind of `` grey - body '' hawking radiation : the temperature is @xmath408 , but its density or flux decreases as the universe expands ( @xmath618 increases as the universe expands ) . the sum of all @xmath617 contributions is @xmath619 in the case of @xmath337 ( the self - consistent case ) , we have @xmath620 ( @xmath621 ) and thus @xmath622 . then @xmath623 where @xmath624 . as @xmath625 , the contribution of all @xmath617 terms decreases exponentially to zero . thus , at events far from the cauchy horizon in @xmath435 , the particle detector perceives pure hawking radiation given by the @xmath157 term . as @xmath626 ( near the cauchy horizon ) , we have @xmath627 this is a `` grey - body '' hawking radiation with @xmath628 . near the cauchy horizon the total radiation is the sum of a pure hawking radiation ( given by the @xmath157 term ) and a `` grey - body '' hawking radiation ( given by all @xmath617 terms ) . the total intensity of the radiation near the cauchy horizon is a factor of @xmath629 that of regular hawking radiation , but its spectrum is the same as the usual hawking radiation . _ 3 . a particle detector moving along a co - moving worldline in the steady - state coordinate system_. suppose the detector moves along the geodesic @xmath630 ( such a worldline is a timelike geodesic passing through @xmath434 and into @xmath435 ) where @xmath631 and the proper time @xmath267 are related to the static radius @xmath632 by @xmath633 the cauchy horizon is at @xmath479 , or @xmath634 . on the worldline of the detector the hadamard function is @xmath635 where @xmath575 , @xmath407 , @xmath636 , @xmath329 is given by @xmath637 and @xmath638 is related to @xmath639 and @xmath292 by @xmath640 by analytical continuation , eqs . ( [ e107]-[e109 ] ) hold in the whole region covered by the steady - state coordinates in de sitter space . the wightman function @xmath280 is equal to one half of @xmath333 with @xmath292 replaced by @xmath577 [ eq . ( [ e26 ] ) ] . the response function is @xmath641 where @xmath642 } \right\}_{\delta\tau\rightarrow\delta \tau - i\epsilon}. \label{e110}\end{aligned}\ ] ] the poles of the integrand in the complex-@xmath292 plane are given by @xmath643 this together with eq . ( [ e108 ] ) and eq . ( [ e109 ] ) leads to @xmath644 the roots of eq . ( [ e112 ] ) are @xmath645 where @xmath646 where @xmath647 . by carefully checking @xmath648 in eq . ( [ e113 ] ) , as we did in case 2 , we find that : ( 1 ) for @xmath649 ( or @xmath650 , i.e. , in region @xmath434 ) , only @xmath651 solve eq . ( [ e111 ] ) ; ( 2 ) for @xmath652 ( or @xmath653 , i.e. , in region @xmath435 ) , only @xmath654 solve eq . ( [ e111 ] ) . ( here it is assumed that @xmath655 and the self - consistent case with @xmath337 obviously satisfies this condition . ) all other @xmath292 s in eq . ( [ e113 ] ) are not roots of eq . ( [ e111 ] ) , though they solve eq . ( [ e112 ] ) . therefore the poles are ( where @xmath583 is returned ) @xmath656 obviously in region @xmath434 all poles are in the upper - half plane of complex @xmath292 . therefore @xmath657 when the particle detector is in region @xmath434 . so the particle detector sees nothing while it is in region @xmath434 . in region @xmath435 , only the poles with @xmath658 are in the lower - half plane of complex @xmath292 . the residues of the integrand at poles @xmath659 are @xmath660 @xmath661 where @xmath662 and @xmath663 . by the residue theorem , we have that @xmath664 has the same value as that in eq . ( [ e101 ] ) , which represents hawking radiation with the gibbons - hawking temperature ; the contribution of all @xmath617 terms ( note that @xmath665 ) is @xmath666 which represents a `` grey - body '' hawking radiation . as @xmath625 ( or @xmath667 ) , @xmath668 exponentially drops to zero ; therefore , at events far from the cauchy horizon in @xmath435 , the particle detector only perceives pure hawking radiation ( the same as that in case 2 ) . as @xmath669 ( approaching the cauchy horizon ) , we also have @xmath670 . thus as the cauchy horizon is approached from the side of region @xmath435 , the particle detector comoving in the steady - state coordinate system perceives pure hawking radiation with gibbons - hawking temperature . from the above discussion , we find that in our multiply connected de sitter space with the adapted rindler vacuum , region @xmath434 is cold ( where the temperature is zero ) but region @xmath435 is hot ( where the temperature is @xmath408 ) . similarly , region @xmath449 is cold but @xmath448 is hot , the above results can be easily extended to these regions . this gives rise to an arrow of increasing entropy , from a cold region to a hot region ( fig . [ f5 ] ) . in classical electromagnetic theory , it is well known that both the retarded potential @xmath671 and the advanced potential @xmath672 ( and any part - retarded - and - part - advanced potential @xmath673 with @xmath674 ) are solutions of maxwell s equations . but from our experience , we know that all the electromagnetic perturbations we see are propagated only by the _ retarded _ potential . ( for example , if at some time and some place , a light signal is emitted , it can only be received by a receiver at another place sometime _ later _ ) . this indicates that there is an _ arrow of time _ in the solutions of maxwell s equations , though maxwell s equations themselves are time - symmetric . this arrow of time is sometimes called the electromagnetic arrow of time , or the causal arrow of time . how this arrow of time arises is a mystery . many people have tried to solve this problem by attributing it to a boundary condition of the universe @xcite ( for review of the arrows of time , see @xcite ) . in this subsection we argue that the principle of self - consistency @xcite naturally gives rise to an arrow of time in our multiply connected de sitter space . first let us consider the arrow of time in misner space . suppose at an event e in region f in misner space [ by boost and translation , assume we have moved e to @xmath675 , a spherical pulse of electromagnetic wave is created . if the potential is retarded [ here `` retarded '' and `` advanced '' are defined relative to the direction of @xmath676 ( @xmath38 is the time coordinate in the global cartesian coordinates of the covering space minkowski space ) ] , the pulse will propagate in the future direction as a light cone originating from e. at any point on the light cone , the energy - momentum tensor of the wave is @xmath677 where @xmath678 is a scalar function and @xmath679 is a null vector tangent to the light cone , and the energy density measured by an observer with four - velocity vector @xmath676 ( whose ordinary three - velocity is zero ) is @xmath680 ( thus @xmath681 measures the energy density of the electromagnetic wave . ) by einstein s equations , the back - reaction of @xmath682 on @xmath256 and @xmath683 ( where @xmath684 is the ricci tensor and @xmath685 is the ricci scalar curvature ) is @xmath686 , @xmath687 . the riemann tensor can be decomposed as @xmath688 , where @xmath689 is the weyl tensor and @xmath690 is constructed entirely from the ricci tensor @xmath691b}-g_{b[c}r_{d]a}-{1\over3}rg_{a[c}g_{d]b } , \label{wel}\end{aligned}\ ] ] where square brackets denote antisymmetrization @xcite . the weyl tensor describes the part of the curvature that is due to pure gravitational field , whereas the ricci tensor describes the part that , according to einstein s equations , is directly due to the energy - momentum tensor of matter @xcite . therefore , in some sense , the values of @xmath692 and @xmath693 determine the influence of matter fields on the stability of the background spacetime . an infinite @xmath692 or @xmath693 implies that the spacetime is unstable against this perturbation and a singularity may form ; on the other hand , if @xmath692 and @xmath693 are finite , the spacetime may be stable against this perturbation . self - consistent solutions should require that @xmath692 and @xmath693 do not blow up . if they did , the starting geometry on the basis of which @xmath692 and @xmath693 were calculated would be greatly perturbed and the @xmath692 and @xmath693 calculation itself would be invalid , and thus it would not be a self - consistent solution . for electromagnetic fields we always have @xmath694 , so we need only consider @xmath693 . for @xmath682 in eq . ( [ e122 ] ) , we also have @xmath695 thus significant perturbations ( indicated by a non - vanishing @xmath693 ) can only occur when the light cone `` collides '' with its images under the boost transformation . at any point @xmath144 on the intersection of the light cone @xmath696 and its @xmath250-th image @xmath697 ( suppose @xmath212 ) , the energy - momentum tensor is @xmath698 where @xmath699 is the null vector tangent to the light cone @xmath696 at @xmath144 , @xmath700 is the null vector tangent to the light cone @xmath697 at @xmath144 ; @xmath681 measures the energy density in light cone @xmath696 , @xmath701 measures the energy density in light cone @xmath697 . from eq . ( [ e125 ] ) we have @xmath702_p , \label{e126}\end{aligned}\ ] ] the index @xmath144 denotes that the quantity is evaluated at the point @xmath144 . since the point @xmath144 on @xmath697 is obtained from some point @xmath703 on @xmath696 by boost transformation , @xmath144 and @xmath703 must have the same timelike separation from the origin @xmath704 ( remember that @xmath144 is on the intersection of @xmath696 and @xmath697 , see fig . [ f6]a ) . if we take the @xmath700 at @xmath144 being transported from the @xmath705 at @xmath703 , we have @xmath706 . because the light cone @xmath696 is spherically symmetric , we have @xmath707=@xmath708 . therefore we have @xmath709 and at @xmath144 we have @xmath710 . under the boost transformation @xmath472 , we have @xmath711={k^\prime}^0\left[\cosh nb~\left({\partial\over\partial t}\right)^a+\sinh nb~\left ( { \partial\over\partial x}\right)^a\right]+ \nonumber\\ & & { k^\prime}^1\left[\cosh nb~\left({\partial\over\partial x}\right)^a+\sinh nb~\left({\partial\over\partial t}\right)^a\right]+ { k^\prime}^2\left({\partial\over\partial y}\right)^a+{k^\prime}^3\left({\partial \over\partial z}\right)^a , \label{e127}\end{aligned}\ ] ] where @xmath712 . due to the spherical symmetry , we have @xmath713 . define @xmath714 by @xmath715 , @xmath716 , and @xmath717 . then we have @xmath718 , @xmath719 , @xmath720 ( `` @xmath721 '' means `` at @xmath703 '' ) , and @xmath722 and @xmath723 then @xmath724 , \label{e130}\end{aligned}\ ] ] and @xmath725 ^ 2 . \label{e131}\end{aligned}\ ] ] it is easy to find that @xmath726 reaches a maximum at @xmath727 and @xmath728 where @xmath729 is the energy density from @xmath696 as measured in a frame at event @xmath144 with ordinary velocity @xmath730 . @xmath731 is always finite [ less than @xmath732 since @xmath250 is positive . if @xmath733 we have @xmath734 . so if we have a retarded potential in region f , even considering the infinite number of images , @xmath693 is always finite . if the potential is advanced however , the pulse wave will propagate backward in the past direction as a light cone originating from e. and , within a finite time , it will hit the cauchy horizon . by an analysis similar to the above arguments , we find that in this case @xmath735 ^ 2 , \label{e133}\end{aligned}\ ] ] which reaches a maximum at @xmath727 and @xmath736 since @xmath729 is finite ( the past light cone from e at @xmath727 hits the cauchy horizon in a finite affine distance ) , thus @xmath737 as @xmath738 . as @xmath738 , @xmath696 and @xmath697 collide at the cauchy horizon [ as @xmath738 the point @xmath739 approaches the cauchy horizon ] ( see fig . thus @xmath731 diverges as the cauchy horizon is approached and the cauchy horizon may be destroyed . therefore the advanced potential is _ not _ self - consistent in region f of misner space . it is easy to see that any part - retarded - and - part - advanced potential is also _ not _ self - consistent in f. the _ only _ self - consistent potential in region f is the _ retarded _ potential . similarly , in region p the only self - consistent potential is the _ advanced _ potential ( see fig . [ [ note that here `` advanced '' and `` retarded '' are defined relative to the global time direction in minkowski spacetime ( the covering space ) . an observer in p will regard it as `` retarded '' relative to his own time direction . ] in region r , by boost and translation , we can always move the event e ( where a spherical pulse of electromagnetic waves is emitted ) to @xmath740 . either pure retarded or pure advanced potentials are self - consistent in this region because the light cone never `` collides '' with the images of itself and thus we always have @xmath741 ( see fig . but , for a part - retarded - and - part - advanced potential , the retarded light cone ( @xmath742 ) propagates forward while the advanced light cone ( @xmath743 ) propagates backward , both originating from e. the forward part of the light cone will collide with images of the backward part of the light cone and _ vice versa _ ( see fig . we find that at a point @xmath144 on the intersection of @xmath742 and @xmath744 ( or @xmath743 and @xmath745 ) @xmath746 ^ 2 , \label{e135}\end{aligned}\ ] ] where @xmath747 is the energy density from @xmath742 observed in a frame on @xmath742 with time coordinate @xmath38 and with ordinary velocity @xmath730 and @xmath748 is the energy density from @xmath743 seen in a frame on @xmath743 with time coordinate @xmath749 and with ordinary velocity @xmath730 . @xmath726 reaches a maximum at @xmath750 , and @xmath751 where @xmath38 is the global time coordinate in the covering minkowski space . as @xmath144 approaches the cauchy horizon , where @xmath738 , @xmath747 and @xmath748 are both finite , since in the @xmath750 direction the future and past light cones of e both hit the cauchy horizon in a finite affine distance . thus @xmath737 as @xmath144 approaches the cauchy horizon ( where @xmath738 ) . therefore in region r both the retarded and the advanced potential are self - consistent , but the part - retarded - and - part - advanced potential is _ not _ self - consistent . this conclusion also holds for region l. furthermore , there must be a correlation between time arrows in region l and region r : if we choose the retarded potential in r , we must choose the advanced potential in l ( see fig . [ f6]b ) ; if we choose the advanced potential in r , we must choose the retarded potential in l. otherwise the collision of light cones from r and light cones from l will destroy the cauchy horizon . as another treatment for perturbations in misner space , consider that at an event e in region f two photons are created @xcite [ we choose e to be at @xmath752 as before ] . one photon runs to the right along the @xmath753 direction , the other photon runs to the left along the @xmath754 direction . they have the same frequency ( thus the same energy ) . the tangent vector of the null geodesic of the right - moving photon is chosen to be @xmath755 , where @xmath756 is an affine parameter of the geodesic , @xmath757 is a constant and @xmath758 , @xmath759 . the tangent vector of the null geodesic of the left - moving photon is chosen to be @xmath760 , where @xmath761 is an affine parameter of that geodesic . the null vectors @xmath762 and @xmath763 are invariant under boost transformations . at any point where a photon with null wave - vector @xmath699 is passing by , the frequency of the photon measured in a frame of reference passing by the same point with the four - velocity @xmath764 is @xmath765 . if @xmath766 ( i.e. , the frame of reference has ordinary three - velocity @xmath730 ) and @xmath767 or @xmath763 , we have @xmath768 ( thus @xmath757 measures the frequency of the photon ) . at any point where the @xmath250-th image of the right - moving ( left - moving ) photon is passing by , using the boost transformation we can always find a frame of reference in which the frequency of the photon is @xmath769 . but at a point @xmath144 where the right - moving ( left - moving ) photon passes the @xmath250-th image of the left - moving ( right - moving ) photon , we can not find a frame of reference such that the two `` colliding '' photons both have frequency @xmath769 . in such a case we should analyze it in the center - of - momentum frame . the four - velocity of the center - of - momentum frame is @xmath770 where @xmath771^{-1}=uv / q^2=\tilde{\eta}^2/q^2 $ ] where @xmath772 is the proper time separation of @xmath144 from the origin @xmath704 . therefore the total energy of the two oppositely directed photons in the center - of - momentum frame is @xmath773 ( for all other frames the total energy would be greater . ) if the potential is retarded , so photons move in the future direction , all points where photons and their images `` collide '' are in the future of the hypersurface @xmath774 . therefore we have @xmath775 and @xmath776 , so the total energy in the center - of - momentum frame is always bounded . but , if the potential is advanced , photons move in the past direction ; thus all points where photons and oppositely directed image photons `` collide '' are in the past of the hypersurface @xmath774 . in particular , the right - moving ( left - moving ) photon collides with the @xmath397-th ( @xmath224-th ) image of the left - moving ( right - moving ) photon at the cauchy horizon , where @xmath777 and thus @xmath778 . thus , the cauchy horizon may be destroyed by these photon pairs . therefore in agreement with our earlier argument , the advanced potential is _ not _ self - consistent in region f. the _ retarded _ potential is self - consistent in region f. similarly , the _ advanced _ potential is self - consistent in region p. in region r and region l , both the retarded potential and the advanced potential are self - consistent , because the photons and their images will not collide with each other and at any point a photon is passing by we can always find a frame for whom the frequency of this photon is @xmath769 . and , the potentials in region r and region l must be correlated in the following way : if the potential in r is retarded , the potential in l must be advanced ; if the potential in r is advanced , the potential in l must be retarded ( we would call them `` anti - correlated '' ) . otherwise the photons from l and photons from r passing in opposite directions would be measured to have infinite energy in center - of - momentum frames as the cauchy horizon is approached and this may similarly destroy the cauchy horizon . these conclusions are consistent with those obtained from the analysis of the perturbation of a pulse wave discussed above . our multiply connected de sitter space is conformally related to misner space via eqs . ( [ e58]-[e61 ] ) . because light cones and chronological relations are conformally invariant @xcite ( thus regions @xmath435 , @xmath448 , @xmath434 , and @xmath449 in multiply connected de sitter space correspond respectively to regions f , p , r , and l in misner space under the conformal map , as discussed in section [ ix.b ] ) , maxwell s equations are also conformally invariant @xcite , so it is easy to generalize the results from misner space to our multiply connected de sitter space . under the conformal transformation @xmath779 , the energy - momentum tensor of the electromagnetic field is transformed as @xmath780 @xcite . thus @xmath726 is transformed as @xmath781 . from the above discussion of @xmath726 in misner space , we know that @xmath726 is zero everywhere except at the intersection of two light cones . thus , in multiply connected de sitters pace , @xmath726 is also zero everywhere except at the intersection of two light cones . at the intersection of two light cones in multiply connected de sitter space , it is easy to show that the maximum value of @xmath726 is at the points with @xmath727 or @xmath750 on the intersection . from eq . ( [ e60 ] ) and eq . ( [ e60a ] ) we find that for @xmath727 or @xmath750 , @xmath477 is non - zero except at the points with @xmath727 on the cauchy horizon ( where @xmath479 or @xmath782 ) . also because @xmath477 is finite everywhere on the cauchy horizon ( i.e. it is never infinite ) , we have that : ( 1 ) if @xmath726 diverges on the cauchy horizon in misner space , the corresponding @xmath726 also diverges on the cauchy horizon in our multiply connected de sitter space ; ( 2 ) if @xmath726 is finite in some region ( except at the cauchy horizon ) in misner space , the corresponding @xmath726 is also finite in the corresponding region ( not at the cauchy horizon ) in the multiply connected de sitter space ; ( 3 ) if @xmath726 is zero in some region ( not a single point ) in misner space , the corresponding @xmath726 is also zero in the corresponding region in the multiply connected de sitter space . under the conformal transformation @xmath779 , the affine parameter of a null geodesic is transformed as @xmath783 where @xmath784 is a constant @xcite and thus the null vector @xmath785 is transformed as @xmath786 . then @xmath787^{-1/2}$ ] is transformed as @xmath788 and the total energy of the photon pairs in the center - of - momentum frame is transformed as @xmath789 and the constant @xmath790 can be absorbed into @xmath769 . therefore , we can transplant the above results for misner space directly to our multiply connected de sitter space : _ in region @xmath435 the only self - consistent potential is the retarded potential ; in region @xmath448 the only self - consistent potential is the advanced potential ; in regions @xmath434 and @xmath449 both the retarded potential and the advanced potential are self - consistent , but they must be anti - correlated _ ( fig . [ f6 ] ) . the cauchy horizon @xcite separating a region with ctcs from that without closed causal curves is also called a chronology horizon @xcite . a chronology horizon is called a _ future _ chronology horizon if the region with ctcs lies to the future of the region without closed causal curves ; a chronology horizon is called a _ past _ chronology horizon if the region with ctcs is in the past of the region without closed causal curves . it is generally believed that a future chronology horizon is classically unstable unless there is some diverging effect near the horizon @xcite . the argument says that a wave packet propagating in the future direction in this spacetime will pile up on the future chronology horizon and destroy the horizon due to the effect of the infinite blue - shift of the frequency ( and thus the energy ) seen by a timelike observer near a closed null geodesic on the horizon @xcite . but if there is some diverging mechanism ( like the diverging effect of a wormhole in a spacetime with ctcs constructed from a wormhole @xcite ) near the horizon , the amplitude of the wave packet will decrease with time due to this mechanism , and this may cancel the effect of the blue - shift of the frequency , making the energy finite and thus rendering the future chronology horizon classically stable . unfortunately , in our multiply connected de sitter spacetime ( as also in misner space ) there is no such diverging mechanism . a light ray propagating in de sitter space will focus rather than diverge . this can be seen from the focusing equation @xcite @xmath791 where @xmath792 is the cross - sectional area of the bundle of rays , @xmath22 is the affine parameter along the central ray , the null vector @xmath699 is @xmath785 , and @xmath59 is the magnitude of the shear of the rays . for de sitter space we have @xmath793 and thus we have @xmath794 , so the ray will never diverge . ( in fact this always holds if the spacetime satisfies either the weak energy condition or the strong energy condition and it is called the focusing theorem @xcite . ) hawking @xcite has given a general proof along the above lines that any _ future _ chronology horizon is classically unstable unless light rays are diverging when they propagate near the chronology horizon . you could cause this instability by shaking an electron in the vicinity of the future chronology horizon . the retarded wave would then propagate to the future causing the instability . however , in hawking s proof @xcite , if we replace a future chronology horizon with a _ past _ chronology horizon , then the proof breaks down because , in such a case , a wave packet propagating toward the future near the past chronology horizon will suffer a red - shift instead of a blue - shift . therefore a _ past _ chronology horizon , according to hawking s argument , is classically stable in a world with retarded potentials . if the universe started with a region of ctcs , but there are no ctcs now , that early region of ctcs would be bounded to the future by a past chronology horizon , and that horizon would be classically stable in a world with retarded potentials which is what we want . in our multiply connected de sitter space , this is realized , since the arrow of time in region @xmath435 is in the future direction and the arrow of time in region @xmath448 is in the past direction [ here `` future '' and `` past '' are defined globally by the direction of @xmath795 , where @xmath267 is the time coordinate in the global coordinate system @xmath422 of the de sitter covering space ] . @xmath435 and @xmath434 can have retarded potentials , while @xmath448 and @xmath449 have advanced potentials , as we have noted . in this case the cauchy horizons separating @xmath498 from @xmath434 and @xmath448 from @xmath449 are classically stable , as indicated by our detailed study of @xmath726 as these cauchy horizons are approached . what about the cauchy horizons separating @xmath448 from @xmath434 and @xmath435 from @xmath449 ? in region @xmath448 , the potentials are advanced , so hawking s instability does not arise as one approaches the cauchy horizon separating it from @xmath434 . in region @xmath434 , the potentials are retarded , so by hawking s argument , one might think that there would be an instability as the cauchy horizon separating @xmath434 from @xmath448 is approached from the @xmath434 side . but , as we have shown , with retarded potentials in @xmath434 , @xmath726 does not diverge as the cauchy horizon separating @xmath434 from @xmath448 is approached from the @xmath434 side , indicating no instability . because one can always find frames where the passing photon energies are bounded as the cauchy horizon is approached . hawking s argument works only if one can pick a particular frame like the frame of a timelike observer crossing the cauchy horizons and observe the blow up of the energy in that frame . ( thus hawking s approach is observer - dependent , while our approach with @xmath726 is observer - independent . ) hawking s timelike observer would be killed by these photons . but , as we have shown , @xmath434 is in a pure vacuum state in our model , so there are no timelike observers in this region , and no preferred frame . if there were timelike particles of positive mass crossing from @xmath448 to @xmath434 through the cauchy horizon , we have shown ( li and gott @xcite ) that these would cause a classical instability ; but there are none . there are , as we shall show in the next subsection , no real particles in regions @xmath449 and @xmath434 ( because these are vacuum states ) and no real particles in region @xmath435 and @xmath448 until the vacuum state there decays by forming bubbles at a timelike separation @xmath796 from the origin ( @xmath797 will be given in the next subsection ) . thus , there are no particles crossing the cauchy horizons separating @xmath448 from @xmath434 and @xmath435 form @xmath449 . thus , there is no instability caused by particles crossing the cauchy horizons ; and since there are no timelike observers in region @xmath434 to be hit by photons as the cauchy horizon separating @xmath434 from @xmath798 is approached , there is no instability , as indicated by the fact @xmath726 does not blow up as that cauchy horizon is approached . as indicated in fig . [ f4]b , region @xmath799 is one causally connected region which can be pictured as partially bounded to the future by the future light cone of an event e@xmath78 and bounded to the past by the future light cone of an event e ; but e and e@xmath78 are identified by the action of the boost , so these two light cones are identified , creating a periodic boundary condition for region @xmath799 . as our treatment using @xmath693 with images indicates , retarded photons created in @xmath799 cause no instability . particles with timelike worldlines crossing the cauchy horizons separating @xmath799 from @xmath800 would cause instability by crossing an infinite number of times between the future light cones of e and e@xmath78 , thus making an infinite number of passages through the region @xmath799 ( also @xmath801 ) shown in fig . however , as we have shown , there should be no such particles with timelike worldlines crossing the cauchy horizons separating @xmath799 from @xmath800 , and no photons crossing these horizons either , since the potentials in @xmath799 are retarded , while the potentials in @xmath801 are advanced . thus , we expect @xmath802 and @xmath801 to both be stable , and causally disconnected from each other . ( see further discussion in the next subsection ) . thus , the principle of self - consistency @xcite produces classical stability of the cauchy horizons and naturally gives rise to an arrow of time in our model of the universe . from the above discussion we find that in the multiply connected de sitter space region @xmath435 and region @xmath448 are causally independent in physics : the self - consistent potential in @xmath435 is the retarded potential , while the self - consistent potential in @xmath798 is the advanced potential , thus an event in @xmath435 can never influence an event in @xmath448 , and _ vice versa_. @xmath435 and @xmath448 are physically disconnected though they are mathematically connected . if we choose the potential in @xmath434 to be retarded , then the potential in @xmath449 must be advanced . ( note that here `` advanced '' and `` retarded '' are defined relative to the global time direction in de sitter space the covering space of our multiply connected de sitter space . ) then region @xmath803 ( including the cauchy horizon separating @xmath435 from @xmath434 ) forms a causal unit , and region @xmath804 ( including the cauchy horizon separating @xmath448 from @xmath449 ) forms another causal unit . ( see fig . 4b , where the two null surfaces partially bounding the grey @xmath799 region to the past and future are identified . similarly for the null surfaces partially bounding the @xmath801 region . ) an event in @xmath803 and an event in @xmath804 are always causally independent in physics : they can never physically influence each other though they may be mathematically connected by some causal curves ( null curves or timelike curves ) . though @xmath803 and @xmath804 are connected in mathematics , they are disconnected in physics . they are separated by a cauchy horizon . when we consider physics in @xmath799 , we can completely forget region @xmath804 ( and _ vice versa _ ) . though in such a case the cauchy horizon separating @xmath799 from @xmath801 is a null spacetime boundary , we do not need any boundary condition on it because the topological multi - connectivity in @xmath799 has already given rise to a periodic boundary condition ( which is a kind of self - consistent boundary condition ) . ( in fig . 4b this is shown by the fact that the null curves partially bounding @xmath799 to the past and future are identified . ) this periodic boundary condition ( the self - consistent condition ) is sufficient to fix the solutions of the universe . for example , in our multiply connected de sitter space model , the stability of the cauchy horizon requires that the regions with ctcs ( @xmath434 and @xmath449 ) must be confined in the past and in these regions all quantum fields must be in vacuum states ( as we have already remarked , the appearance of any real particles there seems to destroy the cauchy horizon @xcite ) . this gives rise to an arrow of time and an arrow of entropy in this model . @xmath799 is a hausdorff manifold with a null boundary , and thus @xmath803 is geodesically incomplete to the past . but , the geodesic incompleteness of @xmath799 may _ not _ be important in physics because in the inflationary scenario all real particles are created during the reheating process after inflation within bubbles created in region @xmath435 and these particles emit only retarded photons which never run off the spacetime because here the geodesic incompleteness takes place only in the past direction . on the other hand , we can smoothly extend @xmath799 to @xmath801 so that the total multiply connected de sitter space @xmath471 is geodesically complete but at the price that it is not a manifold at a two - sphere ( section [ ix.a ] ) . this model describes two physically disconnected but mathematically connected universes . [ the analogy between the causal structures in region @xmath799 and region @xmath801 might motivate us to identify antipodal points in our multiply connected de sitter space , as we did for the simply connected de sitter space ( section [ viii ] ) . the spacetime so obtained is a hausdorff manifold everywhere . it is geodesically complete but not time orientable . for computing the energy - momentum tensor of vacuum polarization , we must take into account the images of antipodal points in addition to the images produced by the boost transformation . further research is needed to find a self - consistent vacuum for this spacetime . ] now we consider formation of bubbles in @xmath799 in multiply connected de sitter space . [ the results ( and the arguments for @xmath799 in the previous paragraph ) also apply to region @xmath801 , except that while in @xmath799 bubbles expand in the future direction , in @xmath801 they expand in the past direction ; here `` future '' and `` past '' are defined with respect to @xmath795 where @xmath267 is the time coordinate in the global coordinates of de sitter space . ] region @xmath434 ( for its fundamental cell see fig . [ f4 ] ) which is multiply connected has a finite four - volume @xmath805 ( here @xmath576 , @xmath353 is the de sitter boost parameter ) . if the probability of forming a bubble per volume @xmath70 in de sitter space is @xmath279 , then the total probability of forming a bubble in @xmath806 is @xmath807 . region @xmath435 ( its fundamental cell is shown in fig . [ f4 ] ) has an infinite four - volume and thus there should be an infinite number of bubble universes formed @xcite . the metric in region @xmath435 is given by eq . ( [ e50 ] ) with @xmath808 , @xmath809 , @xmath427 , and @xmath428 ( see fig . 4a ) ; it is multiply connected ( periodic in @xmath810 with period @xmath353 ) . in order that the inflation proceeds and the bubbles ( which expand to the future as expected with the retarded potential in region @xmath435 ) do not percolate , it is required that @xmath71 where @xmath811 @xcite . gott and statler @xcite showed that in order that we on earth today should not have witnessed another bubble colliding with ours within our past light cone ( with @xmath109 confidence ) @xmath279 must be less than @xmath812 for @xmath813 ( for @xmath75 gott @xcite found @xmath74 ) . in our multiply connected de sitter space , for inflation to proceed , there should be the additional requirement that bubbles do not collide with images of themselves ( producing percolation ) . a necessary condition for a bubble formed in @xmath435 not to collide with itself is that from time @xmath267 when the bubble forms to future infinity ( @xmath814 ) a light signal moving along the @xmath810 direction [ where @xmath267 and @xmath810 are defined in eq . ( [ e47 ] ) and eq . ( [ e49 ] ) ] propagates a co - moving distance less than @xmath815 , which leads to the condition that @xmath816 . in fact this is also a sufficient condition , which can be shown by the conformal mapping between region @xmath435 in the multiply connected de sitter space and region f - o in misner space defined by eqs . ( [ e59a]-[e61 ] ) . if the collision of two light cones in f occurs beyond the hyperbola @xmath817 in misner space ( i.e. , in the region o ) , the corresponding two light cones ( and thus the bubbles formed inside these light cones ) in @xmath435 will never collide because @xmath501 in f corresponds to @xmath814 in @xmath435 . it is easy to show that the condition for a light cone not to collide with its images within f - o is that @xmath818 , where @xmath295 is the event where the light cone originates . by eq . ( [ e59a ] ) this condition corresponds to @xmath819>1+({{\tilde t}\over r_0})^2 - 2{{\tilde t}\over r_0}\cos\theta$ ] . since @xmath820 and @xmath821 , a _ sufficient _ condition is @xmath819>1+({{\tilde t}\over r_0})^2 + 2{{\tilde t}\over r_0}$ ] , i.e. @xmath822 which is equivalent to @xmath823 . therefore all bubbles formed after the epoch @xmath797 in @xmath435 in the multiply connected de sitter space will never collide with themselves . the @xmath824 part of the fundamental cell in @xmath435 has a finite four - volume @xmath825 . the total probability of forming a bubble in @xmath826 is @xmath827 . for @xmath337 we have @xmath828 , @xmath829 , and thus @xmath830 . for the case of @xmath337 , in order that there be less than a @xmath73 chance that a bubble forms in @xmath806 ( and thus less than @xmath831 chance in @xmath826 ) , @xmath279 should be less than @xmath832 . this should be no problem because we expect that this tunneling probability @xmath279 should be exponentially small . thus it would not be surprising to find region @xmath434 and region @xmath435 for epochs @xmath833 clear of bubble formation events ( and clear of real particles ) , which is all we require . also note that there may be two epochs of inflation , one at the planck scale caused by @xmath834 [ eq . ( [ e75 ] ) ] which later decays in region @xmath435 at @xmath835 into an inflationary metastable state at the gut scale produced by a potential @xmath65 , which , still later , forms bubble universes . inflationary universes can lead to the formation of baby universes in several different scenarios . if one of these baby universes simply turns out to be the original universe that one started out with , we have a multiply connected solution in many ways similar to our multiply connected de sitter space . there would be a multiply connected region of ctcs bounded by a past cauchy horizon which would be stable because of the self - consistency requirement as in the previous section , and this would also engender pure retarded potentials . thus , in a wide class of scenarios , the epoch of ctcs would be long over by now , as we would be one of the many later - formed bubble universes . also , the model might either be geodesically complete to the past or not . this might not be a problem in physics since we would in any case have a periodic boundary condition ; and because with its pure retarded potentials , no causal signals could be propagated to the past in any case . there are several different baby universe scenarios any one of which could accommodate our type of model . first , there is the farhi , guth , and guven @xcite method of creation of baby universes in the lab . at late times in an open universe , for example , an advanced civilization might implode a mass ( interestingly , it does not have to be a large mass a few kilograms will do ) with enough energy to drive it up to the gut energy scale , whereupon it might settle into a metastable vacuum , creating a small spherical bubble of false vacuum with a @xmath836 metastable vacuum inside . this could be done either by just driving the region up over the potential barrier , or by going close to the barrier and tunneling through . the inside of this vacuum bubble would contain a positive cosmological constant with a positive energy density and a negative pressure . this bubble could be created with an initial kinetic energy of expansion with the bubble wall moving outward . but the negative pressure would pull it inward , and it would eventually reach a point of maximum expansion ( a classical turning point ) , after which it would start to collapse and would form a black hole . but occasionally , ( probability @xmath837 for typical gut scales @xcite ) when it reaches its point of maximum expansion it tunnels to a state of equal energy but a different geometry , like a doorknob , crossing the einstein - rosen bridge @xcite . the `` knob '' itself would be the the interior of the bubble , containing the positive cosmological constant , and sitting in the metastable vacuum state with @xmath836 . the `` knob '' consists of more than a hemisphere of an initially static @xmath1 closed de sitter universe , where the bubble wall is a surface of constant `` latitude '' on this sphere . at the wall , the circumferential radius is thus decreasing as one moves outward toward the external spacetime . just outside the wall is the einstein - rosen neck which reaches a minimum circumferential radius at @xmath838 , and then the circumference increases to join the open external solution . this `` doorknob '' solution then evolves classically . the knob inflates to form a de sitter space of eventually infinite size . it is connected to the original spacetime by the narrow einstein - rosen bridge . but an observer sitting at @xmath838 in the einstein - rosen bridge will shortly hit a singularity in the future , just as in the schwarzschild solution . so the connection only lasts for a short time . the interior of the `` knob '' is hidden from an observer in the external spacetime by an event horizon at @xmath838 . eventually the black hole evaporates via hawking radiation @xcite , leaving a flat external spacetime ( actually part of an open big bang universe ) with simply a coordinate singularity at @xmath480 as seen from outside . ( see fig . [ f7 ] . ) from the point of view of an observer sitting at the center of @xmath836 bubble , he would see himself , just after the tunneling event , as sitting in a de sitter space that was initially static but which starts to inflate . centered on this observer s antipodal point in de sitter space , he would see a bubble of ordinary @xmath133 vacuum surrounding a black hole of mass @xmath174 . the observer sees his side of the einstein - rosen bridge and an event horizon at @xmath838 which hides the external spacetime at late times from him . from the point of view of the de sitter observer , the black hole also evaporates by hawking radiation , eventually leaving an empty @xmath133 bubble in an ever - expanding de sitter space . this infinitely expanding de sitter space , which begins expanding at the tunneling event , is a perfect starting point ( just like vilenkin s tunneling universe ) for making an infinite number of bubble universes , as this de sitter space has a finite beginning and then expands forever . now suppose _ one _ of these open bubble universes simply turns out to be the _ original _ open universe where that advanced civilization made the baby de sitter universe in the first place ( fig . now the model is multiply connected , with no earliest event . there is a cauchy horizon ( @xmath839 , see fig . [ f7 ] ) separating the region of ctcs from the later region that does not contain them . this cauchy horizon is generated by ingoing closed null geodesics that represent signals that could be sent toward the black hole , which then tunnel across the euclidean tunneling section jumping across the einstein - rosen bridge and then continuing as ingoing signals to enter the de sitter space and reach the open single bubble in the de sitter space ( that turns out to be the original bubble in which the tunneling event occurs ) . a retarded photon traveling around one of those closed null geodesics will be red - shifted more and more on each cycle , thus not causing an instability . another novel effect is that although these null generators are converging just before the tunneling event , they are diverging just after the tunneling event , having jumped to the other side of the einstein - rosen bridge . thus , converging rays are turned into diverging rays ( as in the wormhole solution ) during the tunneling event without violating the weak energy condition . these closed null geodesics need not be infinitely extendible in affine distance toward the past . it would seem that it can be arranged that the renormalized energy - momentum tensor does not blow up on this cauchy horizon so that a self - consistent solution is possible . using the method of images , note that the @xmath239-th image is from @xmath239 cycles around the multiply connected spacetime . the path connecting an observer to the @xmath239-th image will have to travel @xmath239 times through the hot big bang phase which occurs in the open bubble after the false vacuum with @xmath840 dumps its false vacuum energy into thermal radiation as it falls off the plateau and reaches the true vacuum @xmath841 . thus , to reach the @xmath239-th image one has to pass through the hot optically thick thermal radiation of the hot big bang @xmath239 times . and this will cause the contribution of the @xmath239-th image to the renormalized energy - momentum tensor to be exponentially damped by a factor @xmath842 where @xmath843 ( where @xmath250 is the number density of target particles , @xmath367 is the thickness of hot material , @xmath844 is the total cross - section ) . li @xcite has calculated the renormalized energy - momentum tensor of vacuum polarization with the effect of absorption . li @xcite has estimated the fluctuation of the metric of the background spacetime caused by vacuum polarization with absorption , which is a small number in most cases . if the absorption is caused by electron - positron pair production by a photon in a photon - electron collision , the maximum value of the metric fluctuation is @xmath845 , where @xmath846 is the planck length , @xmath847 is the classical radius of electron , @xmath696 is the spatial distance between the identified points in the frame of rest relative to the absorber @xcite . if we take @xmath696 to be the hubble radius at the recombination epoch ( @xmath848 cm ) , we have @xmath849 . thus , we expect that the renormalized energy - momentum tensor will not blow up at the cauchy horizon @xcite , so that a self - consistent solution is possible . the tunneling event is shown as the epoch indicated by the dashed line in fig . [ f7 ] . during the tunneling event , the trajectory may be approximated as a classical space with four spacelike dimensions solving einstein s equations , with the potential inverted , so that this euclidean section bridges the gap between the two classical turning points . ( in such a case , the concepts of ctcs and closed null curves should be generalized to contain a spacelike interval . thus , there are neither closed null geodesics nor closed timelike geodesics with the traditional definitions . according to li @xcite , this kind of spacetime can be stable against vacuum polarization . ) as farhi , guth , and guven @xcite note , the probability for forming such a universe is exponentially small , so an exponentially large number of trials would be required before an intelligent civilization would achieve this feat . if the metastable vacuum is at the planck density , the number of trials required is expected to be not too large ; but if it is at the gut density which turns out to be many orders of magnitude lower than the planck density , then the number of trials becomes truly formidable ( @xmath850 ) @xcite . thus , farhi , guth , and guven @xcite guess that it is unlikely that the human race will ever succeed in making such a universe in the lab at the gut scale . gott @xcite , applying the copernican principle to estimate our future prospects , would come to similar conclusions . however , if our universe is open , it has an infinite number of galaxies , and it would likely have some super - civilizations powerful enough to succeed at such a creation event , or at least have so many super - civilizations ( an infinite number ) that even if they each tried only a few times , then some of them ( again an infinite number ) would succeed . in fact , if the probability for a civilization to form on a habitable planet like the earth and eventually succeed at creating a universe in the lab is some finite number greater than zero ( even if it is very low ) , then our universe ( if it is an open bubble universe ) should spawn an infinite number of such baby universes . this notion has caused harrison @xcite to speculate that our universe was created in this way in the lab by some super - civilization in a previous universe . he noted correctly that if super - civilizations in a universe can create many baby universes , then baby universes created in this way should greatly outnumber the parent universes , and that you ( being not special ) are simply likely to live in one of the many baby universes , because there are so many more of them . here he is using implicitly the formulation of gott @xcite that according to the copernican principle , out of all the places for intelligent observers to be , there are , by definition , only a few special places and many non - special places , and you are simply more likely to be in one of the many non - special places . thus , if there are many baby universes created by intelligent supercivilizations in an infinite open bubble universe , then you are likely to live in a baby universe created in this way . harrison uses this idea to explain the strong anthropic principle . the strong anthropic principle as advanced by carter @xcite says that the laws of physics , in our universe at least , must be such as to allow the development of the intelligent life . because we are here . it is just a self - consistency argument . this might lead some to believe , particularly with inflationary cosmologies that are capable of producing an infinite number of bubble universes , that these different universes might develop with many different laws of physics , given a complicated , many - dimensional inflationary potential with many different minima , and many different low energy laws of physics . if some of these did not allow the development of intelligent life and some of these did , well , which type of universe would you expect to find yourself in ? one that allowed intelligent observers , of course . ( by the same argument , you are not surprised to find yourself on a habitable planet earth although such habitable planets may well be outnumbered by uninhabitable ones mercury , venus , pluto , etc . ) thus , there may be many more universes that have laws of physics that do not allow intelligent life you just would not find yourself living there . it has been noticed that there are various coincidences in the physical constants like the numerical value of the fine structure constant , or the ratio of the electron to proton mass , or the energy levels in the carbon nucleus which , if they were very different , would make intelligent life either impossible , or much less likely . if we observe such a coincidence , according to carter @xcite , it simply means that if it were otherwise , we would not be here . harrison @xcite has noted that if intelligent civilizations made baby universes they might well , by intelligent choice , make universes that purposely had such coincidences in them in order to foster the development of intelligent life in the baby universes they created . if that were the case , then the majority of universes would have laws of physics conducive to the formation of intelligent life . in this case , the reason that we observe such coincidences is that a previous intelligent civilization made them that way . one might even speculate in this scenario that if they were smart enough , they could have left us a message of sorts in these dimensionless numbers ( a theme that resonates , by the way , with part of carl sagan s thesis in _ contact _ ) . however , it is unclear whether any super - civilization would be able to control the laws of physics in the universes they created . all , they might reasonably be able to do would be to drive the baby universe up into a particular metastable vacuum @xcite . but then , such a metastable vacuum inflates in the knob , and an infinite number of bubble universes form later , with perhaps many different laws of physics depending on how they tunnel away from the metastable vacuum and which of the many potential minima they roll down into . controlling these phase transitions would seem difficult . thus , it would seem difficult for the super - civilization that made the metastable state that later gave rise to our universe to have been able to manipulate the physical constants in our universe . harrison s model could occur in many generations , making it likely that we were produced as great , great , ... , great grandchildren universes from a sequence of intelligent civilizations . harrison @xcite was able to explain all the universes by this mechanism except for the first one ! for that , he had to rely on natural mechanisms . this seems to be an unfortunate gap . in our scenario , suppose that `` first '' universe simply turned out to be one of the infinite ones formed later by intelligent civilizations . then the universe note capital u would be multiply connected , and would have a region of ctcs ; all of the individual universes would owe their birth to some intelligent civilization in particular in this picture . all this may overestimate the importance of intelligent civilizations . it may be that bubbles of inflating metastable vacuum are simply produced at late times in any big bang cosmology by natural processes , and that baby universes produced by natural processes may vastly outnumber those produced by intelligent civilizations . such a mechanism has been considered by frolov , markov , and mukhanov @xcite . they considered the hypothesis that spacetime curvature is limited by quantum mechanics and that as this limit is approached , the curvature approaches that of de sitter space . then , as any black hole collapses , the curvature increases as the singularity is approached ; but before getting there it will convert into a collapsing de sitter solution . this can be done in detail in the following way . inside the horizon , but outside the collapsing star the geometry becomes schwarzschild which is a radially collapsing but stretching cylinder . this can be matched onto a radially collapsing and radially shrinking cylinder in de sitter space as described by the metric in eq . ( [ e50 ] ) with the time @xmath267 being negative and the coordinate @xmath810 being unbounded rather than periodic . both surfaces are cylinders with identical intrinsic curvature , but with different extrinsic curvature . this mismatch is cured by introducing a shell of matter which converts the stretching of the schwarzschild cylinder to collapsing as well which then matches onto the collapsing de sitter solution . this phase transition may occur in segments which then merge as noted by barabes and frolov @xcite . the de sitter solution then bounces and becomes an expanding de sitter solution which can in turn spawn an infinite number of open bubble universes . this all happens behind the event horizon of the black hole . within the de sitter phase , one finds a cauchy horizon like the interior cauchy horizon of the reisner - nordstrom solution , but this inner cauchy horizon is not unstable because the curvature is bounded by the de sitter value so the curvature is not allowed to blow up on the inner horizon . ( this is an argument that one could also rely on to produce self - consistent multiply connected de sitter phases with ctcs if needed . ) this model thus produces , inside the black hole , to the future , and behind the event horizon , an expanding de sitter phase that has a beginning , just like vilenkin s tunneling universe . if one of those bubble universes simply turns out to be the original one in which the black hole formed , then the solution is multiply connected with a region of ctcs . this would make every black hole produce an infinite number of universes . this would be the dominant mechanism for making new bubble universes , since the number of black holes in our universe would appear to greatly outnumber the number of baby universes ever produced by intelligent civilizations , since the tunneling probability for that process to succeed is exceedingly small . smolin @xcite has proposed that this type of mechanism works and furthermore that the laws of physics ( in the bubble universes ) are like those in our own but with small variations . then , there would be a darwinian evolution of universes . universes that produced many black holes would have more children that would inherit their characteristics with some small variations . soon , most universes would have laws of physics that were fine - tuned to produce the maximum number of black holes . smolin @xcite points out that this theory is testable , since we can calculate whether small changes in the physical constants would decrease the number of black holes formed . in this picture we should be near a global maximum in the black hole production rate . one problem is that the laws of physics that maximize the number of black holes and those that simply maximize the number of main sequence stars may be rather similar , and the laws that maximize the number of main sequence stars might well simply maximize the number of intelligent observers , and the anthropic principle alone would suggest a preference for us observing such laws , even if no baby universes were created in black holes . another possible problem with this model , pointed out by rothman and ellis @xcite , is that if the density fluctuations in the early universe had been higher in amplitude , this would form many tiny primordial black holes ( presumably more black holes per comoving volume than in our universe ) , so , we well might wonder why the density fluctuations in our universe were so small . one way out might be that tiny black holes do not form any baby universes , but this seems a bit forced since the de sitter neck formed can be as small as the planck scale or gut scale and it would seem that even primordial black holes could be large enough to produce an infinite number of open bubble universes . another possibility is the recycling universe of garriga and vilenkin @xcite . in this model there is a metastable vacuum with cosmological constant @xmath851 , and a true lowest energy vacuum with a cosmological constant @xmath852 . @xmath851 is at the gut or planck energy scale , while @xmath852 is taken to be the present value of @xmath12 ( as might be the case in a flat-@xmath12 model ) . as long as @xmath853 , then garriga and vilenkin assert that there is a finite ( but small ) probability per unit four volume that the @xmath852 state could tunnel to form a bubble of @xmath851 state , which could therefore inflate , decaying into bubbles of @xmath852 vacuum , which could recycle forming @xmath851 bubbles , and so forth . they point out that depending on the coordinate system , a bubble of @xmath852 forming inside a @xmath851 universe could also be seen as a @xmath851 bubble forming inside of a @xmath852 universe . take two de sitter spaces , one with @xmath851 and one with @xmath852 , and cut each along a vertical slice ( @xmath854 ) in the embedding space . they can then be joined along an appropriate hyperbola of one sheet representing a bubble wall , with the @xmath852 universe lying to the @xmath855 side and the @xmath851 universe lying to the @xmath856 side . slicing along hyperplanes with @xmath857 gives a steady - state coordinate system for a @xmath851 universe in which a bubble of @xmath852 vacuum appears . slicing along hyperplanes with @xmath858 , however , gives a steady - state coordinate system for a @xmath852 universe in which a bubble of @xmath851 appears . so , one can find a steady - state coordinate system in which there is a @xmath851 universe , with bubbles of @xmath852 inside it , and bubbles of @xmath851 inside these @xmath852 bubbles , and so forth . if the roll down is slow , within the @xmath852 bubble as it forms , as in gott s open bubble universe @xcite , then it will have at least 67 @xmath28-folds of inflation with @xmath859 before it falls off the plateau into the absolute minimum at @xmath852 , and this will be an acceptable big bang model which will have the usual big bang properties except that it will eventually be dominated by a lambda term @xmath852 . being bubble universes , they will all be open with negative curvature as in gott s model @xcite but they will be asymptotically open de sitter models at late times with @xmath48 and @xmath860 . garriga and vilenkin @xcite wondered whether such a recycling model could be geodesically complete toward the past . such a outcome , they pointed out , would violate no known theorems and should be investigated . they hoped to find such a geodesically - complete - to - the - past model so that it could be eternal without a need for a beginning . however , in the special case , where @xmath861 , one can show that the recycling steady state solution becomes a simple single de sitter space geometry with @xmath851 and the usual steady - state coordinate system in a single de sitter space is not geodesically complete to the past . now take this recycling model where it turns out that one of the @xmath851 bubbles formed inside an @xmath852 bubble inside a @xmath851 region is , in fact , the @xmath851 region that one started out with . in this case , we would have a multiply connected model such as we are proposing which would include a region of ctcs ( fig . ( if @xmath861 , this model is just the multiply connected de sitter space we have considered . ) if our multiply connected model was geodesically complete to the past , so would the covering space ( a simply connected garriga - vilenkin model ) be . if our multiply connected model was geodesically incomplete to the past , so would the covering space ( a simply connected garriga - vilenkin model ) be also . in our model , there would be a strong self - consistency reason for pure retarded potential , whereas in the garriga - vilenkin recycling model , there would be no such strong reason for it . with pure retarded potentials throughout , the issue of whether the spacetime was geodesically complete to the past is less compelling , as we have argued above , and our model , having a periodic boundary condition , would not need further boundary conditions , unlike a simply connected recycling model that was geodesically incomplete to the past . thus , there are a number of models in which baby universes are created which can be converted into models in which the universe creates itself , if one of those created baby universes turns out to be the original universe that one started with . since these models are all ones in which there are an infinite number of baby universes created , this multiply connected outcome must occur unless the probability for a particular multiple connectivity to exist is exactly zero . in other words , it should occur , unless it is forbidden by the laws of physics . given quantum mechanics , it would seem that such multiple connectivities would not be absolutely forbidden , particularly in the planck foam era . we should note here that , in principle , there might even be solutions that are simply connected in which there was an early region of ctcs bounded to the future by a cauchy horizon followed by an inflationary region giving rise to an infinite number of bubble universes . the models considered so far have all obeyed the weak energy condition , and these models have all been multiply connected ; in other words , they have a genus of @xmath862 , like a donut , since one of the later baby universes is connected with the original one . consider an asymptotically flat spacetime with two connected wormhole mouths that are widely separated . the existence of the wormhole connection increases the genus by one . instead of a flat plane , it becomes a flat plane with a handle . to do this , the wormhole solution must violate the weak energy condition @xcite . it must have some negative energy density material , for it is a diverging lens ( converging light rays entering one wormhole mouth , diverge upon exiting the other mouth ) . for a compact two dimensional surface , the integrated gaussian curvature over the surface divided by @xmath863 is equal to @xmath862 minus the genus . thus , the integrated gaussian curvature over a sphere ( genus=0 ) is @xmath863 , while the integrated gaussian curvature over a donut ( genus=1 ) is zero , and the integrated gaussian curvature over a figure 8 pretzel ( genus=2 ) is @xmath864 . negative curvature is added each time the genus is increased . conversely , positive curvature can be added to reduce the genus by @xmath862 . when a donut is cut , so that it resembles a letter `` c '' , the ends of the letter `` c '' are sealed with positive curvature ( two spherical hemispherical caps would do the job , for example ) . our solutions are already multiply connected , so they might in principle be made simply connected by the addition of some extra positive mass density , without violating the weak energy condition . an example of this is seen by comparing grant space @xcite with gott s two - string spacetime @xcite . grant space is multiply connected , has @xmath865 everywhere , and includes ctcs . it can be pictured as a cylinder . gott s two - string spacetime is simply connected , but is identical to grant space at large distances from the strings . it also contains ctcs . it can be pictured as a cardboard cylinder that has been stepped on and then stapled shut at one end , like an envelope . there are two corners at the closed end , representing the two strings , but the cylinder continues outward forever toward its open end ( so it is like a test tube , a cylinder closed on one end ) . the two strings provide positive energy density ( i.e. they do not violate the weak energy condition ) . ctcs that wrap around the two strings far out in the cylinder ( which is identical to a part of grant space ; see laurence @xcite ) can be shrunken to points by slipping them through the strings but they become spacelike curves during this process . thus , gott space represents how a multiply connected spacetime with ctcs ( grant space ) can be converted into a simply connected spacetime with ctcs by adding to the solution material that obeys the weak energy condition . a similar thing might in principle be possible with these cosmological models . since our multiply connected versions already obey the weak energy condition , so would the associated simply connected versions . the question of first - cause has been a troubling one for cosmology . often , this has been solved by postulating a universe that has existed forever in the past . big bang models supposed that the first - cause was a singularity , but questions about its almost , but not quite , uniformity remained . besides , the big bang singularity just indicated a breakdown of classical general relativity , and with a proper theory - of - everything , one could perhaps push through to earlier times . inflation has solved some of these problems , but borde and vilenkin have shown that if the initial inflationary state is metastable , then it must have had a finite beginning also . ultimately , the problem seems to be how to create something out of nothing . so far , the best attempt at this has been vilenkin s tunneling from nothing model and the similar hartle - hawking no - boundary proposal . unfortunately , tunneling is , as the name suggests , usually a process that involves tunneling from _ one _ classical state to _ another _ , thus , with the wheeler - dewitt potential and `` energy '' @xmath866 that hartle and hawking adopted , the universe , we argue , should really start not as nothing but as an @xmath1 universe of radius zero a point . a point is as close to nothing as one can get , but it is not nothing . also , how could a point include the laws of physics ? in quantum cosmology , the wave function of the universe is treated as the solution of a schrdinger - like equation ( the wheeler - dewitt equation ) , where the three - sphere @xmath1 radius @xmath21 is the abscissa and there is a potential @xmath154 with a metastable minimum at @xmath867 , and a barrier with @xmath868 for @xmath869 , and @xmath870 for @xmath871 . thus , the evolution can be seen as a particle , representing the universe , starting as a point , @xmath127 , at the bottom of the metastable potential well , with @xmath866 . then it tunnels through the barrier and emerges at @xmath872 with @xmath866 , whereupon it becomes a classically inflating de sitter solution . it can then decay via the formation of open single bubble universes @xcite . the problem with this model is that it ignores the `` zero - point - energy '' . if there is a conformal scalar field @xmath90 , then the `` energy '' levels should be @xmath873 . even for @xmath157 there is a `` zero - point - energy '' . the potential makes the system behave like a harmonic oscillator in the potential well near @xmath127 . a harmonic oscillator can not sit at the bottom of the potential well the uncertainty principle would not allow it . there must be some zero - point - energy and the particle must have some momentum , as it oscillates within the potential well when the field @xmath90 is included . thus , when the `` zero - point - energy '' is considered , we see that the initial state is not a point but a tiny oscillating ( @xmath874 ) big bang universe , that oscillates between big bangs and big crunches ( though the singularities at the big bangs and big crunches might be smeared by quantum effects ) . this is the initial classical state from which the tunneling occurs . it is metastable , so this oscillating universe could not have existed forever : after a finite half - life , it is likely to decay . it reaches maximum radius @xmath875 , and then tunnels to a classical de sitter state at minimum radius @xmath876 where @xmath877 . the original oscillating universe could have formed by a similar tunneling process from a contracting de sitter phase , but such a phase would have been much more likely to have simply classically bounced to an expanding de sitter phase instead of tunneling into the oscillating metastable state at the origin . in this case , if one found oneself in an expanding de sitter phase , it would be much more likely that it was the result of classical bounce from a contracting de sitter phase , rather than the result of a contracting de sitter phase that had tunneled to an oscillating phase and then back out to an expanding de sitter phase . besides , a contracting de sitter phase would be destroyed by the formation of bubbles which would percolate before the minimum radius was ever reached . in this paper , we consider instead the notion that the universe did not arise out of nothing , but rather created itself . one of the remarkable properties of the theory of general relativity is that in principle it allows solutions with ctcs . why not apply this to the problem of the first - cause ? usually the beginning of the universe is viewed like the south pole . asking what is before that is like asking what is south of the south pole , it is said . but as we have seen , there remain unresolved problems with this model . if instead there were a region of ctcs in the early universe , then asking what was the earliest point in the universe would be like asking what is the easternmost point on the earth . there is no easternmost point you can continue going east around and around the earth . every point has points that are to the east of it . if the universe contained an early region of ctcs , there would be no first - cause . every event would have events to its past . and yet the universe would not have existed eternally in the past ( see fig . thus , one of the most remarkable properties of general relativity the ability in principle to allow ctcs would be called upon to solve one of the most perplexing problems in cosmology . such an early region of ctcs could well be over by now , being bounded to the future by a cauchy horizon . we construct some examples to show that vacuum states can be found such that the renormalized energy - momentum tensor does not blow up as one approaches the cauchy horizon . for such a model to work the universe has to reproduce at some later time the same conditions that obtained at an earlier time . inflation is particularly useful in this regard , for starting with a tiny piece of inflating state , at later times a huge volume of inflating state is produced , little pieces of which look just like the one we started with . many inflationary models allow creation of baby inflationary universes inside black holes , either by tunneling across the einstein - rosen bridge , or by formation as one approaches the singularity . if one of these baby universes simply turns out to be the universe we started with , then a multiply connected model with early ctcs bounded by a cauchy horizon is produced . since any closed null geodesics generating the cauchy horizon must circulate through the optically thick region of the hot big bang phase of the universe after the inflation has stopped , the renormalized energy - momentum tensor should not blow up as the cauchy horizon is approached . as a particularly simple example we consider a multiply connected de sitter solution where events e@xmath878 are topologically identified with events e@xmath879 that lie inside these future light cones via a boost transformation . if the boost @xmath337 , we show that we can find a rindler - type vacuum where the renormalized energy - momentum tensor does not blow up as the cauchy horizon is approached but rather produces a cosmological constant throughout the spacetime which self - consistently solves einstein s equations for this geometry . thus , it is possible to find self - consistent solutions . when analyzing classical fields in this model , the only self - consistent solution without a blow up as the cauchy horizon is approached occurs when there is a pure retarded potential in the causally connected region of the model . thus , the multiply connected nature of this model and the possibility of waves running into themselves , ensure the creation of an arrow of time in this model . this is a remarkable property of this model . interestingly , this model , although having no earliest event and having some timelike geodesics that are infinitely extendible to the past , is nevertheless geodesically incomplete to the past . this is not a property we should have thought desirable , but since pure retarded potentials are established automatically in this model , there are no waves propagating to the past and so there may be no problem in physics with this , since there are never any waves that run off the edge of the spacetime . the region of ctcs has a finite four - volume equal to @xmath880 and should be in a pure vacuum state containing no real particles or hawking radiation and no bubbles . after the cauchy horizon for a certain amount of proper time ( depending on the bubble formation probability per four volume @xmath70 ) no bubbles ( or real particles ) form , but eventually this model expands to infinite volume , creating an infinite number of open bubble universes , which do not percolate . at late times in the de sitter phase a particle detector would find the usual hawking radiation just as in the usual vacuum for de sitter space . there are a number of problems to be solved in this model . the chronology projection conjecture proposes that the laws of physics conspire so as to prevent the formation of ctcs . this conjecture was motivated by hiscock and konkowski s result that the energy - momentum tensor of the adapted minkowski vacuum in misner space diverges as the cauchy horizon is approached . but as we have shown @xcite , the adapted rindler vacuum for misner space has @xmath881 throughout the space if @xmath337 ; thus , this is a self - consistent vacuum for this spacetime since it solves einstein s equations for this geometry . it s true that @xmath882 remains formally ill - defined on the cauchy horizon itself [ @xmath348 in eq . ( [ e69 ] ) with @xmath337 ] , a set of measure zero . but it is not clear that this creates a problem for physics , since continuity might require that this formally ill - defined quantity be defined to be zero on this set of measure zero as well , since it is zero everywhere else . in fact , a treatment in the euclidean section shows this is the case , for in the euclidean section , if @xmath337 , @xmath881 everywhere , including at @xmath348 . other counter - examples to the chronology protection conjecture have also been found , as discussed in section v. hawking himself has also admitted that the back - reaction of vacuum polarization does not enforce the chronology protection conjecture . one of the remarkable properties of general relativity is that it allows , in principle , the formation of event horizons . this appears to be realized in the case of black holes . just as black hole theory introduced singularities at the end , standard big bang cosmology introduced singularities at the beginning of the universe . now , with inflation , we see that event horizons should exist in the early universe as well @xcite . inflationary ideas prompt the suggestion that baby universes may be born . if one of the baby universes simply turns out to be the one we started with , then we get a model with an epoch of ctcs that is over by now , bounded toward the future by a cauchy horizon . we have argued that the divergence of the energy - momentum tensor as one approaches the cauchy horizon does not necessarily occur , particularly when the cauchy horizon crosses through a hot big bang phase where absorption occurs . if the energy - momentum tensor does not diverge as the cauchy horizon is approached , other problems must still be tackled . the classical instability of a cauchy horizon to the future ( a future chronology horizon ) in a spacetime with ctcs is one . but this problem is solved in a world with retarded potentials for a cauchy horizon that occurs to our past ( a past chronology horizon ) and which ends an epoch of ctcs . it thus seems easier to have a cauchy horizon in the early universe . at the microscopic level , quantum mechanics appears to allow acausal behavior . indeed the creation and annihilation of a virtual positron - electron pair can be viewed as creation of a small closed loop , where the electron traveling backward in time to complete the loop appears as a positron . so , why should the laws of physics forbid time travel globally ? indeed one of the most remarkable properties of the laws of physics is that although they are time ( cpt ) symmetric , the solutions we observe have an arrow of time and retarded potentials . without this feature of the solutions , acausal behavior would be seen all the time . interestingly , in our model , the multiply connected nature of the spacetime geometry forces an arrow of time and retarded potentials . thus , it is the very presence of the initial region of ctcs that produces the strong causality that we observe later on . this is a very interesting and unexpected property . an entropy arrow of time is automatically produced as well , with the region of ctcs in the simplest models sitting automatically in a cold vacuum state , with the universe becoming heated after the cauchy horizon . recently , cassidy and hawking @xcite have proposed yet another supposed difficulty for ctcs , in that the formally defined entropy appears to diverge to negative infinity as the cauchy horizon is approached . yet , in the early universe this may turn out to be an advantage , since to produce the ordinary entropy arrow of time we observe in the universe today , we must necessarily have some kind of natural low - 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the question of first - cause has troubled philosophers and cosmologists alike . now that it is apparent that our universe began in a big bang explosion , the question of what happened before the big bang arises . inflation seems like a very promising answer , but as borde and vilenkin have shown , the inflationary state preceding the big bang could not have been infinite in duration it must have had a beginning also . where did it come from ? ultimately , the difficult question seems to be how to make something out of nothing . this paper explores the idea that this is the wrong question that _ that _ is not how the universe got here . instead , we explore the idea of whether there is anything in the laws of physics that would prevent the universe from creating itself . because spacetimes can be curved and multiply connected , general relativity allows for the possibility of closed timelike curves ( ctcs ) . thus , tracing backwards in time through the original inflationary state we may eventually encounter a region of ctcs giving _ no _ first - cause . this region of ctcs may well be over by now ( being bounded toward the future by a cauchy horizon ) . we illustrate that such models with ctcs are _ not _ necessarily inconsistent by demonstrating self - consistent vacuums for misner space and a multiply connected de sitter space in which the renormalized energy - momentum tensor does not diverge as one approaches the cauchy horizon and solves einstein s equations . some specific scenarios ( out of many possible ones ) for this type of model are described . for example : a metastable vacuum inflates producing an infinite number of ( big - bang - type ) bubble universes . in many of these , either by natural causes or by action of advanced civilizations , a number of bubbles of metastable vacuum are created at late times by high energy events . these bubbles will usually collapse and form black holes , but occasionally one will tunnel to create an expanding metastable vacuum ( a baby universe ) on the other side of the black hole s einstein - rosen bridge as proposed by farhi , guth , and guven . one of the expanding metastable - vacuum baby universes produced in this way simply turns out to be the original inflating metastable vacuum we began with . we show that a universe with ctcs can be stable against vacuum polarization . and , it can be classically stable and self - consistent if and only if the potentials in this universe are retarded which gives a natural explanation of the arrow of time in our universe . interestingly , the laws of physics may allow the universe to be its own mother . # 1@xmath0#1

the deviation of an active region ( ar ) magnetic field from its potential configuration can arise due to photospheric footpoint motions and/or during flux emergence . however , studies with the high - quality data from space and ground based observatories over the last decade suggest that the latter is more dominant mechanism @xcite . the deviation of magnetic field from potential configuration is called non - potentiality ( np ) of the field . the magnetic energy of an active region ( ar ) in excess of the potential magnetic energy is called free magnetic energy @xcite . the free magnetic energy of an ar , or a portion of it , is released when flare and/or coronal mass ejections ( cmes ) is triggered due to the instability or non - equilibrium @xcite . the energy released in the flare is then limited by the amount of free energy or magnetic np of the ar . therefore , it is important to characterize the np of the ar magnetic field in order to predict the intensity of the flare and/or cme . traditionally , the so - called magnetic shear i.e. , the angle between the observed ( @xmath1 ) and the potential ( @xmath2 ) field azimuths , is used to characterize the np of the active region magnetic field and is measured as @xmath3 where @xmath4 and @xmath5 are the observed and potential transverse field vectors . the term magnetic shear " , used here and in similar studies on solar - magnetism , really refers to observational shear " , which is not to be confused with the actual shear " a microscopic property in fluid dynamics . in what follows , we will use the term magnetic shear for traditional reasons and by this term we would always mean the observational shear . various forms of this parameter are used namely , mean shear @xcite , weighted mean shear @xcite , spatially averaged signed shear @xcite , most probable shear @xcite etc . it is well known that the polarity inversion line ( pil ) in active regions bearing highly sheared magnetic fields are the potential sites for flares @xcite . these pils are generally characterized by dark filaments in the images taken in chromospheric hydrogen alpha line @xcite . it may be noted that all of the shear parameters mentioned above measure the twist - shear component of the shear which is measured in the horizontal plane . however , one can also measure the shear of a magnetic field in the vertical plane . this type of shear can be called as the dip - shear " ( we choose the term dip " because it is synonymous to the dip angle measured in geomagnetism ) , and can be measured by taking the difference between the observed ( @xmath6 ) and the potential field inclination angle ( @xmath7 ) i.e. , @xmath8 . physically , the dip - shear can be understood in terms of azimuthal currents , in the same way as the twist - shear is understood in terms of axial currents . in our knowledge , the dip - shear has not been studied earlier in solar active regions . henceforth , we will call the parameter @xmath9 and @xmath10 and as twist - shear and dip - shear , respectively . the larger the value of these angles the larger will be the np of the ar . it may be noticed that , unlike twist - shear , the dip - shear is not affected by the 180 degree azimuth ambiguity , provided that the active region is observed close to the disk center . this is because the dip - shear depends upon the inclination angle of the magnetic field which can be measured without ambiguity @xcite . in this _ letter _ , we study the evolution of twist - shear and dip - shear in a penumbral region located close to the flaring site in ar noaa 10930 . we use sequence of high - quality vector magnetograms observed by the _ hinode _ space mission . this active region was in a @xmath0-sunspot configuration which led to a x3.4 flare and a large cme during 02:20 ut on 13 december 2006 . the flare was quite powerful and the white light flare ribbons along with impulsive lateral motion of the penumbral filaments were observed @xcite . evolution of the twist - shear and dip - shear together shows interesting patterns which can be distinguished in the pre - flare and post - flare stages . in general we find that ( a ) the regions with high twist - shear also exhibit high dip - shear , ( b ) the penumbral region close to the flare site shows high twist - shear and dip - shear , and ( c ) twist - shear and dip - shear studied together can be used to study flare related changes in the active regions . the paper is organized as follows . the observational data and the methods of analysis are described in section 2 . the results are presented in section 3 and the discussions and conclusions are made in section 4 . a sunspot with @xmath0 configuration was observed in ar noaa 10930 during 12 - 13 december 2006 by the spectro - polarimeter ( sp ) instrument @xcite with solar optical telescope ( sot ) @xcite onboard _ _ satellite @xcite . the sp obtains the stokes profiles , simultaneously in fe i 6301.5 and 6302.5 line pair . the spectro - polarimetric maps of the active region are made by scanning the slit across the field - of - view . this takes about an hour to complete one scan . we choose a sequence of six sp scans from 12 december 2006 03:50 ut to 13 december 2006 16:21 ut when the sunspot was located close to the disk center with heliocentric distance ( @xmath11 ) of 0.99 and 0.97 , respectively . the scans were taken in fast map " observing mode with following characteristics : ( i ) field - of - view ( fov ) 295 x 162 arc - sec , ( ii ) integration time of 1.8 seconds and ( iii ) pixel - width across and along the slit of 0.32 and 0.29 arc - sec , respectively . the stokes profiles were then fitted to an analytic solution of unno - rachkovsky equations @xcite under the assumptions of local thermodynamic equilibrium ( lte ) and milne - eddington model atmosphere @xcite with a non - linear least square fitting code called _ helix _ @xcite . the physical parameters of the model atmosphere retrieved after inversion are the magnetic field strength , its inclination and azimuth , the line - of - sight velocity , the doppler width , the damping constant , the ratio of the line center to the continuum opacity , the slope of the source function and the source function at @xmath12 = 0 . we fit a single component model atmosphere along with a stray light component . the inversion code _ helix _ is based upon a reliable genetic algorithm @xcite . this algorithm , although slower , is more robust than the classical levenberg - marquardt algorithm in the sense that the global minimum of the merit function is reached with higher reliability @xcite . the vector magnetograms obtained after inversion , were first solved for 180 degree azimuth ambiguity by using the acute angle method @xcite and then transformed from the observed frame ( image plane ) to local solar frame ( heliographic plane ) using the procedure described in @xcite . the potential field was computed from the line - of - sight field component by using the fourier transform method @xcite . the idl routine used for potential field computation is ` fff.pro ` which is available in the nlfff ( non - linear force free field ) package of the solarsoft library . the continuum intensity images of the sunspot , corresponding to the sequence of scans used , are shown in figure 1 , with the transverse field vectors overlaid on it . the two magnetograms were aligned using the cross - correlation technique applied to the continuum image of the sunspot . a black rectangle is overlaid on these images to show the location of the region where we study the evolution of twist - shear and dip - shear . the location of this black rectangle is chosen with the help of a co - aligned g - band filtergram observed from hinode filtergraph ( fg ) instrument during flare . this g - band image is shown in the left panel of figure 2 . the flare ribbon is marked by ` + ' symbols and the black rectangle of figure 1 is shown here also . the flare ribbons sweep across the rectangular box during 02:20 to 02:26 ut . this indicates that the rectangle is chosen such that it samples the penumbra which is very close to the flaring site . the right panel of figure 2 shows the longitudinal magnetogram in order to indicate the location of rectangle ( shown here with white color ) with respect to the pil . the maps of dip - shear @xmath10 and twist - shear @xmath13 for the sequence of vector magnetograms are shown in figure 3 and 4 respectively . in figure 5 we show the distribution of dip - shear @xmath14 and twist - shear @xmath9 inside the black rectangle and its evolution with time . the maps of dip - shear @xmath10 for the entire sequence of vector magnetograms covering the pre - flare ( panels ( a)(c ) ) and post - flare ( panels ( d)(f ) ) phases are shown in figure 3 . the value of field inclination @xmath15 is measured with respect to local solar vertical direction and ranges from 0 to 180 degrees . for purely vertical positive ( negative ) polarity field the value of @xmath15 corresponds to 0 degrees ( 180 degrees ) . the black rectangle in figure 3 corresponds to negative polarity field . therefore , the positive value of dip - shear @xmath10 inside this rectangle means that the observed field is more vertical than potential field . the magnitude of dip - shear @xmath10 can be judged with the aid of colorbar at the bottom of figure 3 . it may be noticed that : + ( i ) the value of dip - shear @xmath10 is large inside the rectangle as compared to other penumbral locations . + ( ii ) in the pre - flare ( panels ( a)(c ) ) phase the dip - shear @xmath10 consistently has a large magnitude which decreases in the post - flare phase ( panels ( d)(f ) ) . the field azimuth has been solved for 180 degree azimuth ambiguity by using the acute angle method @xcite and the projection effects have been removed by the application of vector transformation @xcite . this azimuth ambiguity resolution method works well in the regions where the angle @xmath9 is less or greater than 90@xmath16 . for regions where @xmath9 reaches value close to 90@xmath16 such as along parts of polarity inversion line ( pil ) in flaring active regions , the accuracy of the method is poor . this is why we choose a rectangular box for studying the evolution of twist - shear and dip - shear sample the penumbra close to flaring site and at the same time stay away from the pil where the acute angle method may have problems in resolving the azimuth ambiguity . the maps of twist - shear @xmath13 for the entire sequence of vector magnetograms covering the pre - flare ( panels ( a)(c ) ) and post - flare ( panels ( d)(f ) ) phases are shown in figure 4 . the value of field azimuth @xmath17 is measured with respect to the positive x - axis and is positive in the anti - clockwise direction . the magnitude of twist - shear @xmath13 can be judged with the aid of colorbar at the bottom of figure 4 . however , for the present study the sign of shear angle is not important , so we shall focus on its magnitude . it may be noticed that the value of twist - shear @xmath13 is large inside the rectangle and adjacent pil as compared to other penumbral locations . however , the flare related changes are not so discernable to eye as compared to dip - shear in figure 3 . in figure 5 we show the scatter between the dip - shear @xmath14 and twist - shear @xmath9 for the pixels within the black rectangle shown in previous figures . the panels ( a ) to ( c ) correspond to pre - flare while the panels i.e. , ( d ) to ( f ) correspond to post - flare phase . it may be noticed that : + ( i ) the distribution of dip - shear @xmath14 and twist - shear @xmath9 in panels ( a)-(c ) is different from the distribution in panels ( d)-(f ) . + ( ii ) the dip - shear increases before the flare ( panels ( a)-(c ) ) but twist - shear tends to decrease at the same time . + ( iii ) the dip - shear and twist - shear are in general correlated , i.e. the pixels with large dip - shear also have large twist - shear . + ( iv ) the most important change can be noticed after the flare , i.e. , between panels ( c ) and ( d ) . after the flare , ( panel ( d ) ) the dip - shear decreases significantly while twist - shear increases . however , now both shear components show less dispersion i.e. , follow a tight correlation . + ( v ) in panels ( d)-(f ) the two shears maintain smaller dispersion but dip - shear starts to increase once again . this increase suggests that the non - potentiality was building up again in the active region . it may be noted that the flaring activity continued in this region on the next day i.e. , 14 december 2006 also , with another x - class flare occurring at about 22:00 ut . @xcite conjectured that the free - energy is stored in non - potential magnetic loops that are stretched upwards and the free - energy release during the flare must be accompanied by sudden shrinkage or implosion in the field . also , it is predicted that after the flare the field should become more horizontal @xcite . using coronal images during flares , there are observational reports about detection of the loop contraction during flares @xcite . further , it was shown by @xcite that in force - free fields a high non - potentiality implies weaker magnetic tension , which in turn implies a larger vertical extension of the field due to lower magnetic pressure gradient . conversely , the release of free magnetic energy during flare implies a loss of magnetic non - potentiality leading to a decrease in the vertical extension of the field or shrinkage @xcite . the non - linear force - free field ( nlfff ) extrapolations of the noaa 10930 active region by @xcite show the non - potentiality of this active region in the form of a twist flux rope structure . as suggested by @xcite and @xcite , such a structure will have larger vertical extension in pre - flare as compared to the post - flare configuration . the closer the post - flare field approaches to the potential field configuration the smaller is the value of inclination difference @xmath14 expected . this may give an explanation for the decrease of dip - shear @xmath14 after the flare . however , in contrast , the increase in the twist - shear @xmath18 after the flare also needs an explanation . the opposite behaviour of twist - shear and dip - shear in relation to the flare can be understood in the following way . the twist - shear is dependent on sub - photospheric / photospheric forces , so the twist - shear will continue to increase independent of coronal processes like flare . however , the plasma @xmath19 decreases rapidly above the photosphere and thus there is no non - magnetic force or shear that is strong enough to change the inclination of the field lines . hence inclination will be more responsive to coronal processes . this may explain why inclination became more potential after the flare . hence , dip - shear could be a better diagnostic of np above the photosphere . in summary , we studied the evolution of twist - shear and dip - shear in a flaring @xmath0-sunspot using a sequence of high - quality vector magnetograms spanning the pre - flare and post - flare phases and found that : ( i ) the penumbra located close to the flaring site has high twist - shear and dip - shear as compared to other parts of the penumbra , ( ii ) the twist - shear increases after the flare which was earlier reported by @xcite also , ( iii ) the dip - shear however shows a decrease after the flare , ( iv ) the twist - shear and dip - shear are correlated , i.e. pixels with high twist - shear exhibit high dip - shear , and this correlation is much tighter after the flare , and ( v ) distribution of twist - shear and dip - shear and its evolution ( in figure 5 ) clearly shows different patterns before and after the flare . this type of behaviour in the twist - shear and dip - shear parameters will need to be evaluated further in more flares before it can be understood physically . we plan to carry out more extensive study of the dip - shear and twist - shear in existing _ _ datasets . however , a high - cadence study of these shear parameters would be possible only with the upcoming observations from helioseismic and magnetic imager ( hmi ) onboard solar dynamics observatory ( sdo ) @xcite . the present study is important in the sense that it points the way to a vector - field follow - up to the results of @xcite , which established the line - of - sight field changes during powerful flares . in the context of present study , one should bear in mind that the vector magnetograms derived from the _ hinode _ sot / sp scans , although polarimetrically very precise are very noisy geometrically . an unwanted consequence of the geometric noise could be that the flows , specially on long time scales , would tend to create an appearance of non - potentiality , even if there was none . this is an important issue which needs to be addressed sooner than later , considering the widespread use of sot / sp magnetic maps as the vector magnetograms " . we plan to carry out a detailed study of this effect using simultaneously observed spectro - polarimeter ( sp ) scan from _ hinode _ sot and vector - magnetograms from hmi onboard sdo . we thank the anonymous referee for his / her valuable comments and suggestions , specially for pointing out the geometric noise in sot / sp scans and its side effects . we thank dr . ron moore and dr . pascal d@xmath20moulin for reading the manuscript . hinode is a japanese mission developed and launched by isas / jaxa , collaborating with naoj as a domestic partner , nasa and stfc ( uk ) as international partners . scientific operation of the hinode mission is conducted by the hinode science team organized at isas / jaxa . this team mainly consists of scientists from institutes in the partner countries . support for the post - launch operation is provided by jaxa and naoj ( japan ) , stfc ( u.k . ) , nasa , esa , and nsc ( norway ) . we also would like to thank dr . andreas lagg for providing his helix code used in this study . alissandrakis , c. e. 1981 , , 100 , 197 charbonneau , p. 1995 , , 101 , 309 dmoulin , p. , mandrini , c. h. , van driel - 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shear and dip - shear . ] -sunspot in noaa ar 10930 during 12 december 2006 02:30 ut , with the location of the flare ribbon marked by ` + ' symbols . the flare ribbons sweep across the rectangular box during 02:20 to 02:26 ut . the right panel shows the map of the longitudinal field component for this sunspot . the black ( white ) rectangle in left ( right ) panel marks the region where we monitor the evolution of twist - shear and dip - shear . , title="fig:"]-sunspot in noaa ar 10930 during 12 december 2006 02:30 ut , with the location of the flare ribbon marked by ` + ' symbols . the flare ribbons sweep across the rectangular box during 02:20 to 02:26 ut . the right panel shows the map of the longitudinal field component for this sunspot . the black ( white ) rectangle in left ( right ) panel marks the region where we monitor the evolution of twist - shear and dip - shear . , title="fig : " ]

the non - potentiality ( np ) of the solar magnetic fields is measured traditionally in terms of magnetic shear angle i.e. , the angle between observed and potential field azimuth . here , we introduce another measure of shear that has not been studied earlier in solar active regions , i.e. the one that is associated with the inclination angle of the magnetic field . this form of shear , which we call as the dip - shear " , can be calculated by taking the difference between the observed and potential field inclination . in this _ letter _ , we study the evolution of dip - shear as well as the conventional twist - shear in a @xmath0-sunspot using high - resolution vector magnetograms from _ hinode _ space mission . we monitor these shears in a penumbral region located close to flare site during 12 and 13 december 2006 . it is found that : ( i ) the penumbral area close to the flaring site shows high value of twist - shear and dip - shear as compared to other parts of penumbra , ( ii ) after the flare the value of dip - shear drops in this region while the twist - shear in this region tends to increase after the flare , ( iii ) the dip - shear and twist - shear are correlated such that pixels with large twist - shear also tend to exhibit large dip - shear , and ( iv ) the correlation between the twist - shear and dip - shear is tighter after the flare . the present study suggests that monitoring twist - shear during the flare alone is not sufficient but we need to monitor it together with dip - shear .

holding a fishing rod at one end ( @xmath0 ) and starting from the rod at rest , is it possible to achieve any desired displacement at the other ( @xmath1 ) by prescribing the motion of the held end ? this problem has a long history and is of practical relevance for the control of flexible robotic manipulators , a field of research that has attained a high degree of mathematical sophistication . the modest objective of this note , on the other hand , is to shed some light on the feasibility of solutions of the problem from the point of view of the elementary theory of partial differential equations . the analysis is conducted within the realm of small deflections of a timoshenko beam . two separate situations are envisioned . in the first , that we call the 2 - 2 problem , to achieve any desired displacement and rotation functions at the free end it is required to prescribe both the displacement and the rotation of the held end . since initially the rod is at rest , the motion at the free end will start with a delay dictated by the slower speed of wave propagation in the rod . the existence of a solution to this problem is argued on the basis of the existence of a solution to the cauchy problem when the roles of time and space are interchanged . the second situation , called the 1 - 1 problem , consists of attaining a desired displacement evolution at the free end while keeping the displacement of the held end fixed and prescribing only its rotation . the solution in this case is based upon the solution of a recursive functional equation . under fairly general assumptions , namely , 1 . cross sections remain plane , though not necessarily perpendicular to the deformed axis of the beam , 2 . both translational and rotational inertia terms are included 3 . transverse deflections @xmath2 are very small compared with the length of the beam 4 . the material abides by hooke s law timoshenko derived the dynamic equations of an ideal elastic beam , which bears his name . for the free motion ( no external forces ) of a beam of constant cross section and uniform material properties , timoshenko s equations can be expressed in terms of the transverse deflection @xmath2 and the cross - section rotation @xmath3 as @xmath4 and @xmath5 where @xmath6 are positive constants and where subscripts indicate partial derivatives with respect to the natural body - time coordinates @xmath7 . it is possible to eliminate @xmath3 between the two equations so as to obtain a single fourth - order pde for the transverse displacement . the result is @xmath8 where @xmath9 . the constants @xmath10 , @xmath11 and @xmath12 represent , respectively , relative measures of the shear compliance , rotational inertia and translational inertia . in the limiting case when the stiffness compliance and the rotational inertial approach zero values , we recover the classical bernoulli beam equation . an interesting intermediate case is that for which only the shear compliance vanishes . it is also possible to express the equations of motion as a symmetric system of four first - order pdes . the original formulation in terms of two second - order equations , however , is most suitable for the imposition of physically meaningful boundary conditions . the characteristic polynomial associated with this differential equation is @xmath13 the roots of this polynomial are @xmath14 and @xmath15 discarding the unlikely case in which @xmath16 , . ] equation ( [ beam1 ] ) is a totally hyperbolic pde , with @xmath17 as the four distinct characteristic speeds . physically , @xmath18 and @xmath19 correspond , respectively , to the forward and backward bending waves , while @xmath20 and @xmath21 pertain to the shear waves . notice that in the case of the bernoulli beam all roots @xmath22 of the characteristic polynomial vanish , which indicates an infinite speed of propagation of all signals . in the intermediate case of a bernoulli beam endowed with rotational inertia ( @xmath23 ) , bending waves propagate at a finite speed . the bernoulli beam equation stands with respect to the timoshenko beam in a relation somewhat analogous to the relation between the heat equation and the wave equation . the fact that all characteristic roots vanish implies not merely the loss of hyperbolicity but also that , the characteristic line being purely spatial , the initial value problem specifies initial data on a characteristic line , just as in the case of the classical heat equation . although considerations of well - posedness can be derived for these non - hyperbolic characteristic - initial problems , the theory for totally hyperbolic problems is better established . for this reason , we will limit our considerations to the timoshenko beam and adduce physical arguments to claim that the results obtained for a flexible manipulator based on the timoshenko beam are also applicable to the bernoulli beam . let a @xmath24-th order totally hyperbolic linear pde for a function @xmath2 of the two independent variables @xmath25 and @xmath26 be given . consider a smooth line @xmath27 in the @xmath7 plane with the property of not being anywhere tangent to a characteristic direction . assume , moreover , that on this line we have stipulated sufficiently regular values of the function @xmath2 and of its derivatives up to and including the order @xmath28 in a direction transversal to @xmath27 . ) and ( [ beam0 ] ) the data to be stipulated are the two functions @xmath2 and @xmath3 and their first derivatives in a direction transversal to @xmath27 . ] the main result of the theory states that the solution at a point @xmath29 is completely and uniquely determined by the given data within the _ domain of dependence _ of @xmath29 obtained as the portion of @xmath27 comprised between the two extreme intersections with the characteristic lines through @xmath29 . for an equation with constant coefficients , the characteristics constitute @xmath24 independent systems of parallel straight lines . figure [ fig : domain ] shows the typical picture of the domain of dependence for the timoshenko beam equation . ( -2,0 ) to ( 10,0 ) ; ( 0,-2 ) to ( 0,10 ) ; at ( 10,0 ) @xmath25 ; at ( 0,10 ) @xmath26 ; ( -2,1 ) to [ out=30,in=190 ] ( 10,5 ) ; at ( 10,5 ) @xmath27 ; at ( 4,7.5 ) @xmath29 ; ( 4,7.5 ) to ( 1,2.5 ) ; ( 4,7.5 ) to ( 6.1,4 ) ; ( 4,7.5 ) to ( 2.5,3 ) ; ( 4,7.5 ) to ( 5.2,3.9 ) ; at ( 1,2.5 ) @xmath30 ; at ( 6.1,4)@xmath31 ; if the data and the solution are confined between two other non - characteristic lines , as shown in figure [ fig : boundary ] for the typical case of boundary conditions at the two ends of a finite beam , then additional conditions need to be specified for the function and its transverse derivatives on those lines . these conditions , however , can not be prescribed arbitrarily . it can be shown that the number of conditions to be imposed on any such line must be equal to the number of intersections with this line of the characteristic lines issuing from @xmath29 toward @xmath27 . in the case of the timoshenko beam , this implies that at each boundary exactly 2 boundary conditions must be specified . if 3 conditions were specified at one end and only 1 at the other end , this situation would in general imply an inconsistency of the initial data ( overdetermined near one end and underdetermined near the other ) . ( -2,0 ) to ( 10,0 ) ; ( 0,-2 ) to ( 0,10 ) ; ( 6,0 ) to ( 6,10 ) ; at ( 10,0 ) @xmath25 ; at ( 0,10 ) @xmath26 ; ( 0,0 ) to ( 6,0 ) ; at ( 3,0 ) @xmath27 ; at ( 4,8 ) @xmath29 ; ( 4,8 ) to ( 0,4 ) ; ( 4,8 ) to ( 6,6 ) ; ( 4,8 ) to ( 0,6 ) ; ( 4,8 ) to ( 6,7 ) ; at the free end ( @xmath1 ) we have zero shear force and zero bending moment . the corresponding boundary conditions are @xmath32 and @xmath33 suppose now that the beam has been at rest for all times @xmath34 and that for some time @xmath35 we want to have that @xmath36 and @xmath37 where @xmath38 and @xmath39 are sufficiently regular functions defined for all @xmath26 and vanishing identically for @xmath40 . the 2 - 2 problem consists of finding two inputs ( such as @xmath2 and @xmath3 ) at the held end so that the desired outputs ( @xmath41 and @xmath42 ) are obtained at the free end while starting from vanishing initial conditions at all @xmath34 . the proof that the solution to this problem exists and the actual procedure to construct it are based on the fact that the line @xmath1 is non - characteristic . interchanging the roles of space and time , the problem effectively becomes an ` initial ' value ( cauchy ) problem . indeed , since we have specified 4 independent data on this line , according to the fundamental theorem we can obtain a unique solution for the region @xmath43\times \mathbb{r}$ ] . this solution vanishes identically in the lower trapezoidal subregion shaded in figure [ fig : twoplustwo ] , namely the region below the ` slow ' characteristic line issuing from the point @xmath44 . in particular , we obtain well - defined functions @xmath45 and @xmath46 which vanish for all @xmath47 . the initial - boundary - value problem with the initial conditions @xmath48 and with the boundary conditions ( [ beam5 ] ) and ( [ beam6 ] ) at @xmath1 and @xmath49 is a well - posed problem . its unique solution must perforce satisfy the desired displacement @xmath41 and rotation @xmath42 at @xmath1 . ( 6,4 ) ( 0,-2 ) ( 0,-4 ) ( 6,4 ) ; ( -2,0 ) to ( 10,0 ) ; ( 0,-4 ) to ( 0,10 ) ; ( 6,-4 ) to ( 6,10 ) ; at ( 10,0 ) @xmath25 ; at ( 0,10 ) @xmath26 ; at ( 6,4 ) @xmath44 ; at ( 0,-2 ) @xmath50 ; at ( 6,0 ) @xmath51 ; ( 6,4 ) to ( 0,-2 ) ; ( 6,4 ) to ( 0,1 ) ; it is convenient to obtain a non - dimensional version of the equations of motion ( [ beam-1 ] ) and ( [ beam0 ] ) . defining the non - dimensional variables @xmath52 by @xmath53 we obtain @xmath54 and @xmath55 where @xmath56 and @xmath57 is a measure of the slenderness ratio of the beam . in most applications ( that is , when the rotational inertia equals the mass density times the moment of inertia and when the shear stiffness is governed by a shear factor @xmath24 ) we obtain @xmath58 and @xmath59 where @xmath60 is poisson s ratio and @xmath61 is the radius of gyration of the cross section . interchanging the roles of space and time , we give as ` initial ' conditions on the line @xmath62 the following desired deflection @xmath63 where @xmath64 is the heaviside step function , a vanishing rotation ( for lack of a better choice ) , namely , @xmath65 and zero shear and bending moment as stipulated by conditions ( [ beam5 ] ) and ( [ beam6 ] ) . notice that we have chosen our desired deflection at the free end ( @xmath62 ) so that it starts to manifest itself , according to equation ( [ beam20 ] ) , at @xmath66 . to run this problem in a commercial software , we stipulate artificial vanishing ` boundary ' conditions of displacement and rotation at ` time boundaries ' located far enough from the domain of interest ( at @xmath67 in our example ) . for the values @xmath68 , the solution was obtained with mathematica^^. the code is shown in figure [ fig : mathcode ] . notice that the non - dimensional wave speeds are @xmath69 and @xmath70 , respectively , for the shear and the bending waves . accordingly , the non - dimensional time for starting to impose the displacement and rotation on the held end ( @xmath71 ) , should be exactly equal to @xmath72 . and this is indeed the case in the numerical solution ( which does not make use of characteristics ) , as can be observed in the graphs . the solution converges for a very wide range of values of the slenderness parameter @xmath73 . on intuitive grounds it seems that , if the proverbial fishing rod is hinged to a support so as to permit only the variation of the angle at the held end , it should be possible to achieve any desired displacement at the free end provided one does not insist on imposing the rotation thereat . we call this the 1 - 1 problem . the intuitive reasoning is likely driven by a principle of conservation of the number of data necessary to solve a problem . it seems reasonable to conclude that all we are doing is exchanging the imposition of a desired rotation at the free end with a zero displacement at the other end . in the 2 - 2 case , the solution was attainable because we managed to convert this problem into a 0 + 4 formulation , namely , when all 4 data were available along a single non - characteristic line ( @xmath62 ) . this is the so - called cauchy problem , whose solution is guaranteed . to solve the 1 - 1 problem , on the other hand , we would have to resort to a 1 + 3 formulation . in other words , we would have to specify 3 data on the line @xmath62 ( namely , the desired displacement and the vanishing of the bending moment and the shear force ) and 1 on the line @xmath71 ( namely , the vanishing of the displacement ) . thus , it appears that , as we already pointed out in section [ sec : wellposedness ] , there will be a conflict between the initial conditions and the boundary conditions on either side of the rod . that this conflict may be resolvable should not come as a surprise , since we already encountered a similar conflict by imposing all 4 boundary conditions on a single end in the solution of the 2 - 2 problem . the key to the resolution resides in the fact that an arbitrary time delay is at our disposal . the details in the 1 - 1 case , however , are more subtle and perhaps less convincing . let a point @xmath29 be located on the line @xmath0 , as shown in figure [ fig : theoneoneproblem ] . if data are to be specified on the line @xmath1 , the domain of dependence thereat is dictated by the segment between the intersections , @xmath30 and @xmath31 , of this line with the two slower characteristics issuing form @xmath29 . ( -2,0 ) to ( 10,0 ) ; ( 0,-2 ) to ( 0,10 ) ; ( 6,0 ) to ( 6,10 ) ; at ( 10,0 ) @xmath25 ; at ( 0,10 ) @xmath26 ; ( 0,0 ) to ( 6,0 ) ; at ( 3,0 ) @xmath74 ; ( 0,5 ) to ( 6,8 ) ; ( 0,5 ) to ( 6,2 ) ; ( 0,5 ) to ( 6,6.5 ) ; ( 0,5 ) to ( 6,3.5 ) ; at ( 0,5 ) @xmath75 ; at ( 6,8 ) @xmath76 ; at ( 6,2 ) @xmath77 ; at ( 6,6.5 ) @xmath78 ; at ( 6,3.5 ) @xmath79 ; the value of the solution at @xmath29 , for the case of the homogeneous equation , depends exclusively on the data @xmath80 for @xmath81 . of these data , we have specified the conditions of vanishing bending moment ( @xmath82 ) and vanishing shear force ( @xmath83 ) . moreover , we also specify the function @xmath84 under the condition that it vanishes identically for all times prior to some value @xmath85 , just as in the 2 - 2 problem . we have left the rotation @xmath86 unspecified . the value of @xmath2 at point @xmath29 , however , is known to vanish identically for all times ( pinned end ) . consequently , we have a functional equation for the missing function @xmath86 . differentiating this equation with respect to @xmath26 , we may obtain a more explicit expression in terms of the values of the functions @xmath84 and @xmath86 at the 4 points @xmath87 only . the argument just presented , which will be illustrated in the next section by means of an example , should be strengthened . in particular , the issue of continuous dependence of the solution ( i.e. , the rotation ) on the boundary data ( i.e. , the displacement ) deserves special attention . to obtain an explicit solution of the general 1 - 1 problem we would need a corresponding explicit expression of the solution of the cauchy problem with ` initial ' data on the line @xmath88 . this expression not being available for general values of the parameters @xmath10 , @xmath11 and @xmath12 in equation ( [ beam1 ] ) , we will solve the problem only for the case @xmath89 , namely , for the equation @xmath90 whose physical meaning is of less practical interest , but whose characteristic polynomial is identical to that of the general case . from the point of view of the original pair of equations ( [ beam-1 ] ) and ( [ beam0 ] ) , we have a weaker coupling resulting from neglecting the shear term in the second equation , that is , @xmath91 and @xmath92 the explicit solution of the cauchy problem for these equations under the ` initial ' conditions stipulated by equations ( [ beam5 ] , [ beam6 ] , [ beam7 ] , [ beam8 ] ) is given by @xmath93 and @xmath94 where we have shifted the origin to @xmath1 by introducing the variable @xmath95 we still need to determine the function @xmath96 . to achieve this aim , we set the deflection to zero at @xmath97 , giving us the desired recursive functional equation . more explicitly , setting @xmath98 we obtain @xmath99 having thus determined the missing ` initial ' condition , we have a legitimate cauchy problem which can be solved for @xmath2 and @xmath3 . the resulting function @xmath100 should clearly vanish , while the function @xmath101 provides the desired angle to be prescribed at the held end to produce the desired deflection at the free end . notice that the fact that , by definition , the desired displacement @xmath102 vanishes identically for all @xmath103 , guarantees the existence of a unique solution of equation ( [ beam37 ] ) , provided we impose the same condition on @xmath96 . similar considerations apply to the original timoshenko equations , except for the fact that the recursion formula is not available explicitly .

the existence of solutions to the boundary tracking of the displacement at one end of a linear timoshenko beam is discussed on the basis of the cauchy problem with time and space interchanged .

the launch of the wfc3/ir camera in 2009 signified a major milestone in our ability to observe galaxies within the cosmic reionization epoch at @xmath16 . thanks to its @xmath17 times higher efficiency for detecting galaxies in the near - infrared ( nir ) compared to previous cameras on the hubble space telescope ( hst ) we have pushed the observational frontier to within only @xmath18 myr from the big bang . in its first year of operation wfc3/ir resulted in the detection of @xmath19 new galaxies at @xmath20 ( see e.g. * ? ? ? * ) . three years of science operations of wfc3/ir and several deep extra - galactic surveys have now resulted in a large sample of more than 200 galaxies in the reionization epoch , primarily at @xmath21 and @xmath2 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) from these samples it has become clear that the build - up of galaxies during the first gyr was a gradual process at @xmath22 . the end of reionization left no noticeable imprint on the galaxy population ( at least down to the current limits of @xmath23 corresponding to a star - formation rate of sfr@xmath24yr@xmath11 ) . the build - up of the uv luminosity function ( lf ) progresses smoothly across the @xmath25 reionization boundary , following a constant trend all the way from @xmath2 to @xmath26 . galaxies typically become brighter by @xmath27 per unit redshift accompanied by a proportional ( but somewhat larger ) increase in the average star - formation rate of galaxies ( see e.g. * ? ? ? * ; * ? ? ? . given the large samples of galaxies discovered at @xmath28 , the current observational frontier is at @xmath3 and at earlier times . this is a period when significant evolution of the galaxy uv lf is expected from models ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the observational evidence has been suggestive of a significant drop on the uv luminosity density ( ld ) at @xmath0 , but has not been conclusive @xcite . as a result , the extent to which the luminosity function and the star formation rate density are evolving at @xmath29 has been the subject of some debate ( see , e.g. , * ? ? ? * ; * ? ? ? * ) . at these early epochs , current galaxy samples are still very small as hst is approaching its limits . initially , only one @xmath30 galaxy candidate was identified ( udfj-39546284 ) , even in extremely deep wfc3/ir imaging of the hubble ultra deep field ( hudf ) as part of the hudf09 survey @xcite . when combined with all the existing data over the chandra deep field south ( cdfs ) , this one source suggested that the galaxy population is changing quickly , building up very rapidly from @xmath4 to @xmath2 . in galaxies with sfr @xmath31yr@xmath11 ( equivalent to @xmath23 mag ) , the inferred uv luminosity density ( ld ) was found to increase by more than an order of magnitude in only @xmath15 myr from @xmath4 to @xmath2 @xcite . this is a factor @xmath32 times larger than what would have been expected from a simple extrapolation of the lower redshift trends of the uv lf evolution to @xmath4 . several datasets have allowed us to improve these first constraints . the multi - cycle treasury program clash ( pi : postman ) has provided four sources at @xmath33 @xcite . in particular , the detections of three @xmath3 galaxies around clash clusters by @xcite have provided a valuable estimate of the luminosity density in this key redshift range . estimating volume densities from highly - magnified sources found behind strong lensing clusters is challenging , involving systematic uncertainties due to the lensing model . @xcite used a novel technique of comparing the @xmath3 source counts to those at @xmath2 in the same clusters and obtained a good relative luminosity density estimate . since the @xmath34 density is well - established from the field ( e.g. * ? ? ? * ; * ? ? ? * ) , this gave a more robust measure than trying to infer source densities directly using lensing models . interestingly , the three @xmath35 candidates from @xcite are completely consistent with the observed drop in the uv luminosity density and an accelerated evolution of the galaxy population that was previously seen at @xmath36 in the hudf and cdfs by @xcite . two other high redshift sources detected in the clash dataset ( one of which is in common with and proceeded the sample of * ? ? ? * ) have added to the available constraints . @xcite and @xcite discovered two highly magnified @xmath30 ( @xmath37 and @xmath38 ) galaxies in the analysis of the clash cluster data . the luminosity densities inferred from these galaxies are somewhat higher , but are very uncertain . the large errors on the luminosity density from these two detections encompass a wide range of possible trends from @xmath34 to earlier times . however , as we show later in this paper , taken together , the sources from the clash dataset along with the latest sources and constraints from the hudf / cdfs region , are consistent with our earlier estimates of substantial accelerated change in the luminosity density from @xmath4 to @xmath2 . additional progress in exploring the galaxy population at @xmath0 has been made through gamma - ray burst ( grb ) afterglow observations . the current record holder of an independently confirmed redshift measurement was achieved at @xmath39 for grb090423 @xcite , and grb redshifts were photometrically measured out to @xmath40 @xcite . these measurements can provide additional constraints on the total star - formation rate density in the very early universe , since grb rates are thought to be an unbiased tracer of the total star - formation rate density ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? while the initial results are encouraging , it is clear that our understanding of the galaxy population at @xmath29 is still far from complete . given the very small number of sources in each study , it is perhaps not surprising that the uv luminosity density measurements at @xmath29 are currently all within @xmath41 of each other . a next step forward in exploring the @xmath33 universe can now be taken thanks to the 128 orbit hudf12 campaign ( pi : ellis , go12498 ) . while the critical @xmath42 observations to discover @xmath4 galaxies only reach deeper by @xmath430.2 mag compared to the previous hudf09 image , the hudf12 survey adds deep f140w ( @xmath44 ) imaging . this allows for lyman break galaxy sample selections at @xmath3 and @xmath45 ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? lcccccccccc hudf12/xdf & 4.7 & 29.8 & 30.3 & 30.4 & 29.1 & 29.4 & 29.7 & 29.7 & 29.7 & 29.8 + hudf09 - 1 & 4.7 & & 29.5 & 29.3 & & 29.3 & 29.0 & 29.2 & & 29.0 + hudf09 - 2 & 4.7 & 29.5 & 29.9 & 29.5 & & 29.2 & 29.0 & 29.2 & & 29.3 + ers & 41.3 & 28.4 & 28.7 & 28.2 & 28.5 & 28.0 & 27.8 & 28.2 & & 28.0 + goodss - deep & 63.1 & 28.4 & 28.7 & 28.2 & 29.0 & 28.1 & 28.3 & 28.5 & & 28.3 + goodss - wide & 41.9 & 28.4 & 28.7 & 28.2 & 28.5 & 28.0 & 27.5 & 27.7 & & 27.5 in a first analysis of their proprietary hudf12 data , @xcite compiled a sample of six extremely faint @xmath46 galaxy candidates based on a photometric redshift technique . one of these sources was already in an earlier @xmath47 sample of @xcite based on the hudf09 data set . @xcite also re - analyzed our previously detected @xmath4 candidate udfj-39546284 @xcite . from the three years of wfc3/ir @xmath42 data , it is completely clear now that xdfjh-39546284 is a real source as it is significantly detected in all three major sets of data taken in 2009 , 2010 and 2012 @xcite . very surprisingly , however , udfj-39546284 appears not to be detected in the new f140w image , indicating that this source is either a very extreme emission line galaxy at @xmath48 or that it lies at @xmath49 with the spectral break of the galaxy at @xmath431.6@xmath50 ( see also * ? ? ? * ) . with all the hudf12 data publicly available , we can now extend our search for @xmath33 galaxies to even deeper limits and to higher redshifts than previously possible . in this paper , we perform a search for @xmath51 galaxies over the hudf based on the lyman break technique . this makes use of the fact that the hydrogen gas in the universe is essentially neutral at @xmath20 , which results in near - complete absorption of rest - frame uv photons short - ward of the redshifted ly@xmath52 line . star - forming galaxies at @xmath20 can therefore be selected as blue continuum sources which effectively disappear in shorter wavelength filters . in section [ sec : lbgvszphot ] we outline our reasons to use a lyman break selection instead of photometric redshift selection , as is frequently adopted to identify very high - redshift galaxies in the literature , e.g. , in @xcite . this paper is organized as follows : we start by describing the data used for this study in section [ sec : data ] and define our source selection criteria in section [ sec : selection ] , where we also present our @xmath33 galaxy candidates . these are subsequently used to constrain the evolution of the uv luminous galaxy population out to @xmath1 in section [ sec : results ] , where we present our results . in section [ sec : summary ] , we summarize and discuss further possible progress in this field before jwst . throughout this paper , we will refer to the hst filters f435w , f606w , f775w , f850lp , f098 m , f105w , f125w , f140w , f160w as @xmath53 , @xmath54 , @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath44 , @xmath42 , respectively . we adopt @xmath60 kms@xmath11mpc@xmath11 , i.e. @xmath61 . magnitudes are given in the ab system @xcite . ab mag in several bands . the parallel fields hudf09 - 1 and hudf09 - 2 ( also dark red ) are only @xmath62 mag shallower . the wider area data covering the whole goods - s field are from the ers ( yellow ) and the candels programs ( orange ) . all these fields include imaging in @xmath58 ( or @xmath57 over the ers ) , as well as @xmath59 and @xmath42 , which makes it possible to search for @xmath4 galaxies . the hudf12/xdf field is additionally covered by very deep @xmath44 imaging , which we exploit to select @xmath3 lyman break galaxies and obtain some of the first limits on the galaxy population at @xmath1 . ] the core dataset of this paper is the combination of ultra - deep acs and wfc3/ir imaging over the hudf09/hudf12/xdf field . we enhance this deep dataset by using wfc3/ir and acs data over both hudf09 parallel fields , as well as all candels and ers data over the goods - south field ( see figure [ fig : fields ] ) . these datasets provide valuable constraints on the more luminous sources , particularly by providing limits over a larger area than is covered by the small hudf09/hudf12 field . all these datasets include @xmath59 and @xmath42 imaging in addition to deep , multi - band optical acs data , which allows for reliable @xmath4 galaxy selections ( see section [ sec : z10 ] ) . all the wfc3/ir and acs data are reduced following standard procedures . we subtract a super median image to improve the image flatness and we register to the goods acs frames . for wfc3/ir data we mask pixels affected by persistence using the maps provided by stsci . the acs data are corrected for charge transfer losses when necessary using the public code provided by stsci . all images are drizzled to a final pixel scale of @xmath63 , and the rms maps are rescaled to match the actual flux fluctuations in circular apertures of 035 diameter , dropped down on empty sky positions in the images . the spatial resolution of the data is @xmath64 and @xmath65 for the acs and wfc3/ir data , respectively . the individual datasets used for our analysis are described in more detail in the following sections . they are furthermore summarized in table [ tab : data ] and are shown in figure [ fig : fields ] . the hubble ultra - deep field ( hudf ; * ? ? ? * ) was imaged with wfc3/ir as part of two large hst programs now . the hudf09 ( pi : illingworth ; * ? ? ? * ) provided one pointing ( 4.7 arcmin@xmath66 ) of deep imaging in the three filters @xmath58 ( 24 orbits ) , @xmath59 ( 34 orbits ) , and @xmath42 ( 53 orbits ) . these data were extended recently with the hudf12 campaign @xcite , which imaged the hudf further in @xmath58 ( 72 orbits ) and @xmath42 ( 26 orbits ) , and additionally added a deep exposure in @xmath44 ( 30 orbits ) . these are the deepest nir images ever taken , resulting in a final 5@xmath67 depth of @xmath68 mag ( see also table [ tab : data ] ) . since the acquisition of the original optical hudf acs data , several programs have added deeper acs coverage to this region , mainly as part of parallel imaging . we combined all the available acs data over the hudf , which allows us to improve the backgrounds and also to push photometry limits deeper by @xmath69 mag . these data , along with the matched wfc3/ir data from all programs , will be released publicly as the extreme deep field ( xdf ) and are discussed in more detail in illingworth et al . ( in preparation ) . for longer wavelength constraints we also include the ultra - deep spitzer / irac data from the 262 h iudf program ( pi : labb ; see also * ? ? ? * ) , which reach to @xmath70 mag ab ( 5@xmath67 total ) in both [ 3.6 ] and [ 4.5 ] channels . these data are extremely important for eliminating lower redshift contaminating sources , particularly intermediate redshift dusty and/or evolved galaxies ( see section [ sec : dustycontamin ] ) . in addition to the hudf data , we also include the two additional deep parallel fields from the hudf09 program , as well as all the wfc3/ir data over the goods south field . the latter were taken as part of the early release science ( ers ) program @xcite and the multi - cycle treasury campaign candels ( pi : faber / ferguson ; * ? ? ? * ; * ? ? ? these data were already used for a @xmath4 search in our previous analysis from @xcite . we therefore refer the reader to that paper for a more detailed discussion . however , since our previous analysis the acquisition of an additional 4 epochs of candels deep data was completed , resulting in deeper data by about 0.2 mag . these are included now in this paper , which will allow us to further tighten our constraints on the @xmath4 lf . in the optical , we make use of all acs data taken over the goods south field , which includes additional imaging from supernova follow - up programs . these images reach @xmath71 mag deeper than the v2.0 reductions of goods , in particular in the @xmath56-band . we also reduce and include all the @xmath72 data , which was taken over this field . by combining all these datasets we have produced what is the deepest optical image to date over the goods - s field . such deep optical data is very important for excluding lower redshift interlopers in lbg samples . for constraints from spitzer / irac , we use the public data from the goods campaign . these exposures are 23 h deep and reach to @xmath4326 mag ( m. dickinson et al . in prep ) . all these fields are also outlined in figure [ fig : fields ] . . due to the high neutral fraction in the igm , essentially all photons shortward of the redshifted ly@xmath52 line are absorbed for galaxies at @xmath20 . this effect is used to select such high redshift sources based on broad - band photometry . the redshift of ly@xmath52 is indicated on the top axis . the vertical black dotted lines indicate the location of the break at @xmath73 and 10.5 . ] source catalogs are obtained with sextractor @xcite , which is run in dual image mode with a specific detection image , depending on the galaxy sample we are interested in . for all samples at @xmath74 , we use a @xmath75 detection image @xcite based on the @xmath44 and @xmath42 bands . for @xmath76 @xmath44-dropout selections we use the @xmath42-band for source detection . all images are matched to the same psf when performing photometry measurements . colors are based on small kron apertures ( kron factor 1.2 ) , typically 02 radius , while magnitudes are derived from large apertures using the standard kron factor of 2.5 , typically 04 radius . an additional correction to total fluxes is performed based on the encircled flux measurements of stars in the @xmath42 band to account for flux loss in the psf wings . this correction depends on the size of individual galaxies and is typically @xmath77 mag . in this paper , we adopt a lyman break galaxy selection to identify galaxies at @xmath78 . the major advantages of this approach over a photometric redshift selected sample as is used , e.g. , in @xcite are simplicity and robustness . the lyman break technique provides a straightforward and robust selection , which is easily reproducible by other teams ( e.g. * ? ? ? furthermore , the simplicity of the lyman break color - color criteria also allows for a straightforward estimate of the selection volumes based on simulations . in contrast , the photometric redshift likelihood functions are heavily dependent on the assumed template set and even on the specific photometric redshift code that is used . additionally , the photometric redshift likelihood functions depend on largely unknown priors which are needed to account for the number density of intermediate redshift passive or dusty galaxies . a particular problem is that most high - redshift photometric analyses give equal weight to all templates at all redshift ( i.e. they adopt a flat prior ) . this includes faint galaxies with extreme dust extinction at intermediate redshift or passive sources at @xmath79 , which are unlikely to be very abundant in reality . a further uncertainty , in particular for high redshift sources , is how undetected fluxes are treated in the fitting process . this can have significant influence on the lower redshift likelihood estimates . given all these advantages , we will therefore select high - redshift galaxies using the lyman break technique and we will determine their photometric redshifts a posteriori using standard template fitting on this pre - selected sample of lbgs . the addition of deep f140w imaging data over the hudf gives us the ability to select new samples of @xmath3 galaxies over that field . as can be seen in figure [ fig : filters ] , the absorption due to the inter - galactic neutral hydrogen shifts in between the @xmath58 and @xmath59 filters at @xmath33 . for a robust lyman break selection , we thus combine the @xmath58 and @xmath59 filter fluxes in which galaxies start to disappear at @xmath3 . our adopted selection criteria are : @xmath80 @xmath81 @xmath82 these criteria ( shown in figure [ fig : colsel ] ) are chosen to select sources at @xmath83 . we additionally use a @xmath84 criterion to cleanly distinguish our @xmath3 and @xmath4 samples ( see next section ) . we only include sources which are significantly detected in the @xmath42 and @xmath44 images with at least 3@xmath67 in each filter and with @xmath85 in at least one of the two . as a cross - check we selected sources based on an inverse - variance weighted combination of the @xmath59 , @xmath44 , and @xmath42 images at @xmath86 . both selections resulted in the same final sample of high - redshift sources , i.e. all selected candidates are @xmath87 detections . in addition to the color selections , we require sources to be undetected at shorter wavelengths . in particular , we use @xmath88 non - detection criteria in all optical bands individually . furthermore , we adopt an optical pseudo-@xmath75 constraint . this is defined as @xmath89 . the summation runs over all the optical filter bands @xmath53 , @xmath54 , @xmath55 , @xmath72 , and @xmath56 , and sgn is the sign function , i.e. @xmath90 if @xmath91 and @xmath92 if @xmath93 . this measure allows us to make full use of all information in the optical data . we only consider galaxies with @xmath94 . this cut reduces the contamination rate by a factor @xmath8 , while it only reduces the selection volume of real sources by 20% ( see also * ? ? ? this is a powerful tool for providing source lists with low contamination rates ( see also section [ sec : scattersim ] ) . these selection criteria result in seven @xmath3 galaxy candidates in the hudf12/xdf dataset . these sources are listed in table [ tab : phot ] and their images are shown in figure [ fig : stampszgtr8 ] . in figure [ fig : sedfits ] , we additionally show the spectral energy distribution ( sed ) fits and redshift likelihood functions for these sources . for comparison , these are derived from two photometric redshift codes , zebra @xcite as well as eazy @xcite . as is evident , the vast majority of sources does show a prominent peak at @xmath95 together with a secondary , lower likelihood peak at @xmath48 . the best - fit photometric redshifts of these candidates range between @xmath96 , with the exception of one source ( xdfyj-39446317 ) , which has a zebra photometric redshift of only @xmath97 . however , using the eazy code and template set the best - fit redshift is found at @xmath98 . the photometric redshift likelihood function for this source is very wide using both codes . from photometric scatter simulations ( see section [ sec : scattersim ] ) , we expect to find @xmath99 low redshift contaminants in our @xmath3 sample due to photometric uncertainties . therefore , finding a lbg candidate with such a low redshift is not necessarily unexpected . we will thus list it as possible candidate in table [ tab : phot ] however , we will exclude xdfyj-39446317 from our determination of the uv lf at @xmath3 . galaxies at redshifts approaching @xmath4 start to disappear in the @xmath59 filter . following @xcite and @xcite , we select @xmath4 galaxies based on very red @xmath100 colors and we use spitzer / irac photometry to guard this selection against intermediate redshift extremely dusty and evolved galaxies in a second step . this selection process also used @xmath44 data when available ( i.e. over the hudf12/xdf field ) , and was used for all the datasets shown in figure [ fig : fields ] . the hst selection criteria are : @xmath101 @xmath102 in addition to at least 3@xmath67 detections in both @xmath42 and @xmath44 and @xmath103 in one of these . all sources in our final list also satisfy a @xmath87 detection criterion in the combined @xmath104 image . the @xmath105 color criterion was introduced to distinguish @xmath4 from @xmath1 galaxies over the hudf12/xdf field . the other fields , which do not have deep @xmath44 imaging , do not include this criterion . we account for this difference in our analysis of the selection functions ( section [ sec : selfun ] ) . when applying these selection criteria to the wfc3/ir+acs data over goods - s , we previously identified 16 galaxies which satisfied these criteria . however , these are all extremely bright in the spitzer / irac bands and are even detectable in the shallow [ 5.8 ] and [ 8.0 ] channel data over goods - s ( having @xmath106=2.4 - 4.0 $ ] mag ) . these sources were therefore excluded from our @xmath4 analysis , as their @xmath42 to irac colors were too red for a genuine @xmath4 galaxy . these are most likely @xmath107 galaxies with significant extinction and possibly evolved stellar populations ( see * ? ? ? even taking advantage of the deeper wfc3/ir data that became available over the candels - south field subsequent to the @xcite analysis , no new credible @xmath4 source could be found . however , our selection revealed three potential sources in the hudf12/xdf data . unfortunately , two of these are very close to a bright , clumpy foreground galaxy . their photometry is therefore very uncertain , and it is unlikely that they are real high - redshift sources . we nevertheless list these as potential sources in table [ tab : phot ] . however , we will not use them in the subsequent analysis . this leaves us with only one likely @xmath4 galaxy candidate in all the fields we have analyzed here . this is xdfj-38126243 , which we had previously identified in the first - epoch data of the hudf09 as a potential @xmath4 candidate @xcite . however , it was not detected at a significant enough level in the subsequent second epoch @xmath42 data to indicate at high confidence that it was real . as a result it was not included in our final sample of @xmath4 sources from the hudf09 data . the source xdfj-38126243 is now clearly detected both in the new @xmath42 and in the @xmath44 data from the hudf12 survey , which clearly confirms its reality . this is demonstrated in figure [ fig : epochstamps ] . as can also be seen from that figure , the source is extremely compact , consistent with being a point source . we can therefore not exclude that this source is powered by an agn , which could also explain the possible variability over a timescale of 1 year ( see lower panel of fig . [ fig : epochstamps ] ) . however , the low flux measurement in the second - year hudf09 data is still consistent with expectations from gaussian noise . taken together , the flux measurements of all three epochs are consistent with the source showing no time variability ( @xmath108 ) . the source is not detected in our ultra - deep irac data , and its colors place it at a photometric redshift of @xmath109 ( see also table [ tab : phot ] and figure [ fig : sedfitz10 ] ) . for completeness , we also list three additional potential candidates in the appendix in table [ tab : additional ] . these sources lie very close to bright foreground galaxies . formally , they show colors consistent with being at very high redshift . however , they are significantly blended with their neighbors , such that the fluxes of these sources can not be accurately measured with sextractor , without a more sophisticated neighbor subtraction technique . additionally , the close proximity to very bright foreground sources casts significant doubt on the reality of these sources , and we therefore do not include these in our analysis . candidate xdfj-38126243 which is likely the highest - redshift source over the hudf12/xdf . the top two stamps are the @xmath44 and @xmath42 observations . the @xmath42 stack is also split in observations at three different epochs , shown in the bottom three stamps . ` epoch 1 ' corresponds to the hudf09 year 1 data ( 28 orbits ) , ` epoch 2 ' are the hudf09 year 2 data ( 25 orbits ) , and ` epoch 3 ' are the remaining 31 orbits from the hudf12 and candels observations . the s / n in each band is listed in the lower left . this source was initially selected as high - redshift candidate after the first year hudf09 data @xcite . however , as can be seen , the galaxy was only very weakly detected ( 1.4@xmath67 ) in the hudf09 year 2 data . nevertheless , the source is clearly visible at @xmath110 in all other epochs , as well as in the @xmath44 data ( only taken from the hudf12 program ) . the source is therefore clearly real . the lower signal detection in the ` epoch 2 ' data is still consistent with the expectation from a gaussian noise distribution . _ bottom _ the @xmath42 magnitude measurement for the three different epochs . the number of orbits going into each image is indicated close to each datapoint . the fluxes are measured in a circular aperture of 035 diameter . the magnitude from the total 84-orbit @xmath42 image is indicated by the gray line , with errorbars represented by the filled gray area . the flux measurements are consistent with no variability in this source ( @xmath108 ) . however , an agn contribution to its uv flux can not be excluded.,title="fig : " ] candidate xdfj-38126243 which is likely the highest - redshift source over the hudf12/xdf . the top two stamps are the @xmath44 and @xmath42 observations . the @xmath42 stack is also split in observations at three different epochs , shown in the bottom three stamps . ` epoch 1 ' corresponds to the hudf09 year 1 data ( 28 orbits ) , ` epoch 2 ' are the hudf09 year 2 data ( 25 orbits ) , and ` epoch 3 ' are the remaining 31 orbits from the hudf12 and candels observations . the s / n in each band is listed in the lower left . this source was initially selected as high - redshift candidate after the first year hudf09 data @xcite . however , as can be seen , the galaxy was only very weakly detected ( 1.4@xmath67 ) in the hudf09 year 2 data . nevertheless , the source is clearly visible at @xmath110 in all other epochs , as well as in the @xmath44 data ( only taken from the hudf12 program ) . the source is therefore clearly real . the lower signal detection in the ` epoch 2 ' data is still consistent with the expectation from a gaussian noise distribution . _ bottom _ the @xmath42 magnitude measurement for the three different epochs . the number of orbits going into each image is indicated close to each datapoint . the fluxes are measured in a circular aperture of 035 diameter . the magnitude from the total 84-orbit @xmath42 image is indicated by the gray line , with errorbars represented by the filled gray area . the flux measurements are consistent with no variability in this source ( @xmath108 ) . however , an agn contribution to its uv flux can not be excluded.,title="fig : " ] , which is represented by the blue line . the gray sed corresponds to the best low - redshift solution at @xmath111 . open blue squares and gray circles show the expected magnitudes of these seds . 1@xmath67 upper limits to undetected fluxes are shown as red arrows . the inset shows the redshift likelihood function as estimated with zebra ( blue ) and eazy ( orange ) . both photometric redshift codes consistently find a prominent peak at @xmath112 and a lower - significance peak at @xmath113 . ] the addition of @xmath44 imaging from the hudf12 data also allows for the selection of @xmath114 galaxies based on a red @xmath105 color , as the igm absorption shifts through the center of the @xmath44 filter ( see figure [ fig : colsel ] ) . we therefore search for galaxies in the hudf12/xdf data satisfying the following : @xmath115 @xmath116 in order to ensure the reality of sources in this single - band detection sample we require @xmath87 detections in @xmath42 . only one source satisfies these criteria : xdfj-39546284 . this is our previous highest redshift candidate from the hudf09 data ( see * ? ? ? surprisingly , with @xmath117 it has an extremely red color in these largely overlapping filters . it is by far the reddest source in the hudf12/xdf data , and its rather dramatic color raises some questions . at face value , the extreme color , together with the non - detection in the optical data and in our deep irac imaging , results in a best - fit redshift of @xmath118 for this source . however , such a high redshift would imply that the source is @xmath119 brighter than expected for @xmath49 sources at the same number density ( see * ? ? ? while a strong ly@xmath52 emission line could reduce the remarkably high continuum brightness , the lack of ly@xmath52 seen in galaxies within the reionization epoch at @xmath16 , indicates that the high fraction of neutral hydrogen in the universe at early times absorbs the majority of ly@xmath52 photons of these galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? seeing strong ly@xmath52 emission from a source at @xmath49 during the early phase of reionization is particularly unexpected ( although perhaps not impossible ) . alternatively , the source could be a @xmath48 extreme line emitter . an emission line at @xmath120 would have to produce the majority of its @xmath42 flux , which would require extreme equivalent widths . a possible example of such a source is presented in @xcite . for more extensive discussions of these alternative options for this source see @xcite and @xcite . given the uncertain nature of this source , we will treat its detection as an upper limit of @xmath121 source in the following analysis , and we will only derive upper limits on the luminosity and star - formation rate densities at @xmath38 from this single - source sample . in the following sections we discuss possible contamination of our @xmath0 lbg samples . as already pointed out in @xcite , galaxies with strong balmer breaks , or with high dust obscuration are a potential source of contamination for @xmath122 galaxy searches . in particular , in fields with limited depth in the wfc3/ir and optical data , such extremely red sources can remain undetected shortward of @xmath42 and can thus satisfy the hst selection criteria . fortunately , with the availability of spitzer / irac over all the search fields in this study , such sources can readily be excluded from the samples based on a @xmath123<2 $ ] color criterion . as shown in @xcite , the candels data contains 16 intermediate brightness sources which satisfy our @xmath4 @xmath59-dropout selections . however , these could all be excluded based on the irac constraints . they are all found at @xmath124 mag , which suggests that such red , lower redshift galaxies show a peaked lf . therefore , it is expected that they would be much less of a problem as contaminants in our fainter samples . given this expectation , it is particularly interesting that we actually did not find the lower luminosity counterparts of such sources in any of the three deep fields , even though our irac data are sensitive enough thanks to the iudf program . although the survey volume of our deep data is limited , this further suggests that such red galaxies are indeed very rare at lower luminosities . it remains an open and interesting question as to the nature and redshift of these red @xmath124 mag galaxies ( the redshift is expected to be low , i.e. @xmath125 , but exactly over what redshift range they are seen is still quite uncertain ) . after excluding contamination from intermediate redshift , red galaxies , the next most important source of contamination is photometric scatter . photometric scatter can cause faint , low - redshift sources to have colors and magnitudes such that they would be selected in our sample . we estimate the magnitude of this effect with simulations using real galaxies based on our photometric catalogs . in particular , we select all sources with @xmath42 magnitudes in the range 24 to 25 and we rescale their fluxes and apply the appropriate amount of photometric scatter as observed for real sources at fainter luminosities . we then apply our selection criteria to these simulated catalogs in order to estimate the contamination fraction . this is repeated 5000 times , which results in reliably measured contamination fractions . as expected , contamination due to photometric scatter is most significant at the faint end of our sample . the above simulations show that we do not expect to see any contaminant at @xmath127 mag from the detection limit . in the hudf12/xdf @xmath3 galaxy sample , we find that 0.9 contaminants are expected per simulation . given that we find 7 sources , this signifies a @xmath128 contamination fraction . note that this would have been a factor 3@xmath9 higher ( 2.6 contaminants expected ) had we not included our optical @xmath75 measurement . again this shows clearly the power of having deep shorter wavelength data , and the effectiveness with which data over a range of wavelengths can be used . for the higher redshift samples , we estimate 0.2 and @xmath129 contaminants in the hudf12/xdf lbg selection at @xmath4 and @xmath38 , respectively , from analogous simulations . overall , our extensive simulations show that the contamination due to photometric scatter is thus expected to be @xmath130 for all these samples . as an additional test of the contamination rate in our samples , we can use the best - fitting low - redshift seds for our @xmath3 candidates in order to estimate with what probability such types of galaxies would be selected as lbgs . in particular , we use the expected magnitudes of the @xmath48 sed fits shown in figure [ fig : sedfits ] , perturb these with the appropriate photometric scatter , and apply our @xmath3 lbg selection . we repeat this simulation 10@xmath131 times for each of our @xmath3 galaxy candidates , which allows us to estimate the probability for our @xmath3 sample to contain a certain number of contaminants . we find that at 65% confidence , our sample contains zero or one contaminant , while @xmath132 contaminants are found in 90% of the realizations . finally , the chance that the majority ( i.e. , @xmath133 ) of these @xmath3 candidates lie at @xmath48 is estimated to be @xmath134 . note that the average number of contaminants per realization is found to be 1.1 , i.e. , very similar to our previous estimate of 0.9 contaminants based on using the real , bright galaxy population . it should be noted that the sed - based test makes no assumptions about the relative abundance of faint , star - forming @xmath0 galaxies and the possible intermediate redshift passive sources at the same observed magnitude ( @xmath135 mag ab ) . both observationally and theoretically , the number density of low - mass , passive galaxies at @xmath136 is still very poorly understood , making it very difficult to gauge the contamination rates for our @xmath0 samples . in particular , these passive @xmath48 galaxies would need to have only @xmath137 @xmath10 and @xmath138 to @xmath139 mag . nevertheless , these tests show that while we can not completely exclude intermediate redshift contamination in our samples , the majority of our candidates are clearly expected to lie at @xmath0 . as we noted earlier , to account for this contamination we exclude the @xmath3 candidate xdfyj-39446317 from the subsequent analysis , in agreement with its ambiguous photometric redshift at @xmath140 or @xmath98 . galactic dwarf stars can be a significant concern for @xmath28 galaxy samples , due to strong absorption features in their atmospheres , which causes their intrinsic colors to overlap with the high - redshift galaxy selection criteria . however , this is not as much of a concern at @xmath122 . stellar spectra are significantly bluer in our selection colors than high - redshift galaxies . this can be seen in figure [ fig : colsel ] , where we plot the location of the stellar sequence including m , l , and t dwarfs . stars with intrinsically red colors are therefore not expected to be a significant contaminant in our samples . the only possibility for such stars to contaminate our selection is due to photometric scatter , which we implicitly accounted for in our photometric simulations in the previous section . additionally , we can exclude contamination by supernovae . we verified that all galaxies in our sample are detected at statistically - consistent s / n levels in the images taken over a time baseline of about 3yr as part of the hudf09 and hudf12 campaigns . for a more extensive review of possible contamination in @xmath141 samples , see also @xcite and @xcite . -dropout samples . the new @xmath142-dropout selection has a mean redshift @xmath143 , while the @xmath144-dropout sample is expected to lie at a mean @xmath145 . although the @xmath146 color is only satisfied for @xmath147 galaxies , the @xmath44-dropout sample extends to significantly lower redshift , and peaks only at @xmath148 . this is mainly due to photometric scatter and due to the relatively slow change in @xmath105 color from @xmath149 ( see figure [ fig : colsel ] ) . ] the expected redshift distributions of our lbg samples are estimated based on extensive simulations of artificial galaxies inserted in the real data which are then re - selected in the same manner as the original sources ( see also * ? ? ? * ; * ? ? ? in particular , we estimate the completeness @xmath150 and selection probabilities @xmath151 as a function of @xmath42 magnitude @xmath152 and redshift @xmath126 . following @xcite , we use the profiles of @xmath26 lbgs from the hudf and goods observations as templates for these simulations . the images of these sources are scaled to account for the difference in angular diameter distance as well as a size scaling of @xmath153 . the latter is motivated by observational trends of lbg sizes with redshift across @xmath154 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the colors of the simulated galaxy population are chosen to follow a distribution of uv continuum slopes with @xmath155 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and are modulated by the igm absorption model of @xcite over a range of redshifts @xmath156 to @xmath157 . @xmath158 galaxies are simulated for each redshift bin in steps of @xmath159 , which allows for a reliable estimate of the completeness and selection probability taking into account the dispersion between input and output magnitudes . the result of these simulations enables us to compute the redshift distribution of galaxies after assuming a lf @xmath160 . @xmath161)\ ] ] we assume a baseline uv lf evolution with @xmath162 and @xmath163 , consistent with the trends found across @xmath26 to @xmath2 @xcite . the normalization is not relevant for the relative distributions . for the k - correction in the conversion from absolute to observed magnitude , we use a 100 myr old , star - forming template of @xcite . the results of this calculation are shown in figure [ fig : zselection ] . it is clear that the redshift selection functions are significantly wider than the target redshift range based on the simple lbg color tracks shown in figure [ fig : colsel ] . the reason for this is simply photometric scatter . the mean redshift ( and the width ) of our samples are 9.0 ( @xmath164 ) , 10.0 ( @xmath164 ) , and 10.7 ( @xmath165 ) for the @xmath142- , @xmath144- and @xmath166-dropout samples , respectively . candidate hudf12 - 4106 - 7304 of @xcite appears to be significantly boosted by a diffraction spike . _ top _ a 15view around a the bright foreground neighbor of the source hudf12 - 4106 - 7304 . the source clearly lies exactly along the direction of the diffraction spike caused by the bright neighbor . _ bottom _ stamps ( 4 by 4 ) of the @xmath167 candidate hudf12 - 4106 - 7304 of @xcite _ ( left ) _ next to an image of a diffraction spike of a nearby star _ ( center ) _ , with the rescaled flux of the diffraction spike subtracted from the udf12 - 4106 - 7304 image _ ( right)_. the latter was derived by fitting the core of the bright galaxy to the right ( west ) of hudf12 - 4106 - 7304 . this was done for both f140w and f160w . the images are centered at the same pixel offset from the nearby bright object causing the diffraction spike . the bright object lies 5 arcsec to the right ( w ) in each case . the location of the putative @xmath168 candidate is marked as red circle . it clearly lies extremely close to where the peak flux of the diffraction spike is seen . a hint of a linear spike is seen in the @xmath44 image , running across the source location . while a source is still seen in the subtracted image , its estimated flux is reduced by a factor @xmath169 , making this a @xmath170 detection only . this very weak detection evidence , together with the near - blending of the source with another foreground galaxy , strongly suggests that this is not a real source . such faint higher - order diffraction spikes are a well - known pitfall when pushing the data to its limits , in particular in ultra - deep imaging data , which is mostly taken at the same rotation angle . , title="fig : " ] candidate hudf12 - 4106 - 7304 of @xcite appears to be significantly boosted by a diffraction spike . _ top _ a 15view around a the bright foreground neighbor of the source hudf12 - 4106 - 7304 . the source clearly lies exactly along the direction of the diffraction spike caused by the bright neighbor . _ bottom _ stamps ( 4 by 4 ) of the @xmath167 candidate hudf12 - 4106 - 7304 of @xcite _ ( left ) _ next to an image of a diffraction spike of a nearby star _ ( center ) _ , with the rescaled flux of the diffraction spike subtracted from the udf12 - 4106 - 7304 image _ ( right)_. the latter was derived by fitting the core of the bright galaxy to the right ( west ) of hudf12 - 4106 - 7304 . this was done for both f140w and f160w . the images are centered at the same pixel offset from the nearby bright object causing the diffraction spike . the bright object lies 5 arcsec to the right ( w ) in each case . the location of the putative @xmath168 candidate is marked as red circle . it clearly lies extremely close to where the peak flux of the diffraction spike is seen . a hint of a linear spike is seen in the @xmath44 image , running across the source location . while a source is still seen in the subtracted image , its estimated flux is reduced by a factor @xmath169 , making this a @xmath170 detection only . this very weak detection evidence , together with the near - blending of the source with another foreground galaxy , strongly suggests that this is not a real source . such faint higher - order diffraction spikes are a well - known pitfall when pushing the data to its limits , in particular in ultra - deep imaging data , which is mostly taken at the same rotation angle . , title="fig : " ] in the following sections , we compare our lbg samples with previous selections over these fields in the literature . in @xcite , we already identified three possible sources at @xmath172 . these sources were identified based on their red @xmath173 colors . with the advent of the hudf12 data , all these source are confirmed as valid high - redshift candidates . however , they are all weakly detected in the @xmath58 filter , which results in a somewhat lower estimate on their redshift . nevertheless , one of these sources ( udfy-38135540 ) is included in our present @xmath3 sample . the other two ( udfy-37796000 and udfy-33436598 ) do have @xmath174 colors of @xmath175 , which are too blue to be included in our sample . their photometric redshifts are 7.8 and 7.7 , respectively . the photometry of both these sources are also listed in the appendix in table [ tab : additional ] . as deeper data becomes available it is not unusual to find that the photometric redshifts undergo small shifts to lower values , also due to the larger number of sources at lower redshifts @xcite . the original bias to higher redshifts results from the larger photometric scatter in shallower data , resulting in an overestimate of the lyman break amplitude . similar biases also affect the photometric redshift samples , e.g. , compare the redshift estimates of @xcite with @xcite . this is a well - known and well - understood effect and should be expected to affect all redshift estimates derived from photometric data , regardless of the procedures used . this effect explains the question raised by @xcite regarding the slightly lower redshift for these sources . in our previous analysis of the full hudf09 data over the hudf , we already identified the source xdfjh-39546284 as a probable high - redshift source . based on those data and on a plausible evolution of the uv lf to higher redshift , we expected this source to lie at @xmath176 . surprisingly however , xdfjh-39546284 was not detected in the new @xmath44 data and so its redshift can not be @xmath176 ( see also * ? ? ? its nature is now unclear . the best - fit @xmath177 solution is quite unlikely , given what we now know about the evolution of the lf at redshifts @xmath178 . xdfjh-39546284 is @xmath119 brighter than expected for a @xmath49 galaxy at its number density ( see figure 4 of * ? ? ? dramatic changes to higher luminosity densities at @xmath147 are unlikely and so this object presents us with an interesting conundrum . this is discussed in detail in @xcite and @xcite . after the first half of the hudf09 data was taken over the hudf in the first year of observations , we had identified three potential @xmath4 sources ( see the supplementary information / appendix a of * ? ? ? these sources were selected as @xmath59-dropouts , very similar to the candidates selected in the present analysis . however , the three candidates were not detected at sufficient significance in the second year wfc3/ir @xmath42 data , which raised the possibility that they were spurious detections . with the advent of additional @xmath42 data from the hudf12 survey , we can now confirm that all these three sources are in fact real . they are all significantly detected in the full @xmath42 and @xmath44 data . however , only one of these sources is now in our @xmath0 galaxy sample . two sources show photometric redshifts of @xmath2 , given their very faint detections in the @xmath58 data of 0.5@xmath67 and 2.3@xmath67 , respectively . however , we remark that one of these two sources may still be at @xmath179 given the tentative nature of its @xmath58 band detection ( i.e. @xmath180 ) . the last source ( xdfj-38126243 ) remains in our new @xmath4 @xmath59-dropout sample . for this source , we find a photometric redshift of @xmath181 . with the possible exception of the enigmatic @xmath182 redshift candidate ( xdfjh-39546284 ) , this is therefore the highest redshift galaxy candidate in the hudf12/xdf field . in figure [ fig : epochstamps ] , we show the @xmath44 and @xmath42 stamps of the source xdfj-38126243 , including splits of the data by epoch . it is clear that the source is real , as it is now detected at @xmath183 in @xmath42 and at 3.4@xmath67 in @xmath44 . as can be seen , the second year hudf09 data ( epoch 2 ) , only contains a weak , though statistically - consistent , signal of this source . given that there were just two epochs available at that time and the overall s / n of the source was below our threshold , we did not include this source in the @xcite and @xcite analyses . also shown in figure [ fig : epochstamps ] is the best - fit template and photometric redshift distribution for this source . with both photometric redshift codes zebra @xcite and eazy @xcite , we find a consistent best - fit photometric redshift at @xmath184 with uncertainties of @xmath185 ( see table [ tab : phot ] ) . as expected for such a faint source , the redshift likelihood function shows a lower redshift peak around @xmath113 ( gray sed ) . the integrated low - redshift ( @xmath186 ) likelihood is 18% . this is consistent with our estimate of lower redshift contamination due to photometric scatter in our @xmath59-dropout sample . taken together the data are consistent with this being a viable and likely @xmath4 candidate . note that @xcite do not include this source in their analysis , but they specifically discuss it . they state the source is not significantly detected ( @xmath187 ) in their summed @xmath104 image , for which we see two main reasons . ( 1 ) with a photometric redshift of @xmath4 , this galaxy is largely redshifted out of the @xmath59 band , greatly reducing ( by @xmath188 ) its detection significance in a @xmath104 stack . use of a @xmath189 stack is better for such cases and is what we do here . ( 2 ) the source is very compact , and therefore it is detected at higher significance in the small apertures we use here for s / n measurements ( 035 diameter ) compared to the @xcite analysis ( @xmath190diameter ) . moreover , we stress that this source is significantly detected in several independent sub - sets of the data , and is therefore certainly real ( see figure [ fig : epochstamps ] ) . the hudf12 team has recently published a sample of seven @xmath78 galaxy candidates identified in the hudf12 data in @xcite . these sources were based on a photometric redshift selection technique ( see also * ? ? ? * ) , with a @xmath87 detection in the @xmath59+@xmath44+@xmath42 summed image . however , as shown in figure [ fig : filters ] galaxies start to disappear in @xmath59 at @xmath122 , which is why we could expect to find additional sources in our sample compared to @xcite . furthermore , we use smaller apertures for s / n measurements than @xcite , which in most cases are more optimal for such very small @xmath191 sources . this results in a small additional gain of @xmath192 in s / n . in general , our sample is in very good agreement with the selection of @xcite . with the exception of two , we include all their sources in our @xmath0 samples . the discrepant ones are udf12 - 3895 - 7114 and udf12 - 4106 - 7304 , which we discuss below . their photometry is additionally listed in the appendix in table [ tab : additional ] . _ udf12 - 3895 - 7114 : _ this source certainly show colors very similar to a @xmath0 candidate . however , we measure @xmath193 , which is bluer than our selection color for the @xmath3 sample . hence it is not included in our @xmath3 sample . while @xcite find a best - fit photometric redshift of @xmath194 , the source is not present in the ` robust ' sample of @xcite , and we find a photometric redshift distribution function which is very broad , with a best - fit at @xmath195 ( using both zebra or eazy ) . this different result compared to the @xcite redshift estimate may be caused by small uncertainties in the photometry measurements ( given that we also use different apertures ) . additionally , we perform irac flux measurements on a source by source basis . these include an additional uncertainty due to the subtraction of neighboring sources ( see e.g. * ? ? ? * ; * ? ? ? this is not the case in the @xcite analysis , who note that they use constant upper limits on the irac fluxes . as we have discussed , photometric redshifts are very uncertain for sources this faint and so there is a chance that this source is still at @xmath0 . nevertheless , our analysis raises significant doubt about its high - redshift nature . _ udf12 - 4106 - 7304 : _ the wfc3/ir psf shows significant diffraction spikes , which are caused by the mount of the secondary mirror . while typically only seen around bright stars , these diffraction spikes are so strong that they can also emanate from compact foreground galaxies , particularly in the redder wfc3/ir filters . the source udf12 - 4106 - 7304 of @xcite is located at the edge of such a diffraction spike for both the @xmath42 and the @xmath44 filters ( the only filters wherein udf12 - 4106 - 7304 is significantly detected ) . this is shown in figure [ fig : spike ] . the photometry of this source is clearly significantly enhanced by the diffraction spike . the detection significance of udf12 - 4106 - 7304 is critically reduced once the diffraction spike signal is removed . the profile of the bright foreground galaxy is non - trivial to model , but fortunately only its core is relevant for causing the diffraction spikes . we therefore use galfit @xcite to model the center of this source and subtract the diffraction spikes that were scaled to match the core flux . doing so results in the flux of udf12 - 4106 - 7304 being reduced by a factor @xmath196 , both in @xmath44 and @xmath42 , which makes it only a 2.8@xmath67 total nir detection . this is too low to be included in a robust sample . additional uncertainty about the reality of this source arises due to its different morphology in the @xmath42 and @xmath44 images . the ` source ' also lies close to another faint foreground galaxy . it is therefore not clear whether the diffraction spike and the neighboring galaxy conspired to lead to the detection of this potential candidate . in any case , for these reasons , and for the low detection flux , the reality of udf12 - 4106 - 7304 remains in question and we do not include this source in our analysis . candidates per bin of 0.25 mag in the different fields considered in our analysis , assuming that the uv lf evolves steadily from @xmath2 to @xmath4 , consistent with the well - established trends from @xmath34 to @xmath197 . with this assumption the hudf12/xdf alone should have contained 4.6 candidate galaxies at @xmath30 . in our whole survey area , we would have expected to see @xmath198 sources now . given that only one candidate galaxy could be identified ( shown by the arrow ) , this provides strong , direct evidence that the uv luminosity function and ld are evolving rapidly from @xmath4 to @xmath2 . ] the sample of nine @xmath0 galaxy candidates we compiled in the previous sections allows us to make some of the first estimates of the @xmath51 uv lfs . although limited in area , the hudf12/xdf data alone provide very useful constraints already at @xmath3 and limits at @xmath1 . additionally , due to the deeper data over the hudf and the candels goods - s field compared to our previous analysis in @xcite , we are able to improve our constraints on the @xmath4 lf . these new constraints at @xmath51 will allow us to test whether the galaxy population underwent accelerated evolution at @xmath0 as previously found in @xcite and @xcite , or whether the uv lf trends from lower redshift continue unchanged to @xmath0 ( the preferred interpretation of , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? in order to test for such accelerated evolution , we start by estimating the number of galaxies we would have expected to see in our @xmath51 lbg samples , if the lower redshift trends were to hold unchanged at @xmath0 . by comparing the observed number of sources with those expected from the extrapolations we derive a direct estimate of any changes in the evolution of galaxies at @xmath0 . to derive the expected numbers we use our estimates of the selection function and completeness measurements described in section [ sec : selfun ] . this allows us to compute the number of sources expected as a function of observed magnitude . for an assumed lf @xmath160 , this is given by : @xmath199)\ ] ] for the uv lf evolution , we adopt the relations of @xcite : @xmath200mpc@xmath201mag@xmath202const , @xmath203const and @xmath204 . note that we assume constant values for the faint - end slope @xmath52 and the normalization @xmath205 . these relations are used as a baseline , when extrapolated to higher redshifts , to test whether the observed galaxy population at @xmath0 is consistent with the trends at later times , i.e. , at lower redshifts . with these assumptions , we find that we would expect to detect a total of @xmath206 galaxies in our @xmath3 @xmath142-dropout sample over the hudf12/xdf field alone . yet , after correcting for one potential contaminant ( see section [ sec : scattersim ] ) , we only detect six sources . this is @xmath207 fewer than expected from the trends from the lower redshift lfs . ( top ) and @xmath4 ( bottom ) uv lf from the hudf12/xdf data as well as from the additional fields for @xmath4 . lower redshift lfs are shown as gray solid lines for illustration of the lf evolution trends . these are the most recent determinations from @xcite at @xmath208 and @xcite at @xmath2 . _ ( top ) _ our step - wise @xmath3 lf ( dark blue circles ) is computed in bins of 1 mag , which contain 1 , 2 and 3 sources , respectively . these measurements are consistent with ( but consistently below ) the expected lf given an extrapolation from lower redshift ( dashed blue line ) . the best - fit lf based on luminosity evolution is shown as a solid blue line . this is derived from the expected number of sources in bins one magnitude wide , and is a factor @xmath209 below the extrapolated lf . also shown as green squares is the step - wise @xmath3 determination from @xcite who use a photometric redshift sample derived from hudf12 data . their determination is in very good agreement with our measurement , although it is unclear why they only constrain the lf at the very faint end . the dot - dashed line represents the best @xmath3 lf estimate of @xcite over the magnitude range where it is constrained by their lensed candidates from the clash dataset . their determination is based on scaling the normalization of the @xmath2 lf to account for the low number density of @xmath3 lbg candidates found over the first 19 clusters . within the current uncertainties , all three determinations of the @xmath3 lfs are in good agreement , finding accelerated evolution compared to the lower redshift trends . _ ( bottom ) _ at @xmath4 , our analysis includes several additional fields , which is why we can probe to much lower volume densities than for our @xmath3 lf . nevertheless , since we only find one potential @xmath4 galaxy candidate in our data , we can mostly only infer upper limits on the lf . again , these are consistently below the extrapolated lf ( dashed red line ) , indicating that the galaxy population evolves more rapidly at @xmath29 than at lower redshift . , title="fig : " ] ( top ) and @xmath4 ( bottom ) uv lf from the hudf12/xdf data as well as from the additional fields for @xmath4 . lower redshift lfs are shown as gray solid lines for illustration of the lf evolution trends . these are the most recent determinations from @xcite at @xmath208 and @xcite at @xmath2 . _ ( top ) _ our step - wise @xmath3 lf ( dark blue circles ) is computed in bins of 1 mag , which contain 1 , 2 and 3 sources , respectively . these measurements are consistent with ( but consistently below ) the expected lf given an extrapolation from lower redshift ( dashed blue line ) . the best - fit lf based on luminosity evolution is shown as a solid blue line . this is derived from the expected number of sources in bins one magnitude wide , and is a factor @xmath209 below the extrapolated lf . also shown as green squares is the step - wise @xmath3 determination from @xcite who use a photometric redshift sample derived from hudf12 data . their determination is in very good agreement with our measurement , although it is unclear why they only constrain the lf at the very faint end . the dot - dashed line represents the best @xmath3 lf estimate of @xcite over the magnitude range where it is constrained by their lensed candidates from the clash dataset . their determination is based on scaling the normalization of the @xmath2 lf to account for the low number density of @xmath3 lbg candidates found over the first 19 clusters . within the current uncertainties , all three determinations of the @xmath3 lfs are in good agreement , finding accelerated evolution compared to the lower redshift trends . _ ( bottom ) _ at @xmath4 , our analysis includes several additional fields , which is why we can probe to much lower volume densities than for our @xmath3 lf . nevertheless , since we only find one potential @xmath4 galaxy candidate in our data , we can mostly only infer upper limits on the lf . again , these are consistently below the extrapolated lf ( dashed red line ) , indicating that the galaxy population evolves more rapidly at @xmath29 than at lower redshift . , title="fig : " ] similarly , with the same assumptions about the evolution of the uv lf from @xmath26 to @xmath4 , we would expect to see a total of @xmath210 sources in our @xmath4 @xmath144-dropout sample . we only find one such source , which suggests that beyond @xmath3 , the decrement compared to the baseline evolution is even larger than we found previously from the hudf09 and the 6-epoch candels data . we now expect to see three more sources compared to the six that we expected to see in the earlier analysis of @xcite . yet , no additional @xmath4 sources are found . the expected magnitude distribution for the @xmath4 sample is shown in figure [ fig : nexp ] . as indicated in the figure , our search for @xmath4 sources in the candels and ers fields of goods - s should have resulted in two detections . only 0.5 sources were expected in these fields from our previous analysis using somewhat shallower data @xcite . depending on the assumptions about halo occupation , the expected cosmic variance for a single wfc3/ir field is @xmath211 @xcite . in order to estimate the significance of our finding of a large offset ( i.e. , decrement ) between the expected number of sources and that seen , we have to combine the poissonian and cosmic variance uncertainties . we estimate the chance of finding @xmath121 source in our full search area using the appropriate expected number counts and cosmic variance estimates for the individual search fields . the latter are based on the cosmic variance calculator of @xcite . using simple monte - carlo simulations , we derive that given that we expected to find 9 sources , finding @xmath121 occurs at a probability of only 0.5% . therefore , our data are inconsistent with a simple extrapolation of the lower redshift lf evolution at 99.5% . this new estimate reinforces the conclusion of @xcite and @xcite that the evolution in the number density of star forming galaxies between @xmath4 and @xmath2 is large , and larger than expected from the rate of increase at later times , i.e. , the evolution was accelerated in the @xmath43 200 myr from @xmath4 to @xmath2 . ccc @xmath212 & @xmath213 + @xmath214 & @xmath215 + @xmath216 & @xmath217 + @xmath218 & @xmath219 + @xmath220 & @xmath221 + @xmath222 & @xmath223 + @xmath224 & @xmath225 + @xmath226 & @xmath227 + @xmath228 & @xmath229 + @xmath230 & @xmath231 + @xmath232 & @xmath233 lcccccccc this work & 11 & @xmath234 ( fixed ) & @xmath235 ( 1@xmath67 ) & @xmath236 ( fixed ) + this work & 10 & @xmath234 ( fixed ) & @xmath237 & @xmath236 ( fixed ) + this work & 9 & @xmath234 ( fixed ) & @xmath238 & @xmath236 ( fixed ) + @xcite & 10 & @xmath239 ( fixed ) & @xmath240 & @xmath241 ( fixed ) + @xcite & 9.2 & @xmath242 & @xmath243 ( fixed ) & @xmath244 ( fixed ) the above calculations of the expected number of sources can be used directly to constrain the uv lfs at @xmath51 . since the number of candidates are small , we need to make some assumptions about how to characterize the evolution . the uv lfs at later times provide a valuable guide . the parameter that evolves the most is the characteristic luminosity @xmath245 ; the normalization ( @xmath246 and the faint - end slope ( @xmath52 ) are relatively unchanged from @xmath2 to @xmath26 ( although we do have evidence for evolution toward steeper faint - end slopes ) . this suggests that we should estimate what evolution in the characteristic luminosity best reproduces the observed number of sources , while keeping both the normalization and the faint - end slope fixed . doing so results in a best - estimate of the luminosity evolution of @xmath247 from @xmath25 to @xmath3 and @xmath248 to @xmath4 . the characteristic magnitudes at these redshifts are thus expected to be @xmath249 and @xmath250 . the uncertainties on these measurements are still quite large , given the small sample sizes and the small area probed , in particular for the @xmath3 search . we perform the same calculation for the @xmath38 @xmath44-dropout sample . however , given the uncertain nature of the single candidate source in that sample ( xdfjh-39546284 ) , we treat the estimate at @xmath38 as an upper limit . we therefore compute the evolution in @xmath251 which is needed to produce one source or fewer in the sample . this is found to be @xmath252 , which is a less stringent constraint than that for our @xmath4 estimate , due to the much smaller area probed by our @xmath38 search . the inferred constraint on the characteristic magnitude is @xmath253 mag . all our estimates of the uv lfs are summarized in table [ tab : lfcomparison ] . the above estimates for the characteristic luminosity of the uv lf can also be compared with the characteristic luminosity from the step - wise determination of the lf using the observed galaxies and limits . the step - wise luminosity function is derived using an approximation of the effective selection volume as a function of observed magnitude @xmath254 . the lf is then simply @xmath255 . this derivation is only valid as long as the absolute magnitude varies slowly with observed magnitude . this is not the case for @xmath147 , since the igm absorption affects the @xmath42 band such that the relation between luminosity and the observed magnitude becomes strongly redshift dependent . we therefore restrict our analysis of the step - wise lfs to the @xmath256 samples . in any case , the small area probed by the hudf12/xdf data does currently not significantly constrain the uv lf at @xmath1 . the step - wise @xmath3 and @xmath4 lfs are tabulated in table [ tab : z910lf ] . figure [ fig : lfevol ] shows our constraints on the uv lf at @xmath3 and @xmath4 . the expected uv lfs extrapolated from the lower - redshift trends are shown as dashed lines . clearly , at both redshifts , the observed lf lies significantly below this extrapolation , as expected from our analysis of the observed number of sources in the previous section . the best - fit @xmath3 lf using @xmath251 evolution is a factor @xmath257 below the extrapolated lf at @xmath258 mag . at the bright end , the small area probed by the single hudf12/xdf field limits our lf constraints to @xmath259 mag@xmath11mpc@xmath201 , which is clearly too high to be meaningful at @xmath260 . it will therefore be very important to cover several fields with f140w imaging in the future in order to push the @xmath3 selection volumes to interesting limits . at @xmath4 , the use of deeper data both on the hudf and on the goods - s field allows us to push our previous constraints on the uv lf to fainter limits . since we only detect one @xmath4 galaxy candidate , however , our constraints mainly consist of upper limits . nevertheless , it is evident that at all magnitudes @xmath258 mag these upper limits are consistently below the extrapolated uv lf , up to a factor @xmath261 lower . additionally , the limits are also clearly below the best - fit @xmath3 lf , showing that the uv lf continues to decline at @xmath122 . contributed by all galaxies brighter than @xmath262 mag . our new measurements from the hudf12/xdf data are shown as red squares . measurements of the ld at @xmath263 are derived from the uv lfs from @xcite . no correction for dust extinction has been applied . the measurements are plotted at the mode of the redshift distributions shown in figure 6 . for the highest redshift @xmath38 @xmath166-dropout sample , we only show an upper limit given that the single source we find in this sample is either at an even higher redshift ( where the selection volume of our data is very small ) or is a low redshift extreme line emitter . the dark gray line and shaded area represent an extrapolation of the redshift evolution trends of the @xmath264 uv lf . our ld estimates at @xmath191 are clearly lower than this extrapolation . however , the observed rapid build - up of galaxies at @xmath30 to @xmath34 is not unexpected , since it is consistent with a whole suite of theoretical models . some of these are shown as colored lines . they are halo occupation models ( blue solid and dashed , * ? ? ? * ; * ? ? ? * ) , a semi - analytical model ( orange dashes , * ? ? ? * ) , and two hydrodynamical simulations @xcite . these different models uniformly predict a steepening in the ld evolution at @xmath0 . the conclusion to be drawn is that the shape of the trend from @xmath4 to @xmath21 is mainly due to the rapid build - up of the underlying dark - matter halo mass function , rather than any physical changes in the star - formation properties of galaxies . ] the evolution of the uv luminosity density ( ld ) at @xmath0 has received considerable attention in recent papers , triggered by our initial finding of a significant drop in the ld from @xmath2 to @xmath4 ( i.e. , a rapid increase in the ld within a short period of time ) . with the new hudf12/xdf data , it is now possible to refine this measurement by adding a @xmath3 and a @xmath1 point , while also allowing us to improve upon our previous measurements at @xmath4 . the uv lds inferred from our @xmath0 galaxy samples are shown in figure [ fig : ldevol ] . the measurements show the ld derived by integrating the best - fit uv lf determined in the previous section . the integration limit is set to @xmath265 mag , which is the current limit probed by the hudf12/xdf data . for comparison , we also show the lower redshift ld measurements from the compilation of @xcite . these were computed in the same manner as the new @xmath0 values , and were not corrected for dust extinction . a summary of our measurements for the ld are listed in table [ tab : ldsfrd ] . as can be seen , our new measurements at @xmath0 lie significantly below the @xmath2 value . the decrement in ld from @xmath2 to @xmath3 is @xmath12 dex , and it is even larger at @xmath266 dex to @xmath4 . therefore , our data confirms our previous finding of more than an order of magnitude increase of the uv ld in the short time period , only 170 myr , from @xmath4 to @xmath2 . the gray line and shaded area show the expected ld evolution when extrapolating the @xmath264 schechter function trends to higher redshift . all our measurements at @xmath0 lie below the extrapolation . although the offsets individually are not large ( they are @xmath267 ) , the consistent offset to lower ld supports a hypothesis that significant changes are occurring in the ld evolution at @xmath0 . it is interesting to note that this offset to lower ld is not unexpected , as it is also seen in several theoretical models . in figure [ fig : ldevol ] , we compare our observational results to two conditional luminosity function models from @xcite and @xcite , the prediction from a semi - analytical model of @xcite , as well as the results from two hydrodynamical simulations of @xcite and @xcite . all these models are in relatively good agreement with the lower redshift ( @xmath22 ) measurements . as can be seen from the figure , essentially all models do show a steeper evolution at @xmath0 than a purely empirical extrapolation of the uv lf further into the epoch of reionization . since these models are all very different in nature , this strongly suggests that the rapid build - up we observe in the galaxy population is mainly driven by the build - up in the underlying dark matter halo mass function , which is also evolving very rapidly at these epochs . cccc yj & 9.0 & @xmath268 & @xmath269 + j & 10.0 & @xmath270 & @xmath271 + jh & 10.7 & @xmath272 & @xmath273 + b & 3.8 & @xmath274 & @xmath275 + v & 5.0 & @xmath276 & @xmath277 + i & 5.9 & @xmath278 & @xmath279 + z & 6.8 & @xmath280 & @xmath281 + y & 8.0 & @xmath282 & @xmath283 our results combined with those of others now provide a substantially larger sample at @xmath141 for estimate of the sfr density than was available for @xcite and @xcite . the sfr density at @xmath0 was recently estimated based on four high - redshift galaxies identified in the clash survey @xcite , and from seven galaxies identified in the hudf12 data @xcite . we present all these results , together with our own measurements in figure [ fig : sfrdevol ] , where we plot the sfr density as a function of redshift in star - forming galaxies with sfr @xmath284yr@xmath11 ( corresponding to a magnitude limit of @xmath285 mag ) . the sfr densities are derived from the uv ld estimates after correction for dust extinction . we use the most recent determinations of the uv continuum slopes @xmath286 as a function of uv luminosity and redshift from @xcite together with the @xcite @xmath286-extinction relation . the dust - corrected lds are then converted to sfr densities using the conversion factor of @xcite , assuming a salpeter initial mass function . across @xmath287 to @xmath2 , the star - formation rate density clearly evolves very uniformly . the evolution is well reproduced by a power law @xmath288 , which is shown as dark gray line in figure [ fig : sfrdevol ] . interestingly , all measurements lie below the extrapolation of this trend to higher redshift . again , individual measurements are within @xmath267 of the trend , but the offset to smaller sfr densities in the mean is very clear . note that each of the three groups find a consistent decrement of the sfrd from @xmath2 to @xmath3 . specifically , from our data , we find a drop by @xmath12 dex . this is @xmath289 below the simple extrapolation of the lower redshift trends . at @xmath3 our sfrd estimate is in excellent agreement with the measurement of @xcite based on a photometric redshift selection . at @xmath4 , however , we find a significantly lower value , mainly due to our inclusion of a larger dataset covering a larger area , in which we do not find any additional candidate . nevertheless , our measurement is within @xmath290 of the result of @xcite , in particular , after we correct their measurement down by a factor two to account for the source that is likely the result of a diffraction spike in their @xmath291 sample ( figure [ fig : spike ] ) . also at @xmath38 , our upper limit is significantly below the @xmath292 estimate of @xcite . this is likely due to the wide redshift range probed by our @xmath44-dropout sample , which extends from @xmath167 to @xmath182 ( see figure [ fig : zselection ] ) . @xcite consider a strict boundary of @xmath293 to @xmath294 . in order to compute the best - fit evolution of the sfrd at @xmath295 , we combine all @xmath0 measurements from clash with our improved estimates at @xmath51 , together with the previous @xmath2 sfrd measurement @xcite . the best - fit evolution falls off very rapidly , following @xmath296 . this is shown as the black line in figure [ fig : sfrdevol ] . as can be seen , this is significantly steeper than the @xmath264 trends . by @xmath4 , the best - fit evolution is already a factor @xmath297 below the lower redshift trend . therefore , the combined constraint on the sfrd evolution from all datasets in the literature clearly points to an accelerated evolution at @xmath0 . we have used the new , ultra - deep wfc3/ir data over the hudf field as well as the optical xdf data to provide a reliable selection of galaxies at @xmath0 . the new observations from the hudf12 program push the depth of the @xmath42 imaging deeper by @xmath430.2 mag compared to our previous data from the hudf09 survey , and they provide additional @xmath44 imaging . the @xmath44 data are very useful for selecting some of the first @xmath3 and @xmath1 galaxy samples using the lyman break technique . furthermore , we extended our previous search for @xmath4 galaxies @xcite to fainter limits by including this new hudf12/xdf data set . our analysis is the most extensive search for @xmath0 galaxies to date . from our full dataset , we find a total sample of nine @xmath0 galaxy candidates . seven of these lie in our @xmath3 selection . contamination is always a central concern for high redshift samples and after careful analysis we expect that the contamination fraction is small , being only about @xmath298 . we found that one of the @xmath3 sources has a very wide photometric redshift likelihood distribution , with an ambiguous best - fit at @xmath299 ( using zebra ; @xmath300 using eazy ) . we therefore exclude this source from the subsequent analysis . we discover a new @xmath4 source ( at @xmath5 ) , making it one of the very few galaxies known at this very high redshift , just 460 myr after the big bang . the highest redshift candidate in our sample is xdfjh-39546284 that was previously identified at @xmath176 . however , these new data ( the @xmath44 in particular ) constrain this galaxy to be at @xmath182 , if it is at high redshift . this interpretation is problematic and has led to discussion about it being a possible lower - redshift @xmath48 object , and so its true nature remains quite uncertain at this time ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? this sample of @xmath0 galaxy candidates proves to be very important for setting a number of constraints on galaxy build - up at very early times , allowing us to derive an estimate of the uv lf at @xmath3 , to improve our constraints at @xmath4 , and to set limits at @xmath1 . the main result from our analysis is a confirmation of our previous finding that the galaxy population , as seen down to @xmath265 mag , evolves much more rapidly at @xmath0 than at lower redshift ( identified as accelerated evolution ; * ? ? ? * ; * ? ? ? this is seen in ( 1 ) the expected number of galaxies when extrapolating the lower redshift trends to @xmath0 ( figure [ fig : nexp ] ) , ( 2 ) in the direct constraints on the uv lf ( figure [ fig : lfevol ] ) , and ( 3 ) in the evolution of the luminosity and star - formation rate densities down to our current completeness limits ( figures [ fig : ldevol ] , [ fig : sfrdevol ] ) . all measurements consistently point to accelerated evolution at early times , beyond @xmath2 . specifically , if the lower redshift trends of the uv lf are extrapolated to @xmath4 , we would have expected to see @xmath210 candidate sources in our full data set , @xmath32 of which only in the hudf12/xdf data alone . however , only one such candidate is found in the hudf12/xdf data , which suggests that the galaxy population evolves more rapidly than at lower redshift at 99.5% significance ( see section [ sec : z8abundance ] ) . from @xmath2 to @xmath3 , the luminosity density in star - forming galaxies with sfr@xmath301yr@xmath11 ( i.e. @xmath14 ) decreases by @xmath12 dex . this decrement is fully consistent with previous estimates from clash @xcite and from the hudf12 data alone @xcite . these results on the galaxy evolution at @xmath0 contrast with the conclusions drawn by several recent papers , who argue that the uv ld evolution at @xmath0 is consistent with the lower redshift trends ( e.g. * ; * ? ? ? * ; * ? ? ? * ) . however , the small sample sizes of @xmath0 galaxies in these individual analyses resulted in large uncertainties on the ld and sfrd evolution . we show here that once all these measurements are combined self - consistently , they do indeed point to accelerated evolution at @xmath0 , consistent with theoretical expectations . note that the steep fall - off we find in the uv ld at @xmath0 is not at odds with galaxies driving reionization . our measurements only reach to @xmath302 at @xmath4 ( i.e. to @xmath265 mag ) . however , with the steep faint - end slopes that are consistently found for @xmath303 uv lfs , the total luminosity density is completely dominated by galaxies below this threshold ( see e.g. * ? ? ? * ; * ? ? ? with wfc3/ir we are now in a similar situation in studying @xmath304 as we were three years ago with nicmos at @xmath21 . galaxy samples are still small , and the conclusions are uncertain . however , over the next few years the @xmath0 frontier will be explored more extensively . in particular , the additional deep field observations to be taken as part of the deep fields initiative ( a large director s discretionary program ) , will significantly increase sample sizes and should allow for improved constraints on the @xmath3 and @xmath4 lf at @xmath305 mag . this will enable more precise constrains on the accelerated evolution that we now see in the galaxy population from the data over goods - south . xdfyj-38135540 & 03:32:38.13 & -27:45:54.0 & @xmath306 & @xmath307 & @xmath308 & @xmath309 & 13.1 & 16.0 & 9.8 & 0.2 + & @xmath310 & @xmath311 & + xdfyj-39478076 & 03:32:39.47 & -27:48:07.6 & @xmath312 & @xmath313 & @xmath314 & @xmath315 & 8.7 & 9.7 & 5.0 & -0.6 + & @xmath316 & @xmath317 & + xdfyj-39216322 & 03:32:39.21 & -27:46:32.2 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & 5.1 & 4.9 & 3.2 & -1.4 + & @xmath322 & @xmath323 & + xdfyj-42647049 & 03:32:42.64 & -27:47:04.9 & @xmath324 & @xmath325 & @xmath326 & @xmath321 & 4.4 & 4.9 & 2.1 & -0.5 + & @xmath327 & @xmath328 & + xdfyj-40248004 & 03:32:40.24 & -27:48:00.4 & @xmath329 & @xmath330 & @xmath331 & @xmath332 & 3.5 & 3.3 & 2.2 & 0.0 + & @xmath322 & @xmath333 & + xdfyj-43456547 & 03:32:43.45 & -27:46:54.7 & @xmath334 & @xmath330 & @xmath335 & @xmath332 & 3.1 & 3.5 & 1.8 & -3.8 + & @xmath336 & @xmath337 & + xdfyj-39446317 & 03:32:39.44 & -27:46:31.7 & @xmath338 & @xmath339 & @xmath340 & @xmath341 & 3.8 & 3.7 & 1.3 & 1.3 + & @xmath342 & @xmath343 & + xdfj-38126243 & 03:32:38.12 & -27:46:24.3 & @xmath344 & @xmath345 & @xmath346 & @xmath347 & 5.8 & 3.4 & 1.2 & -0.6 + & @xmath109 & @xmath348 & + & & & + xdfjh-39546284 & 03:32:39.54 & -27:46:28.4 & @xmath349 & & @xmath350 & @xmath350 & 7.3 & 0.2 & -1.6 & 0.8 + & @xmath351 & @xmath352 & + 42126501 & 03:32:42.12 & -27:46:50.1 & @xmath353 & @xmath354 & @xmath355 & @xmath356 & 22.2 & 15.8 & 5.2 & -0.1 + & @xmath357 & @xmath358 & + 43246481 & 03:32:43.24 & -27:46:48.1 & @xmath359 & @xmath360 & @xmath361 & @xmath362 & 5.4 & 3.0 & 2.8 & -1.9 + & @xmath363 & @xmath364 & + 43286481 & 03:32:43.28 & -27:46:48.1 & @xmath365 & @xmath366 & @xmath345 & @xmath367 & 6.0 & 2.5 & 0.5 & -0.2 + & @xmath368 & @xmath369 & + udf12 - 4106 - 7304 & 03:32:41.06 & -27:47:30.4 & & & & & & & & + & & & + & & & + udf12 - 3895 - 7115 & 03:32:38.95 & -27:47:11.5 & @xmath370 & @xmath371 & @xmath372 & @xmath373 & 3.5 & 3.7 & 3.7 & 1.2 + & @xmath374 & @xmath375 & + & & & + udfy-37806001 & 03:32:37.80 & -27:46:00.1 & @xmath376 & @xmath377 & @xmath378 & @xmath309 & 10.1 & 13.7 & 9.4 & 2.7 + & @xmath379 & @xmath380 & + udfy-33446598 & 03:32:33.44 & -27:46:59.8 & @xmath381 & @xmath382 & @xmath383 & @xmath384 & 5.6 & 6.0 & 4.7 & -1.9 + & @xmath385 & @xmath386 &

we present a comprehensive analysis of @xmath0 galaxies based on ultra - deep wfc3/ir data . we constrain the evolution of the uv luminosity function ( lf ) and luminosity densities from @xmath1 to @xmath2 by exploiting all the wfc3/ir data over the hubble ultra - deep field from the hudf09 and the new hudf12 program , in addition to the hudf09 parallel field data , as well as wider area wfc3/ir imaging over goods - south . galaxies are selected based on the lyman break technique in three samples centered around @xmath3 , @xmath4 and @xmath1 , with seven @xmath3 galaxy candidates , and one each at @xmath4 and @xmath1 . we confirm a new @xmath4 candidate ( with @xmath5 ) that was not convincingly identified in our first @xmath4 sample . the deeper data over the hudf confirms all our previous @xmath6 candidates as genuine high - redshift candidates , and extends our samples to higher redshift and fainter limits ( @xmath7 mag ) . we perform one of the first estimates of the @xmath3 uv lf and improve our previous constraints at @xmath4 . extrapolating the lower redshift uv lf evolution should have revealed 17 @xmath3 and 9 @xmath4 sources , i.e. , a factor @xmath8 and 9@xmath9 larger than observed . the inferred star - formation rate density ( sfrd ) in galaxies above 0.7 @xmath10yr@xmath11 decreases by @xmath12 dex from @xmath2 to @xmath3 , in good agreement with previous estimates . the low number of sources found at @xmath0 is consistent with a very rapid build - up of galaxies across @xmath4 to @xmath2 . from a combination of all current measurements , we find a best estimate of a factor 10@xmath9 decrease in the sfrd from @xmath2 to @xmath4 , following @xmath13 . our measurements thus confirm our previous finding of an accelerated evolution beyond @xmath2 , and signify a rapid build - up of galaxies with @xmath14 mag within only @xmath15 myr from @xmath4 to @xmath2 , in the heart of cosmic reionization .

the radiative properties of emitters ( e.g. atoms or quantum dots ) located inside a dielectric or conducting material can be significantly modified compared to those in vacuum . the modification results from the variation of the density of modes of the em field which can be adjusted by changing the geometric shape or space period structure of a material @xcite . the radiative properties of emitters can also be modified by locating the atoms close to the surface of a dielectric or conducting material @xcite . in this case , the modification results from the presence of surface em modes known as plasmon guided ( pg ) field @xcite . a new category of materials has been proposed , so - called meta - materials @xcite , characterized by specifically designed geometrical structures which drastically modify the density of the em modes , so the field propagation , also yielding to the pg field @xcite . owing to the high local density of modes , emitters may interact strongly with the surface field which affects their radiative properties @xcite . for example , when quantum dots are placed at a distance about several tens of nanometers above two dimensional metal surface , strong coupling could be generated between the quantum dots and the collective mode reflected in the presence of the rabi oscillations @xcite . in the structure composed of zero index and left hand materials , maximum quantum interference and a suppression of the atomic decay rate can be achieved between zeeman levels due to the anisotropy of the em modes @xcite . it has been shown that by changing the strength of the driving field and adjusting position of the quantum dot , the plasma mode on the surface of a metal - nanoparticles material can introduce asymmetrical features into the spectrum @xcite . the study of entanglement between distant atoms and controlled the transmission of information between them are vital to the development of quantum information technology @xcite . a key model to investigate the creation and storage processes of entanglement often follows with a system composed of two - level quantum emitters @xcite . when atoms coupled to the same pg field , the incoherent spontaneous exchange of photons could occur and results in collective damping @xcite . it has been revealed when the decay through one of the collective states is deeply depressed , long lived entanglement of the system could be achieved , which only depends on the distance between atoms @xcite . by applying nanowire structure , the pg field could be well guided and thus entanglement between quantum dots still exists at several vacuum wave lengths @xcite . xu _ et al . _ @xcite have shown that the entanglement between two atoms can exist over distances much larger than the resonant wavelength if the space between the atoms is filled with a thin membrane made of single negative ( @xmath0 or @xmath1 ) and left hand materials . however , most of the studies are focus on metal - dielectric structure , and the entanglement only maintains for a small time scale . corresponding similarities should also then be expected in the dynamics of atoms located close to the interface between two meta - materials . in this paper , we present an analytical treatment of the dynamics of two independent atoms located close to the interface of two meta - materials , one of a negative permeability ( mn ) and the other of a negative permittivity ( en ) . we assume that the atoms are located in the mn material . surface plasmon polaritons ( spp ) , nonradiative electromagnetic excitations associated with charge density waves propagating along the interface are generated . the spp propagate in the @xmath2- and @xmath3-directions along the interface between the meta - materials , and rapidly decay in the @xmath4-direction . as illustrated in figure [ fig1a ] , two atoms located close to the interface can excite the spp and the excitation depends on the distance of the atoms from the interface and the polarization of the atomic dipole moments . for a polarization of both atomic dipole moments in the @xmath5 plane , the atoms could excite the spp which would effectively propagate in the @xmath2 direction . in other words , the polarization of the atomic dipole moments determines the direction of propagation of the spp on the interface . hence , the interface would behave as a directional guiding plasma field mode propagating in the @xmath2-direction , formally analogy of a plasmonic waveguide @xcite . we shall demonstrate below that then it would be possible to achieve a strong interaction between two atoms located close to the interface through their coupling to the spp . and @xmath6 close to the interface between two meta - materials . each atom excites the spp propagating along the interface . the excitation size of the spp depends on the distance of the atom from the interface and polarization of its dipole moment . as illustrated by green and purple ellipses , the excitation area of the surface plasmon polaritons decreases with an decreasing distance @xmath7 and their shapes extend more in the @xmath2 direction than in the @xmath3 direction if the atomic dipole moments are polarized in the @xmath5 plane.,width=321 ] the mathematical approach we adopt here is based on the green s function method @xcite . our focus is on how the plasma field induced at the interface between the materials changes the dynamics of the atoms , in particular , the population transfer and entanglement . the remarkably simple analytical expressions are derived for the probability amplitudes valid for an arbitrary initial state , arbitrary strengths of the coupling constants of the atoms to the plasma field , and arbitrary distances between the atoms . we find a number of interesting general results . in the first place , we distinguish two different time scales of the evolution of the atomic states , one corresponding to the evolution of the collective symmetric state and the other to the antisymmetric state . secondly , we find a threshold behavior of the coupling constants which separate the non - markovian behavior of the system from the markovian one @xcite . the markovian evolution is usually attributed to a weak coupling of an atom to the field . we show that the collective effects may result in the markovian evolution even in the limit of a strong coupling of the atoms to the field . inversely , a non - markovian evolution can be seen even in the regime of a weak coupling . thirdly , we find that the plasma field does not appear as a common reservoir to the atoms . in order to explain the behavior of the atoms we adopt the image method and show that the dynamics of the two atoms are completely equivalent to those of a four - atom system . finally , we consider the case in which the plasma field frequency is off - resonant with the atoms and find that in this case the dynamics resemble those of two atoms coupled to a common reservoir . the atoms interact through the exchange of virtual photons resulting in the absence of the images . the plan of this paper is as follows . in section [ sec2 ] we introduce the model and present the explicit analytic expressions for the time dependence of the probability amplitudes of the atoms . detailed dynamics of the atoms are studied in section [ sec3 ] . we assume that a single excitation is present initially in the system and demonstrate how the evolution of the system can be simply understood in terms of the evolution of the atoms and their corresponding images . we then demonstrate in section [ sec4 ] the collective behavior of the atoms in both markovian and non - markovian regimes of the evolution . entangled properties of the atoms are discussed in section [ sec5 ] , where we calculate the concurrence for different initial states and different coupling strengths of the atoms to the plasma field . the effect of an off - resonant coupling of the atoms to the plasma field on the collective dynamics and entanglement is discussed in section [ sec6 ] . we summarize our results in section [ sec7 ] . the paper concludes with two appendices in which we give details of the derivation of the integro - differential equations for the probability amplitudes and the calculations of the integral kernels . both longitudinal and transverse parts of the green function are considered in the evaluation of the kernels . perfectly conducting materials are known to generate a strong pg field which is refined in a short regime near the surface @xcite . a plasma field can also be generated at the surface of a meta - material with either negative permittivity ( @xmath0 ) or negative permeability ( @xmath1 ) . however , the density of the plasma field near surface only originates from one or several discrete modes , which can be derived by applying the continuous conditions on the boundaries . recently , tan _ et al . _ @xcite have shown that the density of the em modes can be significantly enhanced at the interface of two meta - materials , one with negative @xmath8 and the other with negative @xmath9 . especially , when the materials are perfectly paired , i.e. @xmath10 and @xmath11 , the effective permittivity and permeability , defined as @xmath12 are both zero when the slabs have the same thickness , @xmath13 . in this case , the band gap disappears and the density of modes of the em field becomes continuous . this means that a large density of the modes exists at the interface between the two perfectly paired negative meta - material slabs , and can be treated as optical topological material @xcite . we consider a system composed of two identical atoms located at a distance @xmath14 from the interface between two different negative index material slabs , @xmath9-negative ( mn ) and @xmath8-negative ( en ) slabs , as shown in figure [ fig1 ] . we assume that the atoms are located in the mn slab and the distance between the atoms , @xmath15 , is large compared to the atomic wavelength , @xmath16 , so there is no direct interaction between the atoms . each atom is represented by its ground state @xmath17 , an excited state @xmath18 , the atomic transition frequency @xmath19 , and the atomic transition dipole moment @xmath20 . axis is taken normal to the interface between mn and en slabs with its origin at the interface . the slabs have thickness @xmath21 and @xmath22 , respectively , and are assumed to have infinite extents in the @xmath23 plane . two atoms are embedded in the mn slab at fixed positions @xmath24 and @xmath25 , where @xmath14 is the distance of the atoms from the interface between the materials . the atomic transition dipole moments @xmath26 and @xmath27 are parallel to each other and oriented in the @xmath5 plane.,width=321 ] the atoms interact with an electromagnetic field via a dipole interaction according to the hamiltonian @xcite @xmath28 where @xmath29 is the unperturbed hamiltonian of the atoms and the field , and @xmath30 \label{h3}\end{aligned}\ ] ] is the interaction of the atoms with the field . here , @xmath31 and @xmath32 are the creation and annihilation operators which can be viewed as collective excitations of the electromagnetic field , @xmath33 represents noise polarization of the en material , @xmath34 represents noise magnetization of the mn material @xcite , @xmath35 and @xmath36 are the raising ( lowering ) and the energy difference operators of atom @xmath37 . the positive frequency part of the electric field operator at the position @xmath38 of the @xmath37th atom is given by @xmath39 } \stackrel{\leftrightarrow}{\bf g}({\bf r}_{j},{\bf r},\omega)\cdot \hat{{\bf f}}_e({\bf r},\omega)\right . \nonumber\\ & + & \left . \!\sqrt{-\im[\kappa({\bf r},\omega)]}{\bf \nabla}\times\!\stackrel{\leftrightarrow}{\bf g}\!\!({\bf r}_{j},{\bf r},\omega)\!\cdot\!\hat{{\bf f}}_m({\bf r},\omega)\!\right\},\end{aligned}\ ] ] where @xmath40 $ ] is the imaginary part of permittivity , @xmath41 $ ] is the imaginary part of reciprocal of permeability ( @xmath42 ) , respectively , and @xmath43 is the green tensor of the field , which characterizes the density of the field modes at the location @xmath38 of atom . for the permittivity and permeability of the slabs , we assume that @xmath44 and @xmath45 are positive constants , but @xmath46 and @xmath47 are negative and strongly depend on frequency of electromagnetic field which have following forms @xcite @xmath48 where @xmath49 and @xmath50 are plasma frequencies of the electric and magnetic materials , respectively , @xmath51 and @xmath52 are resonance frequencies of the materials , and @xmath53 are dissipation ( losses ) parameters of the materials . for clarity of the notation we have omitted the spatial argument . it is clear from eq . ( [ e1 ] ) that in the frequency region above the resonance , @xmath54 and @xmath55 , the en ( mn ) slab is a single - negative material . thus , the structure of a meta - material can be designed @xcite . to study the dynamics of the atoms , we consider the wave function of the system whose the time evolution is governed by the schrdinger equation @xmath56 if the field was initially at @xmath57 in the vacuum state and the atoms shared a single excitation , the wave function of the system at time @xmath58 , written in the interaction picture is of the form @xmath59 where @xmath60 is the probability amplitude of the state in which atom @xmath61 is in its excited state @xmath62 , atom @xmath63 is in the ground state @xmath64 , and the field is in the vacuum state @xmath65 , @xmath66 is the probability amplitude of the state in which atom @xmath61 is in its ground state @xmath67 , atom @xmath63 is in the excited state @xmath68 , and the field is in the vacuum state @xmath65 , and @xmath69 is the probability amplitude of the state in which both atoms are in their ground states , @xmath70 , and there is an excitation of the medium - assisted field @xmath71 . with the interaction ( [ h3 ] ) , the schrdinger equation ( [ es ] ) transforms into four coupled equations of motion for the probability amplitudes . when the amplitudes @xmath72 are eliminated we arrive , as shown in appendix a , into two coupled integro - differential equations for the probability amplitudes of the atoms @xmath73 in which @xmath74 is the integral kernel determined by the imaginary part of the one - point green tensor , @xmath75 , whereas @xmath76 is the integral kernel determined by the two - point green tensor , @xmath77 . the kernels can be evaluated explicitly , and straightforward but lengthly calculations ( for details see appendix b ) lead to the following explicit expressions @xmath78 where @xmath79 is the detuning of the atomic transition frequency from the plasma field frequency , @xmath80\right|\!\left(2z_{0}/\lambda_{s}\right)^2\right]}{64\pi^{3}(2z_{0}/\lambda_{s})^3}\right\}^{1/2 } \label{om0}\end{aligned}\ ] ] is the coupling strength of the atoms to the surface plasma field , and @xmath81 \nonumber\\ & + \frac{1}{3 + 4\pi^{2}\!\left|\re\,[\mu_{1}(\omega_s)]\right|\!\left(2z_{0}/\lambda_{s}\right)^{2 } } \nonumber\\ & \times \left\{f\!\left[\frac{3}{2},2,2;-\frac{x_{21}^2}{(2z_{0})^2}\right]\!+\!2f\!\left[\frac{3}{2},2,1;-\frac{x_{21}^2}{(2z_{0})^2}\right]\right . \nonumber\\ & -\left . \nonumber\\ & -\left . 3\frac{x_{21}^2}{(2z_{0})^2}f\left[\frac{5}{2},3,3;-\frac{x_{21}^2}{(2z_{0})^2}\right]\right\ } \label{u}\end{aligned}\ ] ] determines the strength of the interaction between the atoms resulting from the coupling of the atoms with the same plasma field . here , @xmath82 is the spontaneous emission rate of the atoms in free space , assumed the atoms are identical , @xmath83 . on separation between the atoms @xmath84 for several different distances of the atoms from the interface : @xmath85 ( solid black line ) , @xmath86 ( dashed red line ) , @xmath87 ( dashed - dotted blue line ) , and @xmath88 ( solid green line).,width=340 ] the function @xmath89 depends on the separation between the atoms , @xmath90 , and also their distance @xmath14 from the interface . it determines the strength of the coupling between the atoms . in the limit of @xmath91 , @xmath92 , while for @xmath93 , @xmath94 . thus , for large @xmath90 , the effects of the coupling between the atoms become negligible and the atoms evolve independently . notice that at @xmath95 the function @xmath89 is always unity independent of the value of @xmath14 , the distance of the atoms from the interface . however , the variation of @xmath89 with @xmath90 depends strongly on @xmath14 . this is illustrated in figure [ fig4 ] which shows @xmath89 as a function of @xmath90 for several different values of @xmath14 . it is seen that for large @xmath14 , the function @xmath89 varies slowly with @xmath90 . in that case , the indirect coupling between the atoms which is provided by the plasma field is effectively quite strong even at large separations . on the other hand , for small @xmath14 , @xmath96 , the function @xmath89 is different from zero only over very small distances @xmath90 and decays rapidly to zero as @xmath90 increases . thus , a strong coupling of the atoms to the plasma field _ destroys _ the collective behavior of the atoms . in the physical terms , the location of the atoms at a small distance @xmath14 from the interface leads to a strong spatial confinement ( localization ) of the surface plasmon fields around @xmath97 and @xmath98 resulting in a weak overlap of the surface fields produced by the atoms . having available the explicit expressions for the integral kernels , we can now solve eqs . ( [ e11u ] ) and ( [ e12u ] ) and study the time evolution of the atomic system . if we introduce symmetric and antisymmetric combinations of the probability amplitudes , @xmath99 and @xmath100 , corresponding to collective symmetric and antisymmetric states of the two - atom system , we readily find that eqs . ( [ e11u ] ) and ( [ e12u ] ) simplify to @xmath101 where @xmath102 here , @xmath103 $ ] and @xmath104 $ ] are coupling strengths of the symmetric and antisymmetric states to the plasma field , respectively . clearly , the coupling strengths of the collective states are altered by the atomic interaction with @xmath105 enhanced and @xmath106 reduced by @xmath89 . this may have an interesting effect on the dynamics of the atoms that at small distances between the atoms , at which @xmath94 , the antisymmetric state could be completely decoupled from the interaction with the plasma field leaving only the symmetric state to be strongly coupled to the field . it is seen from eqs . ( [ a3 ] ) and ( [ a4 ] ) that the equations of motion for the probability amplitudes @xmath107 and @xmath108 are independent of each other and are similar in form . we therefore need to obtain the solution for @xmath107 , and then the solution for @xmath108 can be obtained simply by replacing @xmath105 by @xmath106 . from eq . ( [ a5 ] ) it follows that the kernel @xmath109 is a function only of the time difference @xmath110 . therefore , the integro - differential equation ( [ a3 ] ) can be solved exactly by laplace transformation . thus if @xmath111 we get from eq . ( [ a3 ] ) @xmath112 or @xmath113 by inverse laplace transformation we then have the result @xmath114e^{-\left(\frac{1}{4}\gamma -\tilde{\omega}_{s}\right ) t } \right . \nonumber\\ & + \left . \left[1-\frac{(\gamma\!+\!2i\delta)}{4\tilde{\omega}_{s}}\right]e^{-\left(\frac{1}{4}\gamma + \tilde{\omega}_{s}\right ) t}\right\ } , \label{a42}\end{aligned}\ ] ] where @xmath115 is the effective rabi frequency of the interaction of the atoms with the plasma field , @xmath116 and @xmath117 . similarly , for @xmath108 , we get @xmath118e^{-\left(\frac{1}{4}\gamma -\tilde{\omega}_{a}\right ) t}\right . \nonumber\\ & + \left . \left[1-\frac{(\gamma\!+\!2i\delta)}{4\tilde{\omega}_{a}}\right]e^{-\left(\frac{1}{4}\gamma + \tilde{\omega}_{a}\right ) t}\right\ } , \label{a43}\end{aligned}\ ] ] where @xmath119 . two important features of the results ( [ a42 ] ) and ( [ a43 ] ) should be noted . firstly , the effective rabi frequencies @xmath120 and @xmath121 exhibit a threshold effect that depending upon @xmath122 or @xmath123 , the rabi frequencies can be either purely real or purely imaginary . in the other words , the time evolution of the probability amplitudes could be either exponential or sinusoidal . secondly , we note that below threshold the time evolution of both @xmath107 and @xmath108 involves two decaying exponentials with a reduced ( subradiant ) decay constant , @xmath124 , and an enhanced ( superradiant ) decay constant , @xmath125 . the involvement of the fast and slow decay rates in the evolution of both superradiant and subradiant states seems to contradicts our expectation since , according to dicke @xcite ( see also refs . @xcite ) , each of the collective amplitudes of a two atom system should decay with a single rate : the symmetric superposition @xmath107 should decay with the fast ( superradiant ) rate while @xmath108 should decay with the slow ( subradiant ) rate . from the interface between two materials and their images located at a distance @xmath14 behind the interface . the atoms are not directly coupled to each other , but can be coupled by the radiation reflected from the interface . a photon emitted by atom @xmath61 and reflected from the interface towards atom @xmath63 can be viewed as being emitted from the image of the atom @xmath61.,width=302 ] a qualitative understanding of the involvement of both fast and slow decay rates in the evolution of @xmath107 and @xmath108 may be obtained by considering the interaction of the atoms with the plasma field as the interaction with images of the atoms located at a distance @xmath14 behind the interface . this is illustrated in fig . [ fig3 ] , which shows that the two - atom system interacting with the surface plasma field can be seen as a four - qubit system , the two atoms plus two images . the radiation field emitted by either atom @xmath61 or atom @xmath63 and reflected from the interface in the direction normal to the interface can be regarded as the radiation from an image located at a distance @xmath14 behind the interface . the radiation field emitted by atom @xmath61 and reflected from the interface towards atom @xmath63 can be viewed as being emitted by the image of the atom @xmath61 located at a distance @xmath14 behind the interface . to study the evolution of the atoms in terms of the interaction with their images , we write eqs . ( [ e11u ] ) and ( [ e12u ] ) in the following form @xmath126 where @xmath127 $ ] and @xmath128 $ ] . following the fig . [ fig3 ] , the second term on the right - hand side of eq . ( [ e11w ] ) may be interpreted as arising from the coupling of the atom @xmath61 to its image , whereas the third term may be interpreted as arising from the coupling of the atom @xmath61 to the image of the atom @xmath63 . thus , we can immediately write eqs . ( [ e11w ] ) and ( [ e12w ] ) as @xmath129 where @xmath130 are the probability amplitudes of the images of the atom @xmath61 and @xmath63 , respectively . the equation of motion for the probability amplitudes of the corresponding images are @xmath131 we focus on the evolution of the symmetric combinations of the probability amplitudes , which obey the equations @xmath132 it is then straightforward to show that the solution of eq . ( [ e29 ] ) for @xmath107 is of the same form as eq . ( [ a42 ] ) . the involvement of the images allows us to write the general solution for @xmath107 as a sum of two amplitudes @xmath133\!e^{-\left(\frac{1}{2}\gamma + i\delta\right)t } , \end{aligned}\ ] ] where @xmath134t}\sin\phi , \nonumber\\ \tilde{d}_{a}(t ) & = \tilde{c}_{s}(t)\cos\phi -i\tilde{c}_{si}(t)\sin\phi \nonumber\\ & = c_{s}(0)e^{\left[\frac{1}{4}(\gamma + 2i\delta)+\tilde{\omega}_{s}\right]t}\cos\phi , \end{aligned}\ ] ] are symmetric and antisymmetric superpositions of the probability amplitudes of the atomic and image states , with @xmath135 a similar treatment can be applied to @xmath108 , which can be written in the form @xmath136\!e^{-\left(\frac{1}{2}\gamma + i\delta\right)t } , \end{aligned}\ ] ] where @xmath137t}\sin\psi , \nonumber\\ \tilde{g}_{a}(t ) & = \tilde{c}_{a}(t)\cos\psi -i\tilde{c}_{ai}(t)\sin\psi \nonumber\\ & = c_{a}(0)e^{\left[\frac{1}{4}(\gamma + 2i\delta)+\tilde{\omega}_{a}\right]t}\cos\psi , \end{aligned}\ ] ] with @xmath138 the reason for the presence of both the superradiant and subradiant terms in eqs . ( [ a42 ] ) and ( [ a43 ] ) is now clear : the superradiant terms are associated with the decay of the symmetric superpositions involving the atomic and image states , @xmath139 and @xmath140 , whereas the subradiant terms are associated with the decay of the antisymmetric superpositions @xmath141 and @xmath142 . the slowest decay rate in the system , @xmath143 , is the decay rate of the antisymmetric superposition @xmath142 whereas the fastest decay rate , @xmath144 , is the decay rate of the symmetric superposition @xmath139 . we may conclude that the interaction of the atoms with the surface plasma field can be viewed as the interaction between the atoms and their corresponding images . we have already seen that different locations of the atoms lead to two different rabi frequencies @xmath120 and @xmath121 determining the evolution of the system and though two different time scales of the evolution . the forms of @xmath120 and @xmath121 show a threshold effect for the rabi frequencies that depending upon @xmath122 or @xmath123 , the time evolution of the probability amplitudes could be either exponential or sinusoidal . since @xmath145 , the threshold conditions for the evolution of the symmetric and antisymmetric states do not coincide with each other , and thus we can distinguish between three regions of @xmath105 and @xmath106 : ( a ) @xmath146 and @xmath147 , ( b ) @xmath148 and @xmath147 , ( c ) @xmath148 and @xmath149 . physically , the threshold values of @xmath120 and @xmath121 separate what we can identify as the non - markovian ( memory preserved ) regime from the markovian ( memoryless ) regime of the evolution @xcite . the case ( a ) , in which @xmath120 and @xmath121 are real so the rabi frequencies contribute to the decay rates , that the exponentially decaying amplitudes of the symmetric and antisymmetric states are a manifestation of a markovian evolution . in the case ( b ) , the dynamics of the system are partly markovian and partly non - markovian . the symmetric state undergoes a non - markovian whereas the antisymmetric state undergoes a markovian evolutions . in the case ( c ) , the dynamics of the system are fully non - markovian that the amplitudes of both symmetric and antisymmetric states undergo an oscillatory evolution , which is a manifestation of a non - markovian evolution . a non - markovian evolution is a reversible process characterized by a flow ( oscillation ) of the information between the atoms and the field , but a markovian evolution is a irreversible process of a flow ( decay ) of the information to the field . a possibility to control the non - markovian dynamics is essential in quantum information technology since it plays a crucial role in preserving quantum memory . let us specialize eqs . ( [ a42 ] ) and ( [ a43 ] ) to the case of exact resonance , @xmath150 , and first examine the situation when the coupling of the atoms to the plasma field is weak , @xmath151 . in this case , both @xmath120 and @xmath121 could be below threshold . if @xmath122 , then @xmath120 and @xmath121 are real and we see from eqs . ( [ a42 ] ) and ( [ a43 ] ) that the rabi frequencies contribute to the damping rates of the probability amplitudes . physically , this is the kind of behavior corresponding to a markovian evolution . since @xmath145 , we see that the weak coupling of the atoms to the plasma field may result in the decay of the probability amplitudes @xmath60 and @xmath66 with four rates , two enhanced and two reduced rates . ( black solid line ) and @xmath152 ( red dashed line ) for @xmath153 , @xmath154 , and @xmath155 , corresponding to both @xmath105 and @xmath106 below the threshold of @xmath156 , @xmath157 and @xmath158 . the atoms were initially in the state @xmath159 . , width=340 ] figure [ fig4n ] shows the time evolution of the populations @xmath160 and @xmath161 for both @xmath120 and @xmath121 below threshold . at early times , the population @xmath162 decreases whereas @xmath163 increases until the populations become equal . at that time , the populations began to decay monotonically . the rate they decay is equal to @xmath164 , the slowest decay rate of the antisymmetric state . since the atoms are very strongly coupled to each other , @xmath155 , the decay rate @xmath165 . consequently , the effective decay time of the populations can be very long . the decay of the populations is irreversible so the evolution of the system is markovian . at large values of @xmath89 it may happen that @xmath166 and @xmath167 even if the atoms are weakly coupled to the plasma field , i.e. @xmath168 . for example , when @xmath94 , we have @xmath169 and @xmath170 . hence , @xmath105 can be larger than @xmath171 even if @xmath168 and at the same time @xmath106 can be smaller than @xmath171 . for @xmath148 the rabi frequency @xmath120 is purely imaginary , and then the time evolution of the probability amplitude @xmath107 takes the form @xmath172 , \label{a45n}\ ] ] where @xmath173 . the temporal evolution of the @xmath107 is sinusoidal whereas the temporal evolution of the amplitude @xmath108 , which is below threshold , is exponential and is given in eq . ( [ a43 ] ) . in this case , the symmetric mode evolves in the non - markovian regime whereas the antisymmetric mode evolves in the markovian regime . it follows that in this case each atom evolves under the simultaneous influence of markovian and non - markovian mechanisms . ( black solid line ) and @xmath152 ( red dashed line ) for @xmath153 , @xmath174 , and @xmath175 corresponding to @xmath105 above and @xmath106 below the threshold of @xmath156 , @xmath176 and @xmath177 . the atoms were initially in the state @xmath159 . , width=340 ] figure [ fig5n ] shows the evolution of the populations for @xmath105 above and @xmath106 below the threshold of @xmath178 . at early times , the oscillations of the populations with the rabi frequency of the symmetric mode are clearly visible . at early times the populations oscillate with the rabi frequency of the symmetric mode . in other words , the evolution of the populations is reversible but the reversibility occurs in a restricted time range @xmath179 . beyond @xmath180 the populations decay monotonically that the evolution is irreversible . thus , we can clearly distinguish between the non - markovian and markovian regimes of the evolutions . we see that the upper limit on time of the reversible evolution results from the presence of the interaction between the atoms . clearly , it is a collective effect . physically , it is a consequence of the fact that a large part of the population is trapped in the asymmetric state determined by the amplitude @xmath142 thereby lowering the strength of the coupling of the atoms to the plasma field . it is easy to see , since for small distances between the atoms @xmath94 , we have @xmath169 , which means that the antisymmetric states decouple from the plasma field . this example also shows that the atoms when behaving collectively can be weakly coupled to the field even if individually they are strongly coupled to the field . above the thresholds , @xmath120 and @xmath121 are purely imaginary . the time evolution of the probability amplitudes is then given by @xmath181 , \label{a45}\\ c_{a}(t ) & = c_{a}(0)e^{-\frac{1}{4}\left(\gamma + 2i\delta\right)t}\!\left[\cos\bar{\omega}_{a } t + \frac{(\gamma + 2i\delta)}{4\bar{\omega}_{a}}\sin\bar{\omega}_{a } t\right ] , \label{a46}\end{aligned}\ ] ] where @xmath182 . in this case , the time evolution of the probability amplitudes becomes sinusoidal . such dynamics reflect the reversible property of the system that the evolution is non - markovian . and ( b ) @xmath152 for @xmath153 , @xmath183 , and @xmath184 . the atoms were initially in the state @xmath159 . , width=340 ] figure [ fig5 ] shows the time evolution of the populations @xmath160 and @xmath152 when the atoms are strongly coupled to the plasma field with @xmath185 , @xmath150 , but are weakly coupled to each other , @xmath184 . note the presence of two characteristic time scales of the oscillations associated with the presence of two slightly different rabi frequencies . at short times , @xmath186 , the initially excited atom @xmath61 periodically exchange the excitation with the plasma field at the rabi frequency @xmath187 . the population of the atom @xmath63 builds up with the oscillation of frequency @xmath187 . the amplitudes of the populations are modulated with frequency @xmath188 causing collapses and revivals of the atomic populations . when the atoms are close to each other the collapses and revivals of the populations are absent . instead , a periodic localization of the excitation is observed . this is illustrated in figure [ fig6 ] which shows the evolution of the populations for a small distance between the atoms at which @xmath189 . the manner the populations oscillate is different for @xmath162 and @xmath163 . we see a periodic localization of the excitation that even at long times the memory effects are still evident . the explanation of this feature follows from the observation that at small distances between the atoms , the time scale of the oscillations of the antisymmetric state is very large , approaches infinity when @xmath190 . therefore , the system effectively evolves with a single time scale determined by the rabi frequency of the symmetric state , @xmath191 . and ( b ) @xmath152 for @xmath153 , @xmath192 , and @xmath189 . the atoms were initially in the state @xmath159 . , width=340 ] given the time evolution of the probability amplitudes , we now proceed to evaluate the concurrence , a measure of entanglement between two qubits @xcite . following the definition of the concurrence , we find that in terms of the probability amplitudes , @xmath60 and @xmath66 , the concurrence is given by an expression @xmath193 in terms of the amplitudes of the symmetric and antisymmetric combinations , the concurrence can be written as @xmath194\!\left[c^{\ast}_{s}(t ) + c^{\ast}_{a}(t)\right]\right| .\label{e47}\end{aligned}\ ] ] a positive value of the concurrence , @xmath195 , indicates entanglement between the atoms , and @xmath196 corresponds to maximally entangled atoms . it is clear from eq . ( [ e47 ] ) that the atoms are entangled whenever @xmath197 . otherwise , the atoms are separable . thus , to examine the occurrence of entanglement between the atoms we must look at differences in the evolution of the amplitudes @xmath107 and @xmath108 . if initially , @xmath198 , then according to the solutions eqs . ( [ a42 ] ) and ( [ a43 ] ) , the amplitudes will evolve differently only if @xmath199 . it then follows that the coupling between the atoms through the plasma field is necessary to create entanglement between the atoms from an initial separable state . and ( b ) @xmath200 with @xmath201 , @xmath150 and different @xmath89 : @xmath175 ( solid black line ) , @xmath202 ( dashed red line ) , @xmath203 ( dashed - dotted blue line).,width=340 ] the features of the concurrence for the three regions of @xmath105 and @xmath106 are illustrated in figs . [ fig9 ] - [ fig12 ] . figure [ fig9 ] shows the effect of increasing interaction strength between the atoms on the concurrence for a weak coupling of the atoms to the plasma field , both @xmath105 and @xmath106 below threshold , which corresponds to a markovian evolution of the system . in figure [ fig9](a ) the system starts from the separable state @xmath204 , whereas in figure [ fig9](b ) the initial state of the system is the maximally entangled state @xmath205 . we see that even in the weak coupling regime , a large and long living entanglement can be created between the atoms . the entanglement created from the initial separable state increases with an increasing @xmath89 and attains the maximal value of @xmath206 for @xmath94 . the behavior of the concurrence is similar to that noted in the decay of two atoms into a common markovian reservoir @xcite . when the system starts from a maximally entangled state , either @xmath205 or @xmath207 , the initial entanglement always decays to zero with no entanglement present at long times , as illustrated in figure [ fig9](b ) . this is readily understood if it is recalled that the symmetric and antisymmetric states evolve independently in time . thus , if the population of the antisymmetric state was initially zero it will remain zero for all times . in this case , the system of two atoms effectively behave as a single two - level system with the upper state @xmath205 and the ground state @xmath208 . then , the initial population of the state @xmath205 decays exponentially to the ground state with the rate @xmath209 . , @xmath150 , ( a ) @xmath184 and ( b ) @xmath189 . the atoms were initially in the state @xmath159 . , width=340 ] turning now to the case of a strong coupling of the atoms to the plasma field at which @xmath105 and @xmath106 are above their thresholds , we show in figure [ fig10 ] the evolution of the concurrence for weakly @xmath210 and strongly @xmath211 interacting atoms . the interaction creates a small difference between the frequencies of the oscillation of the symmetric and antisymmetric modes that @xmath212 . the frequency difference induces beating oscillations of the populations of the atoms , as was seen in figure [ fig5 ] , and one can see from figure [ fig10 ] that these beating oscillations are rendered visible as beats in the concurrence . interesting features of the entanglement also appear when the symmetric mode evolves at rabi frequency which is above the threshold , @xmath148 , and simultaneously the antisymmetric mode evolves at rabi frequency which is below the threshold , @xmath147 . under this circumstance , the probability amplitude of the symmetric mode is determined by eq . ( [ a45 ] ) whereas the amplitude of the antisymmetric mode is given by eq . ( [ a43 ] ) . , @xmath150 and @xmath94 corresponding to the case of @xmath105 above threshold but @xmath106 below the threshold . frame ( a ) shows the concurrence for @xmath155 ( solid black line ) and @xmath175 ( dashed red line ) . the atoms were initially in a separable state @xmath159 . frame ( b ) shows the concurrence for @xmath155 and two different initial states , the maximally entangled symmetric state @xmath200 ( solid black line ) and the maximally entangled antisymmetric state @xmath213 ( dashed red line).,width=340 ] figure [ fig12 ] shows the evolution of the concurrence for this special case . we see that the concurrence is zero only at the initial time @xmath57 . as time progresses the concurrence develops to a nonzero value . the concurrence never becomes zero as time develops , and thus no periodic entanglement quenching occurs . this feature is associated with the fact that with the rabi frequency @xmath147 , the population of the antisymmetric state does not evolve in time leading to a trapping of a part of the atomic populations in their energy states . this is illustrated in figure [ fig13 ] , which shows the time evolution of the population @xmath162 and @xmath163 for the same parameters as in figure [ fig12 ] . we see from figure [ fig13](a ) that the initial population is periodically transferred between the atoms . however , the transfer is not complete that the populations of the atoms never become zero during the evolution . a part of the population is trapped in the atoms and is not transferred between them . figure [ fig13](b ) shows the evolution of the population @xmath162 for two initial maximally entangled states , @xmath205 and @xmath207 . since in this case @xmath214 , we clearly see that the concurrence , if starts from maximally entangled state , it follows the evolution of the population of the atoms . . frame ( a ) shows the populations @xmath162(solid black line ) and @xmath163 ( dashed red line ) for @xmath215 , @xmath150 and @xmath155 . the atoms were initially in the state @xmath159 . frame ( b ) shows the time evolution of the population @xmath162 for two different initial states , @xmath200 ( solid black line ) and @xmath213 ( dashed green line ) . not shown is @xmath163 since in this case @xmath216.,width=340 ] to this end we have discussed the collective effects induced by the resonant interaction of the atoms with the plasma field . we have established the importance of the images of the atoms in the atomic dynamics . moreover , we have demonstrated the equivalence of the system with that of four interacting atoms . we now turn to the off - resonant case of the atomic transition frequencies strongly detuned from the plasma frequency , @xmath217 . under such condition , the effective rabi frequency @xmath120 can be approximated by @xmath218 thus , if in eq . ( [ a42 ] ) the effective rabi frequency is replaced by ( [ f42 ] ) , we get , up to terms of order @xmath219 , @xmath220 a similar expression with @xmath221 gives @xmath108 . we see that @xmath107 is composed of fast and slow oscillating terms varying in time with frequencies @xmath222 and @xmath223 , respectively . of the two terms it is the one of the small magnitude @xmath224 arising from the presence of the images . thus , the evolution of @xmath107 is well determined without much contribution of the images . it is particularly well seen from eq . ( [ c1 ] ) that in the limit of @xmath225 , @xmath226 so that the superposition amplitude @xmath227 and @xmath141 is reduced to the atomic amplitude @xmath228 . in other words , two atoms significantly detuned from the plasma field are coupled each other by exchanging virtual photons through a short interaction time with the plasma field . and ( b ) @xmath152 for @xmath229 , @xmath192 , and @xmath184 . the atoms were initially in the state @xmath159 . , width=340 ] the above considerations are illustrated in figure [ fig7 ] , which shows the time evolution of the atomic populations for @xmath230 . we see that the atoms exchange the population with frequency @xmath231 . the fast oscillations seen in the early time of the evolution occur at frequency @xmath222 and can be attributed to the involvement of the images in the dynamics of the system . the presence of the fast oscillations only at very early times of the evolution is also consistent with energy - time uncertainty arguments . it is easy to understand . at short times the uncertainty of the energy of the atoms and the plasma field is very large , so one can not distinguish between @xmath19 and @xmath232 . this results in the presence of the fast oscillation of frequency @xmath222 . as time progresses , the frequencies become more distinguishable resulting in the disappearance of the fast oscillations . it is interesting to contrast the entanglement created at @xmath233 with that created in the resonant case of @xmath150 . we have seen in sec . [ sec5 ] that in the resonant case the maximal entanglement which can be created between the atoms from an initial separable state can not exceed @xmath206 . for off - resonant case @xmath234 , however , the entanglement can be significantly enhanced . this is shown in figure [ fig11 ] , which illustrates the time evolution of the concurrence for a large detuning @xmath222 . clearly , at early times of the evolutions the concurrence is larger than @xmath235 , increases with an increasing @xmath89 and becoming as large as @xmath236 . this behavior can be explained in terms of the energy - time uncertainty relation . at short times a large uncertainty in the energy results in a large uncertainty in the localization of the excitation . we see that the increased possibility to distinguish between the frequencies of the atoms and the plasma field results in an enhanced entanglement between the atoms . , @xmath230 and different @xmath89 : ( a ) @xmath184 and ( b ) @xmath155 . the atoms were initially in the state @xmath159 . , width=340 ] as a final remark , we would like to comment about a potential experimental system in which the collective dynamics of the atoms could be observed . it could be done in experiments similar to that of refs . @xcite , where a strong coupling between an artificial atom embodied into a material structure composed of mn and en meta - materials was observed . the strong coupling was observed as the rabi oscillations of the temporal evolution of the electric field inside the artificial atom after being excited by a short pulse . the experimental setup could be modified by embodying two artificial atoms into the composed meta - material structure and observe the rabi oscillation of the electric field of the atoms . the presence of the second atom could lead to the modulation of the rabi oscillations of the population of the first atom , as seen in figures [ fig4n]-[fig6 ] , which would be the clear evidence of the collective behavior of the atoms . to clarify the role of the spp in the collective behavior of the atoms , we now consider the emission properties near the interface of an atom , represented by its oscillating dipole @xmath237 , and calculate the electric field at position @xmath238 emitted by the atom located at @xmath239 . the field is given by @xcite @xmath240 suppose that the dipole moment @xmath237 is polarized in the @xmath5 plane , @xmath241 . then the coupling of atom @xmath37 , located at an arbitrary position @xmath238 , to the field produced by the atom @xmath242 is maximal if the dipole moment @xmath243 is parallel to @xmath237 . in this case , the magnitude of the field is @xmath244_{xx } + \ , \im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r},\omega)]_{xz}\right . \nonumber\\ & + \left . \im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r},\omega)]_{zx } + \im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r},\omega)]_{zz}\right\}\nonumber\\ & = \frac{\hbar\varepsilon_{0}}{2p_{a}}\omega^{2}_{0}u(x - x_{i},z_{i } ) , \label{eq45}\end{aligned}\ ] ] where @xmath245 and @xmath246 are given in eqs . ( [ om0 ] ) and ( [ u ] ) , respectively . clearly , the function @xmath246 determines the distribution of the field produced by atom @xmath242 . it is seen from the expression ( [ eq45 ] ) that the field is distributed in the @xmath5 plane and the distribution depends not only on the distance @xmath247 along the interface but also on the distance @xmath7 of the radiating dipole from the interface . as illustrated in figure [ fig4 ] , the variation of the function @xmath246 with @xmath247 depends strongly on the distance @xmath248 that @xmath246 , so that the field distribution , decreases with an decreasing distance @xmath248 . hence , @xmath14 should not be too small in order to achieve a strong coupling between distant atoms through their interaction with the spp propagating along the interface . we have studied the dynamics of two two - level atoms located near to the interface of two meta - materials ; one of negative permeability and the other of negative permittivity . we have derived analytical expressions for the probability amplitudes of the atomic states valid for an arbitrary initial state , arbitrary strengths of the coupling constants of the atoms to the plasma field , and arbitrary distances between the atoms . we have shown that the effect of the plasma field is to produce several interesting features , such as ( 1 ) two different time scales of the evolution of the atomic states , one corresponding to the evolution of the collective symmetric state and the other to the antisymmetric state . the existence of the two evolution time scales results in an entanglement between the atoms even in long times . ( 2 ) a threshold behavior of the coupling constants of the atoms to the plasma field which distinguishes between the non - markovian and markovian regimes of the evolutions . we have shown that the collective behavior of the atoms may lead to three different regimes of the evolution ; fully markovian , simultaneous markovian and non - markovian , and fully non - markovian evolutions . the three regimes determines three different time scales of the evolution of the memory effects and entanglement . ( 3 ) in the case of the resonant coupling of the plasma field to the atoms , the plasma field does not appear as a common reservoir to the atoms . we have adopted the image method and showed that in the resonant case the dynamics of the two atoms are completely equivalent to those of a four - atom system . ( 4 ) in the limit of a strong detuning of the plasma field from the atoms the dynamics resemble those of two atoms coupled to a common reservoir . this work is supported by the national natural science foundation of china ( grant no . 61275123 and no.11474119 ) and the national basic research program of china ( grant no . 2012cb921602 ) . in this appendix we give details of the derivation of the integro - differential equations ( [ e11u ] ) and ( [ e12u ] ) for the probability amplitudes of the atomic states . equations of motion for the probability amplitudes are obtained from the schrdinger equation ( [ es ] ) , which with the interaction hamiltonian ( [ h3 ] ) gives @xmath249}{\bf p}_{1}\cdot\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_1,{\bf r},\omega)\cdot{\bf c}_{e}({\bf r},\omega , t)\right . \nonumber\\ & + \left . \sqrt{-\im[\kappa({\bf r},\omega)]}{\bf p}_{1}\cdot[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_1,{\bf r},\omega)\times{\bf \nabla}]\cdot { \bf c}_{m}({\bf r},\omega , t)\right\ } , \label{e8}\\ & \dot{c}_2(t ) = -\frac{1}{\sqrt{\pi\varepsilon_{0}\hbar}}\int_{0}^{\infty } d\omega\ , \frac{\omega}{c}e^{-i(\omega-\omega_{a})t } \nonumber\\ & \times \int d{\bf r } \left\{\frac{\omega}{c}\sqrt{\im[\varepsilon({\bf r},\omega ) ] } { \bf p}_{2}\cdot\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_2,{\bf r},\omega)\cdot{\bf c}_{e}({\bf r},\omega , t)\right . \nonumber\\ & + \left . \sqrt{-\im[\kappa({\bf r},\omega)]}{\bf p}_{2}\cdot[\stackrel{\leftrightarrow}{\bf g}({\bf r}_2,{\bf r},\omega)\times{\bf \nabla}]\cdot{\bf c}_{m}({\bf r},\omega , t)\right\ } , \label{e9}\end{aligned}\ ] ] and @xmath250}e^{i(\omega-\omega_{a})t}\nonumber\\ & \times\!\left[\stackrel{\leftrightarrow}{\bf g}^{\ast}\!\!\!({\bf r}_1,{\bf r},\omega)\cdot{\bf p}^{\ast}_{1}c_1(t ) + \stackrel{\leftrightarrow}{\bf g}^{\ast}\!\!\!({\bf r}_2,{\bf r},\omega)\cdot{\bf p}^{\ast}_{2}c_2(t)\right ] .\label{e10}\\ & \dot{{\bf c}}_{m}({\bf r},\omega , t ) = \frac{1}{\sqrt{\pi\varepsilon_{0}\hbar } } \frac{\omega}{c}\sqrt{-\im[\kappa({\bf r},\omega)]}e^{i(\omega-\omega_{a})t}\nonumber\\ & \times\!\left[{\bf \nabla}\times\!\stackrel{\leftrightarrow}{\bf g}^{\ast}\!\!\!({\bf r}_1,{\bf r},\omega)\!\cdot\!{\bf p}^{\ast}_{1}c_1(t)\ ! + \ ! { \bf \nabla}\times\!\stackrel{\leftrightarrow}{\bf g}^{\ast}\!\!\!({\bf r}_2,{\bf r},\omega)\!\cdot\!{\bf p}^{\ast}_{2}c_2(t)\right ] .\label{e10a}\end{aligned}\ ] ] we may eliminate the amplitudes for the field components by solving the equations for @xmath251 and @xmath252 . integrating eqs . ( [ e10 ] ) and ( [ e10a ] ) , and substituting the solutions into equations of motion for @xmath60 and @xmath66 , we obtain @xmath253\cdot{\bf p}^{\ast}_{1}\ , c_1(t')\right . \nonumber\\ & + \left . { \bf p}_{1}\cdot\im[\stackrel{\leftrightarrow}{\bf g}({\bf r}_1,{\bf r}_{2},\omega)]\cdot{\bf p}^{\ast}_{2}\,c_2(t')\right\ } , \label{e11}\\ \dot{c}_2(t ) & = -\frac{1}{\pi\varepsilon_{0}\hbar c^{2}}\int^t_0dt'\int^\infty_0d\omega\ , \omega^2 e^{-i(\omega-\omega_{a})(t - t')}\nonumber\\ & \times \left\{{\bf p}_{2}\cdot\im[\stackrel{\leftrightarrow}{\bf g}({\bf r}_{2},{\bf r}_{2},\omega)]\cdot{\bf p}^{\ast}_{2}\ , c_2(t')\right . \nonumber\\ & + \left . { \bf p}_{2}\cdot\im[\stackrel{\leftrightarrow}{\bf g}({\bf r}_2,{\bf r}_1,\omega)]\cdot{\bf p}^{\ast}_{1}\ , c_1(t')\right\ } , \label{e12}\end{aligned}\ ] ] where we have used the following property of the green s tensors @xcite : @xmath254\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_{i},{\bf r},\omega)\stackrel{\leftrightarrow}{\bf g^{\ast}}\!({\bf r},{\bf r}_{j},\omega)-\right . \nonumber\\ & \left . \im[\kappa({\bf r},\omega)][\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_{i},{\bf r},\omega)\times\overleftarrow{\nabla}][\overrightarrow{\nabla}\times\stackrel{\leftrightarrow}{\bf g}^{\ast}\!({\bf r},{\bf r}_{j},\omega)]\right\ } \nonumber\\ & = \im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_{i},{\bf r}_{j},\omega ) ] , \end{aligned}\ ] ] in which @xmath255 $ ] is the imaginary part of @xmath256 . if we introduce the notation @xmath257\cdot{\bf p}^{\ast}_{j } , \ i , j = 1,2 , \label{e12w}\end{aligned}\ ] ] we then easily find that eqs . ( [ e11 ] ) and ( [ e12 ] ) simplify to eqs . ( [ e11u ] ) and ( [ e12u ] ) . in this appendix , we evaluate the integral kernels of the integro - differential equations ( [ e11u ] ) and ( [ e12u ] ) . the kernels involve the imaginary part of the green tensor @xmath258 . the tensor when evaluated at an arbitrary space point @xmath238 , distance @xmath259 from the atom located at @xmath239 , can be written as @xcite @xmath260 where @xmath261 represents the wave number , @xmath44 and @xmath47 are permittivity and permeability of the mn slab in which the atoms are located , and @xmath262 is the unit dyadic . using the weyl s expansion @xcite @xmath263 in which @xmath264 is the @xmath4 component of the propagation vector , we may express the green tensor in terms of a two dimensional fourier transform @xmath265 where @xmath266 is the green tensor on a plane @xmath267 with constant @xmath4 coordinate , @xmath268 and @xmath269 are components , respectively , of the wave vector and the position vector in a plane parallel to the interface , the @xmath270 plane . the expression ( [ w19 ] ) allows us to evaluate the the green tensor in terms of plane waves incident on and reflected from the boundaries between different materials , including the interface between the mn and en materials and the boundaries between the materials and the exterior regions ( vacuum ) on either side . since the atoms are located in the mn material , so that their radiative properties are modified by the field existing inside the material , we will evaluate the green tensor only at points @xmath238 inside the mn material . we follow the procedure of toma @xcite in evaluating the green tensor . the presence of the boundaries results in the field inside the mn material consisting of waves propagating in both the @xmath271 and @xmath272 directions . therefore , @xmath266 can be written in terms of functions @xmath273 defined by imposing boundary conditions in the @xmath4 direction @xmath274e^{i{\bf k_{\parallel}}({\bf \rho}-{\bf \rho_{0 } } ) } , \end{aligned}\ ] ] where the functions @xmath273 describe the electric field in mn slab , with unit strength incident from its upper side ( by taking symbol @xmath275 ) or lower side ( by taking symbol @xmath276 ) , that can be categorized into tm @xmath277 and te @xmath278 types of even @xmath279 and odd @xmath280 symmetries in the @xmath281 directions , and @xmath282 is the unit step function . the forms of the functions @xmath273 for the field inside the mn material can be represented in terms of a sum of incident and reflected waves as @xmath283 where @xmath284 and @xmath285 are orthonormal polarization vectors of the electric field of @xmath286 and @xmath287 polarized waves , respectively ; @xmath288 is the unit vector in the direction of @xmath289 @xmath290 , @xmath291 and @xmath292 are unit vectors in the cartesian coordinates , @xmath293 are reflection coefficients of the waves propagating in the @xmath281 directions , and @xmath294 results from summing the geometrical series due to the multiple reflections from the boundaries between different materials . thus , if we apply the explicit forms of the polarization vectors , and use the polar representation for @xmath298 , the diagonal components of the green tensor @xmath258 evaluated at a point @xmath238 near the position @xmath239 of the @xmath242th atom are then @xmath299e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \nonumber\\ & g_{yy}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{2(2\pi)^{2}}\int dk_\parallel\ , k_\parallel \nonumber\\ & \times \int_{0}^{2\pi}\!\!d\phi\left[\frac{\beta_{1}\!\sin^{2}\!\phi}{k^2_1}r^{(p)}_{-}(z ) + \frac{\cos^{2}\!\phi}{\beta_{1}}r^{(s)}_{+}(z)\right]e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \nonumber\\ & g_{zz}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{2(2\pi)^2}\!\int{\!dk_\parallel}\frac{k_\parallel^3 } { \beta_1k^2_1}r_{+}^{(p)}(z)\!\int_{0}^{2\pi}\!{d\phi}\ , e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \label{e17}\end{aligned}\ ] ] and the off - diagonal components are @xmath300e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \nonumber\\ & g_{xz}({\bf r},{\bf r}_{i},\omega ) = -g_{zx}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{2(2\pi)^{2}}\!\!\int\!\!dk_\parallel\ , k^{2}_\parallel \nonumber\\ & \times\!\!\int_{0}^{2\pi}\!d\phi \frac{\cos\phi}{k_{1}^{2}d_{p}}\!\left[r^{p}_{+}e^{-i\beta_1(z+z_0 - 2d_1 ) } -r^{p}_{-}e^{i\beta_1(z+z_0)}\right]\!e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \nonumber\\ & g_{yz}({\bf r},{\bf r}_{i},\omega ) = -g_{zy}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{2(2\pi)^{2}}\int dk_\parallel\ , k^{2}_\parallel \nonumber\\ & \times\!\!\int\!d\phi \frac{\sin\phi}{k_{1}^{2}d_{p}}\bigg\{r^{p}_{+}e^{-i\beta_1(z+z_0 - 2d_1 ) } - r^{p}_{-}e^{i\beta_1(z+z_0)}\bigg\}e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \label{e18}\end{aligned}\ ] ] where @xmath301 .\label{e18a}\end{aligned}\ ] ] the four terms appearing in eq . ( [ e18a ] ) represent waves propagating inside the mn material . figure [ fig2 ] illustrates the source of those four terms in the expression ( [ e17 ] ) . the term @xmath302 $ ] represents a wave propagated a distance @xmath303 from the source atom located at @xmath14 . this term is independent of the presence of the interface and the boundaries . physically , it corresponds to the source field produced by an atom located at @xmath14 . the term @xmath304 $ ] represents a wave propagated the distance @xmath305 after the reflection from the bottom interface of the mn material . the term @xmath306 $ ] represents a wave propagated the distance @xmath307 after the reflection from the upper interface of the mn material . the final term @xmath308 $ ] represents a wave propagated the distance @xmath309 after two reflections , one from the bottom and the other from the upper interfaces . at this point it should be stressed that the expressions ( [ e17 ] ) and ( [ e18 ] ) , although evaluated in the presence of the boundaries , they contain terms which are independent of the boundaries . the reason is in the fact that the field is not completely bounded into the area inside the materials . the slabs have finite sizes in the @xmath4 direction and in the derivation of eqs . ( [ e17 ] ) and ( [ e18 ] ) it has been assumed that there are nonzero transmission coefficients at the boundaries with the exterior vacuum regions . hence , we may consider the green tensor as a sum of two terms , a source - field part @xmath310 and a scattered - field part @xmath311 as @xcite @xmath312 in particular , @xmath310 is the green tensor for the field which would exist in the material if there were no boundaries present , whereas @xmath311 is the green tensor for the field scattered from the interface and boundaries . the source term has the same properties that would apply to the free ( unbounded ) field . we may extract the source part @xmath313 from @xmath314 simply by putting @xmath315 in eqs . ( [ e17 ] ) and ( [ e18 ] ) . this gives for the diagonal components @xmath316e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \nonumber\\ g^{s}_{yy}({\bf r},{\bf r}_{i},\omega ) & = \frac{i\mu_{1}}{2(2\pi)^{2}}\int dk_\parallel\ , k_\parallel e^{i\beta_1(z - z_0)}\nonumber\\ & \times \int_{0}^{2\pi}\!\!d\phi\left[\frac{\beta_{1}\!\sin^{2}\!\phi}{k^2_1 } + \frac{\cos^{2}\!\phi}{\beta_{1}}\right]e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \nonumber\\ g^{s}_{zz}({\bf r},{\bf r}_{i},\omega ) & = \frac{i\mu_{1}}{2(2\pi)^2}\int{dk_\parallel}\frac{k_\parallel^3 } { \beta_1k^2_1 } e^{i\beta_1(z - z_0)}\nonumber\\ & \times \int_{0}^{2\pi}{d\phi}\ , e^{i{\bf k}_{\parallel}\cdot ( { \bf r}- { \bf r}_{i } ) } , \label{e17s}\end{aligned}\ ] ] and for the off - diagonal components @xmath317 to evaluate the integrals over @xmath318 , which appear in eqs . ( [ e17 ] ) and ( [ e18 ] ) and involving @xmath319 $ ] , we assume , for simplicity , that @xmath320 has only @xmath2 component so that we can write the dot product in the form @xmath321 , where @xmath322 . hence , we arrive at the following expressions for the diagonal components @xmath323r^{(p)}_{-}(z ) + \frac{j_{1}(\alpha_{i})}{\beta_{1}\alpha_{i}}r^{(s)}_{+}(z)\right\ } , \nonumber\\ & g_{yy}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{4\pi}\int dk_\parallel\ , k_\parallel \nonumber\\ & \times \left\{\frac{\beta_{1}j_{1}(\alpha_{i})}{k^2_1\alpha_{i}}r^{(p)}_{-}(z ) + \frac{1}{\beta_{1}}\left[\frac{j_{1}(\alpha_{i})}{\alpha_{i } } - j_{2}(\alpha_{i})\right]r^{(s)}_{+}(z)\right\ } , \nonumber\\ & g_{zz}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{4\pi}\int{dk_\parallel}\frac{k_\parallel^3 } { \beta_1k^2_1}j_{0}(\alpha_{i})r_{+}^{(p)}(z ) , \label{e18p}\end{aligned}\ ] ] and for the off - diagonal components @xmath324 .\label{e18q}\end{aligned}\ ] ] for the source part , we get that only the diagonal elements are different from zero and are given by the following expressions @xmath325 + \frac{j_{1}(\alpha_{i})}{\beta_{1}\alpha_{i}}\right\}e^{i\beta_1(z - z_0 ) } , \nonumber\\ & g^{s}_{yy}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{4\pi}\!\!\int\!\!dk_\parallel\ , k_\parallel \nonumber\\ & \times \left\{\frac{\beta_{1}j_{1}(\alpha_{i})}{k^2_1\alpha_{i } } + \frac{1}{\beta_{1}}\!\left[\frac{j_{1}(\alpha_{i})}{\alpha_{i } } - j_{2}(\alpha_{i})\right]\right\}e^{i\beta_1(z - z_0 ) } , \nonumber\\ & g^{s}_{zz}({\bf r},{\bf r}_{i},\omega ) = \frac{i\mu_{1}}{4\pi}\int{dk_\parallel}\frac{k_\parallel^3 } { \beta_1k^2_1}j_{0}(\alpha_{i})e^{i\beta_1(z - z_0 ) } .\label{e18w}\end{aligned}\ ] ] we see that , in general , the expressions for the diagonal elements of the source part of the green tensor have complex values . however , for a lossless negative index material with @xmath326 and @xmath327 , the @xmath4 component of the propagation vector @xmath328 is pure imaginary . it is easily verified that the resulting expressions , evaluated at @xmath329 are then real numbers , thereby leading to @xmath330 = 0 $ ] . in practice , the material could posses some losses and then @xmath264 would not be a pure imaginary number . in our case , the losses in the materials are determined by the parameter @xmath331 . however , in typical materials the losses are small , @xmath332 , and usually neglected . thus , for a material in which @xmath8 and @xmath9 have opposite signs , the imaginary part of the green tensor is solely determined by the imaginary part of @xmath333 , which physically depicts the interaction between atoms and meta slabs . therefore , we are left with the integral equations ( [ e18p ] ) and ( [ e18q ] ) to be evaluated . in order to evaluate the integrals we need explicit expressions of the reflection coefficients . they are determined from the fresnel s law and the boundary conditions that we can easily obtain reflection coefficients of the surface between different materials @xmath334 where @xmath335 indicates the direction of propagation of the wave , from the material @xmath242 to @xmath37 . for a multiple - reflection case , the reflection coefficient is given by @xmath336 to evaluate the expression @xmath337\cdot{\bf p}^{\ast}_{j}$ ] , we have to specify the orientation of the atomic dipole moments . if we assume that the atomic dipole moments are parallel to each other and are oriented in the @xmath5 plane @xmath338 , \quad { \bf p}_{2 } = \frac{1}{\sqrt{2 } } |{\bf p}_{2}|[1,0,1 ] , \label{w31}\end{aligned}\ ] ] we then find that the term @xmath337\cdot{\bf p}^{\ast}_{j}$ ] becomes @xmath339\cdot{\bf p}^{\ast}_{j } \nonumber\\ & = |{\bf p}_{i}||{\bf p}_{j}|\left(\bar{\bf x}+ \bar{\bf z}\right)\cdot\im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r}_j,\omega)]\cdot(\bar{\bf x } + \bar{\bf z } ) \nonumber\\ & = |{\bf p}_{i}||{\bf p}_{j}|\left\{\im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r}_j,\omega)]_{xx}+\im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r}_j,\omega)]_{xz}\right . \nonumber\\ & \left . + \ , \im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r}_j,\omega)]_{zx}+\im[\stackrel{\leftrightarrow}{\bf g}\!({\bf r}_i,{\bf r}_j,\omega)]_{zz}\right\ } .\label{w31}\end{aligned}\ ] ] since @xmath340 , the term @xmath341\cdot{\bf p}^{\ast}_{j}$ ] therefore becomes independent of the off - diagonal elements . thus , with the choice of the orientation of the atomic dipole moments given by eq . ( [ w31 ] ) nonvanishing contributions to @xmath341\cdot{\bf p}^{\ast}_{j}$ ] can come only from the diagonal @xmath2 and @xmath4 components of the green tensor . although the choice of the dipole polarization in the @xmath5 plane affects the contribution of the diagonal elements of the green tensor , it has no effect on the contribution of the off diagonal elements since independent of the atomic polarization all off - diagonal elements involving the @xmath3 component are zero . the expressions for the components of the green tensor , eqs . ( [ e18p ] ) , are exact and as such could be evaluated numerically for any values of the parameters involved . however , for many of the situations of interest the atoms will be located close to the interface , at a distance @xmath14 small compared to the radiation wavelength @xmath342 . thus , we may limit ourselves then to considering the case , @xmath343 . furthermore , we may assume that the thickness of the material slab is much larger than the localization length , @xmath344 . in this case , the factors @xmath345 $ ] in the integrants of eqs . ( [ e18p ] ) can be discarded leaving only the terms with factor @xmath346 $ ] to contribute to these integrants . moreover , in this limit the module of the term @xmath347 in the expressions for @xmath348 and @xmath349 is much smaller than 1 and thus can also be neglected . we can then approximate @xmath350 , as given in eq . ( [ e21 ] ) , by @xmath351 and when this result is inserted in eq . ( [ e20 ] ) , we obtain for the reflection coefficients @xmath352 before proceeding further we would like to point out that the reflection coefficient at the interface between the en and mn materials , eq . ( [ e23 ] ) , differs significantly from the reflection coefficient at the interface between an ordinary dielectric or a metal material @xcite . according to eq . ( [ e20 ] ) , the dispersion relation for tm - polarized mode propagating along the interface can be approximated by @xmath353 . when @xmath354 and @xmath355 $ ] , i.e. two slabs are perfectly paired , the dispersion relation reduces to @xmath356 . in this case , the propagation of the plasma mode of frequency @xmath357 is independent of the parallel component of the wave vector . this property is different from that of the plasma mode propagating at the interface of ordinary materials . in this case , the resonant plasma frequency depends on @xmath358 @xcite . hence , adopting the results of eq . ( [ e1 ] ) , and assuming the frequency region of @xmath359 , the reflection coefficients at the interface between the en and mn slabs take the form @xmath360 where @xmath361 and , for simplicity , we have assumed that the dissipation parameters of the two slabs are equal , @xmath362 . if we now substitute eqs . ( [ e24 ] ) and ( [ e25 ] ) into eqs . ( [ e18p ] ) , we can perform the integration and arrive at the analytical expressions for the imaginary parts of the components of the green tensor . following the result ( [ w31 ] ) , we evaluate only the diagonal @xmath2 and @xmath4 components of the green tensor . thus , when setting the parameter values @xmath363 and @xmath364=-\mu_2 $ ] , @xmath365=-\varepsilon_1 $ ] , the explicit expressions for the imaginary parts of the diagonal @xmath4 component of the one- and two - point green tensors are @xmath366 = \frac{\gamma\omega_s}{12\pi k^{2}\left(\delta\omega^2 + \frac{1}{4}\gamma^{2}\right)(2z_{0})^3 } , \label{w37}\end{aligned}\ ] ] and @xmath367 & = \frac{\gamma\omega_s}{12\pi k^{2}\left(\delta\omega^{2}+\frac{1}{4}\gamma^{2}\right)(2z_{0})^3 } \nonumber\\ & \times f\left[\frac{3}{2},2,1;-\frac{x_{21}^2}{(2z_{0})^2}\right ] .\label{w38}\end{aligned}\ ] ] where @xmath368 , @xmath15 is the distance between the atoms , and @xmath369 is the hypergeometrical function . similarly , for the diagonal @xmath2 component of the one- and two - point green tensors , we find @xmath370 = \im[g_{xx}({\bf r}_{2},{\bf r}_{2},\omega ) ] \nonumber\\ & = \frac{\gamma c^{2}\omega_{s}\!\left\{1-k_{s}^{2}\re[\mu_1(\omega_s)](2z_{0})^2\right\}}{24\pi \omega^{2}\left(\delta\omega^2 + \frac{1}{4}\gamma^{2}\right)(2z_{0})^3 } , \label{w39}\\ & \im[g_{xx}({\bf r}_2,{\bf r}_1,\omega ) ] = \im[g_{xx}({\bf r}_{1},{\bf r}_{2},\omega ) ] \nonumber\\ & = \frac{\gamma c^{2}\omega_{s}}{24\pi \omega^{2}\left(\delta\omega^2 + \frac{1}{4}\gamma^{2}\right)(2z_{0})^3 } \nonumber\\ & \times \left\{\!f\!\left[\frac{3}{2},2,2;-\frac{x_{21}^2}{(2z_{0})^2}\right ] -3\frac{x_{21}^2}{(2z_{0})^2}f\!\left[\frac{5}{2},3,3;-\frac{x_{21}^2}{(2z_{0})^2}\right]\right . \nonumber\\ & \left . -\re[\mu_1(\omega_s)](2z_{0}k_{s})^{2 } f\!\left[\frac{1}{2},1,2;-\frac{x_{21}^2}{(2z_{0})^2}\right]\right\ } .\label{w40}\end{aligned}\ ] ] these expressions show that the imaginary parts of the green tensor , when evaluated as a function of @xmath371 , are of the form of a lorentzian centered at the plasma frequency @xmath372 and possesses a bandwidth @xmath373 . thus , for small @xmath331 we expect that the largest contributions to the field come from @xmath374 . therefore , when substituting eqs . ( [ w37])-([w40 ] ) into eq . ( [ e12w ] ) , we can replace @xmath375 by @xmath376 and extend the lower limit in the integration over @xmath371 to @xmath377 . then after a simple algebra we obtain eqs . ( [ ek1 ] ) and ( [ ek2 ] ) . i. s. gradshteyn and i. m. ryzhik , _ table of integrals , series , and products _ ( academic , orlando , 1980 ) . i. avrutsky , i. salakhutdinov , j. elser , and v. podolskiy , link:\doibase 10.1103/physrevb.75.241402[phys . b * 75 * , 241402 ( 2007 ) . ] e. n. economou , link:\doibase 10.1103/physrev.182.539[phys . rev . * 182 * , 539 ( 1969 ) . ]

we study the dynamics of two two - level atoms embedded near to the interface of paired meta - material slabs , one of negative permeability and the other of negative permittivity . the interface behaves as a plasmonic waveguide composed of surface - plasmon polariton modes . it is found that significantly different dynamics occur for the resonant and an off - resonant couplings of the plasma field to the atoms . in the case of the resonant coupling , the plasma field does not appear as a dissipative reservoir to the atoms . we adopt the image method and show that the dynamics of the two atoms are completely equivalent to those of a four - atom system . moreover , two threshold coupling strengths exist , one corresponding to the strength of coupling of the plasma field to the symmetric and the other the antisymmetric mode of the two - atom system . the thresholds distinguish between the non - markovian ( memory preserving ) and markovian ( memoryless ) regimes of the evolutions that different time scales of the evolution of the memory effects and entanglement can be observed . the markovian regime is characterized by exponentially decaying whereas the non - markovian regime by sinusoidally oscillating contributions to the evolution of the probability amplitudes . the solutions predict a large and long living entanglement mediated by the plasma field in both markovian and non - markovian regimes of the evolution . we also show that a simultaneous markovian and non - markovian regime of the evolution may occur in which the memory effects exist over a finite evolution time . in the case of an off - resonant coupling of the atoms to the plasma field , the atoms interact with each other by exchanging virtual photons which results in the dynamics corresponding to those of two atoms coupled to a common reservoir . in addition , the entanglement is significantly enhanced under the off - resonant coupling .

in cases when solving the lippmann - schwinger or bethe - salpeter type of equation is numerically involved , one often resorts to a partial - wave decomposition ( pwd ) in the center - of - mass ( cm ) frame . in doing so one can exploit the spherical symmetry of the interaction and perform the integration over the two - dimensional solid angle of the intermediate momentum analytically . while this reduces the equation s dimension by two , one has to deal with summing the partial - wave series , and hence this procedure is beneficial when only a few partial waves dominate . in the case when many partial waves must be taken into account , when restriction to the cm frame is not desirable , or when the potential is not spherically - symmetric , the partial - wave expansion is not helpful and one has to face the complexity of three- or four - dimensional integral equations . fortunately , as had been noted by glckle and collaborators @xcite in the context of the nucleon - nucleon ( @xmath1 ) interaction , the dependence on the intermediate momentum azimuthal angle factorizes and can still be performed analytically without employing any kind of expansion or truncation . while this procedure has been successfully applied a number of times to the @xmath1 situation @xcite , here we would like to examine general conditions which potentials must satisfy to factorize the azimuthal integration . we then apply it to solve a specific example of relativistic potential scattering in the pion - nucleon ( @xmath0 ) system and compare with the usual method of using the partial - wave expansion . in section ii we give the general requirements on the potential that allow one to remove the azimuthal angle dependence in the integral equation . in section iii we focus on the bethe - salpeter equation for @xmath2n scattering with one - nucleon - exchange potential and show in detail how the azimuthal - angle dependence can be integrated out in this case . furthermore , in section iv , we solve the resulting equation using a quasipotential approximation and compare the solution to the one obtained using the partial - wave expansion . in section v we examine an extension of this approach to the calculation of pion electroproduction from the nucleon including the @xmath0 final state interaction . our conclusions are summarized in section vi . the starting point in calculating observables of a two - body scattering process is an equation for the scattering amplitude . we shall assume relativistic scattering , in which case the equation is a 4-dimensional integral equation of the bethe - salpeter type : @xmath3 where @xmath4 is the sought t - matrix , @xmath5 is the two - particle propagator , and @xmath6 is the two - particle - irreducible potential . moreover , throughout the paper , @xmath7 , @xmath8 , @xmath9 stand for the relative 4-momenta of the incoming / intemediate / outgoing channel while @xmath10 is the total 4-momentum with @xmath11 , @xmath12 , @xmath13 and @xmath14 , @xmath15 , @xmath16 the incoming / intermediate / outgoing momenta of particle one and particle two , respectively . in order to investigate the conditions under which the above equation can be integrated over the intermediate azimuthal angle we work in the helicity basis and only display the dependence on the azimuthal angle and helicity : @xmath17 an important point here is that the two - particle propagator @xmath5 can always be made independent of the intermediate angle @xmath18 by choosing the total three - momentum along the @xmath19-axis , i.e. , choosing the _ co - linear frame _ : @xmath20 . furthermore , we shall observe that in the case when only spin-0 and spin-1/2 particles are involved , the azimuthal - angle dependence of the fully off - shell potential in the co - linear frame is given as follows @xmath21 where @xmath22 and @xmath23 stand for the combined helicities of the initial and final state , respectively . the half - off - shell potential then takes a very simple form : . v_( , ) @xmath22 is the helicity of the on - shell state . it is in this case , when conditions and are met , the exact integration over the azimuthal - angle can readily be done . first , by using in , we see that the azimuthal dependence of the t - matrix is given by : t_( , ) = e^-it_(- ) e^i . since @xmath24 and @xmath25 only depend on difference @xmath26 , we expand them in a simple fourier series : v_ ( ) = _ m v^(m)_ e^im , t_ ( ) = _ m t^(m)_ e^im . it is straightforward to show that their fourier transforms , v^(m)_ = _ 0 ^ 2 v_ ( ) e^-i m , t^(m)_ = _ 0 ^ 2 t_ ( ) e^-i m , satisfy the following equation which does not involve the @xmath27-integration : t^(m)_ = v^(m)_ + _ v^(m)_g t^(m ) _ . in principle , @xmath28 runs to infinity and so we have an infinite number of equations to solve even though they are not coupled . fortunately , since only the half off - shell potential is needed to solve the equations and it obeys condition , the corresponding fourier transform is non - vanishing only for @xmath29 : .v^(m)_|_half - off - shell = ._-m v_(0)|_half - off - shell . the scalar system is the simplest one where this procedure can be demonstrated . in that case the potential is a scalar function of scalar products of relevant 4-momenta : v(q,q ; p)=v(qq , pq , pq , q^2 , q^2 , p^2 ) given @xmath30 and similarly for @xmath9 , we easily convince ourselves that , in the co - linear frame , the azimuthal dependence enters only through the product : qq = q_0 q_0 - ||||and hence it is of the necessary form given in . furthermore , in the half - off - shell case the momentum of the on - shell state , say @xmath31 , can always be chosen along the @xmath19-axis , i.e. , such that @xmath32 . hence the half - off - shell potential is independent of azimuthal angles which fulfills condition for the spinless case . the two - particle propagator @xmath33 is of course independent of @xmath27 in the co - linear frame . once we have found that conditions and are satisfied , while @xmath5 is independent of @xmath27 , the integration over @xmath27 can be done immediately . we will now show this more explicitly for the more complicated case of a scalar - spinor system . consider the bethe - salpeter equation for the case of elastic scattering of a scalar with mass @xmath34 the `` pion '' on a spinor with mass @xmath35 the `` nucleon '' . we attribute the momenta @xmath14 , @xmath16 to the nucleon and @xmath11 , @xmath13 to the pion . the relative 4-momentum of the incoming channel is conveniently defined by @xmath36 , where lorentz scalars @xmath37 and @xmath38 are given by & = & pp / s= ( s+m_n^2-m_)/2s , + & = & kp / s=(s - m_n^2+m_)/2s , with @xmath39 . similarly one defines @xmath40 and @xmath41 as the relative 4-momenta of the outgoing and intermediate state , respectively . in terms of these variables , the two - body @xmath2n green s function of is : @xmath42 projecting the equation onto the basis of the nucleon helicity spinors ( defined in appendix a ) , we obtain @xmath43 where the helicity amplitudes are defined as t^ _ ( q,q , p)=(1/4 ) @xmath6 , while the defining equation for @xmath44 is , with @xmath45 and @xmath46 . the most general lorentz structure of the fully off - shell potential in the helicity basis can be written in the form ] ] v^_ ( q,q;p ) = the dot - products of the relevant momenta , i.e. , a_i = a_i(qq , pq , pq , q^2 , q^2 , p^2 ) . considering the dependence of these functions on the azimuthal angles of @xmath31 and @xmath9 , we see that in the _ co - linear frame _ it is given by the difference @xmath26 , for the reason described below . the rest of the @xmath27-dependence resides in the nucleon spinors . according to , in the co - linear frame we need to consider only @xmath47 where @xmath48 s are the pauli spinors ( cf . appendix a ) , @xmath49 , @xmath27 and @xmath50 , @xmath51 define the orientation of @xmath52 and @xmath53 , respectively . since , ^_ ( , ) _ ( , ) & = & e^-ie^i , + ^_ ( , ) _ 3 _ ( , ) & = & e^-ie^i , we observe that the @xmath27-dependence of these elements is of the desired form . and for the half - off - shell situation , where we can choose @xmath32 ( hence @xmath54 , in the co - linear frame ) and use @xmath55 , we find the form , ^_ ( , ) _ ( 0 , ) & = & e^-i(-) d^1/2_( ) , + ^_ ( , ) _ 3 _ ( 0 , ) & = & e^-i(-)(-1)^1/2- d^1/2_( ) , which obeys the necessary half - shell condition . therefore , we have demonstrated that the azimuthal - angle dependence of a pion - nucleon potential in the co - linear frame always satisfies conditions and . it is also apparent from that the two - particle green s function does not have any azimuthal dependence in that frame . thus the integration over @xmath27 can exactly be done in the bethe - salpeter equation for @xmath0 system by means of the procedure of sec . similar arguments apply in the case when both particles have spin 1/2 , _ e.g. _ , the nucleon - nucleon ( nn ) scattering . it should only be noted that in this case the potential satisfies conditions and with @xmath56 , @xmath57 . in other words , helicities of the two particles must be combined . the standard route to solution of a potential scattering equation such as is to decompose it into an infinite set of equations for partial - wave amplitudes , see e.g. @xcite . the advantage of doing a partial wave decomposition is that the equation for each partial wave is of 2 lesser dimensions than the original equation , while the partial - wave series is usually rapidly converging , hence only the first few partial - wave amplitudes need to be solved for . on the other hand , solving for the full amplitude directly has its own important benefits . and if the exact azimuthal - angle integration can be done _ a priori _ , the numerical feasibility of this approach becomes comparable to the pwd method . in this section we would like to compare the two methods for the example of solving a relativistic equation for the @xmath0 system . for our toy - calculation potential we take the one - nucleon exchange , , and use the _ instantaneous _ approximation , thus neglecting retardation effects in the potential . the latter approximation allows us to perform the relative - energy ( @xmath58 ) integration such that we are left with a relativistic 3-dimensional salpeter equation : t_^(,;p ) = v^_(,;p ) + _ v^_ ( ,;p ) g_et^()(;p)t_^ ( , ; p ) , where the equal - time two - particle propagator in the cm system is given by @xmath59 this 3-dimensional equation for @xmath0 has been described in detail and solved using a pwd in the cm system by pascalutsa and tjon @xcite . we , on the other hand , solve this equation by using the framework of the two previous sections to reduce the @xmath27-integration analytically and solve numerically the resulting 2-dimensional integral equation for the @xmath28-th fourier component of the full amplitude : @xmath60 where , without loss of generality , we have also assumed the cm frame . the explicit form of the fourier transform of the one - nucleon - exchange potential is worked out in appendix b. let us emphasize that it is necessary to solve for only one of the fourier components ( either @xmath61 or @xmath62 ) , the other ones either vanish or can be obtained by relations due to the parity and time - reversal invariance . we solve by the pade approximants as in refs . @xcite thus maintaining exact elastic unitarity . the numerical integrations are performed by the gauss - legendre method . the integral over @xmath63 in contains the cut singularity at @xmath64[s-(m_{n}+m_{\pi})^{2}]/4s}\equiv \hat q$ ] , which is handled by the well - known identity : @xmath65 where @xmath66 denoted the principal - value integral . when computing the latter the integration region is divided into two intervals : @xmath67 $ ] , and @xmath68 . the gaussian points are then distributed separately for each interval to make use of the property that an even number of gaussian points falls symmetrically with respect to the middle of the interval hence the singularity in the middle of the first interval is avoided . the polar angle integration is straightforward for both the principal value term and the imaginary contribution . we find it sufficient to use 16 gauss points for the momentum integration and 8 points for the polar - angle integration . upon increasing the number of points to 32 and 16 respectively , the results change by less than @xmath69 in the considered energy range . in all cases we found that 6 iterations combined with the use of pad approximants works extremely well . after we solve to find the full @xmath0 @xmath4-matrix , we can of course also find the partial wave amplitudes : @xmath70 where @xmath71 is the angle between @xmath72 and @xmath73 . we then investigate the convergence of the partial wave series : @xmath74 in particular , in fig.[angle300f3 ] and fig.[energy1pif4 ] we plot the on - shell values of @xmath75 compared with the truncation of the partial - wave series for 3 terms and 5 terms ( i.e. , @xmath76 and @xmath77 respectively ) . angular dependence for @xmath78 at @xmath79 mev . solid line is the full calculation , dashed and dotted are the resumming of partial terms . ] energy dependence for @xmath80 at @xmath81 . the lines are defined the same as in fig.[angle300f3 ] ] in order to compare the computational efficiency of the two methods , we compare the number of partial waves needed to achieve convergence in the pwd method with the number of gauss points for the polar - angle integration which appear in the `` w / o pwd '' method . the figures show that the effect of truncations of the partial - wave series increases with the angle ( fig . [ angle300f3 ] ) the energy of the incoming @xmath2 ( fig . [ energy1pif4 ] ) . in our particular case of one - nucleon exchange computing @xmath82 or more partial wave amplitudes is sufficient to reproduce the full result to a 1 per cent accuracy in a broad energy domain . thus , in this case , the efficiency of the two methods is comparable since we need 5 multipoles versus 8 gauss points of the polar - angle integration . it is important to emphasize that the ability to do the azimuthal - angle integration analytically is necessary to achieve comparable efficiency . we have checked that it usually takes at least 16 gaussian points for the azimuthal integration which slows down the calculation by more than an order of magnitude . our procedure for performing the analytic @xmath27-integration is applicable in the photo- or electro - meson production to first order in the electromagnetic coupling . here we describe the extension to the case of @xmath2 photoproduction within a simple final - state - interaction model @xcite . the model begins with the following coupled channel equation : ( cc t _ + t _ & t _ ) =( cc v _ & v _ + v _ & v _ ) + ( cc v _ & v _ + v _ & v _ ) ( cc g _ & 0 + 0 & g _ ) ( cc t _ + t _ & t _ ) where @xmath4 and @xmath6 are the amplitudes and driving potentials of the @xmath2n scattering @xmath83 , pion photo - production @xmath84 , absorption @xmath85 , and the nucleon compton effect @xmath86 , respectively . the above equations are solved up to first order in the electromagnetic coupling @xmath87 , hence preserving two body unitarity to this order only . in solving the photoproduction scattering equation we calculate first @xmath88 as described for @xmath2n scattering and we then iterate in the following manner : @xmath89 where we used @xmath90 from time - reversal invariance . this solution procedure is obviously suitable for our case since the half - shell @xmath91 has a simple azimuthal angle dependence similar to the case of @xmath88 ( see ) . the reduced kernel ( see ) has two terms rather than the one term in the @xmath2n case due to the complication " of having to couple a spin 1 photon to spin 1/2 as opposed to coupling a spin 0 meson to spin 1/2 . for example , if one considers the nucleon u - channel exchange ( compare to the @xmath2n case in ) the half - shell photoproduction potential can be written as : v_^(q,q)= _ 1v_^(q_0,|*q*,q_0,|*q*|,)e^-i(--) + _ 2v_^(q_0,|*q*,q_0,|*q*|, ) e^-i(+) where @xmath92 represents the helicity of the incoming photon . one sees that when is iterated in two de - coupled scattering equations are obtained ( each corresponding to @xmath93 or @xmath94 ) . for each of these equations , one can show that the corresponding @xmath51 dependence re - appears after doing the @xmath18 integration and therefore once again we can perform the azimuthal - angle integration analytically . as in the @xmath2n case , the resulting reduced kernels obey 2-d integral equations . as a check of our procedures we calculated the @xmath95-channel contribution to pion photoproduction using the analytic azimuthal - angle integration along with 2-d numerical integration and compared to the results of refs . @xcite obtained using the multipole expansion . at @xmath96 of @xmath97 mev with five multipoles we found agreement to better than 1% over a wide angular range . in recent years glckle and collaborators @xcite introduced a method which greatly simplifies the numerical integration of two - body scattering equations without performing the partial - wave expansion . the method exploits a certain azimuthal symmetry of the potential thus allowing exact integration of the azimuthal dependence . in this paper we have established general form of the azimuthal - dependence of the kernel which allows for this procedure to go through . we have argued that these conditions is in general applicable to any system of spin- 0 and/or spin- @xmath98 particles we have applied this method to the case of @xmath2n system and . with some extra effort it can be applied to higher spin systems , however the procedure becomed increasingly complex with the increase of the spin of the involved particles . we have successfully applied the method to pion photo- and electro - production from the nucleon , however only to the leading order in electromagnetic coupling . even though we have used the salpeter equation for or numerical exercises , the method can of course be applied to the full 4-d bethe - salpeter equation , which for the @xmath0 system has so far been solved in partial waves only @xcite . performing the azimuthal - angle integration analytically greatly facilitates finding the full solution and makes the numerical feasibility of this approach comparable to finging the solution using the partial - wave expansion . we define the four - component nucleon helicity spinors as follows : u_(e_p,*p*)= ( c + 2 ) _ ( , ) where @xmath99 is the helicity , @xmath100 is the energy , @xmath71 and @xmath27 are the spherical angles of the 3-momentum @xmath101 , and @xmath102 is the two - component pauli spinor . the positive- and negative - energy nucleon spinors in the convention of kubis @xcite are defined as follows : u_^()(*p * ) = u_(e_p,*p * ) , they satisfy the following orthogonality and completeness conditions : & & u_^ ( ) ( ) u_^( ) ( ) = _ _ , + & & _ , u_^ ( ) ( ) u_^ ( ) ( ) = 1 . the pauli spinors along the @xmath19-axis are given by @xmath103 while along an arbitrary direction @xmath104 they can be obtained using the wigner rotation functions : @xmath105 or , explicitly , @xmath106 as an example , we consider the @xmath95-channel nucleon exchange potential given by the graph in and the following expression , @xmath107 ^ 2-m_n^2+i\veps } \gamma^{5 } \gamma \cdot ( \beta p - q ) ] , \end{aligned}\ ] ] where @xmath37 and @xmath38 are defined in . for simplicity we choose the cm frame , where the potential in the helicity basis takes the form : @xmath108u_{\rho''}^{\lambda''}({\bf q''})\end{aligned}\ ] ] with @xmath109\end{aligned}\ ] ] @xmath110 \nonumber\\ & & + ( p'^{2}+m_{n}^{2})(p''_{0}-\rho '' e_{p '' } ) \nonumber \\ & & + ( p''^{2}+m_{n}^{2})(p'_{0}-\rho ' e_{p'}).\end{aligned}\ ] ] the azimuthal dependence @xmath111 arises from dirac spinors and from various scalar products involving the the four vector @xmath8 . choosing the vector part of the total momentum @xmath112 to be along the @xmath19-axis ( or to be zero in the cm frame ) allows the @xmath113 dependence , for the fully _ off - shell _ potential , to be displayed in the form : @xmath114 where @xmath115 , @xmath116 , and @xmath117 are factors which depend on the type of the diagram and of the exchanged particle , but are independent of the azimuthal angle . the quantities , @xmath118 @xmath119 are factors which result from the helicity spinors . in we have employed the usual trigonometric relation between two arbitrary directions defined by @xmath120 and @xmath73 : @xmath121 it is easy to see that the fully off - shell potential in has the azimuthal dependence of . furthermore , in iterating the quantization axis is defined by the _ on - shell _ relative momentum @xmath72 ( i.e. @xmath122 ) , hence reduces to @xmath123 , therefore the _ half - off - shell _ potential reduces to : @xmath124 which if of the form of the result in . therefore , the azimuthal angle dependence can be removed from the bethe - salpeter equation for this case . we achieved this result by explicitly displaying the azimuthal angle dependence and align @xmath125 with the z - axis so that only @xmath126 and @xmath127 appear in @xmath128 . the presence of @xmath129 or @xmath130 would introduce additional azimuthal angle dependence in the spinor matrix elements and make the algebra much more complicated . for the @xmath95-channel nucleon exchange the coefficients @xmath117 are : @xmath131 @xmath132 from these relations one can exactly identify the angular dependence of potential given in in the 4-product , @xmath133 the relative momenta @xmath9 and @xmath8 , defined in section ii , are to be introduced in - by @xmath134 and @xmath135 where @xmath136 after applying standard trigonometric manipulations : @xmath137,\ ] ] and @xmath138.\ ] ] the integral over the azimuthal angle of the intermediate momentum in can be reduced to integrals of the following type : @xmath139 for values @xmath140 , this definite integral can be evaluated analytically to obtain : @xmath141\end{aligned}\ ] ] where @xmath142 . the results given above in work for all standard particle exchanges in the s , t , or u channels . furthermore , it should be noted that additional azimuthal angle dependences introduced by various form factors can easily be handled by simple algebraic methods . the maximum power of @xmath143 needed for a particular diagram may increase ( for example , n=2 for u - channel @xmath144 exchange ) . in addition , @xmath145 will , in general , contain a sum of various terms corresponding to each diagram included . however , all of these terms can be evaluated using . in addition as noted earlier , this procedure is not at all affected by the _ equal time _ approximation and can be applied in the same manner to the full 4-d bethe salpeter equation . this work was performed in part under the auspices of the u. s. department of energy , under the contract no . de - fg02 - 93er40756 with ohio university and the national science foundation under grant nsf - sger - 0094668 . i. fachruddin , ch . elster , and w. glockle , phys . c62 , 044002 n. k. devine and s. j. wallace , phys . c * 48 * , r973 ( 1993 ) ; n. k. devine , phd thesis ( university of maryland , 1992 ) . g. ramalho , a. arriaga and m. t. pena , nucl . a * 689 * , 511 ( 2001 ) . m. jacob and g.c . wick , ann . * 7 * , 404 ( 1959 ) . kubis , phys . d * 6 * , 547 ( 1972 ) . v. pascalutsa and j. a. tjon , in preparation . nieland and j.a . tjon , phys . * 27b * , 309 ( 1968 ) . a. d. lahiff and i. r. afnan , few body syst . * 10 * , 147 ( 1999 ) ; phys . rev . c * 60 * , 024608 ( 1999 ) .

considering two - body integral equations we show how they can be dimensionally reduced by integrating exactly over the azimuthal angle of the intermediate momentum . numerical solution of the resulting equation is feasible without employing a partial - wave expansion . we illustrate this procedure for the bethe - salpeter equation for pion - nucleon scattering and give explicit details for the one - nucleon - exchange term in the potential . finally , we show how this method can be applied to pion photoproduction from the nucleon with @xmath0 rescattering being treated so as to maintain unitarity to first order in the electromagnetic coupling . the procedure for removing the azimuthal angle dependence becomes increasingly complex as the spin of the particles involved increases . = 10000

the sloan digital sky survey ( sdss ) ( york , 2000 ) and sloan extension for galactic understanding and exploration ( segue ) ( yanny , 2009 ) have spectral observations of @xmath1 stars as of data release 9 ( dr9 ) ( ahn , 2012 ) . these spectral observations include tens of thousands of stars that fall in the color range of the rrl instability strip , including several thousand rrl variables . these spectra are very useful in determining the rrl radial velocity and can be used to determine the overall metallicity . a typical spectrum from the sdss database is a composite spectrum produced from multiple single observations . these single observations sometimes fall on the same night but often include observations that occur days and even weeks apart . combining these observations allows for easy removal of cosmic ray events and helps to increase the signal - to - noise ratio for the star . this is a very useful technique if the star is non - variable . for variable stars , and in particular the rrl stars , combining single spectra can distort the spectral line profiles , which change as a function of pulsation phase . the sdss - dr9 includes the flux- and wavelength - calibrated , single - epoch spectra ( ses ) that were used to create the final composite spectrum . with this data in hand we now have the ability to measure line strengths that are not distorted from the combining effect and to use the changes in line strength from multiple ses to identify variables and estimate pulsation phase at the time of observation . as rrls pulsate and their surface temperature changes , there is significant change to the strength of absorption lines . in particular , the hydrogen balmer lines and the caii k line undergo substantial changes as a function of pulsation phase . in general , the strength of the balmer lines are anti - correlated to the strength of the caii k line , which shrinks as the temperature increases . this anti - correlation can be exploited in the ses spectra to identify rrl variables . furthermore , the amount of change between ses observations can be used to estimate the phase information for the variable . the phase is of crucial importance for metallicity determination , because at certain pulsation phases particularly during light - rise time shocks in the photosphere cause substantial distortions in the balmer lines , making it difficult to determine precise metallicity . in this talk we present results for our identification and phase estimation technique by comparing to known sdss variables in stripe 82 ( sesar , 2010 ) . here we constrain the test to rrab variables and to ses that are taken in a single night . future testing will include rrc variables and stars with ses spectra taken on multiple nights . we have chosen eleven rrab variables from stripe 82 , with phase information reported in the sesar paper . we also chose five stars that are non - variable to test our ability to detect variability . all test stars have ses taken on a single night with observations taken consecutively . the number of ses range from a minimum of three spectra up to seven . table 1 lists these stars , their period , the time base line of observation ( t - base ) , the percent of the phase covered by the ses , and the magnitude of the star . the ability to detect variation will be a function of t - base line , the period of the variable , and the signal - to - noise of the ses spectra . a further constraint is the pulsation phase during the observation because changes in line strengths occur more rapidly at certain pulsation phases . for the non - variable stars , we chose ses that covered similar ranges in t - base and signal - to - noise to test for null detections . rccccc 0989 - 52468 - 0032 & ab & 0.580767811 & 0.7139 & 0.0512 & 18.388 + 0702 - 52178 - 0100 & ab & 0.649984725 & 0.7522 & 0.0482 & 17.845 + 0402 - 51793 - 0117 & ab & 0.797848004 & 0.6364 & 0.0332 & 15.763 + 1918 - 53240 - 0121 & ab & 0.512230079 & 0.6528 & 0.0531 & 14.782 + 1507 - 53763 - 0132 & ab & 0.722990487 & 0.6308 & 0.0364 & 17.946 + 1502 - 53741 - 0170 & ab & ... & 1.7386 & 0.1200 & 16.383 + 0380 - 51792 - 0237 & ab & 0.569410914 & 0.8797 & 0.0644 & 18.204 + 2636 - 54082 - 0248 & ab & 0.542616712 & 2.2861 & 0.1755 & 17.998 + 0371 - 52078 - 0334 & ab & 0.622160962 & 0.6355 & 0.0425 & 16.913 + 0685 - 52203 - 0526 & ab & 0.793004216 & 1.0889 & 0.0572 & 18.850 + 0701 - 52179 - 0570 & ab & 0.557696402 & 0.6147 & 0.0459 & 17.693 + 1903 - 53357 - 0185 & nv & ... & 1.3319 & ... & ... + 1918 - 53240 - 0241 & nv & ... & 0.6528 & ... & ... + 1084 - 52591 - 0422 & nv & ... & 0.7633 & ... & ... + 1138 - 53228 - 0444 & nv & ... & 1.1444 & ... & ... + 1133 - 52993 - 0463 & nv & ... & 1.0608 & ... & ... -0.4 in to test for variability , we use empirical templates obtained from observations of metallicity standard rrab stars . these spectra were taken at mcdonald observatory on the 2.1meter telescope using the es2 , low - resolution spectrograph . these stars are being used in our metallicity calibration , which has been reported at this meeting by e. spalding . the spectra were taken over large portions of the pulsation phase . currently none of the stars have complete phase coverage , so for this study we have used a combination of templates from the variable stars x ari , tt lyn , tw lyn and tu uma . the combination of these stars gives full phase coverage with a phase resolution of @xmath2 . in our technique we make use of the quotient spectrum , the division of one spectrum by another , to characterize the changes in spectral line strength in a given phase range . because our current template sample is constructed from stars with various metallicities , we only divide spectra for a given star to construct the phase range . this minimizes effects of metallicity , particularly on the caii k line , for the quotient templates . figures 1 and 2 show two quotient templates at different phases . the quotient spectra would have a value of unity , across all wavelengths , if there were no spectral changes . in figure 1 the hydrogen balmer lines can be seen in `` absorption '' ( decreasing line strength ) while the caii k line shows an `` emission '' peak ( increasing line strength ) . the blend of caii h / h@xmath3 displays broad wings and an `` emission '' peak at the line center , clearly showing the anti - correlated nature of the caii lines and the balmer lines . in figure 2 , during the light - rise phase , the effect is reversed with the balmer lines growing while the caii lines are shrinking . comparing these two figures , it becomes evident how the phase can be estimated , based on the sign and strength of the spectral features from the quotient templates . to test our technique we make a quotient spectrum from the division of the first and final ses spectra for a given star . a particularly good result can be seen in figure 3 . here the black line is the quotient spectrum for a stripe82 variable , while the red line is the best matched quotient template . it is clear in this figure that the star is a variable and at a phase approaching 1.0 . to quantify the signal for the variable candidates we perform a fft ( iraf task fxcor ) to correlate the template and observed quotient spectra . we use a welch fourier filter with a wavenumber range of @xmath4 , where the low frequency cutoff removes large scale variations produced by the flux normalization and a high frequency limit to remove high frequency noise . we further constrained our correlation peak to the velocity range of @xmath5 km / s , to eliminate correlation peaks that would have unrealistic velocities for stars in the galaxy . the tonry - davis ratio , which is the ratio of correlation height to anti - symmetric noise ( tonry & davis , 1979 ) , is used to determine the significance of the correlation peak and to select the best quotient template to match the observed data . figure 4 shows the tdr value as a function of various phase templates . the filled black circles are the stripe82 variable and the black - hatched squares are the auto - correlation for the @xmath6 template relative to all template spectra . the red circles and red squares are the anti - correlation responses for the stripe82 star and the template , respectively . the anti - correlation is found to be inverting the templates , and thus the greater the negative peak , the stronger the anti - correlation . it is clear that the stripe82 star has virtually the same template response as the template with @xmath7 . this increases the likelihood that the tdr peak is real and not a random variation . as a final test , we use the velocities from the star s spectra , and test if the velocity from the peak correlation of the quotient spectra , yield a consistent result . for this strong signal star it is clear that the tdr peak occurs at the velocities of the individual spectra ( figure 5 ) . in cases where the tdr peak is weak and only due to noise variations , it is less likely that the velocity of the correlation peak will match the velocity of the single spectra . this constraint adds more certainty to the significance of the tdr peak . both the auto - correlation and the radial velocity constraints must be met in order for the variability result to be considered positive . of the eleven rrab stars that we tested , only four were found to have a significant tdr peak . the other seven variables failed either the auto - response criteria , the radial velocity criteria , or both . of the six non - variable stars that we tested , all six failed the variability test . figure 6 is a plot of the tdr peak height as a function of the time baseline . the black circles are the positive detections and the red circles are the null results . the green squares represent the non - variable stars . in general the null rrab detections are for time baselines that are less than 45 minutes , indicating that the time baseline length is an important criteria for variability detection . the low peak height of the non - variable stars shows that the height of the tdr peak is an indicator of variability , as expected . figure 7 shows the results for the predicted phase versus the phase computed from the ephemeris data of sesar . the four positive detections are close to the one - to - one correspondence line , although there appears to be a systematic shift for the two stars with the smallest phase . the null detections appear more randomly distributed in the plot . this suggests that for positive detections it will be possible to estimate the phase to within @xmath2 or so . it should be noted , however , that the positive detections were found to be during the phases close to maximum light , when the spectral changes occur the most rapidly . this suggests that detections are most likely to be found during phases of rapid spectral line change . this is again expected , given the short time baseline for most of these stars . we find that using our empirical templates it is possible to detect variability and estimate phase for rrab stars . when the ses are all constrained to a single night , the success rate is mainly determined by the time baseline and , in particular , the percentage of the pulsation period that is represented by the data . for all the positive detections , a time baseline that is greater than @xmath8 of the pulsation period is needed to return a positive result . it is also most likely that positive results will occur for phases near the maximum light , when the spectral changes are occurring rapidly . we see no strong relation between positive detections and the signal - to - noise ratio of the spectra . this suggests that the technique will work for very low signal - to - noise spectra . we have not tested the rrc variables at this point , but the shorter periods will make even the shortest time baselines a large percentage ( @xmath9 ) of the pulsation phase . this should increase in the number of positive detections for single - night , ses , although the spectral changes are not as extreme as the rrab stars . we have also not tested the @xmath10 of the stars that have ses taken on multiple nights . for these we expect a much higher frequency of positive detections , but the phase determination will be more difficult due to an unknown number of pulsation phases between the observations . this july we are continuing the observation of standard rrl variables . our goal is to observe an rrab and rrc through the entire pulsation period and at high time resolution ( @xmath11 of pulsation period ) this should increase the precision of the phase estimation .

the sdss has been a gold mine for understanding properties of the milky way . below , in the watershed , there remain small nuggets to be found flowing from the deepest recesses of the mine . the sdss - dr9 included the release of the flux- and wavelength - calibrated , individual spectra which were subsequently combined to form the composite spectra found in previous sdss data releases . these single - epoch spectra ( ses ) can be analyzed to find flux and spectral line variability , and to probe aspects of phase variations for objects such as rrl stars . for @xmath0 of the spectra in the rrl color range , ses were taken on separate nights , sometimes weeks apart . the remaining dataset have a time baseline of 0.75 to 1.5 hours and consist of 4 - 7 separate exposures . in my talk i will present details of our project to detect variability and to constrain the pulsation phase at the time of observation . i will discuss our auto - detection technique that uses division of all ses for a given star to search for variations in the flux , the hydrogen balmer lines and the caii k line . this procedure is being tested against known variables and non - variable stars within stripe 82 to determine the identification effectiveness as a function of signal - to - noise , rrl type , and time baseline length . i will also discuss the use of empirical standard star templates to predict pulsation phase from the combination of hydrogen line strengths and radial velocity variations . our ultimate goal is to combine the ses analysis with our new metallicity calibration in order to increase the number of rrl stars that have reliable metallicity determinations and which can then be used to probe the structure of the milky way halo . [ 1996/06/01 ]

the violation of cp invariance and the existence of tree level flavor changing neutral currents ( fcnc ) are generic features of @xmath1 theories with two higgs doublets . indeed the couplings @xmath2 , @xmath3 , @xmath4 and @xmath5 in the higgs potential of the two higgs doublet model ( 2hdm ) @xmath6 + \frac{\lambda_1}{2 } |h_1|^4 + \frac{\lambda_2}{2}|h_2|^4+\lambda_3 |h_1|^2 |h_2|^2\\ + \lambda_4|h_1^{\dagger}h_2|^2 + \biggl[\frac{\lambda_5}{2}(h_1^{\dagger}h_2)^2 + \lambda_6 |h_1|^2 ( h_1^{\dagger}h_2)+\lambda_7 |h_2|^2(h_1^{\dagger}h_2)+h.c . \biggr ] \end{array } \label{1}\ ] ] can be complex inducing cp violation . although one can eliminate the violation of cp invariance and tree level fcnc transitions by imposing a discrete @xmath7 symmetry , such a symmetry leads to the formation of domain walls in the early universe which would create unacceptably large anisotropies in the cosmic microwave background radiation . here we consider the multiple point principle ( mpp ) @xcite as a possible mechanism for the suppression of fcnc and cp violation effects . mpp postulates the existence of the maximal number of phases with the same energy density which are allowed by a given theory @xcite . when applied to the 2hdm the multiple point principle implies the existence of a large set of degenerate vacua at some high energy scale @xmath8 ( mpp scale ) that can be parametrised as @xmath9 where @xmath10 , @xmath11 , @xmath12 while @xmath13 is an arbitrary parameter . according to the mpp vacua at the electroweak and mpp scales must have the same vacuum energy density . the degeneracy of the mpp scale vacua can be achieved only if the lagrangian for the higgs fermion interactions is invariant under symmetry transformations ( see @xcite ) : @xmath14 where @xmath15 , @xmath16 , @xmath17 are quark and lepton eigenstates which couple to @xmath18 while @xmath19 , @xmath20 , @xmath21 are fermion eigenstates that interact with @xmath22 . in eq . ( [ 3 ] ) the subscript @xmath23 denote right - handed fermion fields . two global @xmath24 symmetries ( [ 3 ] ) forbid non diagonal flavour transitions at the tree level . moreover these symmetries lead to the vanishing of @xmath3 , @xmath4 and @xmath5 in the higgs potential ( [ 1 ] ) that cause cp nonconservation in the 2hdm . the mixing term @xmath25 , which is not forbidden by the mpp , softly breaks custodial global symmetries but does not create new sources of cp violation or fcnc transitions . the degeneracy of the physical and mpp scale vacua requires higgs and yukawa couplings to be adjusted so that an appropriate cancellation among the quartic terms in the higgs potential ( [ 1 ] ) takes place . such cancellation becomes possible only if @xmath26 and the combination of the higgs self couplings @xmath27 as well as its beta function @xmath28 go to zero at the scale @xmath8 @xcite . at moderate values of @xmath29 where @xmath30 and @xmath31 are vacuum expectation values of @xmath22 and @xmath18 in the physical vacuum , the last two mpp conditions can be written as follows @xcite @xmath32 where @xmath33 is the top quark yukawa coupling . thus @xmath34 and @xmath35 are not independent parameters in the considered model . as a result the mpp inspired 2hdm has less free parameters than the 2hdm of type ii and therefore can be regarded as a minimal non supersymmetric two higgs doublet extension of the sm . it is also worth noting here that the mpp conditions ( [ 4 ] ) are satisfied identically in the mssm above the supersymmetry breaking scale . the mpp conditions ( [ 4 ] ) must be supplemented by the vacuum stability requirements , i.e. @xmath36 and @xmath37 , which have to be fulfilled everywhere from @xmath38 to @xmath39 . otherwise another minimum of the higgs potential ( [ 1 ] ) with negative vacuum energy density arises at some intermediate scale , preventing the consistent realisation of the mpp in the 2hdm . the renormalisation group ( rg ) flow of all couplings in the mpp inspired 2hdm is determined by @xmath40 , @xmath41 and @xmath33 . when @xmath42 the solutions of the rg equations for the top quark yukawa coupling are concentrated in the vicinity of the quasi fixed point at the electroweak scale . the value of @xmath43 that corresponds to the quasi fixed point scenario depends mainly on the mpp scale @xmath8 . it varies from @xmath44 to @xmath45 when @xmath8 changes from @xmath46 to @xmath47 @xcite . at large values of @xmath33 the mpp and vacuum stability conditions constrain @xmath48 very strongly . our numerical studies show that for @xmath49 and @xmath50 the ratio @xmath51 can vary only within a very narrow interval from @xmath52 to @xmath53 if @xmath54 . this ensures the convergence of the solutions of the rg equations for @xmath55 to the quasi fixed points . + absolute values of the relative couplings @xmath56 of the higgs scalars to the top quark in the quasi fixed point scenario as a function of @xmath57 for @xmath58 . solid and dashed dotted curves correspond to the lightest and heaviest cp even higgs states . , title="fig : " ] + absolute values of the relative couplings @xmath56 of the higgs scalars to the top quark in the quasi fixed point scenario as a function of @xmath57 for @xmath58 . solid and dashed dotted curves correspond to the lightest and heaviest cp even higgs states . , title="fig : " ] + + the higgs spectrum of the two higgs doublet extension of the sm contains two charged and three neutral scalar states . because in the mpp inspired 2hdm cp invariance is preserved one of the neutral higgs bosons is always cp odd while the two others are cp even . the qualitative pattern of the higgs spectrum depends very strongly on the mass of the pseudoscalar higgs boson @xmath57 ( see fig . 1 ) . when @xmath59 the masses of charged , cp odd and heaviest cp even higgs bosons are almost degenerate around @xmath57 . in the considered limit the lightest cp even higgs boson mass @xmath60 attains its maximal value which is determined by @xmath61 and @xmath43 . for each mpp scale @xmath8 one can compute the values of the higgs self couplings at the electroweak scale and @xmath43 near the quasi fixed point . because at large values of the pseudoscalar mass @xmath60 is almost independent of @xmath57 , the upper bound on the lightest higgs scalar mass depends predominantly on the scale @xmath8 . when the mpp scale is high such dependence is relatively weak . if @xmath62 the mass of the lightest higgs particle in the quasi fixed point scenario does not exceed @xmath63 @xcite . however at low mpp scales @xmath0 the theoretical restriction on @xmath60 reaches @xmath64 . the lightest higgs scalar in the considered case is predominantly a sm like higgs boson , since its relative coupling to a @xmath65 pair is rather close to unity . nevertheless at low mpp scales the quasi fixed point scenario leads to large values of the relative coupling of the lightest higgs scalar to the top quark resulting in the enhanced production of this particle at hadron colliders ( see fig . 2 ) . thus the analysis of production and decay rates of the sm like higgs boson at the lhc should make possible the distinction between the quasi fixed point scenario in the mpp inspired 2hdm with low scale @xmath8 , the sm and the mssm even if extra higgs states are relatively heavy , i.e. @xmath66 . we have argued that the mpp assumption allows us to suppress fcnc and cp violating phenomena in the 2hdm . when @xmath67 the solutions of the rg equations in the mpp inspired 2hdm converge to the quasi fixed point , leading to stringent restrictions on the lightest higgs scalar mass . in the quasi fixed point scenario the higgs couplings to the @xmath68quark can be significantly larger than in the sm , which allows us to test this scenario at hadron colliders . rn acknowledge support from the shefc grant hr03020 supa 36878 . 9 bennett d l and nielsen h b 1994 _ int . phys . _ a * 9 * 5155 bennett d l , froggatt c d and nielsen h b 1995 _ perspectives in particle physics94 _ ed d klabucar , i picek and d tadi ( singapore : world scientific ) p 255 ( _ preprint _ hep - ph/9504294 ) froggatt c d , nevzorov r , nielsen h b and thompson d 2008 on the origin of @xmath7 symmetry in the two - higgs doublet extension of the sm _ preprint _ arxiv:0806.3190 [ hep - ph ] froggatt c d , laperashvili l , nevzorov r , nielsen h b and sher m 2004 the two - higgs doublet model and the multiple point principle _ preprint _ hep - ph/0412333 froggatt c d , laperashvili l , nevzorov r , nielsen h b and sher m 2006 _ phys . _ d * 73 * 095005 froggatt c d , nevzorov r , nielsen h b and thompson d 2007 _ phys . _ b * 657 * 95

the multiple point principle ( mpp ) can be used to suppress non diagonal flavour transitions and cp violation in the two higgs doublet extension of the standard model . we discuss the quasi fixed point scenario in the mpp inspired two higgs doublet model which leads to the enhanced production of higgs particles at the lhc if the mpp scale @xmath0 . cern - ph - th/2007 - 193 +

the quantum spin-1/2 dimer - plaquette or orthogonal - dimer chain @xcite represents one of the known examples of partially exactly solvable models with the dimerized ground state @xcite . originally it was suggested as a one - dimensional counterpart of the depleted square lattice @xcite or the shastry - sutherland lattice @xcite . despite its specific structure the shastry - sutherland model is related to a number of magnetic compounds ( srcu@xmath2(bo@xmath3)@xmath2 , tmb@xmath4 , tbb@xmath4 , etc . ) having either almost isotropic heisenberg or highly anisotropic ising interactions ( see ref . @xcite for a recent review ) . magnetization curves of these compounds exhibit the set of fractional plateaux , which has not been firmly explained yet @xcite . a consistent explanation of the series of plateaux in a low - temperature magnetization curve of srcu@xmath2(bo@xmath3)@xmath2 remains the hot topic of the current research @xcite . the comprehensive study of the model is in general quite difficult and the only known exact result concerns the dimerized ground state for the sufficiently strong intra - dimer ( diagonal ) interactions @xcite . since the orthogonal - dimer chain or two coupled orthogonal - dimer chains @xcite can approximate to some extent the shastry - sutherland lattice , their study may reveal some basic features which are typical also for this frustrated two - dimensional model . for instance , the quantum spin-1/2 heisenberg orthogonal - dimer chain shows an infinite series of the magnetization plateaux @xcite . at the same time the heisenberg model of two coupled orthogonal - dimer chains considered in ref . @xcite exhibits a number of fractional plateaux and some of them are identical to the ones observed in the shastry - sutherland model . recently , a quite specific quantum spin-1/2 orthogonal - dimer chain with triangular @xmath1 heisenberg clusters coupled via the intermediate ising spins has been considered by ohanyan and honecker @xcite . this simplified ising - heisenberg orthogonal - dimer chain is exactly soluble by means of the transfer - matrix method and shows under certain conditions surprisingly good correspondence to the pure quantum model with all heisenberg interactions . in the present work , we will study another version of the spin-1/2 ising - heisenberg orthogonal - dimer chain , where the quantum heisenberg interactions are retained on all vertical and horizontal bonds coupled together through the ising interactions . this model preserves the @xmath0-component of the total spin on vertical heisenberg bonds ( dimers ) , which allows us to obtain exactly all ground states and thermodynamic properties . during the preparation of this work , we became aware of the similar work treating the same spin-1/2 ising - heisenberg orthogonal - dimer chain in an absence of the external field using somewhat different approach based on a direct algebraic mapping transformation @xcite . last but not least , let us provide some insight into an experimental background of our work . although the exactly solved ising - heisenberg models with alternating ising and heisenberg bonds could be regarded more as a mathematical curiosity rather than the realistic models of some actual magnetic materials , recent progress in the field of magnetochemistry has opened up new possibilities for a targeted design of magnetic materials with a very specific combination of magnetic interactions . a few eminent ising - heisenberg models have proved their usefulness by an explanation of the magnetic behavior of some real insulating magnetic materials such as [ ( cul)@xmath2dy][mo(cn)@xmath5 @xcite , [ fe(h@xmath2o)(l)][nb(cn)@xmath5[fe(l ) ] @xcite and dy(no@xmath3)(dmso)@xmath2cu(opba)(dmso)@xmath2 @xcite . a series of isostructural 3d-4f coordination polymers [ ln(hfac)@xmath2(ch@xmath3oh)]@xmath2[cu(dmg)(hdmg)]@xmath2 ( ln = gd , dy , tb , ho , er , pr , nd , sm , eu ) involves an unprecedented heterobimetallic motif of the orthogonal - dimer chain @xcite . the dysprosium - based member of this isomorphous series dy(hfac)@xmath2(ch@xmath3oh)]@xmath2[cu(dmg)(hdmg)]@xmath2 to be further abbreviated as [ dy@xmath2cu@xmath2]@xmath7 ( see fig.[fig_dy - cu](a ) ) provides a valuable experimental realization of the spin-1/2 ising - heisenberg orthogonal - dimer chain due to a rather strong magnetic anisotropy of dy@xmath8 ions @xcite . as a matter of fact , the vertical spin-1/2 ising dimers assigned to double oxo - bridged dinuclear entities of dy@xmath8 ions regularly alternate within the polymeric compound [ dy@xmath2cu@xmath2]@xmath7 with the horizontal spin-1/2 heisenberg dimers assigned to the macrocyclic dinuclear entities of cu@xmath9 ions . it turns out that the antiferomagnetic superexchange coupling between dy@xmath8 and cu@xmath9 ions mediated by the oximate bridge is by far the most dominant coupling , whereas the superexchange mechanism for the double oxo - bridged dinuclear entities of dy@xmath8 ions and the macrocyclic dinuclear entities of cu@xmath9 ions transmit presumably much weaker ferromagnetic coupling @xcite . it can be clearly seen from fig.[fig_dy - cu](a ) that the magnetic structure of [ dy@xmath2cu@xmath2]@xmath7 implies a more complex ( asymmetric ) interactions between the spin-1/2 ising and heisenberg dimers , which involve in total four different exchange pathways between dy - dy , cu - cu and dy - cu magnetic ions ( see fig.[fig_dy - cu](b ) ) . for the sake of simplicity , we will restrict our further analysis only to the symmetric particular case with just two different exchange couplings ( see fig . [ od - chain ] ) , whereas the comprehensive analysis of the more general ( asymmetric ) case involving four different exchange couplings will be the subject matter of our future work . the paper is organized as follows . in section [ model_section ] we introduce the spin-1/2 ising - heisenberg orthogonal - dimer chain with the alternating heisenberg and ising interactions and solve it using the transfer - matrix method . in section [ gs_section ] we will consider in detail the ground - state phase diagram . section [ thermodynamics_section ] presents the most interesting results for the thermodynamic quantities and the magnetocaloric effect . the most important findings are briefly summarized in section [ conclusions ] . let us define the quantum spin-1/2 ising - heisenberg orthogonal - dimer chain with the heisenberg intra - dimer and the ising inter - dimer interactions in a magnetic field through the hamiltonian ( see fig . [ od - chain ] ) : @xmath10 \nonumber\\ & & + j({\mathbf s}_{1,2i+1}\cdot{\mathbf s}_{2,2i+1})_{\delta } { -}h(s_{1,2i+1}^z{+}s_{2,2i+1}^z ) , \nonumber\\ h_{2i}&{= } & j({\mathbf s}_{1,2i}\cdot{\mathbf s}_{2,2i})_{\delta}-h(s_{1,2i}^z+s_{2,2i}^z ) , \nonumber\end{aligned}\ ] ] where @xmath11 , @xmath12 denotes spatial projections ( @xmath13 ) of the spin-@xmath14 operator , @xmath15 is the anisotropic heisenberg intra - dimer interaction between spins on vertical and horizontal bonds , @xmath16 is the anisotropy parameter and @xmath17 is the ising inter - dimer interaction between spins from different bonds . in what follows , we will be mainly interested in investigating the particular case of antiferromagnetic interactions @xmath18 , @xmath19 , @xmath20 , which brings the spin frustration into play . further , the periodic boundary condition for spins @xmath21 will be implied for convenience . since @xmath0-component of the total spin on a vertical heisenberg bond is the integral of motion , and it is the only common operator for neighboring local hamiltonians @xmath22 , all @xmath22 commute with each other . hence , it follows that it is quite convenient to use a decomposition of the total hamiltonian ( [ gen_ham1 ] ) into the sum of commuting parts @xmath23 , where @xmath24 consider now the total spin momentum operator on the vertical heisenberg bonds @xmath25 . it is quite apparent that @xmath26 represents the conserved quantity with well defined quantum numbers @xmath27 and @xmath28 , @xmath29 @xcite . the respective eigenstates of this momentum spin operator can be denoted as @xmath30 . using the transfer - matrix method @xcite , the partition function of the model can be written in the form : @xmath31 where the transfer - matrix @xmath32 contains the trace over two spins from the @xmath33-st horizontal heisenberg bond . here , @xmath34 denotes the inverse temperature ( boltzmann s constant is set to unity @xmath35 ) . the straightforward calculation gives the transfer - matrix in the form ( where rows and columns corresponds to the following set of states @xmath36 , @xmath37 , @xmath38 , @xmath39 ) : @xmath40 @xmath41 { + } \mbox{e}^{\frac{\beta\delta j}{4 } } \cosh\left(\frac{\beta j}{2}\right ) \right\ } , \nonumber\\ & & a_2{=}2\left\ { \mbox{e}^{{-}\frac{\beta\delta j}{4 } } \cosh(\beta h ) { + } \mbox{e}^{\frac{\beta\delta j}{4 } } \cosh\left ( \frac{\beta } { 2}\sqrt{j^2{+}4j_1 ^ 2}\right ) \right\ } , \nonumber\\ & & a^{\pm}_3{=}2\left\ { \mbox{e}^{{-}\frac{\beta\delta j}{4 } } \cosh\left[\beta \left(\frac{j_1}{2}{\pm}h\right)\right ] \right . \nonumber\\ & & \left.{+}\mbox{e}^{\frac{\beta\delta j}{4 } } \cosh\left ( \frac{\beta } { 2}\sqrt{j^2{+}j_1 ^ 2}\right ) \right\ } , \nonumber\\ & & a_4{=}2\left\ { \mbox{e}^{{-}\frac{\beta\delta j}{4 } } \cosh(\beta h ) { + } \mbox{e}^{\frac{\beta\delta j}{4 } } \cosh\left ( \frac{\beta j}{2}\right ) \right\ } , \nonumber\\ & & b^{\pm}_1{=}\mbox{e}^{{-}\frac{\beta}{2}\left(\frac{\delta j}{4}{\pm}h\right)},\ : b_2{=}\mbox{e}^{\frac{\beta j(2{+}\delta)}{8 } } , \ : c{=}\mbox{e}^{{-}\frac{\beta j}{2}}. \nonumber\end{aligned}\ ] ] since the first and third row of the transfer matrix ( [ t - matrix ] ) are linearly dependent , the transfer - matrix @xmath42 is the degenerate matrix , and at least one of the eigenvalues equals zero . in the case of zero external field @xmath43 , @xmath44 , @xmath45 and consequently , it is possible to find the eigenvalues of the transfer matrix as two simple roots and two roots of quadratic equation @xcite . in the case of non - zero external field @xmath46 , one eigenvalue of the transfer matrix still equals to zero and additional three eigenvalues are given by the roots of cubic equation @xmath47 , where @xmath48b_2 ^ 2(1+c^2 ) % \nonumber\\ & & + ( a_1 ^ -a_1^+-a_2 ^ 2)(b_1 ^ -)^2(b_1^+)^2 , \nonumber\\ & & c= \big[-(a_1 ^ -a_1^+ - a_2 ^ 2)a_4 + a_1 ^ -(a_3^+)^2 + a_1^+(a_3 ^ -)^2 - 2a_2a_3 ^ - a_3^+ \big ] ( b_1 ^ -)^2(b_1^+)^2b_2 ^ 2(1+c^2 ) . \nonumber\end{aligned}\ ] ] the roots can be calculated using trigonometric solution of cubic equation ( see e. g. @xcite ) : @xmath49 the free energy per site in the thermodynamic limit is obtained within the transfer - matrix method @xcite as @xmath50 where @xmath51 denotes the maximal eigenvalue of the transfer matrix ( [ t - matrix ] ) . let us start by examining the ground - state properties of the spin-1/2 ising - heisenberg orthogonal - dimer chain . to get the ground state , we have to find first the lowest - energy eigenstate of the local hamiltonian ( [ local_ham ] ) . for one - dimensional system it is then always possible to extend this state to the whole chain , which can be afterwards proven to be the global ground state using the variational principle ( see e.g. @xcite ) . using this procedure , we have found the following six ground states : * the unique _ singlet dimer _ ( sd ) phase @xmath52 with the energy @xmath53 , * the two - fold degenerate _ modulated antiferromagnetic _ ( maf ) phase @xmath54 with the energy @xmath55 and the quantum reduction of the staggered magnetization of the spins residing on the horizontal heisenberg bonds : @xmath56 , * the two - fold degenerate _ antiferromagnetic _ ( af ) phase @xmath57 with the energy @xmath58 , * the two - fold degenerate _ modulated ferrimagnetic _ ( mfi ) phase @xmath59 with the energy @xmath60 and the quantum reduction of the staggered magnetization of the spins residing on the horizontal heisenberg bonds : @xmath61 , * the two - fold degenerate _ staggered bond _ ( sb ) phase @xmath62 with the energy @xmath63 , * the _ ferromagnetic _ ( fm ) phase @xmath64 . + the zero - field ground - state phase diagram shown in fig . [ gs_pd1 ] contains just three different ground states sd , maf and af . obviously , the sd phase becomes the ground state in the particular case of the antiferromagnetic heisenberg coupling and sufficiently weak ising inter - dimer interaction . when the ising inter - dimer interaction @xmath17 strengthens , the maf phase is favored with a peculiar quantum antiferromagnetic order of the spins from the horizontal heisenberg bonds accompanied with the alternating character of the fully polarized triplets on the vertical heisenberg bonds ( [ maf ] ) . finally , the fully polarized triplet states on the vertical and horizontal heisenberg dimers become favorable for the special case of ferromagnetic intra - dimer @xmath65 coupling ( @xmath66 ) , whereas the nearest - neighboring vertical and horizontal heisenberg bonds are polarized in opposite direction . the boundary between the relevant phases can be readily calculated by comparing the ground - state energies @xmath67 of individual phases : * sd - af : @xmath68 , * sd - maf : @xmath69 , * maf - af : @xmath70 . it is noteworthy that all curves meet at one triple point with the coordinates @xmath71 , @xmath72 . as it will be shown below , the ground - state boundary between maf and af has macroscopic degeneracy @xmath73 . the effect of non - zero magnetic field on the overall ground - state phase diagram is shown in fig . [ gs_pd2 ] for several values of the exchange anisotropy @xmath16 . consider first the particular case of antiferromagnetic heisenberg intra - dimer interaction with @xmath74 . under this condition , the zero - field ground state can be either sd or maf phase depending on a mutual competition between the heisenberg intra - dimer and ising inter - dimer interactions @xmath15 and @xmath17 . however , the magnetic field of moderate strength destroys both sd and maf states due to energetic stabilization of the mfi phase , which is characterized by the alternating character of the non - magnetic singlets and fully polarized triplets on the vertical heisenberg bonds . consequently , this latter ground state manifests itself in a magnetization process as the intermediate one - quarter plateau , because there is no contribution to the total magnetization from the spins on the horizontal heisenberg bonds displaying a quantum antiferromagnetic order . a further increase in the external magnetic field leads to a presence of the sb phase , in which the non - magnetic singlets and the fully polarized triplets regularly alternate on the horizontal and vertical heisenberg bonds . as a result , the sb phase will cause a presence of the additional plateau at one - half of the saturation magnetization . finally , an extremely strong magnetic field naturally flips all spins to the external - field direction , which results in the fully polarized fm phase . to summarize our findings for the magnetization process , the zero - temperature magnetization curve displays two different intermediate plateaux at one - quarter and one - half of the saturation magnetization , which provide a macroscopic manifestation of two striking mfi and sb phases of purely quantum character . it should be noted that this picture does not qualitatively change for a more general case of the antiferromagnetic heisenberg intra - dimer interactions with @xmath19 . on the other hand , several topologies of the ground - state phase diagram are possible for the special case @xmath66 corresponding to the ferromagnetic heisenberg intra - dimer interaction . if @xmath75 , the new af phase can be detected in a low - field region of the ground - state phase diagram for sufficiently high values of the ising inter - dimer interaction @xmath17 . as a consequence , for sufficiently strong inter - dimer interactions the model exhibits a direct field - induced transition from the af phase towards the one - half plateau sb phase omitting the one - quarter plateau mfi phase . if the ising inter - dimer interaction @xmath17 is of moderate strength or sufficiently weak , the effect of external magnetic field is quite similar to the case discussed previously . the increasing magnetic field changes at the first transition field the ground state sd , maf or af to the one - quarter plateau mfi phase , then at the second transition field the mfi phase to the one - half plateau sb phase and finally , the sb phase at the saturation field to the fully polarized fm phase . accordingly , the one - quarter and one - half fractional plateaux are still present in the relevant zero - temperature magnetization curves . next , the vertical stripe corresponding to the maf phase completely disappears from the ground - state phase diagram for @xmath76 . finally , the ground - state phase diagram becomes quite simple for @xmath77 when the ferromagnetic heisenberg intra - dimer interaction of the easy - axis type supports the polarized triplet states only . the ground state in an absence of the magnetic field is created by the af phase and this order breaks at the saturation field to the fm phase . it is quite clear from the above argument that both intermediate fractional plateaux at one - quarter and one - half of the saturation magnetization vanish from the zero - temperature magnetization curve . for completeness , let us provide the expressions for boundaries between individual phases , which have been obtained by comparing the ground - state energies : * sd - mfi : @xmath78 , for @xmath79 , macroscopic degeneracy @xmath80 in the limit @xmath81 ; * maf - mfi : @xmath82 , for @xmath83 , @xmath84 ; * mfi - sb : @xmath85 , @xmath73 ; * sb - fm : @xmath86 , @xmath87 ; * af - mfi : @xmath88 , for @xmath89 , @xmath80 ; * af - sb : @xmath90 , for @xmath89 , @xmath73 ; * af - fm : @xmath91 , for @xmath92 , @xmath87 . it is worthwhile to remark that the ground state at some phase boundaries is highly degenerate . to get the degeneracy , we can apply the notion of counting the dimer coverings on a chain as used in ref . consider for instance the phase boundary between the sb and fm phases . the fm phase can be represented as a one - dimensional lattice of the gas where all sites are empty [ @xmath93 , while the sb phase corresponds to the configuration where each second site is occupied [ @xmath94 . here , the magnetized state of the heisenberg dimer is set to be an empty state , while the non - magnetic singlet state of the heisenberg dimer is represented by a filled state . as a result , the ground state at the relevant phase boundary between these two phases can be constructed from all configurations of particles with the following restriction : particles can not occupy neighboring sites , i.e. there is an infinitely strong nearest - neighbor repulsion between particles . such a system can be reformulated as a dimer problem , and the calculated degeneracy in the thermodynamic limit is given by @xmath95^n$ ] @xcite . it is noteworthy that the same value of the degeneracy is obtained at the boundary between the fm and af phases if the @xmath96 ( @xmath97 ) bond state is treated as an occupied site in latter case . similar representation can be constructed also at the sd - mfi and af - mfi boundaries . the only difference is that only vertical heisenberg bonds are involved into the lattice - gas picture . therefore , the degeneracy is @xmath95^{n/2}$ ] . all other boundaries have the degeneracy of monomers on a chain of size @xmath98 or @xmath99 or of the free ising spins . for instance , let us consider all possible configurations at the boundary between the maf and mfi states . the maf phase is consistent with the antiferromagnetic order of the vertical heisenberg bonds and can be schematically presented as [ @xmath100 . on the contrary , the mfi phase shows on the vertical heisenberg dimers a regular alternation of the fully polarized triplet and non - magnetic singlet bonds [ @xmath101 . here , @xmath102(@xmath97 ) denotes the @xmath103 state and @xmath104 refers to the singlet state . both phases are two - fold degenerate . in this particular case , the bond state on odd vertical heisenberg bonds coincides for both phases , while two different states ( either polarized or singlet ) are available for any even vertical heisenberg bond . two degrees of freedom of any even vertical heisenberg bond can be then identified as the fictitious ising spin @xmath14 and therefore , the ground state at the boundary between the maf and mfi phases has the macroscopic degeneracy @xmath105 . the special case of the af - sb boundary has the same picture as the maf - mfi boundary . the only difference is that the discussion concerns not only vertical , but all the bonds . thus , the macroscopic degeneracy at the af - sb phase boundary is proportional to @xmath106 . the maf and af phases also have classical orderings ( antiferromagnetic and ferromagnetic ) for the vertical bonds , i.e. [ @xmath100 and [ @xmath107 , [ @xmath108 correspondingly . at the ground - state boundary the energy of any random ordering of vertical bond configurations will be the same , which means that the ground - state degeneracy equals to @xmath106 for this particular case . the similar case is the phase boundary between the mfi and sb phases . the only difference with respect to the previous case is that @xmath97 on the vertical bonds must be changed to the singlet state @xmath104 . the rest of analysis is analogous and gives the degeneracy @xmath106 . comparing the ground - state phase diagram of the spin-1/2 ising - heisenberg orthogonal - dimer chain with the analogous result for the spin-1/2 heisenberg orthogonal - dimer chain obtained by means of the extensive dmrg calculations ( see fig . 9 of ref . ) , we observe that both phase diagrams contain the regions with the zero , one - quarter and one - half magnetization plateaux . the difference between phase diagrams becomes fundamentally distinct for the sufficiently strong inter - dimer coupling @xmath17 , when the dimer - plaquette phase evolves in the heisenberg model instead of the maf phase maintained by the ising - heisenberg model . this discrepancy is caused by an infinitely strong anisotropy of the ising inter - dimer interaction . in addition , the spin-1/2 heisenberg orthogonal - dimer chain exhibits the infinite series of magnetization steps between one - quarter and one - half magnetization plateaux @xcite , as well as , the continuous change of the magnetization from the one - half plateau to the saturation magnetization . on the other hand , the spin-1/2 ising - heisenberg orthogonal - dimer chain displays a high degeneracy at the saturation field that is accompanied with the respective magnetization jump instead . the macroscopic degeneracy found in the ground state may manifest itself in the low - temperature behavior of basic thermodynamic quantities such as entropy , specific heat or magnetization . at first one should notice that the entropy can take finite values at zero temperature whenever the ground state is macroscopically degenerate due to a phase coexistence . the entropy per site can be easily obtained using the thermodynamic relation @xmath109 , while the residual entropy on phase boundaries is related to the macroscopic degeneracy of the ground - state manifold according to the boltzmann s equation @xmath110 @xcite . thus , the residual entropy at the ground - state boundaries between different phases can be straightforwardly calculated from the results presented in section [ gs_section ] . bearing all this in mind , the residual entropy takes the value @xmath111 at the sb - fm and af - fm ground - state boundaries , @xmath112 at the sd - mfi and af - mfi boundaries , @xmath113 at the mfi - sb and af - sb boundaries , @xmath114 at the maf - mfi boundary . as one can see from fig . [ fig_entropy1 ] , the field dependence of the entropy indeed shows remarkable peaks at transition fields whose heights is in accordance with the reported values of the residual entropy ( the particular case shown in fig . [ fig_entropy1 ] exhibits three successive field - induced transitions sd - mfi , mfi - sb and sb - fm ) . it can be also clearly seen from fig . [ fig_entropy1 ] that even small temperature smooths the field dependence of entropy and thus destroys its distinct profile . besides , it is well known that quantum frustrated spin models may exhibit an enhanced magnetocaloric effect near the field - induced quantum critical point @xcite . we have therefore studied also the adiabatic demagnetization of the model under investigation , which can be easily understood from the density plot of entropy depicted in fig . [ fig_entropy2 ] . note that the curves of constant entropy determine the change of the temperature with the magnetic field during the adiabatic process . since the spin-1/2 ising - heisenberg orthogonal - dimer chain may have up to three critical fields accompanied with the macroscopic degeneracy of the ground state , the temperature rapidly decreases near a critical field whenever the entropy is selected close enough to the corresponding value of the residual entropy . this behavior may evidently promote a high adiabatic magnetocaloric rate @xmath115 . the obtained exact solution allows us to examine the effect of spin frustration and external field on the specific heat , which can be obtained from the thermodynamic relation @xmath116 . some typical thermal variations of the specific heat are presented in fig . [ fig_heat1 ] for different values of the interactions and external magnetic field . the temperature dependencies of zero - field specific heat are displayed in fig . [ fig_heat1](a ) . the investigated spin system is far from the degenerate ground state for the special case @xmath74 , @xmath117 , @xmath118 and hence , the specific heat exhibits just one broad peak of schottky type . on the other hand , the zero - field specific heat gains an additional low - temperature peak by changing the ising inter - dimer coupling @xmath119 sufficiently close to the sd - maf boundary . the set of parameters driving the investigated spin chain close to the macroscopically degenerate maf - af boundary shows even more complex temperature dependence with rapidly increasing specific heat at low temperature and several round maxima . [ fig_heat1](b ) and ( c ) illustrate thermal variations of the specific heat when the external field is selected close to critical fields . in fig . [ fig_heat1](b ) , the applied magnetic field @xmath120 is sufficient to stabilize the one - quarter plateau with a quite small gap between the ground state and first excited state . when the external field achieves the critical value , the ground state becomes macroscopically degenerate . the specific heat then shows a sharp peak at very low temperature and quite broad nearly flat region between two peaks . this unusual dependence indicates the existence of a large number of states with energies quite close to the ground state energy . the specific heat near the mfi - sb border ( case of @xmath121 in fig . [ fig_heat1](b ) ) shows even more striking temperature dependence with three peaks , whereas all three peaks are of the same order . the specific heat for the other particular case of the ferromagnetic heisenberg intra - dimer coupling shown in fig . [ fig_heat1](c ) has similar features . it should be nevertheless mentioned that the zero - field specific heat shows a very sharp low - temperature peak for the interaction parameters driving the investigated spin chain close to the maf - af ground - state boundary . the external field generally broadens this peak and shifts it towards slightly higher temperatures . let us also briefly comment on a magnetization process of the spin-1/2 ising - heisenberg orthogonal - dimer chain at low temperatures . it is quite evident from the ground - state phase diagram shown in fig . [ gs_pd2 ] that the low - temperature magnetization curve may contain fractional plateaux at one - quarter and one - half of the saturation magnetization . as one could expect , the intermediate plateaux and magnetization jumps gradually become smoother as temperature increases . however , it is quite surprising how fast a step - like magnetization curve is demolished by even very low temperature . for illustration , we present in fig . [ fig_mag](a ) the relevant low - temperature magnetization curves for the isotropic heisenberg intra - dimer interaction . the width of both intermediate plateaux is nearly the same for the particular case @xmath118 and it actually turns out that even a rather small temperature @xmath122 makes the step - like structure almost indistinguishable within the relevant magnetization curve . contrary to this , the other particular case @xmath123 seems to be more robust against thermal fluctuations when the intermediate plateaux and magnetization jumps can not be discerned in the relevant magnetization curve just at slightly higher temperature . similar trends can be observed in the low - temperature magnetization curves of the investigated model with the ferromagnetic heisenberg intra - dimer interaction depicted in fig . [ fig_mag](b ) . under this condition , the magnetization curve may contain either one or two intermediate plateaux depending on an interplay between the interaction parameters . in general , it could be concluded that the rather rapid thermal smoothing can be attributed to the huge degeneracy of the ground state at critical fields . in the present work we considered the orthogonal - dimer chain with the heisenberg intra - dimer and ising inter - dimer interactions by means of a rigorous approach based on the transfer - matrix method . we have obtained the exact expressions for the partition function and analyzed the ground state and thermodynamic properties of the model quite rigorously . the ground - state phase diagram of the model in a magnetic field has been obtained and it was shown that two fractional plateaux at one - quarter and one - half of the saturation magnetization are present . we have also studied the effect of the exchange anisotropy in the heisenberg coupling . it has been shown that the ferromagnetic @xmath65 intra - dimer interaction may lead to the appearance of new phases in zero field and may substantially change the ground - state phase diagram in a non - zero magnetic field . in general , this kind of interaction leads to the vanishing of one - quarter and one - half plateaux . the ground state at the border between different phases may exhibit a high macroscopic degeneracy , which leads to the non - zero residual entropy . we have calculated the degeneracy and the residual entropy at all boundaries using the notion of the monomer or dimer covering of a chain @xcite . the degenerate or nearly degenerate ground state has turned out to basically affect the low - temperature thermodynamics of the model . we have calculated the entropy as a function of temperature and magnetic field , which evidences an enhanced magnetocaloric effect close to critical fields . the effect of spin frustration and magnetic field on temperature dependence of specific heat has been examined in detail . it has been found that the interplay of all factors may lead to the complex low - temperature behavior of the specific heat with several more or less separated maxima . the exact results for the magnetization curves have proved that even small temperature may destroy the step - like field dependence of the magnetization . we have also found that the ising - heisenberg model on the orthogonal - dimer chain exhibits some common features with the analogous pure heisenberg model , for instance a presence of the one - quarter and one - half magnetization plateaux . the main discrepancy between both models is as follows : when the heisenberg model shows step - like magnetization between one - quarter and one - half plateaux and a continuous change of the magnetization above the one - half plateau , the ising - heisenberg model can not capture those features as it possesses macroscopic degeneracy at critical fields only and shows just two intermediate plateaux . it could be expected , however , that the treatment of the quantum @xmath124 part of inter - dimer interaction within the perturbation theory for degenerate states could restore some features of the magnetization curve of the pure heisenberg model when starting from the exactly solved ising - heisenberg model . finally , it should be mentioned that there exist an extensive series of heterobimetallic coordination polymers [ ln(hfac)@xmath2(ch@xmath3oh)]@xmath2[cu(dmg)(hdmg)]@xmath2 @xcite with the magnetic structure similar to the considered model . in addition , the dysprosium - based member [ dy@xmath2cu@xmath2]@xmath7 of this series provides an interesting experimental realization of the spin-1/2 ising - heisenberg orthogonal - dimer chain owing to a strong magnetic anisotropy of dy@xmath8 ions . although a more complete description of the polymeric coordination compound [ dy@xmath2cu@xmath2]@xmath7 would require an analysis based on the more general ( asymmetric ) spin-1/2 ising - heisenberg orthogonal - dimer chain with four different exchange couplings , the af ground state reported for the symmetric spin-1/2 ising - heisenberg orthogonal - dimer chain with just two different exchange couplings already correctly reproduces the ferrimagnetic spin arrangement observed experimentally due to the antiferromagnetic inter - dimer and ferromagnetic intra - dimer interactions @xcite . moreover , the procedure elaborated in the present work can be rather straightforwardly adopted also for a theoretical treatment of the more general ( asymmetric ) spin-1/2 ising - heisenberg orthogonal - dimer chain , which would ensure a more correct description of the heterobimetallic complex [ dy@xmath2cu@xmath2]@xmath7 . in this direction we will continue our further efforts . miyahara , exact results in frustrated quantum magnetism . in : introduction to frustrated magnetism , c. lacroix , ph . mendels , f. mila ( eds . ) , springer series in solid - state science * 164 * , springer - verlag berlin heidelberg ( 2011 ) , p.513 . m. takigawa and f. mila , magnetization plateaus , in : introduction to frustrated magnetism , c. lacroix , ph . mendels , f. mila ( eds . ) , springer series in solid - state science * 164 * , springer - verlag berlin heidelberg ( 2011 ) , p.241 . b. wolf , y. tsui , d. jaiswal - nagar , u. tutsch , a. honecker , k. removi - langer , g. hofmann , a. prokofiev , w. assmus , g. donath , and m. lang , proceedings of the national academy of sciences * 108 * , 6862 ( 2011 ) . note that we have assumed in the present work equal @xmath125-factors for the horizontal and vertical spin-1/2 dimers , what consequently leads to the af ground state with zero total magnetization in a parameter space with the antiferromagnetic inter - dimer and ferromagnetic intra - dimer interactions . even though dy@xmath8 and cu@xmath9 magnetic ions can be treated as the spin-1/2 entities at low enough temperatures , the difference in the relevant @xmath125-factors ( @xmath126 vs. @xmath127 ) is responsible in [ dy@xmath2cu@xmath2]@xmath7 for the ferrimagnetic ground state with a non - zero total magnetization that however exactly coincides with the spin arrangement reported for the af ground state .

the quantum spin-1/2 orthogonal - dimer chain with the heisenberg intra - dimer and ising inter - dimer interactions in a magnetic field is considered by a rigorous approach . the model conserves the @xmath0-component of total spin on vertical heisenberg bonds and this property is used to calculate exactly the partition function using the transfer - matrix method . we have found the ground - state phase diagram of the given model in a magnetic field as well as the macroscopic degeneracy along field - induced transitions accompanied with the magnetization jumps . the model exhibits two intermediate fractional plateaux at one - quarter and one - half of the saturation magnetization . we have examined the effect of the exchange anisotropy in the @xmath1 heisenberg intra - dimer interaction on the ground state . it is shown that the one - quarter and one - half plateaux may disappear from the magnetization curve for the ferromagnetic heisenberg intra - dimer interaction . we have also studied rigorously the effect of frustrated interactions on the thermodynamic and magnetic properties of the model and show how the macroscopic degeneracy of the ground state is reflected in the low - temperature behavior of the magnetization , entropy and specific heat . a possibility of observing enhanced magnetocaloric effect during the adiabatic demagnetization is discussed in detail .

only simple graphs are considered in this paper . let @xmath0 be a graph with vertex set @xmath8 and edge set @xmath9 . proper edge-@xmath2-coloring _ is a mapping @xmath10 such that any two adjacent edges receive different colors . the graph @xmath0 is _ edge-@xmath2-colorable _ if it has an edge-@xmath2-coloring . the _ chromatic index _ @xmath11 of @xmath0 is the smallest integer @xmath2 such that @xmath0 is edge-@xmath2-colorable . a proper edge-@xmath2-coloring of @xmath0 is called _ acyclic _ if there are no bichromatic cycles in @xmath0 , i.e. , the union of any two color classes induces a subgraph of @xmath0 that is a forest . the _ acyclic chromatic index _ of @xmath0 , denoted @xmath1 , is the smallest integer @xmath2 such that @xmath0 is acyclically edge-@xmath2-colorable . let @xmath12 ( @xmath5 for short ) denote the maximum degree of a graph @xmath0 . by vizing s theorem @xcite , thus , it holds trivially that @xmath14 . fiam@xmath3ik @xcite and later alon , sudakov and zaks @xcite made the following conjecture : [ con1 ] for any graph @xmath0 , @xmath15 . using probabilistic method , alon , mcdiarmid and reed @xcite proved that @xmath16 for any graph @xmath0 . this bound has been recently improved to that @xmath17 in @xcite , that @xmath18 in @xcite , and that @xmath19 in @xcite . in @xcite , n@xmath20set@xmath21il and wormald proved that @xmath22 for a random @xmath5-regular graph @xmath0 . the acyclic edge coloring of some special classes of graphs was also investigated , including subcubic graphs @xcite , outerplanar graphs @xcite , series - parallel graphs @xcite , 2-degenerate graphs @xcite , and planar graphs @xcite . in particular , conjecture [ con1 ] was confirmed for planar graphs without 3-cycles @xcite , or without 4-cycles @xcite , or without 5-cycles @xcite . it was shown in @xcite that every planar graph @xmath0 has @xmath23 . this upper bound was recently improved to that @xmath24 @xcite . basavaraju and chandran @xcite showed that if @xmath0 is a graph with @xmath6 and @xmath25 , then @xmath26 . equivalently , every non - regular graph @xmath0 with @xmath6 satisfies conjecture 1 . in this paper , we will settle the case where @xmath0 is 4-regular . our result and the result of @xcite confirm conjecture 1 for graphs with @xmath6 . in this section , we discuss the acyclic chromatic indices of @xmath27-regular graphs . before establishing our main result , we need to introduce some notation . assume that @xmath28 is a partial acyclic edge-@xmath2-coloring of a graph @xmath0 using the color set @xmath29 . for a vertex @xmath30 , we use @xmath31 to denote the set of colors assigned to edges incident to @xmath32 under @xmath28 . if the edges of a cycle @xmath33 are alternatively colored with colors @xmath34 and @xmath35 , then we call such cycle an _ @xmath36-cycle_. if the edges of a path @xmath37 are alternatively colored with colors @xmath34 and @xmath35 , then we call such path an _ @xmath36-path_. in the following figures , black spots are pairwise distinct , whereas the others may be not . for simplicity , we use _ @xmath38 _ to express that all edges @xmath39 , @xmath40 , @xmath41 , @xmath42 are colored or recolored with the color @xmath43 . in particular , when @xmath44 , we write simply @xmath45 . moreover , we use @xmath46 to denote the fact that @xmath47 for @xmath48 . let @xmath49 denote the fact that @xmath50 is colored or recolored with the color @xmath51 for @xmath52 . [ 12 ] suppose that a graph @xmath0 has an edge-6-coloring @xmath28 such that @xmath53 , where @xmath54 is an arbitrary edge of @xmath0 . if @xmath54 can be recolored properly with the color @xmath55 , then @xmath0 contains no @xmath56-path under @xmath28 . [ exchange ] suppose that a graph @xmath0 has an edge-6-coloring @xmath28 . let @xmath57 be an @xmath58-path in @xmath0 with @xmath59 and @xmath60 . if @xmath61 , then there does not exist an @xmath62-path in @xmath0 under @xmath28 . * otherwise , assume that there is an @xmath63-path @xmath64 in @xmath0 , where @xmath65 and @xmath66 . note that some @xmath67 may be identical to some @xmath68 , @xmath69 , @xmath70 . since @xmath71 and @xmath61 , it is easy to see that there exists a vertex @xmath68 and some vertex @xmath67 such that @xmath72 and @xmath73 have the same color @xmath34 or @xmath35 , which contradicts the fact that @xmath28 is a proper edge coloring . @xmath74 * proof . * by vizing s theorem @xcite , @xmath0 admits a proper edge-6-coloring @xmath28 using the color set @xmath75 . let @xmath76 denote the number of bichromatic cycles in @xmath0 with respect to the coloring @xmath28 . if @xmath77 , then @xmath28 is an acyclic edge-@xmath78-coloring of @xmath0 , hence we are done . otherwise , @xmath79 . let @xmath80 be a bichromaic cycle of @xmath0 . we are going to show that there is a proper edge-6-coloring @xmath81 , formed by recoloring suitably some edges of @xmath0 , such that @xmath80 is no longer a bichromatic cycle and on new bichromatic cycles are produced . namely , @xmath82 . repeating this process , we finally obtain an acyclic edge-6-coloring @xmath83 of @xmath0 . to arrive at our conclusion , assume , w.l.o.g . , that @xmath80 is a @xmath84-cycle with @xmath85 . let @xmath86 be the neighbors of @xmath87 different from @xmath32 , and @xmath88 the neighbors of @xmath32 different from @xmath87 . since @xmath0 contains no triangles , @xmath89 are pairwise distinct . clearly , we may assume that @xmath90 . since @xmath91 , the proof is split into the following three cases , as shown in fig.1 . assume that @xmath98 . if @xmath0 contains no @xmath99-path , then let @xmath100 . otherwise , @xmath0 contains a @xmath101-path and let @xmath102 . otherwise , assume that @xmath103 and @xmath104 . if @xmath105 and @xmath0 contains no @xmath106-path , then let @xmath107 . if @xmath108 and @xmath0 contains no @xmath109-path , then let @xmath110 . by lemma [ exchange ] , no new bichromatic cycles are produced . otherwise , assume that : by ( @xmath111 ) , @xmath113 . if @xmath114 , then let @xmath115 . if @xmath108 , then let @xmath116 . thus , @xmath117 and assume that @xmath118 and @xmath119 since @xmath120 by ( @xmath111 ) . if @xmath121 , then let @xmath122 . if @xmath123 , then let @xmath124 . otherwise , @xmath125 . suppose that we can recolor some edges in @xmath126 such that @xmath127 , or @xmath128 and @xmath129 , and no new bichromatic cycles are produced in @xmath130 . by lemma [ exchange ] , @xmath80 does not exist even if @xmath131 . if @xmath0 contains a @xmath132-cycle for some @xmath133 , then let @xmath134 and no new bichromatic cycles are produced . next , we claim that : if @xmath0 contains no @xmath138-path for some @xmath139 , then let @xmath140 by lemma [ exchange ] . otherwise , @xmath0 contains a @xmath141-path for every @xmath142 and a @xmath143-path for every @xmath144 similarly . thus , @xmath145 and @xmath146 . however , @xmath147 , a contradiction . if @xmath0 contains neither a @xmath154-path nor a @xmath155-path , then let @xmath156 by ( @xmath135 ) . otherwise , @xmath0 contains a @xmath157-path and @xmath158 . assume that @xmath159 . then first let @xmath160 . next , if @xmath0 contains no @xmath161-path , then let @xmath162 ; otherwise , @xmath0 contains a @xmath163-path , we let @xmath164 . assume that @xmath165 and @xmath166 similarly ( it follows that @xmath167 if @xmath0 contains a @xmath168-path ) . then , let @xmath169 by ( @xmath135 ) . if @xmath0 contains no @xmath154-path , then let @xmath156 . otherwise , @xmath0 contains a @xmath154-path and @xmath170 . if @xmath171 , then let @xmath172 . otherwise , @xmath165 . if @xmath173 and @xmath0 contains no @xmath174-path , then let @xmath175 . otherwise , @xmath176 or @xmath0 contains a @xmath177-path . if @xmath173 and @xmath0 contains a @xmath177-path , i.e. , @xmath178 , @xmath179 , then let @xmath180 otherwise , @xmath181 . if @xmath182 , then let @xmath183 ; otherwise , @xmath184 , let @xmath185 and we are done by ( @xmath135 ) or lemma [ exchange ] . since @xmath188 by ( @xmath111 ) , we conclude that @xmath189 and @xmath190 , or @xmath191 and @xmath192 . let @xmath193 be the other three neighbors of @xmath194 and @xmath195 . note that @xmath0 contains a @xmath196-path and @xmath197 by ( @xmath111 ) . if @xmath0 contains no @xmath201 -path , then let @xmath202 . otherwise , @xmath0 contains a @xmath203-path . if @xmath0 contains no @xmath204-path , then let @xmath122 ; otherwise , assume that @xmath0 contains a @xmath205 and @xmath206 . if @xmath207 for some @xmath208 , then let @xmath209 and recolor @xmath210 with a color in @xmath211 . otherwise , @xmath212 and @xmath213 . then , we switch the colors of @xmath214 and @xmath215 , and let @xmath216 . note that @xmath0 contains a @xmath220-path by ( @xmath111 ) and then @xmath206 . assume that @xmath221 . if @xmath0 contains no @xmath222-path , then let @xmath100 ; otherwise , @xmath0 contains a @xmath223-path and then let @xmath224 . otherwise , @xmath225 and @xmath226 similarly . thus , @xmath227 . assume that @xmath228 and first let @xmath229 . next , if @xmath0 contains no @xmath230-path , then let @xmath231 ; otherwise , @xmath0 contains a @xmath232-path , we let @xmath233 . otherwise , @xmath234 and @xmath235 similarly . thus , @xmath236 . then , switch the colors of @xmath214 and @xmath215 and let @xmath216 . it follows that @xmath0 contains a @xmath245-path . if @xmath257 , then let @xmath107 . if @xmath258 , then let @xmath259 . otherwise , @xmath260 and first let @xmath261 . if @xmath0 contains no @xmath262-path , let @xmath263 ; otherwise , let @xmath252 . next assume that @xmath268 and @xmath269 . if @xmath270 , then let @xmath271 . so assume that @xmath272 , i.e. , @xmath273 . let @xmath274 . if @xmath0 contains no @xmath275-path , further let @xmath266 ; otherwise , @xmath276 . if @xmath0 contains no @xmath280-path , then let @xmath281 . otherwise , @xmath0 contains a @xmath280-path and @xmath282 . if @xmath283 , then by case 1.1 , we can break that @xmath284-cycle by setting @xmath281 and hence break the bichromatic cycle @xmath80 such that no new bichromatic cycles are produced . thus , @xmath285 . assume that there exists @xmath286 . if @xmath0 contains no @xmath287-path , then let @xmath288 ; otherwise , let @xmath289 . thus , assume that @xmath290 and @xmath291 , @xmath292 ( if @xmath293 and @xmath294 , we can let @xmath281 and have a similar discussion ) . if @xmath0 contains neither a @xmath154-path nor a @xmath155-path , then let @xmath295 , @xmath296 , @xmath297 @xmath298 @xmath299 . otherwise , @xmath0 contains either a @xmath154-path or a @xmath168-path . if @xmath301 @xmath302 , then let @xmath162 and @xmath303 ; otherwise , @xmath304 . if @xmath173 and @xmath0 contains no @xmath305-path , then let @xmath306 ; otherwise , @xmath176 , or @xmath0 contains a @xmath177-path and @xmath178 , @xmath307 . if @xmath173 , then let @xmath308 . otherwise , @xmath181 . if @xmath0 contains no @xmath309-path , then let @xmath308 ; otherwise , @xmath0 contains a @xmath310-path . if @xmath182 , then let @xmath311 . otherwise , @xmath184 and let @xmath312 , @xmath313 . first , assume that @xmath181 . if @xmath0 contains no @xmath315-path , let @xmath308 ; otherwise , @xmath0 contains a @xmath310-path . if @xmath182 , let @xmath316 . if @xmath317 , let @xmath312 and @xmath318 . otherwise , @xmath319 and let @xmath320 . by ( @xmath363 ) , @xmath0 contains a @xmath201-path , and moreover @xmath0 contains a @xmath407-path when @xmath408 , a @xmath409-path when @xmath410 . if @xmath411 for some @xmath412 , then let @xmath413 . otherwise , @xmath125 . since @xmath414 and noting that @xmath415 , we have to consider the following two possibilities : if @xmath416 and @xmath417 , let @xmath418 . if @xmath419 and @xmath403 , let @xmath420 . suppose that we can recolor some edges in @xmath0 such that @xmath430 , and no new bichromatic cycles are produced in @xmath130 . then , by the discussion of case 1.1 , if @xmath0 contains an @xmath431-cycle for some @xmath432 , then we can break this @xmath36-cycle and hence break the cycle @xmath80 such that no new bichromatic cycles are produced . that is , we have the following statement : if @xmath434 and @xmath0 contains no @xmath435-path , let @xmath261 ; if @xmath415 and @xmath0 contains no @xmath436-path , let @xmath437 . clearly , @xmath93 and no new bichromatic cycles are produced in @xmath130 . by ( @xmath433 ) , we complete the proof . similarly , if @xmath438 and @xmath0 contains no @xmath439-path , we can let @xmath440 . hence , assume that : by ( @xmath427 ) and ( @xmath441 ) , @xmath0 contains a @xmath429-path with @xmath459 , or a @xmath443-path with @xmath460 and @xmath461 , or a @xmath462-path with @xmath186 and @xmath463 . if @xmath244 and @xmath0 contains no @xmath464-path , let @xmath465 . otherwise , @xmath466 , or @xmath0 contains a @xmath109-path and @xmath467 , @xmath335 . we need to consider the following subcases . if @xmath112 and @xmath0 contains no @xmath473-path , let @xmath474 and @xmath475 . by ( @xmath427 ) and lemma [ exchange ] , no new bichromatic cycles are produced . otherwise , @xmath186 or @xmath0 contains a @xmath473-path and @xmath476 . if @xmath0 contains no @xmath482-path , let @xmath483 . so assume that @xmath0 contains a @xmath484-path . it follows that @xmath444 by ( @xmath441 ) and the fact that @xmath149 . if @xmath485 , let @xmath486 . otherwise , we derive that @xmath487 . if @xmath0 contains no @xmath488-path , then let @xmath489 . thus , assume that @xmath0 contains a @xmath490-path and @xmath158 . if @xmath0 contains no @xmath491-path , then let @xmath156 ; otherwise , @xmath0 contains a @xmath492-path , which implies that @xmath493 and @xmath165 . if @xmath494 , let @xmath495 otherwise , @xmath496 and then @xmath173 since @xmath497 . if @xmath323 , then let @xmath498 ; otherwise , let @xmath499 . if @xmath244 and @xmath461 , then @xmath502 , let @xmath503 . if @xmath468 and @xmath415 , then @xmath504 , let @xmath505 . otherwise , assume that @xmath244 and @xmath415 . then @xmath506 and @xmath507 . if @xmath0 contains no @xmath508-path , let @xmath509 . otherwise , it follows that @xmath444 by ( @xmath441 ) and @xmath149 . let @xmath510 . if @xmath512 , then the proof is reduced to case 2.1 . thus , @xmath513 and @xmath149 . by ( @xmath427 ) and ( @xmath441 ) , @xmath0 contains a @xmath514-path with @xmath459 , or @xmath0 contains a @xmath439-path with @xmath460 and @xmath461 , or a @xmath515-path with @xmath246 and @xmath186 . if @xmath244 and @xmath0 contains no @xmath109-path , then let @xmath516 . otherwise , @xmath468 , or @xmath0 contains no @xmath517-path and @xmath187 , @xmath335 . we need to consider the following two subcases . if @xmath0 contains no @xmath525-path , let @xmath526 . so assume that @xmath0 contains a @xmath525-path and @xmath165 . if @xmath0 contains no @xmath154-path , let @xmath156 . thus , assume that @xmath0 contains a @xmath527-path and @xmath158 , @xmath528 . note that @xmath529 or @xmath0 contains a @xmath530-path , and @xmath531 by ( @xmath441 ) and then @xmath323 . if @xmath173 , then let @xmath532 . otherwise , @xmath176 . assume that @xmath123 and then @xmath533 , @xmath534 by ( @xmath441 ) . if @xmath535 , let @xmath536 ; otherwise , @xmath153 , let @xmath537 . next , assume that @xmath538 . if @xmath121 , let @xmath539 ; otherwise , @xmath534 . if @xmath540 , then let @xmath536 ; otherwise , @xmath153 , let @xmath541 . if @xmath244 and @xmath461 , then @xmath502 , let @xmath503 . if @xmath244 and @xmath415 , then @xmath506 , @xmath542 , and let @xmath543 . otherwise , assume that @xmath468 and @xmath415 . then @xmath544 . if @xmath0 contains no @xmath545-path , then let @xmath543 ; otherwise , let @xmath546 . @xmath136 assume that @xmath0 contains no @xmath492-path . let @xmath156 . if @xmath0 contains no @xmath572-path , then we are done ; otherwise , @xmath0 contains a @xmath572-path , we can break this @xmath572-cycle by case 1 or case 2.1 as @xmath272 . @xmath136 assume that @xmath0 contains a @xmath492-path . if @xmath573 and @xmath0 contains no @xmath574-path , then let @xmath575 . so assume that @xmath158 or @xmath0 contains a @xmath574-path . if @xmath576 and @xmath0 contains a @xmath574-path , let @xmath577 . if @xmath323 and @xmath0 contains no @xmath578-path , let @xmath498 . if @xmath0 contains no @xmath579-path , we are done ; otherwise , @xmath0 contains a @xmath579-path , we can break this @xmath572-cycle by cases 1 or 2.1 as @xmath272 . otherwise , @xmath158 , and @xmath580 or @xmath0 contains a @xmath578-path . together with ( @xmath441 ) , we have @xmath145 , @xmath581 , and then @xmath582 since @xmath583 . let @xmath584 , @xmath585 , and @xmath586 . by ( @xmath433 ) , we complete the proof . first , assume that @xmath112 , and hence @xmath446 by ( @xmath441 ) . if @xmath329 , let @xmath330 ; otherwise , @xmath587 . if @xmath105 and @xmath0 contains no @xmath588-path , then let @xmath521 and we are done by ( @xmath433 ) ; otherwise , @xmath246 or @xmath0 contains a @xmath473-path . assume that @xmath589 and then @xmath590 since @xmath0 contains no @xmath473-path and @xmath591 . then , let @xmath344 . otherwise , assume that @xmath592 . if @xmath257 , let @xmath593 ; otherwise , @xmath187 . assume that @xmath105 . if @xmath0 contains no @xmath594-path , let @xmath107 ; otherwise , @xmath0 contains a @xmath106-path , let @xmath470 and we are done by ( @xmath433 ) . otherwise , @xmath595 . if @xmath0 contains no @xmath203-path , then let @xmath259 ; otherwise , @xmath0 contains a @xmath201-path , let @xmath596 and we are done by ( @xmath433 ) . assume that @xmath244 . note that @xmath0 contains a @xmath598-path and a @xmath138-path for any @xmath556 by ( @xmath427 ) and ( @xmath441 ) . if @xmath453 , then let @xmath599 and @xmath600 . otherwise , @xmath601 , let @xmath602 , @xmath296 , @xmath297 @xmath298 @xmath603 . it remains to consider the case @xmath604 and @xmath605 since @xmath450 by ( @xmath441 ) . if @xmath171 , let @xmath606 . so assume that @xmath165 . if @xmath607 , then @xmath608 , let @xmath609 . otherwise , @xmath610 and @xmath323 . if @xmath173 , let @xmath609 ; otherwise , @xmath176 and @xmath582 , let @xmath611 . hence , @xmath612 . if @xmath613 and @xmath171 , let @xmath614 , @xmath615 , @xmath297 @xmath298 @xmath616 . otherwise , @xmath327 or @xmath165 and assume that @xmath617 and @xmath618 by ( @xmath441 ) . let @xmath619 . first , let @xmath281 . if @xmath0 contains no @xmath551-path , we are done . otherwise , @xmath0 contains a @xmath552-path and @xmath557 by ( @xmath441 ) . if @xmath625 , then by cases 1 , 2.1 - 2.3 , we can break this @xmath284-cycle and hence break the cycle @xmath80 so that no new bichromatic cycles are produced . thus , assume that @xmath626 . note that @xmath0 contains a @xmath627-path and a @xmath628-path by ( @xmath427 ) and ( @xmath441 ) . if @xmath558 , let @xmath162 . otherwise , @xmath629 . if @xmath566 , let @xmath567 ; otherwise , @xmath630 . if @xmath569 , let @xmath570 ; otherwise , @xmath562 , assuming that @xmath571 , @xmath569 , let @xmath561 . if @xmath648 @xmath649 or @xmath650 , then @xmath651 or @xmath652 and we reduce the proof to cases 1 or 2 . thus , @xmath653 . if @xmath0 contains neither a @xmath654-path nor a @xmath655-path for some @xmath656 , then let @xmath657 . otherwise , for any @xmath656 , @xmath0 contains an @xmath36-path for some @xmath133 , and @xmath0 contains a @xmath658-path , @xmath272 . if @xmath0 contains no @xmath659-path , let @xmath660 . otherwise , @xmath0 contains a @xmath661-path and a @xmath391-path similarly . it follows that @xmath327 and @xmath662 similarly . if @xmath105 and @xmath0 contains no @xmath594-path , then let @xmath663 . hence , @xmath246 or @xmath0 contains a @xmath594-path . it follows that @xmath664 , @xmath244 , and if @xmath105 , then @xmath461 . if @xmath332 , let @xmath665 otherwise , @xmath666 and @xmath667 . let @xmath668 . @xmath74 * proof . * the proof is proceeded by induction on the vertex number of @xmath0 . if @xmath670 , then @xmath0 is the complete graph @xmath671 , and it is easy to show that @xmath669 . let @xmath0 be a 4-regular graph with @xmath672 . obviously we may assume that @xmath0 is 2-connected by lemma [ 4 ] . if @xmath0 contains no 3-cycles , then @xmath26 by theorem [ triangle - free ] . so assume that @xmath0 contains at least one 3-cycle . for any graph @xmath673 with @xmath674 and @xmath675 , by induction hypothesis or lemma [ 4 ] , @xmath673 admits an acyclic edge-6-coloring @xmath28 using the color set @xmath75 . let @xmath30 be a vertex with neighbors @xmath676 that lies in at least one 3-cycle . to extend @xmath28 to the original graph @xmath0 , we split the proof into the following cases . for @xmath678 , let @xmath679 be the neighbor of @xmath680 different from @xmath681 , where the indices are taken modulo @xmath27 . since @xmath672 , @xmath682 . we only need to consider the following subcases by symmetry : since @xmath0 is @xmath685-connected , @xmath686 . let @xmath687 . without loss of generality , assume that @xmath688 and @xmath689 . to extend @xmath28 to the graph @xmath0 , we let @xmath690 , @xmath691 , @xmath692 , @xmath693 , and @xmath694 . by case 1 , @xmath762 . let @xmath763 and assume that @xmath764 . then , first , let @xmath765 , @xmath766 . next , if @xmath767 and assume that @xmath768 , @xmath769 , then let @xmath770 and @xmath771 . otherwise , @xmath772 and @xmath773 . then , let @xmath774 and @xmath775 . since @xmath673 has no bichromatic cycles , it is obvious that @xmath673 has no @xmath776-path and thus @xmath0 contains no @xmath777-cycle . ( see fig.5 . we neglect to explain why such bichromatic cycles can not exist in the following similar cases . ) if @xmath785 and @xmath182 , then let @xmath786 , @xmath787 , and @xmath788 . otherwise , @xmath773 . then , first let @xmath789 , @xmath790 . and next , if @xmath769 , then let @xmath791 . otherwise , @xmath792 and assume that @xmath793 , then let @xmath794 . if @xmath769 , then let @xmath789 , @xmath796 , and @xmath797 . since @xmath673 has no bichromatic cycles and @xmath798 , it is obvious that @xmath673 has no @xmath799-path for any @xmath800 and thus @xmath0 contains no @xmath801-cycle . ( see ( a ) , ( b ) in fig.6 . ) if @xmath792 , then let @xmath787 , @xmath802 . if @xmath323 , then let @xmath803 . otherwise , @xmath580 and @xmath804 similarly . it follows that @xmath805 . then , let @xmath806 . since @xmath798 in @xmath673 , it is obvious that @xmath317 and thus @xmath0 contains no @xmath807-cycle . ( see ( a ) , ( c ) in fig.6 . ) let @xmath699 be the neighbor of @xmath759 different from @xmath811 . let @xmath706 be the neighbor of @xmath736 different from @xmath737 . let @xmath812 be the neighbor of @xmath742 different from @xmath743 . by cases 1 and 2 , we can assume that @xmath813 are pairwise distinct . let @xmath825 and assume that @xmath826 , @xmath827 . then , first let @xmath828 , @xmath829 . next , if @xmath830 and @xmath182 , then let @xmath831 ; otherwise , @xmath773 and let @xmath832 . if @xmath833 , then let @xmath834 and @xmath835 ; otherwise , @xmath836 and let @xmath837 . let @xmath699 be the neighbor of @xmath736 different from @xmath737 . let @xmath838 be the other two neighbors of @xmath759 different from @xmath839 . note that @xmath813 are pairwise distinct by the previous argument . let @xmath840 and assume that @xmath841 , @xmath842 , @xmath843 , @xmath844 @xmath781 @xmath816 . then , first let @xmath845 . if @xmath847 , then let @xmath864 and @xmath865 @xmath298 @xmath866 . if @xmath0 contains no @xmath867-path , then let @xmath868 . otherwise , @xmath0 contains a @xmath869-path and hence @xmath870 . we let @xmath871 @xmath298 @xmath872 . let @xmath892 be the set of the other neighbors of @xmath680 for @xmath893 . by cases 1 to 3 , @xmath894 and @xmath895 . let @xmath896 , where @xmath87 and @xmath897 are new vertices added . assume that @xmath898 , @xmath899 , @xmath900 , @xmath901 @xmath781 @xmath902 , @xmath685 , @xmath903 , @xmath904 , @xmath905 @xmath781 @xmath906 , @xmath907 , where @xmath908 , see fig.8 . then , first let @xmath909 for @xmath910 , and @xmath911 for @xmath912 , @xmath913 . next , by symmetry , we need to consider the following . if @xmath0 contains no @xmath928-path for some @xmath929 and @xmath930 , then let @xmath931 , @xmath932 , and @xmath933 . otherwise , @xmath0 contains a @xmath928-path for any @xmath208 and @xmath930 and @xmath934 . it follows that @xmath935 and assume that @xmath936 . first , let @xmath937 , @xmath938 , and @xmath939 . next , if @xmath940 , then we are done ; otherwise , @xmath941 and let @xmath942 , @xmath943 . if @xmath926 and @xmath944 , let @xmath945 @xmath298 @xmath903 and @xmath295 , @xmath615 , @xmath946 , @xmath947 @xmath298 @xmath902 , @xmath27 , @xmath822 , @xmath948 . if @xmath914 and @xmath949 , let @xmath295 , @xmath950 , @xmath615 , @xmath951 @xmath298 @xmath952 . if @xmath953 and @xmath954 , let @xmath295 , @xmath950 , @xmath955 , @xmath956 @xmath298 @xmath952 . if @xmath953 and @xmath957 , let @xmath958 @xmath298 @xmath822 and @xmath959 . if @xmath926 and @xmath960 , let @xmath961 and @xmath962 . if @xmath914 and @xmath963 , let @xmath961 and @xmath964 . let @xmath967 be defined similarly as in case 4 . by cases 1 to 4 , @xmath968 . let @xmath969 and assume @xmath970 , as shown in fig.9 . then , first let @xmath971 for @xmath972 . next , since @xmath973 or @xmath974 , we need to consider the following . if @xmath0 contains neither @xmath975-path nor @xmath976-path , let @xmath977 @xmath978 . otherwise , @xmath0 contains either a @xmath979-path or a @xmath980-path . if @xmath981 , then let @xmath982 and @xmath983 , @xmath984 . if @xmath985 , then let @xmath986 and @xmath987 , @xmath988 . otherwise , @xmath989 , @xmath990 similarly and then @xmath182 . first , let @xmath991 and @xmath992 . next , if @xmath0 contains no @xmath993-path , then we are done ; otherwise , @xmath0 contains a @xmath993-path , which can not pass through @xmath742 and @xmath994 . we switch the colors of @xmath615 and @xmath950 . otherwise , assume that @xmath165 and @xmath995 similarly . first , assume that @xmath0 contains no @xmath996-path and let @xmath997 . if @xmath0 contains no @xmath998-path for some @xmath999 , then let @xmath1000 and @xmath1001 . otherwise , @xmath0 contains a @xmath1002-path for any @xmath999 and @xmath1003 , @xmath1004 . if @xmath1005 and @xmath849 , then let @xmath1006 ; otherwise , @xmath1007 , @xmath1008 and let @xmath1009 . next , assume that @xmath0 contains a @xmath996-path and @xmath792 , @xmath849 . then , let @xmath1010 . otherwise , @xmath145 and @xmath1011 . by symmetry , we may assume that @xmath123 . if @xmath0 contains no @xmath1015-path for some @xmath912 , then let @xmath1016 . otherwise , @xmath0 contains a @xmath1015-path for any @xmath912 and @xmath1017 . if @xmath1018 , then let @xmath1019 . otherwise , @xmath1020 . if @xmath0 contains no @xmath1022-path , then let @xmath1023 and @xmath1024 . otherwise , @xmath0 contains a @xmath1022-path and @xmath1025 and @xmath1026 , @xmath1027 . if @xmath0 contains no @xmath1028-path for some @xmath1029 , then let @xmath1030 and @xmath1031 . otherwise , @xmath0 contains a @xmath1028-path for any @xmath1032 and @xmath1033 . if @xmath1034 , then let @xmath1035 , @xmath1036 @xmath298 @xmath1037 . if @xmath1038 , then let @xmath1039 and @xmath1040 . otherwise , @xmath1041 and @xmath769 . then , let @xmath923 , @xmath1042 , and @xmath1043 .

an acyclic edge coloring of a graph @xmath0 is a proper edge coloring such that no bichromatic cycles are produced . the acyclic chromatic index @xmath1 of @xmath0 is the smallest integer @xmath2 such that @xmath0 has an acyclic edge coloring using @xmath2 colors . fiam@xmath3ik ( 1978 ) and later alon , sudakov and zaks ( 2001 ) conjectured that @xmath4 for any simple graph @xmath0 with maximum degree @xmath5 . basavaraju and chandran ( 2009 ) showed that every graph @xmath0 with @xmath6 , which is not 4-regular , satisfies the conjecture . in this paper , we settle the 4-regular case , i.e. , we show that every 4-regular graph @xmath0 has @xmath7 . * keywords : * acyclic edge coloring ; 4-regular graph ; maximum degree * ams subject classification . * 05c15 [ theorem]corollary [ theorem]proposition = 0.5 cm

recently , the sno collaboration has presented the first direct measurement of the total active flux of @xmath0b neutrinos comming from the sun @xcite . sno detects solar neutrinos by means of three different reactions : charged current reaction ( cc ) @xmath1 , neutral current reaction ( nc ) @xmath2 , and elastic scattering reaction ( es ) @xmath3 . the cc reaction is sensitive exclusively to @xmath4 , while he nc and es reactions are sensitive to all flavors , with less sensitivity to @xmath5 ( @xmath6 ) in the case of es . using the integrated rates above the threshold of 5 mev for the three reactions , they have determined both , the electron and the active non-@xmath4 component of the @xmath0b neutrino flux . the latter is @xmath7 grater than zero , yielding strong evidence for neutrino flavor transformation . this result has been obtained under the assumption that the shape of the @xmath0b neutrino spectrum is the same as predicted by the standard solar model ( ssm ) @xcite . the absence of a significant distortion of the spectrum has been observed by super - kamiokande ( sk ) @xcite and confirmed by sno . the impact of recent sno results on the global oscillations solutions , including all solar neutrino data , have been analyzed by several authors @xcite . in this work we address the question of how the forthcoming sno rates unconstrained by the standard @xmath0b shape can be used to test the presence of non - electron active neutrinos in the solar neutrino flux . in sec . [ primera ] we establish the relation between the fractional flavor components of the spectrum @xmath8 and the quantities @xmath9 that determine the cc , nc , and es fluxes in terms of the measured experimental rates . this relation involves an average over the appropriate experimental response functions and are presented in sec . [ segunda ] . in this section we also illustrate to what extent the sno rates unconstrained by the standard @xmath0b shape could play a role in testing the active oscillations hypothesis . a model independent analysis of the data of sno and sk incorporating the nc measurement of sno is given in sec . [ ultima ] . the count - rate per energy interval at sno for the nc events is related to the true ( and unknown ) spectrum of solar neutrinos arriving at the earth @xmath10 , as follows : = , where @xmath11 is the cross section for the nc process and @xmath12 . the quantities @xmath13 ( @xmath14 ) satisfy @xmath15 , with @xmath16 denoting a sterile neutrino . according to the ssm @xcite the only neutrinos produced in the sun are @xmath4 , therefore the neutral current count rate at sno should be = _ , where @xmath17 is the energy spectrum of the @xmath4 given by the ssm . let @xmath18 be the ratio of the _ true _ total neutrino flux @xmath19 to the predicted total flux @xmath20 : = f _ . we say that there is _ no deformation of the neutrino spectrum _ produced in the sun with respect to the ssm prediction if @xmath21 . in a more general situation , we could have = 1 , _ _ with @xmath22 a certain positive function of @xmath23 , that satisfies @xmath24 . we have assumed that only @xmath4 are produced in the sun . in general , the ratio @xmath25 of the observed to the theoretical neutral current spectra will be energy - dependent : r^_= = f , where _ x= p_x , and @xmath26 . if the neutrino spectrum produced in the sun has no deformation , then the function @xmath22 in ( [ deformation_def ] ) is equal to one for all energies . in this case , @xmath27 , and @xmath28 . let @xmath29 denote the normalized solar neutrino spectrum predicted by the ssm . this quantity satisfies the relation @xmath30 , where @xmath31 is the true ( and unknown ) normalized solar neutrino spectrum . the integrals over the relevant energy range of the normalised spectra are equal to one d= d_= 1 . from the fact that @xmath32 , we have d _ = + d 1 . and therefore , if @xmath33 is a constant , we have @xmath34\leq 1 $ ] . in addition , if all the @xmath35 are constant then @xmath36 . the ratio of the observed to the predicted charged current spectra can also be written as r^_&= & + & = & = f , where @xmath37 is the cross - section for the cc reaction . relations ( [ ratio_nc ] ) and ( [ ratio_cc ] ) are model independent . they make no assumption on @xmath18 or neutrino oscillations , nor require the quantities @xmath38 to be considered as probabilities . the elastic scattering event rate is also available from sno . this rate , normalised to the ssm prediction is given by r^_&= & + & = & f ( + ) , where @xmath39 for @xmath40 mev . using eqs . ( [ deformation_def ] ) and ( [ def_peacal ] ) , the @xmath4 component of the solar neutrino flux @xmath41 can be written as = f _ . we will say that the electron neutrino spectrum has _ no deformation at the earth _ whenever @xmath42 is proportional to @xmath17 . then , from eq . ( [ phi_f_pcal ] ) we see that a constant @xmath43 would imply that there is no distortion of the @xmath4 spectrum at the earth , and viceversa . according to sk @xcite and sno @xcite the ratios @xmath44 , @xmath25 , and @xmath45 are practically constant for @xmath46 5 mev . as a consecuence , @xmath38 are constants as can be seen by taking any combination of two equations among ( [ ratio_nc ] ) , ( [ ratio_cc ] ) , and ( [ ratio_es ] ) . for example , from eqs . ( [ ratio_nc ] ) , and ( [ ratio_cc ] ) we have = , = ( r^ _ - r^ _ ) , with @xmath47 , @xmath48 , and @xmath9 constants . therefore , the present experimental evidence indicates that no significant distortion of the @xmath0b neutrino spectrum has been observed at the earth . in principle , in eq . ( [ phi_f_pcal ] ) the energy dependence of the _ true _ survival probability @xmath49 could be approximately compensated by @xmath22 in order to explain the observed energy independence of the neutrino spectrum at the earth . therefore , a distortion of the neutrino spectrum produced in the sun remains as an unlikely speculation . the elastic scatering rate measured by sno can be written in the form r^ _ = ^ _ ^ _ , with ^_&= & , + ^ _ & = & de _ _ ^ _ , + ^ _ & = & de _ _ ^_p___e , a . _ _ here , @xmath50 is the measured elastic scattering flux . with similar definitions , the cc event count - rate is given by r^ _ = ^ _ ^ _ , where ^_&= & ^ _ , + ^ _ & = & de _ _ ^ _ , + ^ _ & = & de _ _ ^_p___e . _ _ in eq . ( [ ff ] ) , @xmath51 is the flux measured by sno through the cc reaction . the electron neutrino component of the flux seen by sno through the elastic scattering reaction is ( ^ _ ) _ = ^ _ . from ( [ prob_av_cc ] ) and ( [ esta ] ) we get = . the event count - rate for the nc can be written as follows : r^ _ = ^ _ ^ _ , where we have defined ^_&= & , + ^ _ & = & de _ _ ^ _ , + ^ _ & = & de _ _ ^_p___e , a . _ _ here , @xmath52 represents the flux measured by sno through the nc reaction . we must keep in mind that the cross sections @xmath53 , @xmath54 , and @xmath55 , that appear in eqs . ( [ prob_av_es ] ) , ( [ prob_av_cc ] ) , and ( [ prob_av_nc ] ) depend on the response functions of the sno detector . if @xmath56 is the electron neutrino component of the flux seen by sno through the nc reaction , then from ( [ prob_av_cc ] ) it is clear that = . a ratio @xmath57 less than one necessarily implies the presence of a non-@xmath4 active neutrino in the solar neutrino flux . what can actually be done with the experimental measurements is to calculate the ratio @xmath58 . as eq . ( [ aa ] ) shows , in principle it could be possible to have the ratio @xmath59 equal to one , and still be in agreement with the experimental results from sno by having @xmath60 . however , given the observed non - dependency of the quantities @xmath61 on the energy , we have that the averages defined in eqs . ( [ prob_av_es ] ) and ( [ prob_av_cc ] ) are approximately equal : @xmath62 . when this result is combined with eq . ( [ aa ] ) , gives irrefutable evidence that there are @xmath63 and/or @xmath64 arriving at the detector . a similar conclusion can be drawn by comparing the cc and nc fluxes . the experimental evidence suggests that @xmath65 , from where we see that @xmath66 implies @xmath67 . the @xmath68 ratio of rates given by the sno collaboration has been derived assuming the ssm @xmath0b spectral shape . up to now sno has not released the information for the corresponding unconstrained ratios . when this information becomes available the absence of active neutrino flavor transformations could be ruled out even for a non constant @xmath61 . to see this , let us assume for a moment that @xmath69 . then , we have = = 1 , and from eqs . ( [ aa ] ) and ( [ otra ] ) , we could write = = 1 , where @xmath70 with @xmath71 . taking into account the equality in eq . ( [ 3eq ] ) we find that the following condition should be met d _ _ = 0 . where @xmath72 . using the values calculated by bahcall @xcite for the cc and nc cross sections which take into account the resolution and threshold used in sno , it can be seen that @xmath73 , for @xmath74mev , whenever the ratio @xmath75 . since @xmath76 is positive then , if the measured ratio @xmath77 is greater than @xmath78 , the condition stated in eq . ( [ nueva ] ) can not be met , leading to the conclusion that @xmath79 can not be equal to zero . for reference , @xmath80 mev corresponds to an average recoil electron kinetic energy of @xmath81 mev , according to @xcite . then , the integrand in eq . ( [ nueva ] ) is negative definite in the relevant neutrino energy range if @xmath82 ( see fig . [ sigmas ] ) . it is possible to estimate the unconstrained rates of sno using the information that has been published by the collaboration @xcite . the es unconstrained rate can be taken to be the same as that constrained by the @xmath0b standard shape , since it is determined essentialy from energy independent observations ( @xmath83 distribution ) . the nc unconstrained rate @xmath84 can be estimated in terms of the constrained rate @xmath85 and the corresponding total fluxes that have been reported by the collaboration : @xmath86 where @xmath87 and @xmath88 are the total unconstrained and constrained nc fluxes , respectively . finally , the cc unconstrained rate is calculated considering that the total number of signal events is the same as for the constrained analysis . taking these considerations properly into account , we estimate the ratio of unconstrained rates to be @xmath89 the error is large because the error in the estimate of the nc unconstrained rate in terms of the unconstrained total nc flux is large . nontheless , the central value is well above the lower limit of 2.31 given above , and indicates that the need for active oscillations is favored . if the forthcomming results from sno confirm that @xmath77 is actually larger than the limit we found using the estimates of @xcite for the ( response - averaged ) cross - section , then the probability transition of solar @xmath4 into an active neutrino must be different from zero . consequently , it is not possible to explain the experimental cc and nc results of the collaboration claiming only spectral distortion at the earth and/or oscillations into sterile neutrinos . it is important to notice that we arrived to this conclusion without assuming that @xmath38 are constant . a systematic calculation of the shape of the @xmath0b neutrino spectrum has been presented in @xcite , together with an estimation of the theoretical and experimental uncertainties . no such precise knowledge has been required in our approach , based in the analysis of the negativeness of the integrand in eq . ( [ nueva ] ) . in this section , we will use the elastic scattering measurement of sk instead of the corresponding measurement of sno because it has a smaller error . equivalently to eq . ( [ gg ] ) , we have r^ _ = ^ _ ^ _ , with definitions like those given in eq . ( [ prob_av_es ] ) . as noted by fogli _ @xcite , the response functions of sno and sk behave quite similarly if appropiate thresholds are used . in this way the equality of @xmath90 and @xmath91 can be ensured . as discussed in the previous section and noticed in ref . @xcite , this equality can also be stablished independently of the kinetic energy threshold if the energy independence of the @xmath38 is adopted . here we follow this approach . accordingly , eqs . ( [ ff ] ) , ( [ ee ] ) , and ( [ jj ] ) can be rewritten as follows : r _ & = & x + y , + r _ & = & x , + r _ & = & x + y , where @xmath92 , @xmath93 , and @xmath94 are the total rates normalised to the ssm predictions : r_^ & = & ^ _ _ , + r_^ & = & ^ _ _ . we have introduced the variables @xmath95 and @xmath96 , which represent the relevant degrees of freedom of the problem . since @xmath9 are constants , then @xmath97 . from eq . ( [ las_ecs ] ) , @xmath94 can be expressed in terms of @xmath93 and @xmath92 : r _ = , which is valid for any value of @xmath98 and @xmath99 @xcite . we define the @xmath100 function ^2 = _ x , where @xmath101 are given in eq . ( [ las_ecs ] ) . here , @xmath102 and @xmath103 are the experimental values for the normalised rates and their errors respectively @xcite : r^ _ & = & 0.459 0.017 + r^ _ & = & 0.349 0.021 + r^ _ & = & 1.008 0.123 . letting @xmath98 and @xmath99 vary as free parameters , we find the minimum value of @xmath100 ( @xmath104 ) , and the @xmath105 , and 9 contours for these parameters as shown in fig . [ xy ] . the projection of these contours on the @xmath98 , and @xmath99 axes ( @xmath106 in each case ) , give their @xmath107 , @xmath108 , and @xmath109 ranges @xcite . the best fit values along with their @xmath107 errors are x & = & 0.350.02 , + y & = & 0.66 0.11 . the previous values times the ssm total @xmath0b flux ( @xmath110 ) , give the @xmath4 and @xmath5 components of the flux which are consistent with the values reported by sno @xcite . when @xmath111 , _ i.e. _ , there is no discrepancy between the ssm and the true total @xmath0b neutrino flux , ec.([x_y_best_fit ] ) gives the 1@xmath112 ranges for the quantities @xmath43 and @xmath113 . in this case the sum @xmath114 is consistent with being equal to one . let us now assume that there exist oscillations only among active states . then , we have @xmath115 , there is also no deformation of the spectrum produced in the sun ( @xmath116 ) , and @xmath117 . we obtain the @xmath107 , @xmath108 , and @xmath109 ranges for @xmath18 and @xmath43 from the contours in fig . [ fpe ] , built by mapping the contours of fig . [ xy ] to the plane @xmath118 using the constriction @xmath119 . these contours coincide with those found in ref @xcite directly from eq . ( [ las_ecs ] ) , with @xmath99 replaced by @xmath120 . from fig . [ fpe ] it can be seen that , by including the nc measurement in the analysis , a significant improovement has been achieved in the @xmath107 error bar of @xmath18 with respect to the one obtained using only the sk and the sno cc data @xcite . the best fit values and their @xmath107 ranges are f&=&1.01^+0.11_-0.09 + & = & 0.34^+0.05_-0.04 impossing the less stringent condition @xmath121 , with @xmath122 the value of @xmath18 will be bounded by x+yf , from where we see that allowing for a non vanishing probability to oscillate into a sterile neutrino ( @xmath123 ) , we have larger upper bound for @xmath18 . assuming that @xmath124 , we have that = 1 - . from the dispersion of @xmath98 and @xmath99 we can find allowed regions in the @xmath125 plane , corresponding to the 68 , 95 , and 99 @xmath126 confidence levels . as shown in fig . [ fps ] , these regions are not bounded and hence it is not possible to determine @xmath18 and @xmath127 with the existing data @xcite . in this work we have examined the relation between the observed quantities @xmath128 , @xmath58 with the flavor fractional @xmath4 content of the fluxes measured through the es and nc reactions . when combined with the hypothesis of a non distorted @xmath0b spectrum the measurement gives a clear signal of active flavor transformation . as we also show , when available , the sno experimental rates unconstrained by the @xmath0b standard shape , combined with the cross - section as calculated in ref.@xcite , could give conclusive evidence for active oscillations , even for a non constant @xmath61 . finally a model independent analysis including the latest sk and sno data is performed under the assumption of constant @xmath38 , with and without the condition @xmath115 . our result agrees with ref . @xcite in the sense that no conclusion can be drawn with the present data about the sterile neutrino content of the solar neutrino flux .

we discuss the relation between the observed cc , es , and nc fluxes with the flavor fractional content of the solar neutrino flux seen by sno . by using existing estimates of the cross sections for the charged and neutral current reactions which take into account the detector resolution , we show how the forthcoming sno rates unconstrained by the standard @xmath0b shape could test the oscillations into active states . we perform a model independent analysis for the super - k and sno data , assuming a non distorted spectrum . # 1#1 # 1#1

point defects can affect the properties of solids in a dramatic way.@xcite they determine , for example , the conductivity of semiconductors , the color of natural crystals , and the mechanical properties of materials . equally important , defects influence or govern the performance and the long - term stability of a wide range of semiconductor devices , such as metal - oxide - semiconductor field - effect transistors , photovoltaic cells , solid fuel cells , to name a few . the theoretical characterization of defects , especially in wide band - gap materials , has become increasingly important in the attempt to understand and control the performance of these devices.@xcite in the last decades , density functional theory ( dft ) has grown into the standard theoretical model to describe the electronic and atomic structure of solids . the common approximations to dft , _ viz . _ the local density approximation ( lda ) and the generalized gradient approximation ( gga ) , systematically underestimate band gaps of semiconductors and insulators . since the band gap is the relevant energy scale in the study of defects , this so - called `` band - gap problem '' of lda and gga severely affects the predictive power of these approximations when applied to defect levels . recently there have been lots of efforts to assess the importance of band gap corrections @xcite and to use theoretical models giving a much more appropriate description of the bulk band structure . the choice of methods is large and includes the lda+@xmath0 method,@xcite approximate self - interaction correction schemes,@xcite hybrid density functionals , @xcite the use of modified pseudopotentials,@xcite empirical schemes,@xcite and more advanced theoretical tools , such as the many - body perturbation theory within the @xmath1 and higher approximations . @xcite it appears evident to assume that a good theoretical model must at least satisfy two conditions , namely _ ( i ) _ give an accurate electron density of the defect system and _ ( ii ) _ yield a good band gap of the host material . while these two requirements form a necessary prerequisite to obtain reliable results concerning defect formation energies and associated charge transition levels,@xcite it has recently become apparent that it is by no means sufficient . this is best exemplified in the case of defect energy levels in zno.@xcite this is a particularly severe case , because the lda and the gga yield a bulk band - gap of 0.6 - 0.8 ev , severely underestimating the experimental value of 3.44 ev . for the case of the ( + 2/0 ) charge transition level of the oxygen vacancy ( v@xmath2 ) theoretical models yield levels either as low as 0.6 ev above the valence band maximum ( vbm ) or as high as 2.4 ev above vbm . these results differ significantly despite the fact that in all these theoretical models the `` band - gap problem '' was accounted for . in addition , other critical issues , such as finite - size effects associated to the supercell treatment , were presumably under control in these studies . furthermore , the first condition concerning the accuracy of the electron density was clearly also fulfilled since the involved electronic state corresponds to the fully symmetric @xmath3 state which is already correctly described via a semilocal functional . the second condition concerning the band gap was fulfilled by construction . recently , lany and zunger provided a very detailed overview of the way various theoretical and computational approximations affect the determination of defect formation energies and charge transition levels.@xcite they concluded that , in addition to the two requirements discussed above , a reliable theoretical model should correctly describe the relative positions of all relevant electronic states . for zno , this condition mainly concerns the position of the zn 3@xmath4 states with respect to the conduction and valence band edges . the importance of this requirement becomes evident when considering shallow defects , the wavefunctions of which can be always thought as arising from a linear combination of bulk bands . in this work , we show that that there is yet another crucial requirement that the theoretical model must fulfill . in order to yield an appropriate description of defect formation energies and associated charge transition levels , the positions of the vbm and the conduction band minimum ( cbm ) with respect to a suitably defined reference potential should also be accurately described . to demonstrate this , we first calculate the ( + 2/0 ) charge transition level of the v@xmath2 in zno and compare our result with those available in the literature . our study adds to a series of studies,@xcite in which conflicting results were found . however , we show that these seemingly incompatible findings agree reasonably with each other when an alternative alignment scheme is used . we provide theoretical arguments to rationalize this finding . similar results are expected for other atomically localized defects and for other materials in which the `` band - gap problem '' of semilocal calculations is particularly severe . our investigation thus leads to a deeper understanding of the `` band - edge problem '' in the theoretical study of defect levels and provides a requirement for the theoretical model in addition to the conditions mentioned above . this paper is organized as follows . in sec . [ comp ] , we summarize our computational approach for calculating defect formation energies and charge transition levels . the obtained results are discussed and compared to other calculations in sec . [ ov ] . an alignment scheme with respect to the average electrostatic potential is introduced and found to bring all the calculated results in good agreement with each other . the significance of this alignment of bulk band structures is discussed in more detail in sec . [ alignment ] . to understand our findings about defect charge transition levels , fundamental differences between localized and extended states in approximate dft formulations are discussed in sec . [ states ] . in sec . [ edge ] , two different theories reproducing the experimental band gap but differing in the positions of the bulk band edges with respect to the vacuum level are taken under consideration to complete our rationale . we summarize our work and draw conclusions in sec . [ conclusions ] . in the present calculations , the electronic structure was treated using two different functionals . first , we employed the gga functional proposed by perdew , burke , and ernherhof ( pbe ) . @xcite for comparison with previous calculations in the literature , we obtained for bulk zno a band gap of 0.83 ev , to be compared with the experimental value of 3.44 ev . to obtain an improved band gap , we used a hybrid density functional @xcite defined by a single parameter @xmath5 corresponding to the fraction of nonlocal fock exchange admixed to the gga exchange : @xmath6 a hybrid functional with @xmath7 and with the pbe for the gga part @xcite is referred to as pbe0 , pbeh , or pbe1pbe . for zno , we obtained a band gap of 2.82 ev using this functional . the experimental band gap is reproduced with @xmath8 . in the following , we refer to this functional as to pbeh-32 . while this adjustment of @xmath5 is empirical , it can be justified to a certain extent.@xcite it can be shown that the optimal value of @xmath9 , i.e. the one which reproduces the experimental band gap , is approximately given by @xmath10 . here , @xmath11 is the electronic part of the static dielectric constant . for a large number of materials this relationship is approximately fulfilled.@xcite the adjustment of @xmath5 can also be justified in some cases by comparison with more accurate @xmath1 calculations.@xcite the main quantity that needs to be calculated is the formation energy of the oxygen vacancy in a charge state @xmath12 , which is given as:@xcite @xmath13 here @xmath14 is the total energy of the defect system containing a single o vacancy in the supercell , @xmath15 is the total energy of the host material without any defect , @xmath16 is the atomic chemical potential of oxygen , and @xmath17 is the electron chemical potential . the latter is referred to the vbm @xmath18 . except for semiconductors with degenerate doping , @xmath17 varies between zero and the band gap of the material @xmath19 . the atomic chemical potentials @xmath16 and @xmath20 are bound by the condition that zno is in thermal equilibrium with the reservoir of o and zn atoms , i.e. @xmath21 . oxygen - rich conditions are defined by the onset of spontaneous formation of o@xmath22 molecules , i.e. by @xmath23 . oxygen - poor ( zn - rich ) conditions are correspondingly defined by the onset of spontaneous formation of bulk zn crystallites , i.e. via @xmath24 . the formation of oxygen vacancies in zno is hindered in o - rich conditions , and facilitated in o - poor conditions . the calculation of the o chemical potential in o - poor conditions poses some difficulties when hybrid density functionals are used , because this involves the calculation of the total energy of bulk zn . in hartree - fock theory the description of metals leads to divergences , and the same problem is also found with hybrid functionals . to overcome this problem , we assume that the cohesive energy of bulk zn , which is well described in the gga , does not change significantly in the hybrid functional calculation.@xcite alternatively , one could define the o chemical potential in o - poor conditions by assuming that the separation between the o - rich and o - poor chemical potentials in gga is preserved in the hybrid functional calculation ; this condition corresponds to assuming equal formation energies for zno in gga and in the hybrid functional scheme . these two ways of determining the o chemical potential in o - poor conditions lead to formation energies differing by about 0.4 ev . charge transition levels correspond to the specific value of the electron chemical potential for which two charge states have equal formation energies . the ( + 2/0 ) charge transition level is thus given by : @xmath25 charge transition levels do not depend on atomic chemical potentials . the calculations were performed within a plane - wave pseudopotential formulation . soft norm - conserving pseudopotentials @xcite were generated at the pbe level and used in all subsequent calculations . the plane - wave kinetic energy cutoff , determined by the much harder o pseudopotential , was set to 80 ry . the calculations in the present paper were performed with the code cpmd.@xcite we explicitly treated the singularity in the nonlocal exchange potential.@xcite we used the experimental lattice parameters for bulk zno , since these were found to be very close to theoretical lattice parameters obtained with hybrid functionals.@xcite we also used experimental lattice constants in our gga calculations , finding results which did not differ in any significant way from gga calculations performed with theoretical lattice parameters.@xcite upon defect formation , geometry relaxations were performed with both the gga and the pbeh-32 functionals . the defect structures achieved in the two cases were found to be very similar : in pbeh-32 , for example , pbe - optimized defect structures are only 0.08 ev higher in energy than those optimized consistently at the pbeh-32 level . hence , geometry optimization at the pbeh-32 level has no effect on the position of the ( + 2/0 ) charge transition level [ eq . ( [ ctl1 ] ) ] . vs inverse number of atoms contained in the supercell @xmath26 , ( a ) for the pbe calculation ( @xmath27 and @xmath28 @xmath29-point meshes ) and ( b ) for the pbeh-32 calculation ( @xmath27 mesh ) . @xmath30 is referred to the respective vbm . [ conv ] , width=264 ] for the defect structures we used the supercell approach . this gives rise to finite - size effects which need to be accounted for . first , as suggested by van de walle and neugebauer , @xcite the total energies of charged defects were corrected by @xmath31 , @xmath32 being the shift needed to align the local potential of the _ neutral _ system far from the defect to that of a separate unperturbed bulk calculation , which was used to determine @xmath18 . this term was found to be quite small for the supercells employed in our calculations . second , the total energies of charged defect states are subject to spurious electrostatic contributions associated to the periodic boundary conditions and to the compensating background charge in our supercell calculations . to evaluate these effects , we used an extrapolation scheme based on supercell calculations of increasing size , containing 96 , 192 , and 384 atoms , as shown in fig . when using the pbe functional , the convergence of formation energies and charge transition levels is accelerated when using the @xmath28 monkhorst - pack mesh instead of a sampling at the sole @xmath33 point [ fig . [ conv](a ) ] . hence , finite - size corrections are sizeable for the pbe calculation and a careful extrapolation of the results is needed , as previously shown by oba _ _ @xcite at variance , a denser @xmath29-point mesh turned out to be unnecessary for a calculation with the hybrid functional pbeh-32 [ fig . [ conv](b ) ] . indeed , in the latter case , the bulk band gap is substantially larger and the dispersion of the defect state is already negligible for the smallest supercells considered . this behavior is in line with observations in a previous study on defects in zno.@xcite a notable difference between finite size effects in pbe and pbeh-32 calculations suggests that unphysical defect - defect interactions mediated by bulk bands could be operative in the former case.@xcite for the largest supercell considered here , we obtain a conservative estimate of 0.20 ev for the residual finite - size error by considering the monopole correction proposed by makov and payne.@xcite for the neutral oxygen vacancy , we obtained , at the pbe level , formation energies of 3.17 ev in o - rich conditions and of 0.50 ev in o - poor conditions . in the pbeh-32 calculation , the corresponding value is 3.57 ev in o - rich conditions . in o - poor conditions , we found 0.50 and 0.90 ev depending on whether the cohesive energy of zn or the formation energy of zno is taken from the gga , respectively . our values agree well with the value of 0.8 ev found in ref . and that of 0.9 - 1.0 ev in ref . thus , our results confirm that the formation energy of the o vacancy in o - poor conditions is small enough to lead to a noticeable concentration of these defects . at variance with these results , janotti and van de walle reported much higher formation energies for the neutral v@xmath2.@xcite they used an extrapolation procedure based on lda+@xmath34 and an additional assumption about the behavior of the formation energy of the _ charged _ vacancy upon the band - gap correction . while the former extrapolation has been criticized due to the unphysical values to which the @xmath34 parameter extrapolates to,@xcite we argue here that it is the latter assumption that is inconsistent with the hybrid functional calculations . indeed , the extrapolation procedure adopted in ref . leads to charge transition levels that agree well with those obtained with hybrid functionals . the dependence of the formation energy on the electron chemical potential is shown fig . [ format ] for oxygen - poor conditions . for simplicity , the oxygen chemical potential was set to the average value derived from the two definition schemes described above . the ( + 2/0 ) charge transition level occurs at @xmath35 ev . this result agrees well with other calculations based on hybrid functionals . et al_. found the ( + 2/0 ) charge transition level at @xmath36 ev,@xcite using the heyd - scuseria - ernzerhof ( hse ) hybrid functional based on screened exchange@xcite in which the fraction of non - local exchange was set to 0.375 ( hse-37.5 ) . using the same functional but with @xmath5 set to 0.40 ( hse-40 ) , clark _ et al_. obtained the transition level at @xmath37 ev.@xcite thus , it appears that when @xmath5 in either pbeh or hse functionals is tuned to reproduce the experimental band gap , one consistently obtains the ( + 2/0 ) charge transition level at 2.23@xmath382.38 ev from the vbm . the occurrence of such an agreement has recently been rationalized in general terms.@xcite janotti and van de walle,@xcite who adopted an extrapolation method based on lda+@xmath34 , found this charge transition level at 2.17 ev , in a fair agreement with the hybrid functional calculations . , width=188 ] as already noted in the literature , @xcite the charge transition level at @xmath39-@xmath40 ev is in stark disagreement with calculations based on other methods for correcting the band gap . for example , adopting a lda@xmath41 scheme,@xcite lany and zunger obtained the charge transition level at @xmath42 ev.@xcite in the lda@xmath41 method , the hubbard @xmath34 term acts on the zn 3@xmath4 states and the band - gap problem is not fully corrected . when one tunes the @xmath34 parameter so that the position of zn 3@xmath4 states are correctly positioned with respect to the vbm , one obtains a band gap of 1.5 ev , considerably smaller than the experimental one . the remaining band - gap error was corrected by an upward shift of the cbm.@xcite in another study , paudel and lambrecht adopted a lda@xmath43 scheme , in which the hubbard @xmath0 term was applied to both zn 3@xmath4 and zn 4@xmath44 states.@xcite while this scheme brings the theoretical band gap in agreement with experiment , the ( + 2/0 ) charge transition level is found at @xmath45 ev . some of the results obtained in ref . have recently been reviewed and improved by boonchun and lambrecht.@xcite we here mainly elaborate on the original results , but the conclusions that we draw are independent of this choice . using a similar method as that of paudel and lambrecht , lany and zunger have obtained a level at @xmath46 ev . @xcite the charge transition levels obtained with different methods are compared in fig . [ ov](a ) . we note that the observed differences do not stem from different electron densities of the defect state , as the oxygen vacancy is characterized by a fully symmetric state of @xmath3 symmetry which is well described in all schemes . the origin of this apparent disagreement between various methods has lately been discussed to some extent.@xcite however , it remains unclear whether the observed differences originate from failures of some specific methods or whether they point to a more fundamental problem common to all approximate electronic structure methods . a clue to the understanding why different methods seemingly differ so much is provided by the realization that the band edges of bulk zno calculation undergo drastically different shifts when going from lda / gga calculations @xcite to band - gap corrected schemes . such shifts between two different electronic structure calculations are properly defined through the alignment of the average electrostatic potential . for example , the lda@xmath41 method of ref . yields a shift in the vbm , @xmath47 ev . the lda@xmath43 method of ref . gives a shift of @xmath48 ev , while our calculations yield @xmath49 ev . in fig . [ ov](b ) we show the comparison of the ( 2+/0 ) charge transition level obtained with various methods , when the vbms in the lda / gga calculations are aligned . this is equivalent to aligning the electrostatic potential of all calculations ( see sec . [ alignment ] ) . with this alignment , the various methods yield charge transition levels differing by at most 0.4 ev . this is to be contrasted to the variation of up to 1.8 ev achieved when the electronic structures are aligned via their respective vbm [ fig . [ ov](a ) ] . thus , these theoretical calculations do not in fact differ as much as has been previously claimed . our conclusion is that , when a suitably defined common reference level is adopted , the charge transition levels are more accurately described than the bulk band edges.@xcite in secs . [ states ] and [ edge ] below , we give a detailed explanation of this behavior and address its general consequences for theoretical studies of defects . the previous discussion relied on the assumption that the bulk band structures of two theoretical calculations can be aligned with respect to each other , as done in fig . [ ov](b ) . this alignment allows one to determine the shifts in the valence band @xmath50 and in the conduction band @xmath51 for a given theoretical scheme with respect to another one . in this section , we discuss the meaning of such an alignment.@xcite the alignment between the electronic structures of the same bulk material within different theoretical schemes could in principle be achieved through the identification of a common reference potential . for instance , the vacuum level could serve this purpose , requiring the explicit consideration of the surface between the considered material and vacuum within both theoretical schemes . since the surface dipole depends on the specific crystal surface which is considered , the same orientation has to be chosen for both theoretical schemes . in this way , properly defined bulk levels in the two schemes , such as @xmath18 and @xmath52 , can be positioned with respect to the vacuum level and thus aligned . by constructon , the alignment achieved in this way is _ not _ an intrinsic bulk property of the two theoretical schemes . indeed , differences between the surface dipoles in the two surface calculations directly affect the alignment . while such a procedure can always be carried out , we note that the alignment between different electronic structures for the same bulk material is a meaningful concept only as long as their associated electron densities are identical ( or very close ) . indeed , different electron densities at surfaces of the material could yield different surface dipoles and thus the achieved alignment would depend on the particular surface adopted and give rise to ambiguity . moreover , different surface dipoles could result from different electron densities in the bulk , for instance because of different theoretical equilibrium lattice parameters . in such a case , the alignment with respect to the vacuum level would again be surface dependent . when comparing electronic structures of bulk materials as achieved within different theoretical schemes , we will thus additionally assume that their electron densities do not differ essentially . in practical calculations involving semilocal and hybrid density functionals , this condition is close to being satisfied . indeed , surface and interface dipoles in a variety of cases were found to differ by at most a few tenths of an ev.@xcite under the assumption of yielding close electron densities , two different theoretical schemes can be expected to give similar surface dipoles . this implies that an alignment to the vacuum level is equivalent to an alignment to the average electrostatic potentials within the bulk of the materials.@xcite this consequence is particularly convenient and allows us to compare different bulk calculations without the necessity of performing surface calculations.@xcite note , however , that it is implicitly understood that alignment shifts resulting from slight differences in the electron density are negligible when compared to the shifts undergone by the band edges . to produce fig . [ ov](b ) , we relied on shifts @xmath50 and @xmath51 calculated in the respective papers . indeed , the position of the vbm and the cbm in the more advanced theory were generally given with respect to the ( semi-)local density functional calculation ( lda or gga ) for an alignment with respect to the average electrostatic potential . for instance , the lda@xmath43 and lda band structures obtained in ref . , corresponding to the left column in fig . [ ov](b ) , were aligned through the average electrostatic potential in the bulk . in refs . , corresponding to the results in the middle column in fig . [ ov](b ) , the authors determined the shifts of the bulk bands in the lda@xmath41 with respect to the lda by referring the energies to o 2@xmath44 states which do not directly couple to the @xmath4 states on which the hubbard correction was applied . this is again equivalent to the alignment to the average electrostatic potential in the bulk . in our own calculations , presented in the right column in fig . [ ov](b ) , we aligned the two band structures through the average electrostatic potential in the bulk . unfortunately , the reported data did not allow us to establish the relative alignment for all the studies referred to in fig . [ ov](a ) . however , we can assume that similar theories yield close @xmath50 and @xmath51 . for instance , the lda@xmath43 calculations of lany and zunger @xcite are expected to yield similar shifts as those found by paudel and lambrecht @xcite [ fig . [ ov](a ) ] . as far as the screened hybrid functionals are concerned [ fig . [ ov](a ) ] , a recent study has shown that these functionals yield very similar shifts as the unscreened functionals used in our calculations , as long as the fraction of nonlocal exchange is tuned to reproduce the experimental band gap.@xcite hence , although the results in fig . [ ov](b ) are restricted to those studies which explicitly give the shifts in the band edges , the present considerations are expected to carry a much broader validity and to equally hold for all other calculations reported in fig . [ ov](a ) . we showed above that different theoretical models give quite consistent results concerning the description of the ( + 2/0 ) charge transition level of the o vacancy in zno provided they are aligned through the average electrostatic potential , taken as a common reference level . to understand why this happens , we first discuss fundamental differences between localized ( atomic - like ) and extended ( bulk - like ) states in approximate density functional schemes . for ( approximate ) density functionals janak s theorem @xcite applies : @xmath53 i.e. the derivative of the total energy with respect to the change of occupation number @xmath54 of the highest - occupied state @xmath55 is equal to the single - particle eigenvalue of this state @xmath56 , when the latter is referred to the average local potential.@xcite the integral form of janak s theorem is @xmath57 where @xmath58 is the total energy of the system with @xmath59 electrons . in the above expressions , we suppressed the spin variable . while in the original derivation of janak s theorem the functionals were implicitly assumed to be continuous , eq . ( [ janak1 ] ) equally applies to functionals which possess a discontinuity @xcite at integer number of electrons . in this case one has to distinguish between left and right derivatives and the corresponding single particle eigenvalues . the integral form of janak s theorem , eq . ( [ janak2 ] ) , applies to discontinuous functionals without modifications . in the case of localized states , such as , e.g. , in molecules and atoms , the single particle eigenvalue in approximate density functional schemes depends sensitively on the fractional occupation . accordingly , total energy differences pertaining to the change of number of electrons are given by eq . ( [ janak2 ] ) . in particular , the ionization potential ( ip ) of a system is given by @xmath60 where @xmath61 is the highest occupied orbital of the @xmath59-electron system . similarly , the electron affinity ( ea ) can be expressed as @xmath62 where @xmath63 is the lowest unoccupied state of the @xmath59-electron system . it has been known for some time that total energy differences pertaining to the change of charge state of a localized state are quite accurately described in approximate density functional schemes , both in semilocal and hybrid ones . for example , curtiss _ et al_. calculated ips and eas for a large set of molecules using gga and hybrid functionals.@xcite they calculated these quantities via total energy differences ( @xmath64scf method ) , yielding an average deviation with respect to experiment lower than @xmath65-@xmath66 ev for both gga ( blyp ) and hybrid ( b3lyp ) functionals . this accuracy is achieved despite the fact that the single - particle eigenvalues of the highest - occupied molecular orbital ( homo ) @xmath67 and of the lowest - unoccupied molecular orbital ( lumo ) @xmath68 are substantially different in the gga and in hybrid functional schemes . a similar agreement with experiment also holds for screened hybrid functionals.@xcite however , plain lda yields slightly larger errors , of the order of @xmath69-@xmath70 ev for the same quantities.@xcite we illustrate this property in the case of the pentacene ( c@xmath71h@xmath72 ) molecule in fig . [ molecules].@xcite pentacene is a convenient example because , unlike several smaller acenes , it possesses a positive electron affinity . the single - particle homo and lumo levels , calculated with the semilocal pbe functional ( left , solid lines ) , do not agree well with the negative of the experimental ip and ea ( right ) . in particular , the single - particle gap @xmath73 of 1.12 ev is severely underestimated with respect to the experimental gap @xmath74 of 5.29 ev . the use of the hybrid pbe0 ( i.e. pbeh-25 ) functional ( left , dashed lines ) gives some improvement , but the calculated single - particle homo - lumo gap of 2.34 ev remains much smaller than the experimental one . at variance , when calculated via total energy differences , the ips and eas in both pbe and pbe0 are much closer to their corresponding experimental values , 6.64 ev@xcite and 1.35 ev,@xcite respectively . the two theoretical values ( fig . [ molecules ] ) differ by less than 0.10 ev , with the hybrid functional calculation in slightly better agreement with the experimental results . the residual differences between calculated and measured values ( @xmath750.45 ev for the ip and @xmath750.25 ev for the ea ) can be accounted for by the quite large electron correlation effects in the pentacene molecule.@xcite in any case , the present result shows that these theoretical schemes yield total energy differences in good agreement with experiment and with each other , while the single - particle levels in the two schemes are very different . this is consistent with the general trend found by curtiss _ for a large set of smaller molecules.@xcite thus , we conclude that total energy differences pertaining to the change of charge state of localized states are accurately described with approximate density functionals . approximating the integrals appearing in eqs . ( [ janak_ip ] ) and ( [ janak_ea ] ) through the trapezoidal rule , we arrive at the following expressions for the ip and the ea : @xmath76 and @xmath77 here , @xmath78 and @xmath79 . electronic states at half - filling correspond to slater - transition states.@xcite since eqs . ( [ slater_ip ] ) and ( [ slater_ea ] ) apply equally well to various semilocal and hybrid functionals , the generally good agreement with experiment implies that the respective eigenvalues @xmath80 defined as a function of filling all approximately cross at half - filling . this has indeed already been observed.@xcite the reason for this good performance of approximate density functionals should be ascribed to the fact that such functionals fulfill several exact constraints of the many - body fermionic system.@xcite in particular , the most relevant in this context is the generalized sum - rule of the exchange - correlation hole . this rule holds for systems with an _ integer _ number of electrons , i.e. for closed systems in which no exchange of electrons with the environment occurs.@xcite this condition is enforced for most approximate functionals , including the lda and various ggas . furthermore , since this constraint is naturally fulfilled in the hartree - fock theory , it also holds for any hybrid functional with an exchange energy of the type given in eq . ( [ hb ] ) . the situation is very different in the case of infinitely extended bulk - like states . indeed the band - gap problem pertaining to ( generalized ) kohn - sham eigenvalues _ can not _ be overcome by considering total - energy differences . when a fraction @xmath81 of an electron or even a full electron is added to or removed from an extended state , the total electron density changes negligibly . thus , the local potential , both the hartree and the approximate exchange - correlation potential remain unaffected . as a result , the single - particle eigenvalues do not depend on the filling @xmath81 of this state . using the integral form of janak s theorem given in eq . ( [ janak2 ] ) , we get for the valence band maximum : @xmath82 and for the conduction band minimum : @xmath83 to illustrate this property , we show in fig . [ extended ] the vbm and cbm of @xmath84-quartz calculated via total energy differences as a function of the supercell size . the considered cells contain 72 , 144 , 288 , and 576 atoms , and their brillouin zones are sampled at the sole @xmath33 point . the semilocal pbe functional was used . in the case of @xmath84-quartz , total energy differences are very close to single particle eigenvalues already for the smallest cells . for the 72-atom cell , the difference is 0.015 ev for the vbm and 0.035 ev for the cbm , while for the 576 cell these are 0.003 ev and 0.004 ev , respectively . the particular case of hybrid functionals has been addressed in detail in ref . . hence , unlike for the localized states in fig . [ molecules ] , the consideration of total - energy differences in the case of extended states is not useful to improve the comparison with experiment and the same limitations pertaining to the single - particle eigenvalues ( band - gap problem ) are encountered . @xcite a similar comparison involving extended states of gaas and localized states of the f atom can be found in ref . . ( @xmath84-quartz ) calculated via total energy differences as a function of @xmath85 , where @xmath86 is the total number of atoms in the supercell . calculations have been performed with the semilocal pbe functional.,width=321 ] the above discussion highlights an important difference between localized and extended states as described within approximate density functional schemes . while the band - gap problem associated to single particle eigenvalues can be circumvented by considering total - energy differences for localized states , such a solution does not apply to extended states for which the band - gap problem remains a fundamental obstacle . recently , a clear explanation has been put forward for justifying this different behavior.@xcite the inaccurate total energies for large systems with _ number of electrons stems from the failure of approximate density functionals in describing small systems with _ charges.@xcite indeed , approximate functionals generally do not reproduce the property of the exact density functional by which the total energy depends linearly on the number of electrons . there is at present an on - going effort to achieve improved descriptions on the basis of these ideas.@xcite the degree of localization required for achieving an accurate description with current density functionals is still to a large extent an open question . we refer the reader to the interesting debate on this issue in refs . . having stressed the different properties of localized and extended states with respect to a change in electron occupation , we return in this section to the discussion of charge transition levels . for the sake of simplicity , let us consider the @xmath87 transition of a point defect characterized by an atomically localized wave function . using eq . ( [ ted_ev ] ) , we write the charge transition level @xmath88 as the difference between two terms , each of them corresponding to a total - energy difference : @xmath89 the second term clearly describes the total energy difference pertaining to a delocalized bulk state , while the first term can to a very good approximation be related to the total energy difference pertaining to a localized state . formally , the first term describes the total energy of the whole manifold of states involving both defect and bulk states , but can be related to the localized defect state through the slater transition - state approximation.@xcite for atomically localized defect states , this is a very good approximation.@xcite in view of the following discussion , it is convenient to rewrite eq . ( [ ted2 ] ) as @xmath90 where the charge transition level @xmath91 and the vbm @xmath92 are referred to the average electrostatic potential @xmath93 of the unperturbed bulk material . let us assume that we study the @xmath87 charge transition level of the same defect using two different theories : theory i and theory ii . the first theory severely underestimates the band gap , while the second one gives a band gap in a much closer agreement with experiment . the two theories differ only by the exchange - correlation potential . according to eq . ( [ ted3 ] ) , the corresponding charge transition levels referred to the respective valence band maxima are : @xmath94 and @xmath95 we further assume that the two theories produce a sufficiently accurate representation of the electron density so that it is justified to align the two bulk band structures through the average electrostatic potential @xmath93 in the two theories , as discussed in sec . [ alignment ] . under the assumption that the defect wave function @xmath96 differs very little in the two theories , we can express the difference between the two charge transition levels @xmath91 making use of the slater transition state:@xcite @xmath97 where the exchange - correlation potentials are evaluated with the defect state at half occupation . only the difference in the exchange - correlation potentials enters the expression in eq . ( [ janak_defect ] ) . indeed , if the electron density and the single - particle wave functions are very similar in the two calculations , the interaction between the defect and the ionic cores , the long - range electrostatic electron - electron interaction , and the kinetic energy are the same in the two theories and cancel . to understand the behavior of defect levels , it is convenient to focus first on defects with extremely localized wave functions . hence , according to eq . ( [ janak_defect ] ) , the difference @xmath98 can then be expressed in terms of an expectation value involving the sole localized defect state.@xcite however , we know from sec . [ states ] that total energy differences pertaining to localized states , or , equivalently , slater transition - state eigenvalues of localized states , are almost the same , independent of the functional . thus , we get : @xmath99 this means that charge transition levels for such defects are almost equal in the two theories , when the energy scales are aligned through the average electrostatic potential @xmath93 . at variance , the charge transition levels are substantially different when the energy scales in the two theories are aligned through the respective valence band maxima , i.e. through @xmath100 in eqs . ( [ rw1 ] ) and ( [ rw2 ] ) , because of the different positions of the bulk band edges with respect to the potential @xmath93 . this scenario pertaining to a defect with an extremely localized wave function is illustrated in fig . [ th2 ] by the defect state @xmath5 . the validity of the ideal alignment illustrated by this type of defect has been demonstrated for a wide class of defects encompassing various host materials.@xcite schematic illustration of energy levels of various type of defect states differing by the extent of their wave function : ( a ) defect level with an atomically localized wave function , ( b ) an intermediate case , and ( c ) an effective - mass - like defect . the results of two electronic structure theories ( theory i and theory ii ) giving different band gaps are compared to illustrate the band - gap problem . the alignment is made through the average electrostatic potential.,width=321 ] figure [ th2 ] also illustrates the shifts of other type of defects . in the opposite limit , defect @xmath101 corresponds to an effective - mass - like defect with a spatially extended wave function . in this case , the defect level is anchored to the bulk band to which it pertains and rigidly follows the band edge upon the opening of the band gap in theory ii . defect @xmath102 has an intermediate extension compared to defects @xmath5 and @xmath101 , and is partially affected by the shift of the band edges . the relation between the departure from ideal alignment and the spatial extension of the defect wave functions has been documented for various defects and host materials in ref . . however , the detailed behavior of such defects is intrinsically system - dependent , and no universal considerations can be made . in this section , we limited the discussion to defects states occurring in the band gap for both theories . more complex situations occur when defect states are resonant with the band states for one of the theories.@xcite however , the physical description of the defect state is altered in such cases . the main motivation of the present work is to understand the effect of the band gap opening under the assumption that the defect wave function remains essentially unmodified . in the previous section , we compared defect charge transition levels as obtained within two different theories giving different band gaps . we found that the energy levels of defects states described by atomically localized wave functions are already well described in theories with a pronounced `` band - gap problem '' , provided those levels are referred to a relevant reference level . for such defects , the problem of finding the defect level is essentially decoupled from that of finding the band edges . schematic illustration of energy levels of various type of defect states differing by the extent of their wave function : ( a ) a deep defect level with an atomically localized wave function , ( b ) an intermediate case , and ( c ) an effective - mass - like defect . the results of two electronic structure theories ( theory i and theory ii ) giving the same band gap but different band edge positions are compared to illustrate the `` band - edge problem '' . the alignment is made through the average electrostatic potential.,width=321 ] let us thus consider two different theories , theory i and theory ii , yielding this time the same band gap ( taken to coincide with the experimental one ) , but giving different positions of the vbm and the cbm with respect to the average electrostatic potential @xmath93 of the bulk . we assume that the two theories are sufficently accurate yielding in particular close electron densities , so that the energy scales of the two theories can be aligned through @xmath93 , as discussed in sec . [ alignment ] . for instance , theory i could be lda+@xmath0 in which the remaining band - gap underestimation is corrected by a rigid shift of the conduction band , while theory ii could be a hybrid functional scheme in which the fraction of fock exchange is tuned to reproduce the experimental band gap . for an atomically localized defect , the same argument holds as in the previous section and the charge transition levels obtained within the two theories are expected to fall very close to each other , as illustrated in fig . [ th3 ] for defect @xmath5 . in fig . [ th3 ] , a departure from the ideal alignment is seen for defects @xmath102 and @xmath101 , corresponding to defect wave functions of intermediate and effective - mass - like extensions , respectively . figure [ th3 ] summarizes the principal finding of the present work . in a condensed form , the following statement can be formulated concerning the comparison of charge transition levels of atomically localized defects . despite the good description of the experimental band gap in both theories , the defect levels differ when referred to their respective vbm , because the band edges in the two theories are located differently with respect to the common electrostatic potential @xmath93 . this occurs even when the defect wave function is almost identical in the two theories . this alignment property deteriorates with the extension of the defect wave function . thus , the correct description of band edges relative to the average electrostatic potential is a crucial prerequisite for an accurate location of charge transition levels within the band gap . we refer to this issue as to the `` band - edge problem '' for the calculation of defect levels . in other words , there is not only a `` band gap problem '' related to the underestimation of the band gap but also a `` band - edge problem '' related to the position of the band edges with respect to the average electrostatic potential , ultimately corresponding to an absolute alignment with respect to an external vacuum level . as far as the determination of the ( + 2/0 ) charge transition level of the oxygen vacancy in zno is concerned , the present considerations appear confirmed [ cf . [ ov](b ) ] . this defect level behaves like the defect state @xmath102 in fig . [ th3 ] , showing a shift which does not depart in a significant way from the case of ideal alignment ( defect state @xmath5 ) . indeed , when referred to a common reference level , all previous calculations yield the ( + 2/0 ) level within 0.4 ev,@xcite which corresponds to just one ninth of the band gap of bulk zno . hence , contrary to previous claims , we find that all previous defect calculations agree quite well with each other . in fact , these calculations differ in the positions of the bulk band edges with respect to the average electrostatic potential . these considerations lead to the question about which theoretical description should be adopted for positioning the band edges . this corresponds to determining the shift @xmath50 of the valence band and the shift @xmath51 of the conduction band , when taking the lda or the gga as a starting point . a direct comparison between theory and experiment is in principle possible . the bulk band structure can for instance be referred to the vacuum level through a surface calculation . the vbm and the cbm determined in this way could then be compared with ionization potentials and electron affinities , as obtained by means of photoelectron and inverse photoelectron spectroscopy . however , such measurements are often shrouded by very pronounced effects associated to charged native defects and impurities which influence the electrostatics and alter the alignment . more practically , the validity of a given theoretical scheme can be examined addressing band offsets at interfaces.@xcite band offsets are well - defined quantities and can generally be measured through a large set of experimental techniques . the comparison between theoretical and experimental band offsets then allows one to determine the overall accuracy with which such shifts are obtained within various theoretical schemes . in the absence of experimental data , the validity of the shifts @xmath50 and @xmath51 could also be assessed by comparing with electronic structure calculations of higher accuracy , such as those based on many - body perturbation theory ( mbpt ) in the @xmath1 approximation or beyond.@xcite indeed , such calculations not only provide improved relative positions of bulk bands , but also shifts of those bands with respect to theoretical schemes of lower level . however , recent work has shown that the shifts of the band edges with respect to the average electrostatic potential are more difficult to converge than relative positions of bands.@xcite furthermore , such shifts are sensitive to various levels of approximation , such as , e.g. , the use of different models for the plasmon pole to describe the frequency dependence of the dielectric function , the inclusion of vertex corrections @xmath33 , and various levels of self - consistency on @xmath103 , @xmath104 , @xmath33 , and the electron wave functions . @xcite to illustrate this point , we quote a recent work , @xcite in which the relative shift of the valence band with respect to the overall band gap correction , i.e. @xmath105 , was found to range from @xmath380.68 to @xmath380.42 in the case of sio@xmath22 , depending on the level of approximation in the @xmath1 scheme . even for a material as simple as si the value of @xmath105 as predicted by different @xmath1 schemes ranges from @xmath380.75 to + 0.17.@xcite thus , clearly more work is needed to clarify these issues . a systematic study of the effects of different levels of approximation in mbpt on the shifts in the band edges is thus vital for the study of defect levels . in this work , we carried out a theoretical analysis of the ( + 2/0 ) charge transition level of the oxygen vacancy in zno . in recent years , this defect has grown into a benchmark case to assess the quality of various advanced electronic - structure theories . indeed , common approximations to density functional theory , such as the lda and the various ggas , severely underestimate the band gap of bulk zno , and the treatment at a more advanced level thus becomes crucial even for drawing qualitative conclusions . however , different advanced theoretical methods applied hitherto yielded conflicting results regarding the position of the defect level in the band gap . we here showed that apparently conflicting theoretical results are in a much better agreement with each other when the charge transition levels are aligned with respect to the average electrostatic potential rather than to the respective valence band maximum . we showed that the former alignment is equivalent to the choice of a common external potential such as the vacuum level , provided the electron densities are sufficiently accurately described . we have rationalized our finding by considering fundamental differences between the ways approximate density functionals describe localized and delocalized states . for localized states , the `` band - gap problem '' can generally be overcome through the consideration of total energy differences . on the other hand , such a solution is not applicable to delocalized states , for which the `` band - gap problem '' remains an intrinsic shortcoming . in particular , the present study highlights a specific criterion that needs to be fulfilled in order to properly describe charge transition level and formation energies of defects . we clearly demonstrated that the band structure of the host material needs to be correctly positioned with respect to an external potential , such as the vacuum level . when the electron density is accurately described , this alignment condition can equivalently be replaced by the alignment with respect to the average electrostatic potential in the bulk . this condition is additional with respect to the accurate reproduction of the band gap . our analysis of the oxygen vacancy in zno shows that conflicting theoretical results arise for theories yielding an accurate band gap , but differing positions for the band edges . we particularly thank p. broqvist for his contribution to a wider research project from which the present study takes its origin . we also acknowledge fruitful interactions with a. carvalho , h .- komsa , and o. a. vydrov . partial financial support from the swiss national science foundation is acknowledged under grant no . 200020 - 111747 . we used computational resources at dit - epfl ( bluegene ) , csea - epfl , and cscs . in the case of zno lda yields a band gap of about 0.6 ev , which is yet smaller that the one in the gga calculation . however , for the sake of simplicity we assume that bulk band edges are described similarly in lda and gga . the error this assumption introduces is of the order of 0.1 ev ( ref . ) and thus unsubstantial for our consideration . the calculations of the pentacene molecule were performed with cubic supercells of increasing side ( 30 , 40 , and 50 ) and extrapolated to infinity . single particle eigenvalues were aligned to the vacuum level far from the molecule . for charged systems , this vacuum level was preserved at the same energy and the total energies were consistently corrected . in addition , the electrostatic monopole - monopole correction reflecting the interaction of an array of charges with the neutralizing background was included .

calculations of formation energies and charge transition levels of defects routinely rely on density functional theory ( dft ) for describing the electronic structure . since bulk band gaps of semiconductors and insulators are not well described in semilocal approximations to dft , band - gap correction schemes or advanced theoretical models which properly describe band gaps need to be employed . however , it has become apparent that different methods that reproduce the experimental band gap can yield substantially different results regarding charge transition levels of point defects . we investigate this problem in the case of the ( + 2/0 ) charge transition level of the o vacancy in zno , which has attracted considerable attention as a benchmark case . for this purpose , we first perform calculations based on non - screened hybrid density functionals , and then compare our results with those of other methods . while our results agree very well with those obtained with screened hybrid functionals , they are strikingly different compared to those obtained with other band - gap - corrected schemes . nevertheless , we show that all the different methods agree well with each other and with our calculations when a suitable alignment procedure is adopted . the proposed procedure consists in aligning the electron band structure through an external potential , such as the vacuum level . when the electron densities are well reproduced , this procedure is equivalent to an alignment through the average electrostatic potential in a calculation subject to periodic boundary conditions . we stress that , in order to give accurate defect levels , a theoretical scheme is required to yield not only _ band gaps _ in agreement with experiment , but also _ band edges _ correctly positioned with respect to such a reference potential .

although of fundamental importance to stellar astrophysics , precise measurements of angular radii are generically difficult to acquire routinely and in a model - independent way . classical direct methods of measuring stellar radii include lunar occultations , interferometry , and eclipsing binaries . lunar occultation measurements yield precise angular radii ( see richichi et al . 1999 and references therein ) , but the number of stars to which this technique can be applied is limited . the number of direct measurements using interferometers has recently increased dramatically with advent of , e.g. the palomar testbed interferometer ( van belle et al . 1999a , colavita et al . 1999 ) , and the navy prototype optical interferometer ( armstrong et al . 1998 , nordgren et al . 1999 ) , and is likely to continue to increase as more technologically advanced interferometers come online . unfortunately , both lunar occultation and interferometric angular diameter measurements have traditionally been primarily limited to nearby , evolved stars . angular radii of main - sequence stars can be determined using detached eclipsing binaries ( i.e. popper 1998 ) , however the large amount of data ( both photometric and spectroscopic ) required to yield accurate radii determinations makes this method prohibitive . thus , of the @xmath17 direct , precise angular diameter measurements compiled by van belle ( 1999 ) , the overwhelming majority , @xmath18 , are of evolved stars . finally , it will be difficult to acquire a large sample of angular radii determinations of stars with metallicity considerably smaller than solar using these methods , due to the paucity of metal - poor stars in the local neighborhood . here we present a method , based on a suggestion by paczyski ( 1998 ) , of measuring angular radii of stars that overcomes some of the difficulties inherent in the classical methods . this method employs the extraordinary angular resolution provided by caustics in gravitational microlensing events , and as such is yet another in the growing list of applications of microlensing to the study of stellar astrophysics ( see gould 2001 for a review ) . the original suggestion of paczyski ( 1998 ) was to invert the method of gould ( 1994 ) for measuring the relative source - lens proper motion @xmath19 in microlensing events . if the lens transits the source in a microlensing event , precise photometry can be used to determine the time it takes for the lens to transit one source radius , @xmath20 , where @xmath1 is angular radius of the source . an estimate of @xmath1 , using an empirical color - surface brightness relation , together with a measurement of the flux of the source , can then be used to estimate @xmath19 , which gould ( 1994 ) argued could be used to constrain the location of the lens . however , as paczyski ( 1998 ) pointed out , it is possible to independently measure the angular einstein ring radius of the lens , @xmath21 by making precise astrometric measurements of the centroid shift of the source during the microlensing event using , i.e. , the _ space interferometry mission _ ( sim ) . here @xmath22 is the mass of the lens , @xmath23 is defined by , @xmath24 , and @xmath25 , @xmath26 , and @xmath27 are the distances between the observer - source , observer - lens , and lens - source , respectively . since @xmath28 , by combining the measurement of @xmath5 with the einstein timescale @xmath3 of the event determined from the light curve , it is possible to measure @xmath1 for the source stars of microlensing events . we show that , with reasonable expenditure of resources , it should be possible to measure angular radii of a significant sample ( @xmath29 ) of giant stars in the bulge to an accuracy of @xmath30 , or @xmath10 main - sequence stars to an accuracy of @xmath31 . limb - darkening determinations should also be possible for the majority of the sources , and most will be relatively metal poor as compared to those for which angular radii determinations are currently available . although measurements of @xmath1 can be made using single - lens events , in [ sec : bvs ] we argue that this method is better suited to caustic - crossing binary - lens events , which are more common , easier to plan for , and considerably less resource - intensive than source - crossing single - lens events . we describe in some detail the basic method of measuring @xmath1 for the source stars of caustic - crossing binary - lens events in [ sec : method ] , including a discussion of the expected errors on the individual parameters that enter into the measurement . we discuss various subtleties , complications , and extensions to the method in [ sec : discussion ] , and also present an estimate of the number of @xmath1 measurements that might be made in this way . finally , we summarize and conclude in [ sec : conclusion ] . the primary requirement to be able to measure @xmath5 in a microlensing event is that the source should be resolved by the gravitational lens . this effectively means that the source must cross a caustic in the source plane . caustics are the set of positions in the source plane where the determinant of the jacobian of the lens mapping from source to lens plane vanishes , and where the magnification therefore is formally infinite . large gradients ( with respect to source position ) in the magnification exist near caustics , enabling the resolution of the source . generically , microlenses come in two classes : single and binary lenses . here ` binary lens ' means a lens systems composed of two masses with angular separation of order the angular einstein ring radius of the system . very close and very wide binaries act essentially as single lenses . single lenses have a caustic that consists of a single point at the position of the lens . in these cases , the magnification close to the caustic diverges as the inverse of the distance to the caustic . in contrast , the caustics of binary lenses are extended , and can cover a significant fraction of the einstein ring . the caustics of binary lenses generically consist of two types of singularities , folds and cusps .. however , folds and cusps are the only stable singularities of any lens system ( petters , levine , & wambsganss 2000 ) . beak - to - beak singularities are unstable in the sense that , for infinitesimally small variations in the lens parameters , a beak - to - beak singularity disintegrates into two cusp - type singularities . thus the set of parameters where beak - to - beak singularities are expected is formally sparse , and practically small . interestingly , alcock et al . ( 2000a ) suggested that an observed event may have been due to a source crossing a beak - to - beak singularity in a binary - lens , although alcock et al . ( 2000b ) seem to favor the interpretation that this event is due to a background supernova . ] near a fold , the caustic is well - described by generic linear fold singularities , for which the magnification locally diverges inversely as the square - root of the distance to the caustic ( schneider & weiss 1986 , schneider , ehlers , & falco 1992 , gaudi & petters 2002a ) . cusps are points where two fold caustics meet , and the magnification for cusps locally diverges roughly inversely as the distance to the cusp point , similar to the magnification pattern near the point caustic of a single lens ( schneider & weiss 1992 , schneider , ehlers , & falco 1992 , gaudi & petters 2002b ) . the fact that the magnification near the point caustic of a single lens or near a cusp diverges inversely as the distance to the singularity , rather than as the square root of the distance to the singularity as with folds , means that for a given source size , the ` resolving power ' of fold caustic crossings is less than single lens or cusp crossings . although single - lens events are more well suited to studies of stellar atmospheres , they are much less useful for measuring the sizes of stars for three main reasons . they are generally rarer than fold caustic crossings , their crossings can not be predicted in advance , and their centroid motion is more difficult to measure . caustic - crossing binaries comprise roughly @xmath32 of all events toward the galactic bulge ( alcock et al . 2000a ; udalski et al . 2000 ) . of the remaining @xmath33 events , which we will conservatively assume due to single lenses , only a fraction @xmath34 will exhibit caustic crossings . therefore , the expected ratio of binary - to - single events for which the source is resolved ( and thus measurement of @xmath1 is possible ) is @xmath35 where @xmath36 is the physical radius of the source , and we adopted @xmath37 , @xmath38 , and @xmath39 for the scaling relation on the extreme right hand side . although not overwhelming for giant sources , for main sequence sources we expect at least an order of magnitude more binary lensing events for which the source is resolved . the ratio of the number of fold - to - cusp crossings is roughly @xmath40 where @xmath41 is the number of cusps , which is either 6 , 8 , or 10 for a binary lens , depending primarily on @xmath42 . thus the overwhelming majority of events for which it will be possible to measure @xmath1 will be fold caustic crossing binary - lens events . binary - lens fold caustic - crossing events also have the advantage that the second caustic crossing can be predicted in advance . this is because fold caustic crossings always come in pairs , and it is typically easy to tell , even with sparse sampling , that the first caustic crossing has occurred . then more frequent sampling can be used to monitor the rise to the second caustic , in principle enabling the prediction of the time of the second crossing a day or more in advance ( jaroszy ' nski & mao 2001 ) , and allowing the marshaling of the resources necessary to obtain the dense coverage of the second crossing needed to measure the crossing time @xmath2 of the source ( see [ sec : dt ] ) . in contrast , a single lens caustic crossing can only reliably be ` predicted ' at about the time it begins . finally , it is considerably harder to measure @xmath5 for caustic - crossing single - lens events than binary - lensing events . single lens events have a maximum absolute centroid shift relative to the unlensed source position of @xmath43 , whereas binary lensing events can exhibit large variations of size @xmath44 or more when the source crosses the caustic ( han , chung , & chang 1999 ) . therefore considerably more time will generally be required to determine @xmath5 to given accuracy for single - lens events than for binary - lens events . since the astrometric measurements essentially require the capabilities of sim ( or some similarly precious instrument ) , it is highly desirable to minimize the amount of time spent on this step . given the above arguments , we conclude that fold caustic crossing binary lensing events are the most suitable for use in routinely measuring @xmath1 . we will therefore focus on this case for the remainder of the discussion , however we will briefly revisit single - lens and cusp - crossing events in [ sec : cusp ] . consider a binary - lens event in which the source crosses a simple linear caustic . , and when the angle of incidence of the source trajectory to the caustic is not small . this approximation will break down when the source is large compared to the overall size of the caustic , when the source crosses near a cusp , or when the source ` straddles ' the caustic for a long time due to a small incidence angle . although it will still be possible to measure @xmath1 for such events , the relation between the observables and @xmath1 is less straightforward . we discuss such cases in [ sec : cusp ] . ] defining @xmath2 as the ( one - half ) the time it takes for the source to completely traverse the caustic , and @xmath4 as the angle between the source trajectory and the tangent to the caustic at the crossing point , then the time for the lens to cross the angular radius of the source is @xmath45 . however , we also have that @xmath46 , where again @xmath19 is the relative source - lens proper motion . combining these expressions , we have that @xmath47 . using the definition of @xmath19 , we can write the angular source radius @xmath1 as the following function of observables , @xmath48 the process of @xmath1 measurement can therefore be subdivided into three basic steps : ( 1 ) : : measurement of the caustic crossing timescale @xmath2 from a single photometrically well - resolved caustic crossing . ( 2 ) : : measurement of the angle @xmath4 and timescale @xmath3 from the global fit to the binary - lens light curve . ( 3 ) : : measurement of the angular einstein ring radius @xmath5 using precise astrometric measurements of the source centroid . each of these steps are illustrated schematically in figure 1 . in the following subsections , we consider each of these steps in more detail . we outline the basic requirements for the measurement of each of the four parameters ( @xmath49 , and @xmath5 ) , and the expected accuracy with which each can be determined assuming reasonable expenditure of observing resources . when the source is interior to a binary - lens caustic , five images are created . as the source approaches a fold caustic , two of these images brighten and merge in a characteristic way , and eventually disappear when the source completely exits the caustic . in contrast to the significant brightening of the two images associated with the fold , the remaining three images generally vary only slowly over the timescale of the crossing . all fold caustics locally have this generic behavior , and the magnification @xmath50 of the source near a caustic crossing as a function of time is typically well - fitted by the functional form ( albrow et al . 1999b ) , @xmath51 where @xmath52 is the effective rise time of the caustic , which is related to the local derivatives of the lens mapping ( petters et al.2001 , gaudi & petters 2002 ) , @xmath53 is the time when the center of the source crosses the caustic , @xmath54 is the magnification of all the images unrelated to the fold caustic at @xmath55 , @xmath56 is the slope of the magnification of these images as a function of time , and @xmath57 can be expressed in terms of complete elliptic integrals of the first and second kind ( schneider & weiss 1987 ) . note that is only formally appropriate for a simple linear fold caustic . thus , for a well - sampled fold caustic crossing , @xmath2 can be determined essentially independently of the global geometry of the event , and indeed without reference to the photometric data away from the crossing itself . in practice , the magnification is not directly observable , but rather the flux @xmath58 as a function of time . the form for @xmath58 takes on a similar form as , but with a slightly different parameterization ( see albrow et al . 1999b ) . equation ( [ eqn : foft ] ) assumes a uniform source . this will likely be a poor approximation in optical bands , and assuming uniform source in the presence of limb - darkening may result in a systematic underestimate of @xmath2 , and therefore @xmath1 since , the effect of limb - darkening can partially compensated for by a smaller ( dimensionless ) source size , at least for poorly - sampled light curves . however , for well - sampled caustic crossings and @xmath59 photometric accuracy , it should generally be possible to accurately measure both the source size and limb - darkening coefficient(s ) ( rhie & bennett 1999 ) . such data quality is readily achievable ; indeed , independent limb - darkening and ( dimensionless ) source size measurements have been made for the source stars of at least five microlensing events ( albrow et al . 1999a , afonso et al . 2000 , albrow et al . 2000 , albrow et al . 2001 , an et al . a generalized form of that includes simple limb - darkening can be found in albrow et al . ( 1999b ) and afonso et al . ( 2001 ) . by fitting ( or a generalized form of it ) to the light curve near a well - sampled fold caustic crossing , one can derive the parameters @xmath60 , and @xmath2 . the parameters @xmath61 , and @xmath52 can subsequently be used to constrain the global solution to the entire light curve ( see albrow et al . however , of primary interest here is the parameter @xmath2 , whose value is essentially independent of the global solution . see figure 1(b ) . this means that @xmath2 and the parameters determined from the global solution , @xmath3 and @xmath4 , will be essentially uncorrelated . in order to be able to determine @xmath2 from the caustic crossing data alone , the caustic crossing must be well - sampled , so that the parameters in be well - constrained . practically , this requires forewarning of the caustic crossing . fortunately , this is generally possible with only photometry available from the collaborations which survey the galactic bulge and find the alert the microlensing events real - time ( eros , afonso et al . 2001 ; moa , bond et al . 2001 ; ogle , udalski et al . 2000 , woziak et al . 2001 ) , although improved predictions would be possible with continuous photometry ( jaroszy ' nski & mao 2001 ) . thus no additional resources need to be invested to predict caustic crossings . however , as we discuss in [ sec : teandphi ] , some additional sampling of the overall light curve may be needed to constrain @xmath3 and @xmath4 and determine a unique global solution . the accuracy with which @xmath2 can be determined for a given caustic crossing will depend not only on the intrinsic parameters of the caustic crossing , but also on the sampling rate and photometric accuracy near the caustic crossing , which in turn will depend on weather , blending , etc . , which tend to vary in a stochastic manner . therefore it is not very useful to attempt to quantify the expected errors on @xmath2 for an idealized observing setup . however we can obtain an order - of - magnitude estimate for @xmath62 by examining measurements of @xmath2 from published analyses of observed caustic - crossing events . we discuss these determinations more thoroughly in [ sec : errors ] . typically , well - covered caustic crossings yield fractional errors of @xmath63 . the parameters @xmath3 and @xmath4 are determined from the global solution to the overall geometry of the binary - lens light curve . the relationship between the salient features of an observed light curve , and the canonical parameters of a binary - lensing event , are generally not obvious or straightforward . this fact generally makes fitting an observed light curve , and thus inferring the parameters @xmath3 and @xmath4 , quite difficult ( see albrow et al . 1999b for a thorough discussion ) . although many methods have been proposed to overcome these difficulties , the lack of obvious correspondence between these parameters of interest and the light curve morphology generally implies that it is difficult to make general statements about the kinds of observations that are needed to reliably measure @xmath3 and @xmath4 . however , we can make some generic comments . for caustic - crossing binary lenses , one potential observable is the time @xmath64 between caustic crossings . between caustic crossings , the magnification is typically considerably larger than outside the caustic , and therefore even with sparse ( @xmath65 day ) sampling , it should be possible to determine to reasonable precision the time of the first caustic crossing , provided that data exist before the first crossing . the requisite photometry will generally be acquired by the survey collaboration(s ) . once the binary - lens event is alerted , more frequent photometry can be acquired by follow - up collaborations ( planet , albrow et al . 1998 ; mps , rhie et al . 2000 ) , thus mapping the `` u''-shaped curve between the caustic crossings . this shape , combined with information from cusp - approaches ( or lack thereof ) just outside the caustic , provides information about the shape of the caustic and the trajectory of the source through it . from this , the angle @xmath4 of the trajectory with respect to the caustic can be derived , and also the distance @xmath66 between the caustic crossings in units of @xmath5 . then , the timescale is given by @xmath67 . although the above analysis is highly trivialized , it does suggest that the following steps should be taken to ensure an accurate measurement of @xmath3 and @xmath4 . first , it is important to constrain the time of both caustic crossings to reasonable precision . this means that the survey collaborations should sample on timescales no less than a few days . second , followup photometry should be initiated relatively soon after the first crossing , to measure the shape of the intra - caustic light curve reasonably well . this is also necessary in order to predict the second caustic crossing ( jaroszy ' nski & mao 2001 ) . furthermore , the follow - up photometry should continue past the second caustic crossing , to detect cusp - approaches or cusp - crossings ( or the lack thereof ) . due to the difficulties inherent in fitting binary - lens light curves , it would be extremely difficult to attempt to predict the expected errors on @xmath3 and @xmath4 for a hypothetical observing scenario . furthermore , due to the complicated relation between observables and parameters implies that these errors are likely to depend strongly on the geometry of the event and light curve coverage , and therefore such ` predictions ' would not be very useful . however , we would like to have an order - of - magnitude estimate for the expected errors given reasonable light curve coverage . from determinations of these parameters in published caustic - crossing events , we can expect fractional accuracies of a few percent , provided that the light curve is well - covered , in the sense outlined above . however , if the light curve coverage is incomplete , then errors of @xmath68 are expected . see [ sec : errors ] . cccccccc & & & & & & & + macho 98-smc-1 & @xmath69 & @xmath70 ( @xmath71 ) & ... & @xmath72 & 22.1 & 1,2 & fold crossing + ogle-1999-bul-23 & @xmath73 & @xmath74 ( @xmath75 ) & ... & @xmath76 & 18.1 & 3 & fold crossing + macho 95-blg-30 & ... & ... & @xmath77 & ... & 13.4 & 4 & single lens + macho 97-blg-28 & ... & ... & @xmath78 & ... & 15.6 & 5 & cusp crossing + macho 97-blg-41 & ... & ... & @xmath14 & ... & 16.8 & 6 & rotating binary + eros 2000-blg-5 & ... & ... & @xmath79 & ... & 16.6 & 7 & parallax effects + fractional errors on parameters from observed events . + [ tbl : table1 ] a global solution to the entire light curve effectively requires the specification of the vector position of the source @xmath80 as a function of time in units of @xmath5 , and the topology of the lens , i.e. the mass ratio @xmath81 and projected separation @xmath42 in units of @xmath5 . these parameters yield not only the total magnification @xmath82 of all the images as a function of time . but also the individual image positions @xmath83 and magnifications @xmath84 as a function of time , therefore , it is also possible to predict the centroid @xmath85 of all the individual microimages , @xmath86 note that @xmath85 is the centroid shift with respect to the lens position . is it customary to consider the centroid shift with respect to the unlensed source position , @xmath87 . the unlensed source position @xmath88 , which is comprised of the parallax and proper motion of the source in some astrometric frame , can be determined via measurements of the unlensed motion of the source . since the astrometric effects falls off very slowly ( as @xmath89 ) , these must be obtained many @xmath3 after the event is over . in fact , such measurements are not strictly needed in order to measure @xmath5 ( but may be desirable for other reasons , see [ sec : dos ] ) . rather , one can simultaneously fit astrometric measurements during the course of the event for the relative position and proper motion of the source ( in order to establish a local astrometric reference frame ) , and the offset induced by microlensing . the centroid shift @xmath85 is in units of @xmath5 , and its components are oriented with respect to the projected binary - lens axis , whose orientation @xmath90 on the sky is unknown . the observable centroid is @xmath91 alternatively , one can combine @xmath90 and @xmath5 and simply consider the vector @xmath92 . thus the global solution to the photometric light curve yields not only @xmath3 and @xmath4 , but also yields a prediction for the astrometric curve , up to an unknown orientation and scale @xmath5 .. ] therefore , by making a series of astrometric measurements @xmath93 at several different times during the course of the event , one can determine @xmath5 and @xmath90 via , and using the predictions for @xmath94 from the photometric solution . the accuracy with which @xmath5 can determined will depend on the geometry of the event , the time of the astrometric measurements , and the time span between the measurements . furthermore , as we discuss , there are , in reality , + additional parameters that must be determined from the astrometric data . we therefore perform a monte carlo simulation to estimate the accuracy with which @xmath5 can be recovered . we explore whether the various parameters can be measured independently , or are degenerate with the measurement of @xmath5 , and study the range of fractional uncertainties in measuring @xmath5 for a full ensemble of binary lenses . our simulation consists of the following elements : a galactic model and model of the ensemble of lenses to generate an ensemble of microlensing events , an observational strategy , and a fit of these observations to a set of parameters including @xmath5 . our ensemble of lenses is identical to that of graff & gould ( 2002 ) . briefly , we draw sources and lenses from self - lensing isothermal sphere 8 kpc from the observer , and with two - dimensional velocity dispersion of @xmath95 . both masses in the binary lens are chosen from the remnant mass function of gould ( 2000 ) . we pick a flat distribution in @xmath96 , the logarithm of the dimensionless binary separation . the path of the source through the lens geometry is chosen randomly with a uniform distribution of angular impact parameter @xmath97 , and we only consider paths that cross caustics . , but are weighted towards large @xmath5 events by multiplying the mass function by @xmath98 . ] as discussed in graff & gould ( 2002 ) , in this ensemble of events there are many more events with a short time between caustic crossings , the _ caustic interior time @xmath99 _ , than is observationally detected by the macho and ogle observing groups . this led these authors to suggest that most events with a short caustic interior time are not detected as caustic crossing binaries , and to define a caustic crossing detection efficiency @xmath100 akin to the standard single lens detection efficiency @xmath101 . the observational strategy is relatively unimportant , as long as there are astrometric measurement on either side of a caustic . although it is conceivable that observations might be scheduled at particularly favorable times , such as times of maximum magnification or maximum displacement of the image centroid , it is likely that the telescope measuring the astrometric displacement will be oversubscribed . a simpler strategy would be to schedule periodic observations in advance . we have assumed that observations will be made every four days , with a 24 hour delay after the event is recognized as a caustic crossing binary , i.e. , after the first caustic crossing . that is , the first observation comes @xmath102 days after the first caustic crossing . we assume observations are made for a total of 36 days , which corresponds to approximately @xmath103 for the median event timescale . practically , observations should continue until after the second caustic crossing , and the total number of observations should be at least as large as the number of parameters to be constrained . as we discuss in [ sec : errors ] ( see also table 1 ) , well covered binary lenses can be fit photometrically with small errors on the parameters . thus , we have assumed that all the parameters which can be fit from a single photometric telescope are determined . given a binary microlensing event from our ensemble , we use our observational strategy to create a series of photometric and astrometric measurements , respectively @xmath50 and @xmath93 which we can combine into a single list of measurements @xmath104 each with uncertainty @xmath105 . using the fisher matrix technique , e.g. , gould & welch ( 1996 ) , we determine the covariance matrix @xmath106 of the errors @xmath107 here the @xmath108 are the various parameters being fit . the error in parameter @xmath108 is simply @xmath109 . we assumed that the photometric uncertainty , @xmath105 , of the interferometric telescope , is photon - noise - dominated , and that the total telescope time , aperture , efficiency , filter width , and source brightness are such that a total of @xmath110 photons would be detected from an unmagnified source for a total photometric signal to noise of 250 . and diameter @xmath111 collects @xmath112 photons per second at @xmath113 . ] we assume that the fractional photometric accuracy is simply @xmath114 , and that the astrometric uncertainty is @xmath115 , where @xmath116 is the width of the point spread function , or in the case of an interferometer , the fringe separation . here we have assumed @xmath117mas . it is trivial to scale our results to brighter sources or larger telescopes : the fractional uncertainty in @xmath5 is simply proportional to @xmath114 . we always fit for the four parameters required to establish a local astrometric frame , and @xmath5 and @xmath90 , the size scale and orientation of the microlensing excursion . we also assumed that the satellite which measures the astrometric motion is 0.2 au from the ( ground based ) photometric measurements which fix the lens parameters . thus , we can simultaneously fit for @xmath118 , the projected einstein ring radius , in the manner of graff & gould ( 2002 ) . in addition to this basic fit , we also considered blending from luminous lenses and binary sources , which requires several additional parameters . we discuss parallax and blending in [ sec : parallax ] and [ sec : lumlens ] , respectively . our basic results are summarized in the leftmost curve in figure 2 . we see that in the absence of blending , @xmath5 can be determined with @xmath119 uncertainty in 85% of events . the median error is @xmath120 . this is comparable to the uncertainty found by gould & salim ( 1999 ) for single star events , but with 50 times as many photons as we have assumed here . thus , it is much easier to measure @xmath5 for caustic crossing binary lenses than for single lenses . from the expression for @xmath1 ( eq . [ eqn : thetas ] ) , and assuming the errors in @xmath5 , @xmath3 , @xmath2 , and @xmath4 are small and uncorrelated , the fractional error in @xmath1 is given by , @xmath121^{1/2 } \label{eqn : errths}\ ] ] where @xmath122 , @xmath123 , @xmath62 , and @xmath124 are the uncertainties in @xmath5 , @xmath3 , @xmath2 and @xmath4 , respectively , and @xmath124 is in radians . to date , there exist in the published literature 10 microlensing events for which the source star was well - resolved . unfortunately , for four of the events , those presented in alcock et al . ( 2000a ) , no estimate of the errors of the derived parameters is given . we therefore can not use these events to explore the expected magnitudes of @xmath123 , @xmath62 , and @xmath124 . for the six events for which errors on the relevant fit parameters were given or derivable , only two of them are generic binary - lens fold caustic crossing events . the errors @xmath123 , @xmath62 , and @xmath124 for these two events , macho 98-smc-1 and ogle-1999-bul-23 , are presented in table 1 . for macho 98-smc-1 , these errors have been determined from the ensemble of solutions presented in albrow et al . ( 1999b ) , which were fits to the planet collaboration photometry , which only covered the last half of the event , whereas the analysis of the combined photometry of the eros , macho , mps , ogle , and planet collaborations ( afonso et al . 2000 ) yields considerably smaller errors . we concentrate on the results of albrow et al . ( 1999b ) here in order to demonstrate the kinds of errors that result from incomplete light curve coverage . note that @xmath125 , which is not surprising , since the accuracy with which @xmath2 can be determined depends almost exclusively on the photometric coverage near the caustic crossing . however , the errors on the global parameters @xmath3 and @xmath4 are quite large , @xmath126 and @xmath127 . this is due to the fact that the planet photometry only covered the latter half of the event , and contained no data prior to or during the first caustic crossing . therefore the global geometry was quite poorly constrained with their data alone . such incomplete coverage would clearly jeopardize a precise measurement of @xmath1 . this is in contrast to ogle-1999-bul-23 , for which both planet and ogle obtained data before the first caustic crossing . in this case , it was possible to measure @xmath2 , @xmath3 , and @xmath4 to better than @xmath76 ( albrow et al . 2001a ) . the remaining four include a single lens event ( macho 95-blg-30 ; alcock et al . 1997 ) , a binary - lens event in which the source crossed a cusp ( macho 97-blg-28 ; albrow et al . 1999a ) , a binary - lens event in which the rotation of the binary was detected ( macho 97-blg-41 ; albrow et al . 2000 ) , and a binary - lens event for which both rotation and parallax effects were detected ( eros blg-2000 - 5 ; an et al . 2002 ) . in general , the previous discussion is not directly applicable to these events , as the information in the light curves is not easily decomposed into the parameters @xmath2 , @xmath4 and @xmath3 . nevertheless , in all four cases the dimensionless source size @xmath128 was determined . in three of the cases , @xmath129 . in one case , macho 97-blg-41 , @xmath130 , due primarily to the fact that there exists only a handful of data points in which the source was resolved . thus , although it is difficult to draw any general conclusions from these unique events , it is does seem likely that errors of @xmath131 are achievable for most types of events in which the source is resolved . also presented in table 1 are the determined values of the @xmath132-magnitude of the source , for all six events . at @xmath133 , the source star for macho 98-smc-1 would be too faint to target with most upcoming interferometers , including sim . the other events are primarily bulge clump giants ( @xmath134 ) , for which accuracies of @xmath135 should be achievable with a reasonable amount of exposure time with upcoming interferometers , and in particular with @xmath136 minutes of sim time ( see [ sec : number ] for a estimate of the required exposure times for sim ) . in these cases , the expected error on @xmath1 is typically dominated by @xmath122 , and therefore we can expect @xmath137 . the source of ogle-1999-bul-23 is g / k subgiant ( @xmath138 ) , and thus dimmer . the time required to achieve an accuracy of @xmath139 will be larger by a factor of @xmath140 , or @xmath141 for sim . our goal in [ sec : method ] was to capture the essence of the method of measuring @xmath1 , and the discussions were therefore somewhat oversimplified , and glossed over several important points . in particular , we concentrated on fold caustic crossing binary - lens events toward the bulge , whereas measurements of @xmath1 should be possible in other , rarer , types of events , such as cusp - crossings and single - lens events , and possibly events toward the magellanic clouds . we also ignored various higher - order effects which could , in principle , complicate the measurements . we therefore briefly discuss some of these complications and extensions . we also discuss the prospects for measuring the spectral type of the source , and also its distance , in order to convert from angular radius @xmath1 to physical radius @xmath36 . finally , we present an example observing campaign aimed at measuring angular radii for a significant sample of sources , outlining the resources required , and estimating the number of @xmath1 measurements that might be made per year for such a campaign . although we have focussed on fold caustic - crossing binary - lens events toward the bulge , it is important to emphasize that @xmath1 can , in principle , be measured for other types of caustic - crossing events such as cusp - crossing events , single - lens events , and all types of caustic - crossing events toward the magellanic clouds . indeed , in [ sec : errors ] we discussed examples in the literature of a single - lens and two cusp - crossing events for which a @xmath11 measurement of @xmath1 would have been feasible . in general , isolated cusp crossings , such as in macho 98-blg-28 , can not be predicted in advance , and thus planning for such events is difficult , if not impossible . however , one will still have some advance warning of those cusp events which occur just after or in place of second fold caustic crossings . for such events , sufficient photometric coverage of the crossing should routinely be possible . in all cases , it is more difficult to disentangle the information arising from the cusp itself with the information from the global light curve . this generally implies that the analysis of these light curves will be more complicated , however this does not necessarily preclude an accurate measurement of @xmath1 . single - lens events are less desirable simply because they require a factor of @xmath9 times more astrometric observing time to achieve the same fractional accuracy in @xmath5 as binary - lens events . since the astrometric observations are likely to be the most limited resource , this makes single lens events considerably less attractive . if it were possible to measure angular radii of stars in the magellanic clouds ( mcs ) , this would be quite interesting , due to the metal - poor nature of the stars . unfortunately , there are several major hindrances to measuring @xmath1 for a substantial number of stars in the mcs . first , the event rates toward both the mcs are small , and a large number of stars must be monitored just to detect a few events per year . therefore , the number of caustic - crossing events is quite low . to date , there have been only two caustic crossing events toward the mcs : macho 98-smc-1 , which we discussed in [ sec : errors ] , and macho lmc-9 . these events have source magnitudes of @xmath142 ( afonso et al . 2000 ) and @xmath143 ( alcock et al . 2000a ) , respectively , which brings up a second difficulty : sim can not follow source stars fainter than @xmath144 , so these two events could not have been used to measure the angular radii of their source stars . in fact , even if the entire lmc were monitored for microlensing , only @xmath65 event per year would have @xmath145 , and this event would be from an evolved star . the probability of a caustic - crossing event ( either binary or single lens ) is smaller by at least an order of magnitude . the paucity of events and faintness of the source stars might be circumvented if sufficiently rapid target of opportunity times are available . in this case , it might be possible to use intrinsically fainter source stars , for which caustic - crossing event will be more common , and measure the astrometric displacement during the brief period of time when the source is highly magnified as it crosses the caustic . the maximum magnification of a source of dimensionless size @xmath146 crossing a fold caustic is @xmath147 . for main - sequence sources , @xmath148 , or more than three magnitudes , and thus sources with @xmath149 can briefly be brightened to sim detectability . for example , the source star of macho 98-smc-1 was brighter than @xmath150 for about 7 hours during the second caustic crossing . finally , even if the source does attain a sufficient brightness to be measurable by sim , it remains to be seen whether the centroid varies sufficiently during this time to provide an accurate measurement of @xmath5 . this is especially difficult in light of the fact that typical value of @xmath5 for self - lensing events toward the mcs are only an order of magnitude larger than sim s accuracy ( paczy ' nski 1998 , gould & salim 1999 ) . in summary , it appears that it will be quite difficult to measure angular radii of stars in the mcs using this method , especially if the majority of the events seen toward these targets are due to self - lensing ( sahu 1994 ) . the method we have presented here is only interesting if it can feasibly be used to make precise @xmath1 measurements for a large number of sources with reasonable expenditure of resources . since the requisite astrometric instruments are likely to be the most limited resource , it is crucial that accurate and unambiguous determinations of @xmath5 be generically possible using a few astrometric measurements , when combined with the photometric light curve solution . we have explained how a complete photometric solution _ generally _ leads to a prediction for the astrometric centroid shift up to an unknown scale @xmath5 and orientation @xmath90 on the sky . however , this is true only under a number of simplifying assumptions , including uniform motion of the observer , source , and lens , dark lenses , isolated sources , and unique global solutions . if one or more of these assumptions are violated , then the prediction for shape of the astrometric curve may not be unique , and thus the measurement of @xmath5 may be compromised . we therefore discuss each of these complications and under what conditions they may be important . binary lenses are characterized by two quantities : @xmath81 , the mass ratio , and @xmath42 , the instantaneous projected separation in units of @xmath5 . it has been demonstrated both theoretically ( dominik 1999a ) and observationally ( afonso et al . 2000 , albrow et al . 2002 ) that certain limiting cases of binary lenses can exhibit extremely similar observable properties . in particular , dominik ( 1999 ) showed that the binary - lens equation can be approximated by an single lens with external shear , or chang - refsdal ( cr ) lens ( chang & refsdal 1979,1984 ) , near the individual masses for widely - separated binaries ( @xmath151 ) , and near the secondary ( least massive ) lens when @xmath152 . furthermore , near the center - of - mass of a close binary , the lens equation is well - approximated by a quadrupole lens , and both the quadrupole lens and cr - lens can exhibit extremely similar magnifications when the quadrupole moment is equated to the shear ( albrow et al . 2002 ) . thus there can exist multiple degenerate solutions to an observed photometric light curve , even with extremely accurate photometry . however the astrometric behavior of these degenerate solutions is very similar both in the shape and overall scale of the astrometric curves , at least for the close / wide degeneracy ( gould & han 2000 ) . therefore this degeneracy should not affect the determination of @xmath5 using the prediction from the light curve . it is likely that the other intrinsic denegeracies will also not affect the determination of @xmath5 , since the degeneracy arises from the lens equation itself , and thus affects both the photometric and astrometric curve in the same manner . note that it is important that the normalization of @xmath5 be consistent for the two degenerate solutions . for example , consider the case of the close / wide degeneracy in macho 98-smc-1 ( afonso et al 2000 , gould & han 2000 ) . if one normalizes to the total mass of the binary , the close solution implies a value of @xmath153 , whereas the wide solution has @xmath154 . since the astrometric curves are essentially identical ( both in shape and scale ) , one might therefore suspect the inversion of this process would yield two equally likely values of @xmath1 that differed by a factor of @xmath155 of course , this ` ambiguity ' is wholly artificial , and arises because the value of @xmath156 for the wide binary is normalized to the entire mass of the binary , whereas the lensing effects are basically caused by the least massive lens , since @xmath157 . normalizing to the mass of the single lens , @xmath158 , where here @xmath159 , and thus @xmath160 , essentially identical to the close - binary solution . note that as @xmath42 approaches unity , the identification of the ` proper ' @xmath5 normalization becomes more nebulous , since the lenses can no longer be considered independent . however , the degeneracies also become less severe as @xmath161 . dominik ( 1999b ) has also shown that poorly - sampled binary lens light curve can also yield distinct degenerate solutions . note that these solutions are ` accidental ' in the sense that they do not arise from degeneracies in the lens equation itself . thus han et al . ( 1999 ) found that such degenerate photometric light curves yield astrometric curves which are widely different . such degeneracies would prohibit the measurement of @xmath5 using a few astrometric measurements . therefore well - sampled photometric light curves are essential for reliable measurements of @xmath1 . if the two observers are displaced by a significant fraction of @xmath162 , the angular einstein ring radius projected onto the observer plane , then the source position @xmath88 as seen by the two observers will be significantly different . since sim will be in an earth - trailing orbit , it will drift away from the earth at a rate of @xmath163 . thus after 2.5 years ( half way through the sim mission ) , it will be displaced from the earth by @xmath164 , which corresponds to a displacement in the einstein ring of , @xmath165 where @xmath166 is the angle between the the line of sight and the earth - sim vector . for typical bulge parameters , @xmath167 and therefore @xmath168 . this implies that both the magnification @xmath82 and the centroid shift @xmath169 will be significantly different ( at a given time ) as seen from the earth and sim . since the value of @xmath170 is not known _ a priori _ , @xmath171 can not be predicted from the ground - based photometry alone , and must be estimated from the astrometric data itself . fortunately , sim will likely have excellent photometric capabilities ( see gould & salim 1999 ) , and thus the relative magnifications between the light curves from the ground and sim will provide additional constraints . indeed , in our monte carlo simulations , we assumed that the astrometric observer was displaced by @xmath172 from the earth , and simultaneously fit for both @xmath5 and @xmath170 ( among other parameters ) , and found that @xmath5 could still be constrained quite accurately . this is because the information about @xmath170 comes primarily from the photometry , while the information about @xmath5 comes primarily from the astrometry . therefore , the two parameters are not degenerate . note that a ` byproduct ' of these measurements is a determination of the total mass of the binary lens ( gould & salim 1999 ; han & kim 2000 ; graff & gould 2002 ) there are two additional parallax effects . one is due to the motion of the earth ( or sim ) around the sun , and will become significant on timescales that are a substantial fraction of a year , which corresponds to many @xmath3 for typical bulge events . there is also a second order effect that arises from the difference in projected velocities between the earth and sim . this effect is @xmath173 , where @xmath174 is the transverse velocity of the lens projected on the observer plane , and is @xmath175 for typical bulge self - lensing events . since the velocities and positions of the earth and sim will be known , both of these effects can easily be included in the fit for the microlensing parallax , and so do not present any additional difficulties . with its planned 10 meter baseline , sim will have a resolution of @xmath176 , sufficient to resolve the majority of unassociated nearby stars that are blended with the source in ground - based photometry ( han & kim 1999 ) . since the photometric blending is well - constrained in binary - lens events , unambiguous prediction of the unblended astrometric behavior of the source is possible . thus , blending will typically not affect the measurement of @xmath5 . however , luminous lenses and companions to the source star with separations @xmath177 will not be automatically resolved by sim ( jeong , han & park 1999 ) . dalal & griest ( 2001 ) have shown that , using two pointings , this limit may be lowered to @xmath178 , however it is essentially impossible to resolve multiple sources with separations below this limit ( e.g. binary source companions ) . in these cases , all that will be measured is the total centroid of all the sources in the resolution element . the centroid in the presence of luminous lenses , @xmath179 , is related to the centroid in the absence of blending , @xmath85 , by @xmath180 \left(1 + \frac{f_t}{a}\right)^{-1 } , \label{eqn : dbpclt}\ ] ] where @xmath181 is the sum of the flux of all unlensed sources ( blends ) relative to @xmath182 , the baseline flux of the lensed source , @xmath82 is the magnification of the source , and @xmath183 is the centroid of the blends relative to the origin of the lens . from , it is clear blending is more complicated in astrometric microlensing than in photometric microlensing : whereas photometric blending can be described by one parameter , the blend fraction @xmath184 , astrometric blending requires two additional parameters , the centroid of light of the blend @xmath183 . a special case of astrometric blending is bright lens blending . in the single lens case , this eliminates the blend location parameters , the location of the centroid of light is the moving lens ( i.e. @xmath185 ) . in bright binary lens blending , only one parameter is eliminated , the centroid of light is somewhere on the lens axis between the two stars in the lens . however , it will generally not be known _ a priori _ which case one is dealing with , and therefore @xmath183 must be included as a parameter in the astrometric fit . blending is problematic because it effectively ` dilutes ' the astrometric shift between two points in the lightcurve , which is qualitatively similar to the effect of changing @xmath5 . if the event is not well covered , these two effects can be quite degenerate . in order to determine how degenerate blending is with the @xmath5 , we have included in our monte carlo simulations a fixed amount of blended light of @xmath186 , and @xmath187 . we assume that @xmath184 is known ( i.e. from photometry ) , but @xmath183 is not . the results are shown in figure 2 . we find that , if the blending fraction is close to 1 , then the two effects are nearly degenerate , and our fractional uncertainty in @xmath5 increases by two orders of magnitude to of order unity . however , for @xmath188 , the median error increases by less than a factor of two . in most cases , the blending will be known to be small from the photometric data . in these cases , the fractional uncertainty in @xmath5 will not be seriously degraded . the few events with known large blending can be easily jettisoned from the sample . the photometric effects of lens rotation in binary microlensing events has been explored theoretically by dominik ( 1998 ) and has been detected in event macho 1997-bul-41 ( albrow et al . the astrometric effects of rotating binary - lenses have not been explored , and it is therefore difficult to draw any general conclusions as to the importance of this effect . however , to the extent that it is detectable in the photometric light curve , lens rotation poses no difficulties , as its astrometric effect should be predictable from the global solution . effects that are photometrically undetectable but astrometrically significant are potentially problematic . the amount that a binary - lens rotates during @xmath3 is given by , @xmath189 where @xmath190 , and assuming circular , face - on orbits . because the caustic cross - section is maximized for binaries with separations of order @xmath5 , the majority of detected caustic - crossings will have @xmath191 ( baltz & gondolo 2001 ) . therefore , for typical events , the effects of binary rotation should be small if astrometric observations are closely spaced with respect to the event timescale @xmath3 . this is generally also advantageous for the accurate recovery of @xmath5 ( see [ sec : thetae ] ) . in this paper , we have emphasized the measurement of the angular radius @xmath1 , rather than the physical radius , @xmath36 . however , it may also be interesting to measure @xmath36 for some events . in order to do this , the distance to the source star must be measured independently . fortunately , the astrometric accuracy needed to measure @xmath1 is generally sufficient to measure the parallax @xmath192 of the source stars , @xmath193 in order to measure @xmath36 to a similar accuracy as @xmath1 ( @xmath11 ) , @xmath192 must be measured to somewhat better accuracy , which implies an astrometric error of @xmath194 . for sim and an @xmath113 source , this is achievable with @xmath195 hours of integration , which is considerably more time than is needed for the @xmath1 measurement alone . however , it is important to note that these measurements can be made after the microlensing event is over . therefore , it should be possible to employ ground - based interferometers for the measurement of @xmath192 , rather than spend precious sim resources , at least for brighter sources . a measurement of @xmath1 is essentially useless if the spectral type and luminosity class of the source is not known . the source stars of microlensing events can be typed in two ways . the first is to simply measure the color and apparent magnitude of the source . this information is generally acquired automatically from the fit to the photometric data of the microlensing event . by positioning the source star on a color - magnitude diagram of other stars in the field , one can generally type the source to reasonable accuracy . the primary pitfalls of this method are differential reddening and projection effects ( i.e. the source may in the foreground or background of the bulk of the stars in the field ) . a more robust way of typing the source star is to acquire spectra . this is best done when the source is highly magnified as it crosses a caustic , as this minimizes the effects of blended light and increases the signal - to - noise . thus such measurements require target - of - opportunity observations . for highly - magnified events , spectra with @xmath196 per resolution element can be achieved with exposure times of tens of minutes for low - resolution spectra ( lennon et al . 1996 ) , or a couple of hours for high - resolution spectra ( minniti et al . 1998 ) , using 8 m or 10m - class telescopes . although low - resolution spectra are sufficient for accurate + spectral typing , high resolution spectra are desirable for a number of other applications , including resolution of the atmosphere of the source star ( gaudi & gould 1999 , castro et al . 2001 , albrow et al . 2001b ) , detailed abundance analysis ( minniti et al . 1998 ) , and detection of a luminous lens ( mao , reetz , & lennon 1998 ) . note also that as a byproduct , true space velocities of a sample of stars in the bulge will be obtained by combining the proper motions and parallaxes of the sources acquired from astrometric measurements with radial velocities determined from the spectra . in this section , we review the requirements for measuring @xmath1 for the source stars of galactic bulge microlensing events , and outline the resources needed for a campaign aimed at measuring @xmath1 for a significant number of sources . the first requirement is a large sample of caustic - crossing binary - lens events from which to choose targets , which in turn requires an even larger sample of microlensing events . a large sample is important in that it ensures that only interesting and promising sources and events are followed . currently , both the ogle and moa collaborations monitor many millions of stars in the galactic bulge . both reduce their data real - time , enabling them to issue ` alerts , ' notification of ongoing microlensing events ( udalski et al . 1994 , bond et al . 2002a).ftp / ogle / ogle3/ews / ews.html ( ogle ) and http://www.roe.ac.uk/@xmath15iab/alert/alert.html ( moa ) . ] combined , these two collaborations should alert about @xmath197 events per year ( with the majority of alerts from ogle ) . extrapolating from previous results ( alcock et al . 2000a , udalski et al . 2000 ) , approximately @xmath14 of these will be caustic - crossing binaries , or 25 events per year . figure 3 shows the cumulative distribution of apparent ( i.e.uncorrected for redenning ) @xmath132 magnitudes of the 438 independent bulge microlensing alerts in 2002 for which baseline magnitudes were available . of these events , 382 ( @xmath198 ) were alerted by ogle , 61 ( @xmath199 ) were alerted by moa , and 5 were alerted by both collaborations . for the typical colors of sources in the bulge ( @xmath200 ) , only about @xmath201 of the alerts have @xmath202 , and thus would have been accessible to sim . the boundary between dwarfs and giants will occur at an apparent magnitude that depends on the color of the source , the distance to the source , and the reddening . however , for definiteness we will simply assume that the boundary between giants and main sequence stars occurs at roughly @xmath203 . with this assumption , we can therefore expect that approximately @xmath204 of all alerted events will be due to giant stars . therefore , we can expect approximately @xmath205 of all alerts , or @xmath206 events ( assuming 500 alerts ) , to be caustic crossing events with giant sources , and @xmath207 , or @xmath208 events , to be caustic crossing events with main sequence sources , @xmath209 of which will be bright enough to monitor with sim .- magnitude of the source . in fact , deviations from the single - lens form will generally be easier to detect in brighter sources , however this bias is likely to be relatively small for caustic - crossing events , which generally exhibit dramatic and easily - detectable deviations from the point lens form . this may seem in contradiction with the fact that none of the five events toward the bulge presented in table 1 are main sequence sources . however , this almost certainly a selection effect : bright binary - lens events are currently preferentially monitored by the follow - up collaborations , in order to achieve higher signal - to - noise during the second caustic crossing . ] these numbers are likely to remain valid at least for the next several years . however , in the more distant future , and in particular by the time sim is launched , it is likely that the next generation of microlensing survey collaborations will have come online . thus we can expect that , when sim time is operational , a considerably larger sample of caustic - crossing events will be available . survey - type experiments are needed to discover microlensing events toward the bulge , and survey - quality data is generally sufficient to uncover the caustic - crossing nature of the target events . however , as we discussed in [ sec : dt ] , more accurate and densely - sampled photometry is generally needed during the caustic crossing in order to measure @xmath2 . currently , there are several collaborations with dedicated ( or substantial ) access to 1 - 2 m class telescopes distributed throughout the southern hemisphere that closely monitor alerted microlensing events with the goal of discovering deviations from the single - lens form , with emphasis on search for extrasolar planets ( albrow et al . 1998 , rhie et al . 2000 , tsapras et al . 2001 , bond et al . these collaborations have also been quite successful in predicting and monitoring binary - lens caustic crossings . it seems likely that these collaborations , or similar ones , will still be in place when the next generation of interferometers , or even sim , come online . in our monte carlo simulations we derived the expected precisions @xmath210 assuming that the photometric errors were dominated by photon statistics , and that a total of @xmath110 photons where collected over the entire exposure time for each event this corresponds to total exposure time of @xmath211 for sim on an @xmath113 source . we assume the fractional photometric uncertainty is @xmath114 , and that the astrometric uncertainty was related to the photometric uncertainty via the expression , @xmath212 , with @xmath213mas , as appropriate for sim . since we assumed that the photometric and astrometric errors are given simply by photon statistics , it is trivial to scale our results for other total exposure times @xmath214 and source brightnesses assuming the characteristics of sim : @xmath215 , and @xmath216 . for the purposes of planning observations , and providing an order - of - magnitude estimate for the number of angular radii that can be measured for a given amount of sim time , it is useful to derive an expression for the exposure time required to achieve a given median photometric precision . to be conservative , we assume that the blending is small , but non - negligible . specifically , we adopt the median error found for the monte carlo simulations assuming a blend fraction of @xmath217 , which is @xmath218 . then , @xmath219 thus , for giant sources ( @xmath220 ) , @xmath221 minutes are required to achieve @xmath14 precision , whereas for main - sequence sources ( @xmath222 ) , @xmath223 hours are required for @xmath14 precision , or @xmath224 hours for @xmath16 precision . we have focussed here primarily on astrometric observations with sim because its capabilities are well - suited to this application . the basic requirements to be able to measure @xmath5 accurately for the events we have discussed are relatively high astrometric precisions , @xmath225 , and high sensitivity ( via , i.e. large apertures ) , as the sources we are considering are faint , @xmath226 . these faint sources are inaccessible to current ground - based interferometers . upcoming large - aperture , ground - based interferometers , such as the very large telescope interferometer or the keck interferometer , should be able to achieve the requisite astrometric precisions on all of the bright ( @xmath227 ) giant events . if the target microlensing source happens to have a bright star within the isoplanatic angle , it may be possible to employ phase referencing to extend sensitivity to very faint @xmath228 sources . this would allow one to measure @xmath5 for main - sequences sources from the ground as well . finally , it may be possible to determine @xmath5 from single - epoch measurements of the visibility and/or closure phase ( delplancke , g ' orski , & richichi 2001 , dalal & lane 2003 ) . in this way , sensitivity could plausibly be extended to main - sequences sources by making carefully - timed interferometric measurements of the source when it is highly - magnified during a caustic crossing . however , it is not clear if there exists enough structure in the image positions during this time to extract @xmath5 . this remains an interesting topic for future study . non - targeted space - based astrometric surveys , such as the global astrometric interferometer for astrophysics , are generally not well - suited to this application , due to the relatively sparse sampling of the source stars . finally , access to target - of - opportunity time on 8 - 10 m class telescopes would allow for accurate spectral typing of the source stars . several nights per bulge season would likely be adequate to type the @xmath229 caustic - crossing events per year . however , more time would be required to perform some of the auxiliary science discussed in [ sec : dos ] , such as resolution of the source - star atmospheres . thus , by combining alerts from survey collaborations , with comprehensive ground - based photometry from follow - up collaborations with access to dedicated ( or semi - dedicated ) 1m - class telescopes , and a modest allocation of a total 10 hours of sim time , it should be possible to measure the angular radii of @xmath29 giant stars in the bulge to @xmath14 , or @xmath10 main sequence stars to @xmath16 . several nights of target - of - opportunity time on @xmath230 m telescopes should allow for accurate spectral typing of the sources via high or low - resolution spectroscopy . we have outlined a method to measure the angular radii @xmath1 of giant and main sequence source stars of fold caustic - crossing binary microlensing events toward the galactic bulge . our method to measure @xmath1 consists of four steps . first , survey - quality data can be used to discover and alert caustic - crossing binary - lensing events . such data is sufficient to characterize the event timescale @xmath3 and the angle @xmath4 of source trajectory with respect to the caustic . dense sampling of one of the caustic crossings yields the caustic - crossing timescale @xmath231 . the global solution to the binary - lens light curve yields a prediction for the trajectory of the centroid of the source up to an unknown angle @xmath90 , and the scale , @xmath5 . thus a few , precise astrometric measurements during the course of the event yield @xmath5 . the angular source radius is then simply given by @xmath6 . we argued , based on past experience with modeling binary - lens events , that the parameters @xmath231 , @xmath4 , and @xmath3 should be measurable to a few percent accuracy , provided one caustic - crossing is densely and accurately sampled , and the entire event is reasonably well - covered . we then performed a series of monte carlo experiments that demonstrated that astrometric measurements during the course of the binary - lens event should allow for the determination of @xmath5 to @xmath232 accuracy , assuming photon - limited statistics and a total of 60,000 photons per event . this is a factor of @xmath9 fewer photons than are required to measure @xmath5 to the same precision in single - lens events and corresponds to an exposure time of @xmath233 hour with sim on an @xmath113 source . therefore , it should be possible to measure @xmath1 for a significant sample of giant and main - sequence stars in the bulge with reasonable expenditure of resources . we would like to thank neal dalal for helpful conversations . we would also like to thank the anonymous referee for useful comments and suggestions . this work was supported by nasa through a hubble fellowship grant from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 , by jpl contract 1226901 , and by the science research center ( src ) of the korean science and engineering foundation ( kosef ) .

we outline a method by which the angular radii of giant and main sequence stars located in the galactic bulge can be measured to a few percent accuracy . the method combines comprehensive ground - based photometry of caustic - crossing bulge microlensing events , with a handful of precise ( @xmath0 ) astrometric measurements of the lensed star during the event , to measure the angular radius of the source , @xmath1 . dense photometric coverage of one caustic crossing yields the crossing time scale @xmath2 . less frequent coverage of the entire event yields the einstein timescale @xmath3 and the angle @xmath4 of source trajectory with respect to the caustic . the photometric light curve solution predicts the motion of the source centroid up to an orientation on the sky and overall scale . a few precise astrometric measurements therefore yield @xmath5 , the angular einstein ring radius . then the angular radius of the source is obtained by @xmath6 . we argue that the parameters @xmath7 , and @xmath5 , and therefore @xmath1 , should all be measurable to a few percent accuracy for galactic bulge giant stars using ground - based photometry from a network of small ( 1m - class ) telescopes , combined with astrometric observations with a precision of @xmath8 to measure @xmath5 . we find that a factor of @xmath9 times fewer photons are required to measure @xmath5 to a given precision for binary - lens events than single - lens events . adopting parameters appropriate to the _ space interferometry mission _ ( sim ) , we find that @xmath10 minutes of sim time is required to measure @xmath5 to @xmath11 accuracy for giant sources in the bulge . for main - sequence sources , @xmath5 can be measured to @xmath12 accuracy in @xmath13 hours . thus , with access to a network of 1m - class telescopes , combined with 10 hours of sim time , it should be possible to measure @xmath1 to @xmath14 for @xmath1580 giant stars , or to @xmath16 for @xmath157 main sequence stars . we also discuss methods by which the distances and spectral types of the source stars can be measured . a byproduct of such a campaign is a significant sample of precise binary - lens mass measurements . # 1 # 1equation ( [ # 1 ] )

[ [ hyperbranched - stars - with - gaussian - chain - statistics . ] ] hyperbranched stars with gaussian chain statistics . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + regular exponentially growing starburst dendrimers , as sketched in fig . [ fig_sketch_dend ] , and more general starlike hyperbranched chains @xcite with gaussian chain statistics have been considered theoretically early in the literature @xcite and have continued to attract attention up to the recent past @xcite . one reason for this is that hyperbranched stars @xcite with sufficiently large spacer chains between the branching points ( as indicated by the filled circles ) are expected to be of direct experimental relevance under melt or @xmath2-solvent conditions @xcite . assuming a tree - like structure and translational invariance along the contour , the root - mean - square distance @xmath3 between two monomers @xmath4 and @xmath5 , as shown in panel ( a ) , is thus given by @xmath6 being the inverse fractal dimension of the spacer chains , @xmath7 the curvilinear distance along the tree between both monomers and @xmath8 the statistical segment size of the spacer chains @xcite . as a consequence , the typical distance @xmath9 between the root monomer and the end monomers of the most outer generation @xmath10 of spacer chains , as one possible observable measuring the star size @xcite , scales as @xmath11 with @xmath12 being the length of the spacer chains ( assumed to be monodisperse ) . other moments are obtained from the normalized distribution @xmath13 of the distance @xmath14 which , irrespective of the specific topology of the branched structure , is given by @xmath15 with @xmath16 being the spatial dimension @xcite . due to their theoretical simplicity such gaussian chain stars ( including systems with _ short - range _ interactions along the topological network ) allow to investigate several non - trivial conceptual and technical issues , both for static @xcite and dynamical @xcite properties , related to the in general intricate monomer connectivity imposed by the specific chemical reaction history . [ [ aim - of - current - study . ] ] aim of current study . + + + + + + + + + + + + + + + + + + + + + we assume here that _ ( i ) _ the chemical reaction is irreversible ( quenched ) , _ ( ii ) _ all spacer chains are monodisperse of length @xmath12 and _ ( iii ) _ flexible down to the monomer scale and _ ( iv ) _ that the branching at the spacer ends is at most three - fold ( @xmath17 ) as in the examples given in fig . [ fig_sketch_dend ] . our aim is to revisit various experimentally relevant conformational properties in the limit where the total monomer mass @xmath18 and the total number @xmath19 of spacer chains become sufficiently large to characterize the asymptotic _ universal _ behavior and to sketch for different star architectures the regimes where the gaussian spacer chain assumption becomes a reasonable approximation . we focus on the large-@xmath12 limit since this allows under @xmath2-solvent @xcite or melt conditions to broaden the experimentally meaningful range of the generation number @xmath20 of spacer chains . [ [ fractal - dimension . ] ] fractal dimension . + + + + + + + + + + + + + + + + + + one dimensionless property characterizing the star classes considered below is their fractal dimension @xmath21 which may be defined as @xcite @xmath22 with @xmath18 being the mass and @xmath23 the characteristic chain size . ( less formally , this definition is often written @xmath24 @xcite . ) for the regular dendrimers shown in panel ( a ) the number of spacers @xmath25 and , hence , the total mass @xmath18 increase exponentially with the generation number @xmath20 , while the typical chain size @xmath26 only increases as a power law . that the fractal dimension thus must diverge , is denoted below by the shorthand @xmath27 " . in addition we shall consider star classes of _ finite _ fractal dimension @xmath21 , focusing especially on not too dense systems which should be ( at least conceptionally ) of experimental relevance . specifically , we consider _ ( i ) _ marginally compact chains @xcite of fractal dimension @xmath28 and _ ( ii ) _ stars of fractal dimension @xmath1 which might be thought of as being assembled by diffusion limited aggregation ( dla ) @xcite . [ [ power - law - stars . ] ] power - law stars . + + + + + + + + + + + + + + + + as sketched in panel ( b ) , such hyperbranched stars of finite fractal dimension may be constructed most readily by imposing a number of spacer chains @xmath29 per generation @xmath30 such that the power law @xmath31 holds . hence , @xmath32 . the growth exponent " @xmath33 of these so - called @xmath33-stars " is set by the fractal dimension @xmath34 as may be seen using @xmath24 and @xmath35 @xcite . while being a natural generalization of the regular dendrimer case , restricting the branching of star arms does , unfortunately , not lead to a _ self - similar _ tree since the iteration @xmath36 is not a proper self - similar generator acting on _ all _ spacer chains @xcite . we therefore also consider truly self - similar ( multi)fractal stars , called in the following @xmath37- and @xmath38-stars , generated iteratively as shown in panel ( c ) and panel ( d ) of fig . [ fig_sketch_dend ] by the iterative application of a well - defined generator ( or several generators ) on _ all _ the spacer chains as in the recent theoretical work on vicsek fractals @xcite . for the latter architectures one thus expects to observe for the intramolecular coherent form factor @xmath39 the power - law scaling @xcite @xmath40 in the intermediate regime of the wavevector @xmath41 . note that eq . ( [ eq_fqdf ] ) only holds for open or marginally compact self - similar structures @xcite . in fact , gaussian hyperbranched stars with higher fractal dimension , @xmath42 , approach with increasing generation number and mass the gaussian limit @xmath43 as shall be demonstrated below . [ [ outline . ] ] outline . + + + + + + + + the paper is organized as follows : we summarize first in sect . [ sec_algo ] the numerical methods and specify then in sect . [ sec_topo ] the different topologies studied . some real space properties are presented in sect . [ sec_real ] before we turn to the characterization of the intramolecular form factor @xmath39 in sect . [ sec_form ] . while most of this study is dedicated to strictly gaussian hyperbranched stars , i.e. all excluded volume effects are switched off , we investigate more briefly in sect.[sec_weakev ] by means of monte carlo ( mc ) simulations @xcite effects of a weak excluded volume interaction penalizing too large densities . even an exponentially small excluded volume is seen to change qualitatively the behavior of large regular dendrimers . we conclude the paper in sect . [ sec_conc ] . neglecting deliberately the long - range correlations expected as for linear chains @xcite , we sketch the regime where the gaussian approximation for melts of hyperbranched stars should remain reasonable for sufficiently large spacers . [ [ settings - and - parameter - choice . ] ] settings and parameter choice . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we suppose that the monomers are connected by ideal gaussian springs . the spring constant is chosen such that the effective bond length @xmath8 , eq . ( [ eq_rs ] ) , becomes unity . also , both the temperature @xmath44 and boltzmann s constant @xmath45 are set to unity . all gaussian spacers are of equal length @xmath12 ( which comprises one end monomer or branching monomer ) . with @xmath25 being the total number of spacer chains , a hyperbranched star thus consists of @xmath46 monomers . if nothing else is said , @xmath47 is assumed . ( this arbitrary choice is motivated by simulations of dendrimer melts presented elsewhere . ) for @xmath47 we sampled up to a generation number @xmath48 for regular dendrimers and up to @xmath49 for power - law hyperbranched stars of fractal dimension @xmath50 and @xmath51 . ( even larger @xmath20 obtained using smaller @xmath12 are included below where appropriate . ) some properties of the largest system computed for each investigated star architecture are listed in table [ tab_gauss ] . .various properties for different hyperbranched star types of spacer length @xmath47 : fractal dimension @xmath21 , largest generation number @xmath20 , total mass @xmath18 , number of end monomers @xmath52 in the last generation shell @xmath10 , rescaled wiener index @xmath53 with @xmath54 being the largest curvilinear distance between pairs of monomers , relative root mean - square fluctuation @xmath55 ( r.f . ) of the normalized histogram @xmath56 , root mean - square end distance @xmath9 between the root monomer and the end monomers of the generation shell @xmath10 and radius of gyration @xmath57 . [ tab_gauss ] [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] [ [ local - and - collective - mc - moves . ] ] local and collective mc moves . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + due to their gaussian chain statistics many conformational properties can be readily obtained using gaussian propagator techniques @xcite or equivalent linear algebra relations @xcite . however , some interesting properties , such as the eigenvalues @xmath58 of the inertia tensor , can be more easily computed by direct simulation which are in any case necessary if long - range interactions between the monomers are switched on ( see below ) . as shown in fig . [ fig_sketch_algo ] , we use pivot moves with rigid rotations of the dangling chain end ( as shown by the monomers within the thin circles ) below a randomly chosen pivot monomer @xmath59 . the monomers are collectively turned ( using a quaternion rotation @xcite ) by a random angle @xmath2 around an also randomly chosen rotation axis through the pivot monomer . as illustrated in panel ( b ) of fig . [ fig_sketch_algo ] , it is useful to organize the data structure such that arms and monomers which are turned together are also grouped together . this allows to rotate all monomer @xmath60 with @xmath61 . the tabulated monomer @xmath62 , the last monomer to be turned , must be an end monomer . a pivot move does leave unchanged the distances between connected monomers . ( if the connectivity of the monomers is the only interaction , a suggested move is thus always accepted . ) to relax the local bond length distribution simple local mc jumps are added @xcite . the root monomer at the origin never moves . [ [ excluded - volume - interactions . ] ] excluded volume interactions . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + due to excluded volume constraints the volume fraction occupied by a realistic chain can , obviously , not exceed ( much above ) unity . one simple way to penalize too large densities is to introduce an excluded volume energy through the lattice hamiltonian @xmath63 using the monomer occupation number @xmath64 of a simple cubic lattice . for all examples presented below we set @xmath65 , i.e. the grid volume @xmath66 is unity and @xmath67 measures the instantaneous local density . the hamiltonian is similar to the finite excluded volume bond - fluctuation model for polymer melts on the lattice described in @xcite , however , the particle positions are now _ off - lattice _ and only the interactions are described by the lattice . a local monomer or collective pivot move is accepted using the standard metropolis criterion for mc simulations @xcite . note that the collective pivot moves are best implemented using a second lattice for the attempted moves . [ [ introduction . ] ] introduction . + + + + + + + + + + + + + we assume that the hyperbranched star topology is not annealed , i.e. not in thermal equilibrium , but irreversibly imposed by the chemical reaction . the first step for the understanding of such quenched structures is the specification and characterization of the assumed imposed connectivity , often referred to as connectivity matrix " @xcite . a central property characterizing the monomer connectivity is the normalized histogram of curvilinear distances @xmath68 with @xmath69 being the curvilinear distance between the monomers @xmath4 and @xmath5 . trivially , @xmath70 and @xmath71 for @xmath72 since the same monomer pair is counted twice . note that the histogram @xmath56 , sampled over all pairs of monomers of the chain , may differ in general from the similar distribution @xmath73 of the curvilinear distances between the root monomer and other monomers . we remind also that for a linear polymer chain @xcite @xmath74 with @xmath75 . for most of the star architectures considered the largest curvilinear distance @xmath76 is given by @xmath77 . the histogram @xmath56 will be used below for the determination of experimentally relevant properties such as the radius of gyration @xmath57 and the intramolecular form factor @xmath39 . the first and second moments of @xmath56 are given in table [ tab_gauss ] for the different architectures studied . we remind that @xmath78 is sometimes called wiener index " @xmath79 @xcite . [ [ regular - dendrimers . ] ] regular dendrimers . + + + + + + + + + + + + + + + + + + + let us first summarize several simple properties of the regular dendrimers sketched in fig . [ fig_sketch_dend](a ) . as already mentioned above , the number @xmath29 of spacer chains per generation shell @xmath80 increases exponentially as @xmath81 as shown by the bold line in fig . [ fig_irhisto ] . since we assume monodisperse spacer chains of length @xmath12 , this implies @xmath82 for @xmath83 and that the mass @xmath18 at generation number @xmath20 must also increase exponentially , as shown in fig . [ fig_n ] . the histogram @xmath56 of curvilinear distances @xmath7 for dendrimers is given in panel ( a ) of fig . [ fig_ws ] ( bold solid lines ) . the main panel gives a linear representation of the dimensionless rescaled histogram @xmath84 as function of @xmath85 , the inset on the left - hand side a similar half - logarithmic representation . as one expects , the histogram increases exponentially for curvilinear distances @xmath86 due to the exponential increase of alternative paths of length @xmath7 starting from an arbitrary monomer . using simple combinatorics it can be seen that the histogram must become @xmath87 the cutoff observed for large @xmath88 is due to the finite mass of the star and the finite length of its branches , just as the finite length of a linear chain gives rise to eq . ( [ eq_ws_linear ] ) . as seen from table [ tab_gauss ] , the reduced first moment @xmath53 approaches unity for dendrimers and the relative fluctuations are the smallest for all architectures considered . [ [ hyperbranched - alpha - stars . ] ] hyperbranched @xmath33-stars . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + as already noted in the introduction , a simple way to generate stars of a finite fractal dimension @xmath21 is to impose a power law @xmath89 for the number of spacers per generation shell with @xmath90 being a constant @xcite . this is done by randomly attaching @xmath29 spacer chains to the end monomers of generation @xmath91 ( with the constraint that at most two spacers can be attached per end monomer ) . an example for such an @xmath33-star with @xmath92 is given in fig . [ fig_irhisto ] ( open triangles ) . the corresponding total mass @xmath93 as a function of @xmath20 is shown for @xmath94 @xcite , @xmath95 , @xmath96 @xcite , @xmath92 and @xmath97 by open symbols in fig . [ fig_n ] . the histogram @xmath73 of curvilinear distances from the root monomer increases as @xmath98 for @xmath99 as implied by the @xmath29-scaling ( not shown ) . the curvilinear histograms @xmath56 over all pairs of monomers are presented in the main panel of fig . [ fig_ws](a ) . the histograms are again _ non - monotonous _ increasing first due to the branching and decreasing finally due to the finite length of the star arms . the latter decay becomes the more marked the weaker the branching , i.e. the smaller @xmath33 , getting similar for the smallest exponent @xmath97 studied to the linear chain behavior , eq . ( [ eq_ws_linear ] ) , indicated by the dashed line . as better seen from the double - logarithmic representation in panel ( b ) of fig . [ fig_ws ] , @xmath33-stars _ can not _ be described by a simple power law or exponential behavior for @xmath56 @xcite . [ [ self - similar - beta - stars . ] ] self - similar @xmath37-stars . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + this is different for self - similar fractals created starting from a @xmath100 dendrimer of spacer length @xmath101 ( as specified below ) as initiator and iterating a generator as the one shown in fig . [ fig_sketch_dend](c ) . at every iteration step @xmath59 a spacer of length @xmath102 is replaced by @xmath103 spacers of length @xmath104 . hence , @xmath105 , @xmath106 , @xmath107 and @xmath108 for , respectively , the spacer length , the number of spacers , the total mass and the generation number of the star . importantly , the arms added laterally to the original spacer can always be distributed such that the root - mean square end - to - end distance of the original spacer ( filled circles ) still characterizes the typical size of the replaced spacer . since @xmath109 for the curvilinear distance between the root monomer and the end monomers in the largest generation shell @xmath110 , the typical chain size @xmath23 , thus remains _ by construction _ constant as we shall explicitly verify in sect . [ sec_real ] . note that the spacer length @xmath111 of the final iteration step is set by @xmath112 which fixes the mass @xmath113 of the initiator star . using @xmath114 this implies @xmath115 relating thus both numerical constants @xmath116 and @xmath103 . as shown for @xmath50 ( @xmath117 , @xmath118 ) by the small filled triangles in fig . [ fig_irhisto ] , such a self - similar construction leads to a strongly fluctuating number @xmath29 of spacers . however , as shown by the thin solid line the ( logarithmically ) averaged number of arms still increases as @xmath31 with @xmath119 in agreement with eq . ( [ eq_alphadf ] ) . interestingly , the corresponding ( also logarithmically averaged ) root - mean square fluctuations ( as indicated by open circles ) are of the same order , i.e. the relative fluctuations of spacer number @xmath29 per generation shell are of order one . the important point is here that all monomers are _ statistically equivalent _ and that the root monomer does not play any specific role which would break the self - similarity . ( as we have verified , this implies @xmath120 . ) averaging over all spacer chains , the total mass @xmath18 scales , as expected , again as @xmath121 with @xmath122 as shown in fig . [ fig_n ] by filled triangles for @xmath50 and @xmath51 . the latter architecture , constructed using @xmath123 and @xmath124 , is motivated by the fractal dimension @xmath125 which may characterize self - similar stars generated by dla in @xmath16 dimensions @xcite . in our view this is one interesting universal limit of ( at least conceptional ) experimental relevance @xcite . being self - similar all monomers are equivalent and since the number of monomers at a curvilinear distance @xmath7 must increase on average as @xmath126 , one expects for @xmath86 the power - law scaling @xmath127 with @xmath128 this is confirmed by the histograms ( filled symbols ) shown in fig . [ fig_ws](b ) . [ [ stochastic - two - generator - multifractals . ] ] stochastic two - generator multifractals . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + since the dla limit is of some importance we have sampled a second system class of fractal dimension @xmath51 constructed by mixing the generator a for marginally compact stars shown in panel ( c ) of fig . [ fig_sketch_dend ] with the second generator b shown in panel ( d ) . being constructed using more than one generator these so - called @xmath38-stars " are in fact multifractals @xcite . ( we remember that dla clusters are also multifractal @xcite . no multifractal analysis @xcite is required here , however . ) for a given spacer we apply the generator a with a probability @xmath129 and the generator b with a probability @xmath130 . by choosing different values of @xmath129 any fractal dimension between @xmath131 and @xmath50 can be sampled using both generators . by reworking the arguments leading to eq . ( [ eq_alphabeta ] ) it can be seen that @xmath132 corresponds to @xmath51 . while @xmath37-stars are deterministic , the @xmath38-stars have a _ stochastic _ topology due to the random mixing of both generators and an ensemble average over several stars is thus taken . as may be seen from the crosses in fig . [ fig_n ] and fig . [ fig_ws](b ) , the properties of @xmath37- and @xmath38-stars are , however , rather similar . [ [ end - distance - ensuremathr_mathrme . ] ] end distance @xmath9 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + there are several ways to characterize the typical star size @xmath23 all being equivalent from the scaling point of view . a double - logarithmic representation of the reduced mean - square end distance @xmath133 _ vs. _ the reduced mass @xmath134 is presented in panel ( a ) of fig . [ fig_re ] . note that the values of @xmath9 obtained by direct mc simulations are within statistical accuracy identical to @xmath11 . both data sets are lumped together . the regular dendrimer size increases , of course , logarithmically with the mass ( circles and bold solid line ) . the power - law slopes indicated for finite-@xmath21 systems are consistent with the definition @xmath24 . as one measure of the overall density of a star one may define @xmath135 . ( obviously , a suitable order - one geometrical factor , such as @xmath136 , might be included in this definition . ) as can be seen from panel ( b ) of fig . [ fig_re ] , the density for regular dendrimers exceeds already at @xmath137 an unrealistic order of @xmath138 monomers per volume element . as indicated by the various power - law slopes , @xmath139 for power - law stars of finite fractal dimension , i.e. the density increases for @xmath42 and decreases for @xmath140 as it should @xcite . [ [ radius - of - gyration - ensuremathr_mathrmg . ] ] radius of gyration @xmath57 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the radius of gyration @xmath57 presented in fig . [ fig_rg ] has been determined with identical results ( lumped again together ) either from the mc sampled configuration ensembles or by means the formula @xcite @xmath141 using the histogram of curvilinear distances @xmath56 discussed above and the gaussian chain property @xmath142 . measuring thus the first moment of @xmath56 , the radius of gyration is equivalent for gaussian chains to the wiener index @xmath79 . the reduced radius of gyration @xmath143 is plotted as a function of @xmath20 . since the end monomers dominate the mass distribution of regular dendrimers for large @xmath20 , @xmath57 becomes similar to @xmath9 . as expected , @xmath144 approaches unity from below ( circles ) . interestingly , the ratio @xmath144 is constant for the self - similar @xmath37- and @xmath38-stars , i.e. @xmath9 and @xmath57 are similarly rescaled by the iterative application of the generators . this confirms the choice of generators discussed in sect . [ sec_topo ] . we note finally that other observables characterizing @xmath23 , such as the hydrodynamic radius @xcite , have been found to scale similarly as the end distance @xmath9 and the radius of gyration @xmath57 . [ [ density - profiles . ] ] density profiles . + + + + + + + + + + + + + + + + + figure [ fig_pr ] presents various normalized density profiles @xmath145 with @xmath146 being the radial distance from the root monomer . the rescaled distribution @xmath147 is plotted as a function of the reduced distance @xmath148 with @xmath149 in panel ( a ) and @xmath150 in panel ( b ) and panel ( c ) . the distribution of the end monomers for regular dendrimers ( @xmath151 , @xmath47 ) shown in panel ( a ) is a reminder of eq . ( [ eq_grsgauss ] ) , i.e. of the trivial fact that the distances of _ all _ pairs of monomers have a gaussian distribution ( dashed line ) . the rescaled density @xmath152 of all monomers is shown in panel ( b ) of fig . [ fig_pr ] ( using a half - logarithmic representation ) for the largest star of each topology class . note that the distribution @xmath145 has been either obtained for masses up to @xmath153 from our mc simulations or for larger systems using @xmath154 with @xmath73 being the already mentioned normalized histogram of monomers of same curvilinear distance from the root monomer and @xmath13 the size distribution of a subchain of arc - length @xmath7 given by eq . ( [ eq_grsgauss ] ) . since the density distribution of large regular dendrimers ( circles ) is dominated by the end monomers , @xmath145 becomes essentially gaussian ( dashed line ) . we shall come back to this point at the end of sect . [ sec_form ] . the histograms get naturally broader with decreasing @xmath21 . panel ( c ) on the right - hand side gives a double - logarithmic representation of the total monomer density distribution for three topologies with @xmath51 . as explained in de gennes book @xcite , the density should decrease as @xmath155 with @xmath156 being the mass distributed within the volume @xmath157 . the same power - law exponent is obtained using @xmath98 and integrating eq . ( [ eq_pr_wroots ] ) for @xmath140 and @xmath158 . even the not self - similar @xmath33-star ( open triangles ) is seen to follow the predicted slope ( solid lines ) . it is sufficient for this property that @xmath73 has a power - law asymptotics albeit @xmath56 has not . [ [ center - of - mass - fluctuations . ] ] center of mass fluctuations . + + + + + + + + + + + + + + + + + + + + + + + + + + + + albeit spherically averaged density profiles may reasonably characterize _ some _ aspects of the conformational properties of our hyperbranched polymer stars @xcite it is important to emphasize that a given instantaneous configuration may _ not _ be spherically symmetric and depending on the property probed experimentally or in a computer experiment these aspherical fluctuations become crucial . this issue is addressed in fig . [ fig_sphe ] . the main panel compares the true radius of gyration @xmath159 with a spherical approximation of the mass distribution defined by @xmath160 assuming the center of mass @xmath161 of the star to be set by the root monomer at the origin for all configurations , i.e. @xmath162 . the main panel of fig . [ fig_sphe ] presents @xmath163 as a function of @xmath20 for different topologies . the ratio is always smaller than unity . the ratio is seen to approach unity from below for regular dendrimers and @xmath33-stars with @xmath42 . while the spherical approximation @xmath164 becomes thus better with increasing size , stars with an incredible huge molecular mass are required to reach @xmath165 . interestingly , the ratio _ decreases _ for @xmath33-stars with @xmath166 and @xmath51 ( open triangles ) while it is essentially constant for the self - similar ( multi)fractals . for these experimentally most relevant star types the center - of - mass fluctuations remain thus relevant for asymptotically large chains . [ [ asphericity . ] ] asphericity . + + + + + + + + + + + + the asphericity of the stars may be ( also ) characterized by computing the three eigenvalues @xmath167 of the inertia tensor of each star and averaging over the ensemble . since @xmath168 , the rescaled eigenvalue @xmath169 should vanish for perfectly spherical chains with @xmath170 . we have plotted @xmath171 as a function of the inverse mass for several topologies in the inset of fig . [ fig_sphe ] . as expected from the consideration of @xmath172 , @xmath171 is seen to vanish in the large-@xmath18 limit for regular dendrimers and @xmath33-stars with @xmath42 . ( as shown by the solid line , @xmath171 decays only _ logarithmically _ with mass . ) the opposite behavior is found for smaller fractal dimensions as shown by the open triangles . whether for these systems @xmath171 becomes constant for @xmath173 ( as for linear chains ) can not be confirmed yet from our numerical data . [ [ introduction.-1 ] ] introduction . + + + + + + + + + + + + + conformational properties of branched and hyperbranched star polymers can be determined experimentally by means of light , small angle x - ray or neutron scattering experiments @xcite . using appropriate labeling techniques this allows to extract the coherent intramolecular form factor @xmath39 defined as @xmath174 with @xmath175 being the fourier transform of the instantaneous density and @xmath176 the wavevector . the average is sampled over the ensemble of thermalized chains . for sufficiently large @xmath18 and small @xmath177 the radius of gyration @xmath57 , as one measure of the star size , becomes the only relevant length scale . the form factor thus scales as @xcite @xmath178 being the reduced wavevector and @xmath179 a universal scaling function with @xmath180 in the guinier regime " for @xmath181 . the opposite large-@xmath41 limit probes the density fluctuations within the spacer chains and the form factor becomes @xcite @xmath182 for even larger wavevectors correlations on the monomer scale are probed . in the following we shall focus on the intermediate wavevector range @xmath183 between the guinier regime and the large-@xmath41 limit . [ [ dendrimers . ] ] dendrimers . + + + + + + + + + + + focusing on dendrimers , fig . [ fig_fq_dend ] presents a kratky representation @xcite of the form factor @xmath184 as a function of the reduced wavevector @xmath185 . panel ( a ) shows stars of different spacer length @xmath12 for a generation number @xmath151 , panel ( b ) different generation numbers @xmath20 for a fixed spacer length @xmath186 . the increase of the rescaled data for very large wavevectors @xmath187 observed in both panels is caused by the discrete monomeric units used in our simulations ( see below ) . the scaling observed for different @xmath12 in panel ( a ) for the intermediate wavevector regime , where the gaussian spacer chains are probed , is due to the fact that both the mass @xmath18 and the radius of gyration @xmath188 are linear in @xmath12 . the corresponding failure of eq . ( [ eq_fqscal ] ) in panel ( b ) shows that there is more than one characteristic length scale . note that the strong decay after the guinier regime above @xmath189 becomes systematically sharper with increasing generation number @xmath20 . the bold solid lines in both panels indicate the expected asymptotic limit for @xmath190 as discussed at the end of this section . note that the dendrimer with @xmath191 ( large circles ) shown in panel ( b ) is rather close to this limit . the form factor of this huge chain has not been obtained by mc simulations but by computing numerically the equivalent discrete sum @xmath192 with @xmath56 being the curvilinear segment histogram discussed above and @xmath193 the fourier transform of the segment size distribution @xmath13 . since for gaussian chains @xmath194 with @xmath195 , the form factor is readily computed yielding , as one expects , the same results as obtained from the explicitly computed configuration ensembles . this can be seen from the dashed line in panel ( a ) of fig . [ fig_fq_dend ] for a spacer length @xmath47 . to compute numerically the form factor using @xmath56 has the advantage that the already mentioned discretization effect at @xmath187 can be eliminated . to do this the discrete sum eq . ( [ eq_fq_ws ] ) is replaced by a continuous integral for @xmath196 and the @xmath197-contribution to the form factor is added . as shown by the thin solid line in panel ( a ) , this allows to get rid of the irrelevant discretization effect . [ [ marginally - compact - stars . ] ] marginally compact stars . + + + + + + + + + + + + + + + + + + + + + + + + + figure [ fig_fq_df3 ] presents the form factor obtained using the continuous version of eq . ( [ eq_fq_ws ] ) for self - similar fractals of marginal compactness ( @xmath50 ) . as one expects according to eq . ( [ eq_fqdf ] ) , the data approach with increasing generation number the power - law slope @xmath198 ( bold line ) expected for the intermediate wavevector regime . we remind that eq . ( [ eq_fqdf ] ) can be derived from eq . ( [ eq_fq_ws ] ) and the scaling @xmath199 for self - similar fractals . interestingly , eq . ( [ eq_fqdf ] ) does _ not _ hold for the ( not self - similar ) @xmath33-stars as may be seen from fig . [ fig_fq_df ] . note also that the large-@xmath41 plateau of the rescaled form factor in fig . [ fig_fq_df3 ] only decays as @xmath200 extremely slowly with mass . this makes the numerical confirmation of the power - law slope demanding . for real experiments this implies that the determination of a fractal dimension @xmath201 using the power - law scaling of the form factor for self - similar stars will also be challenging . we remind that a similar slow convergence of the intermediate wavevector regime is well - known for other more - or - less compact polymers such as polymers confined to ultrathin slits or melts of polymer rings @xcite . [ [ comparison - of - different - architectures . ] ] comparison of different architectures . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the rescaled form factors for the largest chains considered for each studied topology are compared in fig . [ fig_fq_df ] . as expected , all data sets collapse in the guinier regime below @xmath202 and become again constant for large wavevectors @xmath203 . ( the discretization effect for large @xmath41 is again avoided using the continuous version of eq . ( [ eq_fq_ws ] ) . ) the decay of the reduced form factor in the intermediate wavevector is seen to become systematically stronger with increasing fractal dimension @xmath21 . for the self - similar stars this decay is described by eq . ( [ eq_fqdf ] ) as emphasized by the solid and the dash - dotted power - law slopes for , respectively , @xmath50 and @xmath51 . all other architectures decay stronger than a power law . note that it is the shape of this decay which is the most central property to be tested experimentally to characterize , at least approximatively , the structure of hyperbranched stars . [ [ spherical - preaveraging . ] ] spherical preaveraging . + + + + + + + + + + + + + + + + + + + + + + + as reminded at the beginning of this section , the intramolecular form factor is the ensemble average of the squared fourier transform @xmath175 of the fluctuating instantaneous monomer density . following the recent work by likos _ @xcite , this begs the question of whether in the limit of large and dense stars , where density fluctuations should become sufficiently small , one may replace @xmath175 by the fourier transform @xmath204 of the averaged density profile @xmath205 discussed in sect . [ sec_real ] . due to the spherical symmetry of our stars this suggests using eq . ( 6.54 ) of ref . @xcite the approximation @xmath206 with @xmath207 being known from eq . ( [ eq_pr_wroots ] ) . as seen in fig . [ fig_fq_df3 ] , eq . ( [ eq_fq_preaver ] ) is not useful for open ( @xmath140 ) and marginally open ( @xmath208 ) architectures for which the density fluctuations are yet too large . the approximation becomes systematically more successful , however , with increasing fractal dimension as seen in fig . [ fig_fq_df ] for @xmath33-stars of fractal dimension @xmath209 . note that the striking decay of the rescaled form factor above the guinier regime is accurately described by the approximation . as we have seen in fig . [ fig_pr ] , the distibution @xmath145 becomes systematically more gaussian with increasing star size and fractal dimension since the end monomers of the largest generation shell dominate the total density . since the fourier transform of a gaussian is again a gaussian , this implies finally eq . ( [ eq_fq_asympt ] ) as already stated in the introduction . as seen by comparing the solid bold lines in fig . [ fig_fq_dend ] and fig . [ fig_fq_df ] with the form factors computed using eq . ( [ eq_fq_ws ] ) for our largest dendrimers ( circles ) , the asymptotic behavior eq . ( [ eq_fq_asympt ] ) gives an excellent fit to our numerical data . [ [ introduction.-2 ] ] introduction . + + + + + + + + + + + + + up to now we have only considered effects of the imposed monomer connectivity assuming all other interactions ( persistence length , excluded volume , ) to be switched off . since essentially all properties ( apart the eigenvalues @xmath58 of the inertia tensor ) can be obtained analytically or numerically using the gaussian chain statistics , the presented mc simulations were less crucial . direct simulations are , however , essential for testing the influence of ( albeit weak ) excluded volume interactions computed using the lattice occupation number hamiltonian , eq . ( [ eq_elatt ] ) , described at the end of sect . [ sec_algo ] . [ [ scaling - of - chain - sizes . ] ] scaling of chain sizes . + + + + + + + + + + + + + + + + + + + + + + + figure [ fig_weakev_r ] presents the excluded volume dependence of the radius of gyration @xmath57 for regular dendrimers . ( similar behavior is found for other characterizations of the typical chain size @xmath23 . ) as reveiled in the main panel , the excluded volume effects are the more marked the larger the mass @xmath210 : the radius of gyration increases already at @xmath211 for @xmath212 while it has barely changed at @xmath213 for @xmath214 . a successful data collapse is seen in the inset of fig . [ fig_weakev_r ] where the rescaled radius of gyration @xmath215 is plotted as a function of the reduced excluded volume @xmath216 with @xmath217 being the typical size of the gaussian dendrimer star and @xmath218 the excluded volume @xcite relevant for our model hamiltonian ( @xmath37 denoting the inverse temperature ) . the characteristic excluded volume @xmath219 below which the star should remain gaussian is set by @xmath220 . this scaling is a direct consequence of fixman s general criterion @xcite @xmath221 for the gaussian chain approximation with @xmath222 the overall density for gaussian stars . that the stars remain gaussian for @xmath223 is emphasized by the horizontal asymptote indicated in the inset . the power - law slope @xmath224 ( bold line ) for large reduced excluded volumes is only an _ approximative _ guide to the eye not taking into account logarithmic corrections . this can be seen _ ( i ) _ from the usual power - law ansatz @xcite @xmath225 , _ ( ii ) _ neglecting the weak logarithmic @xmath18-dependence of @xmath226 ( fig . [ fig_re ] ) and _ ( iii ) _ assuming that the dendrimers become essentially marginally compact , @xmath227 , for large @xmath228 in agreement with ref . the latter point has explicitly been checked . for finite-@xmath21 stars a similar scaling has been found ( not shown ) . [ [ spacer - chain - length - criterion . ] ] spacer chain length criterion . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we note finally that in terms of the generation number @xmath20 and the spacer length @xmath12 , fixman s criterion may be rewritten remembering that @xmath229 for dendrimers and @xmath230 for power - law stars @xcite . hence , the gaussian approximation must hold for @xmath231 with an _ upper _ critical spacer length @xcite @xmath232 respectively , for dendrimers ( @xmath233 ) and finite-@xmath21 hyperbranched stars . in both cases @xmath234 in @xmath16 dimensions ( while four - dimensional stars are only marginally swollen ) . [ [ summary . ] ] summary . + + + + + + + + we have revisited by means of direct analytical calculation , using for instance eq . ( [ eq_fq_ws ] ) , and mc simulations ( sect . [ sec_algo ] ) several conformational properties of regular ( exponentially growing ) dendrimers and power - law hyperbranched stars ( fig . [ fig_sketch_dend ] ) assuming gaussian chain statistics ( @xmath235 ) . as emphasized , a central imposed property is the normalized weight @xmath56 of curvilinear distances @xmath7 between monomer pairs ( fig . [ fig_ws ] ) . focusing on experimentally measurable observables such as the radius of gyration @xmath57 ( fig . [ fig_rg ] ) and the intramolecular form factor @xmath39 ( figs . [ fig_fq_dend]-[fig_fq_df ] ) , we investigated the scaling for asymptotically long stars with different fractal dimensions @xmath21 . due to their topological simplicity regular dendrimers ( @xmath233 ) have played a central role in our presentation ( fig . [ fig_fq_dend ] ) as in other recent computational studies @xcite . being ( in our view ) experimentally and technologically more relevant , we have also focused on stochastic architectures with @xmath50 ( marginally compact ) and @xmath1 as expected for stars created by dla @xcite . we compared @xmath33-stars " constructed by imposing @xmath31 arms per generation with truly self - similar so - called @xmath37-stars " and @xmath38-stars " for which @xmath29 becomes a strongly fluctuating quantity ( fig . [ fig_irhisto ] ) . as shown in fig . [ fig_fq_df ] , only the latter two topologies show the power - law decay , eq . ( [ eq_fqdf ] ) , of the form factor in the intermediate wavevector regime expected for open self - similar systems @xcite . while large compact ( @xmath42 ) stars may roughly be seen as dense colloidal spheres in agreement with likos _ @xcite , the instantaneous aspherical fluctuations can not be neglected for experimentally relevant properties for the smaller fractal dimensions studied ( fig . [ fig_sphe ] , dashed line in fig . [ fig_fq_df3 ] ) . we have commented briefly on the effects of gradually switching on an excluded volume potential . coupling the ( off - lattice ) monomers by means of a ( lattice ) mc scheme ( sect . [ sec_algo ] ) , we have sketched for different architectures the regime ( @xmath236 , @xmath231 ) where the gaussian star approximation can be assumed to be reasonable ( fig . [ fig_weakev_r ] ) . as already pointed out , the gaussian star assumption should be relevant under melt conditions assuming a large spacer length @xmath237 . that this holds can be seen by rewriting fixman s gaussian chain criterion , eq . ( [ eq_phase_criterion ] ) , for melts @xmath238 remembering that the bare excluded volume @xmath239 has to be rescaled by the total chain mass @xmath18 @xcite . the hyperbranched stars should thus remain gaussian for interaction energies @xmath240 . since @xmath228 is not a parameter which can be readily tuned experimentally over several orders of magnitude , it is of some importance that eq . ( [ eq_fixman_melt ] ) sets equivalently a _ lower _ bound @xmath241 depending on the generation number @xmath20 . following the discussion at the end of sect . [ sec_weakev ] , this implies @xmath242 this scaling prediction is sketched in fig . [ fig_conc ] for several architectures . hyperbranched stars should remain thus gaussian ( albeit with a renormalized effective statistical segment length @xcite ) as long as @xmath237 , if the interaction parameter @xmath243 is switched on as in the recent study of linear chain polymer melts @xcite . details may differ somewhat , of course , since the spacer chains may not be rigorously gaussian due to long - range correlations related to the overall incompressibility of the melt @xcite . it is thus possible that even self - similar stars of imposed @xmath51 for the gaussian reference ( @xmath244 ) may swell somewhat . we do conjecture , however , that this swelling " for interacting large-@xmath12 hyperbranched stars in the melt remains _ perturbative _ as long as @xmath245 @xcite . considering the dynamical properties of strongly interpenetrating hyperbranched stars for @xmath237 sampled using standard molecular dynamics @xcite , it will be of some interest to characterize the mean - square displacement of the star center of mass or , even better , the associated displacement correlation function @xcite . as for the center of mass motion of linear polymer melts @xcite , strong deviations from the rouse scaling are to be expected @xcite .

conformational properties of regular dendrimers and more general hyperbranched polymer stars with gaussian statistics for the spacer chains between branching points are revisited numerically . we investigate the scaling for asymptotically long chains especially for fractal dimensions @xmath0 ( marginally compact ) and @xmath1 ( diffusion limited aggregation ) . power - law stars obtained by imposing the number of additional arms per generation are compared to truly self - similar stars . we discuss effects of weak excluded volume interactions and sketch the regime where the gaussian approximation should hold in dense solutions and melts for sufficiently large spacer chains .

radio transient sources have primarily been studied through follow - up observations after discovery at optical , x - ray , or @xmath16-ray wavelengths , or through serendipitous discovery ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this is principally due to the high cost of observing time to survey large areas of sky to significant depth at radio wavelengths @xcite . moreover , existing blind surveys have typically been performed at low frequencies , where survey time is shorter but the effect of synchrotron self - absorption may hide certain source classes . the available parameter space for radio transient surveys is extensive : transients have been detected at , and are predicted for all , radio wavelengths ; timescales of transient and variable behavior range from nanoseconds @xcite to the longest timescales probed @xcite ; and transients may originate from nearly all astrophysical environments including the solar system @xcite , star - forming regions @xcite , the galactic center @xcite , and other galaxies @xcite . nevertheless , extensive surveys have been carried out , exploring different volumes of the parameter space , and , in some cases , discovering new phenomena . roughly , one can separate transients surveys into two classes : ( a ) burst searches that probe timescales of less than @xmath17 second , are often performed with low angular resolution instruments , and are most often performed at low frequencies ; and ( b ) imaging surveys conducted with interferometers which typically probe timescales of tens of seconds and longer . examples of burst searches include stare , a 611-mhz all - sky survey @xcite sensitive to timescales of 0.1 s to a few minutes ; a 843-mhz survey with the molonglo observatory synthesis telescope ( most ) sensitive to timescales of 1 @xmath5s to 800 ms @xcite ; and the parkes 1.4-ghz multi - beam pulsar survey . stare used dipole antennas with a sensitivity threshold of 27 kjy simultaneously over a significant fraction of the sky and found no extra - solar transients on timescales of 0.125 s to a few minutes in 18 months of observations . the most survey determined an upper limit to the transient rate of 5 events @xmath18 for 10 ms events at a flux density limit of 1 jy . the parkes multi - beam survey has been used for extensive searches for periodic emission as well as for pulsed emission . the latter search led to the discovery of rotating radio transients ( rrats ) , which appear to be pulsars with erratic emission @xcite . imaging surveys for transients have focused on individual fields and on large - scale surveys . recently , a blind survey of the galactic center at 330 mhz has discovered an unusual radio transient that displays quasi - periodic emission on a timescale of 70 min @xcite . @xcite used @xmath19 hr of observation of the lockman hole to characterize variability on a timescale of 19 d and 17 mon at 1.4 ghz . they found no transients above 100 @xmath5jy and conclude that the transient density on these timescales is fewer than 18 per square degree . similarly , @xcite identified 4 highly variable sources but no transient sources at 5 and 8.4 ghz using vla observations of fields in which gamma - ray bursts are present . they set an upper limit to the variable source population of 6 deg@xmath20 at sub - mjy sensitivity , comparable to the limit set by @xcite . recently , drift scan interferometric observations have identified several transients of unknown origin , including a 3-jy transient with a duration of 72 hr and three @xmath21 jy transients of @xmath22 d duration @xcite . the most extensive imaging search for transients derives from a comparison of the first and nvss 1.4 ghz catalogs @xcite . the search covered 2400 square degrees at a flux density threshold of 6 mjy . this search generated a number of transient candidates . of these , one source was identified as a radio supernova in a nearby galaxy and another was seen only once in the radio and could not be associated with any known optical source . transient identification in this survey suffers from the methodological problem of the extremely mismatched resolutions ( first : @xmath23 , nvss : @xmath24 ) of the two surveys . in addition to radio supernovae ( rsne ) , a number of highly variable and transient sources are expected to be detectable in significant numbers at radio wavelengths . one example is gamma - ray burst ( grb ) afterglows associated with grbs which are beamed in a direction away from earth , and therefore not observable at high - energy wavelengths @xcite . these so - called `` orphan gamma - ray burst afterglows '' ( ogrbas ) are expected to outnumber grbs by a factor of 10 to 1000 , the specific value depending on the characteristics of the relativistic outflow of the grb and the surrounding medium . there may be related phenomena such as those associated with x - ray flashes @xcite , supernovae ( sne ) @xcite , and short - duration grbs @xcite . activity from stars and compact objects in the galaxy may also be detected as transients ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? a range of active galactic nucleus ( agn ) phenomena is also anticipated , including intra - day variability and other short - term variability driven by the effect of interstellar scintillation on compact components of extragalactic radio sources ( e.g. , * ? ? ? * ) . intrinsic agn variability is also expected . sudden increases in the mass accretion rate following the tidal disruption of an orbiting star , for instance , will produce a transient soft x - ray flare and possibly radio emission @xcite . significant numbers of rsne , ogrbas , and tidal flares are predicted for surveys that achieve a sensitivity @xmath25 mjy and cover an area of @xmath26 square degrees . these phenomena all produce self - absorbed synchrotron radiation and , therefore , are more likely to be observed at high radio frequencies . we describe here the results of a search for extragalactic radio transients using 5 and 8.4 ghz observations from the very large array data archives . the survey includes 944 epochs over 22 years consisting of observations of the same field . at a sensitivity threshold of @xmath27 @xmath5jy , the survey has an effective survey area comparable to that of a survey consisting of two epochs of 10 square degrees each , making this thus far the largest - area sub - mjy imaging survey of radio transient phenomena . between 1983 and 2005 , the vla performed calibration and system check observations of the same blank field with a typical interval of @xmath28 d. each epoch typically was 20 min in duration and consisted of 10 s integrations . the field is centered at @xmath29 , @xmath30 ( j2000 ) . the field is out of the plane of the galaxy ( @xmath31 , @xmath32 ) . 626 epochs of observations were obtained at a frequency of 5 ghz between 1983 and 1999 . 599 epochs were obtained at a frequency of 8.4 ghz between 1989 and 2005 . vla observations of this field ceased in 2005 . between 1989 and 1999 , there was overlap between some of the 5 and 8.4 ghz observations . 281 of the 8.4 ghz epochs were simultaneously observed at 5 ghz , giving a total of 944 independent epochs at either frequency . data were all obtained in standard continuum mode with 50 mhz of bandwidth in each of two intermediate - frequency bands ( with centers separated by 100 mhz ) and two circular polarizations . we used a pipeline procedure for flagging , calibrating , and imaging . phase calibration was made based on brief observations of the compact source j1803 + 784 . since no standard flux density calibrator was used in these observations , we set the amplitude scale of the observations by assuming the mean flux density for j1803 + 784 as measured in the university of michigan radio astronomy observatory database over the same period ( 2.2 jy and 2.8 jy at 5 and 8.4 ghz , respectively * ? ? ? * ) . the assumption of constant flux density introduces less than 15% uncertainty in the flux - density scale for individual epochs . images of the target field typically had a root - mean - square ( rms ) error in the flux density of 40 to 50 @xmath5jy . images from two successive epochs ( 19840613 and 19840620 ) are shown in figure [ fig : typicalimage ] . throughout the paper , we use the notation of yyyymmdd to denote epoch date and refer to specific transients as rt yyyymmdd , based on the epoch of detection . images were obtained in all configurations of the very large array . for the extended configurations , bandwidth smearing within the primary beam can become significant . we compensated for this effect by applying a gaussian taper with a full - width at half - maximum ( fwhm ) of 150 @xmath33 to the @xmath34 data before imaging , which reduces sensitivity and angular resolution but provides imaging to large radii without bandwidth distortions . images were made using natural weighting to achieve maximal sensitivity and were deconvolved using the clean algorithm . we imaged a region slightly larger than the region containing a circle of radius equal to twice the half - power radius , which is @xmath35 at 5 ghz and @xmath36 at 8.4 ghz . a deep image of the region at 5 and 8.4 ghz was also constructed through merging of all visibilities and imaging with a @xmath34 taper of 150 @xmath33 using the miriad software package . the resulting images had resolutions of @xmath37 and @xmath38 , and an rms of 2.6 @xmath5jy and 2.8 @xmath5jy at 5 and 8.4 ghz , respectively . the location of steady and transient radio sources on the deep image is shown in figure [ fig : deep ] . fluxes in the deep image are typically lower than the average of detections since our high detection threshold and the low flux densities of the sources bias the mean . further details of the deep images will be published separately . sources were identified using aips task sad ( search and destroy ) , which identifies peaks in the image above a flux - density threshold and fits point sources . for the relatively uncluttered images that we have , sad is efficient and accurate . we identified sources within twice the half - power radius and corrected their flux densities for primary beam losses . the detection threshold for each image was chosen such that the probability of a false detection ( pfd ) in the full imaged region was @xmath39 , or a single false source for 1000 identical epochs , assuming gaussian statistics for noise in the image . since the synthesized beam changes with array configuration while the primary beam remains the same , the number of independent pixels in the field of view varied substantially . detection thresholds were in the range @xmath40 to @xmath41 , dependent on array configuration . the typical flux density threshold at the center of the image was @xmath42 @xmath5jy . for the number of epochs observed and the circular area of the region searched for sources , we have an expectation of @xmath17 false source in the entire survey . inverting the sign of the intensity in each image and searching for sources with the same thresholds and methods revealed one significant source , consistent with our expectations of one or fewer false detections in the entire survey . wider - field images ( @xmath43 at 5 ghz , @xmath44 at 8.4 ghz ) were made for all epochs in which a transient was detected . a number of transients that were detected on the western and northern edges of the individual epoch image were determined to be aliased power from two bright radio sources ( @xmath45 mjy ) at radii greater than two primary beam widths ( nvss j1457 + 7817 , j1500 + 7827 ) . one source above the detection threshold was rejected on the basis of a non - pointlike image and deep negative stripes in the image . detected sources and their measured properties are listed in tables [ tab : steady ] and [ tab : transients ] . table [ tab : steady ] identifies persistent sources above or near the detection threshold and not multiple - epoch transients . six persistent sources at 5 ghz and 2 persistent sources at 8.4 ghz were detected in multiple epochs . the epochs of detection were distributed over the entire survey . of these persistent sources , three radio sources with flux densities @xmath46 @xmath5jy were detected hundreds of times . one radio source ( j150123 + 781806 ) was detected with our pipeline software at 5 ghz in 452 of 626 epochs , and at 8.4 ghz in 159 of 599 epochs ( figure [ fig : standardlightcurve ] ) ; with a reduced source detection threshold of @xmath47 , we find that this source was present in 602 epochs at 5 ghz and in 445 epochs at 8.4 ghz . the results for this source indicate the overall quality of the data . variations in the flux density of this source are caused by the changing sensitivity of the vla in different configurations , variability in the amplitude calibrator j1803 + 784 , and possibly intrinsic variability in the source . seven sources at 5 ghz and 1 source at 8.4 ghz were detected in individual epochs ( table [ tab : transients ] ) . these sources were detected only once and are classified as transients . we plot the light curves of each transient for the year surrounding their detection ( figure [ fig : lct ] ) . we searched for time variability for each transient by splitting the data into four - minute segments and comparing the flux densities . the reduced @xmath48 for the hypothesis of no variability is less than 1.4 for all the transients , implying no strong evidence for variability . we also imaged each transient in stokes v and found no evidence for circular polarization above @xmath49 . finally , we differenced images between the intermediate frequency bands and found no evidence for a large spectral index . the radial distribution of sources is consistent with a real source population rather than imaging defects . imaging defects would be uniformly distributed throughout the image , without regard for the decline in sensitivity due to the primary beam . four of the eight transients are contained within the half - power radius . this number is inconsistent with the expectation of a uniform distribution within the same region , @xmath50 . on the other hand , a source population with a power - law distribution of flux densities @xmath51 in a flux - density limited sample has an expectation of @xmath52 within the half - power radius , in better agreement with the observed number . images were also constructed in a search for faint radiation from the transients and for new transients using all data from two - month , one - year , and one - decade time spans . we found two new transients in the two - month averages ( table [ tab : transientsmonths ] ) and no transients in the year and decade images . light curves for these sources are shown in figure [ fig : lct ] . given the smaller number of epochs , the expected number of false sources in the two - month average data is @xmath25 . further searches on different time scales or with filters matched to light curve shapes are possible with the same data but are beyond the scope of this paper . dual frequency observations were made only after 1989 and not at every epoch . of the remaining 5 ghz transients , only 1 ( rt 19920826 ) is located within the imaged region of the 8.4 ghz survey . unfortunately , there were no 8.4 ghz observations for this epoch . similarly , for rt 19970205 which was detected at 8.4 ghz , there was not a simultaneous 5 ghz observation . on 25 july 2006 ( ut dates are used throughout this paper ) , we observed the central portion of the radio field with the low - resolution imaging spectrometer ( lris ; * ? ? ? * ) on the keck - i 10-m telescope . eight of the radio transient positions ( see figure 6 ) were observed in two separate pointing centers ( each covered by the lris field - of - view of @xmath53 ) . at each pointing center we obtained six dithered 10-min exposures . the instrument is outfitted with a beam splitter , allowing us to observe simultaneously in the @xmath54 and @xmath55 bands . the data were reduced following standard optical imaging reduction procedures . after fitting a world coordinate system ( tied to the usno b1.0 ) to each reduced frame , we made a mosaic for a given filter using swarp @xcite . the final astrometric uncertainty relative to the international celestial reference system ( icrs ) is 250 mas in each axis . on the night of the keck / lris imaging , we observed the landolt standard star field pg1633 + 099 @xcite in both filters at an airmass of @xmath56 . the catalog magnitudes were converted to the @xmath54 band from the @xmath57-band magnitude and the @xmath58 color using the prescription in bilir et al . ( 2005 ) . from the observed photometry of four pg1633 + 099 stars , the zeropoint conversion from flux in both filters was determined in an @xmath59 aperture . the flux of @xmath47-detected sources in both the @xmath55 and @xmath54 stacked image of the radio field were determined in the same - sized aperture . accounting for the average extinction per unit airmass at mauna kea , we determine that the non - detections of some of the counterparts correspond to upper limits of @xmath60 and @xmath61 mag . these limits assume an unresolved source and a color similar to those of the landolt field stars . an optical spectrum of the counterpart of rt 19840613 was taken on 2005 dec . 4 with the deimos spectrograph @xcite mounted on the keck - ii telescope , using a 600 line mm@xmath12 grating and the gg400 order - blocking filter . spectra of the counterparts and nearby companions of rts 19860115 , 19870422 , 19920826 , and 19970528 were taken on 2006 june 26 with lris using the 600/4000 grism , the 400/8500 grating , and the d560 dichroic . all spectral data were reduced using standard techniques ( e.g. , * ? ? ? * ) . standard ccd processing and spectrum extraction were completed with iraf . the data were extracted with the optimal algorithm of @xcite . we obtained the wavelength scale from low - order polynomial fits to calibration - lamp spectra . small wavelength shifts were applied after cross - correlating night - sky lines extracted with the object to a template sky . using our own idl routines , we fit spectrophotometric standard star spectra to flux calibrate our data and remove telluric lines @xcite . we give redshifts for the counterparts and nearby companions in table [ tab : redshift ] . in the cases of rts 19840613 , 19870422 , 19920826 , and 19970528 , the redshift is for the galaxy that the rt is situated in or closest to . rt 19860115 has two possible hosts in the keck image ; objects 1 and 2 are located to the northeast and southwest of the rt , respectively . data were obtained with pairitel @xcite on 16 feb . images were taken simultaneously in the @xmath62 , @xmath63 , and @xmath64 bands in four different pointings with a total integration time of about 1120 s per pointing . additionally , to get a deep infrared ( ir ) image in one portion of the radio field , several hours of imaging in june and july 2006 were obtained centered at j2000 position @xmath65 = @xmath66 , @xmath67 = @xmath68 . the approximate @xmath47 upper limits in the shallow , wide - angle map are @xmath69 mag , @xmath70 mag , and @xmath71 mag ( the limits are significantly worse toward the edges of the mosaic ) . the pointings were stitched together using swarp after finding preliminary wcs solutions in all bands . the result is a @xmath72 pixel set of mosaic images with @xmath73 resolution that overlaps most of the radio image . an astrometric solution for the mosaic images was found through a cross - correlation with 43 stars in common with the usno b1.0 catalog . typical rms uncertainty of the astrometric tie to the icrs is 250 mas . we have detected a total of 8 transient radio sources in single epochs of this survey ( figure [ fig : lct ] ) . additionally , we have detected two transients in two - month averages of the data . our source identification methods described previously argue that no more than one of these sources is a false detection . all of these sources are adequately fit as point sources . none of the transients is detected in epochs other than the detection epoch , or in longer - term averages with the possible exception of rt 19840613 , in which the host galaxy is detected in the full 20-year image . in figures [ fig : deepradio ] and [ fig : keck ] we show images of the transients overlaid on the deep radio image , on multi - color images from the keck imaging , and , where keck data are missing for two objects , on @xmath64-band images from pairitel . we identify several hosts or potential hosts to the rts using the minnesota palomar plate survey ( maps , http://aps.umn.edu ; * ? ? ? maps is a catalog of sources found with and characterized by plate - measuring equipment surveying the palomar sky survey i ( poss i ) . maps coordinates are consistent with the radio reference frame with an accuracy of @xmath74 . we confirmed this by comparison of the position of two bright reference stars in the maps catalog with positions in the tycho catalog . referring the tycho positions to epoch 1950 ( appropriate for poss ) , we find agreement of @xmath75 for gsc 04562 - 00668 and @xmath3 for gsc 04562 - 00458 . one transient is of particular interest because it is coincident with a spiral galaxy at a redshift @xmath10 ( a distance of @xmath76 mpc using @xmath77 ) . the galaxy , identified as maps - p023 - 0189928 , has @xmath78 mag , @xmath79 mag , and a major axis of @xmath80 . the transient is clearly non - nuclear ; it is offset to the southwest of the optical nucleus by @xmath81 , corresponding to a projected distance of @xmath82 kpc . the optical nucleus is coincident with the infrared nucleus as identified in the 2mass catalog at an accuracy of @xmath83 . the probability of a single chance association with an optical source in the poss at a few arcsecond precision is @xmath84 . we calculate an isotropic radio luminosity of @xmath85 erg s@xmath12 hz@xmath12 , or @xmath86 erg s@xmath12 for a bandwidth of 10 ghz . the transient is not detected in any other individual epoch , even with reduced detection thresholds . the transient is also not detected in any two - month average in the year surrounding its detection , nor is it detected in any annual average from the entire data set . it is also not detected in the year prior to detection , nor in the six years following detection . the absence of a detection in other light curves at 5 ghz is consistent with a rapid rise and a decline in the flux density with a power law of @xmath87 . the deep image of the region surrounding the transient shows extended radio continuum emission associated with the spiral galaxy and a companion galaxy ( figure [ fig:840613deep ] ) . the location of the transient coincides with a region of peak emission , but it is not possible to conclude whether that emission is the result of the transient or a peak in the galactic emission . the peak of emission near the position of the transient is @xmath88jy , which corresponds to a star - formation rate of @xmath89 ( e.g. , * ? ? ? the association of the transient with the non - nuclear region of a spiral galaxy , together with its luminosity , suggest that the transient may be a rsn . the transient is very unlikely to be a type ii rsn , as these sources exhibit rise and decay timescales of months to years , implying that a type ii rsn would be detected multiple times in our survey @xcite . type ia rsn have never been detected at radio wavelengths @xcite . type ib / c rsn , on the other hand , evolve on timescales of days to months . the anomalous type ib / c rsn associated with sn 1998bw and grb 980425 doubled its flux density during its rise and halved its flux density following the peak in @xmath90 d @xcite . the light curve declined as @xmath91 with the exception of a brief period of increase attributed to clumpiness in the circumstellar medium . the luminosity of rt 19840613 places it among the brightest sne ib / c , although a factor of a few fainter than sn 1998bw @xcite . the luminosity of sn 2002ap , also a sn ib / c , is four orders of magnitude less than that of sn 1998bw @xcite . the luminosity of rt 19840613 is three orders of magnitude less than the typical radio luminosity of a grb afterglow . rt 19870422 is located within @xmath92 of the centroid of maps - p023 - 0189613 . the keck image shows that this rt is clearly associated with a blue galaxy and is significantly offset from the nucleus . the integrated @xmath55 magnitude of the galaxy is 20.2 , with @xmath93 mag , and the redshift is determined to be @xmath13 ( or a distance of 1050 mpc ) . galaxy - template matching to the keck spectrum shows it to most likely be an sc galaxy with strong [ ] and [ ] emission , indicating current star formation . the inferred isotropic radio luminosity is @xmath94 , or @xmath95 for a bandwidth of 10 ghz . rt 19870422 was detected in a two - month integration , implying a total energy release @xmath96 erg . we see no evidence for detection in other epochs , implying a source that fades with a timescale of two months or less following its peak . in the deep integration , we see that the galaxy is detected with an irregular morphology ( figure [ fig:870422deep ] ) . the faint radio emission , however , is offset from the position of rt 19870422 . rt 19870422 shares many characteristics with rt 19840613 . the rt position is clearly separated from the peak of the optical emission , ruling out the possibility of an agn event . its longer duration and higher luminosity , however , differentiate it . the luminosity is nearly an order of magnitude greater than that of sn 1998bw , and two orders of magnitude more luminous than the bright type ii sn 1979c @xcite . thus , the luminosity falls in between the maximum observed luminosity of a rsn and the typical luminosity of a grb afterglow . the two - month duration of rt 19870422 is more consistent with known rsne of both type ib / c or type ii , as well as grb afterglows . rt 19920826 is located @xmath37 to the southeast of a red galaxy identified in the keck image , which is not identified in the maps catalog . the host is a red object with no clear emission or absorption lines . rt 19970528 is located @xmath37 to the northwest of maps - p023 - 0189499 , which has @xmath97 mag and @xmath98 mag . two other galaxies are identified within a radius of @xmath99 . the nearest one is a red , early - type galaxy at @xmath100 . galaxy - template matching shows it is most likely an elliptical galaxy , but it is also well fit by an s0 spectrum . we computed the probability of false association of these galaxies with transients through monte carlo simulations . we randomly distributed 100,000 radio sources over the field and then measured the distance to the nearest galaxy as identified in the maps catalog . the probability of a false association at a distance of less than @xmath101 is only 2% . the probability of a false association at a distance of @xmath4 is @xmath102% . the probability for any one of the 10 rts being randomly associated with a galaxy at a distance of @xmath101 is @xmath103 , and at a distance of @xmath4 the probability is nearly unity . the probability of multiple sources located within @xmath101 is @xmath104 . thus , the association of either rt 19970528 or rt 19920826 or both rts with a galaxy is probable but not certain . there is no evidence for faint radio , optical , or infrared counterparts for six of our sources : rt 19840502 , 19860115 , 19860122 , 19970205 , 19990504 , 20010331 . there are no sources apparent in the image obtained by averaging the succeeding two months of data , as well as in the year - long and decade - long averages . these results require a rapidly decreasing light curve , proportional to @xmath91 or steeper . rt 19860115 has two possible hosts . the first possible host , object 1 , is located @xmath105 to the northeast of the rt and is a starburst galaxy at @xmath106 with strong emission lines and a blue continuum . object 1 is identified as maps - p023 - 0190130 , a galaxy with @xmath107 mag and @xmath108 mag , and is also detected as a faint , extended radio source . the other possible host , object 2 , is located @xmath80 to the southwest of the rt and is an early - type galaxy at @xmath109 . galaxy - template matching shows the spectrum is best fit by either an elliptical or s0 galaxy spectrum . several other objects , likely to be galaxies , are in the field . the physical separation of rt 19860115 from the two objects is @xmath45100 kpc , which is still within a radius at which the galaxy may contain stars . nevertheless , given the monte carlo simulations discussed above , we conclude that this transient is likely associated only by chance with these galaxies . for the remainder of sources , there are no obvious associations with galaxies or stars . the absence of optical or infrared counterparts only eliminates host candidates ; the non - simultaneity of radio and optical observations implies that we place no limits on optical transients . we can compute rates of transients in several different ways . we are limited in the accuracy of estimating transient rates by our lack of knowledge of the characteristic time scale , @xmath110 , of the transient sources . this characteristic time falls between the duration of the observations ( @xmath102 min ) and the typical separation between epochs ( 7 days ) . the two - epoch survey sensitivity is independent of @xmath110 . the effective area for a two - epoch survey for a given flux - density limit is just the sum of the areas in each epoch in which a source above the flux - density limit is detectable . for a set of @xmath111 images with uniform sensitivity over a solid angle @xmath112 in which @xmath113 transients are detected , the two - epoch source density is @xmath114 . for a sample with varying image properties , we can compute the two - epoch source density given the image noise statistics , the synthesized beam size , and the primary beam shape . the ratio of the primary beam solid angle to the synthesized beam solid angle gives the number of independent pixels in the image . the source detection threshold per image is then set based on the number of independent pixels and the rms noise . we combine the 5 ghz and 8.4 ghz survey limits and transient detections together . the accumulated results as a function of flux - density threshold are listed in table [ tab : areasearched ] . we use a non - parametric kaplan - meier method @xcite , for estimating the cumulative source - count distribution and then compute transient rates ( figure [ fig : snapshotrate ] ) . seven transients are detected above a flux density of 370 @xmath5jy , giving a two - epoch source density of @xmath115 . the transient rates are reasonably fit by a distribution with @xmath116 and @xmath117 , which corresponds to a non - evolving , volume - limited sample . however , the constraints on @xmath16 are not very strong . a larger @xmath16 appears to be required to achieve agreement between the rate determined from this survey and the nvss - first survey @xcite . different event rates at 1.4 and 5 ghz due to synchrotron self - absorption or other spectral index effects may also influence this comparison . additionally , the first - nvss survey may be incomplete due to confusion from the mismatched resolution of these surveys . we detected no transients in the 17 annual average images . the typical flux detection threshold at the half - power radius is 90 @xmath5jy . the total two - epoch area surveyed at this sensitivity is 0.3 deg@xmath118 , implying a @xmath119 upper limit to the two - epoch density for year - long transients at 90 @xmath5jy of @xmath120 . in 96 images composed of two - month averages , we found two transients . at a flux - density threshold of 200 @xmath5jy , the effective area surveyed is 1.9 deg@xmath118 , giving a @xmath119 upper limit to the two - month transient rate of @xmath121 . we can also determine a rate as a function of area and time . the precision of this rate is limited by our lack of measurement of the characteristic time of the transient events . the rate for sources brighter than 370 @xmath5jy is then @xmath122 . if we assume that rts 19840613 and 19870422 belong to a distinct transient class based on their association with blue galaxies , we can estimate a rate for such transients as a function of volume and time . we could detect this transient to a distance of @xmath123 mpc , which gives a total volume of @xmath42 mpc@xmath124 . assuming @xmath125 , we find a rate of @xmath126 . similarly for rt 19870422 , we find a maximum volume of @xmath127 mpc@xmath124 and use @xmath128 to find a rate of @xmath129 . jointly for rt 19840613 and rt 19870422 , the rate is @xmath130 . these rates are consistent within an order of magnitude of the sn ib / c rate determined optically , @xmath131 @xcite . there is some evidence that our vla field contains an overdensity of galaxies at @xmath132 ( table [ tab : redshift ] ) . additionally , the cluster abell 2047 is located within @xmath133 of the center of the vla field @xcite . a2047 is a low - density cluster ( richness class 0 ) with a radius @xmath134 and a distance class of 6 . the distance class is an estimate of distance based on visual inspection of photographic plates . the distance class of 6 is consistent with a redshift of 0.25 for the cluster @xcite . if all the rts are extragalactic , then the rt rates may be overestimated relative to the rate for the field by no more than a factor of a few . two of the transients are reasonably identified as rsne , possibly type ib / c or ii and similar to known rsne associated with grbs . we consider now whether the eight transients not clearly identified with nearby galaxies are examples of known or expected classes of radio transients . these transients are unlikely to be rsne unless they are associated with faint and/or low surface brightness galaxies . as we demonstrate below , the observed transient parameters do not easily fit into any expected class . the known and anticipated sources of radio transients are very broad . since we have no evidence for variability on a timescale of less than 20 min , we exclude from further discussion transient sources that have characteristic timescales of less than 1 s. these include pulsars and related phenomena such as rrats and giant pulses @xcite , as well as flares from extrasolar planets @xcite . information on the x - ray properties of the transients and transient hosts is very limited . the rosat catalog of bright sources does not identify any sources in the field of our survey . there are also no sources in the rxte all sky monitor catalog in the field , indicating an upper limit to the x - ray flux of events between 1997 and 2001 of @xmath26 mcrab @xcite . no observations of the field have been made with the chandra or xmm newton x - ray observatories . higher - energy counterparts to these transients have not been clearly detected . there are no known optically discovered sne in the field of view of our survey , nor have there been any optical variability campaigns in this field . there are no grbs which are uniquely localized to this field , although there are bursts with large positional uncertainties that could have originated in the field of view of our survey . we identify 7 batse events that occurred within @xmath135 of our field center between 1991 and 1999 . uncertainties in the batse positions , however , are @xmath136 , indicating a probability of @xmath137 for even one of these grbs to have occurred within our field of view . of the grbs , one event ( grb 990508 ) occurred 4 d after rt 19990504 . given the low probability of concurrence and the lack ( in other grbs ) of radio emission before gamma - ray emission , we conclude that this is almost certainly a coincidence . there is no systematic relationship between the remainder of the events and the radio transients . in the standard paradigm of grbs and grb afterglows , the gamma - ray emission is highly anisotropic while the afterglow emission is less anisotropic or even isotropic . thus , the observed grbs and their afterglows from earth are a subset of the total number of grbs . afterglows detected without gamma - ray emission are known as orphan gamma - ray burst afterglows . the ratio of ogrbas to grbs is the inverse of the beaming fraction , @xmath138 ( e.g. , * ? ? ? the phenomenon of a grb afterglow without a corresponding gamma - ray burst has not been conclusively detected . detailed models of ogrbas predict @xmath1391 per square degree at sensitivity thresholds of 0.1 mjy , which is consistent with our number of transient detections @xcite . whereas the canonical radio afterglow for the forward shock rises on a timescale of days and decays slowly , the majority of the radio transients found in the present survey occur on timescales less than 1 week . one possible explanation in the context of grb orphans is that the emission is due to the reverse shock , which is known , in the few cases where follow - up observations occurred soon after the grb , to produce a bright and rapid ( @xmath140 d ) radio signature ( * ? ? ? * ; * ? ? ? * e.g. , ) . strong scintillation ( * ? ? ? * e.g. , ) of a standard , faint afterglow could also be responsible for producing short - term amplification that leads to an apparent fast transient , but then we might see marginal evidence for the forward shock in the epochs following the transient yet this is not seen in individual sources . the single long - timescale transient ( rt 20010331 ) could be consistent with having arisen from the long - lived forward shock ( as it transitions from mildly relativistic to nonrelativistic ) . levinson et al . ( 2002 ) @xcite calculate the expected number of forward - shock dominated ogrbas for radio surveys . taking the canonical interstellar medium density of @xmath141 @xmath142 , it is reasonable to expect that we only find 1 viable forward - shock candidate with the nominal beaming of @xmath143 . nevertheless , since for the parameters of our survey we probe only to @xmath144 , the non - detection of an apparent host galaxy in deep keck imaging of rt 20010331 would seem to suggest that our basic model assumptions for the radio afterglow are suspect . thus , the transients that we have detected can be considered consistent with the expectations of ogrbas . there is , however , a large diversity of high - energy phenomena with poorly explored electromagnetic signatures beyond the classical grb afterglow picture , including x - ray flashes , x - ray rich grbs @xcite , grbs associated with sne ib / c @xcite , and short - duration grbs @xcite . the richness of this phenomenology suggests that the radio transients we have discovered could reflect the continuum of stellar collapse events . agns are highly variable and common . with the appropriate selection technique , agns can be found with a number density of @xmath145 per square degree @xcite . if our transients are agns , then the absence of deep , static detections indicates they are associated with radio - quiet objects , which constitute the majority of agns . short - timescale variability of agns is often associated with interstellar scintillation and extreme scattering events , and typically has amplitudes of @xmath146 @xcite . variations of a factor of a few on timescales of hours have been seen in the most extreme cases such as pks 0405 - 385 and j1819 + 3845 , which is not sufficient to account for the observed transients given the very large amplitude of variation observed @xcite . intrinsic variations among luminous agns are observed to have a timescale of months to years at radio frequencies @xcite . variations are rarely larger than a factor of a few in radio - loud objects . a survey of variable background sources in the direction of m31 found a small number of objects with variability of a factor of a few to 10 on timescales of years @xcite . an interesting example is the nucleus of the nearby spiral galaxy iii zw 2 , which has exhibited factor of 20 fluctuations in its radio luminosity on timescales of months to years . the tev blazar mrk 421 shows 50% variations on a timescale of @xmath26 d @xcite . sudden increases in the accretion rate , or shocks in the jet , could lead to a dramatic flare at radio wavelengths ; however , these flares are expected to have timescales on the order of months to a year , inconsistent with our observations . recently discovered evidence for tidal disruption events in agns observed in the ultraviolet indicate the possibility for large - amplitude agn transients @xcite . however , the timescale for the observed uv transients are months to a year rather than days . without a mechanism for a short timescale in the radio , tidal disruption events are unlikely to account for the observed radio transients . we can not exclude a galactic origin for these transients . a wide range of stars and stellar systems are known to flare on timescales of minutes to hours to days . however , no known stellar types closely fit the properties of these transients . our field ( @xmath147 , @xmath148 ) is out of the plane of the galaxy , away from the galactic center and bulge , and is not near any known star cluster or star - forming region . it is likely , therefore , that any stellar origin for these transients would be even more numerous in one of these more favored directions . the luminosity of the observed transients is @xmath149 . this is near the upper limit of the known stellar radio luminosity distribution . rs cvn binaries , fk com class stars , algol - class stars , and t tauri stars have been observed to radiate at this luminosity ( e.g. , * ? ? ? * ) . the total number of known rs cvn , fk com stars , and algol stars discovered at x - ray wavelengths is @xmath150 , indicating that we are unlikely to detect any such objects by chance , even if the distance cutoff in our survey is 10 kpc @xcite . t tauri stars are more common but are unlikely to be found outside regions of active star formation . m - type dwarfs at distances as large as 1 kpc have been identified as an important contribution to optical transient rates @xcite . due to the typical low radio luminosity ( @xmath151 @xmath152 ) of known dme stars and brown dwarfs , they are likely to be detected at distances of 30 pc or less . in an extreme case , the dme star ev lac was observed to undergo a flare of a factor of @xmath153 to a maximum luminosity @xmath154 , with a rise time of minutes and a decay time of hours @xcite . such a flare would be observed in our survey to a distance of 100 pc . the local space density of stars with @xmath155 is 0.035 pc@xmath156 , implying only a small chance of a low - mass star or brown dwarf within the volume to which we are sensitive @xcite . any stellar origin for these transients must come from a faint source , since we have no detection of stellar counterparts in keck , 2mass , or pairitel images . the 2mass magnitude limit of @xmath157 implies that any l dwarf ( @xmath158 mag ) or earlier - type star must be at a distance @xmath145 pc to remain undetected . pairitel limits require such stars to be at a distance @xmath159 pc . the limiting keck magnitude requires late - type m stars to be at a distance @xmath160 pc . the probability distribution of radio activity in low - mass stars is not well quantified . few stars have been observed at radio wavelengths for more than a few hours or a few days . despite the lack of long - term monitoring of dme stars , one can estimate the probability distribution of flares by comparison with the properties of the sun . the probability of a solar flare at centimeter wavelengths of a given flux density @xmath161 scales as @xmath162 @xcite . > from observations of a large sample of late - type stars @xcite , we estimate a probability of @xmath163 for a late - type star to exhibit a luminosity @xmath154 . if the probability of stellar flares scales with the same relation as solar flares , a flare of @xmath164 from a late - type star at a distance of 1 kpc would be visible in our radio survey but missing in the keck image , and it would occur with a probability of @xmath165 . assuming a constant density of stars set to the local value , the total number of m dwarfs out to 1 kpc within our field of view is @xmath166 . given our @xmath167 observations , the expected number of flares of this magnitude detected is @xmath168 . this is an upper limit since the statistics for the observed stars are biased by selection for activity at other wavelengths . within an order of magnitude , though , this matches the number of transients that we detect , suggesting that low - mass stars at 1 kpc or greater distance may contribute to , but are unlikely to dominate , the observed radio - transient rate . very large radio flares might be accompanied by luminous x - ray flares . if the transients follow the radio / x - ray correlation , then a radio flare of @xmath169 at a distance of 1 kpc would have an x - ray luminosity of @xmath170 , which exceeds the rxte asm detection threshold of @xmath171 . however , many sources do not closely follow this relationship , including many low - mass stars ( e.g. , * ? ? ? the transients share some properties with soft gamma - ray repeaters ( sgrs ) . sgrs are magnetars that exhibit dramatic flares at gamma - ray and radio wavelengths ( e.g. , * ? ? ? sgr 180620 produced a flare that reached a luminosity of @xmath172 , had a characteristic timescale of days , and declined as @xmath173 . such a flare would be detectable in our dataset at a distance of @xmath174 kpc , would appear in only a single epoch , would fade away rapidly so that it is not seen in long - term averages , and would have no lasting optical counterpart . if all of our transients are milky way sgrs to a distance of @xmath175 kpc , the implied rate is @xmath176 at high galactic latitude . without making any correction for galactic population distribution , we estimate a total rate of @xmath177 in the disk of the galaxy , which is at least an order of magnitude too high relative to the number of discovered sgrs . x - ray binaries ( xrbs ) can be seen throughout the galaxy ( e.g. , * ? ? ? * ) but are rare @xcite . for the extremely overdense galactic center , the xrb detection rate in the radio is @xmath178 . the comparable observed transient rate would imply a similar overdensity at high galactic latitude , which is implausible . most pulsar variability occurs on time scales much shorter than our observations and is therefore unlikely to account for our observed transients . there have been some examples , however , of longer timescale variability . psr b1931 + 24 , for instance , has periods of activity and inactivity with characteristic times of days to tens of days @xcite . this variability is quasi - periodic , however , indicating that we would detect it and similar pulsars in multiple epochs throughout our survey . we have already referred to the phenomenon of intra - day variability ( idv ) , which in the most extreme cases leads to variations of a factors of a few on a timescale of hours . idv of this scale has been demonstrated to originate from interstellar scintillation of compact structure in the source . identified a factor of 43 fluctuation in the flux density of psr b0655 + 64 at 325 mhz , which they argue originates from strong focusing in the interstellar medium that is , a caustic . it is difficult to estimate the frequency with which such a phenomenon can occur , but observational evidence is scarce . microlensing could also produce a transient amplification of a background source . a simple calculation estimates that to explain all 10 rts as the result of microlensing requires the product @xmath179 , where @xmath180 is the number of sources that can be lensed and @xmath181 is the lensing opacity . since @xmath181 has been measured to be @xmath182 by macho and ogle experiments , then @xmath183 , which is absurdly large for any known population of galactic or extragalactic radio sources . finally , reflected solar flares off objects in the solar system could produce detectable radio flares . a 1 mjy solar flare reflecting off a 1000 km object at a distance of 1 au from the earth would produce a 0.1 mjy flare at earth . our observed field is far out of the ecliptic plane , however , and there are few known solar - system objects of this size outside of the ecliptic . nearer objects could be smaller but would move significantly during the course of the observation . we have conducted a 944-epoch , 20-yr - long survey for radio transients with archival 5 and 8.4 ghz very large array data . ten radio transients are apparent in this data set . eight transients appear in only a single epoch and disappear completely thereafter . two transients appear in two - month integrations and also never reappear . we estimate the rate of radio transients below 1 mjy with a large uncertainty due to the wide range of characteristic times possible for these sources . two of the transients may be peculiar type ib / c and/or type ii radio sne in nearby galaxies . all of these transients share some characteristics with those expected of orphan gamma - ray burst afterglows , but they can not be conclusively identified as such . the diversity of high - energy stellar - death phenomena makes it difficult to place specific limits on the nature of grbs based on these results . the absence of persistent counterparts to these sources indicate that they are unlikely to be agns . galactic sources appear unlikely to be an explanation , but they certainly can not be ruled out . in particular , late - type stars could contribute to the number of radio transients if their luminosity distribution can be extrapolated by several orders of magnitude beyond the most luminous observed m - dwarf flare . deeper multi - wavelength imaging is critical for identifying the hosts of these events . in the near future , the allen telescope array @xcite will be efficient at discovering transients of this sort . the wide field of view of this telescope makes it a powerful instrument for large solid angle radio surveys . given the apparently short timescale of these phenomena , rapid response to transient discovery at radio and other wavelengths will be critical for determining the nature of these transients . these observations made use of archival data from the very large array . the national radio astronomy observatory is a facility of the national science foundation ( nsf ) operated under cooperative agreement by associated universities , inc . this research utilized the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration ( nasa ) . it also used the simbad database , operated at cds , strasbourg , france , as well as data from the university of michigan radio astronomy observatory , which has been supported by the university of michigan and the nsf . some of the data presented here were obtained at the w. m. keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california , and the national aeronautics and space administration . the observatory was made possible by the generous financial support of the w. m. keck foundation . we thank the keck staff for their assistance . the peters automated infrared imaging telescope ( pairitel ) is operated by the smithsonian astrophysical observatory ( sao ) and was made possible by a grant 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, h. , briel , u. , burkert , w. , dennerl , k. , englhauser , j. , gruber , r. , haberl , f. , hartner , g. , hasinger , g. , krster , m. , pfeffermann , e. , pietsch , w. , predehl , p. , rosso , c. , schmitt , j. h. m. m. , trmper , j. , & zimmermann , h. u. 1999 , , 349 , 389 rrrrr + 15 00 11.48 @xmath184 0.10 & 78 12 42.26 @xmath184 00.13 & 26 & 2380 @xmath184 73 & 502 @xmath184 17 + 15 00 44.87 @xmath184 0.10 & 78 18 39.04 @xmath184 00.32 & 4 & 874 @xmath184 36 & 180 @xmath184 7 + 15 01 11.70 @xmath184 0.10 & 78 15 20.23 @xmath184 00.10 & 141 & 504 @xmath184 6 & 307 @xmath184 4 + 15 01 16.19 @xmath184 0.10 & 78 12 45.88 @xmath184 00.10 & 363 & 1146 @xmath184 14 & 741 @xmath184 5 + 15 01 22.67 @xmath184 0.10 & 78 18 05.66 @xmath184 00.10 & 452 & 880 @xmath184 9 & 634 @xmath184 3 + 15 02 51.40 @xmath184 0.10 & 78 18 12.46 @xmath184 00.10 & 23 & 351 @xmath184 19 & 155 @xmath184 3 + 15 03 24.83 @xmath184 0.14 & 78 17 37.62 @xmath184 00.46 & 1 & 318 @xmath184 54 & 166 @xmath184 4 + 15 03 28.07 @xmath184 0.30 & 78 09 21.50 @xmath184 01.04 & 1 & 2045 @xmath184 301 & @xmath185 + + 15 01 11.13 @xmath184 0.12 & 78 15 22.45 @xmath184 00.32 & 1 & 1081 @xmath184 154 & 291 @xmath184 9 + 15 01 22.58 @xmath184 0.10 & 78 18 05.80 @xmath184 00.10 & 159 & 1099 @xmath184 36 & 794 @xmath184 8 + 15 02 51.38 @xmath184 0.10 & 78 18 13.14 @xmath184 00.18 & 3 & 518 @xmath184 31 & 225 @xmath184 4 + 15 03 23.41 @xmath184 0.41 & 78 17 35.74 @xmath184 00.64 & 1 & 840 @xmath184 157 & 212 @xmath184 8 + rrrrrrrrr + 19840502 & 15 02 24.61 @xmath184 0.21 & 78 16 10.08 @xmath184 00.46 & 448 @xmath184 74 & @xmath186 & 7 & @xmath187 & 4.5 & x + 19840613 & 15 01 38.07 @xmath184 0.21 & 78 18 40.75 @xmath184 00.39 & 566 @xmath184 81 & @xmath188 & 7 & @xmath189 & 7.5 & g + 19860115 & 15 02 26.40 @xmath184 0.44 & 78 17 32.39 @xmath184 01.59 & 370 @xmath184 67 & @xmath186 & 7 & @xmath190 & 3.6 & x + 19860122 & 15 00 50.15 @xmath184 0.34 & 78 15 39.37 @xmath184 01.38 & 1586 @xmath184 248 & @xmath191 & 7 & @xmath192 & 6.4 & x + 19920826 & 15 02 59.89 @xmath184 0.35 & 78 16 10.82 @xmath184 02.44 & 642 @xmath184 101 & @xmath193 & 56 & @xmath194 & 6.3 & ? + 19970528 & 15 00 23.55 @xmath184 0.10 & 78 13 01.37 @xmath184 00.17 & 1731 @xmath184 232 & @xmath195 & 7 & @xmath196 & 7.9 & ? + 19990504 & 14 59 46.42 @xmath184 0.56 & 78 20 29.03 @xmath184 00.74 & 7042 @xmath184 963 & @xmath197 & 21 & @xmath198 & 9.1 & x + + 19970205 & 15 01 29.35 @xmath184 0.10 & 78 19 49.20 @xmath184 00.10 & 2234 @xmath184 288 & @xmath199 & 5 & @xmath200 & 8.0 & x + rrrrrrrrr + 19870422 & 15 00 50.01 @xmath184 0.35 & 78 09 45.49 @xmath184 0.97 & @xmath201 & @xmath202 & 96 & @xmath203 & 3.0 & g + + 20010331 & 15 03 46.18 @xmath184 0.10 & 78 15 41.68 @xmath184 00.11 & 697 @xmath184 94 & @xmath204 & 59 & 85 @xmath184 85 & 3.4 & x + lrrrrr 19840613 & p023 - 0189928 & 2.7@xmath205 nw & 16.5 & 1.8 & 0.040 + 19860115 - 1 & p023 - 0190130 & 15@xmath205 ne & 19.2 & 1.3 & 0.130 + 19860115 - 2 & & 20@xmath205 sw & & & 0.242 + 19870422 & p023 - 0189613 & 1.5@xmath205 se & 20.2 & 2.5 & 0.249 + 19920826 & & 5@xmath205 nw & & & unknown + 19970528 & p023 - 0189499 & 5@xmath205 se & 19.6 & 2.4 & 0.245 + rrrrrrr 70 & 9 & 0.16 & 0 & 0.0 & 9 & 0.16 + 100 & 10 & 0.24 & 0 & 0.0 & 10 & 0.24 + 140 & 12 & 0.34 & 0 & 0.0 & 12 & 0.34 + 200 & 35 & 0.50 & 0 & 0.0 & 35 & 0.50 + 280 & 201 & 1.42 & 442 & 1.45 & 435 & 2.19 + 400 & 464 & 4.57 & 560 & 3.08 & 761 & 6.20 + 560 & 561 & 9.14 & 584 & 4.79 & 873 & 11.68 + 800 & 596 & 14.56 & 591 & 6.64 & 910 & 18.09 + 1120 & 606 & 19.85 & 591 & 8.40 & 924 & 24.31 + 1600 & 607 & 25.51 & 598 & 10.28 & 925 & 30.97 + rrrrrrr 70 & 24 & 0.10 & 9 & 0.02 & 26 & 0.1 + 100 & 41 & 0.49 & 38 & 0.22 & 54 & 0.56 + 140 & 55 & 0.91 & 61 & 0.67 & 73 & 1.11 + 200 & 75 & 1.51 & 74 & 1.31 & 99 & 1.91 + 280 & 90 & 2.24 & 78 & 1.97 & 115 & 2.84 + 400 & 95 & 3.11 & 78 & 2.69 & 120 & 3.94 + 560 & 95 & 3.93 & 78 & 3.37 & 120 & 4.97 + 800 & 95 & 4.81 & 78 & 4.09 & 120 & 6.07 + 1120 & 95 & 5.64 & 78 & 4.77 & 120 & 7.11 + 1600 & 95 & 6.52 & 78 & 5.48 & 120 & 8.22 + rrrrrrr 50 & 5 & 0.02 & 10 & 0.02 & 15 & 0.04 + 70 & 13 & 0.09 & 14 & 0.05 & 15 & 0.10 + 100 & 17 & 0.24 & 15 & 0.08 & 20 & 0.26 + 140 & 17 & 0.38 & 15 & 0.14 & 20 & 0.42 + 200 & 17 & 0.53 & 16 & 0.18 & 21 & 0.57 + 280 & 17 & 0.67 & 16 & 0.23 & 21 & 0.73 + 400 & 17 & 0.82 & 16 & 0.27 & 21 & 0.89 + 560 & 17 & 0.96 & 16 & 0.32 & 21 & 1.14 + 800 & 17 & 1.09 & 16 & 0.35 & 21 & 1.19 + 1120 & 17 & 1.15 & 16 & 0.36 & 21 & 1.24 + 1600 & 17 & 1.16 & 16 & 0.36 & 21 & 1.25 +

we report the results of a 944-epoch survey for transient sources with archival data from the very large array spanning 22 years with a typical epoch separation of 7 days . observations were obtained at 5 or 8.4 ghz for a single field of view with a full - width at half - maximum of @xmath0 and @xmath1 , respectively , and achieved a typical point - source detection threshold at the beam center of @xmath2jy per epoch . the angular resolution ranged from @xmath3 to @xmath4 . ten transient sources were detected with a significance threshold such that only one false positive would be expected . of these transients , eight were detected in only a single epoch . two transients were too faint to be detected in individual epochs but were detected in two - month averages . none of the ten transients was detected in longer - term averages or associated with persistent emission in the deep image produced from the combination of all epochs . the cumulative rate for the short timescale radio transients above 370 @xmath5jy at 5 and 8.4 ghz is @xmath6 , where the uncertainty is due to the unknown duration of the transients , @xmath7 . a two - epoch survey for transients will detect @xmath8 transient per square degrees above a flux density of 370 @xmath5jy . one transient is located @xmath9 kpc in projection from the nucleus of a spiral galaxy at a redshift @xmath10 . based on the duration of the transient , its luminosity ( @xmath11 erg s@xmath12 ) , and its location in the galaxy , we suggest that it may be similar to the peculiar type ib / c radio supernova sn 1998bw associated with grb 980428 . the implied type ib / c rate is consistent to an order of magnitude with rates determined optically . a second transient is associated with a blue galaxy at @xmath13 and may also be a luminous radio supernova or gamma - ray burst afterglow . two other transient sources are possibly associated with the outer parts of faint , optically detected galaxies . the remaining six transients have no counterparts in the optical or infrared to limiting magnitudes of @xmath14 and @xmath15 and show no faint persistent radio flux . for the eight transients without clear associations , known source classes of radio transients including radio supernovae , gamma - ray burst afterglows , active galactic nuclei , and stellar flares do not provide natural fits to the observed parameters . the hosts and progenitors of these transients are unknown .

this study focuses on the possibility of creating and studying vortices in a two - component spin - orbit coupled bose - einstein condensate ( bec ) of ultracold atoms . the principal motivation is the theoretical proposal by spielman @xcite , and the subsequent two experimental papers by the nist group : creation of vortices without rotation @xcite and the study of spin - orbit structure in the hamiltonian @xcite . these experiments are modeled by a bec in a thin essentially two - dimensional harmonic trap with tight confinement in the perpendicular direction . for a one - component condensate , many studies have shown the creation of vortices with a single unit of circulation , mostly by stirring the bec to induce rotation @xcite . in a few cases , experiments have been able to study the real - time vortex dynamics for up to 1 s , typically a uniform precession @xcite . the generalization of these ideas to a two - component spin - orbit coupled bec involves the concept of synthetic gauge fields , and sec . [ syn ] contains an introduction to the nist spin - orbit hamiltonian . here the principal emphasis is on a bec in a trap provided by one or more focused laser beams . for completeness , this section also mentions the recent extensive studies of synthetic gauge fields in optical lattices ( for both bosons and fermions ) , but they are not directly relevant to the present study of two - component vortices . section [ vortex1 ] summarizes the experimental procedure @xcite that used a sudden thermal quench from the normal cold atomic gas above the bec transition temperature deep into the one - component bec . roughly 25% of the time , the quench created a singly quantized vortex ( with random orientation ) , and the experiment was able to take 6 - 8 time - lapse images of the dynamical precession over nearly 1 s. as discussed in sec . [ syn ] , the nist procedure to create a spin - orbit coupled bec uses a focused red - detuned laser beam to trap the cold atoms , a magnetic field along @xmath1 to separate the @xmath2 hyperfine levels , and a pair of opposing external raman laser beams , typically aligned along @xmath3 . these external laser beams and the magnetic field are fixed in the laboratory , which makes it difficult @xcite to use the usual experimental approach of rotating the condensate to nucleate vortices in a one - component bec ( in a rotating frame , these external beams and fields become time dependent ) . in contrast , the thermal - quench method @xcite should also work well to nucleate vortices in a two - component bec because no rotation is required . section [ theory ] discusses my theoretical analysis that relies on the time - dependent variational lagrangian formalism @xcite . the two - dimensional vortex position @xmath4 serves as a time - dependent variational parameter , and the resulting dynamical lagrangian equations show that the vortex moves on a contour of constant energy . for a one - component vortex , this picture yields a uniform circular precession . in contrast , a two - component vortex offers a wider set of structures . each component can have its own circulation quantum number @xmath5 and @xmath6 . if both have the same value ( say @xmath7 ) , then the precession should remain uniform . other possibilities exist , however , and a thermal quench should sometimes create a half - quantum vortex with ( say ) @xmath8 and @xmath9 , which has been predicted and observed in exciton - polariton condensates @xcite . my theoretical analysis suggests that a half - quantum vortex in a spin - orbit coupled condensate would have topologically distinct orbits , some of which would remain in the condensate , and others that would move to the edge of the condensate and then disappear . [ sec:1 ] synthetic gauge fields and spin - orbit coupling have generated great excitement over the past decade especially with the initial realization by the nist group @xcite for a cold atomic gas of @xmath10rb . this atom has a nuclear spin @xmath11 and the single valence electron has a spin @xmath12 . thus the vector sum @xmath13 has two values @xmath14 and @xmath15 . the lower manifold @xmath2 has three substates @xmath16 of these , only the state @xmath17 is confined in a typical magnetic trap . in practice , creating the effective spin - orbit coupling requires an applied magnetic field to split the @xmath2 manifold . hence it is necessary to use an optical laser dipole trap ( instead of a magnetic trap ) to confine the cold atoms . for a given electric dipole moment @xmath18 in an external electric field @xmath19 , the energy is @xmath20 . a single neutral atom has an ac polarizability @xmath21 , which yields an induced dipole moment @xmath22 . if the electric field is turned on adiabatically , the resulting energy is @xmath23 ( also known as the ac stark effect ) . for low frequency ( red - detuned " ) laser light , the polarizability is positive . hence the atoms are drawn to regions of large @xmath24 . in this case , a focused infrared laser beam will trap the atoms at the narrow waist , where @xmath24 is largest . as an introduction to the idea of synthetic gauge fields , recall the transformation to a frame rotating uniformly with an angular velocity @xmath25 . in this case , a simple analysis relates the hamiltonian in the rotating frame @xmath26 to @xmath27 in the laboratory ( stationary ) frame @xcite : @xmath28 , where @xmath29 is the angular momentum . rewrite the last term as @xmath30 . combine with free - particle kinetic energy to obtain @xmath31 where last term is a ( negative ) centrifugal potential that opposes any applied trapping potential @xmath32 . i can now interpret the term @xmath33 as an effective gauge potential @xmath34 since it appears in the familiar combination @xmath35 . more generally , whenever the hamiltonian contains a term linear in @xmath36 , the coefficient can be taken to define an effective ( or synthetic ) vector potential @xmath34 . in classical physics , a particle with charge @xmath37 in a magnetic field @xmath38 experiences a lorentz force @xmath39 . for a quantum system , however , the focus is on the vector potential @xmath40 , where @xmath41 . in particular , when a charged particle moves from @xmath42 to @xmath43 along a path @xmath44 , its wave function acquires a phase @xmath45 if it is possible to create such a phase , by whatever means , even a neutral particle can experience a `` synthetic '' gauge field . hence the new perspective is on `` phase engineering '' of the quantum state . spielman at nist has demonstrated synthetic gauge - field effects for trapped neutral atoms of @xmath10rb @xcite . more generally , refs . @xcite review this exciting and rapidly developing subject . recently , various experimental groups have created synthetic gauge fields in optical lattices created by standing waves of interfering laser beams . in the tight - binding model for the lowest band , synthetic gauge fields appear as complex phases associated with the hopping parameters in the single - particle hamiltonian . these effects can arise by shaking an optical lattice @xcite with special time - dependent driving forces . this and related methods have created uniform synthetic flux in a two - dimensional optical lattice with @xmath46 flux quantum per lattice plaquette @xcite , a realization of the topological haldane model @xcite in a distorted two - dimensional hexagonal optical lattice , and chiral edge states in hall ribbons using synthetic dimensions " @xcite . the creation of such synthetic gauge fields for atomic gases usually relies on strong laser fields that couple two or more atomic states . typically , this coupling yields `` dressed '' eigenstates @xmath47 , where the spatial dependence is crucial . when a particle moves adiabatically from @xmath42 to @xmath43 , this spatial dependence yields a berry s phase @xmath48 where @xmath49 is the synthetic gauge field . here , we deal with neutral atoms , and it is convenient to take the effective charge as 1 , so that @xmath50 has the dimension of momentum . a typical normalized wave function has the form @xmath51 \\[.1 cm ] e^{i\eta({\bf r})}\sin[\chi({\bf r})/2 ] \end{pmatrix}\ ] ] where both @xmath52 and @xmath53 depend on the spatial coordinate @xmath54 . the resulting vector potential is @xmath55 so that the state vector must have a spatially dependent phase @xmath53 . correspondingly , the induced synthetic magnetic field is @xmath56 . the essential conclusion is that we need ( 1 ) both @xmath53 and @xmath52 to have spatial dependence , and ( 2 ) their gradients must point in different directions . in practice , the nist group @xcite take @xmath57 and @xmath58 , leading to a synthetic `` landau '' gauge , with @xmath59 and a nearly uniform @xmath60 over a restricted region . the original paper @xcite proposed a technique to create vortices in a non - rotating condensate , where the relevant angular momentum comes from the synthetic electromagnetic field . the experiment @xcite produced remarkable images of vortices shown as dark regions where the cores have reduced density . how does this example @xcite work in detail ? the discussion will lead to the important idea of spin - orbit coupling in an ultracold dilute atomic gas . a bose - einstein condensate is trapped in a red - detuned ( typically infrared ) focused laser . apply two counter - propagating raman laser beams along @xmath3 , with @xmath61 and @xmath62 , taking @xmath63 ( see fig . [ figraman ] ) . in a raman transition , an atom absorbs a photon from one beam and emits a photon into the other beam , while making a transition between two different internal atomic states . the momentum transfer to the atom is @xmath64 because of the recoil . acoustic - optical modulators control the corresponding frequency transfer @xmath65 . and emits another with @xmath66 from opposite raman laser beams , thus acquiring a momentum @xmath67 . in addition , a zeeman magnetic field along @xmath68 splits the @xmath2 manifold into substates , and our model focuses on only the two lower ones . adapted from @xcite with permission.,width=432 ] in addition to the raman laser beams along @xmath69 , apply a zeeman magnetic field along @xmath68 , splitting the three @xmath70 states for the @xmath2 manifold . it is possible to isolate the two states @xmath71 and @xmath72 , which provides a convenient two - component basis . one can think of this pair as a pseudospin-@xmath73 . in this two - component basis , the raman beams act to couple the two pseudospin states . varying the applied magnetic field induces a detuning @xmath74 from the raman resonance . in the present context , the most important effect of the raman lasers is to induce an off - diagonal rabi coupling between the two states @xmath75 and @xmath76 . the relevant matrix element involves the electric dipole energy @xmath77 , and the total electric field @xmath78 has the spatial dependence @xmath79 , leading to the matrix element @xmath80 here the quantity @xmath81 is the rabi frequency ; it is fixed by the strength of the applied raman laser beams . note that i also neglect the effect of the trap potential and the gaussian curvature of the trapping laser . in the two - component basis , the single - particle hamiltonian becomes @xmath82 \hbar\omega e^{-2ik_0 x } & ( p^2/m ) - \hbar\delta \end{pmatrix},\ ] ] where i again omit the trap potential . the presence of spatially varying off - diagonal elements complicates the problem , but this spatial dependence can be removed with a unitary transformation @xcite @xmath83 0 & e^{-ik_0x } \end{pmatrix}.\ ] ] a simple analysis yields a new single - particle hamiltonian @xmath84 that now has a spin - orbit structure@xmath85 & = & \frac{1}{2 } \begin{pmatrix}(\hbar^2/m ) ( -i\bm \nabla + k_0\hat x)^2 + \hbar\delta & \hbar\omega\\[.2 cm ] \hbar\omega & ( \hbar^2/m ) ( -i\bm \nabla -k_0\hat x)^2 - \hbar\delta \end{pmatrix}\\[.2 cm ] & = & { \textstyle\frac{1}{2 } } \left[(\hbar^2/m)(-i\bm\nabla { \cal i } + k_0\hat x \sigma^z)^2 + \hbar\delta\sigma^z + \hbar\omega\sigma^x\right],\label{so}\end{aligned}\ ] ] where @xmath86 denotes the @xmath87 unit matrix and @xmath88 are the usual pauli matrices . in effect , the raman beams shift the minima of the two pseudospin dispersion relations to new and different local minima at @xmath89 . these shifted minima represent the vector gauge fields @xmath90 , with @xmath91 . in addition , the rabi coupling induces the off - diagonal term @xmath92 . to understand the new physics , continue to ignore the nonuniform trap potential so that @xmath84 has one - dimensional plane - wave solutions @xmath93 . use @xmath94as the unit of length and the recoil energy @xmath95 as the unit of energy , leading to the dimensionless spin - orbit coupled single - particle hamiltonian @xmath96 \omega/2 & ( k-1)^2 -\delta/2 \end{pmatrix}.\ ] ] the associated eigenvalues follow immediately @xmath97 and i here focus on the lower band @xmath98 . two cases are of special interest , and the first is the behavior for large rabi frequency ( @xmath99 ) . an expansion of @xmath98 in powers of @xmath100 yields @xmath101 the first two terms are simply an overall downward energy shift , and the factor in front of the quadratic term is an effective mass . note that the minimum in the dispersion relation is shifted from the usual position @xmath102 to the new position @xmath103 , identifying the @xmath104 component of the synthetic vector potential as @xmath105 this is a central result of spielman s analysis @xcite , namely the synthetic gauge field varies linearly with the detuning @xmath106 . if @xmath107 is constant , then there is no synthetic magnetic field because @xmath108 would be constant . to obtain a useful synthetic field , the experiment @xcite uses a magnetic field gradient along @xmath68 , so that @xmath109 , with @xmath110 ( a constant proportional to the field gradient ) as a control parameter . in this way , the synthetic vector potential has the form of landau gauge @xmath111 , familiar from the quantum description of an electron in a uniform magnetic field . its curl yields an effectively uniform synthetic magnetic field along @xmath112 proportional to the control parameter @xmath110 . in this case , the neutral atoms experience an effective lorentz force completely analogous to the real coriolis force observed with a rotating condensate @xcite . note that we are here effectively already in the rotating frame , so that the vortices are at rest in the laboratory frame . reference @xcite used this approach to create vortices in a nonrotating condensate , as seen in fig . [ vortices ] . in landau gauge creates vortices for sufficiently large @xmath110 . adapted from @xcite with permission.,width=432 ] note that these shapes become progressively more distorted with increasing control parameter @xmath110 , which provides additional insight into the idea of a synthetic gauge field @xcite . here , we have @xmath113 , where @xmath114 is the synthetic uniform magnetic field . to record these images , the trap and ( real ) magnetic fields are suddenly turned off , which generates a synthetic electric field @xmath115 ( ignoring any scalar potential ) . the resulting pulsed electric field produces an impulsive shear , as seen in fig . [ vortices ] . the same hamiltonian in eq . ( [ so ] ) exhibits rather different behavior for small values of the rabi frequency @xmath116 ( now in usual units ) , which emphasizes the spin - orbit character of the interaction . the cross term in the kinetic energy is linear in the momentum and has the form @xmath117 , which exhibits the matrix synthetic gauge field @xmath118 note that this interaction is somewhat different from that familiar in atomic physics , which is proportional to @xmath119 . it arises from similar structure in semiconductor physics , known variously as rashba or dresselhaus coupling . since @xmath50 here has only one component , there is no question of non - abelian gauge fields , for that requires two or more noncommuting components of @xmath50 . i now focus on the situation of zero detuning @xmath120 and small dimensionless rabi coupling , in which case the dimensionless eq . ( [ eigen ] ) reduces to @xmath121 if the rabi coupling vanishes ( @xmath122 ) , this expression reduces to two shifted parabolas @xmath123 that intersect at @xmath102 . for finite @xmath81 , however , an avoided crossing splits the dispersion curves into an upper and a lower band . reference @xcite mapped out this behavior in their study of spin - orbit coupling in cold @xmath10rb atoms , as shown in fig . [ eso ] . . the gray intersecting parabolas are for @xmath122 , and successive colored curves show the behavior with increasing @xmath81 . adapted from @xcite with permission.,width=336 ] note the increasing splitting of the two bands with increasing @xmath81 . the two minima in the lower band become shallower and move closer together with increasing @xmath81 . for @xmath124 , the two local minima are at @xmath125 , whereas for @xmath126 , there is only a single minimum at @xmath102 . figure [ min ] verifies these results in great experimental detail @xcite . as a function of the rabi frequency @xmath81 . adapted from @xcite with permission.,title="fig:",width=288 ] + this nist scheme for creating synthetic gauge fields and spin - orbit coupling singles out @xmath127 as a preferred drection . there are many proposals for more symmetric spin - orbit coupling , and a typical case is pure rashba coupling with the hamiltonian @xmath128 where @xmath129 is a coupling constant with the dimension of wave number . this form has two components of synthetic vector potential @xmath130 and @xmath131 . note that @xmath132\neq 0 $ ] because the two pauli matrices do not commute . in such a situation , these gauge fields are generally known as non - abelian , and many unusual and intriguing properties can arise @xcite . the lower band of the eigenvalue spectrum of @xmath133 in eq . ( [ rashba ] ) [ including the free - particle term @xmath134 has a minimum on a circle of radius @xmath135 , with a shape like the mexican - hat potential . this special form has a linear crossing with a dirac cone between the upper and lower bands , like an axisymmetric version of the gray curves in fig . [ eso ] . in practice , the rashba hamiltonian can contain additional control terms , such as a detuning @xmath136 , which plays a role analogous to mass in the dirac theory and splits the upper and lower axisymmetric bands . despite great experimental efforts , no such rashba coupling has yet been achieved . for reasons that will become clear , it is valuable to review a recent experiment @xcite that created vortices in a nonrotating condensate and took nondestructive time - lapse images of the subsequent vortex dynamics . a parabolic magnetic trap confined @xmath10rb atoms in the particular hyperfine state @xmath137 . the experiment started in the normal state and quenched rapidly into the superfluid state to low temperature @xmath138 , with no measurable thermal cloud . roughly 25% of the time , they found a vortex in the condensate with random ( @xmath139 ) orientation in a disk - shaped condensate . they applied a short microwave pulse that transferred @xmath140 of the atoms to the untrapped state @xmath141 . these atoms fall under gravity and expand , allowing a direct image . because these atoms were part of the original bec , they provide a faithful small copy of the whole condensate , including the image of the vortex core . the experiment could repeat this process 6 - 8 times at intervals of @xmath142 ms , allowing a real - time study of the vortex dynamics . figure [ frei ] shows a dramatic set of images of a precessing vortex , with the upper row showing raw data and the second row the smoothed set of images . the lower part shows the fit to a uniform precession over approximately two full cycles . in addition to finding single vortices , they occasionally ( @xmath143 a few % of the quenches ) found a vortex pair ( often called a vortex dipole ) , which is a + vortex and a - vortex close together . for both the single vortex and the vortex pair , they found good agreement between the observed motion and that predicted with the gross - pitaevskii theory . why is this particular experiment relevant for vortex dynamics in spin - orbit coupled condensates ? a few years ago , radi _ et al . _ @xcite discussed vortices in such spin - orbit coupled systems and pointed out that it would be highly challenging to rotate not only the condensate but also the raman laser beams and the magnetic field . at present , there have been no reports of such rotation experiments , and the thermal quench of ref . @xcite in principle provides a simple way to study vortices in such a nonrotating spin - orbit condensate . if this approach can indeed be implemented , it would also provide a detailed description of the associated vortex dynamics @xcite . i rely on the time - dependent variational lagrangian formalism that has proved valuable in studying the dynamics of vortices in trapped bose - einstein condensates @xcite . note that this variational analysis specifically includes both the confining harmonic trap energy and the interaction energy , based on the thomas - fermi approximation . consider first a one - component cold atomic gas that is tightly confined in the @xmath144 direction . in this case , it will form an effectively two - dimensional bose - einstein condensate with a condensate wave function @xmath145 . the lagrangian here is given by @xmath146 where @xmath147 and @xmath148 is the familiar gross - pitaevskii energy functional @xmath149.\ ] ] here , the three terms are the kinetic energy , the confinement energy of the trap , and the interaction energy with effective two - dimensional coupling constant @xmath150 . variation of the lagrangian @xmath151 with respect to @xmath152 readily yields the exact time - dependent gross - pitaevskii ( gp ) equation . as usual with any variational principle , the current @xmath151 also provides a valuable basis for a variational approximation . the strategy is to assume a trial wave function @xmath153 with the two - dimensional position of the vortex @xmath4 as the time - dependent variational parameter . in particular , i assume a normalized trial function @xmath154 where the first factor ensures the normalization @xmath155 , and the second yields the thomas - fermi shape for the condensate density in a two - dimensional harmonic trap with condensate radius @xmath156 . the position of the vortex @xmath157 appears in the phase @xmath158 for the gp energy ( [ egp ] ) , @xmath4 affects only the kinetic energy , through the quantity @xmath159 which is essentially the induced flow velocity around the vortex core at @xmath4 . it is convenient to use plane polar coordinates @xmath160 . a detailed calculation yields the dimensionless energy @xmath161 , where @xmath162 is the dimensionless scaled radial position of the vortex . for the one - component condensate , @xmath163 does not depend on @xmath164 . similarly , the term @xmath165 becomes @xmath166 the usual lagrangian dynamics yields a pair of coupled equations @xmath167\label{dotu } \dot{\phi}_0 & = & -\frac{1}{u_0(1-u_0 ^ 2)}\frac{\partial \tilde e}{\partial u_0}\label{dotphi}\end{aligned}\ ] ] the hamiltonian structure of these two equations ensures that @xmath168 as the vortex executes its dynamical trajectory . thus the vortex quite generally moves on a contour of fixed @xmath163 . for a vortex in a one - component condensate , @xmath163 is independent of @xmath164 , so that @xmath169 and the motion is uniform circular precession with an angular velocity given by eq . ( [ dotphi ] ) : @xmath170 where @xmath171 is the vortex core radius . this result agrees well with experimental observations @xcite . figure [ m=0 - 14 ] shows typical contours of equal energy . a vortex precesses uniformly along such a circular curve with an angular velocity proportional to the radial gradient @xmath172 in a one - component condensate , where @xmath173 is the vortex s squared dimensionless radial position and @xmath156 is the condensate radius . adapted from @xcite with permission.,width=192 ] it is not difficult to generalize the lagrangian to the more interesting case of a spin - orbit coupled condensate . apart from the trap and interaction energies that remain unchanged , the new feature is the single - particle hamiltonian of the form used in the nist experiments @xcite @xmath174 with @xmath175 . use of this hamiltonian yields modified terms in the gp energy @xmath176 . the experiment can control various parameters : the raman laser wavenumber @xmath177 , the detuning @xmath107 and the rabi coupling strength @xmath81 . the trial function now has two components @xmath178 here the first two factors are the same as in the one - component case ( [ trial ] ) , and @xmath179 is a two - component normalized spinor @xmath180 e^{is_2}e^{i\eta}\sin(\chi/2 ) \end{pmatrix}\ ] ] with two separate phases @xmath181 and @xmath182 [ compare eq . ( [ s ] ) ] @xmath183 where @xmath184 is an integer ( typically @xmath185 ) . this structure assumes a vortex located at @xmath186 , with quantized circulation @xmath187 in the upper and lower components , respectively ; in addition , the parameter @xmath188 allows for an induced velocity along the preferred direction @xmath127 . if @xmath189 with equal circulations , the resulting vortex dynamics is the same as for a single - component situation . if @xmath190 ( namely different circulations ) , however , the dynamics of the two - component vortex line is qualitatively different . i propose using a thermal quench like that of ref . @xcite , anticipating that such an experiment would sometimes create a vortex with different circulations . the specific case @xmath8 and @xmath9 is analogous to a `` half - quantum vortex '' that has been predicted in thin films of superfluid @xmath191he - a and observed for both exciton - polariton becs @xcite and chiral @xmath192-wave superconductors @xcite . a detailed analysis shows that the energy @xmath163 now has terms proportional to @xmath193 and @xmath194 , meaning that the vortex dynamical trajectory now involves radial motion as well as azimuthal motion . nevertheless , the vortex continues to move on a contour of constant energy @xmath195 . unlike the previous situation , the vortex can have trajectories that leave the condensate along with those that remain inside . figure [ m=1 - 14 ] shows such contours for two different values of rabi frequency @xmath196 ( left ) and @xmath197 ( right ) . in a two - component condensate with half - quantum vortex with circulations @xmath198 and @xmath9 . here , terms proportional to @xmath193 and @xmath194 shift the center of the energy contours . the first figure shows contours for @xmath199 and the second for @xmath200 . adapted from @xcite with permission.,width=336 ] this article arose from a presentation at a workshop on quantum gases , fluids , and solids , involving both the helium community and the cold - atom community . for that reason , i include a treatment of the recent cold - atom achievements in creating a two - component spin - orbit coupled hamiltonian @xcite , which will not be familiar to the broader low - temperature community . owing to various lasers and magnetic fields that are fixed in the laboratory , it is not possible to transform to a single rotating frame with a time - independent hamiltonian . thus creating a two - component vortex must rely on other approaches , and i here propose a rapid thermal quench from the normal thermal cloud of cold atoms deep into the bec . this method has successfully created singly quantized vortices in a one - component system @xcite , and the same technique should also work in a spin - orbit coupled bec . in addition to conventional two - component vortices with the same circulation in both components , the two - component structure should also allow half - quantum vortices , in which one component has unit circulation and the other has zero circulation . if such half - quantum vortices exist , i find that their dynamics would be distinctive , in that a fraction of the vortex orbits would leave the condensate ( see fig . [ m=1 - 14 ] ) . the variational approach in sec . [ theory ] has several inherent limitations . it assumes a thomas - fermi form for the density and spatially uniform spinor parameters @xmath52 and @xmath53 . for a plane - wave solution , these parameters would depend on the wave vector @xmath201 , and the present spinor yields the best constant variational choice . my wave function also takes the vortex singularity to have the same location for both components . in addition , the resulting vortex dynamics becomes singular near the outer edge of the condensate . to improve the description , a full numerical solution of the two - component gross - pitaevskii equation is probably preferable to a modified variational trial function . i have assumed small rabi frequency @xmath202 to ensure miscibility of the two components @xcite , but experiments for larger values would also be of interest . since it seems necessary to use an optical trap , techniques are needed to release a small coherent fraction of the condensate atoms , but related methods have served well in similar contexts @xcite . the nist group created spin - orbit coupling in a trap , where the raman beams provide a preferred direction . nevertheless , many groups have proposed a more symmetric rashba coupling , as seen in eq . ( [ rashba ] ) . it will be interesting to extend the current study to include the dynamics of vortices in the presence of such symmetric rashba coupling . part of this article was written during a visit to the institute for advanced study , tsinghua university , beijing , and i am grateful to t .- l . ho and h. zhai for their hospitality . i thank w. zheng for a valuable discussion concerning the interpretation of the synthetic electric field in fig . [ vortices ] . i. spielman and d. hall have provided copies of some of their figures , and i thank them for this assistance . i am grateful to g .- q . liu and d. w. snoke for discussions of the half - quantum vortices in exciton - polariton condensates . i. b. spielman , raman processes and effective gauge potentials , phys . a * 79 * , 063613 ( 2009 ) . lin , r. l. compton , k. jimnez - garca , j. v. porto , and i. b. spielman , synthetic magnetic fields for untracold neutral atoms , nature * 462 * , 628 - 632 ( 2009 ) . lin , k. jimnez - garca , and i. b. spielman , spin - orbit - coupled bose - einstein condensates , nature * 471 * , 83 - 87 ( 2011 ) . a. l. fetter and a. a. svidzinsky , vortices in a trapped dilute bose - einstein condensate , j. phys . matter * 13 * , r135-r194 ( 2001 ) . a. l. fetter , rotating trapped bose - einstein condensates , rev . 81 * , 647 - 691 ( 2009 ) . b. p. anderson , p. c. haljan , c. e. weiman , and e. a. cornell , vortex precession in bose - einstein condensates : observations with filled and empty cores , phys . rev . lett . * 85 * , 2857 ( 2000 ) . d. v. freilich , d. m. bianchi , a. m. kaufman , t. k. langin , and d. s. hall , real - time dynamics of single vortex lines and vortex dipoles in a bose - einstein condensate , science * 329 * , 1182 - 1185 ( 2010 ) . j. radi , t. a. sedrakyan , i. b. spielman , and v. galitski , vortices in spin - orbit coupled bose - einstein condensates , phys . a * 84 * , 063604 ( 2011 ) . a. l. fetter , vortex dynamics in spin - orbit coupled bose - einstein condensates , phys . a * 89 * , 023629 ( 2014 ) . y. g. rubo , half vortices in exciton polariton condensates , phys . lett . * 99 * , 106401 ( 2007 ) . k. g. lagoudakis , t. ostatnick , a. v. kavokin , y. g. rubo , r. andr , and b. deveaud - pldran , observation of half - quantum vortices in an exciton - polariton condensate , science * 326 * , 974 - 976 ( 2009 ) . j. dalibard , f. gerbier , g. juzelinas , and p. hberg , colloquium : artificial gauge potentials for neutral atoms , rev . * 83 * , 1523 - 1543 ( 2011 ) . h. zhai , spin - orbit coupled quantum gases , int . j. mod . b * 26 * , 1230001 ( 2012 ) . v. galitski and i. b. spielman , spin - orbit coupling in quantum gases , nature * 494 * , 49 - 54 ( 2013 ) . n. goldman , g. juzelinas , p. hberg , and i. b. spielman , light - induced gauge fields for ultracold atoms , arxiv:1308.6533v2 ( 2013 ) . h. zhai , degenerate quantum gases with spin - orbit coupling , arxiv:1403.8021v1 ( 2014 ) . j. struck , c. lschlger , m. weinberg , p. hauke , j. simonet , a. eckardt , m. lewenstein , k. sengstock , and p. windpassinger , tunable gauge potential for neutral and spinless particles in driven lattices , phys . lett . * 108 * , 225304 ( 2012 ) . m. aidelsburger , m. atala , m. lohse , j. t. barreiro , b. paredes , and i. bloch , realization of the hofstdter hamiltonian with ultracold atoms in optical lattices , phys . * 111 * , 185301 ( 2013 ) . h. miyake , g. a. siviloglou , c. j. kennedy , w. c. burton , and w. ketterle , realizing the harper hamiltonian with laser - assisted tunneling in optical lattices , phys . * 111 * , 185302 ( 2013 ) . g. jotzu , m. messer , r. desbuquois , m. lebrat , t. uehlinger , d. greif , and t. esslinger , experimental realization of the topological haldane model , nature * 515 * , 237 - 240 ( 2014 ) . m. mancini , g. pagano , g. cappellini , l. livi , m. rider , j. catani , c. sias , p. zoller , m. inguscio , m. dalmonte , and l. fallani , observation of chiral edge states with neutral fermions in synthetic hall ribbons , arxiv:1502.02495v1 ( 2015 ) . b. k. stuhl , h .- lu , l. m. aycock , d. genkina , and i. b. spielman , visualizing edge states with an atomic bose gas in the quantum hall regime , arxiv:1502.02496v1 ( 2015 ) . j. jang , d. g ferguson , v. vakaryuk , r. budakian , s. b. chung , p. m. goldbart , and y. maeno , observation of half - height magnetization steps in sr@xmath203ruo@xmath204 , science * 331 * , 186 - 188 ( 2011 ) . a. ramanathan , s. r. muniz , k. c. wright , r. p. anderson , w. d. phillips , k. helmerson , and g. c. campbell , partial - transfer absorption imaging : a versatile technique for optimal imaging of ultracold gases , rev . instrum . * 83 * , 083119 ( 2012 ) .

vortices in a one - component dilute atomic ultracold bose - einstein condensate ( bec ) usually arise as a response to externally driven rotation . apart from a few special situations , these vortices are singly quantized with unit circulation @xcite . recently , the nist group has constructed a two - component bec with a spin - orbit coupled hamiltonian involving pauli matrices @xcite , and i here study the dynamics of a two - component vortex in such a spin - orbit coupled condensate . these spin - orbit coupled becs use an applied magnetic field to split the hyperfine levels . hence they rely on a focused laser beam to trap the atoms . in addition , two raman laser beams create an effective ( or synthetic ) gauge potential . the resulting spin - orbit hamiltonian is discussed in some detail . the various laser beams are fixed in the laboratory , so that it is not feasible to nucleate a vortex by an applied rotation that would need to rotate all the laser beams and the magnetic field . in a one - component bec , a vortex can also be created by a thermal quench , starting from the normal state and suddenly cooling deep into the condensed state @xcite . i propose that a similar method would work for a vortex in a spin - orbit coupled bec . such a vortex has two components , and each has its own circulation quantum number ( typically @xmath0 ) . if both components have the same circulation , i find that the composite vortex should execute uniform precession , like that observed in a single - component bec @xcite . in contrast , if one component has unit circulation and the other has zero circulation , then some fraction of the dynamical vortex trajectories should eventually leave the condensate , providing clear experimental evidence for this unusual vortex structure . in the context of exciton - polariton condensates , such a vortex is known as a half - quantum vortex " @xcite . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

@xmath0-balls represent stationary localized solutions of a complex scalar field theory with a suitable self - interaction in flat space @xcite . the global phase invariance of the scalar field theory is associated with a conserved charge @xmath0 @xcite , which represents the electromagnetic charge , once the theory is promoted to a gauge theory . the simplest type of @xmath0-balls is spherically symmetric . these possess a finite mass and charge , but carry no angular momentum . considering their mass as a function of their charge , there are two branches of @xmath0-balls , merging and ending at a cusp , where mass and charge assume their minimal values @xcite . recently , a new type of scalar potential for @xmath0-balls was considered , leading to the signum - gordan equation for the scalar field @xcite . this potential gives rise to spatially compact @xmath0-balls , where the scalar field vanishes identically outside a critical radius @xmath1 @xcite . when coupled to electromagnetism , a new type of solution appears , @xmath0-shells @xcite . in @xmath0-shells the scalar field vanishes identically both inside a critical radius @xmath2 and outside a critical radius @xmath1 , thus forming a finite shell @xmath3 of charged matter . when gravity is coupled to @xmath0-balls , boson stars arise , representing globally regular self - gravitating solutions @xcite . the presence of gravity has a crucial influence on the domain of existence of the classical solutions . instead of only two branches of solutions joint at a single cusp , the boson stars exhibit an intricate cusp structure , where mass and charge oscillate endlessly . for black holes with scalar fields , on the other hand , a number of theorems exist , which exclude their existence under a large variety of conditions @xcite . here we consider the effect of gravity on the @xmath0-balls and @xmath0-shells of the signum - gordon model coupled to a maxwell field . we construct these charged boson stars and gravitating @xmath0-shells and analyze their properties and their domains of existence . we observe that at certain critical values of the mass and charge , the space - times form a throat at the ( outer ) radius @xmath1 , rendering the respective exterior space - time an exterior extremal reissner - nordstrm space - time . moreover , we show that in this model the black hole theorems can be elluded , that forbid black holes with scalar hair . indeed , the gravitating @xmath0-shells can be endowed with a horizon @xmath4 in the interior region @xmath5 , where the scalar field vanishes and the gauge potential is constant . this schwarzschild - type black hole in the interior is surrounded by a shell of charged matter , @xmath6 , leading to an exterior space - time @xmath7 of reissner - nordstrm type . we analyze the global and horizon properties of these black holes within @xmath0-shells , and present a mass relation . when a throat develops at the outer radius @xmath1 , which renders the respective exterior space - time an exterior extremal reissner - nordstrm space - time , the temperature of the horizon @xmath4 tends to zero . in section 2 we recall the action and anstze for the fields . we present the gravitating @xmath0-balls and @xmath0-shells in section 3 and 4 , respectively , and the black holes within @xmath0-shells in section 5 . we end with a conclusion and an outlook . we consider the action of a self - interacting complex scalar field @xmath8 coupled to a u(1 ) gauge field and to einstein gravity @xmath9 \sqrt{-g } d^4x , \label{action}\ ] ] with field strength tensor @xmath10 covariant derivative @xmath11 curvature scalar @xmath12 , newton s constant @xmath13 , gauge coupling constant @xmath14 , and the asterisk denotes complex conjugation . the scalar potential @xmath15 is chosen as @xmath16 variation of the action with respect to the metric and the matter fields leads , respectively , to the einstein equations @xmath17 with stress - energy tensor @xmath18 and the matter field equations , @xmath19 @xmath20 to construct static spherically symmetric solutions we employ schwarzschild - like coordinates and adopt the spherically symmetric metric @xmath21 with @xmath22 for solutions with vanishing magnetic field the ansatz for the matter fields has the form @xmath23 @xmath24 for notational simplicity , we introduce new coupling constants @xcite @xmath25 and redefine the matter field functions , @xmath26 the latter corresponds to performing a gauge transformation to make the scalar field real and absorbing the frequency @xmath27 of the scalar field into the gauge transformed vector potential . note , that the parameter @xmath28 can be removed by rescaling and will therefore be set to one @xcite . thus the only parameter left is the gravitational coupling @xmath29 . let us now specify the boundary conditions for the metric and matter functions . for the metric function @xmath30 we adopt @xmath31 where @xmath1 is the outer radius , thus fixing the time coordinate . for the mass function @xmath32 we require for globally regular ball - like boson star solutions @xmath33 for globally regular shell - like solutions @xmath34 where @xmath2 is the inner radius of the shell , and for black hole solutions @xmath35 where @xmath36 denotes the horizon . to be able to specify more than four boundary conditions for the matter functions we introduce one or more auxiliary variables . for globally regular boson star solutions we require at the origin and at the outer radius @xmath1 the conditions @xmath37 where the prime denotes differentiation with respect to @xmath38 . in order to choose also the value of @xmath39 as a boundary condition , we make the outer radius @xmath1 an auxiliary ( constant ) variable , and thus add the differential equation @xmath40 , without imposing a boundary condition . for globally regular shell solutions as well as for black holes we require at the inner radius @xmath2 and at the outer radius @xmath1 the conditions @xmath41 in order to choose also the value of @xmath42 as a boundary condition , we now also make the ratio of inner and outer radius @xmath43 an auxiliary ( constant ) variable . alternatively to demanding a certain value for @xmath42 , we may also specify the value of the electric charge @xmath0 . the charge @xmath0 of the solutions is associated with the conserved current @xmath44 i.e. , the charge is obtained as the spatial integral @xmath45 the mass @xmath46 of the stationary asymptotically flat space - times is obtained from the corresponding komar expression . for globally regular space - times like boson stars and shells of boson matter the mass is given by @xmath47 where @xmath48 denotes an asymptotically flat spacelike hypersurface , @xmath49 is normal to @xmath48 with @xmath50 , @xmath51 is the natural volume element on @xmath48 , and @xmath52 denotes an asymptotically timelike killing vector field @xcite . replacing the ricci tensor via the einstein equations by the stress - energy tensor yields @xmath53 for black hole space - times the corresponding komar expression is given by @xmath54 where @xmath55 is the horizon mass of the black hole . the mass of all ( gravitating ) solutions can be directly obtained from the asymptotic form of their metric . in the units employed , we find @xmath56 we first address the set of globally regular ball - like boson star solutions , obtained by coupling the @xmath0-ball solutions @xcite to gravity . we illustrate the physical properties of these solutions in fig . [ bosonstar ] . [ phasediag ] represents the phase diagram for the boson star solutions . here the sets of solutions for a sequence of values of the gravitational coupling @xmath29 are exhibited in terms of the values of the matter field functions at the center of the solutions , i.e. , the value of the scalar field function @xmath57 and the value of the gauge field function @xmath39 . the figure consists of four distinct regions , which we have labelled i , ia , ii and iia . for @xmath58 , the @xmath0-ball solutions form a continuous set , represented by a single curve in the lower part of the figure , denoted as region i. these non - gravitating solutions are bounded by some maximal value of @xmath57 , by some minimal value of @xmath39 , and by the bifurcation point with the shell - like solutions , where @xmath57 reaches zero . as the gravitational coupling constant @xmath29 is increased from zero , the resulting sets of solutions smoothly deform and fill region i. the maximal value of @xmath57 of these sets of solutions ( for each value of @xmath29 given by a single continuous curve ) then increases , the corresponding minimal value of @xmath39 decreases , and the bifurcation point with the shell - like solutions decreases as well . this decrease of the bifurcation point continues until zero is reached at the critical value of the gravitational coupling constant , @xmath59 . beyond @xmath60 no shell - like solutions exist . however , the set of solutions continues to deform smoothly with increasing @xmath29 . this smooth evolution continues , until a second critical value of the gravitational coupling constant @xmath29 is reached , @xmath61 . here a bifurcation with a second set of solutions is encountered , which constitute the boundary of region ia . this second type of solutions is present for each finite value of @xmath62 . for this type of solutions @xmath57 has a minimal value for fixed @xmath29 , which decreases with increasing @xmath29 , until at @xmath63 the bifurcation is reached . at @xmath63 the set i and set ia solutions touch and bifurcate . for @xmath64 they then split into a right and left set of solutions , forming regions ii and iia , respectively . the solutions in region ii correspond to the larger values of @xmath39 , while the solutions in region iia are restricted to the smaller values of @xmath39 . with increasing @xmath29 , the sets of solutions in region iia move towards smaller values of @xmath39 , possibly disappearing at some critical value of the gravitational coupling , whereas the sets of solutions in region ii move towards larger values of @xmath39 . [ qb_r0_vs_b0 ] shows the outer radius @xmath1 for these sets of solutions , and thus the size of the corresponding boson stars . clearly , the biggest size for a given @xmath62 is always reached in region i at the bifurcation point with the shell - like solutions . the oscillations of the gauge field value @xmath39 with increasing scalar field value @xmath57 seen in region ii in fig . [ phasediag ] are reflected in the spirals formed by the outer radius @xmath1 in region ii in fig . [ qb_r0_vs_b0 ] . they are also present in regions iia and ia , whenever the gauge field value @xmath39 exhibits oscillations . the mass @xmath46 and the charge @xmath0 of these sets of boson star solutions are exhibited in figs . [ qb_m_vs_b0 ] and [ qb_q_vs_b0 ] . both show a very similar pattern . again , the biggest mass and charge for a given @xmath62 are reached in region i at the respective bifurcation point with the shell - like solutions , while the oscillations of @xmath39 seen in regions ia , ii and iia lead to spiral patterns for the mass and charge . the corresponding family of curves for the asymptotic value @xmath65 of the gauge field function @xmath66 at infinity ( which can be indentified with the value of the scalar field frequency @xmath27 in the gauge , where the gauge field vanishes at infinity ) is exhibited in fig . [ qb_om_vs_b0 ] . here the overall pattern is different , but spirals occur as well . finally , in fig . [ qb_mvs_q ] we exhibit the ratio of mass and charge @xmath67 versus @xmath0 . we observe a linear increase of @xmath67 with @xmath0 for the larger values of @xmath0 in regions i and iia , where the slope decreases with increasing @xmath29 , making @xmath67 almost constant for larger values of @xmath29 ( e.g. , @xmath68 ) . while the occurrence of spirals is a typical feature of boson star solutions @xcite , the present sets of solutions exhibit a for boson stars new phenomenon , namely the formation of throats . as a throat is formed , the minimum of the metric function @xmath69 tends to zero , and the zero is reached precisely at the outer radius @xmath1 . at the same time the metric function @xmath70 tends to a step function , that vanishes inside @xmath1 , and assumes the asymptotic value @xmath71 outside @xmath1 . . [ fun ] the functions close to throat formation are exhibited in the case of black holes . ) the space - time for @xmath72 then corresponds to the exterior space - time of an extremal reissner - nordstrm ( rn ) black hole . indeed , there the metric function @xmath69 can be expressed as @xmath73 i.e. , @xmath74 for the extremal rn solution ( in the units employed ) . as seen in fig . [ qb_mvs_q ] , this relation is precisely satisfied , when @xmath75 . thus a throat is formed , when in a set of solutions the value @xmath39 of the gauge field function tends to zero . in fact , the function @xmath66 then tends to zero in the whole region @xmath76 , and its derivative @xmath77 does so as well . however , at @xmath1 the derivative @xmath77 jumps to a finite value , necessary for the coulomb fall - off of a solution with charge @xmath0 . finally we note , that the sets of boson star solutions with fixed gravitational coupling constant @xmath29 satisfy a mass relation . this relation is based on the observation , that @xmath78 shown to hold for the regular solutions in flat space @xcite . since ( [ mreg2 ] ) continues to hold for the gravitating solutions , integration yields the mass relation @xmath79 where the mass @xmath80 of a regular solution with charge @xmath81 is obtained by integrating from any regular solution @xmath82 with charge @xmath83 along the curve of intermediate solutions of the set . let us next consider the gravitating shell - like solutions . here the space - time consists of 3 parts . in the inner part @xmath84 the gauge potential is constant and the scalar field vanishes . consequently , it is minkowski - like , with @xmath85 and @xmath86 . the middle region @xmath3 then represents the shell of charged bosonic matter , while the outer region @xmath87 corresponds to part of a reissner - nordstrm space - time , where the gauge field exhibits the standard coulomb fall - off , while the scalar field vanishes identically . this behaviour of the functions is demonstrated in fig . [ fun ] for the shell - like solution with charge @xmath88 and gravitational coupling constant @xmath89 . we exhibit in fig . [ q_shell ] the properties of shell - like solutions . [ phasesdiag2 ] shows the ratio of the inner radius @xmath2 to the outer radius @xmath1 for these solutions . for a given finite value of the gravitational coupling , the branch of gravitating shells emerges at the corresponding boson star solution and ends , when a throat is formed at the outer radius @xmath1 . as this happens , the value of @xmath42 reaches zero ( or equivalently @xmath90 , since @xmath66 is constant in the interior , @xmath91 ) . the exterior space - time @xmath92 then corresponds to the exterior of an extremal rn space - time . thus in contrast to @xmath0-shells in flat space , which grow rapidly in size , mass and charge as the ratio @xmath93 , the growth of gravitating @xmath0-shells is limited by gravity , and the restriction in size , mass and charge is the stronger , the greater the value of the gravitational coupling constant @xmath29 . this is demonstrated in figs . [ qs_r0_vs_b1 ] , [ qbs_m_vs_b1 ] and [ qs_mq_vs_b1 ] , where the outer radius @xmath1 , the mass @xmath46 and the charge @xmath0 are exhibited for a sequence of values of the gravitational coupling constant . in fig . [ qbs_m_vs_b1 ] for comparison also the mass of the corresponding boson star solutions ( resp . @xmath0-ball solutions for vanishing gravitational coupling constant ) are exhibited . the transitions from the ball - like to the shell - like solutions are indicated in the figure by the small asterisks . with increasing @xmath29 the sets of shell - like solutions decrease rapidly , until at the critical value @xmath60 ( see fig . [ phasediag ] ) they cease to exist . [ qs_mq_vs_b1 ] exhibits the scaled mass @xmath94 and the scaled charge @xmath95 for several sets of gravitating @xmath0-shells . together with fig . [ qs_r0_vs_b1 ] the figure demonstrates , that the condition for extremal rn solutions , @xmath96 , is satisfied for the shell - like solutions , as the throat forms at the outer radius @xmath1 . finally we note , that the shell - like solutions satisfy the mass relation ( [ mreg ] ) as well . consequently the mass relation holds for any two globally regular solutions of a set with given gravitational coupling constant , thus relating also ball - like and shell - like solutions . let us finally address black holes in this model . the simplest type of black holes is obtained , when the minkowski - like inner part of the space - time , @xmath91 , of gravitating @xmath0-shell solutions is replaced by the inner part of a curved schwarzschild - like space - time . the metric in the interior region @xmath91 is then determined by the function @xmath97 and a constant function @xmath70 . thus the event horizon resides at @xmath36 . but the presence of the @xmath0-shell outside the event horizon , makes the properties of the black hole differ from those of a pure schwarzschild black hole . since with the event horizon size a further variable appears , which is an important physical quantity , we discuss the black hole properties with respect to the horizon radius @xmath4 in the following . the metric and matter field functions for black holes with charge @xmath88 at gravitational coupling @xmath89 are exhibited in fig . [ fun ] for several values of the horizon radius @xmath4 . to illustrate the domain of existence of such black hole solutions , we again choose a sequence of values for the gravitational coupling constant , but we now keep the charge @xmath0 fixed , as we vary the horizon radius , starting from the corresponding globally regular @xmath0-shell solution . a respective set of solutions is shown in fig . [ bh1 ] for @xmath9810 and 100 . first of all we note , that the horizon radius is always limited in size , where the maximal size grows with the charge @xmath0 . for small @xmath0 , e.g. @xmath9810 , we observe two distinct patterns for the black hole solutions . the first pattern arises when the fixed gravitational coupling constant has a value below a certain critical value . here a maximal horizon size is reached , when the horizon radius @xmath4 gets close to the inner radius of the shell @xmath2 . there a bifurcation occurs and a second branch emerges , which ends at a second bifurcation , where a third branch emerges , etc . this results in a spiralling pattern , where the mass @xmath46 and the temperature @xmath99 of the solutions tend towards finite limiting values . ( the first few branches are apparent in fig . [ bhq10_b1_vs_rh ] , and enlarged in the inlet for a representative value of the gravitational coupling constant , @xmath100 , while the higher branches are too small to be resolved there . ) the second pattern is present above that critical value of the coupling constant . here the set of black hole solutions for fixed gravitational coupling ends , when a throat is formed at the outer shell radius @xmath1 . there the condition for extremal rn solutions , @xmath96 , is satisfied again , as seen in figs . [ bhq10_m_vs_rh ] and [ bhq100_m_vs_rh ] . as the throat forms , the temperature @xmath99 at the event horizon of the schwarzschild - like black hole @xmath36 tends to zero , as seen in figs . [ bhq10_t_vs_rh ] and [ bhq100_t_vs_rh ] . while appearing at first unexpected , the reason for the vanishing of the temperature @xmath99 is the behaviour of the metric function @xmath70 in @xmath101 , since @xmath70 tends to zero in the interior , when the throat is formed , as seen in fig . we recall , that the ratio of the temperature @xmath99 of the black hole within the @xmath0-shell to the temperature @xmath102 of the schwarzschild black hole is given by @xmath103 . for larger ( fixed ) values of the charge we always observe this second pattern , although the throat may either be reached directly after a monotonic increase of the horizon radius @xmath4 to its maximum value , or along a second branch , where the horizon radius is decreasing again ( having passed a bifurcation ) , as seen in fig . [ bhq100_b1_vs_rh ] . as seen in the figure , whenever bifurcations occur , there are two ( or more ) black hole space - times with the same value of the charge @xmath0 and the same horizon radius @xmath4 ( within a certain range of values ) , but different values of the total mass @xmath46 as measured at infinity . surprisingly , however , there are also two ( or more ) black hole space - times with the same value of the charge @xmath0 and the same value of the total mass @xmath46 ( within a certain range of values ) . these black holes thus have the same set of global charges but are otherwise distinct solutions of the einstein - matter equations . consequently black hole uniqueness does not hold in this model of scalar electrodynamics . let us finally consider some mass relations for these black holes space - times possessing @xmath0-shells . we begin by recalling an interesting result obtained in the isolated horizon framework @xcite . it states that the mass @xmath46 of a black hole space - time with horizon radius @xmath4 and the mass @xmath104 of the corresponding globally regular space - time obtained in the limit @xmath105 are related via @xcite @xmath106 where the mass contribution @xmath107 is defined by @xmath108 here @xmath109 represents the surface gravity of the black hole with horizon radius @xmath4 , @xmath110 . accordingly , the mass @xmath46 of a space - time with a black hole with horizon radius @xmath4 within a @xmath0-shell with total charge @xmath0 should be obtained as the sum of the globally regular gravitating @xmath0-shell with charge @xmath0 and the integral @xmath107 along the set of black hole space - times , obtained by increasing the horizon radius for fixed charge from zero to @xmath4 . this relation is demonstrated in fig . [ figiso1 ] for the set of solutions with charge @xmath88 and gravitational coupling constant @xmath100 . the values for the mass @xmath46 obtained from the relation ( [ ihmud ] ) are seen to agree with the values for the black hole mass @xmath46 obtained from the asymptotics ( [ mass ] ) . the set of solutions exhibited has spiralling character , i.e. , it has besides the main first branch a second branch , also exhibited , and further branches , not resolved in the figure . when the charge is allowed to vary , too , one expects a change of the above relation in accordance with ( [ mreg2 ] ) and the first law ( in the units empoyed ) , i.e. , @xmath111 where @xmath112 denotes the area of the horizon and @xmath65 represents the electrostatic potential at infinity . thus we generalize the above relation ( [ ihmu ] ) to read @xmath113 this relation is demonstrated in fig . [ figiso2 ] , where for several values of the gravitational coupling constant and for fixed ratio of inner and outer shell radii @xmath43 , the values for the mass @xmath46 obtained from the relation ( [ ihmudq2 ] ) are shown together with the values for the mass @xmath46 obtained from the asymptotics ( [ mass ] ) . we have considered boson stars , gravitating @xmath0-shells and black holes within @xmath0-shells in scalar electrodynamics with a @xmath114-shaped scalar potential , where the scalar field is finite only in compact ball - like or shell - like regions . the gravitating @xmath0-shells surround a flat minkoswki - like interior region , while their exterior represents part of an exterior rn space - time . when the flat interior is replaced by a schwarzschild - like interior , black hole space - times result , where a schwarzschild - like black hole is surrounded by a compact shell of charged matter , whose exterior again represents part of an exterior rn space - time . these black hole space - times violate black hole uniqueness , in certain regions of parameter space . here for the same values of the mass @xmath46 and the charge @xmath0 two or more distinct solutions of the einstein - matter equations exist . the solutions satisfy certain relations of the type obtained first in the isolated horizon formalism , which connect the mass @xmath46 of a black hole solution with the mass @xmath104 of the associated globally regular solution . the masses of two regular solutions are related in an analogous ( simpler ) manner . this formalism further suggests to interpret the black hole space - times as bound states of schwarzschild - type black holes and gravitating @xmath0-shells @xcite . while we have restricted our discussion here to schwarzschild - type black holes in the interior , there are also black hole space - times with charged , i.e. , reissner - nordstrm - type interior solutions . these more general black hole space - times will be discussed elsewhere . the inclusion of rotation presents another interesting generalization of the solution considered here , since rotating boson stars are well - known @xcite . the construction of the corresponding rotating shells and their black hole generalizations , however , still poses a challenge . r. friedberg , t. d. lee and a. sirlin , phys . rev . d * 13 * , 2739 ( 1976 ) . s. r. coleman , nucl . b * 262 * , 263 ( 1985 ) [ erratum - ibid . b * 269 * , 744 ( 1986 ) ] . h. arodz and j. lis , phys . d * 77 * , 107702 ( 2008 ) [ arxiv:0803.1566 [ hep - th ] ] . h. arodz and j. lis , arxiv:0812.3284 [ hep - th ] . t. d. lee and y. pang , phys . * 221 * , 251 ( 1992 ) . p. jetzer , phys . rept . * 220 * , 163 ( 1992 ) . e. w. mielke and f. e. schunck , nucl . b * 564 * , 185 ( 2000 ) [ arxiv : gr - qc/0001061 ] . f. e. schunck and e. w. mielke , class . grav . * 20 * , r301 ( 2003 ) [ arxiv:0801.0307 [ astro - ph ] ] . j. d. bekenstein , phys . d * 5 * , 1239 ( 1972 ) . j. d. bekenstein , phys . d * 51 * , 6608 ( 1995 ) . a. e. mayo and j. d. bekenstein , phys . d * 54 * , 5059 ( 1996 ) [ arxiv : gr - qc/9602057 ] . r. m. wald , general relativity ( university of chicago press , chicago , 1984 ) a. ashtekar and b. krishnan , living rev . * 7 * , 10 ( 2004 ) [ arxiv : gr - qc/0407042 ] . a. corichi and d. sudarsky , phys . rev . d * 61 * , 101501 ( 2000 ) [ arxiv : gr - qc/9912032 ] . a. ashtekar , a. corichi and d. sudarsky , class . quant . grav . * 18 * , 919 ( 2001 ) [ arxiv : gr - qc/0011081 ] . s. yoshida and y. eriguchi , phys . rev . d * 56 * , 762 ( 1997 ) . b. kleihaus , j. kunz and m. list , phys . d * 72 * , 064002 ( 2005 ) [ arxiv : gr - qc/0505143 ] . b. kleihaus , j. kunz , m. list and i. schaffer , phys . d * 77 * , 064025 ( 2008 ) [ arxiv:0712.3742 [ gr - qc ] ] .

we consider boson stars and black holes in scalar electrodynamics with a v - shaped scalar potential . the boson stars come in two types , having either ball - like or shell - like charge density . we analyze the properties of these solutions and determine their domains of existence . when mass and charge become equal , the space - times develop a throat . the shell - like solutions need not be globally regular , but may possess a horizon . the space - times then consist of a schwarzschild - type black hole in the interior , surrounded by a shell of charged matter , and thus a reissner - nordstrm - type space - time in the exterior . these solutions violate black hole uniqueness . the mass of the black hole solutions is related to the mass of the regular shell - like solutions by a mass formula of the type first obtained within the isolated horizon framework .

the apt was synthesized from high - order harmonics generated in xenon by focusing 35 fs , 796 nm ( 1.56 ev photon energy ) pulses from a 1 khz ti : sapphire laser to an intensity of @xmath15 w@xmath2@xmath3 in a 3 mm long windowless gas cell filled to a static pressure of @xmath16 mbar . the apt was filtered spatially by passing it through a 1.5 mm diameter aperture , and spectrally using a 200 nm thick aluminium filter . the spatial filter removes contributions to the harmonic emission from the longer quantum paths , while the aluminium filter blocks the remaining ir and the intense low - order harmonics @xcite . the spectrum of the apt is shown in fig . [ fig : train]a and consists of harmonics 11 to 17 , with a central energy of 23 ev . the pulses were characterised using the rabitt technique ( reconstruction of attosecond beating by interference of two - photon transitions ) @xcite , and the average duration of the bursts was found to be 370 as with the temporal profile shown in the inset in fig . [ fig : train]a . the dressing ir pulse , a delayed replica of the pulse generating the harmonics , was collinearly overlapped with the apt before both beams were refocused into the spectrometer by a toroidal platinum mirror . recombination was achieved using a mirror with a hole in the centre , through which the apt was sent , and on which the dressing pulse was reflected . the delay of the dressing pulse relative to the apt was finely controlled by a mirror mounted on a piezo - electric translation stage . to obtain a good estimate of the ir intensity , calibration was performed by measuring xe@xmath7 and xe@xmath17 ion yields as a function of intensity . in addition , these estimates were confirmed by comparison with the observed ponderomotive shift @xcite in the photoelectron spectra . the absolute timing between the apt and the ir field was not accessible experimentally , and has been chosen to fit the results of the tdse calculations . a velocity map imaging spectrometer was used for detection , having the advantage of being able to operate either in electron imaging @xcite or in ion time - of - flight mode . for both ion and electron detection , the target gas was injected by means of an atomic beam pulsed at 50 hz . the ion yield measurements were carried out in time - of - flight mode , the signal being collected with a boxcar integrator using active background subtraction based on the laser shots arriving with no target gas present . the 2d projections of the momentum distributions of the photoelectrons were recorded by means of an mcp - assembly and a ccd - camera , and from these the 3d momentum distributions were obtained using the iterative inversion procedure described in @xcite . for the tdse calculations we use the single active electron ( sae ) approximation , in which we assume that only one electron interacts with the field , while the others remain in the ground state . this approximation has been extensively tested for both alkali metal and rare gas atoms and found to produce results which compare well with experiments @xcite . the atomic potentials used in the sae calculations were the standard hartree - fock potential for helium , and a pseudo - potential in argon @xcite . these potentials reproduce the single electron excited states very well . to simulate the experiments we use an ir pulse whose electric field envelope is a cosine function with a fwhm in intensity of 35 fs , and an apt whose electric field envelope is a somewhat sharper @xmath18 function with a fwhm in intensity of 10 fs . the total population excited is calculated as one minus the population remaining in the ground state at the end of the pulse , while the total ionization is calculated either from the photoelectron spectrum @xcite or by running the calculation for 10 additional ir cycles and calculating the probability to remain in the vicinity of the ion . a variety of other field envelopes ( _ e.g. _ apts with varying numbers of pulses and a constant ir pulse envelope ) were used to check details of our analysis . we are grateful to s. thorin , dr . f. lpine and prof . m. j. j. vrakking for help with the imaging spectrometer . this research was supported by the marie curie research training network ( mrtnct-2003 - 505138 , xtra ) , the crafoord foundation , the knut and alice wallenberg foundation , the swedish research council and the national science foundation through grant no . the authors declare that they have no competing financial interests . * experimental pulses and single - photon momentum distributions . * * a * , spectrum of the uv pulses used in the experiment shown in relation to the ionization potentials of helium and argon . for helium , some of the excited states have been indicated for comparison . the corresponding temporal profile is a train of pulses spaced by half the ir laser cycle . the inset shows the temporal profile of the attosecond pulses in the train , each with a duration of 370 as , as reconstructed from the rabitt measurements . panels * b * and * c * show experimental photoelectron momentum distributions from single - photon ionization by the apt in helium and argon , respectively , with the polarization of the light parallel to the @xmath19-axis . * b * , in helium , only a single ring corresponding to ionization by harmonic 17 can be seen , since this is the only spectral component having sufficient energy to overcome the ionization potential . the angular distribution is peaked along the polarization axis , as expected for single - photon ionization from an s - state . * c * , in argon , the full bandwidth of the apt contributes to the ionization and four rings corresponding to harmonics 11 to 17 can be seen . since the ionization starts from a _ p_-state , the resulting angular distribution is a superposition of _ s_- and _ d_-states , with contributions also along the @xmath20-axis . ] * control of the ion yield . * experimentally measured ion yields , @xmath6 , for he@xmath7 ( blue circles ) and ar@xmath7 ( red squares ) as a function of the delay @xmath21 between the attosecond pulses and the ir field , at an intensity of @xmath1 w@xmath2@xmath3 . we use a sine convention for the ir electric field so that delays , @xmath21 , which are multiples of @xmath22 as , where @xmath23 is the ir laser frequency ( @xmath24 ev ) , correspond to the attosecond pulses overlapping the zero - crossings of the ir field . all yields are normalised to those obtained with only the apt present . in helium a clear modulation is observed which is not seen in argon . also shown in the figure are the calculated ion yields at an ir intensity of @xmath1 w@xmath2@xmath3 for he@xmath7 ( blue solid line ) and ar@xmath7 ( red solid line ) . these were obtained using uv and ir fields that closely match the experimental parameters : the apt has a 10 fs fwhm ( full width at half maximum in intensity ) duration and the ir pulse is 35 fs fwhm , and agree well with the experimental results . ] * detailed theoretical study of ionization in helium . * * a * , calculated probabilities for removal of an electron from the ground state ( @xmath9 , blue line ) , ionization ( @xmath6 , red line ) and remaining in an excited bound state ( @xmath10 , green line ) as a function of the phase of the ir field at the time of the attosecond pulses , for an apt with a fwhm of 10 fs and a 35 fs ir field with a peak intensity of @xmath1 w@xmath2@xmath3 . * b * , excitation probability , @xmath9 , versus delay for different apts , normalized to the excitation probability for zero delay in each case . the fwhm of the apt intensity envelope is 1 fs ( blue line ) , 2 fs ( red line ) , 4 fs ( green line ) or 8 fs ( black line ) . the 1 fs envelope corresponds to an isolated attosecond pulse . * c * , contrast in @xmath9 ( defined as the maximum excitation probability divided by the minimum ) for various peak intensities of the ir field ( blue line ) . the apt has a fwhm of 10 fs . for comparison , the contrast obtained with a single attosecond pulse is also shown ( red line ) . ] * experimental photoelectron spectra and momentum distributions . * photoelectron spectra as a function of the delay between the attosecond pulses and the ir field from argon ( * a * ) and helium ( * d * ) . * b * and * c * , momentum distributions obtained at the delays corresponding to the dotted and dashed lines in * a*. * e * and * f * , momentum distributions obtained at the delays corresponding to the dotted and dashed lines in * d*. the polarization directions of the uv and ir fields are parallel to the @xmath19-axis .

attosecond pulses @xcite can be used to initiate and control electron dynamics on a sub - femtosecond time scale . the first step in this process occurs when an atom absorbs an ultraviolet photon leading to the formation of an attosecond electron wave packet ( ewp ) . until now , attosecond pulses have been used to create free ewps in the continuum , where they quickly disperse @xcite . in this paper we use a train of attosecond pulses , synchronized to an infrared ( ir ) laser field , to create a series of ewps that are below the ionization threshold in helium . we show that the ionization probability then becomes a function of the delay between the ir and attosecond fields . calculations that reproduce the experimental results demonstrate that this ionization control results from interference between transiently bound ewps created by different pulses in the train . in this way , we are able to observe , for the first time , wave packet interference in a strongly driven atomic system . the modulation of photon absorption by wave packet interference ( wpi ) has been used in molecular systems as a probe of nuclear dynamics on a femtosecond time scale @xcite , and in rydberg atoms as a probe of electron dynamics on a picosecond time scale @xcite . wpi is a sensitive tool for probing quantum dynamics because it depends on the spatial and temporal behavior of the wave packet in the confining potential . this is most easily appreciated by considering two well separated excitation pulses that create two initially localized wave packets in a manifold of excited states . the total excitation probability will be the simple sum of the integrated population in each wave packet _ unless _ some part of the first wave packet returns to the region of space where it was created during the time when the second wave packet is created , enabling the two to interfere . in this letter we present attosecond wpi experiments using a train of ultraviolet ( uv ) attosecond pulses to ionize either helium or argon atoms in the presence of an ir field . the attosecond pulses are phase locked to the ir field since their spacing in time is precisely one half of the laser period . the central energy of the pulses , @xmath023 ev , is higher than the ionization energy of argon ( 15.8 ev ) , but below that of helium ( 24.6 ev ) , as shown in fig . [ fig : train]a . we demonstrate the ability to control the ion yield in helium through the delay between the two fields , an effect which is absent in argon . we attribute this ionization control to interference between transiently bound ewps created in helium which can modulate the probability that an electron is excited out of the atomic ground state . calculations based on integration of the time - dependent schrdinger equation ( tdse ) show that the contrast in the ionization probability versus the ir - uv delay is an order or magnitude larger that what is achieved with a single pulse , and that the contrast grows as the number of pulses in the train is increased . both of these effects are hallmarks of wpi , seen here in the attosecond domain and for a strongly driven system . details of the experiment can be found in the methods section . the spectral and temporal characteristics of the attosecond pulse train ( apt ) used to excite the atoms are presented in fig . [ fig : train]a , while examples of momentum distributions obtained from uv ionization alone are shown in fig . [ fig : train]b and [ fig : train]c for helium and argon , respectively . the ir laser field , a replica of the laser pulse used to generate the uv pulses , was recombined with the apt after a variable delay line and focused into the detection chamber with an intensity of @xmath1 w@xmath2@xmath3 . a crucial point is that this laser intensity is too low to excite any population out of the ground state by itself . this means that the ground state is connected to the excited bound and continuum states only when an attosecond pulse is present , an essential condition for observing wpi . also of importance is the fact that although the ir laser field is weak from the point of view of an electron in the ground state , it is strong from the point of view of an electron excited out of the ground state . at peak amplitude , the ir field suppresses the coulomb potential by @xmath47 ev at the saddle point , which is enough to unbind all of the single excited bound states of helium . furthermore , this barrier suppression changes very slowly with intensity , scaling as @xmath5 . our method therefore results in creating attosecond ewps in a strong oscillating laser field . figure [ fig : ions ] shows our main experimental result , the delay dependence of the ion yields , @xmath6 , from helium and argon . for ar@xmath7 ( red squares ) , there is no measurable effect of the ir field while for he@xmath7 ( blue circles ) the ion yield is increased by a factor of four when the ir field is present . in addition , the he@xmath7 yield exhibits a modulation as a function of the uv - ir delay . the depth of the modulation is @xmath8 and the period is equal to half the laser period . this modulation is the signature of wpi of attosecond ewps in our experiment . to gain insight into the results presented in fig . [ fig : ions ] , we have performed calculations based on the integration of the tdse @xcite , as explained in more detail in the methods section . the ion yields obtained at the end of the interaction are indicated in fig . [ fig : ions ] as solid red and solid blue lines for argon and helium respectively , showing good agreement with the experiment . in addition , the calculations show that without an ir field , the ionization probability in he is equal to the excitation probability , meaning that no population is left in the excited bound states . as indicated in fig . [ fig : train]a , the spectrum of the uv pulses overlaps poorly with the accessible excited bound states of helium , and the atom is limited to absorbing photons belonging to the 17th harmonic , leading to immediate ionization . in fig . [ fig : ions_theory]a , we show more complete theoretical results for he . shown are both the probability that an electron is excited out of the ground state ( @xmath9 , blue line ) and the ionization probability ( @xmath6 , red line ) . the difference between these probabilities is the probability to remain in an excited state after the ir pulse ends ( @xmath10 , green line ) . two features are immediately apparent . first , the modulation in the he@xmath7 yield is caused by the fact that the amount of population excited out of the ground state by the apt in the presence of the ir field is modulated as a function of uv - ir delay . second , the ionization of the population promoted out of the ground state by the apt is incomplete , leaving 30 - 40% of the promoted population in excited states after the ir field is over . the delay dependence of the he@xmath7 yield has two contributions . first , each pulse in the apt excites population in the presence of an ir field that distorts the atomic potential by an amount that depends on the ir - uv delay . a single attosecond pulse would therefore probe solely the atom s ability to absorb light near the ionization threshold in the presence of an electric field which can be as high as @xmath11 v / cm . our calculations show that the modulation in the ion yield due to such a single attosecond pulse is about 1 - 3% over the intensity range covered by the experiment , 10 times smaller than the observed effect . the other contribution to the delay dependence is from wpi . this temporal interference in the total excitation probability comes about if an ewp created by one pulse in the train has some probability to be near the ion core when a later packet is being excited by a different pulse in the train . this requires that an ewp excited by a single pulse takes more that one half cycle to completely ionize . indeed , at all delays we find that the ewp excited by a single attosecond pulse takes one to several ir cycles to completely ionize , fulfilling this condition for wpi . wpi also causes the excitation probability to scale non - linearly with the number of pulses in the train . in the absence of wpi the relative modulation in the total excitation probability versus delay is the same for different length pulse trains . in fig . [ fig : ions_theory]b , we plot the normalized excitation probability for apts of different length , changing from a single pulse ( the 1 fs envelope ) to two or more . we see that the relative modulation increases as the apt length is increased . we also note that the delay curve reverses its shape when the number of pulses is increased from one to two or more . in argon , by contrast , the total excitation is linear in the length of the pulse train . by its nature , wpi is a very sensitive probe of the electron dynamics in a bound system . in our system these dynamics are most easily altered by changing the ir intensity . in fig . [ fig : ions_theory]c we plot the magnitude of the calculated relative modulation ( _ i.e. _ the contrast ) versus peak ir intensity for intensities ranging from 0.1 to @xmath12 w@xmath2@xmath3 and a 10 fs apt ( blue line ) . as can be seen , the contrast is a very sensitive function of the field amplitude . for comparison , the contrast from using a single 370 as pulse is shown ( red line ) . in this same range of intensities the amount of population ionized after the ir pulse is over ranges from 40 - 100% of the total population excited out of the ground state , and exhibits a very complicated dependence on the ir intensity . additional support for the wpi picture that we present can be found in the experimental measurements of the energy - resolved angular distributions from helium and argon , presented in fig . [ fig : electrons ] . for argon ( fig . [ fig : electrons]a ) the ir field only redistributes the energy of the ionized electrons , depending on the phase of the ir field at the time they enter the continuum . the highest energy electrons are created when the attosecond pulses are timed so that ionization takes place at the zero - crossings of the electric field ( @xmath13 ) , when the momentum transfer from the field to the electronic wave packet is maximum @xcite . the momentum distributions from argon ( fig . [ fig : electrons]b and [ fig : electrons]c ) show the difference between the two delays that results in the greatest and least number of high energy electrons . the angular distributions remain rather broad for all delays . in contrast to this , the photoelectron momentum distributions from helium ( fig . [ fig : electrons]e and [ fig : electrons]f ) are strongly peaked along the polarization axis of the ir field , reflecting the fact that most of the ionization occurs via electrons that escape over the suppressed coulomb barrier along the polarization direction . in addition , at the experimental ir intensity the highest energy electrons are observed when @xmath14 , which corresponds neither to the maxima or zeros of the ir electric field , as seen in fig . [ fig : electrons]d . this illustrates the complex wave packet dynamics discussed previously , which leads to different ratios between bound and free populations as well as between numbers of low and high energy electrons depending on the uv - ir delay and the ir intensity . the wpi that we have observed has a number of similarities and a few important differences as compared to `` traditional '' wpi . in more conventional wpi , the motion takes place on a purely bound potential surface and the wpi is controlled by changing the delay between pulses . in our case , the delay between attosecond pulses is fixed at one half the ir cycle , but the amplitude and phase of the ir field at which the ewps are created is easily changed . also , the ewps are only transiently bound and so both the total population and the energy - resolved angular distributions can be measured as a function of the various parameters in the experiment and compared to theory . wpi offers a unique tool to study the behavior of electrons in a strongly driven atom or molecule , since the ewps are created in the center of the potential well at a well - controlled time . a number of modifications to the experiments we have presented here are accessible in the near future . for instance , the wavelength of the laser field can be varied , perhaps all the way to the mid - infrared , which would allow the time difference between the attosecond pulses to be varied . most importantly perhaps , it should be possible to study the wpi as a function of the apt duration as was done in the theoretical calculations . this could be done in a polarization gating scheme by varying the gate duration @xcite . in conclusion , we have demonstrated that excitation / ionization dynamics can be controlled using an apt in combination with an ir field . previous attosecond experiments have used the uv pulse to control the time at which an ionization process takes place @xcite . the control demonstrated in this experiment is , to the best of our knowledge , the first use of an attosecond pulse to modulate the probability of an ionization event . when coupled to angular - resolved photoelectron distributions it opens the way for future study of the detailed dynamics of ultra broadband ewps in driven atomic and molecular systems .

recently , numerous spin related observations have been published which involve optical and transport experiments on diluted magnetic , semiconducting heterostructures and quantum wells ( qws).@xcite the correct interpretation of these effects requires a detailed knowledge of the underlying band structure . this is especially important for narrow gap semiconductors,@xcite because strong band mixing prevents a simple interpretation of optical and transport results by means of a parabolic band model which might be still applicable for most wide gap materials such as gaas or ingaas.@xcite here , we concentrate on the band structure calculations of hgte / hg@xmath11cd@xmath12te quantum wells . this material has some interesting properties : depending on the qw width ( @xmath13 ) , the qw has either a _ normal _ or _ inverted _ band structure when @xmath14 nm or @xmath15 nm , respectively . in the latter case the conduction band exhibits @xmath16 symmetry which leads to a strong rashba spin - orbit splitting in qw s with an asymmetrical confinement potential.@xcite additionally , the spin splitting of the subbands can be enhanced by introducing magnetic ions ( mn ) in the qw structure , e.g. , hg@xmath17mn@xmath18te / hg@xmath11cd@xmath12te . it should be noted that in ii - vi semiconductors , mn is incorporated into the crystal lattice isoelectrically and does not act as a donor or an acceptor . therefore , mn ions act primarily as a magnetic but not as a coulomb impurity and mobilities achieved for these qw structures with low mn concentrations are comparable with those for non - magnetic structures . this paper is organized as follows : in section ii a detailed description of the model used for the band structure calculations is presented . in subsection ii - a , we consider the model for non - magnetic as well as for magnetic qw s at zero external magnetic field . this model is used to calculate the subband energy dispersion of hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te and hgte / hg@xmath21cd@xmath22te qw s . in subsection ii - b , the band structure model for a hgte / hg@xmath11cd@xmath12te qw in an external magnetic field is described . this model is extended in subsection ii - c in order to take the influence of the @xmath6 exchange interaction on the band structure of magnetic qw s into account . the landau level fan diagram and the density of states of hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te qw are compared with recent transport experiments , and section iii summarizes the results . in the appendices details of the calculations for different growth directions ( a ) and strain as well as piezoelectric effects ( b ) are discussed . the band structure model we use is based on an envelope function approximation introduced by burt.@xcite the total wave function is expanded in terms of band edge ( @xmath24 ) bloch functions @xmath25 : @xmath26 where @xmath27 are the envelope functions . @xmath25 is assumed to be the same in the barrier and the well layers . assuming translation invariance in the plane perpendicular to the growth direction ( @xmath28 axis ) @xmath29 can be represented as follows @xmath30~f_{n}(z),\ ] ] where @xmath31 and @xmath32 are the wave vector components in the plane of the qw . the envelope functions and the energy levels near @xmath33 are determined within the framework of @xmath34 theory by solving a system of coupled differential equations:@xcite @xmath35 where @xmath36 and @xmath37 are the summation indices for the sum over the dimensionality of the chosen basis set , @xmath38 are the respective band edge potentials , and @xmath39 is the self - consistently calculated hartree - potential . the momentum matrix elements @xmath40 , which describe the coupling between the @xmath36 and @xmath37 bands , are treated exactly , while the @xmath41 , which describe the perturbative coupling of the @xmath36 and @xmath37 bands to the remote bands , are calculated using second - order perturbation theory . @xcite in narrow - gap hgte based structures , the strong coupling between @xmath42-like conduction and @xmath43-like valence bands causes mixing of the electronic states and induces nonparabolicity in the conduction bands . these effects were taken into account exactly by kane@xcite in the framework of the @xmath34 theory . in order to consider the coupling between the @xmath44 , @xmath45 and @xmath16 bands we choose the usual eight - band basis set ( see refs . and ): @xmath46 \nonumber\\ u_{5}(\mathitb{r})=|\gamma_8,-1/2\rangle & = & -1/\sqrt{6 } [ ( x - iy)\uparrow + 2z\downarrow ] \\ u_{6}(\mathitb{r})=|\gamma_8,-3/2\rangle & = & -1/\sqrt{2 } ( x - iy)\downarrow \nonumber\\ u_{7}(\mathitb{r})=|\gamma_7,+1/2\rangle & = & 1/\sqrt{3 } [ ( x+iy)\downarrow + z\uparrow ] \nonumber\\ u_{8}(\mathitb{r})=|\gamma_7,-1/2\rangle & = & 1/\sqrt{3 } [ ( x - iy)\uparrow -z\downarrow ] \nonumber.\end{aligned}\ ] ] the total angular momentum is then given by @xmath47 or @xmath48 . for the chosen basis set , the hamiltonian @xmath49 in eq . [ eignprob ] for a two - dimensional system with [ 001 ] growth direction takes the following form:@xcite @xmath50 where @xmath51\right),\nonumber\\ \tilde{s}_{\pm } & = & -\frac{\hbar^{2}}{2m_{0}}\sqrt{3}k_{\pm}\left(\{\gamma_{3},k_{z}\}-\frac{1}{3}[\kappa , k_{z}]\right),\nonumber\\ c & = & \frac{\hbar^{2}}{m_{0}}k_{-}[\kappa , k_{z}]\nonumber.\end{aligned}\ ] ] @xmath52=ab - ba$ ] is the usual commutator and @xmath53 is the anti - commutator for the operators @xmath54 and @xmath55 ; @xmath56 is the kane momentum matrix element ; @xmath57 and @xmath58 are the conduction and valence band edges , respectively ; @xmath59 is the spin - orbit splitting energy ; and @xmath60 , @xmath61 , @xmath62 , @xmath63 and @xmath64 describe the coupling to the remote bands and result in the @xmath65 and @xmath66 parameters according to @xmath67 and @xmath68 . only the terms with non - spherical ( cubic ) symmetry in the hamiltonian are proportional to the warping parameter @xmath65 . the case of @xmath69 corresponds to the axial approximation . the intrinsic inversion asymmetry is neglected in the hamiltonian because this effect is very small in hgte based structures.@xcite the band structure parameters for hgte and cdte at @xmath70 k are listed in table [ parametr ] . the dependence of the band gap ( @xmath71 ) of hg@xmath11cd@xmath12te on the temperature and composition @xmath72 is determined from the empirical expression according to laurenti _ et al._@xcite the valence band offset between hgte and cdte is taken to be equal to 570 mev at @xmath70 k , in agreement with recent experiments,@xcite and is assumed to vary linearly with @xmath72.@xcite .[parametr]band structure parameters of hgte and cdte at @xmath73 k. @xcite [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] so far , we have only considered the case of hgte / hg@xmath11cd@xmath12te qw s with ( 001 ) orientation . however , hgte based structures have also been investigated with orientations other than ( 001 ) , for example , ( 112 ) heterostructures.@xcite the electronic properties of such systems depend strongly on the growth direction . an extension of the model to qw s of a given @xmath74 orientation can be obtained using the approach of los _ _ et al.__@xcite : the set of basis functions [ eqs . ( [ basisset ] ) ] is changed to a set which adopts the symmetry of the problem , and thus the transformed hamiltonian once again has the form of eq . ( [ hamilt ] ) , while the matrix elements [ eqs . ( [ matrel ] ) ] contain additional terms depending on the structure orientation.@xcite the exact formulas for these terms are given in appendix a. also strain and piezoelectric effects can be included , as discussed in appendix b. during the last two decades much attention has been paid to the theoretical and experimental understanding of diluted magnetic semiconductors ( dms ) , both in bulk@xcite as well as in low - dimensional structures.@xcite extensively studied examples of this category are a@xmath75mn@xmath76b@xmath77 alloys , in which the group ii component is replaced randomly by the transition metal mn . @xcite so far , most research on magnetic two - dimensional structures has been done on wide gap dms materials . @xcite in the present work we consider qw s with magnetic ions ( mn ) in narrow gap hg@xmath17mn@xmath18te / hg@xmath11cd@xmath12te qw s . the two - dimensional confinement in the dms based layer combined with the exchange interaction between localized mn magnetic moments and mobile band electrons make such structures quite interesting candidates for the study of their electronic and magnetic properties . the band structure of hg@xmath17mn@xmath18te / hg@xmath11cd@xmath12te qw s in the absence of a magnetic field can be calculated similarly as that of non - magnetic qw s , cf . the description above . the only difference is that the band structure parameters for the well now depend on the mn concentration.@xcite in fig . [ rashba ] the zero - field subband dispersion @xmath78 of @xmath36-type hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te ( a , b , d ) and hgte / hg@xmath21cd@xmath22te ( c ) ( 001 ) qw s are presented . the temperature @xmath79 and the qw s width @xmath13 are set at 4.2 k and 12.2 nm , respectively . [ rashba ] ( a ) corresponds to a hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te qw with symmetrically @xmath36-type modulation doped barriers , while only the barrier on the substrate side is doped for the case presented in fig . [ rashba ] ( b ) . in the calculations it is assumed that ( i ) all donors in the doped layer are ionized , ( ii ) the charge density is constant in this region , and ( iii ) all electrons are transferred to the qw . the depletion charge in the doped layers is taken to be @xmath80 for an asymmetrically and @xmath81 for a symmetrically doped structure , respectively.@xcite ( @xmath82 denotes the charge density of the 2deg in the qw and is chosen to be @xmath83 @xmath84 for the calculations presented here . ) the eigenvalue [ eq . ( [ eignprob ] ) ] and poisson equations for the two - dimensional charge carriers in the qw are solved self - consistently for both cases . the depletion charge in the doped layers , which is assumed to be fixed at the levels indicated above , is included into the boundary conditions to solve the poisson equation.@xcite then the hartree potential is determined according to the charge distribution of electrons in the qw , which is given by the summation over all conduction band states @xmath85 and all components @xmath36 of the envelope functions @xmath86 : @xmath87 where @xmath88 is the electron charge , @xmath89 is the fermi function . here , we assume that @xmath90 and @xmath91 . in analogy , we have for the holes : @xmath92 in this case the summation index @xmath85 runs over all valence band states , and @xmath93 . the self - consistent hartree potential for zero magnetic field is then used to calculate the landau levels of the 2deg ( see description below ) . from fig . [ rashba ] ( a ) and ( b ) one observes that both qw s exhibit an inverted band structure . the conduction band includes two occupied subbands ( labeled as h1 and e2 ) ; the lower conduction subband ( h1 ) exhibits a heavy hole character at @xmath94 . the electron - like e1 subband lies , in this case , below the h2 subband and is now one of the valence subbands . the h1 and h2 subbands are not split for the symmetric qw [ case ( a ) ] ; however , for the asymmetric qw , a spin - orbit splitting ( in the following denoted as a spin splitting as is common in the literature ) of the h2 subband as well as a pronounced splitting of the h1 subband are visible . the splitting of h1 is 33.4 mev at @xmath95 and 30.9 mev at @xmath96 , respectively . the carrier concentrations in the h1@xmath97 and h1@xmath98 subbands for the asymmetric qw , after self - consistency has been obtained , are @xmath99 and @xmath100 @xmath84 , respectively . this large spin - orbit splitting , usually called rashba splitting , of the h1 state in an asymmetric qw was demonstrated to be an unique feature of type - iii qw s in the inverted band regime.@xcite experimentally values up to 30 mev have indeed been observed.@xcite the subband dispersion @xmath78 of hgte / hg@xmath21cd@xmath22te and hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te qw s are shown in fig . [ rashba ] ( c ) and ( d ) , where @xmath101 is taken to be equal to @xmath102 . the band structures of the non - magnetic ( c ) and magnetic ( d ) qw s differ notably in the subband separation at @xmath94 ; the separation between the e2 and h1 subbands is about 24 mev smaller for non - magnetic qw . the rashba splitting of the h1 subband is 15.5 mev for the non - magnetic and 13.1 mev for the magnetic qw s for @xmath103 . the corresponding values are 14.2 mev and 12.2 mev for @xmath104 . this relatively small difference arises because the actual band gap ( @xmath105 ) changes only by @xmath106 upon introduction of mn . [ h ] band structure of @xmath36-type ( a ) symmetrical and ( b , d ) asymmetrical hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te qw s , as well as ( c ) an asymmetrical hgte / hg@xmath21cd@xmath22te qw , all with @xmath107 nm at @xmath108 k. @xmath109 and @xmath110 . here the @xmath111 vector is @xmath103 , however the difference between @xmath112 and @xmath113 is less then 6 mev at @xmath114 and the @xmath115-dependence shows qualitatively the same behavior . the e1 subbands are shown as dashed lines.,title="fig:",width=453 ] in an external magnetic field perpendicular to the plane of the 2deg , the electronic bands are split into a series of landau levels . the effects of a magnetic field @xmath117 can be incorporated by a peierls substitution @xcite in the hamiltonian [ eq . ( [ hamilt ] ) ] as follows : @xmath118 where @xmath119 is the magnetic vector potential , @xmath120 . one possible landau gauge for @xmath121 is @xmath122 . the operator @xmath123 satisfies the following gauge - invariant relation : @xmath124 from now on , we drop the index @xmath125 for simplicity . subsequently , @xmath31 and @xmath32 are rewritten such that@xcite @xmath126 where @xmath127 is the magnetic length ; and @xmath128 and @xmath129 are , respectively , the annihilation and creation operators for the harmonic oscillator functions @xmath130 , where@xcite @xmath131 here , @xmath132 are the eigenvalues of the operator @xmath133 . thus , we can present the hamiltonian in eq . ( [ hamilt ] ) as a function of @xmath128 , @xmath129 , @xmath134 , the band structure parameters and their @xmath28-dependence . additionally , the zeeman term , @xmath135 , has to be included in the hamiltonian [ eq . ( [ hamilt ] ) ] . as shown by weiler,@xcite this leads to the following matrix : @xmath136 in the axial approximation we now assume that the total wave function can be written as@xcite @xmath137 where restrictions on the quantum numbers @xmath138 on the right - hand - side can be derived straightforwardly from eqs . ( [ oscill ] ) . since @xmath132 , the new quantum number @xmath139 . for all quantum numbers @xmath138 a system of ( up to eight ) coupled differential equations has to be solved . for @xmath140 the system is reduced to one equation that corresponds to a state with purely heavy hole character . non - axially symmetric systems can be treated by taking the coupling between the solutions of the axially symmetric problem [ eq . ( [ axial ] ) ] into account.@xcite the form of the coupling depends on the symmetry along the growth direction and can be included by the substitution of a linear combination of the @xmath141 wave functions:@xcite @xmath142 with @xmath143 and @xmath144 for a ( 001 ) oriented structure ( @xmath145 symmetry ) ; @xmath146 and @xmath147 for a ( 111 ) structure ( @xmath148 symmetry ) ; @xmath149 and @xmath150 for a ( 110 ) structure ( @xmath151 symmetry ) ; and @xmath152 and @xmath153 for other growth directions ( @xmath154 symmetry ) . a system of coupled differential equations for the envelope functions @xmath155 has to be solved for each value of the quantum number @xmath156 . the hamiltonian matrix elements @xmath157 are determined using eqs . ( [ creaanni ] ) , ( [ oscill ] ) . the landau level spectra of a hgte / hg@xmath21cd@xmath22te(001 ) qw calculated with and without applying the axial approximation are shown in fig . [ withoutmn ] ( a ) and ( b ) , respectively , for the structure whose corresponding subband dispersion is presented in fig . [ rashba ] ( c ) . as far as the landau level fan diagrams do not show a notable difference we will use the axial approximation in the following . as a result of the inverted band structure the lowest landau level of the h1 conduction subband and the highest landau level of the h2 valence subband cross at @xmath158 t. such a behavior is specific for type - iii qw s and has been examined theoretically and experimentally ( see , for example , ref . ) . the lowest h1 landau level , which corresponds to the quantum number @xmath140 , has purely heavy hole character , while the other landau levels of the h1 subband are mixed states . the h2 landau level with @xmath159 contains both heavy and light states . at @xmath160 t this level becomes the lowest level of the conduction band . [ h ] landau levels of the e2 , h1 and h2 subbands for a @xmath36-type hgte / hg@xmath21cd@xmath22te(001 ) qw as a function of magnetic field ( a ) with and ( b ) without applying the axial approximation . @xmath107 nm , @xmath161 @xmath84 , @xmath162 , and @xmath108 k. the thick line represents the chemical potential.,title="fig:",width=453 ] in the presence of a magnetic field , the @xmath6 exchange interaction of the @xmath42 and @xmath43 band electrons with the 3@xmath163 electrons of mn in hg@xmath17mn@xmath18te layer influences the band structure of the qw . such an interaction can be taken into account by adding an appropriate exchange term ( @xmath164 ) to the hamiltonian [ eq . ( [ hamilt ] ) ] in accordance with refs . and , which leads to @xmath165 where @xmath166 is the spin operator of the band electrons at the position @xmath167 , @xmath168 is the total spin operator of the @xmath36th mn ion at position @xmath169 , and @xmath170 is the electron - ion exchange integral . since the electron wave function is extended , the spin operator @xmath168 can be replaced by the thermal average over all states of mn moments @xmath171 for a magnetic field in the @xmath28-direction ( mean field approximation ) . moreover , within the virtual crystal approximation , @xmath170 can be replaced by @xmath172 , where @xmath173 is mole fraction of mn , and the summation is now carried out over all cation sites . the exchange term in eq . ( [ heisenberg ] ) then becomes@xcite @xmath174 the average @xmath175 of the @xmath28 component of mn spin in the approximation of non - interacting magnetic moments is determined by the empirical expression:@xcite @xmath176 where @xmath177 is the brillouin function for a spin of @xmath178 ; @xmath179 is the @xmath180-factor of mn ; and the effective spin @xmath181 and the effective temperature @xmath182 account for the existence of clusters and antiferromagnetic interaction between mn ions . the values for @xmath181 and @xmath183 are taken from the literature.@xcite the matrix elements of @xmath164 in terms of the bloch functions [ eqs . ( [ basisset ] ) ] have the form : @xmath184 with @xmath185 here , @xmath186 is the number of unit cells per unit volume ; @xmath187 and @xmath188 are constants which describe the exchange interaction according to the @xmath189 and @xmath190 exchange integrals , respectively . experimental values for @xmath187 and @xmath188 can be found , for example , in ref . . the @xmath6 exchange interaction changes the spin splitting of the conduction and valence bands in a magnetic field . in the parabolic approximation the effective @xmath180-factor for the @xmath44 states can be described by the following equation [ cf . ( [ zeeman ] ) , ( [ hex ] ) and ( [ ac ] ) ] : @xmath191 where @xmath192 is the @xmath180-factor of the band electrons ( without exchange term ) . the effect of the exchange interaction on the @xmath16 states can be expressed by replacing the parameter @xmath63 with @xmath193 the influence of the @xmath6 exchange interaction on the band structure is obvious when we compare the landau levels in fig . [ withoutmn ] for the non - magnetic structure with that in fig . [ withmn ] for a magnetic hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te(001 ) qw . the qw width , 2deg density and temperature are the same in both cases . the subband dispersion for the magnetic qw under consideration is given in fig . [ rashba ] ( d ) . the parameters @xmath194 ev , @xmath195 ev , @xmath196 and @xmath197 k are taken from the literature.@xcite due to the exchange interaction the lowest h1 landau level with quantum number @xmath140 ( which contains pure @xmath198 bloch components ) is bent upwards for low magnetic fields . in contrast to the non - magnetic case , pairs of landau levels from the h1 subband cross even at moderate magnetic fields . at high magnetic fields the ordering of the levels is the same as for non - magnetic structures ( fig . [ withoutmn ] ) . such behavior was also reported for @xmath36-type hg@xmath17mn@xmath18te mixed crystals.@xcite the crossing of the lowest landau level of the h1 subband with the @xmath159 landau level of the h2 subband occurs at lower magnetic fields ( @xmath199 t ) due to the exchange enhanced shift towards higher energy of the h2 level . [ h ] landau levels of the e2 , h1 and h2 subbands for a @xmath36-type hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te(001 ) qw as a function of magnetic field ( @xmath107 nm , @xmath83 @xmath84 , @xmath162 , and @xmath79=4.2 k ) . the thick line represents the chemical potential.,title="fig:",width=234 ] in order to compare the calculations with experimental data , the density of states ( dos ) at the fermi level has to be calculated from the landau level spectrum ( fig . [ withmn ] ) , because experimentally the landau level structure becomes visible through the magnetic field dependence of the longitudinal resistance . the shubnikov - de haas ( sdh ) oscillations which are observed in the experiments are directly related to changes of the dos at the fermi energy . assuming a gaussian broadening of the landau levels , the dos is given by@xcite @xmath200 where the summation runs over all landau levels . @xmath201 ( @xmath202=1 t ) is the landau level broadening parameter.@xcite in fig . [ dos ] the calculated dos for the landau level spectrum of fig . [ withmn ] is presented together with the sdh measurement of a sample.@xcite growth and transport characterization parameters ( the qw width , the 2deg density , the doping profile , etc . ) have been used for the band structure calculations . the broadening parameter has been chosen to be @xmath203=1 mev . the main features such as oscillation period , beating nodes and maxima are in good agreement . for a more quantitative comparison the magnetic field dependence of the diffusion constant had to be taken into account.@xcite [ h ] density of states of the @xmath36-type hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te qw at the fermi level ( thick line ) compared with experimental sdh oscillations ( thin line).,title="fig:",width=234 ] a detailed description has been presented of self - consistent band structure calculations within an eight - band @xmath204 model in the envelope function approach with a special emphasis on type - iii hgte / hg@xmath11cd@xmath12te qw structures . this model is an important tool for the interpretation of both optical@xcite and transport@xcite experiments , where strong spin - orbit subband splitting effects are observed . the model has been adopted to account for @xmath205 exchange effects when magnetic ( mn ) ions are introduced into the structures . self - consistently calculated band structure and dos are in good agreement with experimental transport results for hg@xmath19mn@xmath20te / hg@xmath21cd@xmath22te qw s . the calculated band structure of hg@xmath17mn@xmath18te / hg@xmath11cd@xmath12te qw s and the ensuing comparison with experimental data make it possible to understand the mutual influence of the @xmath6 exchange interaction and the two - dimensional confinement effects on the transport properties . moreover , the effect of the qw parameters ( width , doping profile , etc . ) on the values of @xmath187 , @xmath188 , @xmath181 and @xmath183 can now be studied by a direct comparison of experimental data and band structure calculations . we thank r.r . gerhardts , j. sinova and a.h . macdonald for fruitful discussions . financial support from the deutsche forschungsgemeinschaft ( sfb410 and gz:436 wer ) is gratefully acknowledged . the approach of los _ et al . _ @xcite can be used to carry out calculations for ( @xmath207 ) oriented qw s . since the @xmath44 states as well as the coupling between @xmath44 and @xmath16 ( @xmath45 ) bands are spherically symmetric , only the bloch basis functions @xmath208 ( @xmath209 ) [ eqs . ( [ basisset ] ) ] have to be transformed into symmetry adapted basis functions @xmath210 . in addition , the coordinate system is rotated to ( @xmath211,@xmath212,@xmath213 ) such that the @xmath213-axis is oriented along the [ @xmath206 growth direction . the corresponding terms are added to the matrix elements of eqs . ( [ matrel ] ) . the corrections which depend on @xmath115 and @xmath214 ( @xmath215 ) are as follows ( for simplicity the new coordinates are referred as @xmath72 , @xmath173 , @xmath28):@xcite @xmath216 @xmath217\{\mu , k_{z}\}\bigl),\nonumber\end{aligned}\ ] ] @xmath218 the above terms are separated into axial ( index @xmath128 ) and cubic ( index @xmath219 ) components . it can be shown that the axial and non - axial approximations give the same result only for ( 001 ) and ( 111 ) oriented structures at @xmath94 . the effects of strain due to the lattice mismatch between hgte and hg@xmath11cd@xmath12te can be taken into consideration by applying a formalism introduced by bir and pikus.@xcite terms proportional to the strain tensor @xmath220 are added to the matrix elements of the hamiltonian [ eq . ( [ hamilt ] ) ] ; @xmath221 . the bir - pikus hamiltonian @xmath222 is derived from eq . ( [ hamilt ] ) by the following substitution : @xmath223 the strain tensor components ( @xmath224 ) transform as the product @xmath225 and are determined using the model of de caro _ _ et al.__@xcite the band structure parameters have to be replaced by the deformation potentials ; @xmath226 here , @xmath227 and @xmath128 are the hydrostatic , and @xmath228 and @xmath229 the uniaxial deformation potentials . due to the strain , the coupling matrix elements between conduction ( @xmath44 ) and valence ( @xmath16 , @xmath45 ) bands have additional terms which are proportional to the kane momentum matrix element @xmath56.@xcite these elements are actually quite small and consequently are neglected here . the bir - pikus hamiltonian for ( 001 ) oriented qw s can be written as@xcite @xmath230 where @xmath231 @xmath232 is the trace of the strain tensor . for ( @xmath207 ) oriented structures the hamiltonian should be presented in the symmetry adapted set of basis functions as described in appendix a. the transformed hamiltonian has the form of eq . ( [ birpikus ] ) , with appropriate corrections to the matrix elements . these corrections can be derived from eqs . ( [ kklv ] ) , ( [ kklr ] ) and ( [ kkls ] ) by the substitutions indicated in eqs . ( [ tensor ] ) and ( [ potentials ] ) . if the strain tensor has non - zero off - diagonal components ( shear components ) , internal electric fields are generated in the qw due to the piezoelectric effect . we have calculated the strain - induced polarization and electric fields as described in ref . , and have found that the influence of piezoelectric fields on the band structure of fully strained hgte / hg@xmath11cd@xmath12te ( 112 ) heterostructures is negligible.@xcite

the band structure of semimagnetic hg@xmath0mn@xmath1te / hg@xmath2cd@xmath3te type - iii quantum wells has been calculated using eight - band @xmath4 model in an envelope function approach . details of the band structure calculations are given for the mn free case ( @xmath5 ) . a mean field approach is used to take the influence of the @xmath6 exchange interaction on the band structure of qw s with low mn concentrations into account . the calculated landau level fan diagram and the density of states of a hg@xmath7mn@xmath8te / hg@xmath9cd@xmath10te qw are in good agreement with recent experimental transport observations . the model can be used to interpret the mutual influence of the two - dimensional confinement and the @xmath6 exchange interaction on the transport properties of hg@xmath0mn@xmath1te / hg@xmath2cd@xmath3te qw s .

actinide dioxides form a group of important materials from technological viewpoints of a nuclear reactor fuel and a heterogeneous catalyst . on the other hand , this material group has been actively investigated also from a viewpoint of basic science because of its high symmetry of the fluorite structure of the space group @xmath7.@xcite in the circumstance of such high symmetry of crystal structure , it is possible to observe peculiar ordering of multipole higher than dipole , when we change the kind of actinide ions . among several magnetic properties of actinide dioxides , a mysterious low - temperature ordered phase of npo@xmath6 has attracted continuous attention in the research field of condensed matter physics . the phase transition in npo@xmath6 has been confirmed in 1953 from the observation of a peak in the specific heat around 25 k.@xcite due to the behavior of magnetic susceptibility,@xcite it has been first considered that the antiferromagnetic order occurs . unfortunately , no dipole moments have been observed in the low - temperature phase and a puzzling situation has continued . neutron scattering experiments have revealed that the ground state of the crystalline electric field ( cef ) potential is @xmath5 , @xcite which carries multipole moments . then , from several phenomenological works on the ordered phase , a key role of octupole degree of freedom has been focused . @xcite in fact , the octupole ordering has been strongly suggested by @xmath8o - nmr experiment @xcite and also by inelastic neutron scattering study.@xcite as for the microscopic origin of such higher - order multipole ordering , it has been shown that octupole order is stabilized by the orbital - dependent superexchange interaction , obtained by the second - order perturbation of @xmath0 electron hopping in the @xmath5 degenerate hubbard model on a fcc lattice.@xcite recently , significant contribution of dotriacontapole moment has been also pointed out.@xcite since the multipole moments originate from @xmath0 electrons , it seems to be natural to consider the hubbard - like model of @xmath0 electrons . however , from the crystal structure of actinide dioxides , it is also important to include explicitly @xmath1 electrons , since actinide ion is surrounded by eight oxygens and the main hopping process between nearest neighbor sites should occur from the @xmath2-@xmath3 hybridization . in this sense , @xmath2-@xmath3 model is more realistic hamiltonian for actinide dioxides . in fact , the @xmath2-@xmath3 model for actinide dioxides has been analyzed in the fourth - order perturbation theory in terms of @xmath2-@xmath3 hybridization . @xcite then , it has been revealed that octupole order actually occurs even when we include oxygen @xmath1 electrons . however , there has been a peculiar point that the octupole phase appears _ only _ for the small absolute value of @xmath9 , where @xmath10 and @xmath11 are slater - koster integrals between @xmath2 and @xmath3 orbitals . the reason of the sensitivity of the octupole ordered phase concerning the @xmath2-@xmath3 hybridization has not been understood yet . in order to clarify the role of @xmath2-@xmath3 hybridization for the appearance of octupole ordering , maehira and hotta have performed the band - structure calculations for actinide dioxides by a relativistic linear augmented - plane - wave method with the exchange - correlation potential in a local density approximation.@xcite it has been found that the energy bands in the vicinity of the fermi level are mainly due to the hybridization between actinide @xmath0 and oxygen @xmath1 electrons . it has been also pointed out that the electronic structure at the @xmath12 point in the first brillouin zone is not consistent with that of the local cef state . one reason for this inconsistency is that the cef potentials are not satisfactorily included in the calculations , but it is difficult to control the magnitudes of cef potential and @xmath2-@xmath3 hybridization in the band - structure calculations . it is highly requested to reveal the role of @xmath2-@xmath3 hybridization for the simultaneous explanation of the octupole ordering and the local cef states . in this paper , in order to clarify the roles of hybridization between actinide @xmath0 and oxygen @xmath1 electrons for the electronic structure of actinide dioxides , we analyze the tight - binding @xmath2-@xmath3 model in detail . except for the slater - koster integrals of @xmath11 and @xmath10 , we determine the parameters in the model from the comparison with experimental results and band - structure calculations . in order to reproduce the result of the relativistic band - structure calculations and obtain the electronic structure consistent with the local cef state , we find that the slater - koster parameters for @xmath2-@xmath3 hybridization should be limited in a certain range . a typical result is found for @xmath13 and @xmath14 ev , which is consistent with the condition for the appearance of the octupole ordering . the organization of this paper is as follows . in sec . ii , in order to make this paper self - contained , we briefly review the relativistic band - structure calculations for actinide dioxides . it is meaningful to define the problems included in the band - structure calculations . in sec . iii , we explain a way to construct the @xmath2-@xmath3 model in the tight - binding approximation . then , we determine the parameters of the model , except for @xmath10 and @xmath11 , from the comparison with the experimental and band - structure calculation results . in sec . iv , we depict the energy band structure of the @xmath2-@xmath3 model by changing the values of @xmath2-@xmath3 hybridization . we deduce the reasonable regions for @xmath10 and @xmath11 . in sec . v , we discuss some future problems concerning the electronic structure of actinide dioxides . finally , we summarize this paper . throughout this paper , we use such units as @xmath15=@xmath16=1 . let us briefly review the band - structure calculation results in order to clarify the problem in the understanding of electronic structure of actinide dioxides . as for details , readers should consult ref . . we have performed the calculations by using the relativistic linear augmented - plane - wave ( rlapw ) method . we assume that all @xmath0 electrons are itinerant and perform the calculations in the paramagnetic phase . note that we should take into account relativity even in the calculations for solid state physics because of large atomic numbers of the constituent atoms . the spatial shape of the one - electron potential is determined in the muffin - tin approximation . we use the exchange and correlation potential in a local density approximation ( lda ) . the self - consistent calculation is carried out for the experimental lattice constant for actinide dioxides . obtained by the self - consistent rlapw method . note that we pick up only @xmath0 and @xmath1 bands around @xmath17 , which indicate the position of the fermi level.,width=302 ] in fig . 1 , we show a typical result for npo@xmath6 along the symmetry axes in the brillouin zone . in the energy band structure in the vicinity of @xmath17 , there always occurs a hybridization between actinide @xmath0 and oxygen @xmath1 states for actinide dioxides . the lowest six bands originate from the oxygen @xmath1 states and are fully occupied and the width of oxygen @xmath1 band is about 4.76 ev . narrow bands lying in the region 4.5 - 7.5 ev are the @xmath0 bands which are split into two subbands by the spin - orbit interaction . the spin - orbit splitting in the @xmath0 states is estimated as 0.95 ev , which is consistent with that for isolated neutral np atom . note that in the lda calculation , we find the metallic state for npo@xmath6 , not the insulating state . this point will be discussed later . here we remark that @xmath18 doublet and @xmath19 quartet levels appear around @xmath17 at the @xmath12 point . it should be noted that the @xmath4 level is lower than the @xmath5 in our band - structure calculations . however , from the cef analysis on the basis of the @xmath20-@xmath20 coupling scheme , @xmath5 becomes lower than @xmath4 in actinide dioxides . when we accommodate @xmath0 electrons in @xmath5 orbitals , we obtain @xmath21 triplet for @xmath22=2 , @xmath5 quartet for @xmath22=3 , and @xmath23 singlet for @xmath22=4 , as experimentally found in the cef ground states of uo@xmath6 @xcite , npo@xmath6 @xcite , and puo@xmath6 @xcite . note here that @xmath22 denotes the number of local @xmath0 electrons . in order to resolve the problems , it is necessary to improve the method to include the effect of cef potentials beyond the simple estimation of the madelung potential energy . however , it is a difficult task to perform such improvement concerning the formulation of the band - structure calculation . thus , in this paper , we choose an alternative method to exploit the tight - binding @xmath2-@xmath3 model for the purpose to understand the role of @xmath2-@xmath3 hybridization for the change of cef states in the tight - binding model . by changing the parameters in the @xmath2-@xmath3 model , we attempt to clarify the key quantities which characterize the electronic structure of actinide dioxides . before proceeding to the construction of a tight - binding model for actinide dioxides with the fluorite structure , first let us define the unit cell including one actinide ion and two oxygen ions , as shown in the fig . 2 . the basis vectors of the fcc lattice are given by @xmath24 , @xmath25 , and @xmath26 , where @xmath27 is the lattice constant . thus , in fig . 2 , positions of adjacent four actinide ions are given by @xmath28 , @xmath29 , @xmath30 , and @xmath31 , where @xmath28 denotes the position vector for one actinide ion . the positions of eight nearest - neighbor oxygen ions are given by @xmath32 , @xmath33 , @xmath34 , @xmath35 , @xmath36 , @xmath37 , @xmath38 , and @xmath39 . note that the two oxygens , o1 and o2 , in the same unit cell are specified by @xmath40 for o1 and @xmath41 for o2 , respectively . . large solid circles denote actinide ions of which positions are given by @xmath28 , @xmath29 , @xmath30 , and @xmath31 . small open circles indicate oxygen ions . note that two oxygen ions , o1(@xmath28 ) and o2(@xmath28 ) , are included in the unit cell containing one actinide ion specified by @xmath28 . , width=321 ] now we define the basis of @xmath2 electrons when we consider the electronic model for actinide dioxides with the fluorite structure . for the purpose , we solve the problem of one @xmath2 electron in the cef potential . the cef hamiltonian is written as @xmath42 where @xmath43 is the annihilation operator of @xmath2 electron at site @xmath28 with spin @xmath44 in the orbital specified by @xmath45 . note that @xmath45 is the @xmath46-component @xmath45 of angular momentum @xmath47 . we note also that the spin - orbit coupling is not included at this stage . since the fluorite structure belongs to o@xmath48 point group , @xmath49 is given by using a couple of cef parameters @xmath50 and @xmath51 for angular momentum @xmath47 as@xcite @xmath52{15}(b^{0}_{4}-21b^{0}_{6})\nonumber \\ & & b_{2,-2}=300b^{0}_{4}+7560b^{0}_{6}\nonumber,\end{aligned}\ ] ] note the relation of @xmath53 . after performing the diagonalization of @xmath54 , we obtain three kinds of cef states : @xmath55 singlet ( xyz ) , @xmath56 triplet ( @xmath57 , @xmath58 , @xmath59 ) , and @xmath21 triplet ( @xmath60 , @xmath61 , @xmath62 ) . the corresponding cef energies are given by @xmath63 , @xmath64 , and @xmath65 . note that these seven states are elements of cubic harmonics for @xmath47 . in the traditional notation , we express cef parameters @xmath66 and @xmath67 as @xmath68 and @xmath69with @xmath70=15 and @xmath71=180 for angular momentum @xmath47.@xcite note that @xmath72 specifies the cef scheme for @xmath73 point group , while @xmath74 determines an energy scale for the cef potential . @xmath75 , \nonumber \\ e(\gamma_4)=4w[6x-5(1-|x| ) ] , \\ e(\gamma_5)=-4w[2x-9(1-|x| ) ] . \nonumber\end{aligned}\ ] ] the value of @xmath74 and @xmath72 will be discussed later . since we will construct the model in the cubic system , it seems to be natural to use these cubic harmonics as @xmath2-electron basis function . thus , in the following , we define @xmath76 as the index to distinguish the orbitals of cubic harmonics . note that @xmath76 takes @xmath77 and the definitions are as follows : 1 : xyz , 2:@xmath57 , 3:@xmath58 , 4 : @xmath59 , 5:@xmath60 , 6:@xmath61 , and 7:@xmath62 . the corresponding energy @xmath78 is given by the above equations . the hamiltonian is given by @xmath79 where @xmath80 and @xmath81 denote @xmath2- and @xmath3-electron part , respectively , while @xmath82 indicates @xmath2-@xmath3 hybridization term . in the following , we explain the construction of each term . the @xmath2-electron part is given by @xmath83 f^{\dag}_{\bm{k}\mu\sigma}f_{\bm{k}\mu'\sigma } \nonumber\\ & + & \lambda \sum_{\bm{k},\mu,\mu',\sigma,\sigma ' } \zeta_{\mu,\sigma,\mu',\sigma ' } f^{\dag}_{\bm{k}\mu\sigma}f_{\bm{k}\mu'\sigma'},\end{aligned}\ ] ] where @xmath84 is the annihilation operator of @xmath2 electron with spin @xmath44 in the orbital @xmath76 , @xmath85 is the @xmath2-electron dispersion due to the hopping between nearest neighbor actinide ions , @xmath86 is the @xmath2-electron level , @xmath78 denotes the cef potential energy of @xmath76 orbital , @xmath87 is the spin - orbit interaction , and @xmath88 is the spin - orbit matrix element . concerning the expression of the spin - orbit coupling , it is necessary to step back to the basis of the spherical harmonics . on the basis labelled by @xmath45 , the spin - orbit interaction @xmath89 is expressed as @xmath90 and zero for the other cases . by transforming the basis from @xmath45 to @xmath76 , we obtain @xmath91 in eq . ( [ eq : hf ] ) . the @xmath2-electron dispersion in eq . ( [ eq : hf ] ) is expressed as @xmath92 where @xmath93 denotes the vectors connecting twelve nearest neighbor sites of the fcc lattice and @xmath94 indicates the @xmath2-electron hopping amplitude between @xmath76 and @xmath95 orbitals along the direction of @xmath93 . here we note that @xmath93 runs among @xmath96 , @xmath97 , @xmath98 , @xmath99 , @xmath100 , and @xmath101 . the hopping integral @xmath94 is expressed by using the slater - koster table @xcite here we consider only the @xmath2-electron hopping through @xmath44 bond @xmath102 . the @xmath2-@xmath3 hybridization term is written as @xmath103,\ ] ] where @xmath104 is the annihilation operator of @xmath3 electron with spin @xmath44 in the orbital @xmath105 of @xmath20-th oxygen and @xmath20 denotes the label of oxygen ions in the unit cell , as shown in fig . 1 . note that @xmath105 runs among x , y , and z which correspond to @xmath106 , @xmath107 , and @xmath108 orbitals , respectively . the hybridizations @xmath109 and @xmath110 are , respectively , written as @xmath111 and @xmath112 where @xmath113 denotes the hopping amplitude between @xmath2 and @xmath3 orbitals along @xmath114 direction . here we note that @xmath114 runs among @xmath115 . the hopping integral @xmath113 is represented in terms of @xmath10 and @xmath11 by using the slater - koster table.@xcite the @xmath3-electron part is expressed as @xmath116 p^{\dag}_{i \bm{k}\nu\sigma}p_{j \bm{k}\nu'\sigma},\ ] ] where @xmath117 is the @xmath3-electron dispersion , @xmath118 and @xmath20 denote the label of oxygen ions in the unit cell , as shown in fig.1 , and @xmath119 is the @xmath3-electron level . note that we take into account nearest neighbor and next nearest neighbor hoppings for @xmath3 electrons . we also note that the relations of @xmath120 and @xmath121 . the diagonal part is given by @xmath122 where the hopping amplitudes are given by @xmath123 here @xmath124/2 $ ] , where @xmath125 and @xmath126 denote the slater - koster integral of @xmath3 electron among next - nearest neighbor oxygen sites . as for the off - diagonal parts , we obtain @xmath127 \nonumber \\ & + & ( pp\pi)[e^{i\bm{k}\cdot(\bm{a}_1+\bm{a}_2 ) } + e^{i\bm{k}\cdot\bm{a}_3 } ] \nonumber \\ & + & ( pp\pi)[e^{i\bm{k}\cdot(\bm{a}_2+\bm{a}_3 ) } + e^{i\bm{k}\cdot\bm{a}_1}],\end{aligned}\ ] ] @xmath128 \nonumber \\ & + & ( pp\sigma)[e^{i\bm{k}\cdot(\bm{a}_1+\bm{a}_2 ) } + e^{i\bm{k}\cdot\bm{a}_3 } ] \nonumber \\ & + & ( pp\pi)[e^{i\bm{k}\cdot(\bm{a}_2+\bm{a}_3 ) } + e^{i\bm{k}\cdot\bm{a}_1}],\end{aligned}\ ] ] and @xmath129 \nonumber \\ & + & ( pp\pi)[e^{i\bm{k}\cdot(\bm{a}_1+\bm{a}_2 ) } + e^{i\bm{k}\cdot\bm{a}_3 } ] \nonumber \\ & + & ( pp\sigma)[e^{i\bm{k}\cdot(\bm{a}_2+\bm{a}_3 ) } + e^{i\bm{k}\cdot\bm{a}_1}].\end{aligned}\ ] ] other off - diagonal components are all zeros . the tight - binding hamiltonian includes many parameters . here we try to fix some of them from the experimental and band - structure calculations results . _ ( i ) cef parameters . _ it should be noted that it is possible to reproduce the cef states of actinide dioxides , when we accommodate plural numbers of @xmath2 electrons in the level scheme in which @xmath5 is lower than @xmath4 . as already mentioned in sec . ii , we obtain @xmath21 triplet for @xmath22=2 , @xmath5 quartet for @xmath22=3 , and @xmath23 singlet for @xmath22=4 , as experimentally found in the cef ground states of uo@xmath6 @xcite , npo@xmath6 @xcite , and puo@xmath6 @xcite . thus , in the present tight - binding model , we set @xmath130 ev and @xmath131 in order to reproduce that @xmath132 quartet is the ground state and @xmath133 is the excited state with the excitation energy of about 0.2 ev . _ ( ii ) spin - orbit coupling . _ from the relativistic band - structure calculation for actinide atom , the splitting energy between @xmath20=5/2 and @xmath20=7/2 states has been found to be about 1 ev . since the splitting energy is given as @xmath134 with the use of spin - orbit coupling @xmath87 , we fix it as @xmath135 ev . _ ( iii ) @xmath2- and @xmath3-electron levels . _ in this paper , the @xmath2-electron level @xmath136 is set as the origin of energy , leading to @xmath137 . on the other hand , the @xmath3-electron level @xmath119 is considered to be @xmath138 ev from the comparison of the relativistic band - structure calculation results.@xcite _ ( iv ) slater - koster integrals . _ in the model , we use seven slater - koster integrals as @xmath102 , @xmath10 , @xmath11 , @xmath139 , @xmath140 , @xmath125 , and @xmath126 . among them , concerning the @xmath3-electron hoppings , we introduce the ratio @xmath141 between nearest and next nearest neighbor hopping amplitudes , given by @xmath142 . from the ratio of the distances of nearest and next nearest neighbor sites , we set @xmath143.@xcite as for @xmath139 and @xmath140 , we determine them as @xmath139=@xmath144 ev and @xmath140=@xmath145 ev , after several trials to reproduce the structure of the wide @xmath3 bands in the relativistic band structure calculations . concerning @xmath102 , it is related with the bandwidth @xmath74 of @xmath2 electrons in the @xmath20=5/2 states on the fcc lattice . in the limit of infinite @xmath87 , we have obtained @xmath74 as @xmath146@xmath147@xmath148.@xcite note that for the case of finite @xmath87 , the width of @xmath20=5/2 bands is deviated from @xmath74 , but when @xmath87 is large enough as in actual actinide compounds , the bandwidth is found to be almost equal to @xmath74 . from the comparison with the relativistic band - structure calculation results , the width of @xmath20=5/2 bands is 0.5@xmath1490.7 ev , suggesting that @xmath102 is in the order of 0.1 ev . then , we set @xmath102=0.1 ev in the present model . in the following calculations , due to the diagonalization of the hamiltonian , we depict the tight - binding bands by changing @xmath10 and @xmath11 , which are believed to be key parameters to understand the electronic structure of actinide dioxides . = @xmath11=0 and ( b ) @xmath10=@xmath150 ev and @xmath11=@xmath151 ev . , width=302 ] now we show our results of the diagonalization of the tight - binding model . note that in the following figures of the band structure , `` 0 '' in the vertical axis indicates the origin of the energy , not the fermi level @xmath17 . if it is necessary to draw the line of @xmath17 , we set it from the condition of @xmath152=3 for tetravalent np ion in npo@xmath6 , where @xmath152 denotes the average number of @xmath2 electrons per actinide ion . in the present paper , we do not take care of the difference in actinide ions . first we consider the case in which the @xmath2-@xmath3 hybridization is simply suppressed . in fig . 3(a ) , we show the tight - binding bands for @xmath10=@xmath11=0 along the lines in the first brillouin zone . we obtain the @xmath2 and @xmath3 bands which are not hybridized with each other and @xmath2 bands split into @xmath20=5/2 and @xmath20=7/2 . note that @xmath5 becomes lower than @xmath4 at the @xmath12 point due to the effect of local cef potentials . we observe some degeneracy in @xmath3 bands which will be lifted by @xmath2-@xmath3 hybridization . in our first impression , in spite of the simple suppression of the @xmath2-@xmath3 hybridization , the overall structure of @xmath2 and @xmath3 bands seems to be similar to that of the relativistic band - structure calculations in fig . 1 . however , some significant difference is found in the @xmath3-band structure . for instance , we find the level crossing in the @xmath3-band structure of fig . 3(a ) between the l and @xmath12 points , but we do not observe such behavior in fig . 1 . such difference originates from the simplification to consider only actinide @xmath0 and oxygen @xmath1 electrons . the difference in the @xmath3-band structure is not further discussed in this paper . next we include the @xmath2-@xmath3 hybridization as @xmath10=1 ev and @xmath11=@xmath151 ev in fig . due to the effect of @xmath2-@xmath3 hybridization , we find additional dispersion in @xmath2 and @xmath3 bands . in particular , the @xmath3-band structure becomes similar to that in the relativistic band - structure calculations . in this case , we still observe that @xmath5 is lower than @xmath4 at the @xmath12 point . let us now consider the cases of negative @xmath11 by keeping the value of @xmath10=1 ev . in figs . 4(a ) and 4(b ) , we show the results for @xmath11=@xmath153 ev and @xmath154 ev , respectively . for @xmath11=@xmath153 ev , we do not find significant difference in the band structure from the case of @xmath11=@xmath151 ev . however , for @xmath11=@xmath154 ev , we find that @xmath4 is lower than @xmath5 at the @xmath12 point . regarding the cef states at the @xmath12 point , the @xmath2-@xmath3 model with @xmath10=1 ev and @xmath11=@xmath154 ev seems to reproduce the relativistic band - structure calculation results . note that in the @xmath3-band structure , we find the level crossing of two low - energy bands along the line between w and l points , which has not been observed in the band - structure calculation . however , as mentioned above , we do not further pursue the difference in the @xmath3-band structure . = @xmath150 ev and @xmath11=@xmath153 ev and ( b ) @xmath10=@xmath150 ev and @xmath11=@xmath154 ev . , width=302 ] here we turn our attention to the @xmath2-electron states at the @xmath12 point . in the relativistic band - structure calculations for npo@xmath6,@xcite we have already pointed out that the @xmath4 level becomes lower than that of @xmath5 , in sharp contrast to the local cef state in the @xmath20-@xmath20 coupling scheme expected from the experimental results . this is due to the fact that the cef potential is not included satisfactorily in the relativistic band - structure calculation . on the other hand , the cef potential is included in the tight - binding model within the point charge approximation and the change of the level scheme at the @xmath12 point can be explained by the @xmath2-@xmath3 hybridization . when we do not consider the @xmath2-@xmath3 hybridization , we find that @xmath5 level becomes lower than that of @xmath4 , but with the increase of the effect of @xmath2-@xmath3 hybridization , the order of the level at the @xmath12 point is converted . namely , the order of @xmath4 and @xmath5 levels is determined by the competition between the cef potential and the @xmath2-@xmath3 hybridization . in this sense , the cef potential is not included satisfactorily in comparison with the @xmath2-@xmath3 hybridization in the band - structure calculation . in the fluorite crystal structure of ano@xmath6 , actinide ion is surrounded by eight oxygen ions in the [ 111 ] and other equivalent directions . thus , the @xmath4 orbital is penalized from the viewpoint of electrostatic energy , since its wavefunction is elongated along the [ 111 ] directions . however , the wavefunctions of two @xmath5 orbitals are expanded in the directions of axes . namely , it is qualitatively understood that @xmath5 level is lower than @xmath4 one in the actinide dioxides . from the viewpoint of the overlap integral between actinide @xmath0 and oxygen @xmath1 electrons , we expect that the hybridization of @xmath4 orbital is larger than that of @xmath5 . thus , due to the effect of @xmath2-@xmath3 hybridization , the @xmath4 level becomes lower than @xmath5 , even if the local cef ground state is @xmath5 . when the effect of @xmath2-@xmath3 hybridization is relatively larger than that of the cef potential , it is possible to observe that @xmath4 is lower than @xmath5 , as actually found in the relativistic band - structure calculation results . we emphasize that it is one of the key points concerning the @xmath2-@xmath3 hybridization to understand the electronic structure of actinide dioxides . between @xmath4 and @xmath5 states at the @xmath12 point in the @xmath20=5/2 bands as a function of @xmath11 for @xmath10=@xmath155 ev ( solid curve ) , @xmath10=@xmath156 ev ( broken curve ) , and @xmath10=@xmath157 ev ( dotted curve ) . a positive @xmath158 denotes that the energy of @xmath4 is larger than that of @xmath5 . , width=302 ] in fig . 5 , we depict the energy difference @xmath158 between the @xmath5 and @xmath4 states at the @xmath12 point as functions of @xmath11 for several values of @xmath10 . note that @xmath158 is positive when @xmath5 is lower than @xmath4 . for @xmath10=0 , we find that @xmath158 is positive in the region of @xmath159 ev . when we change the value of @xmath10 , @xmath158 is found to be maximum at @xmath160 due to the effective disappearance of the @xmath2-@xmath3 hybridization between actinide @xmath4 and oxygen @xmath1 electrons . readers may consider that the absolute value of @xmath11 should not be so small only for the purpose to keep the order of the local cef states . however , if we increase the absolute value of @xmath11 for @xmath161 ev , we should remark that the @xmath2- and @xmath3-electron bands are significantly changed from those in the relativistic band - structure calculation results . thus , from the viewpoints of the local cef states and the comparison with the band - structure calculations , the reasonable parameters are found in the case of small @xmath162 for @xmath161 ev . in this paper , we have analyzed the tight - binding model for ano@xmath6 in comparison with the local cef states and the result of the relativistic band - structure calculations . we have concluded that @xmath162 should be small for the case of @xmath161 ev in our tight - binding model in order to keep the cef levels at the @xmath12 point . we have also emphasized that such a condition coincides with that for the octupole ordering on the basis of the @xmath2-@xmath3 model.@xcite namely , the condition to keep the local @xmath5 ground state is consistent with the appearance of the ordering of magnetic octupole which is composed of complex spin and orbital degrees of freedom . here we provide a comment on the local cef state in the band - structure calculations . as long as we perform the band - structure calculations with in the lda , it is found that the @xmath4 state becomes lower than the @xmath5 at the @xmath12 point , in contrast to the local cef state expected from the experiment . in this paper , we have proposed the scenario to control the effect of @xmath2-@xmath3 hybridization on the cef state , but it should be remarked that in the lda calculation , we could @xmath163 obtain insulating state corresponding to the multipole ordering for npo@xmath6.@xcite in order to improve this point , we need to consider the effect of the coulomb interactions , but it is a serious problem . one way for this problem is to employ the lda+@xmath164 method . in fact , it has been reported that we the ordered state with octupole and higher multipoles can be reproduced,@xcite suggesting that the @xmath5 state is lower than @xmath4 in the electronic structure . the effective inclusion of the coulomb interaction is an alternative scenario to understand the cef state consistent with the experiments . although we have not discussed the difference in electronic structure due to the change of actinide ions in this paper , it is naively expected that the difference between @xmath136 and @xmath119 becomes small in the order of th , u , np , pu , am , and cm from the chemical trends in actinide ions and the previous band - structure calculations . on the other hand , the change of @xmath2-@xmath3 hybridization among actinide dioxides may play more important role to explain the effect of the difference in actinide ions . it is an interesting future problem to clarify the key issue to understand the difference in electronic structure of actinide dioxides . in summary , we have constructed the @xmath2-@xmath3 model in the tight - binding approximation . we have determined the parameters by the experimental results and the relativistic band - structure calculations . it has been concluded that the absolute value of @xmath11 should be small for @xmath10=1 ev in order to reproduce simultaneously the local cef states and the band - structure calculation results . the small value of @xmath162 is consistent with the condition to obtain the octupole ordering in the previous analysis of the @xmath2-@xmath3 model . we believe that the present tight - binding model will useful to extract the essential point of the electronic structure of actinide dioxides from the complicated band - structure calculation results . the authors thank s. kambe , k. kubo , and y. tokunaga for discussions on actinide dioxides . this work has been supported by a grant - in - aid for for scientific research on innovative areas `` heavy electrons '' ( no . 20102008 ) of the ministry of education , culture , sports , science , and technology , japan .

in order to promote our understanding on electronic structure of actinide dioxides , we construct a tight - binding model composed of actinide @xmath0 and oxygen @xmath1 electrons , which is called @xmath2-@xmath3 model . after the diagonalization of the @xmath2-@xmath3 model , we compare the eigenenergies in the first brillouin zone with the results of relativistic band - structure calculations . here we emphasize a key role of @xmath2-@xmath3 hybridization in order to understand the electronic structure of actinide dioxides . in particular , it is found that the position of energy levels of @xmath4 and @xmath5 states determined from crystalline electric field potentials depends on the @xmath2-@xmath3 hybridization . we clarify the condition on the @xmath2-@xmath3 hybridization to explain the electronic structure which is consistent with the local crystalline electric field state . we briefly discuss the region of the absolute values of the slater - koster integrals for @xmath2-@xmath3 hybridization concerning the appearance of octupole ordering in npo@xmath6 .

the advent of popular manycore systems has made concurrent programming for shared memory systems an important topic . with increasing parallelism available in a single shared - memory system , performance of techniques used to communicate between concurrently executing threads , or to access memory common to many such threads , is becoming one of the most significant components of the performance of an application running on such a system . in many cases of communication - heavy tasks , simple coarse - grained locking is not enough to yield good performance , so programmers need to resort to lock - less communication and concurrent data structures . this work provides a new data structure with an implementation that can be used concurrently and does nt use locks , and uses it to create an implementation of the fetch - and - add object ( a kind of counter ) with improved memory usage . a useful abstraction in a single - threaded system is an object : an entity with a set of methods one can invoke , and a specification of semantics of these methods . naturally , a program can use multiple objects and invoke their methods in any order . we want to use a similar abstraction to model interaction between threads in a multi - threaded system . consider a set of independent , concurrently running threads ( running possibly distinct code ) that can only communicate by calling methods on some objects . we place no assumptions on the speed of execution and delays of the threads they may wait arbitrarily long before a method call . we also assume that method calls on the objects complete instanteously , and that no two of them happen at the same time . this allows us to define semantics of the objects in the same fashion we define them in the single - threaded case : methods of every object are invoked in a known order , so single - threaded semantics suffice to determine the object s behaviour . we will call such objects _ concurrent objects _ to emphasize that they may be accessed concurrently . an object we will predominantly use in this work is a cas ( compare - and - swap ) register . we will specify its semantics by providing pseudocode which correctly implements this object in a single - threaded program ( this will be our method of choice of providing object s semantics ) : * var * @xmath12 : @xmath12 @xmath13 @xmath14 true false intuitively , such a register holds a value that can be read ( by rd ) , modified unconditionally ( by wr ) and modified conditionally ( by cas ) . it s interesting due to its universality properties@xcite and because it is commonly seen in real - world hardware . [ ex - simple ] let s consider two threads , executing following pseudocodes ( x and y are two cas objects , with initial value 0 ) : 1 . x.wr(1 ) y.rd ( ) 2 . y.wr(1 ) x.rd ( ) figure [ fig - ex - simple ] presents a possible execution of such two threads . note that in any correct execution , at least one of the rd ( ) calls returns 1 . ( labx_wr_1 ) at ( 0.5,-0 + 0.5 ) x.wr(1);(0.5,-0-0.1 ) to ( labx_wr_1);(x_wr_1 ) at ( 0.5,-0 ) ; ( laby_rd_1 ) at ( 2.5,-0 + 0.5 ) y.rd()=0;(2.5,-0-0.1 ) to ( laby_rd_1);(y_rd_1 ) at ( 2.5,-0 ) ; ( laby_wr_1 ) at ( 3.5,-1 + 0.5 ) y.wr(1);(3.5,-1-0.1 ) to ( laby_wr_1);(y_wr_1 ) at ( 3.5,-1 ) ; ( labx_rd_1 ) at ( 5.5,-1 + 0.5 ) x.rd()=1;(5.5,-1-0.1 ) to ( labx_rd_1);(x_rd_1 ) at ( 5.5,-1 ) ; ( 0,0 ) to ( 6,0 ) ; ( 0,-1 ) to ( 6,-1 ) ; ( labx_wr_2 ) at ( 8.0,-0 + 0.5 ) x.wr(1);(8.0,-0-0.1 ) to ( labx_wr_2);(x_wr_2 ) at ( 8.0,-0 ) ; ( laby_rd_2 ) at ( 10.0,-0 + 0.5 ) y.rd()=1;(10.0,-0-0.1 ) to ( laby_rd_2);(y_rd_2 ) at ( 10.0,-0 ) ; ( laby_wr_2 ) at ( 8.5,-1 + 0.5 ) y.wr(1);(8.5,-1-0.1 ) to ( laby_wr_2);(y_wr_2 ) at ( 8.5,-1 ) ; ( labx_rd_2 ) at ( 10.5,-1 + 0.5 ) x.rd()=1;(10.5,-1-0.1 ) to ( labx_rd_2);(x_rd_2 ) at ( 10.5,-1 ) ; ( 7.5,0 ) to ( 11,0 ) ; ( 7.5,-1 ) to ( 11,-1 ) ; in many situations it would be convenient to have more complicated concurrent objects : counters that can be incremented , queues that can be pushed to and popped from , etc . alas , real - world hardware usually does nt implement such objects directly . we will thus implement them using cas registers or other simpler objects : we will substitute a set of simpler objects for every such object and for every method call on this object , execute a procedure that implements it instead . however , these prodecures might run concurrently , so we ca nt blindly use a single - threaded implementation of the object . obviously , we will want such substitution to preserve semantics of the program in which it was substituted . the following condition immediately implies this : an implementation of a concurrent object _ serializes _ iff for every execution of a program with the implementation substituted for the object : for every call to object s method that completes , we can define a _ serialization point _ in its execution interval ( between its first and last constituent operation ) , such that all these calls would return the same values if they happened instaneously at their serialization points . an observant reader might note that if all procedures in an implementation never terminate , the definition is vacously true . indeed , we will present some terminations guarantees separately , but first let us consider an example implementation of a simple structure . let s consider an increase - only counter : a shared object with the following semantics : * var * @xmath12 : @xmath15 @xmath12 we will see that the following implementation serializes to this shared object : * var * @xmath16 : @xmath17 [ trivial - faa - rd ] @xmath18 note that every call to inc executes exactly one successful call to @xmath19 . let us claim that the serialization point of inc is that successful call to cas , and that the serialization point of read is the call to @xmath20 . we can easily see that the number returned by rd is exactly the number of incs that serialized before that read . we can also see that this implementation satisfies some global progress property when the cas in inc fails , another inc must have suceeded very recently . in fact , it must have succeeded between the most recent execution of line [ trivial - faa - rd ] and the failed cas . still , a single inc operation can fail to terminate . consider an alternative implementation of an increment - only counter . in the following pseudocode , @xmath9 denotes the number of threads in the system . * var * @xmath21 : [ p]@xmath22 @xmath23.{\textsc{rd}}()$ ] @xmath24.{\textsc{wr}}(x + 1)$ ] @xmath25 @xmath26.{\textsc{rd}}$ ] @xmath27 here we differentiate between threads : the index @xmath28 in inc corresponds to the current thread s i d from range @xmath29 . let us first see that this implementation actually serializes to the increment - only counter . obviously , we need to choose the call to write as the serialization point of inc . from the monotonicity of entries of t we can infer that read returns a value between ( inclusive ) the number of calls to inc that have serialized before read has started and the number of calls serialized before it has finished . thus if a call to read returns @xmath12 , there is a point during its execution when exactly @xmath12 incs had been serialized . we can choose this point to be the serialization point of read . note that it need not correspond to an action executed by the implementation of read . we can see that this implementation satisfies a very strong termination condition the number of steps every procedure takes is bounded and the bound depends only on the number of threads ( and not on the behaviour of the scheduler ) . we will call such implementations _ wait - free_. for a discussion of weaker termination guarantees , see chapter 3.7 of @xcite . every object that has a single - threaded specification has a wait - free implementation@xcite using only cas registers . however , all currently - known such constructions yield objects with step complexities inflated at least by a factor of the number of threads . thus , creating faster wait - free implementations of data structures is interesting , also from a practical point of view . the object implemented in this work is a generalization of a popular building block for multithreaded objects an atomic snapshot object@xcite . its semantics can be defined by the following single - threaded pseudocode : * var * @xmath21 : [ n]@xmath30 @xmath31 \gets x$ ] @xmath32 \gets t[i]$ ] @xmath27 intuitively , this object represents an array that can be modified piecewise and read all at once . there are known wait - free implementations of an @xmath1-element atomic snapshot with @xmath4 update and @xmath33 scan step complexities@xcite . these complexities are obviously optimal . one can try to extend this structure in many ways . two of them are particularly interesting for us . first , we can merge the update and scan operations into an operation that does an update and a scan . this object , which we will call write - and - snapshot@xcite , can be defined by the following single - threaded specification : * var * @xmath21 : [ n]@xmath30 @xmath31 \gets x$ ] @xmath32 \gets t[i]$ ] @xmath27 for the second extension , let us note that in many cases one does nt need the scan operation to return the whole array , but rather some sort of its digest . the f - arrays@xcite allow one to do precisely that for digests expressible as a result of applying an associative operation to all the array s elements ( for example , if we need to retrieve the sum of them ) . if we call the operator @xmath0 , the following is a specification of an f - array : * var * @xmath21 : [ n]@xmath30 @xmath31 \gets x$ ] @xmath34 , t[1 ] , \ldots , t[n-1])$ ] there is an implementation of f - array with step complexities of update and scan being respectively @xmath35 and @xmath4 and the memory complexity being linear in @xmath1 . this work introduces a write - and - f - array : an object that combines these two modifications . its semantics are defined by the single - threaded specification below : * var * @xmath21 : [ n]@xmath30 @xmath31 \gets x$ ] @xmath34 , t[1 ] , \ldots , t[n-1])$ ] the main result of this work is an implementation of a single - writer write - and - f - array : that is , one that disallows concurrent modifications to the same array element . the implementation uses @xmath36 memory and the step complexity of the operation is @xmath37 . we then use this object to implement a fetch - and - add object . fetch - and - add is a generalization of the increment - only counter from previous chapters . its semantics are specified by the listing below : * var * v : @xmath38 @xmath39 @xmath40 fetch - and - add object can be used to produce unique identifiers , implement mutual exclusion , barrier synchronization@xcite , or work queues@xcite . its known wait - free shared memory implementations for @xmath9 processes have @xmath41 memory complexity and @xmath42 step complexity @xcite@xcite . they also need to employ complicated memory management techniques to bound their memory use . our implementation reduces the memory complexity to @xmath43 while maintaining polylogarithmic step complexity . we will first implement a helper object the history object . intuitively , it contains a versioned memory cell and allows retrieval of past @xmath1 values of the cell . the cell holds objects of type . the semantics of this object are precisely specified by the following single - threaded implementation : * var * @xmath44 : [ ] @xmath30 * var * @xmath16 : @xmath16 , @xmath45 $ ] @xmath46 $ ] @xmath46 \gets t$ ] @xmath47 true false our wait - free implementation will impose an additional constraint on its use : every call to publish is parameterized by an integer in range @xmath48 and executions of publish@xmath49 for the same @xmath28 can not run simultaneously . both @xmath1 and @xmath9 must be known when the object is created . our implementation is presented below : * var * @xmath50 : @xmath51 * var * @xmath44 : [ n]@xmath52 * var * @xmath53 : [ p]@xmath52 @xmath54 help ( ) @xmath55.{\textsc{rd}}()$ ] @xmath54 @xmath56.{\textsc{rd}}()$ ] @xmath57.{\textsc{rd}}()$ ] @xmath55.{\textsc{cas}}(h , l)$ ] [ hist - help - cas ] @xmath54 [ hist - get - notyet - sp ] [ hist - get - notyet ] help ( ) @xmath58.{\textsc{rd}}()$ ] @xmath59 [ hist - get - tooold ] @xmath54 [ hist - pub - earlier - sp ] * false * [ hist - pub - earlier - exit ] help ( ) @xmath60.{\textsc{wr}}((v , t))$ ] [ hist - pub - latest ] [ hist - pub - later - sp ] * false * * true * as the comments indicate , @xmath60 $ ] is used to temporarily hold the value being published by publish@xmath49 . the array @xmath44 is used to hold , roughly , the @xmath1 most recently published values ( the most recently published value might be absent ; exact semantics will be given by the lemmas below ) together with their version numbers . the field @xmath50 holds the particulars of the most recently published value ; the successful publish calls will serialize at the moment @xmath50 changes . our implementation contains an additional function help . it is used internally to write the most recently published value to the array @xmath44 , if it is nt stored there already . we will now prove some properties related to exact semantics of @xmath44 and @xmath53 and then use them to prove that our implementation serializes to the specification . we posit that the serialization point of publish is in line [ hist - pub - later - sp ] , unless that line is nt reached . in that case when the call returns false in line [ hist - pub - earlier - exit ] ) the serialization point is in line [ hist - pub - earlier - sp ] . [ hist - cv - latest ] if @xmath61 at time @xmath62 , and at some later point in time @xmath60 = ( v , t)$ ] , then there was a successful call to publish@xmath49@xmath63 with serialization point before time @xmath62 . s must have been modified by a successful cas in line [ hist - pub - later - sp ] . obviously , a successful invocation of publish@xmath49@xmath63 must have executed that cas . let s denote it by @xmath64 . this invocation sets @xmath60 $ ] to @xmath63 . what remains to be proven is that no later invocation of publish@xmath65 will set @xmath60 $ ] to @xmath66 for any x. only invocations of publish@xmath49 modify @xmath60 $ ] and they can start only after @xmath64 has finished . together with a simple observation that @xmath67 is nondecreasing in time this implies that any later invocation of publish that sets @xmath60 $ ] must have been called with a strictly greater @xmath68 . [ obs - hist - is - correct ] if @xmath69 = ( v , t)$ ] at some point then a successful call to publish@xmath65@xmath63 has had its serialization point earlier . [ lem - hist - help - comes ] let @xmath64 be a call to publish@xmath49 . let @xmath70 be a successful call to publish@xmath71 and @xmath72 a call to publish@xmath49 , both with serialization points after serialization point of @xmath64 . then there is a call to help that starts after the serialization point of @xmath64 and ends before the one of @xmath70 and before the execution of line [ hist - pub - latest ] in @xmath72 . the serialization point of @xmath64 must happen before @xmath70 executes line [ hist - pub - earlier - sp ] if s has changed between line [ hist - pub - earlier - sp ] and line [ hist - pub - later - sp ] of @xmath70 , @xmath70 would fail . thus the call to help from @xmath70 will start after the serialization point of @xmath64 and will finish before the serialization point of @xmath70 . the call to help from @xmath72 will start after serialization point of @xmath64 ( @xmath72 may only start after @xmath64 has finished ) and will finish before line [ hist - pub - latest ] of @xmath72 . of these two calls to help , the one that finishes earlier satisfies both conditions . [ lem - hist - is - complete ] if a call to help ( ) has executed fully ( ie . started and finished ) after the serialization point of a successful publish@xmath65@xmath73 , then @xmath74.v \geq v$ ] . let us first note that @xmath74.v$ ] is nondecreasing . it thus suffices to prove that the condition is met at some point before the end of the call to help . we will do so by induction on the time at which the invocation of publish serializes . consider a call to publish@xmath49@xmath73 that serializes at time @xmath62 , and a call to help that starts after @xmath62 . by lemma [ lem - hist - help - comes ] there is a call to help that finishes before the next serialization point of a successful publish@xmath65 ( thus before @xmath50 is modified ) and before @xmath60 $ ] is modified . without loss of generality we can assume that the call to help we are considering satisfies these conditions . by inductive hypothesis and lemma [ lem - hist - help - comes ] , since a successful call to publish@xmath65@xmath75 has occured before time @xmath62 , @xmath76.first \geq v - n$ ] at time @xmath62 . taking into account that @xmath69.v \mod n = i$ ] , at any later point in time one of @xmath76.v = v - n$ ] and @xmath76 \geq v$ ] will hold . thus the cas in line [ hist - help - cas ] either succeeds , which causes the lemma s conclusion to start being satisfied , or fails , which means that the conclusion was already satisfied . the operations in the implementation of the history object serialize to the single - process object with serialization points of publish@xmath65 as posited earlier . note that @xmath67 is at all times equal to the first argument of the latest successful publish . let us first consider a call to get - current that returns @xmath40 . from lemma [ lem - hist - is - complete ] we know that @xmath77 . by observation [ obs - hist - is - correct ] , the successful publish@xmath65@xmath78 happened before the read from @xmath79 $ ] . if @xmath80 , then this call to publish was the most recent successful publish at the time when @xmath50 was read , so we can serialize the operation there . otherwise , it has happened ( had its serialization point ) after that read , so we can serialize the operation just after it . let us consider a calls to publish@xmath65 : * if a call to publish@xmath65@xmath73 fails in line [ hist - pub - earlier - exit ] , then we can see that the most recently serialized ( from the point of view of line [ hist - pub - earlier - sp ] ) successful publish published a version different than @xmath81 . * if a call to publish@xmath65@xmath73 fails by failing the cas in line [ hist - pub - later - sp ] , then a successful publish@xmath65 has occured after line [ hist - pub - earlier - sp ] had been executed , so the most recent successful publish@xmath65 at the time of cas has published a version greater than @xmath81 . * if a call to publish@xmath65@xmath73 succeeds , then at the time of cas in line [ hist - pub - later - sp ] the most recently published version is @xmath81 . this suffices to prove that the value returned by publish is always correct with respect to the posited serialization order . what remains to consider are the calls to get : * if a call to get@xmath82 fails in line [ hist - get - notyet ] , then the most recent successful publish@xmath65 when line [ hist - get - notyet - sp ] executed had a smaller version number than requested , so we can serialize the call to get at line [ hist - get - notyet - sp ] . * if a call to get@xmath82 fails in line [ hist - get - tooold ] because @xmath83 , then ( by observation [ obs - hist - is - correct ] ) a successful call to publish@xmath65@xmath84 has occured by that time and @xmath85 , so if we serialize get at that point , it should fail . * if a call to get@xmath82 fails in line [ hist - get - tooold ] because @xmath86 , then ( by lemma [ lem - hist - is - complete ] ) a call to publish@xmath65@xmath73 has not occured before the call to help in line x. we can thus serialize this get just before the call to help has started . * if a call to get@xmath82 succeeds , then by observation [ obs - hist - is - correct ] it returns a value that was published by a successful call to publish@xmath65@xmath73 . by lemma [ lem - hist - is - complete ] , version @xmath87 was not published before the call to help has started . thus we can serialize this get just after the successfull call to publish@xmath65@xmath73 has occured , or , if it has occured before get started , just before the call to help started . obviously all operations run in @xmath4 time . memory complexity is @xmath88 , where @xmath1 and @xmath9 are the parameters defined at the beginning of this section . we will now use this object in the main result of this work , an implementation of a write - and-@xmath0-array . we will now present our main result an implementation of a write - and - f - array . we will actually present a wait - free implementation of a slightly richer object the additional operations and return values are required for the recursive construction of the concurrent implementation . a single - threaded specification of the structure is shown below . it uses a version function , which is a black - box integer - valued function , subject to following conditions : 1 . subsequent return values of version are nondecreasing . 2 . if a call to version(*false * ) is followed by a call to version(*true * ) , the second call must return a strictly greater integer than the first one . * var * @xmath89 $ ] : [ n]@xmath30 * var * @xmath90 : [ n ] * var * @xmath91 : [ n ] * var * @xmath92 : [ n ] @xmath93 \gets t$ ] @xmath94 , v[1],\ldots , v[n-1])$ ] @xmath95 \gets \text{last\_update}[i ] + 1 $ ] @xmath96 \gets { \textsc{version}}(\textbf{true})$ ] @xmath97 \gets r$ ] @xmath95 , \text{last\_version}[i ] , \text{last\_value}[i]$ ] @xmath95 , \text{last\_version}[i ] , \text{last\_value}[i]$ ] @xmath98 , v[1],\ldots , v[n-1])$ ] one can observe that the version numbers group the calls to write - and - f into groups of consecutive calls with no intervening calls to read . the concurrent implementation will be restricted by disallowing concurrent calls to write - and - f@xmath99 for the same @xmath100 . it is conceptually very similar to jayanti s tree - based f - array implementation . it uses a binary tree structure , with each array element assigned to a leaf and intervals of array elements assigned to internal nodes . we will construct it recursively . the implementation for @xmath101 is presented below . if we note that no concurrent calls to write - and - f may be made in it , we can easily see that it is indeed correct and that all operations take constant time . * var * @xmath50 : @xmath102 @xmath103 @xmath104 @xmath105 @xmath103 @xmath106 @xmath107 the implementation for @xmath108 is presented below ( interspersed with comments ) : * var * @xmath21 : [ 2 ] * var * @xmath68 : [ 2 ] * var * @xmath109 : * var * @xmath68 : * var * @xmath21 : * var * @xmath110 : [ 2]write - and - f - array * var * @xmath44 : * * history object**<node - value > * var * @xmath53 : [ n]last - value c[0 ] and c[1 ] are the subobjects from the recursive construction they are of size @xmath111 and @xmath112 , respectively . array elements of the enclosing structure are mapped bijectively to consecutive array elements of these two subobjects . the mapping is defined by following functions ( the element @xmath100 is mapped to element @xmath113 in c[@xmath114 ) : @xmath115 @xmath116 history object h is used to store the object s current value ( the value that would be returned by read ) and past values . the version numbers exported by the history object correspond to the values returned by version in the specification . the elements of h are nt just values ; they instead contain the values of both children together with their versions . the array l is used to store the values get - last should return , but the values there might be stale ( we will prove bounds on their staleness later on ) . @xmath117 [ get - getownver ] @xmath118 , h.t[1])$ ] @xmath117 [ upd - getown ] @xmath119 , h'.t[0 ] \gets c[0].{\textsc{read}}()$ ] [ upd - getch-0 ] @xmath120 , h'.t[1 ] \gets c[1].{\textsc{read}}()$ ] [ upd - getch-1 ] @xmath121 @xmath122 [ upd - publish ] @xmath123.{\textsc{write - and - f$_{child\_id(i)}$}}(t)$ ] [ upd - chupd ] [ updcall1 ] update@xmath99 ( ) [ updcall2 ] @xmath124 the implementation of read and write - and - f strongly resemble the f - array . write - and - f uses a helper function update . the intuition behind update is that it `` pushes '' new values from @xmath125 $ ] and @xmath126 $ ] to @xmath44 . we will show that it suffices to attempt to call update twice to accomplish that . @xmath127.{\textsc{get - last}}(child\_id(x))$ ] [ get - last - recurse ] @xmath128 binary search @xmath129 for first @xmath68 such that : [ help - binsch ] @xmath130 \geq l_c.v$ ] @xmath131 [ help - getprev ] @xmath132 @xmath133)$ ] @xmath134 , l_c.t)$ ] @xmath135.{\textsc{rd}}$ ] [ upd - last - alreadydone ] @xmath136.\text{cas}(l , ( l_c.n , v , t))$ ] @xmath137 @xmath136.{\textsc{rd}}()$ ] the use of binary search in line [ help - binsch ] warrants explanation . the predicate employed can obviously change its value in time . thus , the binary search will return a @xmath68 such that the predicate held at one point in time for @xmath68 and did nt at another point for @xmath81 . we will see that this is sufficient . indeed , the result of the binary search will be important only in the cases when the value of the predicate does nt change during the search . we will first prove two simple facts about update : [ lem - versions - increase ] for any two subsequent successful calls to @xmath138 with values @xmath139 and @xmath140 , @xmath141 \leq h_2.v[i]$ ] for both @xmath100 . ( labprev_getch ) at ( 1,-1 + 0.5 ) read children;(1,-1 - 0.1 ) to ( labprev_getch);(prev_getch ) at ( 1,-1 ) ; ( labprev_pub ) at ( 3,-1 + 0.5 ) @xmath142;(3,-1 - 0.1 ) to ( labprev_pub);(prev_pub ) at ( 3,-1 ) ; ( labget_ver ) at ( 5,-0 + 0.5 ) @xmath143;(5,-0 - 0.1 ) to ( labget_ver);(get_ver ) at ( 5,-0 ) ; ( labgetch ) at ( 7,-0 + 0.5 ) read children;(7,-0 - 0.1 ) to ( labgetch);(getch ) at ( 7,-0 ) ; ( labpublish ) at ( 9,-0 + 0.5 ) @xmath144;(9,-0 - 0.1 ) to ( labpublish);(publish ) at ( 9,-0 ) ; ( 0,-1 ) to ( 10,-1 ) ; ( 0,0 ) to ( 10,0 ) ; ( 0.5,-1 ) to ( 3.5,-1 ) ; ( 4.5,0 ) to ( 9.5,0 ) ; ( prev_pub ) to ( get_ver ) ; assume otherwise . consider the first pair @xmath145 of consecutive calls to publish that contradicts the lemma ( see figure [ fig - versions - increase ] ) . the only place publish is called is line [ upd - publish ] in update . @xmath144 had its most recent call to get - current ( denoted by @xmath143 ) in line [ upd - getown ] sometime earlier . no successful publish could have occured between @xmath143 and @xmath144 ; otherwise @xmath144 would have failed . thus , @xmath142 must have occured before @xmath143 and so its execution of lines [ upd - getch-0 ] and [ upd - getch-1 ] ( denoted by `` read children '' on the diagram ) must have occured strictly earlier than the same for the second call . this together with monotonicity of versions in children delivers the contradiction . for every execution of write - and - f : during execution of lines [ updcall1 ] and [ updcall2 ] at least one successful call to update occurs . ( labour_gv1 ) at ( 5.5,-0 + 0.5 ) @xmath146;(5.5,-0 - 0.1 ) to ( labour_gv1);(our_gv1 ) at ( 5.5,-0 ) ; ( labbad_pub1 ) at ( 6.5,-1 + 0.5 ) @xmath147;(6.5,-1 - 0.1 ) to ( labbad_pub1);(bad_pub1 ) at ( 6.5,-1 ) ; ( labour_pub1 ) at ( 7.5,-0 + 0.5 ) @xmath148;(7.5,-0 - 0.1 ) to ( labour_pub1);(our_pub1 ) at ( 7.5,-0 ) ; ( our_gv1 ) ( bad_pub1 ) ; ( bad_pub1 ) ( our_pub1 ) ; ( labbad_gv2 ) at ( 7,-2 + 0.5 ) @xmath149;(7,-2 - 0.1 ) to ( labbad_gv2);(bad_gv2 ) at ( 7,-2 ) ; ( bad_pub1 ) ( bad_gv2 ) ; ( labour_gv2 ) at ( 9.5,-0 + 0.5 ) @xmath143;(9.5,-0 - 0.1 ) to ( labour_gv2);(our_gv2 ) at ( 9.5,-0 ) ; ( labbad_pub2 ) at ( 11,-2 + 0.5 ) @xmath150;(11,-2 - 0.1 ) to ( labbad_pub2);(bad_pub2 ) at ( 11,-2 ) ; ( labour_pub2 ) at ( 12,-0 + 0.5 ) @xmath151;(12,-0 - 0.1 ) to ( labour_pub2);(our_pub2 ) at ( 12,-0 ) ; ( our_gv2 ) ( bad_pub2 ) ; ( bad_pub2 ) ( our_pub2 ) ; ( 2.5,0 ) ( 13,0 ) ; ( 3,0 ) ( 12.5,0 ) ; ( 5.5,-1 ) ( 7.5,-1 ) ; ( 6,-1 ) ( 7,-1 ) ; ( 6,-2 ) ( 12,-2 ) ; ( 6.5,-2 ) ( 11.5,-2 ) ; if one of the calls to update from the lines in question suceeds , then the lemma trivially holds . thus we assume that they have both failed . the situation is depicted in figure [ fig - only - two - pubs ] . @xmath152 correspond to these two failed calls to update . in order for @xmath148 to fail , a publish must have succeeded between @xmath146 and @xmath148 let s call it @xmath147 . similarly , a publish must have suceeded between @xmath143 and @xmath151 . alas , this publish ( denoted @xmath150 ) must have had its corresponding call to get - current ( denoted @xmath149 ) after @xmath147 . the call to update that called @xmath149 and @xmath150 satisfies the lemma s conclusion . let us consider an execution of write - and - f@xmath99 . let @xmath153 be the return value of the corresponding call to @xmath123.{\textsc{write - and - f$_{child\_id(i)}$}}$ ] . then by the time the call to write - and - f@xmath99 finishes , the value of the most recently published node has @xmath154 \geq v_c$ ] . we will consider the first call to publish that publishes such a node value as the serialization point of the write - and - f . obviously , many calls to write - and - f can then serialize at the same instant in time . we will order them by taking first these with @xmath155 in the order in which their corresponding calls to write - and - f happened in @xmath125 $ ] and then those with @xmath156 in the corresponding order . the choice of order on elements of @xmath110 is arbitrary , albeit it is reflected in the computation of @xmath21 in help . before we prove that the implementation is correct , we need the following two lemmas about the array @xmath53 . if help is called after the serialization point of the @xmath157 call to write - and - f@xmath99 , then @xmath158.n \geq n$ ] after the call to help completes . [ last - is - complete ] we will prove this lemma by induction on @xmath109 . the proof for the base case will be very similar to the induction step , so we present both of them simultaneously . let @xmath159 be the time the successfull call to publish@xmath73 occured . consider an update@xmath99 that was serialized at @xmath159 . for every @xmath160 , a call to @xmath161 starts and finishes between @xmath162 and @xmath163 . thus , there is such a call that starts and finishes between @xmath159 and @xmath164 . without loss of generality we can assume that the call to help from the lemma s hypothesis executes in this time interval . in that case , the binary search from line [ help - binsch ] will not see any @xmath165 for @xmath166 returning and will return version @xmath68 . for the same reason , both gets will succeed . we want to prove that @xmath158.n \geq n-1 $ ] from @xmath159 on . for the base case this holds , because the initial values satisfy this requirement . for the inductive step , we have to use the hypothesis : at @xmath159 , the @xmath167 call to write - and - f@xmath99 has completed , so a call to @xmath161 has completed between the serialization point of that call to write - and - f@xmath99 and @xmath159 . thus @xmath158.n \geq n-1 $ ] from @xmath159 on . if the condition in line [ upd - last - alreadydone ] is not satisfied or the cas in the next line fails , the lemma hold . otherwise , by the time the cas succeeds the lemma will obviously hold . if a call to @xmath161 sets @xmath158 $ ] to @xmath168 , then by the time the call has finished : [ last - is - correct ] 1 . write - and - f@xmath99 has been called at least @xmath109 times . 2 . let @xmath153 be the return value of the @xmath157 call to @xmath123.{\textsc{write - and - f}}_{child\_id(p)}$ ] . let @xmath169 and @xmath170 be the version and node value published at the serialization point of the @xmath157 write - and - f@xmath99 and @xmath171 be the node value published with version @xmath172 . then @xmath173 and : * if @xmath174 then @xmath175)$ ] , * if @xmath176 then @xmath177 , t_c)$ ] . for the first part , we need to note that by the time line [ get - last - recurse ] executes , the @xmath157 write - and - f@xmath178 has already executed in the appropriate child . if the binary search fails or returns version different from @xmath169 , help will exit early . indeed , if the binary search returns a different @xmath68 , @xmath179 will fail . this immediately proves the first part of the lemma and shows that @xmath171 and @xmath170 in the code have the same values as @xmath171 and @xmath170 in the lemma , which proves the second part . the operations in multi - process write - and - f - array serialize to the single - process specification with serialization point of write - and - f as posited earlier . let us choose the call to get - current as the serialization point of read . we will first prove that these are correct serialization points for write - and - f and read . consider the @xmath157 call to write - and - f@xmath99 and let @xmath180 be its return value . as this triple was read from @xmath158 $ ] , we can use lemmas [ last - is - complete ] and [ last - is - correct ] to show that @xmath181 ( note that executions of write - and - f@xmath99 for the same @xmath100 are disjoint in time ) . from lemma [ last - is - correct ] we get that @xmath169 is equal to the version published at the serialization point of the call to write - and - f . this obviously implies that version is nondecreasing for subsequent calls to write - and - f and read . as only the calls to write - and - f that serialize on a single publish return equal @xmath68 , no read call can serialize in between . this proves both properties required from version . we still need to prove that @xmath182 and the values returned by read are consistent with the posited serialization order . consider a set of write - and - f calls that are serialized together . let @xmath170 and @xmath171 be the history elements published , respectively , at the serialization point and most recently before it . let @xmath183 be the sequence of return values of @xmath125.{\textsc{write - and - f}}$ ] calls corresponding to write - and - f calls in question , in serialization order . let @xmath184 be a similar sequence for @xmath126 $ ] . by lemma [ lem - versions - increase ] these are exactly the calls to @xmath185.{\textsc{write - and - f}}$ ] serialized with versions in @xmath186 , h_{new}.v[j]\right]$ ] for @xmath187 . by lemma [ last - is - correct ] the sequence returned by the write - and - f calls is @xmath188 @xmath189 @xmath190 @xmath191 , b_0 ) , $ ] @xmath191 , b_1 ) , \ldots$ ] in the order of serialization . additionally , @xmath191 , h_{new}.t[1])$ ] is the value that would be returned by any read calls until the next serialization point of write - and - f . this suffices to show that the return values of consecutive calls to write - and - f are actually the results of applying the updates . this proves that values returned by get and update are correct wrt . posited serialization order . we still need to show that we can correctly serialize the calls to get - last . lemma [ last - is - correct ] implies that two calls to @xmath124 will never return different triples with equal @xmath109 . as write - and - f returns the result of a call to get - last , we just need to show that get - last@xmath192 can be serialized so that the @xmath109 in its return value is the number of previously serialized calls to write - and - f@xmath99 . from lemma [ last - is - complete ] we get that @xmath109 is at least the number of calls to write - and - f@xmath99 that serialized before get - last started . from lemma [ last - is - correct ] we know that @xmath109 was at most the number of such calls that serialized before get - last finished . thus , there is a point in the execution interval of get - last when @xmath109 is equal to the number of such calls that have already serialized . we choose any such point in time to be the serialization point of the call to get - last . the structure uses @xmath36 memory . a get - last operation takes @xmath7 time , a read operation takes @xmath4 time and a write - and - f operation takes @xmath3 time . we can construct a fetch - and - add object for @xmath9 threads by using a write - and - f - array of size @xmath9 with addition as the operation @xmath0 . every thread is assigned an element in the array , and modifies only that element . this gives us a fetch - and - add object for @xmath9 threads with @xmath11 step complexity and @xmath43 memory complexity . additionally , the object implements a method that retrieves the current value in @xmath4 time . a fetch - and - add object was implemented using the above - mentioned construction from the write - and - f object . the implementation is written in c++ and uses the c++11 standard library support for atomic operations . it is published on github . direct implementation of previously described algorithm would require cas objects of size larger than the 64-bit intel processors support . however , the way most of these objects are accessed in the implementation makes it possible to split them into multiple cas objects . the only cas objects that do nt afford this transformation are the version number holders from the history object . we use 64-bit versions and thus even these objects are no larger than 128 bits . thus , our implementation can run on 64-bit intel architecture processors , but not on 32-bit ones . unfortunately , the c++ standard library support for atomic operations does nt allow atomic reads of a part of a larger atomic variable ( specifically , reads of 64-bit halves of a 128-bit variable ) . as 128-bit atomic reads on amd64 are very costly ( they use a 128-bit wide cas ) , limiting their number was very important for the efficiency s sake . thus , the implementation makes an unwarranted assumption that an atomic variable has the same memory layout as a normal , non - atomic variable of the same size . this assumption holds for gcc 4.6 on amd64 . the correctness of the implementation was tested experimentally both on real hardware and by using the relacy race detector@xcite . relacy race detector is a framework for testing multi - threaded programs in c++11 . it substitues its own implementation of synchronization primitives and atomic variables and runs user - supplied tests multiple times , with various interleavings of threads . it can detect data races on non - atomic variables , deadlock conditions , and failed user - supplied invariant and assertion checks . one notable feature is the support for atomic operations with reduced consistency relacy can simulate acquire / release semantics , as specified in the c++11 atomic variables library . our fetch - and - add implementation can be compiled both to run on bare hardware and to run in relacy . the use of relacy not only provided confidence about correctness of the implementation , but also allowed us to downgrade consistency guarantees of some writes . the implementation was benchmarked on a machine with 4 12-core amd opteron 6174 processors . the benchmark created a fetch - and - add object and started a preset number of threads . each thread incremented the counter in a loop , counting its operations locally . after 20 seconds elapsed , all threads were signalled to stop and their operation counters were summed . for comparison , the same benchmark was run with the fetch - and - add object implemented by a simple read - modify - write loop . all benchmarks were run when the machine was minimally loaded ( had 5 minute load average smaller than 0.5 at the start of the benchmark ) . the results of the benchmark are shown in figure [ fig - meas ] . the figure contains a plot of average time it takes to complete one operation as a function of the number of concurrently executing threads . the horizontal scale of the figure is proportional to the square of the logarithm of the number of threads . the reason for this is that the optimistic time complexity of our fetch - and - add object is @xmath193 . we can see that the graph for our implementation approximates a straight line up to about 30 threads , where it starts to diverge upward . this peculiar divergence disappears if we assign one core to each thread and force it to run only there ( by setting cpu affinity ) . other affinity settings yielded graphs similar to one of these two . we were unable to explain this behaviour . it might be worth noting that the anomaly in the no - affinity case happens when we start using all 4 physical cpus . it can be easily seen from the graph that our fetch - and - add object is far from practical . it is about 100 times slower than the naive lock - free implementation for small numbers of threads and about 10 times slower for larger numbers of threads . we believe that this implementation is suboptimal , although achieving similar performance as the naive implementation seems impossible . one change that could improve the performance of this implementation is changing the arity of the tree . the construction can easily be adapted to trees with larger arity and this might decrease the number of expensive cas operations at the expense of the number of atomic reads , which are comparatively cheap on intel processors . our implementation of write - and - f - array can be modified by splitting the work done by help into @xmath35 separate pieces ( updates on consecutive levels of the tree ) and running only a single such piece in update . if we then enlarge the history size to @xmath194 , lemma [ last - is - complete ] still holds . this modification increases the memory complexity to @xmath195 and decreases the step complexity of write - and - f to @xmath196 . the structure can be straightforwardly modified to use ll / cs instead of cas , albeit it then requires the ability to have two outstanding lls at the same time . the modified version can be further modified to use bounded version numbers , from a range of size @xmath33 . unfortunately , popular architectures that implement ll / sc ( such as powerpc ) allow only one outstanding ll at a time . the effect of the arity of the tree on the performance seems to be worth investigating . we pose also two purely theoretical problems : * our implementation of write - and - f - array is single - writer , so it is natural to consider the multi - writer write - and - f - array . can we construct a multi - writer write - and - f - array in subquadratic memory with similar step complexities of the operations ? * the memory consumption of write - and - f - array implemented with atomic registers and cas or ll / cs objects is bounded from below by a linear function of the number of processes that can execute write - and - f concurrently@xcite , so it must be @xmath197 . can we achieve lower memory and/or step complexities ? the author would like to thank his advisor , dr . grzegorz herman for many fruitful discussions and help while preparing this work . he would also like to thank szymon acedaski for his help in improving the presentation of this work . 9 herlihy m. , shavit n. `` the art of multiprocessor programming '' , morgan kaufmann publishers inc . , 2008 herlihy m. `` wait - free synchronization '' , acm trans . program . 13 , 1 , pp . 124149 ( 1991 ) riany y. , shavit n. , touitou d. `` towards a practical snapshot algorithm '' , theoretical computer science 269 , pp . 163201 ( 2001 ) afek y. , attiya h. , dolev d. , gafni e. , merritt m. , shavit n. `` atomic snapshots of shared memory '' , journal of the acm 40 , pp . 873890 ( 1993 ) afek y. , weisberger e. `` the instancy of snapshots and commuting objects '' , journal of algorithms 30.1 , pp . 68105 ( 1999 ) chandra t. d. , jayanti p. , tan k. `` a polylog time wait - free construction for closed objects '' , proceedings the 17th podc , pp . 287296 ( 1998 ) `` f - arrays : implementation and applications '' , proceedings of the 21st podc , pp . 270279 ( 2002 ) ellen f. , ramachandran v. , woelfel p. `` efficient fetch - and - increment '' , lecture notes in computer science 7611 , pp . 1630 ( 2012 ) fich f. e. , hendler d. , shavit n. `` linear lower bounds on real - world implementations of concurrent objects '' , foundations of computer science , pp . 165173 ( 2005 ) freudenthal , e. , gottlieb , a. `` process coordination with fetch - and - increment '' , proceedings of asplos - iv , pp . 260-268 ( 1991 ) goodman , j. , vernon , m. , woest , p. `` efficent synchronization primitives for large - scale cache - coherent multiprocessors '' , proceedings of asplos - iii , pp . 64-75 ( 1989 ) vyukov d. , relacy race detector , ` http://www.1024cores.net/home/relacy-race-detector `

we introduce a new shared memory object : the write - and - f - array , provide its wait - free implementation and use it to construct an improved wait - free implementation of the fetch - and - add object . the write - and - f - array generalizes single - writer write - and - snapshot@xcite object in a similar way that the f - array@xcite generalizes the multi - writer snapshot object . more specifically , a write - and - f - array is parameterized by an associative operator @xmath0 and is conceptually an array with two atomic operations : * write - and - f modifies a single array s element and returns the result of applying @xmath0 to all the elements , * read returns the result of applying @xmath0 to all the array s elements . we provide a wait - free implementation of an @xmath1-element write - and - f - array with @xmath2 memory complexity , @xmath3 step complexity of the write - and - f operation and @xmath4 step complexity of the read operation . the implementation uses cas objects and requires their size to be @xmath5 , where @xmath6 is the total number of write - and - f operations executed . we also show , how it can be modified to achieve @xmath7 step complexity of write - and - f , while increasing the memory complexity to @xmath8 . the write - and - f - array can be applied to create a fetch - and - add object for @xmath9 processes with @xmath10 memory complexity and @xmath11 step complexity of the fetch - and - add operation . this is the first implementation of fetch - and - add with polylogarithmic step complexity and subquadratic memory complexity that can be implemented without cas or ll / sc objects of unrealistic size@xcite . keywords : concurrency , wait - free , snapshot , fetch - and - add

in this supplemental material , we study the quasiparticle statistics of the non - abelian moore - read ( mr ) state on the square ( sq ) lattice at filling factor @xmath11 . first , we find convincing evidence that the phase at filling factor @xmath11 manifests the mr state . then we search for the minimal entangled states ( mess ) in the groundstate manifold and determine the modular @xmath0 matrix . versus the @xmath115 ( @xmath116 ) on a @xmath16 sq lattice at @xmath11 . three lowest eigenvalues are labeled by blue square , red cross and green cross . ( b ) low - energy spectra versus boundary phase @xmath117 at a fixed @xmath118 for @xmath107 . ( c ) particle entanglement spectrum ( pes ) for tracing out @xmath17 bosons . there are @xmath119 states below the pes gap ( red dashed line ) for @xmath120 , in good agreement with the counting of quasihole excitations in mr state . , width=316 ] _ low - energy spectrum at @xmath11. _ the hamiltonian has been described in the main text and we select @xmath116 here so that the system is effectively equivalent to a spin@xmath121 system . low - energy spectrum versus different @xmath115 on the @xmath16 sq lattice is shown in fig.[fig : sq : mr : es](a ) . when @xmath122 , we find a groundstate manifold ( gsm ) with three exact degenerate lowest eigenstates . with increasing @xmath115 , the exact degeneracy of gsm is destroyed and two groundstates from the gsm will evolve into the excited spectrum . we have also obtained numerical results from larger lattice size ( @xmath123 and @xmath124 ) and have confirmed that the above picture is qualitatively correct . to confirm the robustness of this fractional quantum hall ( fqh ) phase , we also introduce two boundary phases @xmath117 and @xmath125 for generalized boundary conditions and calculate the chern number ( berry phase in units of @xmath126 ) of gsm . the three groundstates maintain their quasi - degeneracy and are well separated from the low - energy excitation spectrum upon tuning the boundary phases ( fig.[fig : sq : mr : es](b ) ) . moreover , the three - fold gsm is found to share a total chern number @xmath127 . furthermore , we study the particle entanglement spectrum ( pes ) of the three - fold gsm ( fig.[fig : sq : mr : es](c ) ) . pes reveals a gap at @xmath120 and the number of states below this gap exactly agrees with the number of quasihole excitations in a mr state based on the generalized pauli principle that no more than two bosons occupy two consecutive orbitals @xcite . thus we reach the conclusion that the three - fold gsm at @xmath11 mimics non - abelian mr state at @xmath128 . the pes gap disappears for @xmath129 signaling the quantum phase transition from the non - abelian mr state to a topological trivial state @xcite . ) of @xmath41 on @xmath16 sq lattice at @xmath11 . we show entropy profile versus @xmath43 ( @xmath44 ) by setting optimized @xmath130 , @xmath131 . three nearly orthogonal mess are marked by red , green and blue arrows ( dots ) in surface ( contour ) plot . the cyan dashed line represents the states orthogonal to the first mes ( red dot ) . ( b ) entropy for the states along the cyan dashed line as shown in ( a ) . all calculations are for partition along cut - i . , width=153 ] ) of @xmath41 on @xmath16 sq lattice at @xmath11 . we show entropy profile versus @xmath43 ( @xmath44 ) by setting optimized @xmath130 , @xmath131 . three nearly orthogonal mess are marked by red , green and blue arrows ( dots ) in surface ( contour ) plot . the cyan dashed line represents the states orthogonal to the first mes ( red dot ) . ( b ) entropy for the states along the cyan dashed line as shown in ( a ) . all calculations are for partition along cut - i . , width=163 ] _ mess and modular matrix at @xmath11. _ we denote the three groundstates from ed calculation as @xmath47 , ( with @xmath48 ) . now we form the general superposition state as , @xmath49 where @xmath50 , @xmath51 , @xmath52 are real superposition parameters . for each state @xmath53 , we construct the reduced density matrix and obtain the corresponding entanglement entropy . we optimize values of @xmath54 $ ] and @xmath55 $ ] to minimize the entanglement entropy . in fig . [ fig : sq : mr : mes](a - b ) , we show the entropy profile at optimized parameters @xmath56 for mess . in fig . [ fig : sq : mr : mes](a ) , it is found several peaks ( entropy valleys ) in @xmath59 space and the three nearly orthogonal mess are determined as labeled by arrows . in fig . [ fig : sq : mr : mes](b ) , it is shown the entropies of states in the parameter space orthogonal to the first mes ( red arrow ) . the second and the third mess are indeed located in the separated entropy valleys , respectively . here we find that the entropies corresponding to the three mess are different from each other ( as list in table - i in main text ) . the entropy difference between the first mes and the second mes may result from the finite - size effect . to identify the emergence of non - abelian quasiparticles , we calculate the modular matrix and obtain the quasiparticle statistics as below . with the help of mess , we obtain the modular matrix @xmath132 , which is close to the prediction of @xmath71 chern - simons theory : @xmath72 . we extract the three quasiparticle quantum dimension as @xmath133 , @xmath134 , @xmath79 and the total quantum dimension @xmath80 . combined with the fusion rule of the third type of quasiparticle @xmath6 : @xmath135 ( as shown in the main text ) , we confirm that @xmath6 behaves as the non - abelian majarona quasiparticle .

the topological order is equivalent to the pattern of long - range quantum entanglements , which can not be measured by any local observable . here we perform an exact diagonalization study to establish the non - abelian topological order through entanglement entropy measurement . we focus on the quasiparticle statistics of the non - abelian moore - read and read - rezayi states on the lattice boson models . we identify multiple independent minimal entangled states ( mess ) in the groundstate manifold on a torus . the extracted modular @xmath0 matrix from mess faithfully demonstrates the majorana quasiparticle or fibonacci quasiparticle statistics , including the quasiparticle quantum dimensions and the fusion rules for such systems . these findings support that mess manifest the eigenstates of quasiparticles for the non - abelian topological states and encode the full information of the topological order . _ introduction. _ one of the most striking phenomena in the fractional quantum hall ( fqh ) system is the emergent fractionalized quasiparticles obeying abelian @xcite or non - abelian @xcite braiding statistics . interchange of two abelian quasiparticles leads to a nontrivial phase acquired by their wavefunction , whereas interchange of two non - abelian quasiparticles results in an operation of matrix to the degenerating groundstate space and the final state will depend on the order of operations being carried out . the non - abelian quasiparticles and its braiding statistics are fundamentally important for understanding the topological order and also have potential application in topologically fault - tolerant quantum computation @xcite . so far , such quasiparticles have not been definitely identified in nature . however , it is generally believed that they exist in the fqh systems at filling factor @xmath1 @xcite and @xmath2 @xcite , described by the moore - read ( mr ) @xcite and read - rezayi ( rr ) states @xcite , respectively . the topological band model for optical lattices with bosonic particles is another promising platform to realize the non - abelian topological states @xcite . in the mr and rr states , the quasiparticles satisfy the following characteristic fusion rules that specify how the quasiparticles combine and fuse into more than one type of quasiparticles @xcite : @xmath3 where @xmath4 represents the identity particle , @xmath5 the fermion - type quasiparticle , @xmath6 the majorana quasiparticle and @xmath7 the fibonacci quasiparticle . in general , the fusion rule of quasiparticles is encoded in the modular @xmath0 matrix through verlinde formula @xcite . moreover , @xmath0 ( @xmath8 ) also determines the quasiparticle s individual quantum dimension ( @xmath9 ) and total quantum dimension ( @xmath10 ) @xcite . therefore , the @xmath0 matrix plays the central role in identifying topological order and corresponding quasiparticle statistics @xcite . while the berry phase and non - abelian information of quasiparticles moving adiabatically around each other have been studied numerically @xcite , the full modular @xmath0 matrix and the corresponding fusion rules for the microscopic models hosting the non - abelian topological states have been lacking , due to the computational difficulty of directly dealing with and distinguishing different quasiparticles . recently , there is growing interest on characterizing topological order through the quantum entanglement information @xcite . among the recent progresses , the relationship between the entanglement measurement and the modular matrix for topological nontrivial systems has been uncovered @xcite , which may open a new avenue to this challenging issue . the modular matrix and corresponding quasiparticle statistics have been successfully extracted through the mess for chiral spin liquid and the abelian fqh states @xcite , which serve as direct evidences that the mes is the eigenstate of the wilson loop operators with a definite type of quasiparticle @xcite . for non - abelian case , it is more challenging as the quasiparticles usually have different quantum dimensions and topological entanglement entropies , and it is unclear if the mess still represent the quasiparticle eigenstates . a recent variational quantum monte carlo calculation studied the quasiparticle statistics for a projected wavefunction @xcite , however the obtained result does not accurately reproduce the known modular matrix including each quasiparticle quantum dimension for the mr state ; thus it is of critical importance to clarify if the mess can lead to the accurate identification of the modular matrix and the corresponding topological order for microscopic non - abelian quantum states . in this letter , we present an exact diagonalization ( ed ) study of quasiparticle statistics of the possible mr and rr non - abelian states through extracting the modular @xmath0 matrix for topological flat - band models @xcite with bosonic particles at filling numbers @xmath11 and @xmath12 . we map out the entanglement entropy profile for superposition states of the near degenerating groundstates and identify the global mes . we find that all other mess can be obtained in the state space which is orthogonal to the global mes . the obtained mess form the orthogonal and complete basis states for the modular transformation . we extract the modular @xmath0 matrix and establish the corresponding fusion rules , which unambiguously demonstrate the majorana and fibonacci quasiparticles emerging in these systems . we demonstrate that the obtained modular matrix as a topological invariant @xcite of the system remains to be universal in the whole topological phase until a quantum phase transition takes place . as shown by blue dashed line . ( c ) low - energy spectrum @xmath13 versus the @xmath14 ( @xmath15 ) on a @xmath16 sq lattice at @xmath12 . four lowest eigenvalues are labeled by blue square , purple cross , green cross and red circle . ( d ) particle entanglement spectrum ( pes ) for tracing out @xmath17 bosons . there are @xmath18 states below the pes gap ( red dashed line ) for @xmath19 , in good agreement with the counting of quasihole excitations in rr state . , width=144 ] as shown by blue dashed line . ( c ) low - energy spectrum @xmath13 versus the @xmath14 ( @xmath15 ) on a @xmath16 sq lattice at @xmath12 . four lowest eigenvalues are labeled by blue square , purple cross , green cross and red circle . ( d ) particle entanglement spectrum ( pes ) for tracing out @xmath17 bosons . there are @xmath18 states below the pes gap ( red dashed line ) for @xmath19 , in good agreement with the counting of quasihole excitations in rr state . , width=153 ] as shown by blue dashed line . ( c ) low - energy spectrum @xmath13 versus the @xmath14 ( @xmath15 ) on a @xmath16 sq lattice at @xmath12 . four lowest eigenvalues are labeled by blue square , purple cross , green cross and red circle . ( d ) particle entanglement spectrum ( pes ) for tracing out @xmath17 bosons . there are @xmath18 states below the pes gap ( red dashed line ) for @xmath19 , in good agreement with the counting of quasihole excitations in rr state . , width=288 ] we study the lattice boson model with longer - range hoppings , which can be generally written as @xmath20\nonumber + \sum_{n}\frac{u_n}{n!}\sum_{\mathbf{r}}(b^{\dagger}_{\mathbf{r}})^n ( b_{\mathbf{r}})^n \\\end{aligned}\ ] ] where @xmath21 creates ( annihilates ) a boson at site @xmath22 . @xmath23 is an on - site n - body repulsive interaction . here we consider two representative lattice models : the haldane model on the honeycomb ( hc ) lattice @xcite and topological flat band model on the square ( sq ) lattice @xcite . on the hc lattice , we include up to the third nearest - neighbor ( nn ) hopping and a non - zero @xmath24 on the second nn hopping only ( the net flux is zero in one unit cell ) , as shown in fig . [ fig : lattice](a ) . the nn hopping is set to be @xmath25 and the other parameters are defined the same as in ref . @xcite . on the sq lattice @xcite , we select the phase factor @xmath26 corresponding to half flux quanta per plaquette . the amplitude of hopping satisfies a particular gaussian form : @xmath27 , where @xmath28 and @xmath29 as the energy scale here . for both models , we consider a finite size system with @xmath30 unit cells , the filling factor of lower band is @xmath31 , where @xmath32 is boson number and @xmath33 is number of single - particle states in the flat band . we first obtain the low energy spectrum of the sq model at filling number @xmath12 and @xmath11 ( see @xcite ) . for @xmath12 , we set @xmath34 so that only three bosons can go to the same lattice site , which is effectively equivalent to a spin@xmath35 system . due to much larger hilbert space than the conventional hardcore boson systems , the largest size we can deal with is limited to @xmath16 for @xmath12 . we find strong numerical evidence of a possible @xmath12 rr state and @xmath11 mr state ( see @xcite ) on the sq lattice . as shown in the fig . [ fig : lattice](c - d ) , at smaller @xmath14 side , we find robust fourfold degeneracy of groundstates on a torus and the right counting rule of quasiparticle excitations for rr state @xcite . the @xmath11 bosonic mr state on the hc lattice has also been identified in the previous study @xcite . here our focus is to characterize the quasiparticle statistics of the above non - abelian states through calculating the modular @xmath0 matrix . to address the quasiparticle statistics , we first obtain the quasiparticle eigenstates through determining the mess on a torus @xcite . the entanglement entropy is defined as @xmath36 , where the reduced density matrix @xmath37 is obtained through partitioning the full system into two subsystems @xmath38 and @xmath39 and tracing out the subsystem @xmath39 . here we consider two noncontractible bipartitions on torus geometry ( cut - i and cut - ii ) ( fig . [ fig : lattice](a - b ) ) , which is along the lattice vectors @xmath40 , respectively . ) of @xmath41 on @xmath42 hc lattice at @xmath11 . we show entropy profile versus @xmath43 ( @xmath44 ) by setting optimized @xmath45 , @xmath46 . three nearly orthogonal mess are marked by red , green and blue arrows ( dots ) in surface ( contour ) plot . the cyan dashed line represents the states orthogonal to the first mes ( red dot ) . ( b ) entropy for the states along the cyan dashed line as shown in ( a ) . all calculations are for partition along cut - i . , width=144 ] ) of @xmath41 on @xmath42 hc lattice at @xmath11 . we show entropy profile versus @xmath43 ( @xmath44 ) by setting optimized @xmath45 , @xmath46 . three nearly orthogonal mess are marked by red , green and blue arrows ( dots ) in surface ( contour ) plot . the cyan dashed line represents the states orthogonal to the first mes ( red dot ) . ( b ) entropy for the states along the cyan dashed line as shown in ( a ) . all calculations are for partition along cut - i . , width=163 ] _ moore - read state at @xmath11. _ we denote the three groundstates from ed calculation as @xmath47 , ( with @xmath48 ) @xcite . now we form the general superposition states as , @xmath49 where @xmath50 , @xmath51 , @xmath52 are real superposition parameters . for each state @xmath53 , we construct the reduced density matrix and obtain the corresponding entanglement entropy . we optimize values of @xmath54 $ ] and @xmath55 $ ] to minimize the entanglement entropy . in fig . [ fig : hc : mr : mes](a ) , we show the entropy profile at optimized parameters @xmath56 for mess on the hc lattice . here we draw the @xmath57 in the surface and contour plots so that the peaks in entropy show up clearly representing the minimums of @xmath58 . in fig . [ fig : hc : mr : mes](a ) , we find several peaks ( entropy valleys ) in @xmath59 space . the first peak ( red arrow ) relates to the first ( global ) mes @xmath60 . after determining @xmath60 , we search for the states with minimal entropy in the state space orthogonal to @xmath60 as shown in fig . [ fig : hc : mr : mes](b ) . the second and the third mess are shown by green and blue arrows ( the two states labeled by blue arrows are equivalent ) , which are separately located in different entropy valleys as shown in fig.[fig : hc : mr : mes](a ) . we find that the first two mess have almost the same entropy value , indicating they are indeed topological equivalent with the same quantum dimension . we calculate the entropy difference @xmath61 between the third mes and the average of first two mes to extract the information of the quantum dimension of the third quasiparticle . as listed in table-[table : entropy ] , nonzero @xmath62 implies the quantum dimension @xmath63 for the third quasiparticle state @xcite , in agreement with the theoretical expectation for a mr state . we also search for mess for sq lattice @xcite and find similar results as listed in table-[table : entropy ] . .entropy of mess for @xmath11 and @xmath12 . @xmath64 represents the entropy of @xmath65th mes . for @xmath11 , we use @xmath66 . for @xmath12 , @xmath67 . @xmath68 is analytic prediction @xcite . [ cols="<,^,^,^,^,^,^,^,^",options="header " , ] [ table : entropy ] to extract the topological information of the quantum states from mess , we obtain the overlap between the mess for two noncontractible partition directions , which gives rise to the modular matrix @xmath69 @xcite : @xmath70 for the hc lattice and we find very similar result for the sq lattice @xcite . both results are quite close to the theoretical result for mr state based on @xmath71 chern - simons theory @xcite : @xmath72 , which determines the quasiparticle quantum dimension as : @xmath73 , @xmath74 , @xmath75 , @xmath76 and non - trivial fusion rule in eq . ( [ fusiona ] ) . from eq . ( [ mr : hc : modulars ] ) , we obtain @xmath77 , @xmath78 for the hc lattice ( and @xmath79 , @xmath80 for the sq lattice @xcite ) . another striking point is that @xmath81 , which indicates that the two @xmath6 quasiparticles annihilate each other and fuse into different quasiparticles . to demonstrate this non - trivial behavior , we extract the fusion rule related to the third quasiparticle from the numerical @xmath0 matrix through verlinde formula @xcite @xmath82 where @xmath83 : @xmath84 , which agrees excellently with the fusion rule of the mr state ( eq . [ fusiona ] ) . both the @xmath85 and the characteristic fusion rule eq . ( [ mra]-[mrb ] ) unambiguously demonstrate the third quasiparticle @xmath6 representing a majorana fermion . the fusion rule represents two ways to fuse two majorana quasiparticles therefore each pair of majorana quasiparticles can act as a qubit for quantum computation @xcite . at @xmath12 in @xmath86 space by setting optimized @xmath87 . the color of dots represents the magnitude of entropy . the first mes indicated by black arrow . ( b ) the entropy profile versus @xmath88 in the space orthogonal to the first mes . the second , third and fourth mess are labeled by red , green and blue arrows and dots , respectively . the calculation is for bipartition system along cut - i direction . , width=153 ] at @xmath12 in @xmath86 space by setting optimized @xmath87 . the color of dots represents the magnitude of entropy . the first mes indicated by black arrow . ( b ) the entropy profile versus @xmath88 in the space orthogonal to the first mes . the second , third and fourth mess are labeled by red , green and blue arrows and dots , respectively . the calculation is for bipartition system along cut - i direction . , width=144 ] _ read - rezayi state at @xmath12. _ we turn to study the possible non - abelian phase at @xmath12 and detect the fibonacci quasiparticle statistics emerging in this state . following the similar route for mr at @xmath11 , we search for the mess in the space of the groundstate manifold using the following general wavefunctions : @xmath89 where @xmath90 and @xmath91 are the superposition parameters and @xmath47 ( @xmath92 ) are four groundstates from ed calculation . we optimize the superposition parameters @xmath93 to minimize the entanglement entropy . in fig . [ fig : sq : rr : mes](a ) , we show the global mes @xmath60 in parameter space as labeled by the black arrow . the other mess @xmath94 ( @xmath95 ) are determined in the parameter space orthogonal to @xmath60 ( fig . [ fig : sq : rr : mes](b ) ) . the entropies of the last two mess are different from the lowest two mess as list in table-[table : entropy ] , which is consistent with the non - abelian behavior of the quasiparticles . however , we also notice some finite size effect as all the four entropies are different . for the @xmath12 bosonic rr state , the edge conformal field theory is captured by the @xmath96 wess - zumino - witten model @xcite , whose modular matrix can be effectively described by a non - abelian @xmath97 @xmath98-parafermion part coupled by an abelian semion part as : @xmath99 , where @xmath100 is golden ratio number . as a comparison , we obtain the numerical modular matrix by calculating the overlap between the mess along cut i and ii : @xmath101 , which agrees with the analytic prediction ( with a finite size correction of the order of @xmath102 ) . the modular matrix @xmath103 signals fibonacci quasiparticle including the quantum dimension @xmath104 and the related fusion rule as shown in eq . ( [ fusionb ] ) . two fibonacci quasiparticles may fuse into an identity or a fibonacci quasiparticle , which is analogous to two @xmath105 spin-1/2 s combining to either spin-1 or spin-0 total spin @xcite . using this property , fibonacci quasiparticles is capable of universal topological quantum computation @xcite . _ quantum phase transition. _ by tuning the interaction @xmath23 , we can drive a quantum phase transition from the non - abelian state to other quantum phases@xcite . a natural question is how the mess and related modular @xmath0 matrix evolve around the quantum phase transition region . here we study the sq lattice model at @xmath11 as example , in which the quantum phase transition occurs around @xmath106 @xcite . in the mr phase ( @xmath107 ) , we find that the entropy profile of superposition state has three valleys labeled by i , ii and iii , as shown in fig . [ fig : sq : mr : qpt](a ) . the three orthogonal mess are located in the above three valleys , respectively . the resulting modular matrix remains close to the theoretical one for the mr state : @xmath108 . we have also checked that the @xmath0 faithfully represents the quasiparticle information for the whole phase region at @xmath109 . after the quantum phase transition at @xmath110 , we can only find two entropy valleys in entropy map , as labeled by i , ii in fig . [ fig : sq : mr : qpt](b ) , which relates to the first two mess . the third possible mes state ( labeled by white arrow ) determined by the orthogonality relation , actually is not a local minimum . we continue to use these mess as basis states to obtain the modular @xmath0 matrix : @xmath111 , which deviates significantly from the mr @xmath0 matrix . in particular , the quasiparticle fusion rule and statistics has changed with @xmath112 deviating from zero , which demonstrates the disappearance of the mr phase . in @xmath59 space ( @xmath44 ) for ( a ) @xmath107 and ( b ) @xmath113 by setting optimized @xmath114 at @xmath11 on the sq lattice . , width=144 ] in @xmath59 space ( @xmath44 ) for ( a ) @xmath107 and ( b ) @xmath113 by setting optimized @xmath114 at @xmath11 on the sq lattice . , width=144 ] _ summary and discussion. _ we have numerically studied the non - abelian quasiparticle statistics in the lattice boson models which manifest the mr and rr non - abelian states at filling factor @xmath11 and @xmath12 , respectively . our work provides the first convincing demonstration of quasiparticle fusion rules and statistics in microscopic topological band models . the obtained modular @xmath0 matrix faithfully represents majorana and fibonacci quasiparticle statistics including the quantum dimension for each quasiparticle and the fusion rules , which fully support that the each mes is the eigenstate with a definite type of quasiparticle . we are currently developing a numerical method in dmrg simulations to target different mess by projecting out the previously identified lower mess , which we believe will become a useful tool for detecting the full information of the topological order through modular matrix simulation in larger interacting systems @xcite . _ acknowledgements. _ this work is supported by the us nsf under grants dmr-0906816 ( wz , ssg ) , and by the princeton mrsec grant dmr-0819860 ( fdmh ) , and the department of energy office of basic energy sciences under grant no . de - 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a simple model of a josephson junction @xcite proposed by stewart and mccumber @xcite is based , from the mathematical point of view , on the non - linear second order ode @xmath2 here @xmath3 is the unknown function ( called _ phase _ ) representing the difference of phases of the so called order parameter which plays the role of the macroscopic wave function describing the states of two weakly coupled superconducting electrodes constituting josephson junction . the connection of the phase with observable quantities is established by the josephson equation @xcite which connects the voltage across the junction with the time derivative of @xmath4 : @xmath5 here the physical constant @xmath6 ( @xmath7 is the plank constant , @xmath8 is the electron charge ) is called the ( magnetic ) flux quantum , the parameter @xmath9 used in eq . ( [ eq1 ] ) is the rescaled dimensionless time @xmath10 , where @xmath11 is the ` genuine ' ( dimensional ) time , the constant @xmath12 called characteristic frequency is connected with junction properties . it can be defined by the equation @xmath13 , where @xmath14 are the junction characteristics , namely , @xmath15 is its resistance in the normal state , @xmath16 is called critical current ( it estimates the maximal current which can flow through the junction without application of external voltage ) . the positive constant @xmath17 is connected with the junction capacitance , the less the latter , the less the former . the function @xmath18 represents the current ( normalized to @xmath16 and named _ bias _ ) circulating in the junction circuit and supplied by an external source . it is assumed to be specified in advance . here we consider the case of a periodic bias of the known period @xmath19 and arbitrary profile . thus @xmath1 is assumed to satisfy the condition @xmath20 usually @xmath1 is a continuous ( or smooth ) function but a finite number of finite jumps on the period interval is also allowed - like pulses , the problem retains meaningful but such a generalization will not be pursued here . ] . it is worth noting that there is a common practice in the physical literature to divide the periodic bias function @xmath1 into some constant constituent ( ` direct current ' , dc ) @xmath21 and the residual one @xmath22 ( alternating , high frequency , rf current , ac ) : @xmath23 following this convention , we assume for definiteness that the average value of @xmath1 is assigned to @xmath21 . accordingly , by definition , the average value of the residual @xmath22 vanishes . and considers @xmath21 as a free parameter to be varied . in particular , the current - voltage ( i - v ) curve of a josephson junction is understood just as the dependence of the average voltage across it on the dc bias contribution @xmath21 whereas @xmath22 is kept unchanged . ] the important property of a josephson junction , which is perfectly captured by the stewart - mccumber model in spite of its comparative homeliness , is the _ phase - locking _ effect . qualitatively , the phase - locking may be regarded as the relaxation in the course of the phase evolution of the ` loose constituent ' of the phase function reflecting its initial state . then the only surviving dynamical ` process ' proves a steady one ` dragged ' by the periodic bias and not depending of the past phase evolution . said another way , once some lapse depending on the specific rate of relaxation of initial perturbations and the ` magnitude ' of the latter elapses , the phase function profile proceeds with the _ reproducing itself _ , modulo a @xmath24-aliquot contribution , on each subsequent time step of duration @xmath19 . obviously , such a situation would take place , in particular , if , regardless of the initial state the phase starts to evolve from , @xmath25 converges for large @xmath9 to a periodic function of the period @xmath19 . there are also other , non - periodic ( but still fairly specific as we shall see below ) forms of such an asymptotically ` steady ' phase evolution . on the other hand , the phase - locking property is in no way a universal one to be automatically attributed to solutions of eq . ( [ eq1 ] ) . depending on the model parameters , the phase evolution may be completely different , revealing , in particular , no periodicity or any other apparent order @xcite . ( red+green ) , green ) of the single graph describing evolution of the phase function on the time interval @xmath26 $ ] are placed over the common @xmath9-axes segment @xmath27 $ ] by means of appropriate ` horizontal ' shifts ` backwards in time ' ( _ plot segmenting _ ) . the specific parameter value is @xmath28 the values of other parameters are given in the main text . in order to visually resolve the neighbor graphs here ( and in subsequent similar plots ) all the curves , except for a starting one , are uniformly shifted in the ` vertical ' direction by an additional amount of space which is uniformly incremented for each subsequent graph curve ( here by 0.04 units ) . as one can see , the third ( green ) graph segment already coincides with the second one , essentially , yielding therefore the ( approximate ) specimen of the limiting phase evolution function which proves here periodic . ] the plots shown in figure [ f1 ] illustrates the simplest case of the phase - locking in a numerical example . here the result of a straightforward numerical integration of eq . ( [ eq1 ] ) is displayed . for the sake of definiteness , the bias function @xmath29 is chosen to be the periodic rectangular pulse sequence , one of the pulses being shown in the inset . its integral magnitude amounts to 1.5 , the frequency is @xmath30 ( @xmath31 ) , @xmath32 , the width of the pulse peak ( here being rather a plateau symmetric with respect to the plot center ) amounts to 20% of the total pulse period . the ` constant bias constituent ' @xmath21 ( the average of @xmath1 ) equals @xmath33 . the phase evolution displayed in fig . [ f1 ] starts with the null initial conditions @xmath34 chosen for simplicity reasons . is not the most natural choice since it would lead to the ill posedness of the cauchy problem in the limiting case of small @xmath17 . ] the figure displays the phase evolution during _ 3 sequential steps _ of variation of @xmath9 , each of duration @xmath19 . the graph of the first period phase evolution is of red color . the second period plot graph ( @xmath35 $ ] ) is brown . to be placed over the same @xmath9-axes segment as the first graph , it _ is shifted to the left _ ( ` backwards in time ' ) just by @xmath19 . similarly , the graph of the phase function over the third period time step ( @xmath36 $ ] , the green curve ) is shifted to the left by @xmath37 that places it over the same @xmath9-axes segment @xmath27 $ ] . as a result , all the three subsequent portions of the phase evolution graph are displayed over the common segment of the @xmath9-axes becoming more eligible for a visual collation and the monitoring of the @xmath19-scale periodicity . in passing , it is worth emphasizing that due to periodicity of @xmath1 the functions resulted from the above ` translations ' by means of the @xmath9-shifts aliquot @xmath19 _ retain to verify _ eq . ( [ eq1 ] ) . thus , applying such a trick , the ` contiguous ' phase function representing the phase evolution over some time interval , perhaps semi - infinite , can be equally well represented by the sequence of phase functions ( solutions of eq . ( [ eq1 ] ) ) each defined on the finite interval @xmath27 $ ] . besides , for the better clarity , some additional artificial uniformly incrementing _ vertical shifts _ are applied to separate graph segments in fig . [ f1 ] . the point is that we would like to recognize the limiting curve to which the sequence of the displayed ones converges , and the details of such a conversion , provided it takes place . obviously , it is reasonable to add some uniformly incremented vertical space to the vertical positions of separate graph segments in order to visually distinguish the far mutually close curves which otherwise would be seen in the plot overlaid in a messy way . specifically , in the case of fig . [ f1 ] , the vertical shifts amounting to additional 0.04 units per each subsequent graph segment are applied . this allows one to visually distinguish the second ( brown ) and the third ( green ) phase graph segments showing simultaneously that the corresponding functions are very close . in subsequent similar plots , the benefit of the trick will be ever more clear . it turns out that under the conditions assumed , the phase function on the third time step ( the third period ) apparently coincides with the one on the second step and thus already represents , with accuracy characteristic for the plot resolution , the desirable limiting ( asymptotic ) phase time dependence . the @xmath19-scale convergence proves here extremely fast . it is also obvious that in view of the convergence of subsequent graph segments to the common limiting curve the values of @xmath4 on the left and right graph boundaries of its domain coincide , making evidence of the periodicity of the limiting solution of eq . ( [ eq1 ] ) we deal with . it is also of interest to consider the graphs of the corresponding _ voltage _ which coincides , up to the constant factor as it is defined by eq . ( [ eq2 ] ) the ` voltage function ' @xmath38 . the plots of @xmath38 will be referred to below as @xmath39 plots _ in arbitrary units_. ] , with the derivatives of the functions displayed in fig . these are shown in fig . [ f2 ] , the vertical and horizontal shifts ( the part of _ the plot segmenting _ trick ) similar to ones in fig . [ f1 ] being applied . the voltage functions also rapidly converge to a periodic limiting one . . in order to resolve the convergent series of graphs , the graphs except of the starting one are uniformly shifted in the ` vertical ' direction , the trick similar to one applied in fig . [ f1 ] . ] it has to be noted that the phase - locking does not necessarily claim for the asymptotic phase function @xmath25 to be periodic . indeed , keeping in mind the physical aspect of the problem , the actually registered voltage across josephson junction is not momentary but is efficiently averaged over many bias periods @xmath19 . for the sake of simplicity , let us consider the result of the voltage averaging over a _ single period _ ( the shortest time interval making sense in the situation under consideration ) , treating the averaging as the plain integrating followed by the division by the length of the integration interval . then in accordance with eq . ( [ eq2 ] ) the averaging started at the moment @xmath9 has to yield @xmath40 \label{volt}\ ] ] with @xmath41 ( the proportionality relation means here the dropping out of a dimensional universal constant ) . thus the ( asymptotic ) periodicity of @xmath4 means @xmath42 implying that no average voltage across the junction is observed bias current flows through the junction but the [ averaged ] voltage across it is zero . ] . however , instead of @xmath1 periodicity , one may also assume that the increment of @xmath4 across the time step of duration @xmath19 is independent on the moment @xmath43 when the averaging starts : @xmath44 , asymptotically ( for large but otherwise arbitrary @xmath9 ) . this constant may not be arbitrary . the condition above is consistent with eq . ( [ eq1 ] ) if , asymptotically , @xmath45 where @xmath46 is some integer , positive , negative , or zero . this number is named the _ phase - locking order_. evidently , the periodic phase function is the particular case corresponding to the zero order . allowing arbitrary integer @xmath46 , the average voltage proves quantized assuming , in principle , any of discrete uniformly distributed values . is a _ rational _ number , provided the averaging is carried out over sufficiently large number of periods . ] numerical integration of eq . ( [ f1 ] ) allows one to easily confirm the feasibility of non - zero - order phase - locking phase functions . in particular , fig . [ f3 ] displays the example of _ first order _ phase - locking . here the only difference in the parameter setup with the zero order case ( figures [ f1 ] , [ f2 ] ) is the value of the parameter @xmath21 now amounting to 1.1 . the limiting phase function is here _ not _ periodic but its increments on the time steps of duration @xmath19 apparently tends to @xmath24 . on the other hand , subtracting the linearly growing contribution which insures this secular phase incrementing , a periodic function is leaved . , [ f2 ] , except for @xmath47 . the color of subsequent phase function segments uniformly varies from red to green . the uniformly incremented shifts in the ` vertical ' direction are applied including the first order phase - locking responsible contribution @xmath48 for the phase ( @xmath48 + 0.2 units for the top panel and 0.2 units for the lower one per subsequent graph segment , respectively ) . ] similarly , the further increasing of @xmath21 yields the second order phase - locking phase evolution . it is displayed in fig . here @xmath49 , the other parameters being kept unchanged . increasing @xmath21 , the larger order phase - locking phase functions can be produced as well . -[f3 ] , except for @xmath49 . the color of subsequent phase function segments ( top panel ) and voltage function segments ( lower panel , arbitrary units ) uniformly varies from red to green . the uniformly incremented shifts in the ` vertical ' direction are similar to ones applied in fig . [ f3 ] differing in the shift contribution for the phase connected with the non - zero order of the phase - locking which amounts here to @xmath50 instead of @xmath48 applied therein . ] it is seen that in the cases of higher order phase - locking , the convergence of the sequence of segmented phase functions to the common limiting function is slower . indeed , in the case of zero order phase - locking ( fig . [ f1 ] , @xmath28 ) the third iteration apparently coincided with the second one and yield , in fact , the limiting curve ( the phase - locking order can be estimated from the asymptotic relation @xmath51 \label{eq23}\ ] ] taking place for sufficiently large @xmath9 ( formally , in the limit @xmath52 ) , provided the phase - locking is developed ) . in the case of the phase - locking of the first order observed in the case @xmath47 the four @xmath19-lapses are needed for the reaching of a comparable closeness to the limit . for the phase - locking of the order two which takes place , in particular , if @xmath49 the number of necessary iterations increases up to 15 . and it suffices to set up @xmath53 to lose the apparent convergence to any steady asymptotic state at all . the corresponding apparently non - convergent segmented phase and voltage function plots are shown in fig . these reveal no tendency to converge to any limiting curves demonstrating ` irregular ' ( or even apparently ` chaotic ' , as far as one concerns the variations on the scale @xmath19 ) behavior . the value of the expression ( [ eq23 ] ) , irregularly oscillating around 2 , does not reveal apparent tending to any limit as well . -[f4 ] , except of @xmath53 . the uniformly incremented shifts in the ` vertical ' direction ( @xmath54 units for the phase and 0.2 units for the voltage functions per subsequent graph segment , respectively ) are applied . ] it is worth mentioning that the convergence or the divergence of the sequence of phase functions describing a single long - term phase evolution is often difficult to reliably establish by means of straightforward numerical integrating of eq . ( [ eq1 ] ) . indeed , the slower such a sequence converges , the longer fragment of the phase evolution has to be analyzed . accordingly , approaching the hypothetical boundary of the area in the space of the bias functions where the evolution of the phase leads to a steady state ( _ phase - locking area _ in the parameter space ) , one inevitably encounters with insufficiency of the available computational resources . the very boundary , where the convergence reverses to the divergence , is not detectable in this way . besides , in view of the obscurity of the corresponding time scale , the ` strong ' phase divergence ( more exactly , the absence of convergence to any asymptotic steady state ) is even harder to detect through the apparent behavior of numerically generated phase function than the too slow convergence . the formulation and substantiation of the method allowing one to efficiently distinguish the phase - locking property in the important particular case of ` overdamped ' version of equation ( [ eq1 ] ) is the main theme of the present discussion . the term ` overdamped ' refers here the case @xmath55 the parameter @xmath17 scales the ( only ) term involving the second order derivative of the phase function in eq . ( [ eq1 ] ) . it might be supposed that under certain conditions including imposing of limitation on the second derivative values the relevant solutions of eq.([eq1 ] ) may be approximated by the solutions of the equation @xmath56 obtained from eq.([eq1 ] ) by a mere discarding of the second order derivative term . the numerical example illustrating the impact of the above ` simplification ' is shown in fig . [ f6 ] , the discussion of the status of the corresponding approximation for ` overdamped ' systems ( in the case of sinusoidal bias @xmath1 ) can be found in @xcite . in fig . [ f6 ] the black curve represents the phase function computed in accordance with eq . ( [ eq1x ] ) , the red one is the corresponding solution of the general rsj - equation ( [ eq1 ] ) . the ` red ' solution is shifted downward by 0.1 units in order to allow easier visual distinguishing of the close graphs on the common plot panel . the green curve represents the difference of the two phase functions above which is _ magnified by the factor of 10_. the bias pulse is shown in the inset . its form is similar to the ones used above , see fig . [ f1 ] , except for the bias ` dc constituent ' , @xmath21 , which is here chosen vanishing ( i.e. @xmath57 ) , and the integral magnitude of a single pulse which here and in all the numerical examples below amounts to 3.5 . ) ( the red graph ) and to ( [ eq1x ] ) ( the black one ) for the same periodic rectangular pulse bias ( shown in inset ) displayed together with their difference magnified by the factor of 10 ( the green graph ) . the red plot is shifted downward by 0.1 units . the bias is the same as above except the parameter @xmath21 which is now assumed to vanish and the pulse magnitude here equal to 3.5 . ] it is seen that under the conditions assumed the solutions of eqs . ( [ eq1x ] ) and ( [ eq1 ] ) are very close , in total , both in shape and in magnitude , the deviations emerging at several moments of time ultimately relaxing . specifically , the solution difference reveals the splashes originated in three points : at @xmath58 , and on the front and back ` shocks ' of the bias pulse . at right near them , the difference soon reaches a maximum and then quickly die . in total , the distinction of the corresponding solutions of eqs . ( [ eq1 ] ) and ( [ eq1x ] ) may be considered fairly mild . it is easy to see that the splash following the point @xmath58 arises due to the specific choice of the initial conditions for solution of eq . ( [ eq1 ] ) which read @xmath34 . then eq . ( [ eq1 ] ) implies @xmath59 which , for @xmath60 , is a large quantity contrary to assumption required for approximating eq . ( [ eq1 ] ) by ( [ eq1x ] ) . in particular , it leads to the fast growth of the first derivative immediately after the start of the phase evolution . this effect is also observed in fig . [ f2 ] . apart of the apprehensible effect of non - equivalent initial conditions , the cause of the other discrepancy splashes is also quite understandable : they arise due to the discontinuities of the r.h.s . function @xmath1 . indeed , as a pulse front / back jump occurs , the discontinuity in @xmath1 converts to a finite jump in the derivative @xmath38 in solution of eq . ( [ eq1x ] ) which , in turn , emerges the dirac @xmath61 function - like irregularity in the second derivative @xmath62 which again violate the assumption of limitation of @xmath62 magnitude required for the approximating of solutions of ( [ eq1 ] ) by solutions of eq . ( [ eq1x ] ) . evidently , the cause of these peculiarities , leading to major distinctions of solutions of eq . ( [ eq1 ] ) and eq . ( [ eq1x ] ) shown in fig . [ f6 ] , originates in either the admitting somewhat inadequate initial conditions or in a too rough representation ( by a discontinuous function ) of the bias term . since for a more physically realistic bias model no @xmath61-like irregularities of the phase derivative can arise , the agreement of the models based on eqs . ( [ eq1 ] ) and ( [ eq1x ] ) would improve . as compared to more deep eq . ( [ eq1 ] ) whose study is still possible by numerical methods alone , eq . ( [ eq1x ] ) proves more transparent and allows fairly deep analytical treatment . it seems also to be of interest in its own rights as the example of a non - linear problem of a deep physical relevance associated with a simple linear dynamical system in the case of the bias function @xmath63 , where @xmath64 are some constant parameters , which is most important from viewpoint of applications , eq . ( [ eq1x ] ) proves converting to the following remarkable equation : @xmath65 { { \mathbin{\mathrm{d}}}\over{\mathbin{\mathrm{d}}}\zeta } + \omega^{-2}\right\}\upsilon=0\ ] ] ] . ( [ eq1x ] ) possesses therefore a number of remarkable properties which are the subject of the discourse below . the approach to the problem of description of generic properties of solutions of eq . ( [ eq1x ] ) on the timescale exceeding the period @xmath19 including their asymptotic behavior can be based on the following observation : for any piecewise smooth continuous functions @xmath66 , let us define the functions @xmath67({t})=\int\cos\phi_0({t})\,d\,{t } , \label{eq4 } \\ q_0&\equiv&q_0({t})\equiv q_0[\phi_0]({t})=\int e^{-p_0({t})}\,\sin\phi_0({t})\ , d\,{t } , \label{eq5 } \end{aligned}\ ] ] and introduce the notations : @xmath68({t})= \left[-q_0 + { \mathrm{i}}e^{-p_0}{\zeta_0+\zeta\over \zeta_0-\zeta}\right]^{-1}. \label{eq8 } \end{aligned}\ ] ] then the following identity takes place @xmath69 - \zeta d[\zeta_0 ] \label{eq9}\ ] ] with the differential operator @xmath70 $ ] is defined as follows @xmath71\equiv { d z\over d\,{t } } + { 1\over2}(z^2 -1 ) -{\mathrm{i}}f z \label{eq10}\ ] ] for some function @xmath72 and arbitrary piecewise smooth @xmath73 . _ its proof _ reduces to a straightforward computation applying definitions of the functions involved.@xmath74 _ remarks : _ * for the sake of definiteness , the indefinite integrals in eqs . ( [ eq4],[eq5 ] ) can be understood as @xmath75 . another choice of lower integration boundary would lead to a _ linear transformation _ of @xmath76^{-1}$ ] : an overall constant factor is induced by the changing the lower boundary in the @xmath77-integral while a change of the lower boundary in @xmath78-integral definition yields a constant summand . * solving eq . ( [ eq8 ] ) with respect to @xmath79 , one gets the equation @xmath80 assuming @xmath81 to be real , for complex valued @xmath82 both of these variables can be unimodular , @xmath83 ( and @xmath84 real , see ( [ eq6 ] ) ) if and only if @xmath85 is real . the connection of the relationships inferred from the identity ( [ eq9 ] ) with the equation ( [ eq1x ] ) follows from the identity @xmath86 provided @xmath79 and @xmath4 are connected by eq . ( [ eq6 ] ) . in conjunction with master identity , it immediately yields the following @xcite [ c1 ] let @xmath87 verify eq . ( [ eq1x ] ) whereas @xmath88 be defined by eqs . ( [ eq4 ] ) , ( [ eq5 ] ) . then any solution @xmath89 of eq . ( [ eq1x ] ) satisfies either the equation @xmath90 where @xmath85 is some real constant , or the equation @xmath91 the latter being in fact the limiting form of the former as @xmath92 . _ remarks : _ * when regarded as the relationship determining @xmath4 , eq . ( [ eq13 ] ) or ( [ eq13a ] ) specifies it not uniquely but modulo @xmath24 . on the other hand , this is the only arbitrariness involved in such a definition of @xmath25 . * the initial , ` ground ' solution @xmath93 is produced by the same equation ( [ eq13 ] ) in the case @xmath94 when ( [ eq13 ] ) converts to the identical transformation ( modulo @xmath24 ) . that is why just @xmath85 rather than @xmath95 which has a formally simpler representation in terms of @xmath4 s , see ( [ eq8 ] ) , is employed in ( [ eq13 ] ) . * given @xmath96 , the constant @xmath85 can be easily found . indeed , it follows from eq . ( [ eq8 ] ) that , placing the left boundary of the integration interval at @xmath58 , one has @xmath97 , that yields @xmath98 . \label{eq80}\ ] ] thus , given any particular solution of eq . ( [ eq1x ] ) and the integrals @xmath88 , defined from it by eqs . ( [ eq4 ] ) , ( [ eq5 ] ) and obeying the conditions @xmath97 , the solution of the cauchy problem for eq . ( [ eq1x ] ) is explicitly represented ( modulo @xmath24 ) by eqs . ( [ eq13 ] ) and ( [ eq80 ] ) is not covered , formally , by the above formulae , a minor obvious modification corresponding to the transition to the limit @xmath92 allows to treat it as well . ] . * in view of ( [ eq80 ] ) one also gets @xmath99_{|t=0}=-c[\phi,\phi_0]_{|t=0}\ ] ] fig . [ f8 ] illustrates the above statements . here the top panels contain the plots of the two different solutions of eq . ( [ eq1x ] ) . the bias function @xmath1 is the same sequence of periodically repeated rectangular pulses which was used above , see the inset in fig . [ f6 ] , the only difference is the constant constituent which now amounts to @xmath53 ( it is worth reminding also that now @xmath100 ) . the phase function of the first period phase evolution , starting at the coordinate origin , is considered as the ` ground ' solution of eq . ( [ eq1x ] ) denoted above as @xmath93 . it is plotted at the top - left panel . in the lower - left panel , the integrals @xmath101 calculated from @xmath93 are plotted . the top - right panel displays the ` continuation ' of @xmath93 to the second period @xmath102 $ ] which , afterwards , is returned ( ` shifted ' to the left by @xmath19 ) to the @xmath93 domain . ( no ` vertical ' shift is here applied . ) as it had been mentioned above , due to the periodicity of @xmath29 , the ` shifted ' solution is again a solution of eq . ( [ eq1x ] ) . we denote it @xmath25 . ( this function is distinguished by the specific initial condition @xmath103 which however plays no role in the current context . ) finally , the graph displayed in the lower - right panel is the result of straightforward computation of @xmath104({t})$ ] in accordance with definition ( [ eq8 ] ) and ( [ eq6 ] ) . more exactly , the deviation of @xmath85 off its averaged ( on the interval @xmath27 $ ] ) value which is @xmath105@xmath106 is shown . one sees that the functional @xmath107(t)$ ] computed ` from the first principles ' is , after all , a constant up to a small deviation . the latter irregularly oscillates around zero with the amplitude which is at least 6 orders less than the phase magnitudes . it represents the ` numerical noise ' reflecting mostly tolerable inaccuracy of approximate numerical solutions of differential equation . the closely related computation displayed in fig . [ f9 ] illustrates the application of eq . ( [ eq13 ] ) for the generating of new solutions of eq . ( [ eq1x ] ) from a known one . here the black curve shows the ` ground ' solution , @xmath93 , starting in the coordinate origin ; note that any other phase function might be used instead . then we chose , say , @xmath108 and compute @xmath89 by means of eq . ( [ eq13 ] ) . the result is represented by the red curve which was _ shifted downward _ by 0.1 units for the convenience of further visual collations . next , we carry out the straightforward numerical integrating of eq . ( [ eq1x ] ) adopting _ the same initial conditions @xmath109 _ as the just generated solution obeys . the green curve represents the result of the integrating which is also additionally _ shifted downward _ , here by 0.2 units . finally , the blue graph is the difference of the results of computation of the same phase function by these two methods ( viz the application of eq . ( [ eq13 ] ) and the numerical ode integrating ) _ magnified by the factor of @xmath110 . _ again , one sees that eq . ( [ eq13 ] ) ensures the stable accuracy about six true decimal digits with the discrepancy falling in the level of ` numerical noise ' . ) for @xmath108 while the green one is the result of numerical integration of eq . ( [ eq1x ] ) with the same initial conditions as the ` red ' solution obeys . finally , the blue curve shows the difference of the `` red '' and `` green '' functions magnified by the factor of @xmath111 . ] it is important to note that , in the relationships above , the roles of the functions @xmath93 and @xmath4 ( any two solutions of eq . ( [ eq1x ] ) ) should be symmetric by general reasons . however eq . ( [ eq13 ] ) does not reveal , apparently , such a symmetry since it involves the integrals @xmath112 , @xmath113 determined by the solution @xmath93 but no similar contribution connected with @xmath4 is present . besides , eq . ( [ eq13 ] ) is easily solvable with respect to @xmath4 but represents to a nonlinear integral equation with respect to @xmath93 . any inconsistency does not arises here however and the asymmetry mentioned above is actually a fallacious one . the point is that there is a specific algebraic relation which constrains @xmath77- and @xmath78-functionals associated with _ any _ two solutions of eq . ( [ eq1x ] ) . it enables one , in particular , to represent any @xmath114 as a simple elementary function of another @xmath114-pair . namely , the following remarkable relation takes place . let @xmath87 , @xmath89 , @xmath115 , @xmath116 be as in proposition [ c1 ] . let also @xmath117 , @xmath118 be determined from @xmath89 in the same way as @xmath115 , @xmath116 are determined from @xmath87 . then @xmath119 with _ the same _ real constant @xmath85 . _ remarks : _ * inverting ( [ eq14 ] ) , one gets the alike formula @xmath120 which differs from ( [ eq14 ] ) , apart of interchanged roles of @xmath114 and @xmath81 , by the opposite sign of the @xmath85-constant alone . * from viewpoint of the induced transformation group structure , the transformation ( [ eq14 ] ) coincides with the relativistic velocity addition rule . * in view of the relationship above , it seems natural to incorporate the real - valued functions @xmath117 , @xmath118 into the complex - valued one defined as follows : @xmath121 then the transformation described eq . ( [ eq14 ] ) may be named , in a sense , meromorphic since @xmath122 is expressed via @xmath123 as a ( meromorphic ) function of @xmath123 . * the definition of @xmath114-integrals by ( [ eq4 ] ) , ( [ eq5])-like equations is equivalent to the equation @xmath124 indeed , rewriting it as @xmath125 and separating the pure imaginary part , one gets @xmath126 , which , for nonzero @xmath127 , yields @xmath128 , i.e. in accordance with definition ( [ eq : yrtfers ] ) , eq . ( [ eq4 ] ) in fact . next , separating the real part , one gets @xmath129 . integrating it and taking into account the representation of @xmath127 just obtained , one gets eq . ( [ eq5]).@xmath74 * working in terms of @xmath130 instead of @xmath114 , the choice of @xmath131 as the lower integral limit in ( [ eq4 ] ) , ( [ eq5 ] ) , is equivalent to the initial condition @xmath132 for the function @xmath133 verifying eq . ( [ eq85 ] ) . * inverting eq . ( [ eq14 ] ) , @xmath123 can be represented as a function of @xmath122 ( see ( [ eq14a ] ) ) . then eq . ( [ eq1x ] ) can be solved with respect to @xmath134 which is represented as explicit function of @xmath25 , @xmath117 , @xmath118 ( and @xmath85 ) as follows @xmath135 the dual eqs . ( [ eq13]),([eq13a ] ) manifest the symmetric role of the two solutions @xmath136 noted above . _ proposition proof_. the relationship to be proven belongs to the category of ones for which , as the true formula is recorded , the proof reduces to a straightforward computation . indeed , let us rewrite eq . ( [ eq13 ] ) as follows @xmath137 and calculate the @xmath9-derivative of the difference @xmath138 then the straightforward computation yields @xmath139 the following identity takes place @xmath140 it implies , together with the equation just derived , the following series of equalities : @xmath141 the last equation is equivalent to the system of two linear homogeneous first order odes for the two ( real valued ) functions @xmath142 . moreover , in accordance with standard ` normalization ' ( [ eq86 ] ) and @xmath61 definition , one has @xmath143 the null initial condition . thus @xmath144 , i.e. one gets the equation @xmath145 which is nothing but eq . ( [ eq14 ] ) . @xmath74 thus , in accordance with the remarks above , we may consider the expression ( [ eq8 ] ) as the functional over the set of pairs of solutions of eq . ( [ eq1x ] ) ( the direct square of the solution set ) taking it onto the real axis . to be more exact , since the limiting case @xmath92 is quite legitimate , one has to replenish the real axis by the infinitely remote point . the result is the circumference but it appears here as the isomorphic image of the real projective line @xmath146 which is just the natural ` index space ' to be used for the ` enumerating ' of solutions of eq . ( [ eq1x ] ) . specifically , having fixed some @xmath93 , the inverse map from @xmath147 onto the first factor in the direct product yields the ( 1 - 1 non - canonical ) parameterization of the space of solutions by the circumference points . it is worth noting that the @xmath85-map ( [ eq8 ] ) , ( [ eq6 ] ) is _ antisymmetric _ with respect to the interchange of the functional arguments @xmath136 , i.e. @xmath148=-c[\phi_0,\phi ] . \label{eq15}\ ] ] the property ( [ eq15 ] ) can be established as follows . eq ( [ eq9 ] ) implies @xmath149^{-1}\over d\,{t}}= 2 { \mathrm{i}}(\zeta-\zeta_0)^{-2}(\zeta_0 d[\zeta ] - \zeta d[\zeta_0])= -e^{p } { d c[\phi_0,\phi]^{-1}\over d\,{t } } \nonumber\ ] ] and therefore @xmath150 $ ] is automatically a constant , provided @xmath151 $ ] is . further , in accordance with ( [ eq : urtcy ] ) the equality @xmath152=c[\phi,\phi_0]$ ] takes place for @xmath131 . hence these constants coincide up to the opposite signs and ( [ eq15 ] ) holds everywhere . if @xmath136 are solutions of eq ( [ eq1x ] ) then eq . ( [ eq15 ] ) is satisfied . @xmath74 now let us consider the implications of the above relationships in application to the property of the phase - locking . the latter is connected with the specific asymptotic behavior of the phase functions @xmath89 verifying , in our case , eq . ( [ eq1x ] ) . namely , in the case of phase - locking eq . ( [ eq22 ] ) has to be satisfied , asymptotically . to describe efficiently this property , let us define the sequence of functions @xmath153 defined on the segment @xmath27 $ ] as follows . above , it was understood as one of the phase functions from the pair @xmath136 ( see e.g. ( [ eq6 ] ) ) or as the ` ground ' solution obeying the condition @xmath154 as in fig . [ f8 ] , such a usage leading to no interpretation problems . the new interpretation of the same symbol is the particular element of the sequence of functions defined by eq . ( [ eq16 ] ) . it does not match the above . usually , it is clear what is meant . however , if below the both meaning loads of the symbol @xmath93 ( as well as @xmath112 , @xmath113 ) meet in a common context , a slightly modified notation , @xmath155 , will be employed for its first interpretation . ] @xmath156 , \;j=0,1,2,3\dots,\ ; { t}\in{}[0,t ] \label{eq16}\ ] ] where the double brackets @xmath157 $ ] stands for the integer part of the real number enclosed . thus the plot of @xmath158 displays how the ` genuine ' phase function @xmath89 looks like ` on the @xmath159th segment ' of the length ( duration ) @xmath19 with respect to the closest level aliquot to @xmath24 . to that end , its graph is shifted downward or upward by such a number of ` full phase revolutions ' @xmath24 which returns its left - boundary point @xmath160 to the segment @xmath161 . we have already used such a trick in the arrangement of plots of phase functions calling it occasionally ` the segmenting ' . here the explicit transformation of the phase function on the corresponding t - segments of duration @xmath19 yielding a specific sequence of the phase functions defined on the segment @xmath27 $ ] is introduced . all the functions @xmath162 satisfy eq . ( [ eq1x ] ) . the specific ( and characteristic ) property of the sequence of its solutions @xmath162 s associated with a _ single _ solution @xmath4 defined for all @xmath9 is obviously the following : @xmath163 given the sequence @xmath164 of solutions of eq . ( [ eq1x ] ) on the segment @xmath27 $ ] fulfilling the condition ( [ eq16a ] ) , the valid phase function @xmath165 can be reconstructed in the obvious way . it is convenient to represent the phase - locking property in terms of properties of the functional sequence @xmath162 . it can be stated the following : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in the case of phase - locking the sequence @xmath162 uniformly converges on the segment @xmath27 $ ] to some limiting function @xmath166 which satisfies eq . ( [ eq1x ] ) and the condition @xmath167 for some integer k. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ it is natural to select and fix the ` ground ' specimen of phase functions denoting it @xmath87 , @xmath168,$ ] characterizing it by means of the null initial value @xmath154 ( the symbol @xmath169 and its derivates will be also used in cases prone of confusion with the notation introduced in ( [ eq16 ] ) ) . having posed the problem in such a way , @xmath87 is completely determined by the bias function @xmath170 . we also assume the value @xmath58 to be the lower integration boundary for the integrals in the @xmath101 definitions ( [ eq4]),([eq5 ] ) . then eq . ( [ eq13 ] ) implies that there exists a constant @xmath171 such that @xmath172 ( we assume @xmath171 to be finite but the case of ` infinite @xmath171 ' is also tractable , with minor modifications , in the same way . ) taking the above representation of @xmath173 into account , it is easy to see that eq . ( [ eq18 ] ) is satisfied if and only if the equation @xmath174 admits a _ real _ root @xmath171 . in turn , the latter condition is equivalent to the inequality @xmath175&\equiv & -q_0(t ) ( e^{-p_0(t)}+1 ) \sin\phi_0(t ) \nonumber\\ & & -{\mbox{\scriptsize$1\over2$}}\left(-q_0(t)^2+\left(e^{-p_0(t)}+1\right)^2\right ) ( 1-\cos\phi_0(t ) ) \nonumber\\ & & + ( 1-e^{-p_0(t)})^2 \nonumber\\ & \equiv & 4 e^{-p_0(t)}\times \label{eq20a}\\ & & \left ( \left [ -{\mbox{\scriptsize$1\over2$}}e^{{\mbox{\scriptsize$1\over2$}}p_0(t ) } q_0(t)\sin{\mbox{\scriptsize$1\over2$}}\phi_0(t ) + \cosh{\mbox{\scriptsize$1\over2$}}{}p_0(t ) \cos{\mbox{\scriptsize$1\over2$}}\phi_0(t ) \right]^2 -1 \right ) \nonumber\\ & \ge&0 \label{eq20 } \end{aligned}\ ] ] @xmath176 the above is therefore the _ necessary _ condition of phase - locking . it reflects in fact the property of the very function @xmath1 . as we shall show , the similar but _ strict _ inequality is the _ sufficient _ condition of the phase - locking for the phase function described by eq . ( [ eq1x ] ) . ( in the case of the equality , the convergence to some asymptotic limit is observed as well but it is slower and reveals some other specialities . ) the vantage of such a form of the phase - locking ` monitoring ' is that this is the _ property manifesting itself for sufficiently large @xmath9 . generally speaking , one can not forecast in advance how long phase evolution has to be tracked for the detection of the corresponding @xmath19-scale reproducibility of the phase function form which would make evidence of the phase - locking . on the other hand , making use of the condition ( [ eq : yrtred ] ) , the appearance of phase - locking is deduced directly from the properties of a _ single _ solution of eq . ( [ eq1x ] ) computed on the _ finite _ interval @xmath27 $ ] . the function @xmath178 $ ] plays an important role in the problem of description of asymptotic properties of solutions of eq . ( [ eq1x ] ) . we may interpret it as the functional on the set of bias functions @xmath1 since the functions @xmath179 from which it is built upon are uniquely determined by @xmath1 : @xmath93 is the solution of eq . ( [ eq1x ] ) with initial condition @xmath154 , @xmath101 are calculated from @xmath93 in accordance with eqs . ( [ eq4 ] ) , ( [ eq5 ] ) and the initial conditions @xmath180 one may also choose @xmath1 to belong to some more restricted class of functions , for example , a family parameterized by a finite set of parameters . then , one may also regard @xmath181 as the function of these parameters and study its properties . adopting the last interpretation , fig . [ f10 ] shows @xmath181 as the function of a single variable$ ] of @xmath181 , provided this can not lead to a misunderstanding . ] , the constant bias constituent @xmath21 , assuming the very bias function to be the periodic sequence of rectangular pulses shown in the inset in fig . [ f6 ] . one recognizes the three minima on the fragment of the @xmath181 plot seated in fig . [ f10 ] , the left one situating very close to the horizontal coordinate axis and the middle one being fairly steep . more narrowly , the minima above are shown in fig . [ f11 ] , where the plots of their vicinities are displayed with higher resolutions . one sees that in all the three cases the minima lay well below the horizontal coordinate axes . hence each of them is encompassed by some @xmath182 segment where @xmath181 assumes negative values ( and this is the universal property : there are no positive minima of @xmath181 ) . these are separated by segments where @xmath183 is positive . on the constant constitution @xmath21 of the bias with the other parameters held fixed . ] as we shall show , each segment of positive @xmath181 determines the area of the values of the parameter @xmath21 giving rise to the phase - locking phase evolution _ of a common order_. such segments are directly associated with so called shapiro steps strictly constant voltage segments observed on the josephson junction i - v curve @xcite . in particular , the domain of @xmath21 values shown in fig . [ f10 ] , where @xmath184 , contain the steps of the orders , from left to right , @xmath185 ( extending to the left beyond the plot boundary ) , 0 , 1 , and 2 . the values of @xmath21 falling into the segments of negative @xmath181 correspond to the apparently behavior of the phase which is similar to one displayed in fig . we shall see however that these phase evolutions do not correspond to an actual chaos ( pseudo - chaos ) @xcite . rather , these are the manifestations of a ` beating ' produced by two inconsonant frequencies . in particular , such phase evolutions imply quite definite average voltages which are obtained by the averaging of ( [ volt ] ) over large time intervals . this point will be considered in more details later on . the algorithm allowing one to reconstruct the phase functions at any moment of time @xmath9 divides into several steps . at first , one has to solve the quadratic equation ( [ eq19 ] ) with respect to @xmath85 , assuming its roots to be real ( that takes place iff @xmath186 ) . then , if @xmath184 , eq . ( [ eq13 ] ) yields the _ two _ phase functions which are the candidates to the role of possessor of the ` refined ' property of phase - locking ( the limiting function asymptotically approximating generic solutions ) . a somewhat delicate point is the revealing which of the two solutions is the one we search for . the answer is connected with the stability property ( the phase - locking must be stable ) and can be provided , in principle , with the help of the standard perturbational analysis ( the local stability problem , cf . @xcite ) which however yield not the actual problem solution but rather a recipe of its computation . fortunately , there is an attractive opportunity to directly manifest not only the effect of _ infinitesimal _ perturbations but , at one blow , to establish the ` global ' stability property allowing perturbations of arbitrary magnitudes . here we seize upon it . ( starting from the coordinate origin ) . the two red curves are the the candidates to the role of the asymptotic limiting phase function which are constructed by means of eqs . ( [ eq19 ] ) , ( [ eq13 ] ) . the green curves are the phase functions constructed by means of a straightforward numeric integration of ( [ eq1x ] ) obeying the same initial conditions as the ` red ' solutions ; for better clarity , they are shifted , afterwards , in ` vertical ' direction by 0.5 units upward ( the top curve ) and downward ( the lower curve ) , the red graphs being undergone no shifts . ] and obtained by means of eqs . ( [ eq19 ] ) , ( [ eq13 ] ) against the ones obtained by means of straightforward numerical integration of eq . ( [ eq1x ] ) with the same initial conditions as the former functions obey . ] this point is illustrated by fig . [ f12 ] . here the black curve shows the ` ground ' solution @xmath93 ( starting from the coordinate origin ) for the rectangular pulse bias of the profile shown in the inset in fig . [ f6 ] , and @xmath187 ( which yields a positive @xmath181 value , see fig . ( [ f10 ] ) ) . the red curves are the graphs of solutions ( ` phase - locking candidates ' ) one of which is expected to realize the ` refined ' , precisely steady phase - locking evolution obeying the property ( [ eq22 ] ) and representing , in a sense , the asymptotic limit of a generic phase function . the phase - locking candidates are constructed by means of eqs . ( [ eq13 ] ) with the two values of the constant @xmath85 obtained from eq . ( [ eq13 ] ) . the green curves represent the phase functions constructed by means of a straightforward numeric integrations of eq . ( [ eq1x ] ) obeying _ the same initial conditions _ as the ` red ' solutions does . the plot graphs are afterwards shifted ` in vertical direction ' by 0.5 units upward ( the green top curve ) and downward ( the lower one ) making easier the visual collation of the profiles of the functions involved . the apparent qualitative coincidence of the results of the two ways of computation of ` phase - locking candidates ' is numerically confirmed in fig . [ f13 ] where their relative discrepancies are plotted . one sees that the two computations agree at the level one part in @xmath188 which is close to the accuracy of the numerical integrating of the ode involved . it is natural to suppose that only one of the two solutions shown as the red ( or , equivalently , green ) curves in fig . ( [ f12 ] ) is of interest as a model of physical relevance because another one is necessarily _ unstable _ ( it plays the own specific role as the separator of two ` basins ' of the phase - locking in the space of all phase functions , though ) . indeed , the distinction of their stability properties is clearly seen in fig . [ f14 ] . by red graphs are displayed , the perturbation function corresponding to the stable evolution ( the red graph ) being magnified by the factor of @xmath189 . ] here the evolution of perturbations of the two solutions shown in fig . [ f13 ] ( the red curves ) is displayed , the perturbed phase functions are defined as the solutions @xmath190 to eq . ( [ eq1x ] ) with initial conditions distinct from the ones the phase - locking candidates obey by amount of 0.1% : @xmath191 . for definiteness , we chose @xmath192 for the upper curve in fig . [ f13 ] and @xmath193 for the lower one . accordingly , the upper ( green ) graph in fig . [ f14 ] shows @xmath194 for the ` upper ' phase - locking candidate ; it makes evidence of a strong instability since the deviation is permanently ( and , in fact , exponentially ) growing . on the contrary , the lower ( red ) curve in fig . [ f14 ] demonstrate the exponential relaxation of the perturbation ; moreover , to display it more clearly , the perturbation value displayed is magnified by the factor of @xmath189 , i.e. the function @xmath195 is actually plotted . thus we see that only the lower red curve in fig . [ f13 ] corresponds to a stable phase evolution . the evolution described by another , upper , ( red ) curve in fig . [ f13 ] related to another solution of eq . ( [ eq19 ] ) , is unstable . as a matter of fact , the first curve describes an attractor of all the ` neighboring ' solutions while the second one ` repels ' them . above , we have numerically demonstrated the validity of the equation ( [ eq181 ] ) allowing one to build a new solution from the known one . similarly , eq . ( [ eq14 ] ) allows one to determine , knowing @xmath85 , the integrals @xmath117 and @xmath118 associated with the former . figures [ f15 ] and [ f18 ] displays the results of the corresponding computation . there the black curves represent the @xmath77- and @xmath78-integrals corresponding to the ` ground ' phase function @xmath93 which is represented by black graph in fig . the red curves are the integrals for the refined phase - locking candidates which are determined by means of eq . ( [ eq181 ] ) . the green curves ( which are shifted in vertical direction downward ) are the same functions but they are obtained by straightforward numerical integrating in accordance with @xmath77 and @xmath78 definitions . in fig . [ f18 ] , the relative discrepancies of the two ways of computation of the @xmath77- and @xmath78- integrals are displayed . ( note that the relative discrepancy growth observed near the left boundary @xmath58 is the numerical artifact caused by the vanishing of @xmath196 and @xmath197 . ) -integrals corresponding to the stable steady phase - locking are shown . the black curve is the @xmath77-integral for the ground phase function @xmath93 ( the black curve in fig . [ f12 ] ) . the red curve represents @xmath77-integral is associated with the refined phase - locking candidate determined by means of eq . ( [ eq181 ] ) . the green curve represent the same function obtained directly from definition ( [ eq4 ] ) , afterwards it being shifted downward by 0.5 units . ] -integrals corresponding to the same stable phase - locking state as in fig . [ f15 ] are shown . the color meaning is the same as therein . the green graph is shifted downward by 1 unit . ] [ 0.6 ] - and @xmath78-integrals ( by means of eq . ( [ eq181 ] ) and the computation in accordance with definitions from the known phase functions ) corresponding to the stable phase - locking . , title="fig : " ] [ 0.6 ] - and @xmath78-integrals ( by means of eq . ( [ eq181 ] ) and the computation in accordance with definitions from the known phase functions ) corresponding to the stable phase - locking . , title="fig : " ] but it is connected with the unstable steady phase evolution whose phase function is represented by the upper red curve in fig . [ f12 ] . ] but corresponds to the unstable phase - locking candidate . the ` ground ' @xmath78-integral corresponding to the ` ground ' phase function ( the analogue to the black graph in fig . [ f15a ] ) is not shown because of the too large distinction in the magnitudes . ] [ 0.6 ] - and @xmath78-integrals corresponding to the unstable phase - locking candidate . , title="fig : " ] [ 0.6 ] - and @xmath78-integrals corresponding to the unstable phase - locking candidate . , title="fig : " ] as it has been noted , any phase function @xmath89 defined on arbitrary @xmath9 domain can be equivalently described by the sequence of the functions @xmath158 defined on the common interval @xmath198 $ ] , see eq . ( [ eq16 ] ) . at the same time , as a solution of eq . ( [ eq1x ] ) , each function @xmath158 can be represented in the form ( [ eq11 ] ) for the own specific value of the constant @xmath85 which we denote @xmath199 . a simple but important observation reads : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the property of phase - locking equivalent to the existence of the limit @xmath200 of the sequence of _ functions _ @xmath158 ( see eq . ( [ eq17 ] ) ) associated with a generic solution of eq . ( [ eq1x ] ) , is also equivalent to the existence of the limit @xmath201 of the sequence of _ real constants _ @xmath199 , connected with @xmath162 , either finite or infinite . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus @xmath202 must exist , either finite or infinite , provided the phase - locking takes place . on the contrary , the phase - locking does not arise and the phase function reveals apparently behavior if and only if the sequence @xmath199 has no limit . _ remark : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ allowing constants @xmath199 and their limit to assume infinite values , the real projective line @xmath146 isomorphic to the circumference has to be adopted as the space of their ( @xmath85-constants ) originating . this can be realized by means of identifying each @xmath199 with the pair @xmath203 and the interpreting the latter as homogeneous coordinates on @xmath146 . the limit of the sequence @xmath199 has also to be understood as a point in @xmath146 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ which indicates the phase - locking property is shown . the bias is the same periodic rectangular pulse sequence as above besides the specific value of the dc contribution @xmath204 . ] not tending to an apparent limit indicates the absence of the phase - locking ( phase behavior ) . the bias is the same periodic rectangular pulse sequence as above besides the specific value of the dc contribution @xmath205 . ] figures [ f19 ] , [ f20 ] display the two examples of @xmath85-sequences for the phase - locking ( fig . [ f19 ] ) and ( fig . [ f20 ] ) phase evolutions , respectively . the distinction arises because to the slightly different values of the constant bias parameter : @xmath204 for fig . [ f19 ] , and @xmath49 for fig . [ f20 ] , cf . the left - lower panel of fig . the other parameters of the bias function @xmath1 are the same as for the pulse shown in the inset in fig . [ f6 ] . in order to detect the phase - locking without a plain phase simulation , one may , in spite of determination of the constants @xmath199 from the functions @xmath158 by means of eq . ( [ eq80 ] ) , make use of the following recurrence relation which can be derived from eqs . ( [ eq16 ] ) , ( [ eq16a ] ) , ( [ eq13 ] ) and @xmath199 definition : @xmath206 ( it is worth reminding that the subscript ` @xmath207 ' hints at the specific initial conditions the functions @xmath208 obey which read @xmath209 it emphasizes the distinction of the ` ground ' solution and its derivates from the particular element @xmath93 of the sequence @xmath162 ; we shall omit this notation complication whenever possible . ) the relation above is of a notable importance allowing one to obtain , finally , the exhaustive description of the long - term phase behavior on the time scales exceeding @xmath19 by means of pure algebraic manipulations . to that end , one has to mention that the map @xmath210 implied by eq . ( [ eq24 ] ) is fraction linear . it is well known that any such transformation is equivalent to a linear automorphism on a projective space . to make use of much simplification following from such a linear reduction of the problem , let us consider the vector space of 2-element columns @xmath211 $ ] ( using here and below boldface characters for notation of matrix - valued quantities ) with the ratio of the elements equal to @xmath85 , @xmath212 . we consider the columns differing by a non - zero multiplier as equivalent , @xmath213 \sim \left[{a a \atop a b}\right ] , a\not=0 $ ] . let us also introduce the @xmath214 matrix @xmath215which is , as a straightforward check shows , unimodular , @xmath216 . then it is also straightforward to show that the transformation @xmath217 is equivalent to eq . ( [ eq24 ] ) . the map @xmath218 corresponding to eq.([eq26 ] ) is just the linear automorphism of the real projective line @xmath146 mentioned above . since the transformation associated with matrix @xmath219 does not depends on @xmath159 , starting from @xmath220 , after @xmath221 iterations one comes to the equation @xmath222 it is convenient to chose @xmath223 $ ] . the very ` starting ' @xmath85-constant @xmath224 encodes the initial value of the phase function @xmath25 . giving @xmath96 , it can be calculated by means of eq . ( [ eq80 ] ) . ( [ eq27 ] ) is , essentially , the desirable algebraic relationship describing the long - term phase evolution . the time variable @xmath9 is here encoded in the integer variable @xmath159 playing role of a discrete time on the scale @xmath19 and amounting , numerically , to the integer part of @xmath225 . we shall derive below the expanded version of ( [ eq27 ] ) where the power of the operator @xmath219 is given in explicit form . we consider , first , the case of phase evolution when the discriminant @xmath181 , eq . ( [ eq20 ] ) , is negative . in this case we proceed with the problem of calculation of the accumulated phase variation over a long time interval ( the single period averaging does not yield a meaningful result here ) . in accordance with ( [ volt ] ) , it immediately yields us the value of the average voltage across the junction , i.e. the physically measurable quantity . in particular , we shall see that , in spite of apparently phase behavior , the phase evolution manifests actually no signs of chaos . moreover , as a matter of fact , the phase reveals , in a sense , oscillation of a definite frequency and the apparent irregularity of its evolution is manifested because this frequency is in general incommensurable with the bias guiding one ( quasiperiodic behavior ) . as a particular consequence , the average voltage converges to a quite definite value which can be calculated in a general case . elaborating the relationships referred to above , let us calculate , at first , the eigenvalues of the matrix @xmath219 . in view of its role in the phase evolution , one should not be surprised that the discriminant of its characteristic equation @xmath226 coincides , up to the positive factor @xmath227 , with @xmath178 $ ] , see ( [ eq20a ] ) . thus , it is precisely the case of the phase evolution when the matrix ( [ eq25 ] ) has a pair of complex ( and complex conjugated ) roots @xmath228 . furthermore , since their product is the unity , they equal @xmath229 for some real @xmath230 . more concretely , one gets @xmath231 these eigenvalues are connected with the following ( complex valued ) eigenvectors : @xmath232.\ ] ] in other words , @xmath233 further , the following identical decomposition of an arbitrary real - valued 2-element column @xmath234 in terms of the columns @xmath235 takes place : @xmath236 where we make use of the notation @xmath237^{-1}\times \nonumber\\ & & \left [ a\left(q_0(t)\sin{\mbox{\scriptsize$1\over2$}}\phi_0(t ) + ( 1-e^{-p_0(t)})\cos{\mbox{\scriptsize$1\over2$}}\phi_0(t ) \mp { \mathrm{i}}\sqrt{-\delta}\right ) \right . \nonumber\\ & & \left . + 2 b \sin{\mbox{\scriptsize$1\over2$}}\phi_0(t ) \vphantom{\sqrt{-\delta } } \right ] . \end{aligned}\ ] ] the @xmath159-fold ( @xmath238 ) application of the linear matrix operator @xmath219 to the vector @xmath234 expanded in accordance with ( [ eq : otyrfdl ] ) leads to the equation @xmath239 this is in fact the desirable explicit representation of the operator power @xmath240 convenient for our purposes . applying the decomposition ( [ eq : nfhtrse ] ) to eq . ( [ eq27 ] ) , the resulting equation can then be resolved with respect to @xmath199 representing it as a fraction - linear function of @xmath224 , its formula will be given later on ( see eq . ( [ eq42 ] ) ) while here we write down instead the explicit form of the equation @xmath241 following from the definition of @xmath199 ( and making use of the initial values @xmath242 , following , in turn , from definitions of @xmath243 ) . it reads @xmath244 where all the coefficients @xmath245 do not depend on @xmath159 and @xmath246 are real . ( notice that the only ` complex valued ingredient ' in @xmath247 is the factor @xmath248 . ) since @xmath159 represent here the time ` normalized and discretizied on the scale @xmath19 ' ( i.e. the integer part of @xmath225 , in fact ) , eq . ( [ eq : uyrvhyd ] ) manifest , essentially , the specific ` hidden ' periodicity of the phase function which has been occasionally referred to above , the corresponding period amounting to @xmath249 . generally speaking , the latter quantity is incommensurable with the bias period @xmath19 . as a consequence , the discrete parameter @xmath159 never assumes sequential values differed by @xmath250 or a quantity aliquot to it . it is this circumstance which does not allows one to distinguish the oscillations which would be described by eq . ( [ eq : uyrvhyd ] ) if @xmath159 were a continuous variable . now we are in position to apply the relationships derived above for the determination of the _ average voltage _ @xmath251 across junction in the case of phase behavior ( phase - locking absence ) . in view of eqs . ( [ eq2 ] ) , ( [ volt ] ) , @xmath252\ ] ] and , therefore . @xmath253 thus the averaging over a lapse consisting of unboundedly increasing number of bias periods reduces to the computation of the limit @xmath254 for @xmath255 satisfying ( [ eq29 ] ) . it yields a definite result if and only if the latter exists . let us consider how it can be computed . as to the first multiplier involved in eq . ( [ eq29 ] ) , its nonzero base does not depend on @xmath159 and the limit always exists ( obviously , it equals the unity ) . on the contrary , the second multiplier is not a universal constant . it depends on the ratio of the modules of the complex coefficients @xmath256 and @xmath257 . clarifying the latter point , let us calculate the difference @xmath258 . a straightforward algebra yields @xmath259 ( the argument @xmath260 is here omitted , cf . ( [ eq : jfyrser ] ) ) . the sign of the difference coincides therefore with the sign of the factor @xmath261 , apart from the constant parameters determining @xmath262 , @xmath261 also depends on @xmath224 which encodes the starting ( initial ) value of the phase and may assume , in principle , arbitrary real value including the limiting case @xmath92 . however , @xmath261 is itself a second order polynomial in @xmath85 . calculating its discriminant , one finds it to be connected , again , with @xmath177 $ ] , equaling to @xmath263 and being in our case _ negative _ just in view of the phase behavior condition @xmath264 . therefore , as a function of @xmath224 , @xmath261 has no real roots and is either everywhere positive or everywhere negative irrespectively of the initial phase ( connected with the specific value of @xmath224 ) . furthermore , since @xmath265 the sign of @xmath261 ( and @xmath266 ) coincides with the sign of @xmath267 which is determined , ultimately , by the function @xmath1 and may not vanish by virtue of definition of @xmath181 and the ` no convergence ' condition @xmath264 , see ( [ eq20 ] ) . thus we have shown that _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath268 iff @xmath269 + and + @xmath270 iff @xmath271 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ one may employ any of the two decompositions shown below of the second multiplier in the last line of ( [ eq29 ] ) @xmath272 let us arrange to apply the upper representation if @xmath268 and the lower one in the opposite case . then the limit of the last multiplier in the corresponding line of ( [ eq33 ] ) exists and are equal to the unity since its base to be raised to the power @xmath273 is a finite @xmath159-independent gap distant from zero . the existence of limits of the second multiplier is evident , it also equals the unity . thus the limit of the first ( constant ) factor yields the limit value for the whole product . having thus calculated the limit of ( [ eq33 ] ) as @xmath274 , eq . ( [ eq29 ] ) leads to the following conclusion : @xmath275 where @xmath230 is defined in eq . ( [ eq28a ] ) . this simple result yields the explicit representation of the average voltage across junction in the case of phase evolution : @xmath276 for some integer @xmath46 ( we shall discuss the method of its determination later on ) . more exactly , the formula above describes , for a single @xmath46 , a single branch binding the two neighboring shapiro steps , _ i.e. _ ` horizontal ' constant voltage segments on the junction i - v curve whose orders differ by the unity . such branches are sometimes called ` resistive portions of i - v curve ' @xcite although the specific dependence of the average voltage on @xmath21 , as it is described by eq . ( [ eq34 ] ) , may considerably deviate from the simple ohm s law proportionality . in particular , near the edges of these ` resistive portions ' the _ differential resistance _ @xmath277 diverges . after a minor modification , the majority of the above equations is also applicable in the case of phase - locking . here one has to assume @xmath278 then eqs . ( [ eq24]-[eq27 ] ) hold true while the adapted versions of eqs . ( [ eq28a])-([eq28c ] ) read @xmath279 @xmath280 . \ ] ] these are the eigenvalues and the eigenvectors of the matrix @xmath219 ( [ eq25 ] ) , and the following equations take place @xmath281 where in this case @xmath282 are real . it is worth noting that the unimodularity constraint @xmath283 also holds true . then it follows from eq . ( [ eq27 ] ) @xmath284 where @xmath285 and , after some algebra , one gets @xmath286 here all the dependence on @xmath159 ( proportional , up to normalization and discretization , to the evolution time @xmath9 ) is isolated in the factors @xmath287 . more precisely , the @xmath159 dependent terms combine to the powers @xmath288 or @xmath289 but , in view of eq . ( [ eq38 ] ) , these coincide with @xmath290 , respectively . basing on ( [ eq42 ] ) , the explicit equation _ completely determining the phase _ at any moment of time in terms of the ` ground ' phase function @xmath155 specified on the interval @xmath27 $ ] and the functions @xmath291 calculated from @xmath155 and defined on the same interval is easily derived : @xmath292 where @xmath293 @xmath294 is the excess of @xmath9 over the nearest lower @xmath295 , @xmath296 $ ] , for integer @xmath297 $ ] . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ eqs . ( [ eq533])-([eq534a ] ) also apply to the case of phase evolution and exactly in this form . the only difference is in the definitions of @xmath298 ( [ eq35x]),([eq41 ] ) . in case one has to replace the real term @xmath299 by the pure imaginary @xmath300 . then @xmath301 that , in particular , ensure the phase function defined by ( [ eq534 ] ) to be real . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ eq . ( [ eq533 ] ) determines the phase at any moment of time ( up to a constant summand aliquot to @xmath24 which will be computed later on ) through the phase initial value @xmath96 , the solution @xmath302 of eq . ( [ eq1x ] ) vanishing at @xmath131 and the complex valued function @xmath303 both computed on the finite interval @xmath27 $ ] . the exhaustive description of the process of the asymptotic establishing of the steady phase - locking state is its particular byproduct . it allows one to explicitly describe the ` transient processes ' representing the distinction of arbitrary given phase function from the ` refined ' steady one to which the former asymptotically converges . to be more specific , let us now consider in brief some straightforward consequences of eqs . ( [ eq42]),([eq533 ] ) . at first , it should be noted , that the dependence of r.h.s . of ( [ eq533 ] ) on @xmath159 ( the time normalized to the scale @xmath19 and then ` discretizied ' ) disappears in the case of satisfaction of any of the two equations @xmath304 ( it is worth noting that , requiring @xmath224 to be real , each of them implies the condition @xmath305 . thus , neither of eqs . ( [ eq534b ] ) , ( [ eq534c ] ) can be fulfilled in the case . ) then @xmath306 is a periodic function of @xmath9 with the period @xmath19 . in particular , it coincides with the own asymptotic limiting form . such phase evolution can be named a _ steady _ one . notice that the ` steadiness ' does not means the periodicity of @xmath25 but allows , apart from periodic part , the uniformly growing contribution @xmath307 for an integer @xmath46 . it can be shown by a direct check that @xmath224 , satisfying ( [ eq534b ] ) or ( [ eq534c ] ) , also verifies eq . ( [ eq19 ] ) . hence , ( [ eq534b ] ) and ( [ eq534c ] ) determine the @xmath85-constants which correspond to initial data characteristic of the ` refined ' ( steady ) phase - locking states . two choices @xmath308 correspond to two states , stable and unstable . such @xmath85-constants were referred to above as @xmath171 . now let us arrange to reserve this notation for the @xmath85-constant implying the _ stable _ steady phase evolution . in order to determine which of these two states is stable , let us consider a generic case when both @xmath309 . it follows from ( [ eq35x ] ) and ( [ eq38 ] ) that the condition @xmath184 implies either @xmath310 and @xmath311 or @xmath312 and @xmath313 ( and both @xmath314 are either positive or negative ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let us arrange about the following interpretation of the subscripts ` @xmath315 ' and ` @xmath316 ' : @xmath317 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ then one may introduce the equation @xmath318 defining by them the _ positive _ constant @xmath319 . using it , eq . ( [ eq533 ] ) can be recast to the following form : @xmath320 sending here @xmath274 , one finds that the exponent @xmath321 for a ` generic ' solution ] @xmath25 of eq . ( [ eq1x ] ) approaches to the _ periodic _ function @xmath322 defined for @xmath323 $ ] by the equation @xmath324 notice that the mentioned periodicity is not manifest from this formula , it is rather a specific consequence of the properties of its ingredients constituting the essence of the definitions of the latter . the very function of the ` refined ' phase - locking steady phase evolution , @xmath325 , advances at each time step of duration @xmath19 by @xmath326 for some _ fixed _ ( @xmath9-independent ) integer @xmath46 . the formula ( [ eq77 ] ) is nothing else but eq . ( [ eq : ogytgfdr ] ) with @xmath327 this answers the question which of the two equations @xmath328 or @xmath329 describes the stable steady evolution ( asymptotic phase - locking state ) : it is described by the equation @xmath330 the detailed picture of the convergence of a generic phase function @xmath25 to the asymptotic limit @xmath325 can be inferred from the equation ( [ eq647 ] ) recast as follows : @xmath331 which describes what can be called a ` nonlinear exponential ' convergence . if @xmath297 $ ] is sufficiently large to ensure the satisfaction of the condition @xmath332}|z|<e^{2\varkappa j } , \label{eqgvrsde}\]]the simple estimate follows @xmath333 which describes just the exponential convergence . the factor @xmath334 and the exponent coefficient @xmath335 both determine how many periods ( enumerated by the integer @xmath159 ) have to elapse until the phase function approaches the steady asymptotic state with the prescribed accuracy . at the same time , for comparatively small time @xmath9 , the ` transients ' @xmath336 may have no relation to the exponent . ( [ eqgvrsde ] ) allows to estimate the duration of this ` near - zone ' lapse . as to the second possibility @xmath337 ( the solution of the equation @xmath338 ) , it also yields , formally , the steady phase - locking state described by the phase function @xmath339 , which can be computed with the help of equation @xmath340 and is distinct from @xmath341 . this is unstable solution ` repelling ' any neighboring one . indeed , having rewritten eq . ( [ eq647 ] ) as follows @xmath342 one sees that for arbitrary small but non - zero @xmath343 ( which is non - zero if @xmath344 mod @xmath24 ) the function @xmath345 escapes exponentially off the function @xmath346 which , itself , can be revealed only in the case of the exact vanishing of @xmath343 . for any other phase function @xmath345 ultimately approaches , as @xmath159 increases , to @xmath347 . the computations above proves the following statement which has been mentioned above : [ gftdre ] the condition @xmath186 is necessary for the phase - locking to be observed whereas the strict inequality @xmath184 is sufficient.@xmath74 _ remarks : _ * there is a formulation of the criterion above operating with not specific but arbitrary solution @xmath4 of eq . ( [ eq1x ] ) ( instead of @xmath155 ) and not requiring for the functional @xmath130 to obey the specific initial condition ( [ eq86 ] ) may not vanish for any representation . ] . in general form , it reads @xmath348 \over 2\sqrt{\im[{{\cal f}}(t_0+t/2)]\im[{{\cal f}}(t_0-t/2 ) ] } } , \label{eq : iroycd } \\ \delta\phi&=&\phi(t_0+t/2)-\phi(t_0-t/2).\end{aligned}\ ] ] @xmath349 does not depend on @xmath350 . * in terms of @xmath349 , the eigenvalues @xmath351 are represented as follows @xmath352 * it is shown below that for @xmath353 the phase - locking is also observed but its properties reveal some distinction from ones of the case @xmath184 ( the ` weak ' phase - locking against the ` generic ' one ) . for the sake of completeness , let us consider here the specialities of the situation intermediate between the exponentially stable ` generic phase - locking ' taking place if @xmath184 and phase evolutions for which @xmath264 . it is distinguished by the merging of the two eigenvalues of the matrix ( [ eq25 ] ) . zero discriminant condition and eq . ( [ eq20a ] ) imply @xmath354 and then it follows from ( [ eq35x ] ) that the two - fold eigenvalue of @xmath219 which now is represented as follows @xmath355 has also to be equal to either 1 or -1 . in the case @xmath353 eq . ( [ eq : nfhtrse ] ) does not apply . instead , one may employ the following explicit representation of the power of the @xmath214 matrix @xmath356 whose elements obey the constraint @xmath357 ( just meaning that the @xmath219 eigenvalues coincide ) times the 2-element column with arbitrary elements @xmath358 : @xmath359.\end{aligned}\ ] ] ( it can be established , for example , by means of the mathematical induction . ) since the eigenvalue does not vanish , @xmath360 . let us assume also , for a while , that @xmath361 which imply , in view of ( [ eq : rtuyerrr ] ) , @xmath362 . then the following analogue of eq . ( [ eq42 ] ) arises : @xmath363 it implies the existence of the limit @xmath364 which proves independent of @xmath224 . it also follows from ( [ eq : ftreh ] ) that @xmath199 does not depend on @xmath159 ( all the elements of the sequence coincide ) if and only if @xmath365 thus , as opposed to the ` generic phase - locking ' taking place for @xmath184 , there is only a _ single _ initial phase which yields the ` refined steady ' evolution . all the other phase evolutions converge , with the course of time @xmath9 , to the latter which plays therefore the role of attractor . interestingly enough , the same phase function plays simultaneously the role of the repeller and all the distinct phase functions approaching it are simultaneously ` moving away ' . there is not contradiction here since the phase functions live in fact on the closed circumference and ` the one side attracting ' to a point is simultaneously ` the another side repelling ' from the same point . to illustrate the simultaneous attraction / repelling property of a steady phase function , let us compute the leading @xmath159-dependent contribution to @xmath199 . the result is @xmath366 the terms shown do _ not _ depend on the initial phase ( encoded in @xmath224 ) which affects only the higher order contributions . this means , in particular , that all the phase functions approach their common steady limit ` from one side ' ( the leading contribution to the deviation from the limit is common for all of them ) . then , obviously , the closer the initial phase is to the one corresponding to the refined steady state from _ this _ side , the less time is necessary for the reaching , in appropriate sense , the steady limit . on the other hand , the ` very long ' phase evolution until it reaches some fixed vicinity of the ultimate steady state arises when the initial state is ` very close ' to the steady phase function from the opposite , ` wrong ' side . this behavior differs from the ` generic ' phase - locking where the choices of the initial phases closed to the phase of the stable steady evolution always lead to the quick ` monotonous ' convergence to the limiting function irrespectively to the initial relative angular direction . this specialty can be inferred more rigorously from the consideration of the analogue to eq . ( [ eq534 ] ) which follows from ( [ eq : ftreh ] ) and now reads @xmath367 where @xmath368 ( note that , in particular , @xmath369 equals either @xmath370 or @xmath371 and does not vanish ) . it determines the phase function for arbitrary @xmath9 expressing it through the functions @xmath208 specified on the segment @xmath27 $ ] . the most substantial difference with eqs . ( [ eq : uyrvhyd ] ) and ( [ eq533 ] ) is the _ non - exponential _ dependence on @xmath159 . now the convergence to the asymptotic phase function @xmath372 , defined @xmath373 for any @xmath9 by the equation @xmath374},\ ] ] is linear in @xmath375 . ( [ eq : jgudly ] ) can also be rewritten as follows @xmath376 for bounded @xmath159 , the arbitrary constant @xmath224 can always be chosen making @xmath377 so small that the unity is the dominating contribution in the both numerator and denominator in ( [ eq : jgudlyr ] ) . it makes their ratio to be as close to the unity as one desires . if further @xmath159 unboundedly increases , depending on the sign of @xmath377 , the two distinct situations can occur . namely , if @xmath377 is of the same sign as @xmath378 , increasing @xmath159 , the ratio ( [ eq : jgudlyr ] ) ` monotonously ' tend to the unity , the less @xmath377 , the faster the limit is reached . this is the case of the ` true ' situating of @xmath224 with respect to initial phase @xmath171 of the steady phase function . however , if @xmath377 is still small but has the sign opposite to the sign of @xmath378 , increasing @xmath159 , the @xmath159-dependent contribution is subtracted from the leading terms of the numerator and denominator ( here the unity ) and the absolute value of their sum decreases reaching the minimum for @xmath379 $ ] . simultaneously , the ratio ( [ eq : jgudlyr ] ) varies in some way and approaches the initial closeness to the unity only for @xmath380 or greater . the further increasing of @xmath159 already leads to the ` monotonous ' converging to the unity with the rate @xmath381 as above . however , the ` convergence time ' estimated as @xmath382 _ increases _ as the deviation @xmath377 of the initial phase from the phase of the steady function , having ` wrong ' sign , becomes smaller . finally , to abandon the temporal assumptions made above , it has to be noted that the cases @xmath383 or @xmath384 implying @xmath385 which are not covered by the formulae above , reveals qualitatively similar relationships . their analysis is carried out in a similar way , provided the following simple representations of @xmath386 @xmath387 are utilized . they also lead to the convergence of the order @xmath381 . resuming , in the case @xmath353 not covered by the criterium formulated in the form of the theorem [ gftdre ] , the phase - locking understood as the asymptotic convergence of all phase functions to some steady one reproducing the own form @xmath388 on each segment of @xmath9 variation of length @xmath19 takes place as well . however , it proves not exponential in time . rather , the steady limit is approached as @xmath389 . let us now derive yet another important property of phase - locking solutions of ( [ eq1x ] ) following from eq . ( [ eq534 ] ) . it enables one to calculate the integer @xmath46 entering , in particular , equations ( [ eq18]),([eq34 ] ) . to that end , calculating the logarithmic derivative of eq . ( [ eq : ogytgfdr ] ) and applying ( [ eq85 ] ) , one gets the equation @xmath390{\mathbin{\mathrm{d}}}t\end{aligned}\ ] ] integrating it on the interval @xmath27 $ ] , one gets @xmath391{\mathbin{\mathrm{d}}}t . \nonumber\ ] ] up to this point , the manipulations above do not go beyond eq.([eq13 ] ) , definitions and identities . the situation drastically changes if we substitute here in place of arbitrary @xmath85 the constant @xmath171 ( [ eq539 ] ) which , in the case @xmath184 here assumed , converts an arbitrary solution @xmath25 of eq . ( [ eq1x ] ) to the stable refined phase - locking phase function @xmath325 . since the latter advances on each time step @xmath392 $ ] by the strictly fixed increment @xmath393 , one gets let @xmath394 be the solution of ( [ eq1x ] ) obeying the initial condition @xmath395 and determining the complex valued function @xmath122 which , in turn , satisfies eq . ( [ eq85 ] ) with initial condition @xmath396 . let also @xmath177>0 $ ] , where @xmath177 $ ] is defined by eq . ( [ eq20a ] ) . then @xmath397 { \mathbin{\mathrm{d}}}t \label{eq861}\ ] ] is the _ integer _ equal to the total sum ( taking into account the orientation sign ) of the number of full revolutions ( the winding number ) which any phase function , except of @xmath398 , will carry out on the time steps of duration @xmath19 in its asymptotic steady phase - locking state . _ remarks : _ * the formula ( [ eq861 ] ) provides us with the substantiation of the statement made above which concerns the constancy of the order of phase - locking when the bias parameters varies . indeed , in view of ( [ eq861 ] ) the order @xmath46 is continuous with respect to the variables parametrizying the bias function @xmath1 . in particular , @xmath46 continuously depends on @xmath21 as far as @xmath181 retains positive . hence it assumes a constant value on the @xmath21 intervals over which @xmath181 graph shown in fig . [ f10 ] is situated above the horizontal coordinate axes ( more generally , as the bias function @xmath1 varies within the connected component of the phase - locking area ) . * a more general representation of @xmath46 which operates with not special but arbitrary solution of eq . ( [ eq1x ] ) can be derived from eq . ( [ eq861 ] ) , cf . the remark following theorem 1 . however , as opposed to the case of criterion based on the ` universal ' expression ( [ eq : iroycd ] ) , here the _ two _ inequivalent formulae distinct by the opposite roles of the left and right boundary points arise . their difference is a complicated non - linear ( and apparently non - trivial ) functional which vanishes on all solutions of ( [ eq1x ] ) , provided the bias function corresponds to the phase - locking phase evolution . the equation ( [ eq1x ] ) and the properties of its solutions seem to be of a considerable interest in view of several reasons . first of all , this is , of course , their sound physical relevance following from the extensive applications in the applied theory of electric activity of josephson junctions which employs eq . ( [ eq1x ] ) as the base of the efficient model of the junction phase dynamics @xcite in the important case of negligible role of junction capacitance ( overdamped junctions ) @xcite . on the other hand , eq . ( [ eq1x ] ) is of evident interest in its own rights from a pure mathematical point of view . it provides us a remarkable example of apparently supreme simple non - linear ode which prove associated with a linear problem ( page ) is beyond the scope of present discussion . ] and , in view of such a link , allows a deep exploration by analytic methods . at the same time , in spite of its apparent simplicity , it is definitely far of being regarded as a mathematically trivial entity . it suffices to say that the properties of eq . ( [ eq1x ] ) are still not completely understood even for sinusoidal bias function @xmath399 , the case of a primary interest from viewpoint of applications . the relationships considered above does not exhaust the collection of rigorous ones which ( [ eq1x ] ) allows to establish by means of elementary technique . however they are distinguished by the advantage of universality being valid with fairly weak limitations on the class of allowable bias functions . the introduction of the functional @xmath177 $ ] ( [ eq20 ] ) is the central point of the approach . ( [ eq533 ] ) is noteworthy as the explicit representation of the phase function for arbitrary @xmath9 through a single solution of eq . ( [ eq1x ] ) computed on the finite interval @xmath27 $ ] . it is this equation which allows to establish the convergence , in the case @xmath177>0 $ ] , of any phase function ( except of @xmath398 , see eq . ( [ eq77a ] ) ) to an asymptotic limit and to show that this limit coincides mod @xmath24 with @xmath400 defined by eq . ( [ eq77 ] ) . thus eq . ( [ eq533 ] ) yields the rigorous model of the phase - locking property allowing one to compute any of its quantitative characteristic of interest . in combination with eq . ( [ eq : uyrvhyd ] ) describing the long term phase evolution in the opposite case @xmath177<0 $ ] , the above relationships lead to the criterion of the asymptotic property of the phase - locking ( theorem 1 ) which can be reformulated to operate with arbitrary single solution of eq.([eq1x ] ) on a finite segment of the length @xmath19 and its derivates . another important result is the formula ( [ eq861 ] ) determining the integer winding number @xmath46 ( phase - locking order ) through the same easily computable data . in physical terms , the latter integer quantity is directly connected to the average voltage applied across a junction in the phase - locking state which is the supported constant and proves independent of the slow variations , up to a certain extent , of the parameters , provided the period @xmath19 is kept unchanged . this effect lies in the core of the modern dc voltage standards @xcite .

the method of efficient description of long - term behavior of solutions of the non - linear first order ode @xmath0 for arbitrary periodic @xmath1 is discussed . the criterion enabling one to separate and identify the qualitatively different solutions is established . the applications of the method to the modeling of dynamics of overdamped josephson junctions in superconductors are outlined . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ long - term behavior + of solutions of the equation @xmath0 + with periodic @xmath1 + and the modeling of dynamics + of overdamped josephson junctions : + _ unlectured notes _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ s.i . tertychniy + russia , vniiftri

employing electron spin in semiconductor devices could potentially overcome scaling issues with modern charge - based electronics , providing more power efficiency , higher speed , and greater functionality@xcite . in many spintronic device designs , spin precession during transport from injector to detector plays a primary role in determining device output characteristics . this precession is caused by a torque exerted either by effective@xcite or real@xcite magnetic fields oriented perpendicular to the spin direction . because of magnetic field inhomogeneities or transit time uncertainty , transported spins will arrive at the spin detector with a distribution of precession angles . when the angular distribution width approaches 2@xmath0 , the contributions of spin - polarized electrons to the detector output is reduced by partial signal cancellation of oppositely - oriented spins , in a process called `` dephasing '' , or `` hanle effect''@xcite . dephasing in spin - transport devices ( especially when due to transit - time uncertainty caused by spatial diffusion or pathlength variation ) can not be reversed , as it can be with spin - echo techniques in electron spin resonance ( where transport plays no role)@xcite . therefore , spin dephasing is a physical process of fundamental importance to semiconductor spintronic devices . here , we present analysis of a spin drift - diffusion model to quantify the effects of dephasing in a regime where charge transport is dominated by drift , as in recent experiments@xcite . we examine the apparent independence of relative dephasing ( precession fringe visibility in magnetic field spectroscopy ) to injector - detector distance ( transit length ) and quantify the effects of voltage bias across the transport region . these conclusions are bolstered through comparison to experiment , using silicon spin - transport devices@xcite . in addition , the effects of temperature on spin dephasing are discussed . since spin detection is invariably a projection of the electron spin on a fixed measurement axis , the change in output of a spintronics device employing precession will be determined by @xmath1 ( to first order ) , where @xmath2 is spin angle , and @xmath3 is spin precession frequency and @xmath4 is transit time from injector to detector over transit length @xmath5 . clearly , transit - time uncertainty @xmath6 gives rise to precession angle uncertainty @xmath7 . increasing precession frequency ( @xmath8 , where @xmath9 is electron spin g - factor , @xmath10 is the bohr magneton , @xmath11 is the reduced planck s constant , and @xmath12 is perpendicular magnetic field ) by increasing the magnetic field therefore increases the angular uncertainty @xmath13 and causes dephasing . a complete model for dephasing during transport from injector to detector is therefore needed . electron transport in a semiconductor is well - described by the drift - diffusion equation . to model spin transport accurately , finite spin - lifetime must be included within the relaxation - time approximation : @xmath14 where @xmath15 is the spin density , @xmath16 represents the spin relaxation time , @xmath17 is the diffusion coefficient for electrons , and @xmath18 is the electron drift velocity . in the `` ohmic '' regime , @xmath19 , where @xmath20 is mobility and @xmath21 is electric field . [ eqn : spindriftdiffusion ] governs the evolution of spatial distributions of electron spin , and can be solved easily for initial conditions ( at @xmath22 ) of an ensemble of spins which are all spin - polarized in the same direction at the injector ( at @xmath23 ) . in other words , we find the green s function describing the evolution of a dirac delta : @xmath24 the most salient features of this gaussian solution for our subsequent analysis are that the center of the distribution travels with velocity @xmath18 toward the detector , and that the spatial distribution width increases in time as @xmath25 . holding @xmath26 , this solution describes the distribution of arrival times @xmath27 at the detector . therefore , the precession - induced change in total device output signal ( @xmath28 ) comprised from all electrons arriving with different transit times @xmath29 , each contributing @xmath1 due to spin precession , is @xmath30 \cos{\omega t}dt.\ ] ] this integral expression can be easily evaluated to compare to empirical observations . a typical experiment consists of a spectroscopy where perpendicular magnetic field strength @xmath12 is varied ; spin precession oscillations ( where extrema correspond to average precession angles in integer multiples of @xmath0 ) are suppressed in larger magnetic fields due to a larger precession - angle uncertainty . this suppression due to dephasing , or `` hanle effect '' , can be seen in results evaluated from eq . [ eqn : precession2 ] shown in fig . [ fig : fig2](a ) . more relative dephasing increases spin decoherence at the detector and reduces the number of oscillations seen . ( a ) comparison of the spin precession signal output according to eq . [ eqn : precession2 ] for devices with @xmath31 ( black ) and @xmath32 ( light grey ) transport layer thickness , showing identical dephasing . the voltage bias is 2v , mobility is 1400 @xmath33/vs , diffusion coefficient is 36 @xmath33/s , and spin lifetime is 100ns for both simulations . several oscillation extrema are labeled by the average precession angle @xmath0 , @xmath34 , @xmath35 , @xmath36 . ( b ) the independence of dephasing to transport length regardless of oscillation order is shown using eq . [ eqn : dephasing0 ] calculated for @xmath37 m@xmath38 m. ] because the integrand in eq . [ eqn : precession2 ] is dominated by the gaussian term @xmath39 , the exponentially decaying part @xmath40 can be ignored if the spin lifetime @xmath16 is suitably long . then we have @xmath41 making the variable substitution @xmath42 converts eq . [ eqn : temp101 ] to @xmath43 application of the following transformation @xmath44 ( where @xmath45 is a transit - length scaling factor ) converts eq . [ eqn : temp102 ] to @xmath46 other than an overall multiplicative factor of @xmath45 , eq . [ eqn : temp103 ] is identical to eq . [ eqn : temp102 ] ; the spin precession model is therefore virtually invariant to this transformation . in terms of device physics , ohmic transport ( @xmath47 , where @xmath48 is the voltage bias on the transport layer ) automatically accounts for decreasing the drift velocity by a factor of @xmath45 if the length @xmath5 increases by the same proportion at constant voltage ; the first two elements of the transformation are therefore satisfied . the third element of the transformation in eq . [ eqn : transformation ] is accounted for by applying a quadratically weaker magnetic field . the experimental outcome of this result is that in measurements of different devices , we can expect the same number of precession oscillations despite variation in transport lengths , assuming the applied voltage @xmath48 is identical . again , the preceding assumes that the exponential spin - decay term is negligible . this is true when the spin lifetime is much longer than the time it takes the gaussian spin distribution of width @xmath49 to enter into the detector ( transit time uncertainty , @xmath50 ) . since for drift - dominated operation @xmath51 we have @xmath52 if @xmath53 , then finite spin - lifetime can be ignored and variations in @xmath5 make negligible difference to the evaluation of eq . [ eqn : precession2 ] . for characteristic values @xmath54@xmath33/s , @xmath55cm / s , and @xmath56ns , this condition is satisfied for @xmath57 cm , which is certain to hold for semiconductor devices that are typically many orders of magnitude smaller . [ fig : fig2](a ) shows simulation results for different device transport layer thicknesses @xmath5 . the black and light grey curve correspond to @xmath58 m and @xmath59 m , respectively . the simulation is performed using @xmath60@xmath33/vs , @xmath61 @xmath33/s and voltage bias @xmath62 v. the spin lifetime in both simulations is 100ns . for the sake of comparison , the @xmath63 axes are slightly shifted relative to each other . it is very clear that the spectroscopy shape and precession oscillation extrema number is exactly the same , despite the difference in signal magnitude , which is roughly equal to @xmath45 ( a factor of 10 ) . another way to show the dephasing invariance to transit length is to calculate the relative uncertainty in the distribution of final precession angle @xmath64 dephasing ( defined by eq . [ eqn : dephasing0 ] ) for the precession oscillation extrema calculated with eq . [ eqn : precession2 ] as a function of applied voltage across a 350 @xmath20 m transport region with mobility @xmath60@xmath33/vs ( symbols ) . lines are analytically given by eq . [ eqn : length ] with corresponding values of @xmath65 indicated by the legend . note exceptional agreement between these two complementary means of determining dephasing . ] if transport is dominated by drift in the applied electric field@xcite , the transit time is given by @xmath66 ( note the quadratic dependence of transit time on transport length at fixed voltage . ) applying ohmic transport @xmath47 to eq . [ eq : diffbroaden ] gives @xmath67 since the width of the transit time distribution @xmath6 is @xmath68 , the uncertainty in the distribution of final precession angle @xmath65 at the detector is given by application of eq . [ eqn : length0 ] : @xmath69 this result ( valid at precession extrema when the average spin direction is either parallel or antiparallel to the measurement axis ) is independent of the transit length @xmath5 , so that we can expect the same amount of dephasing at the same voltage bias through the transport layer , regardless of the distance from injector to detector for any fixed precession angle ( assuming ohmic behavior , @xmath19 , where @xmath21 is internal electric field ) . although eq . [ eqn : length ] is independent of transit length @xmath5 , it is clearly dependent on voltage bias @xmath48 . specifically , it predicts that dephasing is suppressed with an increase in @xmath48 , which is intuitively expected since diffusion will play a smaller role as drift becomes stronger in a larger electric field . ( a ) side - view and ( b ) schematic conduction band diagram of our 350-micron - thick undoped single - crystal silicon spin transport device used to compare experimental results to the dephasing model presented . ] to further study the dephasing effect , it is necessary to establish a proper quantitative empirical standard for comparison to experimental measurements not dependent on data fitting using evaluation of eq . [ eqn : precession2 ] . if drift dominates and @xmath70 , the factor proportional to @xmath71 in eq . [ eqn : precession2 ] can be approximated as a constant . then , ( assuming again that @xmath72 ) the distribution of precession angles is gaussian : @xmath73 so we have @xmath74 we can use this analytic result to find the angular distribution width from @xmath28 spectroscopy measured empirically . the ratio of the magnitude of the central maximum for @xmath75 ( corresponding to zero precession : @xmath76 and hence @xmath77 ) to the signal value for extrema at @xmath78 ( @xmath79 ) is therefore @xmath80 solving for @xmath13 gives @xmath81 it is therefore possible to compare the dephasing effect in different experiments by studying the signal magnitude decrease of the same order extrema , without having to fit to the complete model ( eq . [ eqn : precession2 ] ) . [ fig : fig2](b ) shows the distribution width @xmath82 for the several precession oscillation extrema based on eq . [ eqn : dephasing0 ] for simulations of devices with different transport layer thickness from @xmath32 to @xmath83 . the same voltage bias , 2v , is applied on the transport layer . it is obvious that , despite an order - of - magnitude change in the transit length @xmath5 , the dephasing is virtually constant , further confirming the conclusions drawn from fig . [ fig : fig2](a ) . voltage - controlled dephasing in a device with @xmath84 from our simulation is shown using eq . [ eqn : dephasing0 ] in fig . [ fig : fig3 ] ( symbols ) . at fixed voltage , the changes in dephasing for successive extrema are approximately equal , because @xmath85 and the magnetic field period of precession oscillations is approximately constant . we compare the dephasing behavior of several precession oscillation extrema from these calculations to the behavior predicted by eq . [ eqn : length ] ( solid lines ) . despite there being no free parameters for fitting , we find excellent graphical agreement between eqs . [ eqn : length ] and [ eqn : dephasing0 ] for all precession oscillation extrema examined , confirming the validity of our obtained expressions . ( a)-(e ) : experimentally - measured spin precession spectroscopy at different accelerating voltages 10v , 14v , 16v , 18v , and 80v , respectively , using spin transport through 350 micron - thick undoped single - crystal si at 150k . unprecedented spin coherence is evident in ( e ) , showing at least 15 full spin rotations . ] we now compare these results to experimental data , using all - electrical hot - electron spin injection and detection to study spin transport through 350-micron - thick undoped single - crystal si(100)@xcite . the device scheme and corresponding band structure are shown in fig . [ fig : fig4](a ) and ( b ) . transport proceeds from top to bottom : equilibrium spin - polarized electrons from the co@xmath86fe@xmath87 cathode tunnel through the al@xmath88o@xmath89 barrier , becoming hot electrons in the nonmagnetic al / cu anode thin film . the electrons coupling with conduction band states over a schottky barrier in the silicon transport layer on the other side forms the injected spin polarized current . drift - dominated transport in the undoped silicon transport layer is controlled by @xmath90 , the `` accelerating voltage '' . after transport through this silicon layer , the electrons are ejected from the conduction band and into the second ferromagnetic metal ( ni@xmath91fe@xmath92 ) and the ballistic component of this current is collected by an n - type silicon substrate below , forming the spin - transport signal @xmath93 . this current is dependent on the projection of electron spin direction onto detector magnetization axis of the ni@xmath91fe@xmath92 layer@xcite . a magnetic field aligned perpendicular to the spin direction and parallel to internal electric field ( and hence drift velocity ) induces spin precession during transport from injector to detector . ( this device is essentially the solid - state and si - based version of the first bare - electron g - factor experiment which used spin precession during ballistic transport of high - energy electrons in vacuum and mott scattering for spin filtering@xcite . ) fig . [ fig : fig5](a)-(e ) shows precession measurements at 150k using this device with accelerating voltage 10v , 14v , 16v , 18v and 80v , respectively . as the accelerating voltage increases , more oscillations are visible ; at the highest voltage applied , more than 15 full spin rotations are evident by the number of oscillation extrema in fig . [ fig : fig5](e ) . the length independence derived earlier ( eq . [ eqn : length ] ) explains the high degree of spin coherence in these precession measurements , despite the 350 microns of si separating injector from detector . using eq . [ eqn : dephasing0 ] , we plot the dephasing at 150k as a function of voltage for several precession oscillation extrema in fig . [ fig : fig6 ] , and compare to the @xmath94 behavior predicted by eq . [ eqn : length ] . although the empirical behavior is qualitatively correct , the remaining discrepancy is likely due to our model assumption of ohmic transport , which at these voltages is not precisely upheld due to onset of velocity saturation@xcite . dephasing ( calculated from experimental data with eq . [ eqn : dephasing0 ] ) for several precession oscillation extrema with different voltage bias across the 350-micron - thick undoped single - crystal si transport layer at 150k . the model - predicted @xmath94 behavior is also plotted for comparison ( dotted line ) . ] ( a)-(c ) temperature dependence on spin dephasing for our si spin transport devices at three temperatures 20k , 60k , and 100k , respectively . panel ( d ) plots the dephasing ( calculated using eq . [ eqn : dephasing0 ] ) for 20k@xmath95t@xmath95100k , showing the unexpected non - monotonic dependence unaccounted for by application of the einstein relation to our model . ] the einstein relation ( or `` fluctuation - dissipation theorem '' ) , valid for non - degenerate charge density , dictates @xmath96 , where @xmath97 is boltzmann s constant and @xmath98 is temperature . because our injected currents are low ( @xmath99a ) , and the spin - injector area is large ( @xmath100m@xmath101 ) , our experimental electron density @xmath102 @xmath103 is clearly non - degenerate in si which has an effective conduction dos @xmath104@xmath105 . if the einstein relation is also valid for the non - equilibrium electrons we generate using ballistic electron injection with our tunnel - junction emitter , eq . [ eqn : length ] implies that the relative dephasing @xmath106 . therefore , more dephasing should be evident at higher temperature . contrary to this expectation , we find a non - monotonic behavior of experimental dephasing as a function of temperature for experimental measurements at fixed accelerating voltage , as shown in fig . [ fig : fig7](a)-(c ) . the dephasing is small at the lowest temperature ( 20k ) , then rises to a maximum at approximately 60k , and drops gradually toward 100k , as shown in fig . [ fig : fig7](d ) . clearly this observation is not compatible with the einstein relation , and a more sophisticated model than the one presented here is required to capture the correct behavior . it is interesting to note , however , that the sharp decrease in dephasing at low temperatures corresponds coincidentally to the onset of a negative differential mobility ( ndm ) regime in si(100 ) below @xmath10740k ( ref . @xcite ) , which in other semiconductors gives rise to the gunn effect : transport is via soliton - like spatial charge domains that suppress longitudinal diffusion and would therefore constrain spin dephasing in these spintronic devices . however , ndm occurs in si(100 ) only in a window of electric field values ( @xmath10750 - 150v / cm ) that are substantially smaller than the ones used here . it is more likely that the observed reduction of dephasing is the result of the consequences of ( unintentional ) doping impurity freeze - out . using a spin precession model based on the green s function of the spin drift - diffusion equation , we determined the expected effect of transit length and voltage changes on spin dephasing . most importantly , we determined that for long spin lifetimes , the relative spin dephasing is independent of the transit length . in future spintronic devices , this length independency may enable multiple gate operations and long distance transport in precession - dependent devices as long as the finite spin lifetime allows . we also constructed an empirical measure of spin dephasing and used it to characterize both the results of simulation and experiment . dephasing is shown to be inversely dependent on the square - root of voltage drop that drives drift , both theoretically and experimentally . for model results closer to experiment , the true 3-dimensional device geometry , magnetic fringe fields from injector and detector contacts , and non - ohmic transport , ( all of which will give other systematic sources of dephasing ) should be taken into account .

a spin transport model is employed to study the effects of spin dephasing induced by diffusion - driven transit - time uncertainty through semiconductor spintronic devices where drift is the dominant transport mechanism . it is found that in the ohmic regime , dephasing is independent of transit length , and determined primarily by voltage drop across the spin transport region . the effects of voltage and temperature predicted by the model are compared to experimental results from a 350-micron - thick silicon spin - transport device using derived mathematical expressions of spin dephasing .

kondo semiconductors ( or insulators ) , e.g. , smb@xmath8 , ybb@xmath9 , and ce@xmath10bi@xmath11pt@xmath10 , are cousins of heavy - fermion compounds.@xcite both classes of materials behave like kondo - impurity systems at high temperatures , _ f_-electron spins independently scattering conduction electrons . as the scattering becomes coherent at low temperatures , the heavy - fermion compounds remain metallic , described as a fermi liquid of heavy quasiparticles , while the kondo semiconductors become semiconducting with a small energy gap opening in the excitation spectrum . since the known kondo semiconductors have an even number of electrons per unit cell , it has been suggested that , despite strong electron - correlation effects , they may be viewed as band insulators.@xcite namely , if one occupied _ f _ state hybridizes with a single half - filled conduction band , a band insulator may result , with the lower hybridized band completely occupied . cenisn , crystallizing in an orthorhombic structure , is a unique variety of the kondo semiconductors , in which the gap formation seems incomplete.@xcite the kondo temperature _ t_@xmath12 of 51 k is deduced from specific heat data above @xmath010 k,@xcite which is comparable to _ t_@xmath12 of 24 k in the representative heavy - fermion compound ceru@xmath13si@xmath13.@xcite although the semiconducting behavior of resistivity observed in early samples ( ref . ) is completely suppressed as the quality of crystals is improved,@xcite there is plenty of evidence for the gap formation below @xmath010 k , e.g. , direct observations of the gap by break - junction tunneling spectroscopy ( ref . ) and the rapid suppression of _ c_/_t _ ( refs . ) and 1/_t_@xmath14,@xcite where _ c _ is the electronic part of specific heat , _ t _ temperature and _ t_@xmath14 the nuclear spin lattice relaxation time . however , _ c_/_t _ levels off at @xmath040 mj / molk@xmath4 below 1 k,@xcite and 1/_t_@xmath14 exhibits the korringa law ( _ t_@xmath14_t _ = constant ) below 0.4 k.@xcite further , the thermal conductivity exhibits _ t_-linear dependence below 0.3 k,@xcite which is characteristic of metallic systems as _ @xmath15 0 . analyses of _ c_/_t _ and 1/_t_@xmath14 suggest that the gap is a v - shaped pseudogap of the width @xmath010 k with finite density of states ( dos ) at the fermi level.@xcite since the width of the gap is small , the magnetic field _ b _ may alter the electronic structure via the zeeman splitting of the up- and down - spin energy bands . there are some experimental indications that the gap may be suppressed by the field applied along the _ a _ axis,@xcite which is the easy axis of magnetization . on the theoretical side , there are two contrasting approaches to cenisn . one is an anisotropic hybridization - gap model.@xcite it is argued that , if the lowest kramers doublet of the ce 4_f _ state has special symmetry , the hybridization between conduction and _ f _ electrons and hence the gap may vanish in some directions in the brillouin zone , leading to finite dos at the fermi level . the other approach takes a view of a two - fluid model , assuming the existence of neutral spin excitations that are decoupled from charged fermi - liquid excitations ( i.e. , charge carriers).@xcite it is claimed that the spin excitations dominate low - temperature thermodynamic properties and that a pseudogap opens only in their spectrum . in a previous paper , we have reported the first observation of shubnikov - de haas ( sdh ) oscillations in cenisn.@xcite in this paper , we present more extensive measurements of low - temperature resistivity and hall effect as well as sdh oscillations in cenisn . we discuss the electronic structure and the influence of the magnetic field on it . single - crystalline ingots of cenisn were grown by the czochralski method and were purified by a solid - state electrotransport technique.@xcite two parallelpipeds ( samples - b1 and -b2 ) of @xmath00.5 mm@xmath4 in cross section and @xmath04 mm in length along the _ b _ axis were cut from an ingot by spark erosion . gold lead wires were spot - welded . the resistivity and hall effect were measured at temperatures down to 35 mk and in magnetic fields up to 20 t by a conventional four - terminal method with a low - frequency ac excitation current ( _ f _ = 17 hz , _ = 100 @xmath2a ) applied along the _ b _ axis . the field was rotated in the _ ac _ and _ bc _ planes , and the field angles @xmath16@xmath17 and @xmath16@xmath18 are measured from the _ c _ axis . both samples were metallic down to the lowest temperature . ( a ) resistivity @xmath19 in cenisn measured along the _ b _ axis as a function of temperature _ t_. ( b ) low temperature part of the same data as a function of _ t_@xmath4 . the broken line shows a fit to @xmath19 = @xmath19@xmath20 + _ a__t_@xmath4 below 0.16 k. ( c ) and ( d ) @xmath19 vs field _ b _ for selected field directions in the _ ac _ and _ bc _ planes , respectively . the field angles @xmath16@xmath17 and @xmath16@xmath18 are measured from the _ c _ axis . note that the field geometry is always transverse in ( c ) , while it changes from transverse at @xmath16@xmath18 = 0 to longitudinal at @xmath16@xmath18 = 90@xmath21 in ( d ) . ( e ) hall resistivity @xmath19@xmath22 vs _ b _ for _ b _ @xmath23 _ a _ and _ b _ @xmath23 _ c_.,width=321 ] figure [ fig:1](a ) shows the temperature dependence of resistivity , which is quite unusual . the resistivity is nearly constant around 1 k , as if a residual resistivity regime is reached , yet it starts to decrease again near 0.8 k. the resistivity exhibits a quadratic temperature dependence at the lowest temperatures [ fig . [ fig:1](b ) ] , which is a characteristic of a fermi liquid . the coherence temperature @xmath24 , i.e. , the upper limit of the _ t_@xmath4 behavior , is very low , @xmath00.16 k. the coefficient _ a _ of the _ t_@xmath4 term is huge , 54 @xmath2@xmath3cm / k@xmath4 . these points may be illustrated by comparison with the corresponding values found in ceru@xmath13si@xmath13 , i.e. , @xmath24 @xmath0 0.6 k and _ a _ = 0.4 @xmath2@xmath3cm / k@xmath4.@xcite figure [ fig:1](c ) shows the magnetoresistance for selected field directions in the _ ac _ plane . the magnetoresistance for _ b _ @xmath23 _ a _ ( @xmath16@xmath17 = 90@xmath21 ) exhibits a broad maximum at 4.4 t and then decreases , showing a bend near 9 t. this behavior is in accord with previous data at 0.45 k.@xcite note that the field geometry is transverse . since cenisn has an even number of electrons per unit cell and hence must be a compensated metal , the negative magnetoresistance at high fields is unexpected . the position of the magnetoresistance maximum moves to higher fields with decreasing @xmath16@xmath17 ( e.g. , 7.6 t at @xmath16@xmath17 = 42@xmath21 ) , going beyond the investigated field range near @xmath16@xmath17 = 8@xmath21 , and then comes back to 15 t for _ b _ @xmath23 _ c _ ( @xmath16@xmath17 = 0 ) . the bend also shifts to higher fields with decreasing @xmath25 and evolves into a dip for @xmath16@xmath17 @xmath26 @xmath060@xmath21 , as seen at 13.5 t at @xmath16@xmath17 = 42@xmath21 . as the field is further rotated , the dip is preceded by an extended region where the resistivity exhibits rapid decrease ; at @xmath16@xmath17 = 27@xmath21 , the rapid decrease starts at 15.5 t , but the dip is not reached up to 20 t. for _ b _ @xmath23 _ c _ , the oscillatory behavior below 15 t is due to the sdh effect and will be described in more detail below . figure [ fig:1](d ) shows the magnetoresistance for selected field directions in the _ bc _ plane . the field geometry changes from transverse at @xmath16@xmath18 = 0 to longitudinal at @xmath16@xmath18 = 90@xmath21 . although this leads to much suppressed magnetoresistance for large @xmath16@xmath18 , the position of the magnetoresistance maximum can be traced . as the field is tilted from the _ c _ axis with increasing @xmath16@xmath18 , the maximum position first moves to slightly lower fields , then goes up , and appears to go out of the field range near @xmath16@xmath18 = 60@xmath21 . the hall resistivity for _ b _ @xmath23 _ a _ is also anomalous [ fig . [ fig:1](e ) ] ; it exhibits a negative peak at 2.5 t and takes small positive values at high fields . similar behavior was previously observed at 1.5 k.@xcite for _ b _ @xmath23 _ c _ , the hall resistivity is linear in _ b _ at low fields and then bends near 15 t in line with the anomaly in the magnetoresistance . the oscillatory behavior below 15 t is ascribed to the sdh effect . although cenisn must have both electron and hole carriers because of the charge compensation , the large negative hall resistivity suggests that electron carriers dominate the electrical conduction . accordingly , if we assume a single - carrier model , the hall coefficient at 2 t , -33@xmath2710@xmath28 @xmath29/c , corresponds to the carrier concentration of 1.2@xmath2710@xmath6 electron / ce . ( a ) resistivity @xmath19 in cenisn for the field parallel to the _ c _ axis and its oscillatory part @xmath30 as functions of the inverse field 1/_b_. to obtain @xmath30 , a polynomial was fitted to the smoothly varying background of @xmath19(1/_b _ ) and subtracted from it . the two samples , b1 and b2 , give essentially identical results . ( b ) the fourier transform of @xmath30 in the sample b1 . spectra at higher temperatures are also shown . the inset shows the temperature dependence of sdh oscillation amplitudes @xmath31 . the solid curve is a fit to the lifshitz - kosevich formula,@xcite which yields the effective mass @xmath32 of 13@xmath331 in the units of the free electron mass _ m_@xmath5 . ( c ) the frequency determined from each one oscillation period ( peak - to - peak or valley - to - valley width ) as a function of the field for the two samples.,width=321 ] in fig . [ fig:2](a ) , we show the oscillatory part of the magnetoresistance measured with _ b _ @xmath23 _ c _ for the two samples . we see that both samples exhibit essentially identical sdh oscillations . the fourier transform of the oscillations indicates a single frequency at 65@xmath335 t [ fig . [ fig:2](b ) ] , where the error is simply set by the inverse of the 1/_b_-width of the original data . the corresponding orbit area is 0.5@xmath34 of the cross - section normal to the _ c _ axis of the brillouin zone . fitting the temperature dependence of the oscillation amplitude to the lifshitz - kosevich formula @xcite ( inset ) , we find the associated effective mass to be 13@xmath331 _ m_@xmath5 , where _ m_@xmath5 is the free electron mass . although the fourier transforms show a single peak , a closer examination of the oscillations reveals that the frequency actually changes with field . figure [ fig:2](c ) shows the frequency determined from each one oscillation period ( peak to peak or valley to valley ) as a function of field . we find that the frequency changes from @xmath070 to @xmath060 t with increasing field . we also examined possible field dependence of the effective mass for the sample b1 by measuring a peak - to - valley height of each oscillation . the masses thus determined are 12.3 , 13.2 , 14.2 , and 13.9 @xmath35 at 7.4 , 8.4 , 9.6 , and 11.5 t , respectively . it appears that the mass slightly increases with field . however , since the errors associated with these estimations are @xmath331 @xmath35 , more precise measurements are necessary to conclusively state the field dependence of the mass . ( a ) resistivity @xmath19 in cenisn for the field angle @xmath16@xmath18 = 60@xmath21 and its oscillatory part @xmath30 as functions of the inverse field 1/_b_. ( b ) the fourier transform of @xmath30 . spectra at higher temperatures are also shown . the inset shows the temperature dependence of sdh oscillation amplitudes @xmath31 . the effective mass @xmath32 is determined to be 19@xmath332 _ m_@xmath5,width=321 ] the field - direction dependence of the sdh frequency in cenisn . the frequencies were determined from fourier transforms of oscillations in the following field intervals : _ b _ = 5.56 - 14.3 t for all @xmath16@xmath18 s and for @xmath16@xmath17 @xmath26 26@xmath21 , and _ b _ = 5.56 - 10.9 t for @xmath16@xmath17 @xmath36 26@xmath21 . for the _ ac _ plane , the measurements on the two samples yielded identical results within experimental error ( compare the circles and squares ) . the effective mass is also shown for selected field directions.,width=226 ] sdh oscillations of this frequency branch are observed in a wide range of field directions ; @xmath16@xmath18 up to 60@xmath21 and @xmath16@xmath17 up to 36@xmath21 . for instance , we show the oscillatory part of the magnetoresistance measured at @xmath16@xmath18 = 60@xmath21 as a function of 1/_b _ in fig . [ fig:3](a ) . the fourier spectrum indicates a frequency of 77@xmath335 t [ fig . [ fig:3](b ) ] , and the effective mass is determined to be 19@xmath332 _ m_@xmath5 ( inset ) . we plot the angular dependence of the sdh frequency in fig . [ fig:4 ] . here the frequencies were determined from fourier transforms of the data in the same field interval from 5.56 to 14.3 t for all @xmath16@xmath18 s and for @xmath16@xmath17 @xmath26 26@xmath21 . for @xmath16@xmath17 @xmath36 26@xmath21 , a narrower window from 5.56 to 10.9 t was used so that the magnetoresistance maximum and bend ( or dip ) will not affect the fourier transforms . the frequency increases faster than 1/cos@xmath25 in the _ ac _ plane , while the angular dependence in the _ bc _ plane is weak . as a rough approximation , the shape of the fermi surface ( fs ) seems a concave lens of which the axis is parallel to the _ a _ axis . we here note evidence that the observed sdh oscillations are intrinsic to high - quality single crystals of pure cenisn , not due to impurities . ( 1 ) we performed measurements not only at positive field angles ( @xmath16@xmath17 or @xmath16@xmath18 @xmath37 0 ) but also at negative ones ( @xmath16@xmath17 or @xmath16@xmath18 @xmath26 0 ) . it was verified that the angular dependences of the sdh frequency and amplitude were symmetric with respect to the _ c _ axis . namely , the sdh oscillations exhibit the appropriate symmetry of the crystal . ( 2 ) the angular dependences in the _ ac _ plane determined for the two samples are identical within error ( fig . [ fig:4 ] ) . ( a ) resistivity @xmath19 in cenisn for the field angle @xmath16@xmath17 = 42@xmath21 and its derivative with respect to the inverse field 1/_b _ as functions of 1/_b_. the magnetoresistance decreases fairly sharply from 12 t and exhibits a dip at 13.5 t. the derivative curve indicates the existence of sdh oscillations above this dip . ( b ) the fourier spectra for field ranges above ( solid curve ) and below ( broken curve ) the magnetoresistance dip . the inset shows the temperature dependence of the amplitude @xmath31 of sdh oscillations observed above the dip.,width=321 ] for a limited range of field directions , we have observed another frequency branch at high fields above the aforementioned magnetoresistance dip . figure [ fig:5](a ) shows the magnetoresistance and its derivative with respect to 1/_b _ for the field angle @xmath16@xmath17 = 42@xmath21 as a function of 1/_b_. the magnetoresistance decreases fairly sharply from @xmath012 t and exhibits a minimum at 13.5 t. in the field range higher than this dip , the magnetoresistance exhibits oscillatory behavior , which is more clearly visible in the derivative curve . the fourier transform of the magnetoresistance in the field range above the dip [ fig . [ fig:5](b ) , solid curve ] shows a clear peak at @xmath0380 t , while the spectrum for the field range below the dip ( broken curve ) does not show any peak . the effective mass for the orbit is determined to be 22@xmath332 _ m_@xmath5 ( inset ) . for the nearby field directions @xmath16@xmath17 = 48@xmath21 and 55@xmath21 , sdh oscillations with a frequency of @xmath0300 t appear similarly on the high - field side of the magnetoresistance dip . for still larger field angles , @xmath16@xmath17 @xmath37 55@xmath21 , no oscillations are observed . this may be attributed to that the resistivity at high fields and hence its oscillatory part are so small for field directions close to the _ a _ axis [ see fig . [ fig:1](c ) ] . we begin with the field - dependence of the sdh frequency shown in fig . [ fig:2](c ) . when a sdh frequency depends on the field , the momentary frequency @xmath38 that is experimentally measured differs from the true frequency @xmath39 that is directly connected with an extremal cross - section of the fs ; @xmath40.@xcite this means that ( 1 ) a linear variation in @xmath39 does not affect @xmath38 , and that , ( 2 ) when @xmath39 has non - linear field dependence , the variation in @xmath38 may be enhanced over that in @xmath39 . a linear zeeman shift of a parabolic energy band leads only to a linear change in a fs cross - section and hence does not affect @xmath38 , i.e. , @xmath38 s of oscillations from up- and down - spin electrons remain the same and constant . the observed field - dependence indicates that the energy band shifts non - linearly with fields and/or that the energy dispersion in the vicinity of the fermi level strongly deviates from quadratic . when theoretical models become available for comparison , further information can be extracted from the present observation . we next discuss the fs responsible for the frequency branch shown in fig . [ fig:4 ] . the size of the observed sdh frequencies , the order of @xmath0100 t , is compatible with the carrier concentration estimated from the hall coefficient with a single - carrier model ; a spherical fs enclosing 10@xmath6 electron / ce would give a sdh frequency of 190 t. the compatibility supports that what we have found from the sdh oscillations is not small structures of a large fs but the major portion of a small but dominant fs . since the hall resistivity is negative , the observed fs is an electron pocket . note that , even if the measured hall resistivity contains some contribution from hole carriers , the above line of argument is still valid . in that case , the true carrier number is smaller than the single - carrier - model estimation , and hence it becomes even more unlikely that larger fs s are hidden . it is interesting to estimate the contribution to @xmath41 (= _ c_/_t _ ) of this electron fs . using isotropic three - dimensional effective - mass approximation , @xmath41 is given by @xmath42 , where @xmath43 and @xmath44 . here , @xmath45 is the fermi energy , @xmath46 the fermi wave number , @xmath47 unit volume , and other symbols as usual . assuming _ n _ to be 10@xmath6 electron / ce and @xmath32 to be 10 - 20 @xmath35 , we have @xmath45 of 26 - 13 k and @xmath41 of 1.5 - 3 mj / molk@xmath4 . the estimated @xmath41 is one order - of - magnitude smaller than the experimental value of 40 mj / molk@xmath4.@xcite having discussed the experimental estimations of the carrier number and @xmath41 , we here compare the two theoretical models of cenisn . the considerable discrepancy between the estimated and measured @xmath41 values may be a support to the two - fluid model,@xcite since it claims that the specific heat is dominated by neutral spin excitations . however , it does not seem to provide clear explanation for the largely reduced carrier number . on the other hand , in the case of the anisotropic hybridization - gap model , small fs s are expected to appear around the nodes in the brillouin zone where the hybridization gap vanishes , if small dispersion of renormalized _ f_-electron levels is taken into account.@xcite thus , the observation of the fs does not conflict with the model . the measured large @xmath41 may be accounted for if we assume that hole carriers have effective mass of @xmath0200 _ m_@xmath5 . this assumption is not so unlikely as it may , at first sight , appear . it is known that the many - body mass enhancement in ce - based heavy fermions varies from fs to fs and hence that light and heavy carriers often coexist in one compound . indeed , effective masses in ceru@xmath13si@xmath13 found by de haas - van alphen measurements range from @xmath01 _ m_@xmath5 to more than 100 _ m_@xmath5.@xcite if _ t_@xmath12 is a measure of possible mass enhancement factor , the existence of carriers with mass of @xmath0200 _ m_@xmath5 in cenisn can not be ruled out , since _ t_@xmath12 is comparable in the two compounds . we now turn to the @xmath1 dependence of resistivity observed below 0.16 k [ fig . [ fig:1](b ) ] . we attribute it to quasiparticle - quasiparticle collisions . namely , we regard it as a low - temperature limiting behavior of a fermi liquid . the two peculiarities in the _ t_@xmath4 dependence , i.e. , the relatively low @xmath24 and anomalously large _ a _ can not be taken as indications of exceptionally strong many - body effects in cenisn , but rather seem to be connected with the extremely low carrier concentration of 10@xmath6 electron / ce . the fermi energy of the electron carriers is estimated to be @xmath020 k as described above , while that of the hole carriers could be one order - of - magnitude smaller because of the larger mass . the suppression of @xmath24 probably relates to these fine energy scales . the huge @xmath48 does not conform to the kadowaki - woods relation , _ a_/@xmath41@xmath4 @xmath0 1@xmath2710@xmath49 @xmath2@xmath3cm(molk / mj)@xmath4 , which is observed in many heavy fermions.@xcite in the theoretical derivation of this relation,@xcite metals with large carrier concentration ( @xmath01 electron / ce ) are assumed . this is , however , not the case with cenisn , in which the formation of the pseudogap suppresses @xmath41 , while it enhances resistivity through reduced carrier number . thus the anomalously large _ a _ can be qualitatively explained . one might argue that electron - phonon scattering could also lead to @xmath1 behavior in semimetals . for example , the resistivity of bismuth varies as @xmath1 between 0.3 and 1 k.@xcite the essence of the electron - phonon scenario of the @xmath1 resistivity is that , if the fs is small and anisotropic , large - angle scattering ( i.e. , scattering angle @xmath0 180@xmath21 ) by phonons may survive down to very low temperatures:@xcite at a low temperature _ t _ , most of excited phonons have wave numbers smaller than @xmath50 , where @xmath51 is a sound velocity . if a minimal caliper of a fs is smaller than @xmath52 , appreciable large - angle scattering can still occur and the temperature dependence of resistivity remains weaker than @xmath53 that is expected from the bloch - grneisen law.@xcite specifically , in the case of a cylindrical fs , it can be shown that the resistivity exhibits @xmath1 dependence in an extended temperature range before the @xmath53 dependence appears.@xcite this scenario , however , does not apply to cenisn . the sound velocity in cenisn is @xmath54,@xcite yielding @xmath55 at @xmath56 , the lowest temperature of the measurements . the observed sdh frequencies , @xmath0100 t , correspond to a fs cross - section of @xmath57 . for simplicity , let us assume the cross - section to be rectangular . the shorter sides of the rectangle must be comparable to @xmath52 so that large - angle scattering can be effective . accordingly , the longer sides will be @xmath58 . such extremely anisotropic fs ( aspect ratio @xmath37 10@xmath59 ) can never be compatible with the anisotropy in the resistivity : the ratio of the resistivities along the most resistive _ c _ and least resistive _ a _ axes is @xmath04 at most.@xcite lastly , we discuss magnetic - field effects on the electronic structure . the negative magnetoresistance and nonlinear hall resistivity for _ b _ @xmath23 _ a _ [ figs . [ fig:1](c ) and ( e ) ] suggest that the electronic structure changes with the field , as was previously pointed out.@xcite although the magnetoresistance maximum and bend ( or dip ) can not necessarily be identified with phase transitions , it seems reasonable to regard them as characteristic fields of the field - induced change . the magnetoresistance maximum probably signals the onset of substantial changes . this view is supported by the fact that the hall resistivity for _ b _ @xmath23 _ c _ deviates from nearly linear field dependence , i.e. , normal behavior , at the field of the magnetoresistance maximum . on the other hand , it is not clear what underlies the bend ( or dip ) . these characteristic fields increase as the field is tilted away from the _ a _ axis . this may be related to that the magnetic susceptibilities along the _ b _ and _ c _ axes are less than half of that along the _ a _ axis.@xcite the sudden appearance of the high - frequency oscillations above the magnetoresistance dip ( fig . [ fig:5 ] ) suggests that a larger fs appears at high fields as a result of the changes in the electronic structure . the quasiparticle mass associated with the high frequency is no smaller than those found for the low - frequency branch . these are basically consistent with that @xmath41 is enhanced in fields along the _ a _ axis.@xcite the variation of @xmath41 is gradual and featureless , compared to the sudden appearance of the high frequency oscillation . the discrepancy could be ascribed to the difference in the field direction and/or the fact that @xmath41 was measured on an early semiconducting sample ( ref . ) or at a much higher temperature of 3.3 k.@xcite we have established that a small number of intrinsic charge carriers with enhanced mass do exist in the ground state of cenisn . the carrier concentration is estimated to be @xmath60 electron / ce , while the quasiparticle mass is @xmath010 - 20 @xmath35 . combining these two parameters , we have estimated the contribution of the observed fs to @xmath41 to be a few mj / molk@xmath4 , which is one order - of - magnitude smaller than the experimental @xmath41 . thus , the essential question raised by the two - fluid model still remains to be solved ; whether or not neutral spin excitations , besides charge carriers , are necessary to explain low - temperature thermodynamic properties in cenisn . the resistivity exhibits @xmath1-dependence below @xmath61 . viewing this dependence as a fermi - liquid behavior , we have argued that the low @xmath24 and huge @xmath48 associated with it may be related to the extremely low carrier concentration . we have observed that the high - frequency sdh oscillations appear above the magnetoresistance - dip anomaly for certain field directions and have discussed these observations in terms of field - induced changes in the electronic structure . we note that the zeeman energy at 20 t is @xmath020 k for a moment of @xmath01 @xmath2@xmath62 and is comparable to ( or much larger than ) the fermi energy of the electron ( hole ) carriers derived in the present work . this suggests that competition between these energies may lead to fundamental change of the electronic structure . therefore field - effects on the electronic structure deserve further studies .

the resistivity and hall effect in cenisn are measured at temperatures down to 35 mk and in magnetic fields up to 20 t with the current applied along the _ b _ axis . the resistivity at zero field exhibits quadratic temperature dependence below @xmath00.16 k with a huge coefficient of the @xmath1 term ( 54 @xmath2@xmath3cm / k@xmath4 ) . the resistivity as a function of field shows an anomalous maximum and dip , the positions of which vary with field directions . shubnikov - de haas ( sdh ) oscillations with a frequency _ f _ of @xmath0100 t are observed for a wide range of field directions in the _ ac _ and _ bc _ planes , and the quasiparticle mass is determined to be @xmath010 - 20 _ m_@xmath5 . the carrier density is estimated to be @xmath010@xmath6 electron / ce . in a narrow range of field directions in the _ ac _ plane , where the magnetoresistance - dip anomaly manifests itself clearer than in other field directions , a higher - frequency ( @xmath7 ) sdh oscillation is found at high fields above the anomaly . this observation is discussed in terms of possible field - induced changes in the electronic structure .

the isotropic distribution of gamma ray bursts ( grbs ) in the sky and their number distribution as function of intensity , measured with the batse instrument aboard the compton gamma ray observatory , provided the first observational evidence that gamma ray bursts ( grbs ) originate at large cosmological distances ( meegan et al . moreover , the rapid variation of their light curves ( bahat et al . 1992 ) indicated that their huge energy is emitted from a very small volume . in the original fireball ( fb ) model of grbs ( e.g. paczynski 1986 ; goodman 1986 ; rees & mszros 1992 ) the emission was spherically symmetric . the implied isotropic energy release of grbs in @xmath0-rays often exceeded @xmath1 , creating an ` energy crisis ' . indeed , such a mighty , abrupt , compact , and @xmath0-ray - efficient source was unforeseen . a simple solution to this puzzle was suggested by shaviv and dar ( 1995 ) : the @xmath0-ray emission is narrowly collimated by the relativistic motion of their _ jetted _ source , which is seen when it points closely enough to the observer . in this view , grbs are not produced by fireballs , but by inverse compton scattering of light by highly relativistic jets of ordinary matter , ejected in violent stellar processes such as supernova explosions , mergers of neutron stars ( paczynski 1986 ; goodman , dar & nussinov 1987 ; dar et al . 1992 ) , or the direct collapse of massive stars to black holes without a supernova ( woosley 1993 ) . the sky localization of grbs by bepposax ( costa et al . 1997 ) led to the discovery ( groot et al . 1997 ; van paradijs et al . 1997 ) of their optical afterglows ( ags ) and their host galaxies ( sahu et al . 1997 ) which were used to extract their cosmological redshifts ( metzger et al . the ags seemed to follow an achromatic power - law decline , as expected from a highly relativistic expanding fireball that drives a blast wave into the circumburst environment ( e.g. , mszros & rees 1997 ) . this prediction of the spherical fireball model ( see e.g. , piran 1999 ) being independent of the assumption of spherical symmetry , it was also argued that the ags , like the grbs themselves , are produced by narrowly collimated jets ( dar 1997 , 1998 ) . the concept of jets was incorporated into the fb model by the substitution of its spherical shells by conical sections thereof , the mechanism for the @xmath0-ray emission still being synchrotron radiation from shock - accelerated @xmath2 pairs in a baryon - poor material ( see , e.g. piran 1999 , 2000 ; mszros 2002 and references therein ) , in spite of the difficulties that such a radiation mechanism encounters ( ghisellini et al . 2000 ) . an elegant and simple way to distinguish between a conical jet and a spherical fireball was suggested by rhoads ( 1997 ) : the ag of a decelerating conical jet will show an achromatic steepening a _ jet break_ in its power - law decline when the relativistic beaming angle of its radiation becomes larger than the opening angle of the jet . soon afterwards , better sampled data on the optical afterglow of grbs showed the existence of what appeared to be such achromatic jet breaks ( harrison et al . 1999 ; stanek et al . 1999 ) , and the spherical fb model was modified into a _ collimated fireball _ model ( e.g. piran , sari & halpern 1999 ) . in this model grb pulses are produced by synchrotron radiation from the collision between conical sections of shells . the collision of the ensemble of shells with the interstellar matter ( ism ) generates the ag by synchrotron radiation from the forward shock propagating in the ism , and/or from the backwards shock within the merged shells . rhoads ( 1999 ) and sari , piran and halpern ( 1999 ) derived a relation between the opening angle of the conical jet and the time of the jet break . this relation has been applied extensively to the pre - swift data to infer the opening angle of the conical jet and to determine the ` true ' energy of grbs , posited to be an approximate standard candle ( frail et al . 2001 ) . since the launch of swift the above generally - accepted ` standard ' paradigm has been challenged , due to the absence of breaks in the ags of many grbs ( panaitescu et al . 2006 ; burrows and racusin 2006 ) , to the chromatic behaviour of the ag of other grbs having the alleged ` jet break ' ( stanek et al . 1999 , harrison et al . 1999 ) , and to the failure of the frail relation ( frail et al . 2001 ) in many swift grbs ( kocevski & butler 2007 ) . in the fireball model , the jet breaks need not be sharp ; they are often parametrized with a varying smoothness ( stanek et al . 1999 ) . allowing for such breaks , covino et al . ( 2006 ) could not identify a swift grb with a fully achromatic break . liang et al . ( 2007 ) have extended this study , and analyzed the swift x - ray data for the 179 grbs detected between january 2005 and january 2007 and the optical ags of 57 pre- and post - swift grbs . they found that not a single burst satisfies all the criteria of a jet break . this brings us fully into the question of the nature and properties of the jets responsible for grbs and their ags and , more specifically in this paper , to the understanding of ` breaks ' and ` missing breaks ' . an alternative to the fireball scenario is offered by the ` cannonball ' ( cb ) model of grbs [ dar & de rjula 2000a , 2004 , hereafter dd2000a , 2004 ; dado , dar & de rjula ( hereafter ddd ) 2002 ; for a recent review see de rjula 2007 ] . in this model _ long - duration _ grbs and their ags are produced by bipolar jets of cbs ( shaviv & dar 1995 ; dar & plaga 1999 ) , ejected in _ ordinary core - collapse _ supernova ( sn ) explosions as matter is accreted onto the newly - formed compact object ( de rjula 1987 ) . the ` cannon - balls ' are made of _ ordinary - matter plasma_. the @xmath0-rays of a single pulse of a grb are produced as a cb coasts through the sn _ glory _ the initial sn light , scattered away from the radial direction by the ` wind ' : the ejecta puffed by the progenitor star in a succession of pre - sn flares . the electrons enclosed in the cb raise the glory s photons to grb energies by _ inverse compton scattering ( ics)_. as a cb coasts through the glory , the distribution of the glory s light becomes increasingly radial and its density decreases rapidly . consequently , the energy of the up - scattered photons is continuously shifted to lower energies and their number decreases swiftly , resulting in a fast softening and decline of the prompt emission ( dd2004 , 2007a , b ) . in the cb model , the ag of a grb is due to _ synchrotron radiation ( sr ) _ from swept - in ism electrons spiraling in the cb s enclosed turbulent magnetic field , generated by the intercepted ism nuclei and electrons ( ddd2002 ) . at x - ray energies , the sr afterglow begins to dominate the ics prompt emission only during the fast - decline phase of the latter ( ddd2006 ) . in the cb model , the _ beau rle _ in the understanding of grbs is played by the doppler factor , @xmath3 , relating times , energies and fluxes in a cb s rest system to those in the observer s system . its form in terms of the observer s angle @xmath4 ( relative to the cb s direction of motion ) and the time - dependent lorentz factor , @xmath5 , of a cb , is : @xmath6}\approx { 2\ , \gamma(t)\over 1+[\gamma(t)\ , \theta]^2}\ ; , \label{delta}\ ] ] where the approximation is excellent for @xmath7 and @xmath8 . the decrease of @xmath5 with time , as a cb encounters the particles of the ism , is calculable on grounds of energy - momentum conservation ( ddd2002 , dar & de rjula 2006 , thereafter dd2006 ) . the energy - integrated energy flux of the ag of a grb , is @xmath9 . let @xmath10 . consider a cb that is observed almost on axis , so that @xmath11 : the observer is _ ab initio _ within the opening cone of the relativistically beamed radiation . as @xmath5 decreases , @xmath3 monotonically decreases and so does the observed ag . consider the same cb , viewed by an observer at a much larger angle , so that @xmath12 is ` a few ' . as @xmath5 decreases , @xmath3 in eq . ( [ delta ] ) _ increases _ , reflecting the fact that the characteristic opening angle of the radiation , @xmath13 , is reaching the observer s direction . past the point @xmath14 , the decrease of @xmath3 is monotonic , as in the first case we considered . the ag radiation parallels again the behaviour of @xmath3 . for observers of the same grb from different angles , as @xmath4 increases at fixed @xmath5 , the ag s flux decreases . all these simple facts , supported by the corresponding explicit derivations , are reflected in fig . [ f1]a , which we have copied from ddd2002 , as it foretells the progressive variety of ag shapes to be studied here . there is more to fig . [ f1]a than what we said . the lorentz factor @xmath5 of a cb only begins to change significantly , in a calculable manner , when the increase in its mass induced by the energy influx of the swept - in ism particles becomes comparable to the cb s initial mass . this happens , as we shall review , at a time @xmath15/\gamma_0 ^ 3 $ ] . at fixed @xmath5 , as reflected in fig . [ f1]a , a larger @xmath4 entails a larger @xmath16 . this achromatic ` deceleration bend ' at @xmath17 , we believe , was often interpreted in fb models as a putative jet break . naturally , the values of @xmath18 and @xmath19 of a given cb also affect the properties of its prompt ics - dominated radiation ( we are presenting this introductory discussion as if there was a single cb generating the prompt and ag radiations , a simplification to be undone when needed ) . in the cb model the ics - dictated @xmath20 dependences of a cb s isotropic energy , peak energy and peak luminosity are @xmath21 , @xmath22 and @xmath23 ( dd2000b ) . the conditions for these quantities to be relatively large ( a relatively small @xmath4 or a large @xmath18 ) are the ones leading to a luminous ag with a small @xmath16 . the basis for one of these expected correlations , studied before in detail in ddd2007c , is illustrated in fig . [ f1]b . if the deceleration bend at time @xmath16 takes place _ after _ the fast - decline phase of the prompt emission , it is observable , and the unabsorbed x - ray light curve is canonical ( ddd2002 ) . in these cases , there is a ` break ' . if @xmath16 takes place earlier , it is hidden under the prompt emission , and only the tail of the canonical behaviour , namely the ` late ' power - law decline of the unabsorbed synchrotron afterglow , is observable . in these cases , the break is missing . the transition from long - plateau , clearly ` broken ' ags , to power - law like ` unbroken ' ags should be anticorrelated with the trend from under-`energetic ' to over - energetic grbs . in the cb model the late - time spectral energy density @xmath24 of the x - ray and optical ag tends to a time and energy - dependence @xmath25 , with @xmath26 the spectral index of the electrons accelerated within a cb and cooled by the emission of the very sr seen as the ag . a prediction that we have not emphasized before is that the temporal power decline should be , grb by grb , half a unit steeper than the spectral decline . in ddd2006 , 2007a we have demonstrated that the most common light curves of the x - ray ag of grbs are well described by the cb model . we have also explained there the various origins of the chromatic behaviours of ags . in ddd2007b we have focused on the fast decline phase of the prompt emission and we have demonstrated that the rapid spectral evolution observed during this phase is also as expected in the cb model . in ddd2007c we have shown , for large ensembles of grbs , how the observed correlations between @xmath27 , @xmath28 [ dar & de rjula 2000b ( dd2000b ) , amati et al . 2002 ] , @xmath29 , and other prompt observables ( pulse rise - time , lag - time and variability ) follow mainly from the same simple geometrical considerations that we have reviewed above on the case - by - case variability of the doppler factor . in the cb model , xrfs are simply grbs seen at relatively large @xmath4 ( ddd2004a ) , even the particularly interesting xrf 060218 is in no way exceptional ( ddd2007a ) . in this paper we focus on the shape of the light curves of the x - ray afterglow of grbs , with and without breaks , measured with the x - ray telescope ( xrt ) aboard swift . we show that the shapes of the x - ray light curves of grbs and xrfs predicted in fig . [ f1]a , and the correlation between @xmath16 and @xmath27 illustrated fig . [ f1]b ( and the consequent apparent presence or absence of breaks in the ag ) agree with the cb - model s expectations . we also analize the @xmath30 correlation on the same light . finally , we investigate the relation between the temporal power - law index of the post - break decline and the photon spectral index , reaching satisfactory results . to do all this , we investigate 16 grbs chosen to reflect the full span of the question of the presence or absence of breaks . the selected grbs range from the faintest known grb ( 980425 , of supernova - association fame ) , which also has the most pronounced plateau and the latest break time , to the brightest swift grb ( 061007 ) , with the most luminous and longest - observed unbroken power - law x - ray ag . in the cb model , the mechanism for the emission of the prompt radiation of grbs and xrfs is inverse compton scattering . the temporal and spectral properties of the prompt phase , including its fast decline , are summarized in a ` master formula ' ( dd2004 ) that we have already contrasted with swift data ( ddd2006 , 2007a , b , c , d ) . we shall not repeat it here as our emphasis in the current study is on breaks in the x - ray light curves of grb afterglows though we shall use it to describe the fading of the prompt emission until the take - over by the synchrotron - ag emission , and the occasional late x - ray flares . neither do we discuss here the optical ags ( ddd2007a ) . the extinction in the optical- and , more so , in the radio- domain ( within the cbs , in the circumburst environment , in the ism of the host galaxy and ours , and in the intergalactic medium ) are difficult to model as reliably as the x - ray extinction . we shall see once again that the x - ray light curves ( corrected for extinction ) carry clear and direct information on the radiation mechanisms that dominate the prompt emission and the ag phase ( ics and sr , respectively , in the cb model ) . during the initial phase of @xmath0-ray emission in a grb , the lorentz factor @xmath0 of a cb stays put at its initial value @xmath31 , for the deceleration induced by the interactions with the ism has not yet had a significant effect . the doppler factor by which the light emitted by a cb is boosted in energy is given by eq . ( [ delta ] ) . since the emitted light is forward - collimated into a cone of characteristic opening angle @xmath32 , the boosted energetic radiation is easiest to detect for @xmath33 . thus , typically , @xmath34 . as a cb ploughs through the ism , fully ionized by the preceding @xmath0 radiation , it gathers and scatters the ism ions , mainly protons . these encounters are ` collisionless ' since , at about the time it becomes transparent to radiation , a cb also becomes ` transparent ' to hadronic interactions . as a consequence of momentum conservation , the scattered and re - emitted protons inevitably exert an inwards ` pressure ' on the cb . we have assumed that the main effect of this pressure is to slow the cb s expansion , posited to be relativistic at the emission time . in the approximation of isotropic re - emission in the cb s rest frame and a constant ism density @xmath35 , one then finds that , typically within minutes of observer s time @xmath36 , a cb reaches a roughly ` coasting ' radius , @xmath37 , which increases slowly until the cb finally stops and blows up ( dd2006 ) . up to the end of the coasting phase , and in a constant density ism , @xmath5 obeys ( ddd2002 ) : @xmath38 where @xmath39 if the ism particles re - emitted fast by the cb are a small fraction of the flux of the intercepted ones . in the opposite limit , @xmath40 . in the cb model of cosmic rays ( dd2006 ) , the observed spectrum strongly favours @xmath39 , used here in our fits . we have also concluded from previous analysis of swift x - ray data that @xmath41 is the right choice . as indicated by first - principle calculations of the relativistic merger of two plasmas ( frederiksen et al . 2004 ) , the ism ions continuously impinging on a cb generate within it turbulent magnetic fields , which we assume to be in approximate energy equipartition with the energy of the intercepted ism , @xmath42 . in this field , the intercepted electrons emit synchrotron radiation . the sr , isotropic in the cb s rest frame , has a characteristic frequency , @xmath43 , the typical frequency radiated by the electrons that enter a cb at time @xmath36 with a relative lorentz factor @xmath5 . in the observer s frame : @xmath44 ^ 3\ , \delta(t)\over 10^{12}}\ , \left[{n\over 10^{-1}\;\rm cm^3}\right]^{1/2 } { \rm hz}. \label{nub}\ ] ] where @xmath45 . the spectral energy density of the sr from a single cb at a luminosity distance @xmath46 is given by ( ddd2003a ) : @xmath47^{-1/2}\ , \left[1 + { \nu\over\nu_b(t)}\right]^{-(p-1)/2}\ , , \label{fnu}\ ] ] where @xmath48 is the typical spectral index ) is only correct for @xmath49 , for otherwise the norm diverges . the cutoffs for the @xmath50 distribution are time - dependent , dictated by the acceleration and sr times of electrons and their ` larmor ' limit . the discussion of these processes being complex ( ddd2003a , dd2006 ) , we shall satisfy ourselves here with the statement that for @xmath51 the ag s normalization is not predicted . ] of the fermi accelerated electrons , @xmath52 is the fraction of the impinging ism electron energy that is synchrotron re - radiated by the cb , and @xmath53 is the attenuation of photons of observed frequency @xmath50 along the line of sight through the cb , the host galaxy ( hg ) , the intergalactic medium ( igm ) and the milky way ( mw ) : @xmath54 . } \label{attenuation}\ ] ] the opacity @xmath55 at very early times , during the fast - expansion phase of the cb , may strongly depend on time and frequency . the opacity of the circumburst medium [ @xmath56 at early times ] is affected by the grb and could also be @xmath36- and @xmath50-dependent . the opacities @xmath56 and @xmath57 should be functions of @xmath36 and @xmath50 , for the line of sight to the cbs varies during the ag observations , due to the hyperluminal motion of cbs . these facts , the different @xmath58 dependences of the ics and sr emissions , and the dependence of the synchrotron ag on @xmath43 , are responsible for the complex observed chromatic behaviour of the ags . to a fair approximation , though , the deceleration bend , if occurring late enough , is achromatic from x - ray energies to the optical domain ( ddd2002 ) but not as far as radio ( ddd2003a ) . the swift x - ray bands are above the characteristic frequency @xmath59 in eq . ( [ nub ] ) at all times . it then follows from eq . ( [ fnu ] ) that the _ unabsorbed _ x - ray spectral energy density has the form : @xmath60 where we have used the customary notation @xmath61 . the functions @xmath62 and @xmath63 of eqs . ( [ delta],[deceleration ] ) evolve slowly , up until a time : @xmath64\;t_0\nonumber\\ \!\!\!\!\!\!\!\!&&\approx ( 130\,{\rm s})\ , [ 1 + 2\,\gamma_0 ^ 2\ , \theta^2]\,(1+z ) \left[{\gamma_0\over 10 ^ 3}\right]^{-3}\ , \left[{n\over 10^{-1}\ , { \rm cm}^{-3}}\right]^{-1 } \left[{r\over 10^{14}\,{\rm cm}}\right]^{-2 } \left[{n_{_{\rm b}}\over 10^{50}}\right ] \ ! , \label{tbreaks}\end{aligned}\ ] ] where we scaled the result to typical cb - model values of @xmath65 and a cb s baryon number , @xmath66 . the combination of the parameters @xmath35 , @xmath65 and @xmath66 appearing in eq . ( [ tbreaks ] ) is best constrained by the excellent x - ray observations discussed here . our previous results on optical and radio ags ( for fixed @xmath65 and @xmath66 ) favoured 10 times a smaller @xmath35 at the much larger sampled times , not an inconsistency , since a cb travels for @xmath67 light - days in one day of grb data . we have chosen to normalize @xmath35 as in eq . ( [ tbreaks ] ) , rather than to reproduce long discussions on the distributions of cb - model parameters ( e.g. de rjula 2007 ) . the quantity @xmath16 in eq . ( [ tbreaks ] ) characterizes the _ deceleration bend - time _ of the cb model ; eq . ( [ deceleration ] ) for @xmath68 describes the gradual character of this ` break ' . at later times ( [ deceleration ] ) implies that @xmath69 , and eq . ( [ delta ] ) that @xmath70 . thus , at @xmath71 , eq . ( [ fnux ] ) yields : @xmath72 with , as announced , a power decay in time half a unit steeper than in frequency . in the cb model , the peak energy of grbs satisfies : @xmath73 where @xmath74 ev is the typical energy of the glory s photons , that of the associated - supernova early light just prior to the ejection of cbs . the isotropic ( or spherical equivalent ) energy of a grb is ( dd2004 ) : @xmath75 where @xmath76 is the mean supernova early optical luminosity , @xmath77 is the number of cbs in the jet , @xmath78 is the comoving early expansion velocity of a cb ( in units of @xmath79 ) , and @xmath80 is the thomson cross section . for @xmath81 ( schaefer 2007 ) , the early sn luminosity required to produce the mean isotropic energy , @xmath82 erg , of ordinary long grbs , is @xmath83 , the estimated early luminosity of sn1998bw . all quantities in eq . ( [ eiso ] ) are normalized to their typical cb - model values . we have normalized to @xmath84 , an adequate mean number of prominent x - ray pulses in the subset of grbs analized here . the results in eqs . ( [ ep],[eiso ] ) are based on the assumption that ics is the mechanism generating the prompt radiation . they depend on @xmath18 and @xmath19 , two parameters also appearing in the description of the sr afterglow . that is why we shall be able to test the implied correlations , grb by grb , between the shape of the ag and the energetics of the prompt radiation , the very strong dependence of the @xmath85 on @xmath4 playing once more the major role . according to eqs . ( [ delta],[ep],[eiso ] ) , cbs with large @xmath18 , and , more so , small @xmath4 , produce the largest values of @xmath28 and @xmath27 : they generate the brightest grbs . according to eq . ( [ tbreaks ] ) , such @xmath86 values entail a small @xmath16 , an expectation that our analysis will validate . in such cases the deceleration bend or ` break ' of the synchrotron ag may take place before the beginning of the xrt observations and/or be hidden under the prompt compton emission . according to eq . ( [ fnux ] ) , these ags must be very luminous at early times , and according to eq . ( [ asymptotic ] ) , they must be well approximated from starters by the asymptotic power law behaviour given by eq . ( [ asymptotic ] ) . our analysis will verify all these predictions . to date , swift has detected and localized nearly 300 long grbs , and for most of them it followed their x - ray emission until it faded into the background . incapable of discussing all of them , we analyze the light curves of the x - ray afterglow of a set of grbs with and ` without ' jet breaks , which represent fairly well the entire spectrum of canonical and non - canonical x - ray afterglows of grbs . they include the most extreme cases of canonical and non canonical behaviour ( grb 0980425 and grb 061126 , respectively ) , the longest - measured canonical and non canonical x - ray light curves ( grb 060729 and grb 061007 ) and a variety of light curves with and without breaks , with and without superimposed x - ray flares . since many cb - model fits to canonical light curves of x - ray afterglows with ` breaks ' were included in previous publications ( ddd2002 , 2006 , 2007a , b , d ) we shall discuss in this paper more cases of grbs with an approximate power - law ag than of grbs with a canonical ag . we start the fits to the x - ray light curves during the transition between the rapid decline phase of the ics - dominated prompt emission to the sr - dominated ag phase . it suffices to include the ics contribution of the last prompt - emission pulse ( or the last two ) , because of an exponential factor in the pulse shape that suppresses very fast the relative contribution of the earlier pulses by the time the data sample the later ones ( ddd2007a , b ) . for the synchrotron contribution , it usually suffices to consider a common emission angle @xmath4 and an average initial lorentz factor @xmath18 for the ensemble of cbs . the ism density along the cbs trajectories is approximated by a constant . we then fit the entire observations of the x - ray ag of the selected grbs by using the ` master formula ' [ dd2004 , eq . ( 11 ) of ddd2007a ] , for the tail of the ics prompt emission contribution , and eq . ( [ fnux ] ) for the sr . many grbs have late x - ray flares , which we interpret as dying pangs of the engine , that is , the emission of cbs in late episodes of accretion into the recently collapsed central compact object . these cbs , whose ics - generated flares can only be seen on the weak background of a decaying sr ` after'-glow ( quote - unquote , since the ` ag ' is observable _ before _ the late ` prompt ' flares ) , are also modeled with the same master formula . the calculated shape of the energy - integrated x - ray ag , eq . ( [ fnux ] ) , depends only on three fit parameters . two of them are the product @xmath87 and the deceleration - bend time , @xmath16 , for an on - axis observer , as given in the first line of eq . ( [ deceleration ] ) . they determine the deceleration - bend time , @xmath16 , observed at a viewing angle @xmath4 , see eq . ( [ tbreaks ] ) . the third fit parameter is the index @xmath26 in the @xmath0 and @xmath85 time - dependent factors of eq . ( [ fnux ] ) . unlike in previous analysis , we let @xmath26 be a free parameter , unrelated to the spectral index @xmath88 , independently extracted by the observers from the shape of the x - ray spectrum . this way we shall be able to test explicitly the cb - model prediction implied by eq . ( [ fnux ] ) , @xmath89 , or by its more readable asymptotic form , eq . ( [ asymptotic ] ) . in all the cases we study , but two ( grbs 071020 and 050416a ) , a single cb or an ` average ' cb suffice to describe the ag . the occasional need for two cbs in the ag light - curve description is not a novelty . the most notable instance is that of grb 030329 ( ddd2003b ) . a comparison between the observed and predicted light curves of the 16 selected grbs is shown in figs . [ f2 ] to [ f5 ] . when well - measured , the ` break ' time , @xmath16 , is indicated in the figure by an arrow . the best fit values , of @xmath26 , @xmath87 , and @xmath16 are listed in table [ t1 ] , along with additional observational information on these grbs ( redshift , peak energy , equivalent isotropic energy , the start - time of the swift xrt observations , the spectral index of the unabsorbed ag , and the @xmath90 per degree of freedom of the fits ) . afterglows which exhibit nearly a pure power - law decline , have a @xmath16 smaller than @xmath91 , the time after trigger when the xrt started its observations of the ag , or a @xmath16 smaller than the time when the afterglow became brighter than the tail of the prompt emission . such ags have a nearly power - law shape , @xmath92 . their fits , however , return upper limits for @xmath93 and @xmath16 , above which the shape of the ag deviates from the data . these limiting values are also reported in table [ t1 ] , but the corresponding limit-@xmath16 location will not be shown in the figures in the case - by - case analysis , as it generally falls off - limits . in most cases ( including many swift ags studied in ddd2006 , 2007a , b , but not shown here ) , the cb model produces good fits with reduced @xmath90 values close to unity . even if the @xmath90 figures are good , we generally have refrained in the past from reporting them . one reason is that it is easy to obtain an excellent @xmath90 for a fit that has many data points , but misses some that clearly reflect a significant structure ( such as a supernova , see ddd2002 ) , or is , even within errors , systematically above the data in one region and below it in another . for that reason , and the occasional local scatter of the data , we consider the eye to be a better judge than any statistical measure . we comment on the @xmath90 values when they are ` bad ' . the values of @xmath12 , @xmath16 ( or @xmath94 ) , and @xmath26 returned by our fits and reported in table [ t1 ] have formal errors of a few percent . the error - correlation matrix has relatively small off - diagonal elements . the reduced-@xmath90 values are very close to unity , once the occasional flares are taken into account , to reveal the presence of a smoother sr background . one reason for all this is that @xmath94 sets the overall time scale , @xmath12 determines the shape of the bend , and @xmath26 is sensitive to the whole sr light curve , playing a major role in its power - law tail . this means that when a light curve is well sampled ( over orders of magnitude in flux and time ) , the fit is very sensitive to its parameters . naturally , the results depend also on the deceleration law , eq . ( [ deceleration ] ) , meant to be an approximation . therefore the extracted parameters have ` systematic ' errors reflecting the approximate nature of eq . ( [ deceleration ] ) . we can argue explicitly why the approximation should be better than it looks at first sight , even case by case ( on average , and independently , it leads to the correct spectrum of cosmic rays from non - relativistic energies up to the ` knee ' at some @xmath95 gev , dd2006 ) . the continuation of this rather formal argument on errors would take us well beyond the scope of this paper . in this section we comment one by one on the 16 grbs or xrfs whose x - ray light curves we discuss . the results of the cb - model fits are shown in figs . [ f2 ] to [ f5 ] , and the parameters relevant to our discussion are listed in table [ t1 ] . the first eight grbs are shown in the order of decreasing @xmath16 . for the next four , only an upper limit on @xmath16 can be extracted from the fits . the last four have very complex ags . the presence or absence of visible breaks in x - ray light curves and their different ` look ' the panoply of possibilities that we illustrate with our grb choices depend not only on @xmath16 , but on its value relative to @xmath91 ( the start - time of xrt observations ) and relative to the duration of the initial period of prompt - radiation dominance over the synchrotron ag . for this reason , it is easier to compare the plethora of looks of our grbs in an order slightly different from that of a decreasing @xmath16 . this we do ( only ) in the next paragraph . grbs 980425 and 060729 have light curves with a complete and simple canonical shape : one or two very clear prompt x - ray flares , a pronounced fast decay , a long plateau , a very visible ` break ' smoothly bending at @xmath16 to become a power - law decay . in grbs 050401 , 060105 , 060418 , 061007 and 050717 , the plateau is becoming less and less pronounced , so that the ag s @xmath16 is hiding better and better under the prompt signal , to the point that the last two are close to a pure power - law tail . in grbs 060813 , 070508 and 050505 , the prompt radiation ended early enough not to be caught by swift s xrt ( in the last case the follow - up started very late ) , but this trio displays very canonical ags , with their neat plateaus softly bending into a late power law . grbs 071025 , 061126 and 070125 are again approximate power - law tails , in which neither the early x - ray flares nor the putative bend are seen . grbs 071020 , 050416a and certainly 060607a , are very complex . the first two require contributions to the ag from two distinct cbs , 050416a having also a late flare . the unsightly x - ray light curve of grb 060607a can be described by the cb model without any new ingredients , but not much is learned from fitting it . * grb 980425 . * the light curve of this memorable single - peak grb , as observed by bepposax ( pian et al . 2000 ) , is shown in fig . the dotted line is the fit in ddd2002 , showing what we called a pronounced ` plateau ' . we have added to it the last ( predicted ) data point , measured with chandra by kouveliotou et al . ( 2004 ) , some 1285 days after burst ! to be consistent with the analysis here , we have re - fit the ensemble of data in the same manner as for all the other grbs to be discussed . the result is the continuous curve in the figure . this grb has , so far , the record large values @xmath96 and @xmath97 s , resulting in a light curve that rises before it falls , as explained in the introduction and illustrated in fig . this is the behaviour expected for far - off - axis grbs ( dd2000a ) . this one barely missed the official classification as an xrf : it s @xmath28 is @xmath98 kev , as opposed to @xmath99 kev ( see dado & dar 2005 for further comment on this point ) . * grb 060729 * and its x - ray light curve were studied in detail by grupe et al . it has a canonical shape , the longest follow - up observations with swift xrt , and the record - high @xmath100s , among the swift grbs . in fig . [ f2]b we show its cb - model description with , superimposed on its prompt decline phase , four ics x - ray flares included in the fit , as discussed in detail in ddd2007a . this grb being ` canonical ' and having a very clear ` break ' as several others also discussed in ddd2007a is included here to illustrate the start of the transition from ` breaks ' to ` missing breaks ' . although the best fit to the x - ray ag appears to be excellent , it yields a large @xmath101 , mostly due to many far - flung isolated data points in the swift data . more accurate data from xmm newton ( grupe et al . 2007 ) do not show such outliers . eliminating their contribution yields the @xmath90 value reported in table 1 . * grb 050505 , * whose x - ray light curve was studied in detail by hurkett et al . ( 2006 ) . at @xmath102 , this grb is amongst the most distant with a known redshift . due to an earth - limb constraint , swift was unable to slew to it until 47 minutes after the grb s trigger , and started measuring its x - ray light curve only 2883s after burst , during the transition of the ag from its shallow - decline phase to a power - law decline . as can be seen from fig . [ f2]c , the cb model describes very well the xrt light curve , except when the counting rate becomes comparable to the background . * grb 050401 , * whose x - ray light curve , studied in detail by de pasquale et al . ( 2006 ) , evolves smoothly from the tail of the prompt emission at around 200s to a short decaying plateau , which suavely breaks into a power - law decline at @xmath103s . its cb - model description is shown in fig . this grb had a very bright x - ray ag , even though it originated at a fairly large redshift , @xmath104 , and had a very large extragalactic absorbing column density along its sight - line , inferred from its x - ray spectrum to be @xmath105 ( de pasquale et al . 2006 ) , or @xmath106 ( watson et al . such a column density implies a very strong extinction of the optical ag and , consequently , an extreme chromaticity : more than 10-magnitude extinction in the v band ( zombeck 1990 ) and more than 30 magnitudes at 1800 @xmath107 . indeed , the optical ag was very dim ( a fitted spectral index , @xmath108 , between the x - band and the optical band , compared to @xmath109 to @xmath110 in ` normal ' grbs . in fact , according to jakobson et al . ( 2004 ) , grb 050401 qualified as a ` dark burst ' . it would not be a good case to discuss ( unattenuated ) chromaticity , or the lack thereof . * grb 070508 . * swift s xrt started to measure the x - ray light curve of this grb 82s after the grb trigger . even at this early time , it already displays the shallow - decay plateau phase of a canonical ag , which later bends into a power - law decline , as shown in fig . [ f3]a . * grb 060813 * , shown in fig . [ f3]b , is a case in which the prompt radiation is not seen by the xrt , and the ag has no obvious flares . in spite of some evidence for local variability , the smoothly bending ag is well described by the cb model ( @xmath111_dof _ @xmath112 for 254 _ dof _ ) . had the break happened a bit earlier , as in other cases , the x - ray ag would look like a power - law . the last data point lies below the fit , it could be due to an overestimated background . * grb 060418 , * whose achromatic ag was studied in detail by molinari et al . its x - ray ag evolves fast into a power - law decline , see fig . the cb - model fit returns an early break at @xmath113s , well hidden under the flaring activity during the fast - decline phase of the prompt emission . the transition from an ics - dominated regime to one in which sr is prevalent is corroborated by the fast spectral softening of the tail of the flare from around @xmath114s ( evans et al . 2007 ) , which suddenly turns , at @xmath115s , into the much harder time - independent power - law spectrum characteristic of the synchrotron ag ( ddd2007b ) . we have checked that the reasonable @xmath111_dof _ @xmath116 for 295 _ dof _ of the fit shown in the figure can be reduced to @xmath111_dof _ @xmath117 by including x - ray flares between 5 to 10 ks , or by replacing the fluctuating data points by average values . * grb 050717 , * studied in detail by krimm et al . it had the largest inferred peak energy of all swift grbs , @xmath118 kev , despite its estimated large redshift , @xmath119 at this @xmath120-limit , @xmath121 erg , and the local peak energy is @xmath122 kev . it also had an initially very bright x - ray ag , after the fast declining prompt emission , with a power - law decline from @xmath123s onwards . the fit in fig . [ f3]d returns an early break - time limit , @xmath124s , well hidden under the prompt - emission tail . the cb - model interpretation of the transition from a prompt ics radiation to a synchrotron ag is supported by the observed rapid spectral softening of the tail of the prompt emission and its sudden change at @xmath123s into the harder time - independent power - law spectrum of the synchrotron ag ( ddd2007b ) . in the case of this grb the best - fit value , @xmath125 , does not satisfy eq . ( [ asymptotic ] ) , with @xmath126 , as inferred from the x - ray spectrum with a fixed column density limited to the galactic one ( krimm et al . however , @xmath125 is consistent with @xmath127 of the ag for @xmath128s , the spectral index reported by zhang et al . ( 2007 ) , after inclusion of host - galaxy and igm absorption . * grb 061126 , * studied in detail by perley et al . ( 2008 ) had two major prompt pulses . due to an earth - limb constraint , swift slewed to the burst s direction only 23 minutes after its localization by its burst alert telescope ( bat ) . its light curve , measured by the xrt between 1.6 ks and 1.88 ms , is shown in fig . the x - ray light curve was reported to be well fit by a power - law in time with index @xmath129 ( sbarufatti et al . 2006 ) . a cb - model fit , with @xmath130 and @xmath131s , is shown in fig . there is a possible indication in the data of a steeper decay between 1.6 ks and 3.6 ks , which might belong to the tail of another cb with a smaller @xmath16 . cases of ags clearly requiring two cbs will be discussed anon . * grb 071025 . * swift s xrt started observations of the x - ray light curve 146s after the bat trigger . the initial relatively hard spectrum ( @xmath132 ) softened beyond 300s and the light curve declined like a single power - law , consistent with the cb - model s asymptotic power - law decline with a power - law index @xmath133 , as shown in fig . the data suggest a flaring activity between 4 ks and 40 ks . the effect of such flares on the cb - model x - ray light curve is illustrated in the figure by adding an ics flare around 40 ks with parameters ( peak time , width and normalization ) chosen , as in all other cases with clear flares , to best fit the data . * grb 070125 , * studied in detail by bellm et al . it was detected by mars odyssey , suzaku , integral , and rhessi . it is one of the swift - era grbs with the largest measured values of @xmath134 erg , @xmath135 , and source - frame @xmath136 kev . the initial detection of this grb occurred while it was not in the bat field of view during the beginning of the prompt emission , and its xrt light curve starts at 46 ks after the burst . as shown in fig . [ f4]c its power - law decline is well described by the cb model . the feature at @xmath137 ks can be interpreted as an x - ray flare , as in the figure . * grb 061007 , * whose ag was studied in detail by schady et al . ( 2006 ) and mundell et al . ( 2007 ) , was the brightest grb detected by swift and was accompanied by an exceptionally luminous x - ray and uv / optical afterglow , which decayed as a power law with an index @xmath138 . it had the largest values of @xmath134 erg , @xmath139 and an emission - point peak energy , @xmath140 kev ( golenetskii et al . this grb is the best example to date of a bright x - ray ag , well - sampled from the start of the xrt observations ( 86s after the bat trigger ) to @xmath141 s. the ag , shown in fig . [ f4]d , is power - law behaved right after the tail of the prompt emission . the cb - model fit returns @xmath142s , below which the @xmath111_dof _ ( a reasonable 1.13 for 1030 _ dof ) _ stays put . * grb 071020 , * measured by swift s xrt between 68s and 1.7 ms after trigger , and shown in fig . holland et al . ( 2007 ) fitted the data with a broken power - law with an initial decay index of @xmath143 , a break at @xmath144s , and a late - time decay index of @xmath145 . the fit is poor between 1.5 ks and 1.5 ms . a cb - model fit with a single cb is also unsatisfactory . the addition of a second cb to the ag s description , as in the fit shown in fig . [ f5]a , greatly improves the fit to @xmath111_dof _ @xmath146 for 174 _ dof _ , acceptable in view of what appears to be evidence for flaring activity , from 1.5 to 15 ks , which we have not endeavoured to describe , given the scarcity of data . * xrf 050416a . * the complex x - ray light curve of this xrf was monitored up to 74 days after the burst ( mangano et al . the late decline rate of the light curve is significantly slower than expected in the cb model from the observed photon spectral index @xmath88 , namely @xmath147 . the prompt signal of xrf 050416a had two clear pulses which , in the cb model , correspond to two separate cbs . the x - ray light curve , modeled with two cbs and shown in fig . [ f5]b , has a sr - component late - power decay that although it is not readable ` by eye ' due to the late - occurring ics flare is compatible with the predicted one . * grb 060105 , * whose x - ray light curve was studied in detail by tashiro et al . ( 2007 ) . following the prompt emission , which ended with a very steep decay , the light curve is canonical , it has a shallow decay after 180s and steepens at around 500s to a fast power - law decline , with a weak flaring activity superposed on it . the deviations from a smooth x - ray light curve may be caused by the flaring activity , not included in this particular fit , whose @xmath111_dof _ @xmath148 for 854 _ dof , _ is not inadequate . * grb 060607a , * was studied by molinari et al . its complex x - ray light curve , like that of quite a few other grbs , is dominated by strong flaring activity , as can be seen in fig . [ f5]d , with its many flares superimposed on the ag of a fitted , single , dominant cb . this fit , which can be improved by splitting the last flare into two , is a very rough description ( @xmath111_dof _ @xmath149 for 440 _ dof _ ) , not a proof of the quality of a prediction . moreover , in cases with such a prominent flaring activity , the photon spectral index of the ag data is an average between the typical index of flares , @xmath150 , and that of synchrotron ag , @xmath151 , i.e. , an average significantly smaller than that of the synchrotron ag . thus , we do not expect such a labyrinthine ag to satisfy the cb - model spectral - index relations , eqs . ( [ fnux],[asymptotic ] ) . we have summarized in eq . ( [ fnu ] ) the predicted form of the spectral energy density of the ag of a grb , in which the time - dependence and the energy - dependence are explicitly concatenated . in the large - frequency limit of the x - ray domain , the expression simplifies to that of eq . ( [ fnux ] ) , implying a predicted relation between the temporal index @xmath26 ( which we fit to the xrt light curve of the x - ray ag ) and the spectral index @xmath88 , independently fitted by the swift team to the x - ray ag spectrum after correcting for attenuation , and reported by zhang et al . the prediction is particularly simple , and is most transparently readable in the late-@xmath36 limit for the ags dependence on @xmath36 and @xmath50 , eq . ( [ asymptotic ] ) , in which both the time and the frequency functional forms are separate power laws . the values of @xmath26 and @xmath88 are listed in table [ t1 ] . notice that @xmath88 varies over a significant range of central values , 1.61 to 2.25 , and that the measurements are not compatible within errors with a common value . to illustrate the prediction in eqs . ( [ fnux],[asymptotic ] ) , we have plotted in fig . [ f6 ] the ratio @xmath152 ( predicted to be unity ) for the various grbs analized in this paper , and added a few other analized in the same fashion . the results are quite satisfactory . the mean value of @xmath153 , for instance , is @xmath154 for the grbs analized here , @xmath155 for the ensemble plotted in fig . [ f6 ] . in the cb model , the functional dependence on @xmath4 and @xmath18 of the deceleration - bend time of the synchrotron ag , @xmath16 , as well as its normalization , are specified by eq . ( [ tbreaks ] ) . this is also the case for the parameters , @xmath28 and @xmath27 of eqs . ( [ ep ] ) and ( [ eiso ] ) , of the prompt ics signal . as we saw in the introduction , this implies explicit correlations between @xmath16 and the prompt observables . the @xmath156 correlation is illustrated in fig . [ f1]b for various choices of @xmath4 and @xmath18 , with the rest of the parameters in @xmath16 and @xmath27 fixed to reference values in eqs . ( [ tbreaks],[eiso ] ) . in fig . [ f7 ] we plot , in the @xmath157 $ ] plane , the values returned by our analysis of the grbs we have discussed , see table [ t1 ] . the grbs represented by arrows reflect the fact that some data are just upper limits . the large shaded contour plot in the figure is the boundary of the domain covered by letting @xmath18 vary from 500 to 1500 , @xmath4 from 0 to 8 mrad , typical ranges encountered in the cb - model analysis of grbs . moreover , the normalization of @xmath16 in eq . ( [ tbreaks ] ) was varied from its central value in eq . ( [ tbreaks ] ) to 1/2 order of magnitude above it , and the normalization of @xmath27 in eq . ( [ eiso ] ) from its central value to 1/2 order of magnitude below and above it . the variability in these normalizations is best ascertained by the current analysis , it has been chosen to make fig . [ f7 ] ` look good ' . we have added to the figure the results for a few grbs which we have previously analyzed along the same lines in ddd2007a , b . there is no reason to expect the data to populate uniformly the region bounded by the contour in fig . [ f7 ] . on the contrary . the relativistically beamed radiation from a point in a cb initially subtends an angle @xmath158 . observers at an angle @xmath4 from the axial direction have a chance @xmath159 of being illuminated . at @xmath160 this chance decreases abruptly , given the fast fall of the doppler factor . all in all , @xmath161 is the optimal observation angle , for _ any _ @xmath18 . most grbs , then , should be seen at @xmath162 . the thick straight line in fig . [ f7 ] is @xmath163 at fixed @xmath12 , for which @xmath164 and @xmath165 . thus : @xmath166 the data follow this trend well , but at the high-@xmath27 end , at which they bend as in fig . [ f1]b . in fig . [ f8 ] we plot , in the @xmath167 $ ] plane , the corresponding results of our analysis . the shaded domain is obtained with the same ranges in @xmath18 , @xmath4 and in the normalization of @xmath16 as in the previous paragraph . the normalization of @xmath28 has been allowed to vary from 1/3 to 1/6 of its value in eq . ( [ ep ] ) of eq . ( [ ep ] ) is the peak energy at the start of a pulse ; set it to @xmath168 . the energy of the radiation is predicted to decrease during the pulse s duration : @xmath169 $ ] , with @xmath170 the width parameter ( the full width at half - maximum , fwhm , is @xmath171 ) . observers usually report @xmath28 at the peak s maximum , expected to be @xmath172 , or its pulse - averaged value : @xmath173 over the fwhm . we have not corrected for these facts , which may explain the choice of the ` best ' domain . ] . the points plotted as ellipses have an unknown @xmath120 , which we have let vary from 0 to 2.75 , the average for swift - era gbrs ( greiner , @xmath174 ) . at fixed @xmath12 , @xmath175 , so that : @xmath176^{-3/2}. \label{naivetbep}\ ] ] the rest of the comments are as in the discussion of the @xmath156 correlation . another direct way to ascertain the variability of the parameters governing the normalizations of @xmath28 and @xmath27 is to study their scatter plot ( dd2000b , amati et al . 2002 , dd2004 , ddd2007c , amati 2006 ) for a large collection of grbs and xrfs . this is done in fig . [ f9 ] , where the varying - power correlation predicted by the cb model ( ddd2007c ) is shown , and to which the grbs with known @xmath120 , @xmath28 and @xmath27 , among those studied here , are added . the figure shows that a total uncertainty of a factor of 2 in the norm of @xmath28 and of one an order of magnitude in the norm of @xmath27 ( as we have adopted ) is adequate to bracket the data . we have also tested elsewhere ( ddd2007d ) the correlation , apparent in fig . [ f1]a , between @xmath16 and the normalization of the ag . willingale et al . ( 2007 ) and nava et al . ( 2006 ) had collected and analyzed a large set of grbs , and made a scatter plot of @xmath16 versus the total ag energy in the 15 - 150 kev x - ray band up to time @xmath16 . to use this available information , we studied this correlation in its willingale - nava form . like the ones in figs . [ f7 ] and [ f8 ] , it turns out to follow the pattern expected in the cb model . the correlations between @xmath16 and @xmath27 or @xmath28 demonstrate that ` sub - energetic ' grbs ( or xrfs ) have large ` break ' times and , consequently , easily observable ` breaks ' . as grbs become more ` energetic ' , @xmath16 decreases and the chances increase to ` miss the break ' , which may be hidden under the prompt radiation , or may precede the swift slew - time minimum , or the start of the observations . a virtue of astrophysical x - ray data is that , in many instances and relative to lower - frequency bands , the corrections for attenuation are simpler and more reliable . the strength of swift in dealing with transient phenomena is , as the satellite s name reflects , the prompt start of its data - taking after an alert . this has made the swift results an excellent testing ground for theories of grbs and xrfs . in particular , the ability to monitor the x - ray flux over a very wide range of times has provided decisive tests of the theoretical predictions . filling the pre - swift gap in the data between ` prompt ' and ` afterglow ' radiations has led to a better understanding of the mechanisms responsible for them . in the cb model they are different : inverse compton scattering and synchrotron radiation , respectively . we have previously argued that the strong case for an ics origin of the ` prompt ' radiation ( dd2004 ) has been reconfirmed by the analysis of the swift x - ray flares , and the fast decay of the prompt signals ( ddd2007b ) . in this respect the study of x - ray and optical data is also particularly meaningful ( ddd2007a ) . the observed correlations between prompt observables @xmath27 , @xmath28 , @xmath29 and pulse rise - time , lag - time and variability also agree with the cb - model ( see ddd2007c and references therein ) . these correlations follow from the same simple considerations , that we have emphasized in this paper , on the dependence of the cited prompt observables on the lorentz and doppler factors of the radiation emitted by a quasi - point - like relativistically moving source . the cb - model s expectations for the interplay between ics and sr were confirmed by the analysis of the ` canonical ' shape of many swift x - ray light curves ( ddd2007a ) . the extreme canonical case is still grb 980425 , shown in fig . ( [ f2]a ) . the trend of the ` hardness ratios ' reflecting the spectral behaviour , and the spectral index itself , also corroborate the expected transitional behaviour ( ddd2007b ) , as the dominant mechanism evolves from ics to sr . in this paper we have shown in detail how the variety of x - ray ag shapes , with and without ` breaks ' , is also to be expected from a decelerating jet of effectively pointlike cannonballs , as in fig . ( [ f1]a ) . that the ag emission mechanism is sr from cbs slowing down in the way approximated by eq . ( [ deceleration ] ) is confirmed by the detailed frequency and time - dependences of eq . ( [ fnu ] ) for the spectral energy density . we have presented a study of the correlation between the synchrotron ags @xmath36- and @xmath50- dependences , specified in the x - ray domain by eq . ( [ fnux ] ) . this results in a relation between the ags spectral index @xmath88 , and the index @xmath26 appearing in their time dependence , a very simple relation at the late times at which the time dependence is also a power - law , see eq . ( [ asymptotic ] ) . the prediction is tested in fig . [ f6 ] . in the cb model , the understanding of ags with breaks or no breaks turns out to be clear : the ` missing ' sr breaks are hiding under the prompt ics radiation , or occur too early to be seen . this sounds like a trivial and model - independent excuse . it is not . it is supported by our case - by - case analysis of ag shapes . moreover , a crucial ingredient the angle of observation of the jet , compared to the beaming angle of its doppler - boosted radiation is validated by the correlations , e.g. the luminous ags are the ones with early or even undetectable breaks , as in fig . [ f1]a , and as in many of the examples we discussed here ( the correlation between @xmath16 and the energy in the x - ray ag was studied in ddd2007d ) . our conclusions are also supported by the correlations between the cbs deceleration - bend ` break - times ' , @xmath16 ( in the synchrotron ags ) , and the values of @xmath27 and @xmath28 ( in the prompt compton signal ) . these correlations , shown in figs . [ f7 ] and [ f8 ] , reconfirm the consistency of the overall picture . we have given no comment in the conclusions to our fits to the grbs and xrfs that we have studied . this is because the point we would like to make is _ not _ that the cb model can be used to fit the data very well . the main issue , in our view , is _ how _ a model , preferably in a predictive manner and in terms of very few concrete concepts like its radiation mechanisms , the aperture of its jets and the angle from which they are viewed can be used to understand the ensemble of long - duration grbs , and xrfs . after all , phenomena that require ever - increasingly complex explanations are of limited scientific interest . 980425 & 0.0085 & 56 & 6.9 e47 & 2.1 @xmath177 ? & 2.20 & 9.17 & 145000 & 36000 & 31/0 + 060729 & 0.54 & & @xmath178 7 e51 & @xmath179 & 2.20 & 2.51 & 8300 & 130 & 1966/207 + 050505 & 4.27 & 214 & & @xmath180 & 2.22 & 1.57 & 1980 & 2833 & 114/95 + 050401 & 2.90 & 132 & 3.5 e53 & @xmath181 & 2.20 & 0.80 & 1660 & 133 & 353/299 + 070508 & 0.82 ? & 188 & 7.0 e52 & @xmath182 & 2.12 & 1.22 & 260 & 82 & 610/469 + 060813 & & 214 & & @xmath183 & 1.70 & 1.13 & 190 & 85 & 256/239 + 060418 & 1.49 & 230 & 9 e52 & @xmath184 & 2.20 & 1.73 & 123 & 84 & 339/280 + 050717 & @xmath185 2.7 ? & 2401 & @xmath185 1 e54 & @xmath186 & 1.67 & ( 0.08 ) & @xmath178 55 & 91 & 114/78 + 061126 & 1.159 & 620 & 1.1 e53 & @xmath187 & 1.89 & ( 1.87 ) & @xmath178 104 & 1604 & 506/261 + 071025 & & & & & 2.20 & ( 0.90 ) & @xmath178 68 & 150 & 330/243 + 070125 & 1.547 & 440 & 9.4 e53 & @xmath188 & 2.38 & ( 1.19 ) & @xmath178 8060 & 47000 & 28/28 + 061007 & 1.261 & 498 & 1.0 e54 & @xmath189 & 2.26 & ( 0.05 ) & @xmath178 89 & 86 & 1147/1015 + 071020 & 2.145 & 322 & 8.0 e52 & @xmath190 & 1.86 & 0.67 & 90 & 68 & 234/154 + @xmath191 `` & & & & & 1.86 & 1.43 & 15100 & 68 & + 050416a & 0.6535 & 15 & 1.2 e51 & @xmath192 & 2.00 & 1.05 & 944 & 85 & 101/92 + @xmath191 '' & & & & & & 2.00 & 14800 & 85 & + 060105 & & 424 & & @xmath193 & 2.33 & 0.53 & 510 & 96 & 1879/839 + 060607a & 3.082&&& & 2.20 & 0.97 & 164 & 73 & 1119/404 + + the values of the peak energy , @xmath28 ( in kev ) and @xmath27 ( in erg ) of the grbs are from gcn reports of data of konus - wind , rhessi and suzaku . the grb redshifts are from gcn reports from ground - based optical telescopes . the start times @xmath91 , of the xrt data after the bat trigger , are from the swift repository ( evans et al . the unabsorbed spectral indices @xmath88 are from swift gcn reports and zhang , liang & zhang ( 2007 ) . the cb - model fits return @xmath26 , @xmath93 and @xmath16 . the parenthesized @xmath93 are for @xmath16 at its upper limit .

the x - ray afterglows ( ags ) of gamma - ray bursts ( grbs ) and x - ray flashes ( xrfs ) have , after the fast decline phase of their prompt emission , a temporal behaviour varying between two extremes . a large fraction of these ags has a ` canonical ' light curve which , after an initial shallow - decay ` plateau ' phase , ` breaks smoothly ' into a fast power - law decline . very energetic grbs , contrariwise , appear not to have a ` break ' , their ag declines like a power law from the start of the observations . breaks and ` missing breaks ' are intimately related to the geometry and deceleration of the jets responsible for grbs . in the frame of the ` cannonball ' ( cb ) model of grbs and xrfs , we analyze the cited extreme behaviours ( canonical and pure power law ) and intermediate cases spanning the observed range of x - ray ag shapes . we show that the entire panoply of x - ray light - curve shapes measured with swift and other satellites are as anticipated in the cb model . we test the expected correlations between the ag s shape and the peak- and isotropic energies of the prompt radiation , strengthening a simple conclusion of the analysis of ag shapes : in energetic grbs the break is not truly ` missing ' , it is hidden under the tail of the prompt emission , or it occurs too early to be recorded . we also verify that the spectral index of the unabsorbed ags and the temporal index of their late power - law decline differ by half a unit , as predicted . -.5 cm